\\zeta;\\nonumber\\\\\nA + \\zeta,~~~\\textup{if } A < -\\zeta;\\nonumber\\\\\n0,~~~~~~~~~\\textup{otherwise}.\\nonumber\n\\end{array}\\right.\\end{aligned}$$ \n3) Update **T** and *κ*:\n \n$$\\begin{aligned}\n\\mathbf{T} = \\mathbf{T} + \\kappa(\\mathbf{P} - \\mathbf{Q}),\\kappa = \\min(\\rho\\kappa,\\kappa\\_{max}),\n\\nonumber\\end{aligned}$$ \nwhere *κ**m**a**x* is a preset maximal value for *κ* and *ρ* is a scaled parameter and can be set as a value greater than 1.\n4) Check convergence:\n \n$$\\begin{aligned}\n\\Vert \\mathbf{P} - \\mathbf{Q} \\Vert\\_\\infty < \\epsilon,\\nonumber\\end{aligned}$$ \nwhere ∞ norm means the maximal value of the element in a matrix and *ε* is the machine epsilon. [alm:nb1]\n### Fix **P** and Update **w**\n \n$$\\begin{aligned}\n\\min\\_{\\mathbf{w}} \\Vert \\mathbf{z} - \\mathbf{A}\\mathbf{w} \\Vert\\_2^2 + \\gamma \\Vert \\mathbf{B}\\mathbf{w}\\Vert\\_2^2 + \\lambda \\Vert \\mathbf{w} \\Vert\\_1,\\nonumber\\\\\n\\textup{s.t.}~\\mathbf{w} \\succeq 0.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\label{eqn:nb12}\\end{aligned}$$ \nHerein, **z** is a vector composed by concatenating the columns of **L***s* one by one, and the *i**t**h* columns of **A** and **B** are a vector by vectorizing their corresponding matrices **P***T***K***i**s*, and $\\mathbf{P}^T(\\frac1{N\\_s}\\mathbf{K}\\_i^s\\mathbf{1}\\_s - \\frac1{N\\_t}\\mathbf{K}\\_i^t\\mathbf{1}\\_t)$, respectively. Eq. ([eqn:nb10]) can be further rewritten as the following standard Lasso problem:\n \n$$\\begin{aligned}\n\\min\\_{\\mathbf{w}} \\Vert \\mathbf{y} - \\mathbf{D}\\mathbf{w} \\Vert\\_2^2 + \\lambda \\Vert \\mathbf{w} \\Vert\\_1,\\nonumber\\\\\n\\textup{s.t.}~\\mathbf{w} \\succeq 0.~~~~~~~~~~~~~~~~~~\\end{aligned}$$ \nwhere $\\mathbf{y} = \\left[\\begin{array}{c} \\mathbf{z} \\\\ \\mathbf{0} \\end{array}\\right]$ and $\\mathbf{D} = \\left[\\begin{array}{c} \\mathbf{A} \\\\ \\mathbf{B} \\end{array}\\right]$. In our optimization, we use the SLEP package to learn the optimal **w**.\n### Check Convergence\nCheck whether the preset maximal iteration steps are reached or the value of the objective function in Eq. ([eqn:nb8]) is less than the preset threshold.\n[optimization]\nRSTR for Solving CDMER Problem\n------------------------------\nOnce the optimal parameters of RSTR are learned based on the labeled source and unlabeled target micro-expression samples using the optimization method given in Section [opt], we can easily predict the micro-expression categories of the testing samples from target database. More specifically, suppose the learned optimal parameters of RSTR are $\\hat\\mathbf{P}$ and $\\hat\\mathbf{w}$. Then, given a testing target micro-expression sample **x***t**e* we are able to estimate its micro-expression label vector by solving the following optimization problem:\n \n$$\\begin{aligned}\n\\min\\_{\\mathbf{l}^{te}} \\Vert \\mathbf{l}^{te} - \\hat\\mathbf{P}^T \\sum\\_{i=1}^M \\hat\\omega\\_i \\mathbf{k}\\_i^{te} \\Vert\\_2^2,\\nonumber \\\\\n\\textup{s.t.}~\\mathbf{l}^{te} \\succeq 0,~\\mathbf{1}^T\\mathbf{l}^{te} = 1,~~~~\n\\label{eqn:nb12}\\end{aligned}$$ \nwhere $\\hat\\omega\\_i$ is the *i**t**h* element of $\\hat\\mathbf{w}$ and **k***i**t**e* = [*ϕ*(**X***i**s*), *ϕ*(**X***i**t*)]*T**ϕ*(**x***i**t**e*) is the kernel matrix and can be computed by the kernel function chosen in training stage. Finally, the micro-expression category of this testing sample is assigned as the one whose corresponding value in **l***t**e* is maximum.\nEvaluation Results\n==================\nImplementation Details\n----------------------\nWe conduct the benchmark CDMER experiments under the designed protocol described in Section [benchmark]. For the evaluated methods, we implement them using the original source codes provided by the authors. To offer a fair comparison, for the experiments of different micro-expression features, the parameters of each spatiotemporal descriptors are fixed throughout the experiments, while for the experiments of DA, we follow the strategy used in current mainstream unsupervised DA evaluation experiments that is reporting the best result (in term of *m**e**a**n* *F*1-*s**c**o**r**e* in our CDMER benchmark experiments) corresponding to the optimal parameter setting for a DA method . Subsequently, we show the detailed parameter settings of both micro-expression features and DA methods used for our benchmark experiments.\n### Parameter Setting for DA Methods\nIn this section, we give the parameter searching space for different DA methods in the CDMER evaluation experiments.\n**SVM** : To offer a fair comparison, we use linear kernel for SVM throughout the evaluation experiments. As to the penalized coefficient *C*, we fix it at *C* = 1. Note that for the experiments of all the DA methods, we all use the linear SVM with *C* = 1 to serve as the classifier (if needed).\n**IW-SVM** : In our experiments, we choose uLSIF , which has shown its excellent performance in CDMER , to learn the importance weights for IW-SVM. Following the suggestion of , we search the trade-off parameter *λ* for uLSIF from a parameter space [1:1:100] × *t* (*t* = 1, 10, 100, 1000, 10000, 100000).\n**TCA** , **GFK** , and **SA** : Among the experiments of these three methods, principal component analysis (PCA) is used to construct the subspace for GFK and SA. For all of them, we search the optimal dimension *k* (the number of eigenvectors for composing the projection matrix) by trying all possible dimensions, i.e., searching *k* ∈ [1, 2, ⋯, *k**m**a**x*].\n**STM** : STM is originally a binary classification model. In our experiments, we extend it to a multi-class version by using one-against-rest strategy. Similar with SVM used in the benchmark evaluation, its penalized coefficient is set as *C* = 1. As to its second trade-off parameter *λ*, which is used to balance the KMM regularization term with the SVM objective function, the searching space is set as [0.001 : 0.001 : 0.009, 0.01 : 0.01 : 0.09, 0.1 : 0.1 : 1, 2 : 1 : 100, 1000, 10000].\n**TKL** : According to the work of , TKL has one important parameter called the eigenspectrum damping factor *ζ*. In the evaluation experiments, we determine its optimal value by searching from the parameter space [0.1 : 0.1 : 5].\n**TSRG** , **DRFS-T** , and **DRLS** : TSRG, DRFS-T, and DRLS have two important trade-off parameters, i.e., *λ* and *μ*. Following the works of , the optimal values of these two parameters are determined by searching from [0.001, 0.01, 0.1, 1, 10, 100, 1000] for *λ* and [0.001 : 0.001 : 0.009, 0.01 : 0.01 : 0.09, 0.1 : 0.1 : 1, 2 : 1 : 10] for *μ*.\n**RSTR**: The proposed RSTR have three trade-off parameters including *λ*, *μ*, and *τ* which control the balance between the loss function and the regularized terms. We search their optimal values from the preset parameter spaces, i.e., *λ* ∈ [0.1, 1, 10, 100, 1000, 10000], *μ* ∈ [0.1 : 0.1 : 5], and *τ* ∈ [0.01 : 0.01 : 0.1].\n### Parameter Setting for Micro-Expression Features\nWe set the parameters for the evaluated spatiotemporal descriptors as follows:\n**LBP-TOP** : LBP-TOP has two important parameters. One is the neighboring radius *R* and the other is the number of the neighboring points *P*. In the evaluation experiments, we set the values of these two parameters as the ones of *R* ∈ {1, 3} and *P* ∈ {4, 8}, respectively, and hence we report FOUR experimental results of LBP-TOP with different parameter setting, i.e., LBP-TOP(*R*1*P*4), LBP-TOP(*R*1*P*8), LBP-TOP(*R*3*P*4), and LBP-TOP(*R*3*P*8). In addition, the uniform pattern is used for LBP coding. We use Hong et al.’s fast LBP-TOP source code to implement the LBP-TOP feature extraction.\n**LBP-SIP** : Only one parameter needs to be set for LBP-SIP, i.e., the neighboring radius *R*. Similar to LBP-TOP, we set *R* as 1 and 3 for LBP-SIP, respectively, and report the experimental results of LBP-SIP(*R*1) and LBP-SIP(*R*3). In the experiments, LBP-SIP is implemented by ourselves.\n**LPQ-TOP** : For LPQ-TOP, we set its parameters following the suggestion of . Specifically, the size of the local window in each dimension is set as the default one ([5, 5, 5]). The parameters [*ρ**s*, *ρ**t*] for a correlation model used in LPQ-TOP are set as [0.1, 0.1] and [0, 0] (without correlation), respectively. The results of LPQ-TOP*d**e**c**o**r**r* = 0.1 and LPQ-TOP*d**e**c**o**r**r* = 0 are reported in the evaluation experiments. The source code of LPQ-TOP is publicly available at http://www.cse.oulu.fi/Downloads/LPQMatlab/.\n**HOG-TOP** and **HIGO-TOP** : HOG-TOP and HIGO-TOP both have one important parameter, i.e., the number of bins *p* to be set, which controls the dimensional histogram of image gradient orientations for three orthogonal planes. In the evaluation experiments, we fix *p* at 4 and 8 for HOG-TOP and HIGO-TOP and they are denoted by HOG-TOP*p* = 4, HOG-TOP*p* = 8, HIGO-TOP*p* = 4 and HIGO-TOP*p* = 8, respectively.\n**C3D** : In the evaluation experiments, we choose the publicly released C3D pretrained on Sports-1M and UCF101 as the feature extractor and use the last two fully connected layers to serve as the micro-expression features, which are denoted by C3D-FC1 (Sports-1M) and C3D-FC2 (Sports-1M), respectively.\nResults and Discussions\n-----------------------\n### Results at A Glance\nIn this section, we report the benchmark results of the evaluated methods including various DA methods and micro-expression features. All the experimental results are depicted in Tables [tab:tab3], [tab:tab4], [tab:tab5], and [tab:tab6]. Before deeply comparing and analyzing the results obtained by different DA methods and micro-expressions, we would like to probe into the CDMER tasks based on the obtained results. We calculate the average results of all the methods in each experiment and all the experiments for each method, which are given in the last line and last column of these tables. From the comparison between them, we are able to reach the following conclusions.\nFirstly, we observe that for our designed benchmark, the second type of experiments (TYPE-II) are significantly more difficult than TYPE-I, which can be clearly revealed by the remarkable differences between the average results of each method in these two types of experiments. For example, the baseline method, GFK (a well-performing representative of DA methods), and LPQ-TOP*d**e**c**o**r**r* = 0.1 (a well-performing representative of micro-expression features) achieve the average *m**e**a**n* *F*1-*s**c**o**r**e* / *A**c**c**u**r**a**c**y* of 0.6003 / 61.62%, 0.7223 / 72.44%, and 0.6157 / 63.79% in the first type of experiments, which are much higher than their achieved results (0.4112 / 45.55%, 0.5161 / 54.31%, and 0.3236 / 38.51) in the second type of experiments. In fact, the differences in difficulty between the TYPE-I and TYPE-II experiments stand to reason. The three datasets (HS, VIS, and NIR) used in TYPE-I involve the same subjects, stimulus materials, and recording environments and just different cameras, which results in the relatively small dataset difference. However, another dataset used in TYPE-II experiments, CASME II, corresponds to substantially different subjects, stimulus material, and recording environments compared with SMIC (HS, VIS, and NIR).\nSecondly, it should be pointed out that the class imbalanced problem existing in the source or target database remarkably degrades the performance of either DA method or micro-expression features in dealing with the CDMER tasks. For example, in the cases with SMIC (NIR) as the target database, i.e., Exp.11, Exp.3, and Exp.5, we can observe that the average performance in terms of *m**e**a**n* *F*1-*s**c**o**r**e* and *A**c**c**u**r**a**c**y* of all the DA methods can reach 0.6881 and 70.42% in Exp.5 whose the source database, SMIC (VIS), is relatively class-balanced. These two metrics drop to 0.6782 and 67.86% in Exp.3 and 0.5011 and 51.39% in Exp.11, where the source databases of Exp.3 and Exp.11 are SMIC (HS) and CASME II, respectively and very class-imbalanced. Similarly, the performance of all the methods is also affected by the class-imbalanced target database. As shown in Exp.6, Exp.3, and Exp.12 whose source database is fixed, i.e., SMIC (NIR), it can be seen that with class-imbalanced database as target one, the average *m**e**a**n* *F*1-*s**c**o**r**e* / *A**c**c**u**r**a**c**y* decrease from the level of 0.7272 / 73.88% (Exp.6: class-balanced) to 0.5668 / 57.26% (Exp.4: class-imbalanced) and 0.3928 / 41.96% (Exp.4: class-imbalanced), respectively.\nThirdly, we can also observe that the heterogeneous problem existing between source and target databases raises the level of difficulty of the CDMER tasks. It is known that the samples in SMIC (NIR) are recorded by a near-infrared camera, whose image-quality is considerably different from the samples recorded by high-speed camera (used in CASME II and SMIC (HS)) and visual camera (used in SMIC (VIS)). Therefore, it is intuitive that the CDMER tasks involving SMIC (NIR) would be more difficult than others. In order to check this point, we first fix the source database as SMIC (HS) and observe two experiments, i.e., Exp.1 and Exp.3, where the target database in Exp.1 is SMIC (VIS) and Exp.3 corresponds to SMIC (NIR). It can be seen that the average results among all the DA methods and micro-expression features (0.8405 / 84.12% and 0.6459 / 67.52%) in Exp.1 (homogeneous) are significantly higher than the results (0.6782 / 67.86% and 0.4420 / 49.65%) in Exp.3 (heterogeneous). We further check the opposite case if the heterogeneous image-quality samples exist in the source database. We observe Exp.2 and Exp.4, whose target databases are the same, i.e., SMIC (HS) and source databases are different, i.e., SMIC (VIS) v.s. SMIC (NIR). From the results, we notice the performance difference between the Exp.2 (homogeneous case) and Exp.4 (heterogeneous case). Specifically, the average performance achieved by DA methods and micro-expressions are 0.5764 / 57.60% and 0.4318 / 45.35% in Exp.2 (homogeneous case) v.s. 0.5668 / 57.26% and 0.3616 / 39.18% in Exp.4 (heterogeneous case).\n[!t]\n[tab:tab3]\n| DA Method | Exp.1: H → V | Exp.2: V → H | Exp.3: H → N | Exp.4: N → H | Exp.5: V → N | Exp.6: N → V | Average |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Baseline | 0.8002 / 80.28 | 0.5421 / 54.27 | 0.5455 / 53.52 | 0.4878 / 54.88 | 0.6186 / 63.38 | 0.6078 / 63.38 | 0.6003 / 61.62 |\n| IW-SVM | 0.8868 / **88.73** | 0.5852 / 58.54 | **0.7469 / 74.65** | 0.5427 / 54.27 | 0.6620 / 69.01 | 0.7228 / 73.24 | 0.6911 / 68.07 |\n| TCA | 0.8269 / 83.10 | 0.5477 / 54.88 | 0.5828 / 59.15 | 0.5443 / 57.32 | 0.5810 / 61.97 | 0.6598 / 67.61 | 0.6238 / 64.01 |\n| GFK | 0.8448 / 84.51 | 0.5957 / 59.15 | 0.6977 / 70.42 | **0.6197 / 62.80** | **0.7619 / 76.06** | 0.8142 / 81.69 | 0.7223 / 72.44 |\n| SA | 0.8037 / 80.28 | 0.5955 / 59.15 | 0.7465 / **74.65** | 0.5644 / 56.10 | 0.7004 / 71.83 | 0.7394 / 74.65 | 0.6917 / 69.44 |\n| STM | 0.8253 / 83.10 | 0.5059 / 51.22 | 0.6628 / 66.20 | 0.5351 / 56.10 | 0.6427 / 67.61 | 0.6922 / 70.42 | 0.6440 / 65.78 |\n| TKL | 0.7742 / 77.46 | 0.5738 / 57.32 | 0.7051 / 70.42 | 0.6116 / 62.20 | 0.7558 / **76.06** | 0.7579 / 76.06 | 0.6964 / 69.92 |\n| TSRG | **0.8869 / 88.73** | 0.5652 / 56.71 | 0.6484 / 64.79 | 0.5770 / 57.93 | 0.7056 / 70.42 | 0.8116 / 81.69 | 0.6991 / 70.05 |\n| DRFS-T | 0.8643 / 85.92 | 0.5767 / 57.32 | 0.7179 / 71.83 | 0.6163 / 61.59 | 0.7286 / 73.24 | 0.7732 / 77.46 | 0.7128 / 71.23 |\n| DRLS | 0.8604 / 85.92 | 0.6120 / 60.98 | 0.6599 / 66.20 | 0.5599 / 55.49 | 0.6620 / 69.01 | 0.5771 / 61.97 | 0.6552 / 66.60 |\n| RSTR | 0.8721 / 87.32 | **0.6401 / 64.02** | 0.7466 / **74.65** | 0.5765 / 57.32 | 0.7506 / **76.06** | **0.8428 / 84.51** | **0.7381 / 73.98** |\n| Average | 0.8405 / 84.12 | 0.5764 / 57.60 | 0.6782 / 67.86 | 0.5668 / 57.26 | 0.6881 / 70.42 | 0.7272 / 73.88 | - |\n[!t]\n[tab:tab4]\n| DA Method | Exp.7: C → H | Exp.8: H → C | Exp.9: C → V | Exp.10: V → C | Exp.11: C → N | Exp.12: N → C | Average |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Baseline | 0.3697 / 45.12 | 0.3245 / 48.46 | 0.4701 / 50.70 | 0.5367 / 53.08 | 0.5295 / 52.11 | 0.2368 / 23.85 | 0.4112 / 45.55 |\n| IW-SVM | 0.3541 / 41.46 | 0.5829 / 62.31 | 0.5778 / 59.15 | 0.5537 / 54.62 | 0.5117 / 50.70 | 0.3456 / 36.15 | 0.4876 / 50.73 |\n| TCA | 0.4637 / 46.34 | 0.4870 / 53.08 | **0.6834 / 69.01** | 0.5789 / 59.23 | 0.4992 / 50.70 | 0.3937 / 42.31 | 0.5177 / 53.45 |\n| GFK | 0.4126 / 46.95 | 0.4776 / 50.77 | 0.6361 / 66.20 | 0.6056 / 61.50 | 0.5180 / 53.52 | 0.4469 / 46.92 | 0.5161 / 54.31 |\n| SA | 0.4302 / 47.56 | 0.5447 / 62.31 | 0.5939 / 59.15 | 0.5243 / 51.54 | 0.4738 / 47.89 | 0.3592 / 36.92 | 0.4877 / 50.90 |\n| STM | 0.3640 / 43.90 | **0.6115 / 63.85** | 0.4051 / 52.11 | 0.2715 / 30.00 | 0.3523 / 42.25 | 0.3850 / 41.54 | 0.3982 / 45.61 |\n| TKL | 0.3829 / 44.51 | 0.4661 / 54.62 | 0.6042 / 60.56 | 0.5378 / 53.08 | 0.5392 / 54.93 | 0.4248 / 43.85 | 0.4925 / 51.93 |\n| TSRG | 0.5042 / 51.83 | 0.5171 / 60.77 | 0.5935 / 59.15 | 0.6208 / 63.08 | **0.5624 / 56.34** | 0.4105 / 46.15 | 0.5348 / 56.22 |\n| DRFS-T | 0.4524 / 46.95 | 0.5460 / 60.00 | 0.6217 / 63.38 | 0.6762 / 68.46 | 0.5369 / **56.34** | 0.4653 / **50.77** | 0.5498 / 57.65 |\n| DRLS | 0.4924 / 53.05 | 0.5267 / 59.23 | 0.5757 / 57.75 | 0.5942 / 60.00 | 0.4885 / 49.83 | 0.3838 / 42.37 | 0.5102 / 53.71 |\n| RSTR | **0.5297 / 54.27** | 0.5622 / 60.77 | 0.5882 / 59.15 | **0.7021 / 70.77** | 0.5009 / 50.70 | **0.4693 / 50.77** | **0.5587 / 57.74** |\n| Average | 0.4323 / 47.45 | 0.5133 / 57.83 | 0.5772 / 59.66 | 0.5638 / 56.85 | 0.5011 / 51.39 | 0.3928 / 41.96 | - |\n[!t]\n[tab:tab5]\n| Spatiotemporal Descriptors | Exp.1: H → V | Exp.2: V → H | Exp.3: H → N | Exp.4: N → H | Exp.5: V → N | Exp.6: N → V | Average |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Baseline | 0.8002 / 80.28 | 0.5421 / 54.27 | 0.5455 / 53.52 | **0.4878 / 54.88** | 0.6186 / 63.38 | 0.6078 / 63.38 | 0.6003 / 61.62 |\n| LBP-TOP(*R*1*P*4) | 0.7185 / 71.83 | 0.3366 / 40.24 | 0.4969 / 49.30 | 0.3457 / 40.24 | 0.5480 / 57.75 | 0.5085 / 59.15 | 0.4924 / 53.32 |\n| LBP-TOP(*R*1*P*8) | 0.8561 / 85.92 | 0.5329 / 53.66 | 0.5164 / 57.75 | 0.3246 / 35.37 | 0.5124 / 57.75 | 0.4481 / 50.70 | 0.5318 / 56.86 |\n| LBP-TOP(*R*3*P*4) | 0.4656 / 49.30 | 0.4122 / 45.12 | 0.3682 / 40.85 | 0.3396 / 40.85 | 0.5069 / 59.15 | 0.5144 / 60.56 | 0.4345 / 49.31 |\n| LBP-SIP(*R*1) | 0.6290 / 63.38 | 0.3447 / 40.85 | 0.3249 / 33.80 | 0.3490 / 42.07 | 0.5477 / 60.56 | 0.5509 / 60.56 | 0.4577 / 50.20 |\n| LBP-SIP(*R*3) | 0.8574 / 85.92 | 0.4886 / 50.00 | 0.4977 / 54.93 | 0.4038 / 42.68 | 0.5444 / 59.15 | 0.3994 / 46.48 | 0.5319 / 56.53 |\n| LPQ-TOP(*d**e**c**o**r**r* = 0.1) | **0.9455 / 94.37** | 0.5523 / 54.88 | 0.5456 / 61.97 | 0.4729 / 47.56 | 0.5416 / 57.75 | 0.6365 / 66.20 | 0.6157 / 63.79 |\n| LPQ-TOP(*d**e**c**o**r**r* = 0) | 0.7711 / 77.46 | 0.4726 / 48.78 | 0.6771 / 67.61 | 0.4701 / 48.17 | **0.7076 / 71.83** | **0.6963 / 70.42** | **0.6325 / 64.05** |\n| HOG-TOP(*p* = 4) | 0.7068 / 71.83 | **0.5649 / 57.32** | **0.6977 / 70.42** | 0.2830 / 29.27 | 0.4569 / 49.30 | 0.3218 / 36.62 | 0.4554 / 48.47 |\n| HOG-TOP(*p* = 8) | 0.7364 / 74.65 | 0.5526 / 56.10 | 0.3990 / 46.48 | 0.2941 / 32.32 | 0.4137 / 46.48 | 0.3245 / 38.03 | 0.4453 / 49.01 |\n| HIGO-TOP(*p* = 4) | 0.7933 / 80.28 | 0.4775 / 50.61 | 0.4023 / 47.89 | 0.3445 / 35.98 | 0.5000 / 53.52 | 0.3747 / 40.85 | 0.4821 / 51.52 |\n| HIGO-TOP(*p* = 8) | 0.8445 / 84.51 | 0.5186 / 53.66 | 0.4793 / 54.93 | 0.4322 / 43.90 | 0.5054 / 54.93 | 0.4056 / 46.48 | 0.5309 / 56.40 |\n| C3D-FC1 (Sports1M) | 0.1577 / 30.99 | 0.2188 / 23.78 | 0.1667 / 30.99 | 0.3119 / 34.15 | 0.3802 / 49.30 | 0.3032 / 36.62 | 0.2564 / 34.31 |\n| C3D-FC2 (Sports1M) | 0.2555 / 36.62 | 0.2974 / 29.27 | 0.2804 / 33.80 | 0.3239 / 36.59 | 0.4518 / 47.89 | 0.3620 / 38.03 | 0.3285 / 37.03 |\n| C3D-FC1 (UCF101) | 0.3803 / 46.48 | 0.3134 / 34.76 | 0.3697 / 47.89 | 0.3440 / 34.76 | 0.3916 / 47.89 | 0.2433 / 29.58 | 0.3404 / 40.23 |\n| C3D-FC2 (UCF101) | 0.4162 / 46.48 | 0.2842 / 32.32 | 0.3053 / 42.25 | 0.2531 / 28.05 | 0.3937 / 47.89 | 0.2489 / 32.39 | 0.3169 / 38.23 |\n| Average | 0.6459 / 67.52 | 0.4318 / 45.35 | 0.4420 / 49.65 | 0.3613 / 39.18 | 0.5013 / 55.28 | 0.4341 / 48.50 | - |\n[!t]\n[tab:tab6]\n| Spatiotemporal Descriptors | Exp.7: C → H | Exp.8: H → C | Exp.9: C → V | Exp.10: V → C | Exp.11: C → N | Exp.12: N → C | Average |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Baseline | 0.3697 / **45.12** | 0.3245 / 48.46 | 0.4701 / 50.70 | 0.5367 / 53.08 | **0.5295 / 52.11** | 0.2368 / 23.85 | 0.4112 / 45.55 |\n| LBP-TOP(*R*1*P*4) | 0.3358 / 44.51 | 0.3260 / 47.69 | 0.2111 / 35.21 | 0.1902 / 26.92 | 0.3810 / 43.66 | 0.2492 / 26.92 | 0.2823 / 37.49 |\n| LBP-TOP(*R*1*P*8) | 0.3680 / 43.90 | 0.3339 / 54.62 | 0.4624 / 49.30 | 0.5880 / 57.69 | 0.3000 / 33.80 | 0.1927 / 23.08 | 0.3742 / 43.73 |\n| LBP-TOP(*R*3*P*4) | 0.3117 / 43.90 | 0.3436 / 44.62 | 0.2723 / 39.44 | 0.2356 / 28.46 | 0.3818 / 49.30 | 0.2332 / 25.38 | 0.2964 / 38.52 |\n| LBP-SIP(*R*1) | 0.3580 / **45.12** | 0.3039 / 44.62 | 0.2537 / 38.03 | 0.1991 / 26.92 | 0.3610 / 46.48 | 0.2194 / 26.92 | 0.2825 / 38.02 |\n| LBP-SIP(*R*3) | 0.3772 / 42.68 | 0.3742 / **56.15** | **0.5846 / 59.15** | **0.6065 / 60.00** | 0.3469 / 35.21 | 0.2790 / 27.69 | **0.4279 / 46.81** |\n| LPQ-TOP(*d**e**c**o**r**r* = 0.1) | 0.3060 / 42.07 | 0.3852 / 48.46 | 0.2525 / 33.80 | 0.4866 / 47.69 | 0.3020 / 35.21 | 0.2094 / 23.85 | 0.3236 / 38.51 |\n| LPQ-TOP(*d**e**c**o**r**r* = 0) | 0.2368 / 43.90 | 0.2890 / 51.54 | 0.2531 / 38.03 | 0.3947 / 40.77 | 0.2369 / 35.21 | **0.4008** / 41.54 | 0.3019 / 41.83 |\n| HOG-TOP(*p* = 4) | 0.3156 / 34.76 | 0.3502 / 47.69 | 0.3266 / 35.21 | 0.4658 / 46.92 | 0.3219 / 35.21 | 0.2163 / 27.46 | 0.3327 / 37.91 |\n| HOG-TOP(*p* = 8) | **0.3992** / 43.90 | **0.4154** / 52.31 | 0.4403 / 45.07 | 0.4678 / 47.69 | 0.4107 / 40.85 | 0.1390 / 20.77 | 0.3787 / 41.77 |\n| HIGO-TOP(*p* = 4) | 0.2945 / 39.63 | 0.3420 / 53.85 | 0.3236 / 40.85 | 0.5590 / 55.38 | 0.2887 / 29.58 | 0.2668 / 31.54 | 0.3458 / 41.81 |\n| HIGO-TOP(*p* = 8) | 0.2978 / 41.46 | 0.3609 / 50.00 | 0.3679 / 43.66 | 0.5699 / 54.62 | 0.3395 / 33.80 | 0.1743 / 22.31 | 0.3517 / 40.98 |\n| C3D-FC1 (Sports1M) | 0.1994 / 42.68 | 0.2394 / 56.15 | 0.1631 / 32.39 | 0.1075 / 19.23 | 0.1631 / 32.39 | 0.2397 / **56.15** | 0.1854 / 39.83 |\n| C3D-FC2 (Sports1M) | 0.1994 / 42.68 | 0.1317 / 24.62 | 0.1631 / 32.39 | 0.1075 / 19.23 | 0.1631 / 32.39 | 0.2397 / **56.15** | 0.1674 / 34.58 |\n| C3D-FC1 (UCF101) | 0.1581 / 31.10 | 0.1075 / 19.23 | 0.1886 / 39.44 | 0.1075 / 19.23 | 0.1886 / 39.44 | 0.2397 / **56.15** | 0.1650 / 34.10 |\n| C3D-FC2 (UCF101) | 0.1994 / 42.68 | 0.1705 / 19.23 | 0.1631 / 32.39 | 0.1075 / 19.23 | 0.1631 / 32.39 | 0.1075 / 19.23 | 0.1414 / 27.53 |\n| Average | 0.2954 / 41.88 | 0.2959 / 44.95 | 0.3060 / 40.32 | 0.3581 / 38.94 | 0.3049 / 37.94 | 0.2277 / 31.80 | - |\n### Results for CDMER Experiments of Using DA Methods\nTables [tab:tab3] and [tab:tab4] show the experimental results of different DA methods corresponding to the TYPE-I and TYPE-II experiments, respectively. From these two tables, it can be seen that in both two types of designed CDMER experiments, nearly all the DA methods can achieve promisingly better results in terms of both mean F1-score and Accuracy than the baseline method (SVM without any DA). More importantly, some well-performing DA methods, e.g., GFK , DRFS-T , and the proposed RSTR, have significant improvements of at least 0.1000 (average *m**e**a**n* *F*1-*s**c**o**r**e*) and 10% (average *A**c**c**u**r**a**c**y*) compared with the baseline results in either TYPE-I or TYPE-II experiments. Based on the above observations, we are able to reach the conclusion that considering CDMER as an DA problem is no doubt an effective solution for the CDMER problem. It is a good choice to develop excellent DA methods to relieve the feature distribution mismatch between the samples from different micro-expression databases.\nMoreover, we observe that our proposed RSTR achieves the best average results among all the methods in the CDMER experiments. Specifically, RSTR achieves the average *m**e**a**n* *F*1-*s**c**o**r**e* / *A**c**c**u**r**a**c**y* of 0.7381 / 73.98% in TYPE-I experiments and 0.5587 / 57.74% for TYPE-II experiments, which are significantly higher than most of DA methods. The superior performance of RSTR may attribute to the consideration of the facial region information in its design. Let us recall the spatial division scheme in Fig. [fig:fig2] adopted by our benchmark, which is used together with spatiotemporal descriptors for describing micro-expressions. In fact, the facial local regions yielded by such spatial division scheme have different contributions to distinguish micro-expressions. Such different contributions are often neglected in most works of micro-expression analysis. Recent work in had shown that it is beneficial for micro-expression recognition if we consider the different contributions of different facial regions. From the advantages of RSTR, it can convincingly verify that taking into consideration the positive cues associated with distinguishing micro-expressions, e.g., different contributions of different facial regions, offers us a new insight to develop the DA methods such that these DA methods are more applicable to CDMER problem.\n### Results of CDMER Experiments by Using Micro-Expression Features\nThe detailed *m**e**a**n* *F*1-*s**c**o**r**e* / *A**c**c**u**r**a**c**y* achieved by various micro-expression features are depicted in Tables [tab:tab5] and [tab:tab6]. From these two tables, it is clear to see that LPQ-TOP with *d**e**c**o**r**r* = 0 and LBP-SIP with *R* = 3 achieve the best average performance in terms of *m**e**a**n* *F*1-*s**c**o**r**e* and *A**c**c**u**r**a**c**y* in TYPE-I and TYPE-II CDMER experiments, respectively. More importantly, we notice that LPQ-TOP(*d**e**c**o**r**r* = 0) obtains the average *m**e**a**n* *F*1-*s**c**o**r**e* / *A**c**c**u**r**a**c**y* of 0.6325 / 64.05% in the first type of experiments, which are even competitive among the results of DA methods. By referring Table [tab:tab3] and Tables [tab:tab5], we can find that LPQ-TOP(*d**e**c**o**r**r* = 0) outperforms TCA (0.6238 / 64.01%) and is very competitive against STM (0.6440 / 65.78%) and DRLS (0.6552 / 66.60%). In addition, we also observe that several micro-expression features achieve very promising results. For example, LPQ-TOP(*d**e**c**o**r**r* = 0.1) achieves the *m**e**a**n* *F*1-*s**c**o**r**e* of 0.9455 and *A**c**c**u**r**a**c**y* of 94.37% in Exp.1, which are significantly better than all the DA methods. In Exp.9 and Exp.10, LBP-SIP(*R* = 3) is a very good competitor against DA methods. Based on the above observations, it is believed that developing database-invariant spatiotemporal descriptors for robustly describing micro-expressions also provide a promising and feasible way to solve the CDMER problem.\nHowever, we have to admit that most of the existing spatiotemporal descriptors including the above well-performing ones are still not satisfactory and cannot completely meet the requirement in CDMER problem. More specifically, several limitations still exist in current micro-expression feature research. Firstly, the performance of most spatiotemporal descriptors is not stable. In other words, one spatiotemporal descriptor performs well in Task A but very poor in Task B. An example is LPQ-TOP. It can be seen that LPQ-TOP with two different parameters both achieve satisfactory results beating the baseline method. But the performance of both two LBP-TOP based micro-expression features decrease sharply and even under the level of baseline method. Secondly, it is clear to see that the performance of most of handcrafted spatiotemporal descriptors is strongly affected by its parameters. As the results of LBP-TOP showed, its performance varies very sensitively in nearly all the experiments of our CDMER benchmark with the changes of its parameter *R* and *P*. Therefore, in the future study of spatiotemporal descriptors used for describing micro-expressions, we should also consider to reduce the sensitiveness of its parameters such that it will be simpler and more convenient to use in dealing with the CDMER problem. Finally, it should be pointed out that at present it is not enough for solving CDMER problem to simply use existing spatiotemporal descriptors as micro-expression features because its average performance is still far from the DA methods. From another point of view, it is believed that there is still very large development space in this direction, i.e., developing robust (database-invariant) micro-expression features.\n[!t]\n[tab:tab7]\n| Deep Features | Exp.1: H → V | Exp.3: H → N | Exp.8: H → C |\n| --- | --- | --- | --- |\n| Baseline | 0.8002 / 80.28 | 0.5455 / 53.52 | 0.3245 / 48.46 |\n| TCA | 0.8269 / 83.10 | 0.5828 / 59.15 | 0.4870 / 53.08 |\n| C3D-FC1 (Sports1M) | 0.1577 / 30.99 | 0.1667 / 30.99 | 0.2394 / 56.15 |\n| C3D-FC2 (Sports1M) | 0.2555 / 36.62 | 0.2804 / 33.80 | 0.1317 / 24.62 |\n| C3D-FC1 (Finetune) | 0.4858 / 56.34 | 0.2751 / 35.21 | 0.3276 / 34.62 |\n| C3D-FC2 (Finetune) | 0.4144 / 47.89 | 0.3758 / 40.85 | 0.3390 / 36.92 |\nLastly, we discuss the deep features evaluated in the benchmark. From the results of deep features extracted by C3D pretrained on Sports1M and UCF101, we can observe that these four deep features perform poorly in nearly all the CDMER experiments and cannot reach the level of most handcrafted features. In fact, this is foreseeable because the Sports1M and UCF101 served for C3D pretraining are both action databases, whose samples are quite different from the micro-expression ones. In fact, to enable the C3D to gain the micro-expression information, we can finetune the pretrained models based on the source database and then use the finetuned model to serve as feature extractor. To this end, we finetune the C3D pretrained on Sports1M based on SMIC (HS) and then extract the deep features to conduct the experiments of using SMIC (HS) as source database including Exp.1, Exp.3, and Exp.8. Note that for finetuning, we augment the samples in SMIC (HS) to 3350[9](#fn9) and divide the samples into a training set whose sample number is 2550 and a validation set containing 800 samples. The experimental results are shown in Table [tab:tab7], where we also list the results of C3D pretrained on Sports1M, the baseline method (LBP-TOP with *R* = 3, *P* = 8), and TCA (a representative DA method). From Table [tab:tab7], it is clear that by resorting to finetuning strategy, the performance of C3D features can be improved significantly compared with the original model. It is also interesting to see that in Exp.8, the deep features extracted by finetuned C3D even outperform the baseline method (LBP-TOP with *R* = 3, *P* = 8) in term of mean F1-score. However, compared with TCA (a representative DA method) and the baseline method, the performance of deep features is still very poor and not satisfactory. We think there may be two possible reasons. Firstly, it may attribute to the problem of lacking enough good data for finetuning the C3D models. It is clear that the training or finetuning of deep learning models requires large numbers of samples while the sample numbers of the CASME II and SMIC are too small. Although we can use some methods to augment the samples, the augmented micro-expression samples are not satisfactory. Therefore, more micro-expression samples need be collected such that a better-performing C3D can be finetuned to deal with CDMER problem. The second reason may be that it seems not enough for learning the database-invariant deep features to simply finetune the deep model based on the source database. Leveraging the idea of domain adaptation methods, i.e., reducing the difference between source and target domains, to finetune deep models may offer a feasible solution to improve the performance of deep features.\nConclusion\n==========\nIn this paper, we have investigated the cross-database micro-expression recognition (CDMER) problem by conducting a standard benchmark evaluation from two different perspectives including domain adaptation (DA) and spatiotemporal features used for describing micro-expressions. First, under a well-designed evaluation protocol, we make use of two widely-used micro-expression databases, i.e., CASME II and SMIC, to set up two types of CDMER experiments. Then, we collect NINE state-of-the-art DA methods and SIX excellent spatiotemporal descriptors to perform benchmark evaluation based on our designed CDMER protocol. Moreover, we also propose a novel DA method called region selective transfer regression (RSTR), which fully considers the different contributions of the facial local regions in recognizing micro-expressions. Finally, comprehensive discussions for the benchmark evaluation results are provided. More importantly, we reach a useful conclusion from the comparison between the proposed RSTR and other state-of-the-art DA methods that it is beneficial for dealing with the CDMER problem if we take full advantage of important cues associated with micro-expressions like facial local region information considered in the design of our RSTR. In addition, all the collected data and codes involving the benchmark evaluation in our paper will be released as soon as possible. We hope our work can advance the micro-expression analysis research by inspiring more researchers to focus on the CDMER problem. In the future, we will insist to collect more well-performing DA methods and micro-expression features for updating the evaluation results in our project page.\n---\n1. Y. Zong, W. Zheng, and C. Tang are with the Key Laboratory of Child Development and Learning Science of Ministry of Education, School of Biological Sciences and Medical Engineering, Southeast University, Nanjing 210096, China. Email: {xhzongyuan, wenming\\_zheng, tcg2016}@seu.edu.cn[↩](#fnref1)\n2. T. Zhang is with the School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China. Email: tony@scut.edu.cn[↩](#fnref2)\n3. X. Hong is with the Xi’an Jiao Tong University, Xi’an 710049, China. Email: hongxiaopeng@ieee.org[↩](#fnref3)\n4. G. Zhao is with the Center for Machine Vision and Signal Analysis, University of Oulu, Finland. Email: {xiaopeng.hong, guoying.zhao}@oulu.fi[↩](#fnref4)\n5. Z. Cui is with the School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China. Email: zhen.cui@njust.edu.cn[↩](#fnref5)\n6. The asterisks indicate the corresponding authors.[↩](#fnref6)\n7. According to the definition of Ekman et al. , FACS contains lots of AUs and action descriptors (ADs). With AUs and ADs, we are able to code human beings’ basic emotions, e.g., happy and disgust.[↩](#fnref7)\n8. For example, LBP-TOP with *R* = 3 requires at least SEVEN frames for a given micro-expression video clip. However, some micro-expression video clip has SIX or less frames, e.g., in VIS subset of SMIC. Therefore, frame number normalization is a necessary preprocessing step when we choose some spatiotemporal descriptors to describe micro-expression.[↩](#fnref8)\n9. Specifically, we first use TIM to normalize the frame number of all the micro-expression samples to 160. Then we divide these 160 frames into 10 parts without overlap. Finally, we randomly extract one frame from all the parts to obtain a micro-expression sample with frame length of 16.[↩](#fnref9)\n"},"data_id":{"kind":"string","value":"arxiv_0003263"}}},{"rowIdx":792,"cells":{"content":{"kind":"string","value":"Constraining Jet/Disk Geometry and Radiative Processes in Stellar Black Holes XTE J1118+480 and GX 339 − 4\n==========================================================================================================\n### Accepted 2009 April 8. Received in original form 2008 June 9\n[firstpage]\nWe present results from modeling of quasi-simultaneous broad band (radio through X-ray) observations of the galactic stellar black hole (BH) transient X-ray binary (XRB) systems XTE J1118+480 and GX 339 − 4 using an irradiated disc + compact jet model. In addition to quantifying the physical properties of the jet, we have developed a new irradiated disc model which also constrains the geometry and temperature of the outer accretion disc by assuming a disc heated by viscous energy release and X-ray irradiation from the inner regions. For the source XTE J1118+480, which has better spectral coverage of the two in optical and near-IR (OIR) wavelengths, we show that the entire broad band continuum can be well described by an outflow-dominated model + an irradiated disc. The best-fit radius of the outer edge of the disc is consistent with the Roche lobe geometry of the system, and the temperature of the outer edge of the accretion disc is similar to those found for other XRBs. Irradiation of the disc by the jet is found to be negligible for this source. For GX 339 − 4, the entire continuum is well described by the jet-dominated model only, with no disc component required. For the two XRBs, which have very different physical and orbital parameters and were in different accretion states during the observations, the sizes of the jet base are similar and both seem to prefer a high fraction of non-thermal electrons in the acceleration/shock region and a magnetically dominated plasma in the jet. These results, along with recent similar results from modeling other galactic XRBs and AGNs, may suggest an inherent *unity in diversity* in the geometric and radiative properties of compact jets from accreting black holes.\nblack hole physics— radiation mechanisms: non-thermal and thermal— X-rays: binaries— stars: winds, outflows– accretion, accretion discs– X-rays: individual: (GX 339 − 4, XTE J1118+480)\nIntroduction\n============\nSoft X-ray transient (SXT) binary systems that occasionally go into episodes of high activity or *outbursts* are some of the best laboratories to study accretion flows near compact (*G**M*/*R* ∼ *c*2) objects. Coordinated multi-wavelength campaigns have been carried out in recent years to observe SXTs simultaneously over a broad frequency range of nine decades or more from radio to X-rays. These campaigns have provided extremely valuable insights about the accretion in/outflows and helped to constrain the theoretical models predicting accretion geometry and physical emission processes. Outbursts of SXTs are usually explained in terms of a disc instability model, postulating a dramatic increase in mass accretion via an accretion disc towards the central compact object. The luminosity of a black hole transient (BHT) can vary by as much as eight orders of magnitude between quiescence and peak outburst, within timescales of weeks to months. Such systems are also ideal for studying the formation of jets, which are a by-product of accretion, yet poorly understood so far.\nMatter spiralling into the gravitational potential well of the compact object forms an accretion disc, and radiates via dissipation of the accumulated gravitational potential energy. The observed UV and soft X-ray spectra of these sources, during epochs of high mass accretion rate, are well modeled by viscous dissipation from the inner regions of a geometrically thin, optically thick “Shakura–Sunyaev disc”.\nHowever in the OIR, the viscously dissipating disc model sometimes significantly underpredicts the observed spectral energy distribution (SED), and the excess is attributed to irradiation of the outer disc by the intense X-ray radiation from the inner region close to the compact object. This irradiation mechanism has been well explored theoretically and their effect observed in optical and near-IR wavelengths. The most prominent, observable effect of irradiation is to heat up the outer regions of the accretion disc to temperatures higher than what would be possible solely by viscous dissipation of a standard Shakura-Sunyaev disc. This leads to a deviation in the shape of the SED of the accretion disc (from that of a standard disc), creating a ‘bump’ in the optical through infra-red (OIR) wavelengths. As the quality of the OIR data improves, the deviation (i.e. the bump) from Shakura-Sunyaev spectrum becomes more and more apparent, and the need to incorporate irradiation in the model becomes important.\nIn Roche lobe overflowing BHTs where the accretor is 10 times or more heavier than the donor, X-ray irradiation of the donor is small. Furthermore irradiation causes the outer disc to flare, shielding most of the donor star from irradiation. While most models assume an azimuthally symmetric accretion disc, warping of the accretion disc can introduce additional uncertainities in shielding the donor from irradiation.\nNot all the mass accreted inwards falls onto the compact object. Outflows, in the form of steady compact jets, or transient, ballistic ejecta are often seen in compact binary systems. Steady, compact jets have been observed from an increasing number of compact binary systems when they are in an *X-ray hard state* (see below). The hallmark signature of such a jet is its flat to slightly inverted spectrum which, following the classic work by, is usually attributed to a superposition of self-absorbed synchrotron emission from segments of a collimated jet. For stellar BHTs the flat spectrum of the jet is expected to extend from radio frequencies up to IR, beyond which the jet synchrotron becomes optically thin. At X-ray frequencies, inverse Compton (IC) upscattering in the jet base of synchrotron as well as external photons from the accretion disc can become comparable and may even dominate the spectrum. As shown in, synchrotron and IC at the jet base can result in spectra very similar to those produced by compact Comptonizing coronal models.\nDuring the course of an outburst, the X-ray spectral and temporal features show an enormous amount of diversity. However extensive observations and their analyses over the past few decades have enabled the classification of spectral and temporal features into relatively few groups known as *X-ray states*. In particular, two canonical states are often seen: a “soft” state characterized by strong thermal emission from the accretion disc and low variability in the light curve, and a “hard” state characterized by non-thermal power law emission, high variability and probably a collimated outflow. Besides these canonical states, a host of intermediate and extreme states with varying ratios of thermal and non-thermal radiation also are seen. Recent OIR observations covering complete outbursts of many XRBs have shown that while irradiation usually plays a dominant role in the evolution of the OIR spectrum during the entire outbursting phase, the contribution from a non-thermal source, e.g. a jet, can become dominant when XRBs (most notably black holes) are in hard state. Thus any model which aims to understand disc-jet coupling must include irradiation.\nWe have developed a new spectral model which calculates continuum emission from an irradiated accretion disc, and in conjunction with a modified version of the compact jet model developed by, allows us to constrain the geometrical and radiative properties not only of the jet, but simultaneously those of the accretion disc. The thermal photons from the irradiated disc are also included in the photon field of the jet for Compton scattering, although due to Doppler redshifting in the jet frame, this external Compton emission is usually much smaller than the synchrotron self Compton (SSC). The new model is fully integrable with the standard spectral analysis software ISIS and parallelizable for using in large cluster computing environments (also see §[sec:parallel]).\nWe have used this new model to analyze broad band quasi-simultaneous observations of two galactic BHT systems XTE J1118+480 and GX 339 − 4. Simultaneous or quasi-simultaneous observations of both sources, during hard X-ray state of their outbursts, have revealed broad band continuum emission ranging from radio to X-rays, strongly suggesting the presence of jets. By studying short time-scale variability in near-IR wavelengths from XTE J1118+480, also suggested a non-thermal origin of the near-IR radiation, with a power law index typical of optically thin synchrotron emission from a jet.\nThe known physical properties of the two sources XTE J1118+480 and GX 339 − 4 have been briefly summarized in §[sec:data]. Since analysis of multi-wavelength data requires careful calibration of data from various ground-based as well as space-borne instruments, we have also presented the data reduction and calibration procedures in §[sec:data]. A brief outline of the model, with somewhat more emphasis on the newly introduced irradiated disc component, is given in §[sec:model], along with a description of the fitting procedure. The results of our broad band SED modeling suggest that despite the differences in size, mass, orbital parameters or accretion state, some of the fundamental physical parameters characterizing the geometrical and radiative properties of the jet are similar for both sources. This similarity may be a global property of jets formed near compact objects and is discussed in §[sec:discussion].\nSources, Observations & Data Reduction\n======================================\nXTE J1118+480 and GX 339 − 4 are two prototype galactic BHTs with markedly different properties. Both binary systems contain high mass compact objects, $8.5\\pm 0.6 \\rm{M}\\_\\odot$ for XTE J1118+480 and $>6\\ \\rm{M}\\_\\odot$ for GX 339 − 4, thus strongly suggesting for the compact objects to be black holes in both cases.\nXTE J1118+480 lies 62∘.3 north of galactic plane, in the halo. It has an orbital period of 4.04 hours. The distance and interstellar column density to XTE J1118+480 are reasonably well known from OIR photometry during quiescence to be 1.72 ± 0.1 kpc and 1.3 × 1020 cm− 2 respectively. We adopted a black hole mass of $8.5\\ \\rm{M}\\_\\odot$ for this source. The orbital inclination of XTE J1118+480, as measured by is 68 ± 2 degrees.\nGX 339 − 4 lies near the galactic plane, only 4∘.3 south of the galactic equator and has an orbital period of 42.14 hours. We adopted a black hole mass of $7\\ \\rm{M}\\_\\odot$ for GX 339 − 4 in this work. The secondary star in the GX 339 − 4 system has eluded detection so far, thus making estimates of masses, distance or line of sight extinction difficult. Here we have used a distance of 6 kpc and $N\\_{\\rm H}=6\\times10^{21}$ cm− 2 for GX 339 − 4, which are consistent with the limits given by and.\nOIR and radio data\n------------------\nXTE J1118+480 went through an outburst during January–February 2005. A discussion of the OIR light curve morphology of the 2005 OIR outburst of XTE J1118+480 is presented by. It was noted by that the 5–12 keV/ 3–5 keV hardness ratio from the *All Sky Monitor* data onboard the *Rossi X-Ray Timing Explorer* satellite suggested that the source was in an X-ray *hard* state during the entire outburst of 2005. For this source we used quasi-simultaneous optical (from the Liverpool telescope), near-IR (from UKIRT) and 15 GHz radio data (from the Ryle telescope). The full set of multi-wavelength observations obtained during this outburst of XTE J1118+480 will be presented in Brocksopp et al. (in prep).\nXTE J1118+480 went through an outburst in 2000 as well. During this outburst the source was extensively observed by many ground based telescopes as well as space-borne missions. In particular, the source was observed quasi-simultaneously around 2000 April 18th using the Ryle radio telescope, UKIRT, HST, EUVE, Chandra and RXTE giving an unprecedented broadband spectral coverage. This excellent data set has been used in several works to infer properties of the source. We revisit this data set to constrain the geometry and radiative properties of both the jet and the disc using our new disc+jet model.\nWe used the OIR and radio SED of GX 339 − 4 reported by, which were obtained during a multi-wavelength campaign to observe its outburst of 2002. For all data sets we used the photometric OIR magnitudes and de-reddened them using the interstellar extinction law of. In the absence of definitive measurement of the ratio of total to selective extinction (*R**V*[ = *A**V*/*E*(*B* − *V*)]) in the direction of either of the sources, we have taken *R**V* = 3.1, the widely used value measured by towards the galactic center, and standardly used for these sources. Thus it must be kept in mind that if the “true” value of *R**V* is significantly different from 3.1 (given neither of the sources are very close to the galactic center), it might introduce additional uncertainity in reddening correction.\nX-ray data\n----------\nFor both sources, we used data from the *Proportional Counter Array* and the *High Energy X-Ray Timing Experiment* onboard the RXTE satellite. The X-ray data were downloaded from the NASA HEASARC’s public archive[2](#fn2) and reduced using an in-house script following the standard reduction procedures outlined in *RXTE cook book[3](#fn3)* using HEASOFT software. To avoid any terrestrial contamination, we considered data only for elevation angles of 10∘ or higher. Also any data with pointing offset more than 0∘.02, or within 30 minutes of SAA passage, or trapped electron contamination ratio more than 0.1 were rejected. Since both the sources were bright (PCU2 count rate > 40 counts/s/PCU), the model *pca\\_bkgd\\_cmbrightvle\\_eMv20051128.mdl* was used for background subtraction. Data from all the layers of the active PCUs during the observations were extracted to create the spectra. Systematic uncertainty of 0.5% was added in quadrature to the PCA data to account for uncertainties in the calibration. For the HEXTE instrument, data from both clusters were added. Both PCA and HEXTE data were binned to achieve a minimum *S*/*N* of 4.5. For the PCA we used the energy range of 3–22 keV and for HEXTE 20–200 keV. The normalization of the OIR and radio data was tied to that of PCA. HEXTE normalization was allowed to vary.\nModel, Analysis and Results\n===========================\nThe continuum jet model used in this work is based on, (henceforth referred to as the agnjet model). Given the importance of irradiation in the observed spectra of XRBs, we have modified the model to include additional physics (see §[sec:irrad] for details) that allow us to calculate the effect of irradiation on the outer disc due to high energy photons from (1) the jet, and (2) the inner regions close to the compact object, impinging on the disc. We also compute inverse Compton scattering of thermal disc photons by the electrons in the jet plasma, although their effect on the total spectrum is negligible, except near the jet base, due to Doppler redshifting of the disc photons in the jet frame.\nJet parameters\n--------------\nThe details of the physics of the jet model as well as a full description of its main input parameters is given in the appendix of. Therefore we only give a brief summary here, and outline the modifications made to the model:\nThe main parameters that determine the properties of the jet are the input jet power (*N**j*), electron temperature of the relativistic thermal plasma entering at the jet base (*T**e*), the ratio of magnetic to particle energy density (a.k.a. the equipartition factor *k*), physical dimensions of the jet base (assumed to be cylindrical with radius *r*0 and height *h*0), and the location of the point on the jet (*z**a**c**c*) beyond which the a significant fraction of the leptons are accelerated to a power law energy distribution. *N**j*, parameterized in terms of the Eddington luminosity, determines the power initially input into the particles and magnetic field at the base of the jets. In the absence of a full understanding of the mechanism for energizing the jets, *N**j* plays a similar role to the compactness parameter in thermal Comptonization models. A typical value of the number density of leptons (*n*) at the base of the jet (derived from fits) is $n=N\\_{\\rm j}L\\_{\\rm Edd}/(4\\gamma\\_{\\rm s}\\beta\\_{\\rm s} m\\_{\\rm p} c^3 \n \\pi r\\_{\\rm 0}^2) \\sim 10^{14-15}$ cm− 3. Here $L\\_{\\rm Edd}$ is the Eddington luminosity of the source, $\\beta\\_{\\rm s}\\sim 0.4$ is the sound speed at the base (see below), $\\gamma\\_{\\rm s}=(1-\\beta\\_{\\rm s})^{-1/2}$, $m\\_{\\rm p}$ is the proton rest mass, and $r\\_{\\rm 0}\\sim 10^7$ cm is the radius of the jet base, implying an optical depth $\\tau=n\\sigma\\_T r\\_{\\rm 0}\\sim 0.001-0.01$. Since the optical depth is small, the probability of multiple Compton scattering is very small and we only consider single scattering. For the non-thermal power law electrons, the lower cutoff in the Lorentz factor is assumed to be equal to peak of the relativistic Maxwell-Juttner distribution, i.e. *γ**e*; *m**i**n* = 2.23*k**T**e*/(*m**e**c*2).\nFollowing the prescription of for maximally efficient jets, we assume that the bulk speed of the plasma (with adiabatic index Γ = 4/3) at the base of the jet is the sound speed given by $\\beta\\_{\\rm s}=\\sqrt{(\\Gamma-1)/(\\Gamma+1)}\\sim0.4$. We do not model the particle acceleration in the jet, but just start at the sonic point. Each jet then accelerates longitudinally along its axis due to pressure gradient and expands laterally with its initial sound speed. The bulk velocity profile, magnetic field strength, electron density and electron Lorentz factor along the jet are calculated by solving the adiabatic, relativistic Euler equation.\nIn the previous versions of the jet model we used the ratio of scattering mean free path to gyroradius of the particles in the plasma through which the shock moves (*f**s**c*), as a model parameter. However given the current lack of physical understanding of the actual shock propagation mechanism, in this paper we fixed the shock speed in the plasma (*β**s**h*) to 0.6 and used the quantity *ε**s**c* = *β**s**h*2/(ratio of scattering mean free path to gyroradius) as a model parameter instead of *f**s**c*. The parameter *ε**s**c* can be physically interpreted to be a parametrization for the shock acceleration rate. The dominant cooling processes within the jet are synchrotron and synchrotron self-Compton radiation. If the inner edge of the accretion disc is sufficiently close to the base of the jet, then the soft photons from the accretion disc, acting as seed photons for external Comptonization, can also cool the particles in the jet. The jet continuum as observed by an observer on Earth is a superposition of spectra from individual segments along the jet axis (taken to be perpendicular to the accretion disc plane), calculated by solving the radiative transfer equation, after taking into account relativistic beaming effects.\nThe irradiated disc\n-------------------\nSince the coupling between disc and jet is not well understood, we assume an independent classical thermal viscous “Shakura-Sunyaev” accretion disc, with a radial temperature profile *T*(*R*) ∝ *R*− 3/4, and modeled by the parameters $T\\_{\\rm in}$, $r\\_{\\rm in}$, the temperature and radius at the inner edge of the disc. If the outer parts of the disc are irradiated, then as discussed below, it is further possible to constrain the temperature and radius at the outer edge of the disc. Given the mounting evidence for the existence of relativistic outflows from black holes, we also explore the influence of the jet acting as a source of irradiation heating of the accretion disc.\n### Can jets influence the energetics of the accretion disc?\nWe computed the radiative jet flux incident on the accretion disc as a function of radial distance on the disc plane, taking into account special relativistic beaming effects. In order to obtain an *upper limit* on the jet-induced heating of the accretion disc, we assumed that the entire incident radiative flux from the jet *f**ν*(*R*) is thermalized upon being absorbed by the disc, thus heating the disc locally at radius *R* at a rate $H\\_{\\rm jet}(R)$, given by\n \n$$H\\_{\\rm jet}(R) = \\int\\_0^\\infty f\\_{\\nu}(R) \\mbox{ } d\\nu\n \\label{jet\\_irrad\\_eqn}$$ \nWe calculated $H\\_{\\rm jet}(R)$ at ten logarithmically spaced radii spanning from $R\\_{\\rm in}$ to $R\\_{\\rm out}$. The ratio of $H\\_{\\rm jet}(R)$ to the local viscous heating ($H\\_{\\rm visc}(R)$) is shown in Fig. [fig:jetirrad]. At large radii (*R* > 100 *R**g*), $H\\_{\\rm jet}(R)$ has a ∼ *r*− 2.4 radial profile, which is somewhat steeper than that expected from irradiation by a static corona or the inner accretion disc (for which the irradiation heating falls as $\\sim r^{-1.6{\\rm~to~}-2.0}$; see, e.g., §[sec:discirrad] and ). However the local viscous heating rate drops outward even faster ($H\\_{\\rm visc}(R) \\propto R^{-3}$), thus causing the ratio $H\\_{\\rm jet}(R)/H\\_{\\rm visc}(R)$ to slowly increase for increasing R. As the disc radius becomes small and approaches that of the jet base, the solid angle subtended by the jet increases rapidly causing the jet heating to saturate. However the viscous heating continues to rise at smaller radii and causes $H\\_{\\rm jet}(R)/H\\_{\\rm visc}(R)$ to drop at small radii. However, as evident from Fig. [fig:jetirrad], even the maximum possible jet-induced heating of the disc is about seven orders of magnitude smaller than the viscous heating. This confirms that relativistic beaming of the jet causes a very small fraction of the flux emitted by the jet to fall back on the disc, and the resulting influence of the jet on the energetics of the disc is negligible.\nLight bending effects near the black hole can enhance the flux incident on the disc, but by not more than a few tens of percent of the numbers calculated above (i.e. by assuming Minkowski metric instead of a more realistic Kerr or Schwarzschild metric).\n### An irradiation source near the compact object\nAnalysis of OIR light curves of XTE J1118+480 by suggests there might have been significant contribution of thermal flux in the optical. The near-IR flux however is reported to be significantly non-thermal in nature from studies of rapid time variability in the light curve, as well as nearly flat IR SED. As discussed in the previous paragraph as well as shown in Fig. [fig:jetirrad], the jet’s contribution in heating the disc is negligible; we have therefore added an irradiation heating term due to a somewhat more static source of irradiating X-rays near the black hole. The physical origin of this irradiating source could be the inner accretion disc near the black hole. Since we assume a radial temperature profile and do not solve the local disc structure, our model cannot discern the source of the irradiating X-rays.\nThe temperature due to irradiation heating ($T\\_{\\rm irrad}$) is usually assumed to have a power law radial dependence of the form *R*− *n*. Depending on the initial assumptions about disc structure and geometry of the irradiating source, theoretical models predict *n* to be in the range of 0.4-0.5. Given the uncertainities in de-reddening and scarcity of data points in the OIR, the exact choice of *n* is not vital for a global understanding of the broad-band SED. We used *n* = 3/7 (vertically isothermal disc with disc height *h* ∝ *r*9/7), mainly for consistency with earlier work.\nWe assume that the total disc heating is a sum of local viscous heating, and heating due to a static source of irradiation near the black hole. The effective temperature of the disc ($T\\_{\\rm eff}$) is therefore related to the viscous temperature ($T\\_{\\rm visc}$) and the irradiation temperature ($T\\_{\\rm irrad}$) as follows:\n \n$$T\\_{\\rm eff}^4(R) = T\\_{\\rm visc}^4(R) + T\\_{\\rm irrad}^4(R){\\rm,}$$ \nwhere $T\\_{\\rm visc} = T\\_{\\rm in}(R/R\\_{\\rm in})^{-3/4}$ and $T\\_{\\rm irrad}= T\\_{out}(R/R\\_{out})^{-3/7}$ with $T\\_{\\rm in/out}$, $R\\_{\\rm in/out}$ as free parameters. Other parameters, viz. masses, inclination and distance to the binary system, are taken from values published elsewhere and kept fixed during the process of fitting. The radius $R\\_{\\rm irrad}$ where irradiation heating becomes equal to viscous heating can be estimated from\n \n$$R\\_{\\rm irrad}=\\left[ (T\\_{\\rm in}/T\\_{\\rm out})\\\n R\\_{\\rm out}^{-3/7}/R\\_{\\rm in}^{-3/4}\\right]^{28/9}.$$ \nSince viscous dissipation dominates for $R < R\\_{\\rm irrad}$, the disc height at $R\\_{\\rm irrad}$ is estimated from Shakura-Sunyaev’s *α*-disc solution as\n \n$$H \\ (R\\_{\\rm irrad})=1.7\\times 10^8 \\ \\alpha^{-1/10} \\ \\dot M\\_{16}^{3/20} \n (M/{M}\\_\\odot)^{-3/8} \\ R\\_{\\rm irrad;10}^{9/8} \\ f^{3/5}\n\\label{eq:H\\_irrad}$$ \nwhere we have $\\dot M\\_{16}=\\dot M/(10^{16} \\ {\\rm g/s})$, $R\\_{\\rm irrad;10}=R\\_{\\rm irrad}/(10^{10} \\ {\\rm cm})$, $f=[1-2GM/(c^2 R\\_{\\rm irrad})]^{1/4}$ and chosen *α* = 0.05. Similarly, since *H* ∝ *R*9/7 when $R > R\\_{\\rm irrad}$, the disc height at $R\\_{\\rm out}$ is given by\n \n$$H \\ (R\\_{\\rm out}) = H(R\\_{\\rm irrad})\\times(R\\_{\\rm out} / R\\_{\\rm irrad})^{9/7}.\n\\label{eq:H\\_out}$$ \nThe solid angle subtended by the outer, irradiation dominated disc as seen from the compact object is computed from equations [eq:Hirrad] and [eq:Hout].\nThe agnjet model computes the broadband continuum from the jet and the irradiated disc but not line emission or reflection features. Therefore we used an additional Gaussian line profile to fit the iron K*α* emission complex near 6.5 keV and the convolution model reflect for the excess in the range of 10–30 keV, both features being usually attributed to reflection by the relatively cold accretion disc. The column density, taken from elsewhere published values, was fixed during the fits. For the reflect model, which accounts for reflection from neutral material in the accretion disc, we used solar abundance and tied the inclination to that of the disc (and also the jet axis). The energy of the iron line in the Gaussian model was allowed to vary between 6–7 keV and its width was fixed to 0.5 keV. The spectral coverage, along with column density ($N\\_{\\rm H}$) towards the sources, distances, and masses of the compact objects are given in Table [tab:data].\nModel fits\n----------\nSpectral fittings were performed using ISIS. Evaluation of confidence intervals for any physical model is computationally expensive, and this is true for agnjet model as well. Therefore we have used the parallelization technique described in and distributed the task of computing confidence intervals[4](#fn4) for *f* free parameters over *f*/2 or (*f* + 1)/2 nodes (depending on whether *f* is even or odd; each node is an Intel Xeon 3.4GHz processor) of a Parallel Virtual Machine running on the LISA cluster in Almere, Netherlands. This reduced the runtime to under 24 hours, a greater than 75% speedup; it also allowed us to discern finer features in parameter space by increasing the tolerance resolution by a factor of 500, while keeping the overall runtime to approximately half the serial runtime of 96 hours. While these are welcome improvements, reducing the model runtime even further would enable us to analyze new data more quickly. For this reason we are experimenting with ways of utilizing more than one processor per parameter, such as partitioning parameter space more finely, and exploring the use of OpenMP to parallelize internal loops within the model source code.\nUsing the distributed parallel computing technique described above, we determined the best-fit parameter values as well as their confidence intervals. Our confidence intervals correspond to Δ*χ*2 = 2.71 for a given parameter (which for *normal* distribution would imply a 90% single parameter confidence limit).\n### XTE J1118+480 during 2005 outburst\nOf the three data sets presented in this work, the data from the 2005 outburst of XTE J1118+480 was the faintest and had a steep power law slope in the X-rays. The flux in the HEXTE bands was extremely low for this data and while regrouping the data we found that no flux with S/N > 4.5 could be detected above 60 keV. Since the single point in the radio frequency does not constrain the length of the jet, we fixed it to 1013.1 cm which is a lower limit on the radiative length of the jet assuming that the radio data point is still on the optically thick synchrotron regime with a flat-to-slightly-inverted spectral index. While the fits require a high fraction of non-thermal electrons in the post-shock jet ( > 0.5), they are not very sensitive to the exact value of this fraction, and we fixed its value to 0.75. The data cannot be described by a non-thermal jet continuum alone and an excess in the OIR data requires additional flux which we modeled using the irradiated accretion disc model described in §[sec:irrad]. This thermal excess in the optical has also been suggested from the OIR coverage of the outburst by and.\nThe X-ray data (3–60 keV) can be well fit (*χ*2/*ν* = 88.8/55) only with a power law, however systematic residuals in the 5–7 keV range suggest that a weak iron line may be present. Adding a Gaussian line to the power law improves the fits to *χ*2/*ν* = 79.5/52, with the line center near 6.6 keV and width of 0.1 keV. Given the weak strength of the lines, we fixed the line energy as well as the width to these values, and did not use any reflection model modeling the broad band SED of XTE J1118+480.\nFits performed by allowing $r\\_{\\rm in}$ and $T\\_{\\rm in}$ of the accretion disc to vary freely results in good fits with quite small values of both $r\\_{\\rm in}$ and $T\\_{\\rm in}$. There are uncertainities in our understanding of the physical properties of the innermost regions of the disc close to ISCO, dereddening of interstellar extinction, and also uncertainities in instrumental response and calibration. These concerns make detection of a cold disc with small $r\\_{\\rm in}$, $T\\_{\\rm in}$ and its physical origin open to debate. Another issue with the XTE J1118+480 data is the jet/disc inclination. While we usually fix the inclination to a value published elsewhere, the extremely flat radio to IR spectrum of XTE J1118+480 requires a much smaller inclination than the published value of ∼ 70∘. Allowing the inclination to vary for this source results in jet inclination of ∼ 25∘.\nTherefore we tried two different models for this data set of XTE J1118+480. In the first model (Model 1; second column in Table [tab:fits]), apart from other regular parameters described above, the jet/disc inclination as well as $r\\_{\\rm in}$ and $T\\_{\\rm in}$ were allowed to vary freely. In the second, somewhat more conservative model (Model 2; third column in Table [tab:fits]), the jet inclination was fixed to 30∘, $r\\_{\\rm in}$, $T\\_{\\rm in}$ were also fixed to 30 *R**g* and 0.1 keV respectively. The best-fit values for the second model also gives similar jet parameters as the first model, although with a slightly higher jet power and steeper electron distribution index. Also the second model favors a somewhat smaller and hotter outer disc edge.\n### XTE J1118+480 during 2000 outburst\nWe used the broadband SED data of XTE J1118+480 obtained around 2000 April 18, published in and. While analyzing this data set a broad dip in the combined EUVE and Chandra spectra was noted between 0.15–2.5 keV (see e.g. the residuals in Fig. [fig:j11182000fit]) which cannot be modeled using the standardly used column density for this source ($1.3\\times 10^{22}\\ \n{\\rm cm}^{-2}$). This feature was been noted by previous works as well, and is attributed to metal absorption in a partially ionized gas,. We have therefore excluded the energy range between 0.15–2.5 keV from our broadband fitting. Results of fitting indicate that during this observation emission from the jet dominates the radio and IR regions. Contribution from the accretion disc starts becoming dominant in the optical. However, as discussed in, the EUVE fluxes are extremely sensitive to the extinction law, and the slope of the dereddened EUVE SED can range between + 2 ≥ *α* ≥ − 4 (*F**ν* ∼ *ν**α*) for the permissible range of N$\\_{\\rm H}$. Here we have used the EUVE fluxes reported by and a column depth of N$\\_{\\rm H}=1.3\\times10^{20}$ cm− 2, which results in a very steep EUVE slope. This cutoff in the EUVE data would imply a cold, truncated inner disc with with r$\\_{\\rm in}\\sim 340$ R*g* and T$\\_{\\rm in}\\sim 2.9\\times 10^5\\ {\\rm K}$. Given the low temperature of the inner disc and the consequent absence of a irradiating soft X-ray photons, it is not unexpected that we do not find any signature of irradiation in this case (i.e. T$\\_{\\rm out}$ is not constrained). As in the case of the 2005 data of this source, the X-ray emission in this model is entirely dominated by optically thin synchrotron emission from the jet. While the fits require a small nozzle (*h*0/*r*0 ∼ 0.2 − 0.6), they are not very sensitive to the exact value of this parameter, and we fixed its value to 0.4. We note that this value is abot a factor of three smaller than what is found typically in other sources, or even during the 2005 outburst of this source.\n### GX 339 − 4 during 2002 outburst\nThe X-ray data of GX 339 − 4 cannot be well fit by a power law alone. Using simple power law+ Gaussian model gives unacceptable fit to the data with reduced *χ*2 of 5.9. Convolving a power-law+gaussian with a reflection model give better fits, but still with rather a large reduced *χ*2 of 2.3. We found that the GX 339 − 4 data can be fitted well without including any thermal component, and the entire broadband SED is well described by the superposition of synchrotron and inverse Compton photons from the pre-shock region near the jet base as well as optically thick synchrotrons from post-shock regions farther from the jet base. From the radio-IR slope, the required inclination is between 45–50 degrees, but not well constrained and hence was fixed at 47 degree. The length of the jet was fixed to 1014.2 cm and fraction of non-thermal electrons fixed to 0.75 for the same reasons given for XTE J1118+480. Since the spectrum of GX 339 − 4 could be entirely dominated by jet emission, and the source was in a bright, hard X-ray state, the input jet power is much larger than that of XTE J1118+480. In fact the jet power for this observation is larger than any of the previous observations of this source. Also the acceleration region starts much farther out for the GX 339 − 4 data set.\nIf the temperature of the thermal electrons ($T\\_{\\rm e}$) at the base is left completely free, as in “Model 1” (fifth column in Table [tab:fits]), then $T\\_{\\rm e}$ for the best-fit model is comes out to be rather high, and suggests that the SED at energies > 10 keV is almost entirely due to SSC emission from the base (see top two panels in Fig. [fig:gx339fit]). The geometry of the base is rather compact and the source is in a bright hard state; therefore a strong SSC emission could imply that the photon density at the base becomes high enough for pair processes to become important. Since pair production is not included in the model, an estimate of the importance of pair processes for the best fit models were done as follows: the pair production rate ($\\dot n\\_{pp}$) was calculated from the photon field at the base of the jet using the angle averaged pair production cross section ; pair annihilation rate ($\\dot n\\_{pa}$) was calculated following. For “Model 1” of GX 339 − 4, $\\dot n\\_{pp}\\sim 10^{17}$ cm− 3*s*− 1 and $\\dot n\\_{pa}\\sim 10^{12}$ cm− 3*s*− 1, suggesting that pair processes could be important in this case. Therefore we explored the parameter space, limiting the electron temperature to be less than 5 × 1010 K and somewhat less compact base (larger r$\\_{\\rm 0}$, h$\\_{\\rm 0}$) to search for solutions which also satisfy the condition $\\dot n\\_{pp}<\\dot n\\_{pa}$ (see “Model 2”, sixth column in Table [tab:fits] and bottom panels in Fig. [fig:gx339fit]). The fit derived number density (*n*), $\\dot n\\_{pa}$ and $\\dot n\\_{pp}$ for the different models presented in Table [tab:fits], are given in Table [tab:pairs].\nIn addition to the continuum jet model, an additional Gaussian line centered at 6.3 keV and reflection from the accretion disc is needed to obtain good fits for the GX 339 − 4 data. Presence of the line and reflection suggests the presence of an accretion disc. However, as in the case for XTE J1118+480, the disc is cold and therefore not detected by the PCA.\nThe best-fit model parameters, associated confidence intervals, *χ*2 statistic and corresponding chance probabilities for both sources are shown in Table [tab:fits]. Given the simplicity of the model it is hardly unexpected that the reduced *χ*2 values obtained from the fitting are not very close to 1 in some cases. For instance, the viscous+irradiated disc model does not include details of disc atmosphere or ionization balance equations, and is therefore not able to predict the spectral breaks in the optical data. Because of its simplicity, this is not meant to be a real model of the disc, although it incorporates the dominant ongoing physical processes and can reproduce the general shape of the broadband continuum. Moreover, while fitting a data set spanning 10 decades in frequency (or energy) space, the long lever arm between radio and X-rays constrains the jet model more strongly than the overall chi-squared. Best fit models along with data and residuals are shown in Figures [fig:j11182005fit], [fig:j11182000fit] and [fig:gx339fit].\nDiscussion and Conclusions\n==========================\nWe have analyzed broad band multi-wavelength observations of the galactic black hole transient systems XTE J1118+480 and GX 339 − 4, during their outbursts in 2000 and 2005 for XTE J1118+480, and during the 2002 outburst of GX 339 − 4. The conclusions from our study are summarized below.\nThe data for XTE J1118+480 cannot be fit well with a jet continuum model alone, due to an excess in the OIR fluxes. During these observations, in OIR the source was at least 3.5 magnitude brighter than quiescence; therefore the contribution from the M1 V secondary star is small. Modeling the data from the 2000 outburst and 2005 outburst of this source we found that the outer disc was strongly irradiated during the 2005 data. It appears that the inner edge of the accretion disc was much closer to the central black hole during the observations of 2005, thus providing an ample source of soft X-ray photons to irradiate the outer disc. On the other hand, the sharp cutoff in the EUVE data and absence of any signature of irradiation from the outer disc suggests a recessed, cold accretion disc for the 2000 data. However, as discussed in §[sec:j11182000], the slope of the EUVE data depends very strongly on the assumed column depth, and a small decrease in N*H* can flatten the EUVE SED significantly. Such a flat EUVE SED would be expected from a disc with small r$\\_{\\rm in}$ and comparatively higher T$\\_{\\rm in}$.\nOur calculations in §[sec:jetirrad] show that at any point on the surface of the accretion disc, the heating of the disc by the jet is seven or more orders smaller than the viscous heating. There are two factors which largely reduce the amount of jet radiation that is incident on the disc: Firstly Doppler de-beaming - since the bulk motion of the jet is relativistic, special relativistic Doppler beaming concentrates most of the emergent radiation from the jet in the direction of its motion, i.e. away from the disc. A second, but significant factor that also contributes to the dimunition of jet flux incident on the disc is the fact that the post-shock jet synchrotron spectra peaks at increasingly longer wavelengths at increasingly larger distances from the base of the jet, thus creating fewer X-ray photons that can heat the disc. This is illustrated in Fig. [fig:jetirrad] where we show the ratio of jet-heating to viscous heating as a function of radius. Since heating by the jet alone cannot produce the OIR excess in Fig. [fig:j11182005fit] we surmised that this the OIR excess is indeed caused by the more conventional form of irradiation, i.e. irradiation by soft X-ray photons from the inner-disc. From an energetics point of view, it is easy to see that the inner accretion disc alone (see Fig. [fig:j11182005fit], at around 1 keV) radiates at least two orders or more energy than the outer region. Furthermore, it is well known that the outer edge of the accretion disc can flare-up or warp, thus making the outer parts of the disc more easily visible from the inner regions.\nFits to the 2005 data of XTE J1118+480 using the jet model+irradiated disc (described in §[sec:discirrad]) suggest a cold disc ($T\\_{\\rm in}\\sim 0.2\\ keV$) with inner radius close to the innermost stable circular orbit (6 *R**g* for a Schwarzschild black hole, and smaller if spinning). Given the uncertainities associated with using a simple *T* ∼ *R*− 3/4 model which ignores correction factors like spectral hardening, full disc structure as well as proper boundary conditions at the inner edge, the errors associated with the estimates of the inner disc parameters are most likely an underestimate. Another source of uncertainity comes from that fact that we assume the jet to be perpendicular to the disc. A misaligned jet, as seems likely for XTE J1118+480, could introduce additional error in the determination of $r\\_{\\rm in}$. Along with the uncertainities in instrument calibration, modeling and jet-disc alignment, the sharp low energy cutoff of PCA sensitivity below 3 keV makes detection of a cold disc with small $r\\_{\\rm in}$ even more difficult than a recessed cold disc. Nevertheless the numbers for $r\\_{\\rm in}$ and $T\\_{\\rm in}$ we obtained for XTE J1118+480 are similar to those obtained by several others for black hole transients in their hard state. If such a cold disc with small r$\\_{\\rm in}$ is indeed realised, then Compton scaterring of the disc photons could be strongly anisotropic, with most of the scattered emission beamed toward the disc. This could compensate partly the beaming effects due to the relativistic motion of the gas, and increase the illumination of the disc (both irradiation and reflection effects). Also in this case there could be a radiative feedback due to the local illumination of the disc by the base of the jet which is similar to the situation in standard accretion disc corona models which would act as a thermostat keeping the temperature at the base lower than what is assumed here. A somewhat more conservative model, where the temperature and radius of the inner edge of the accretion disc were fixed to 0.1 keV and 30 R$\\_{\\rm g}$ respectively (Model 2 in Table [tab:fits]) also describes the data well, showing the difficulty in detecting a cold disc with small r$\\_{\\rm in}$ from the available data.\nThe best-fit outer radii of XTE J1118+480 for both *Model 1* and *Model 2* are smaller than the circularization radius, using the latest values of orbital parameters published by. These results are therefore consistent with the assumption that accretion is occuring via Roche lobe overflow. The observation was made during the decline from outburst peak. Hence the discrepancy between the circularization radius and *R**o**u**t* inferred from fits, in the context of standard disc instability models, could be due to an inward moving cooling wave. A decreasing trend in the outer *radiative disc radius* was also observed by for the system XTE J1859+226. However the value of *R**o**u**t* in our case depends largely on the five OIR data points, all of which lie on the Rayleigh-Jeans region of the irradiated disc. Given the plethora of spectral features seen in OIR spectra of X-ray binaries in outburst, the limitation of using a few bands is obvious.\nAnalysis of the data from the 2000 outburst of XTE J1118+480 by and showed that the UV and higher energy data can be explained by an ADAF model. The flux from the ADAF model alone however falls sharply at energies below optical and underestimates IR and any lower energy data (e.g. radio), thus requiring an additional jet component. In the jet-ADAF model, the X-ray photons have a thermal Comptonization origin, whereas in the current work, the X-ray photons can originate from either synchrotron (e.g in XTE J118+480), or inverse Compton scattering, or a combination of both (e.g. in GX 339 − 4). Better simultaneous broadband coverage and future X-ray polarization studies can help to estimate the relative importance of these two emission components which dominate the X-ray spectra of X-ray binaries in hard state.\nFrom the available data, the X-ray spectrum of XTE J1118+480 during both outbursts show little or no curvature at energies higher than ∼ 10 keV. Reflection features like the iron line complex and the reflection “bump” near 20 keV are also extremely weak. Detailed analysis of the X-ray spectra taken by Chandra and RXTE, during the 2000 outburst of this source by also showed extremely weak reflection features, as would be expected for a jet+recessed disc scenario. From a phenomenological point of view, since the curvature is small, using a broken power law to describe data of X-ray binaries in hard state has been shown to work remarkably well, e.g., for GX 339 − 4 by and for Cyg X − 1 by. Residuals of a power law fit to the X-ray continuum in the 2005 data set of XTE J1118+480 show small systematic variations near 5–7 keV, which improves slightly upon addition of a Gaussian line. However, given the weakness of the line, its energy or width cannot be constrained well. Total contribution from various source of inverse Comptonization (SSC within the jet base and external Comptonization of soft thermal photons from the accretion disc) is also much smaller compared to that of GX 339 − 4. The X-ray region of the SED of XTE J1118+480 is dominated by post-shock synchrotron photons for both 2000 and 2005 outbursts. We note that the electron acceleration region seems to be in the nozzle of the jet in the case of 2005 outburst data of XTE J1118+480 while it is farther out in the jet in GX 339 − 4 as well as the 2000 outburst of XTE J1118+480. While exhaustive searching of the *χ*2 space confirms that this is required by the data, it is not clear why the acceleration region is inside the nozzle for the 2000 data of XTE J1118+480. Another unexpected result is that the base of the jet seems to be quite small (*h*0/*r*0 ∼ 0.4) for the 2000 outburst data of XTE J1118+480, whereas *h*0/*r*0 ∼ 1.5 for other outbursts of XTE J1118+480 as well as other sources (see e.g. ). Given the uncertainties in our understanding of jet formation and the acceleration region, and the simplicity of the model, here we report the best fit parameters as obtained. The particle energy distribution index (*p*) of the relativistic post-shock electrons for data from both outbursts of XTE J1118+480 is ∼ 2.5–2.6. This value is much steeper than what could be expected from an initial distribution of particles accelerated by a diffusive shock process. However it is possible that for XTE J1118+480 the observed X-ray bandpass is at a higher energy than the cooling break energy, where the particle distribution steepens from *p* to *p* + 1 due to enhanced cooling. One of the key assumptions in our model is time independence of particle distributions. However if the steep power law drop-off of the observed photon index is indeed due to enhanced cooling, or more generally a departure from the assumed equillibrium between the heating and cooling processes, a more thorough time dependent analysis of particle evolution is neccessary. We will present time dependent particle evolution in the context of jets in a forthcoming paper.\nThe jet inclination of XTE J1118+480 is still a matter of concern. The optically measured system inclination is near 70∘. However, given the flat radio-to-IR fluxes, jet inclinations higher than ∼ 30∘ would be very difficult to fit which might suggest a large misalignment between between the disc and the jet. Evidence of misalignment between disc and jet has been seen in several stellar X-ray binary systems, e.g., GRO J1655 − 40, SAX J1819 − 2525, XTE J1550 − 564, as well as in several AGNs, e.g., NGC 3079, NGC 1068 and NGC 4258. Allowing the inclination to vary for XTE J1118+480 during the fits gives a best-fit inclination of ∼ 25∘, and it is almost impossible to find statistically good fits (e.g. with *χ*2/*ν* < 2) for inclinations larger than 30∘. Previous fits by also required a similar, small inclination. In our model we assume the lateral expansion and longitudinal acceleration of the jet to be driven solely via the solutions of the relativistic Euler equation for adiabatic expansion. If this is not the case, and e.g. the jet radius and acceleration are determined by some other process(es) leading to a stronger magnetic field and/or smaller bulk velocity along the jet, that could lead to a flatter radio-IR spectrum as well.\nThe data for GX 339 − 4 does not formally require a thermal component (e.g. an accretion disc or emission from the secondary star) and the entire broad band emission from radio to X-rays is satisfactorily well fit by non-thermal photons originating from pre- and post-shock synchrotron, and synchrotron self Compton emission mainly from the jet base. Previous fits to the 3–100 keV X-ray data by also suggested that the disc contribution within this energy range is ∼ 1% of the total luminosity or less. However presence of the line near 6.5 keV and signature of reflection suggests a cold accretion disc near the black hole which cuts off exponentially at an energy significantly below the lower detection limit of the RXTE/PCA. Since the X-ray spectrum has a luminosity of ∼ 5% L$\\_{\\rm Edd}$, the disc luminosity is presumably ∼ 0.05% L$\\_{\\rm Edd}$ or smaller.\nThe data for GX 339 − 4 were taken when the source was in a bright, X-ray hard state. The jet power *N**j* is much higher than that obtained by, who studied data from previous outbursts of the same source. We also obtain a higher equipartition factor (*k*) and shallower particle distribution index. It is interesting to note that *r*0 and *z**a**c**c* for GX 339 − 4 are always found to be higher than those found for other sources. If the electron temperature at the base is left unconstrained (e.g. “Model 1” in the fifth column of Table [tab:fits]), then the density of photon field from best-fit model is high enough that pair processes may become important. In this case many photons are up-scattered above the pair production threshold, and the rate at which pairs are produced is much larger than the rate at which they are annihilated, thereby changing the density at the base of the jet. Constraining the electron temperature to be less than 5 × 1010 K, we present another fit (“Model 2”, sixth column of Table [tab:fits]) where the photon density at the base is small and pair production rate is much smaller than the pair annihilation rate. For this fit the the fractional pair annihilation rate $(\\dot n\\_{\\rm pa}/n)$ is also much smaller than unity, so that pair processes are not important for this fit.\nUsing the irradiated disc+jet model presented in this paper we have shown that in certain cases (e.g. the 2000 outburst data of XTE J1118+480) there are not enough X-ray photons from the inner regions to irradiate the disc appreciably. The 2005 outburst data of XTE J1118+480 presents an intermediate case where the contribution from an irradiated outer disc is comparable to the jet emission in OIR. The 2002 data of GX339 − 4 presents the extreme case where the entire broadband SED is dominated by emission from the jet.\nThe jet model has been used to analyze broad band SEDs of the stellar mass black holes XTE J1118+480, GX 339 − 4, Cyg X-1, A0620-00, GRO J1655-40, as well as that of the supermassive black holes in the nucleus our own galaxy and that of M81. From the analysis of the spectra of these sources we note that the geometry of the jet base, scaled in units of gravitational radius, does not vary largely. The radius of the jet base, *R*0, lies in the range of 10–100 *R**g*, and the ratio of *h*0/*r*0 at the nozzle lies within 0.4–11, both suggesting a relatively compact jet base and the universal nature of mass scaling in accretion onto compact objects. The fraction of particles accelerated to a power law distribution beyond *z**a**c**c*, while not very well constrained by the fits, preferes a value in the range of 0.6–0.9 and is fixed to 0.75. Considering that most of the known acceleration mechanisms in astrophysics do not have such a high efficiency, this number seems rather high and requires further exploration. The particle acceleration index (*p*) lies within 2.2–2.7, which is steeper than that expected from a relativistic shock, suggesting that in these cases the X-ray energies may lie beyond the *cooling break*, so that the power law index of observed SED steepens by an additional amount of 0.5.\nThe magnetic-to-particle energy densities at the jet base (*k*) is often found to be greater than unity. MHD simulations of relativistic jet launching suggest that near the jet base, close to the launching points, the jets are mostly Poynting flux dominated and have small plasma fractions (ratio of gas pressure to magnetic pressure is less than unity). We are focusing on inner regions where this is likely still to be the case. In GRBs and AGN, most data is only relevant to regions well beyond the magnetosonic fast points, where the flow has been driven towards equipartition via conversion of magnetic energy to particle energy.\nTemperature of the thermal electrons at the jet base (*T**e*) is in the range of 2 − 7 × 1010 K for the stellar black holes and ∼ 1011 K for the supermassive black holes. The flow near the jet base could be a radiatively inefficient accretion flow (RIAF), i.e. an ADAF, or in light of the the magnetic domination, an MDAF. In such a case, the ion temperature can be as high as the virial temperature, and electrons at 109 − 10 K. Keep in mind that for an outflow model, the particles are also not required to be gravitationally bound, and processes in the jet launching can heat the particles beyond virial, in theory.\nMost likely the similarity in the derived physical parameters among the diverse sources and their diverse accretion states is not coincindental, and may hint towards an inherent similarity in the process of formation and propagation of compact jets. The results presented above show the importance of quasi-simultaneous, multi-wavelength, broad band SEDs in constraining geometrical and radiative parameters associated with accertion flows near compact objects.\n*Acknowledgements*. This work was supported mainly by Netherlands Organisation for Scientific Research (NWO) grant number 614000530. JW acknowledges DLR grant 50OR0701. We would like to thank Guy Pooley for helping with the radio data, Michelle Buxton and Mickael Coriat for the OIR data reduction. We would also like to thank the anonymous referee for his/her constructive criticisms that have greatly improved the paper. DM thanks Asaf Pe’er for a discussion on pair processes in plasmas. It is a pleasure to acknowledge the hospitality of International Space Science Institute (ISSI) in Bern where a significant amount of this work was carried out.\nThis research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. This work has also made use of the United Kingdom Infrared Telescope (UKIRT) which is operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K. The Liverpool Telescope is operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council. We have also used radio data from the Ryle telescope operated by Mullard Radio Astronomy Observatory. Numerical computations for this work were done on the LISA workstation cluster, which is hosted by Stichting Academisch Rekencentrum Amsterdam (SARA) computing and networking services, Amsterdam.\n| | XTE J1118+480 (2000 outburst) | XTE J1118+480 (2005 outburst) | GX 339 − 4 (2002 outburst) |\n| --- | --- | --- | --- |\n| MJD of observation | 51652 | 53393 | 52367-68 |\n| Spectral Coverage | | | |\n| Radio: | 15 GHz | 15 GHz | 5 GHz |\n| UV,Opt/IR: | M, L, K, H, J-bands, EUVE and HST spectra | K, H, J, V, B-band | H, I, V-band |\n| X-ray: | Chandra (0.24–7 keV), RXTE (3–110 keV) | RXTE (3–70 keV) | RXTE (3–200 keV) |\n| $N\\_{\\rm H}$ (1022 *c**m*− 2) | 0.013 | 0.013 | 0.6 |\n| $M\\_{\\rm BH}$ (*M*⊙) | 8.5 | 8.5 | 7 |\n| Distance (kpc) | 1.7 | 1.7 | 6 |\n[tab:data]\nllllll Parameter (Unit) & XTE J1118+480 & XTE J1118+480 & XTE J1118+480 & GX 339 − 4 & GX 339 − 4 \n& (2005 outburst; Model 1) & (2005 outburst; Model 2) & (2000 outburst) & (2002 outburst; Model 1) & (2002 outburst; Model 2) \n$N\\_{\\rm j}$ (10− 3 ${\\rm L}\\_{\\rm Edd}$) & 2.32− 0.06+ 0.02 & 2.77− 0.05+ 0.01 & 7.2− 0.5+ 0.1 & 55− 9+ 16 & 83.8− 0.2+ 0.2 \n& & & & \n*r*0 (R$\\_{\\rm g}$) & 9.39− 0.17+ 0.18 & 11.8− 0.2+ 0.1 & 20.8− 1.4+ 1.0 & 33− 2+ 3 & 114− 11+ 7 \n& & & & \n*h*0/*r*0 & 1.589− 0.002+ 0.013 & 1.39− 0.02+ 0.03 & 0.4 ⋆ & 1.7− 0.1+ 0.2 & 1.51− 0.03+ 0.02 \n& & & & \n$T\\_{\\rm e}$ (1010 K) & 4.22− 0.02+ 0.11 & 4.04− 0.02+ 0.01 & 3.76− 0.07+ 0.09 & 6.7− 0.5+ 0.4 & 3.8− 0.2+ 0.01 \n& & & & \n$z\\_{\\rm acc}$ (R$\\_{\\rm g}$) & 12.6− 0.0+ 1.5 & 12.6− 0.0+ 2.8 & 8.2− 0.2+ 3.3 & 252− 130+ 212 & 526− 41+ 85 \n& & & & \n*p* & 2.63− 0.01+ 0.01 & 2.71− 0.01+ 0.01 & 2.48− 0.01+ 0.02 & 2.28− 0.03+ 0.03 & 2.20− 0.01+ 0.02 \n& & & & \n$\\epsilon\\_{\\rm sc}$ (10− 3) & 9.9− 0.1+ 0.02 & 29.9− 3.0+ 0.3 & 5.5− 0.8+ 1.4 & 0.16− 0.08+ 0.08 & 44− 1+ 2 \n& & & & \n*k* & 1.9− 0.1+ 0.1 & 2.3− 0.2+ 0.1 & 2.1− 0.1+ 0.1 & 2.2− 0.8+ 1.4 & 2.38− 0.01+ 0.01 \n& & & & \nInclination (deg) & 25.1− 0.0+ 0.1 & 30 ⋆ & 30 ⋆ & 47 ⋆ & 47 ⋆ \n& & & & \n$r\\_{\\rm in}$ (R$\\_{\\rm g}$) & 2.6− 0+ 5 & 30 ⋆ & 341− 22+ 4 & Not constrained & Not constrained \n& & & & \n$T\\_{\\rm in}$ (keV) & 0.254− 0.008+ 0.004 & 0.1 ⋆ & 0.025− 0.002+ 0.001 & Not constrained & Not constrained \n& & & & \n$r\\_{\\rm out}$ (R$\\_{\\rm g}$) & 15700− 350+ 160 & 8850− 1550+ 1800 & 29400− 5750+ 610 & Not constrained & Not constrained \n& & & & \n$T\\_{\\rm out}$ (K) & 18100− 600+ 1000 & 30500− 100+ 200 & 650− 550+ 2150 & Not constrained & Not constrained \n& & & & \n*f* (10− 4) & 1.2 & 1.0 & Not constrained & Not used & Not used \n& & & & \n$E\\_{\\rm Line}$ (keV) & 6.63 ⋆ & 6.63 ⋆ & Not used & 6.3− 0.2+ 0.2 & 6.3− 0.2+ 0.3 \n& & & & \n$W\\_{\\rm Line}$ (eV) & 39.3 & 27.7 & Not used & 93.5 & 58.7 \n& & & & \nΩ/2*π* & Not used & Not used & Not used & 0.22− 0.14+ 0.10 & 0.35− 0.02+ 0.02 \n& & & & \n*χ*2/*ν* [Q] & 70/50 [0.032] & 90/53 [0.0011] & 140/137 [0.406] & 137/112 [0.054] & 157/112 [0.0032] \n[tab:fits]\n| Source/Outburst/Model | log (*n* [cm− 3] ) | log ($\\dot n\\_{pa}$ [cm− 3 s− 1] ) | log ($\\dot n\\_{pp}$ [cm− 3 s− 1] ) |\n| --- | --- | --- | --- |\n| XTE J1118+480/2005/1 | 13.9 | 12.0 | 7.6 |\n| XTE J1118+480/2005/2 | 13.8 | 11.8 | 6.8 |\n| XTE J1118+480/2000 | 13.7 | 11.7 | 5.8 |\n| GX 339 − 4/2002/1 | 14.2 | 12.4 | 17.1 |\n| GX 339 − 4/2002/2 | 13.4 | 11.0 | 6.1 |\n[tab:pairs]\nComparing viscous heating ($H\\_{\\rm visc}$) and *maximum possible* “jet-heating” ($H\\_{\\rm jet}$) for XTE J1118+480. The filled circles are the computed upper limits of the ratio of jet induced heating to viscous heating ($H\\_{\\rm jet}/H\\_{\\rm visc}$), as discussed in §[sec:jetirrad], joined by a smooth spline. $T\\_{\\rm in}$, $R\\_{\\rm in}$ and *R**o**u**t* for the disc were determined from broadband “Model 1” fits to the SED. For *R* > 100 *R**g*, the jet induced irradiation heating term falls more slowly ( ∼ *R*− 2.4) than the viscous heating ( ∼ *R*− 3), which causes the ratio $H\\_{\\rm jet}/H\\_{\\rm visc}$ to slowly increase at increasing radii. For $R\\_{\\rm in} < r\\_0$, the jet heating saturates but the viscous heating continues to increase for smaller radii, causing $H\\_{\\rm jet}/H\\_{\\rm visc}$ to drop sharply for decreasing radii. [fig:jetirrad]\n \n \n \n99 Arnaud, K. A. 1996, in Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, ASP Conf. Ser. No. 101 (San Francisco: ASP), 17 Blandford, R. 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Using our own custom versions of the cl\\_master and cl\\_slave scripts described at http://space.mit.edu/cxc/isis/single\\_param\\_conf\\_limits.html[↩](#fnref4)\n\nConstraining Jet/Disk Geometry and Radiative Processes in Stellar Black Holes XTE J1118+480 and GX 339 − 4\n==========================================================================================================\n### Accepted 2009 April 8. Received in original form 2008 June 9\n[firstpage]\nWe present results from modeling of quasi-simultaneous broad band (radio through X-ray) observations of the galactic stellar black hole (BH) transient X-ray binary (XRB) systems XTE J1118+480 and GX 339 − 4 using an irradiated disc + compact jet model. In addition to quantifying the physical properties of the jet, we have developed a new irradiated disc model which also constrains the geometry and temperature of the outer accretion disc by assuming a disc heated by viscous energy release and X-ray irradiation from the inner regions. For the source XTE J1118+480, which has better spectral coverage of the two in optical and near-IR (OIR) wavelengths, we show that the entire broad band continuum can be well described by an outflow-dominated model + an irradiated disc. The best-fit radius of the outer edge of the disc is consistent with the Roche lobe geometry of the system, and the temperature of the outer edge of the accretion disc is similar to those found for other XRBs. Irradiation of the disc by the jet is found to be negligible for this source. For GX 339 − 4, the entire continuum is well described by the jet-dominated model only, with no disc component required. For the two XRBs, which have very different physical and orbital parameters and were in different accretion states during the observations, the sizes of the jet base are similar and both seem to prefer a high fraction of non-thermal electrons in the acceleration/shock region and a magnetically dominated plasma in the jet. These results, along with recent similar results from modeling other galactic XRBs and AGNs, may suggest an inherent *unity in diversity* in the geometric and radiative properties of compact jets from accreting black holes.\nblack hole physics— radiation mechanisms: non-thermal and thermal— X-rays: binaries— stars: winds, outflows– accretion, accretion discs– X-rays: individual: (GX 339 − 4, XTE J1118+480)\nIntroduction\n============\nSoft X-ray transient (SXT) binary systems that occasionally go into episodes of high activity or *outbursts* are some of the best laboratories to study accretion flows near compact (*G**M*/*R* ∼ *c*2) objects. Coordinated multi-wavelength campaigns have been carried out in recent years to observe SXTs simultaneously over a broad frequency range of nine decades or more from radio to X-rays. These campaigns have provided extremely valuable insights about the accretion in/outflows and helped to constrain the theoretical models predicting accretion geometry and physical emission processes. Outbursts of SXTs are usually explained in terms of a disc instability model, postulating a dramatic increase in mass accretion via an accretion disc towards the central compact object. The luminosity of a black hole transient (BHT) can vary by as much as eight orders of magnitude between quiescence and peak outburst, within timescales of weeks to months. Such systems are also ideal for studying the formation of jets, which are a by-product of accretion, yet poorly understood so far.\nMatter spiralling into the gravitational potential well of the compact object forms an accretion disc, and radiates via dissipation of the accumulated gravitational potential energy. The observed UV and soft X-ray spectra of these sources, during epochs of high mass accretion rate, are well modeled by viscous dissipation from the inner regions of a geometrically thin, optically thick “Shakura–Sunyaev disc”.\nHowever in the OIR, the viscously dissipating disc model sometimes significantly underpredicts the observed spectral energy distribution (SED), and the excess is attributed to irradiation of the outer disc by the intense X-ray radiation from the inner region close to the compact object. This irradiation mechanism has been well explored theoretically and their effect observed in optical and near-IR wavelengths. The most prominent, observable effect of irradiation is to heat up the outer regions of the accretion disc to temperatures higher than what would be possible solely by viscous dissipation of a standard Shakura-Sunyaev disc. This leads to a deviation in the shape of the SED of the accretion disc (from that of a standard disc), creating a ‘bump’ in the optical through infra-red (OIR) wavelengths. As the quality of the OIR data improves, the deviation (i.e. the bump) from Shakura-Sunyaev spectrum becomes more and more apparent, and the need to incorporate irradiation in the model becomes important.\nIn Roche lobe overflowing BHTs where the accretor is 10 times or more heavier than the donor, X-ray irradiation of the donor is small. Furthermore irradiation causes the outer disc to flare, shielding most of the donor star from irradiation. While most models assume an azimuthally symmetric accretion disc, warping of the accretion disc can introduce additional uncertainities in shielding the donor from irradiation.\nNot all the mass accreted inwards falls onto the compact object. Outflows, in the form of steady compact jets, or transient, ballistic ejecta are often seen in compact binary systems. Steady, compact jets have been observed from an increasing number of compact binary systems when they are in an *X-ray hard state* (see below). The hallmark signature of such a jet is its flat to slightly inverted spectrum which, following the classic work by, is usually attributed to a superposition of self-absorbed synchrotron emission from segments of a collimated jet. For stellar BHTs the flat spectrum of the jet is expected to extend from radio frequencies up to IR, beyond which the jet synchrotron becomes optically thin. At X-ray frequencies, inverse Compton (IC) upscattering in the jet base of synchrotron as well as external photons from the accretion disc can become comparable and may even dominate the spectrum. As shown in, synchrotron and IC at the jet base can result in spectra very similar to those produced by compact Comptonizing coronal models.\nDuring the course of an outburst, the X-ray spectral and temporal features show an enormous amount of diversity. However extensive observations and their analyses over the past few decades have enabled the classification of spectral and temporal features into relatively few groups known as *X-ray states*. In particular, two canonical states are often seen: a “soft” state characterized by strong thermal emission from the accretion disc and low variability in the light curve, and a “hard” state characterized by non-thermal power law emission, high variability and probably a collimated outflow. Besides these canonical states, a host of intermediate and extreme states with varying ratios of thermal and non-thermal radiation also are seen. Recent OIR observations covering complete outbursts of many XRBs have shown that while irradiation usually plays a dominant role in the evolution of the OIR spectrum during the entire outbursting phase, the contribution from a non-thermal source, e.g. a jet, can become dominant when XRBs (most notably black holes) are in hard state. Thus any model which aims to understand disc-jet coupling must include irradiation.\nWe have developed a new spectral model which calculates continuum emission from an irradiated accretion disc, and in conjunction with a modified version of the compact jet model developed by, allows us to constrain the geometrical and radiative properties not only of the jet, but simultaneously those of the accretion disc. The thermal photons from the irradiated disc are also included in the photon field of the jet for Compton scattering, although due to Doppler redshifting in the jet frame, this external Compton emission is usually much smaller than the synchrotron self Compton (SSC). The new model is fully integrable with the standard spectral analysis software ISIS and parallelizable for using in large cluster computing environments (also see §[sec:parallel]).\nWe have used this new model to analyze broad band quasi-simultaneous observations of two galactic BHT systems XTE J1118+480 and GX 339 − 4. Simultaneous or quasi-simultaneous observations of both sources, during hard X-ray state of their outbursts, have revealed broad band continuum emission ranging from radio to X-rays, strongly suggesting the presence of jets. By studying short time-scale variability in near-IR wavelengths from XTE J1118+480, also suggested a non-thermal origin of the near-IR radiation, with a power law index typical of optically thin synchrotron emission from a jet.\nThe known physical properties of the two sources XTE J1118+480 and GX 339 − 4 have been briefly summarized in §[sec:data]. Since analysis of multi-wavelength data requires careful calibration of data from various ground-based as well as space-borne instruments, we have also presented the data reduction and calibration procedures in §[sec:data]. A brief outline of the model, with somewhat more emphasis on the newly introduced irradiated disc component, is given in §[sec:model], along with a description of the fitting procedure. The results of our broad band SED modeling suggest that despite the differences in size, mass, orbital parameters or accretion state, some of the fundamental physical parameters characterizing the geometrical and radiative properties of the jet are similar for both sources. This similarity may be a global property of jets formed near compact objects and is discussed in §[sec:discussion].\nSources, Observations & Data Reduction\n======================================\nXTE J1118+480 and GX 339 − 4 are two prototype galactic BHTs with markedly different properties. Both binary systems contain high mass compact objects, $8.5\\pm 0.6 \\rm{M}\\_\\odot$ for XTE J1118+480 and $>6\\ \\rm{M}\\_\\odot$ for GX 339 − 4, thus strongly suggesting for the compact objects to be black holes in both cases.\nXTE J1118+480 lies 62∘.3 north of galactic plane, in the halo. It has an orbital period of 4.04 hours. The distance and interstellar column density to XTE J1118+480 are reasonably well known from OIR photometry during quiescence to be 1.72 ± 0.1 kpc and 1.3 × 1020 cm− 2 respectively. We adopted a black hole mass of $8.5\\ \\rm{M}\\_\\odot$ for this source. The orbital inclination of XTE J1118+480, as measured by is 68 ± 2 degrees.\nGX 339 − 4 lies near the galactic plane, only 4∘.3 south of the galactic equator and has an orbital period of 42.14 hours. We adopted a black hole mass of $7\\ \\rm{M}\\_\\odot$ for GX 339 − 4 in this work. The secondary star in the GX 339 − 4 system has eluded detection so far, thus making estimates of masses, distance or line of sight extinction difficult. Here we have used a distance of 6 kpc and $N\\_{\\rm H}=6\\times10^{21}$ cm− 2 for GX 339 − 4, which are consistent with the limits given by and.\nOIR and radio data\n------------------\nXTE J1118+480 went through an outburst during January–February 2005. A discussion of the OIR light curve morphology of the 2005 OIR outburst of XTE J1118+480 is presented by. It was noted by that the 5–12 keV/ 3–5 keV hardness ratio from the *All Sky Monitor* data onboard the *Rossi X-Ray Timing Explorer* satellite suggested that the source was in an X-ray *hard* state during the entire outburst of 2005. For this source we used quasi-simultaneous optical (from the Liverpool telescope), near-IR (from UKIRT) and 15 GHz radio data (from the Ryle telescope). The full set of multi-wavelength observations obtained during this outburst of XTE J1118+480 will be presented in Brocksopp et al. (in prep).\nXTE J1118+480 went through an outburst in 2000 as well. During this outburst the source was extensively observed by many ground based telescopes as well as space-borne missions. In particular, the source was observed quasi-simultaneously around 2000 April 18th using the Ryle radio telescope, UKIRT, HST, EUVE, Chandra and RXTE giving an unprecedented broadband spectral coverage. This excellent data set has been used in several works to infer properties of the source. We revisit this data set to constrain the geometry and radiative properties of both the jet and the disc using our new disc+jet model.\nWe used the OIR and radio SED of GX 339 − 4 reported by, which were obtained during a multi-wavelength campaign to observe its outburst of 2002. For all data sets we used the photometric OIR magnitudes and de-reddened them using the interstellar extinction law of. In the absence of definitive measurement of the ratio of total to selective extinction (*R**V*[ = *A**V*/*E*(*B* − *V*)]) in the direction of either of the sources, we have taken *R**V* = 3.1, the widely used value measured by towards the galactic center, and standardly used for these sources. Thus it must be kept in mind that if the “true” value of *R**V* is significantly different from 3.1 (given neither of the sources are very close to the galactic center), it might introduce additional uncertainity in reddening correction.\nX-ray data\n----------\nFor both sources, we used data from the *Proportional Counter Array* and the *High Energy X-Ray Timing Experiment* onboard the RXTE satellite. The X-ray data were downloaded from the NASA HEASARC’s public archive[2](#fn2) and reduced using an in-house script following the standard reduction procedures outlined in *RXTE cook book[3](#fn3)* using HEASOFT software. To avoid any terrestrial contamination, we considered data only for elevation angles of 10∘ or higher. Also any data with pointing offset more than 0∘.02, or within 30 minutes of SAA passage, or trapped electron contamination ratio more than 0.1 were rejected. Since both the sources were bright (PCU2 count rate > 40 counts/s/PCU), the model *pca\\_bkgd\\_cmbrightvle\\_eMv20051128.mdl* was used for background subtraction. Data from all the layers of the active PCUs during the observations were extracted to create the spectra. Systematic uncertainty of 0.5% was added in quadrature to the PCA data to account for uncertainties in the calibration. For the HEXTE instrument, data from both clusters were added. Both PCA and HEXTE data were binned to achieve a minimum *S*/*N* of 4.5. For the PCA we used the energy range of 3–22 keV and for HEXTE 20–200 keV. The normalization of the OIR and radio data was tied to that of PCA. HEXTE normalization was allowed to vary.\nModel, Analysis and Results\n===========================\nThe continuum jet model used in this work is based on, (henceforth referred to as the agnjet model). Given the importance of irradiation in the observed spectra of XRBs, we have modified the model to include additional physics (see §[sec:irrad] for details) that allow us to calculate the effect of irradiation on the outer disc due to high energy photons from (1) the jet, and (2) the inner regions close to the compact object, impinging on the disc. We also compute inverse Compton scattering of thermal disc photons by the electrons in the jet plasma, although their effect on the total spectrum is negligible, except near the jet base, due to Doppler redshifting of the disc photons in the jet frame.\nJet parameters\n--------------\nThe details of the physics of the jet model as well as a full description of its main input parameters is given in the appendix of. Therefore we only give a brief summary here, and outline the modifications made to the model:\nThe main parameters that determine the properties of the jet are the input jet power (*N**j*), electron temperature of the relativistic thermal plasma entering at the jet base (*T**e*), the ratio of magnetic to particle energy density (a.k.a. the equipartition factor *k*), physical dimensions of the jet base (assumed to be cylindrical with radius *r*0 and height *h*0), and the location of the point on the jet (*z**a**c**c*) beyond which the a significant fraction of the leptons are accelerated to a power law energy distribution. *N**j*, parameterized in terms of the Eddington luminosity, determines the power initially input into the particles and magnetic field at the base of the jets. In the absence of a full understanding of the mechanism for energizing the jets, *N**j* plays a similar role to the compactness parameter in thermal Comptonization models. A typical value of the number density of leptons (*n*) at the base of the jet (derived from fits) is $n=N\\_{\\rm j}L\\_{\\rm Edd}/(4\\gamma\\_{\\rm s}\\beta\\_{\\rm s} m\\_{\\rm p} c^3 \n \\pi r\\_{\\rm 0}^2) \\sim 10^{14-15}$ cm− 3. Here $L\\_{\\rm Edd}$ is the Eddington luminosity of the source, $\\beta\\_{\\rm s}\\sim 0.4$ is the sound speed at the base (see below), $\\gamma\\_{\\rm s}=(1-\\beta\\_{\\rm s})^{-1/2}$, $m\\_{\\rm p}$ is the proton rest mass, and $r\\_{\\rm 0}\\sim 10^7$ cm is the radius of the jet base, implying an optical depth $\\tau=n\\sigma\\_T r\\_{\\rm 0}\\sim 0.001-0.01$. Since the optical depth is small, the probability of multiple Compton scattering is very small and we only consider single scattering. For the non-thermal power law electrons, the lower cutoff in the Lorentz factor is assumed to be equal to peak of the relativistic Maxwell-Juttner distribution, i.e. *γ**e*; *m**i**n* = 2.23*k**T**e*/(*m**e**c*2).\nFollowing the prescription of for maximally efficient jets, we assume that the bulk speed of the plasma (with adiabatic index Γ = 4/3) at the base of the jet is the sound speed given by $\\beta\\_{\\rm s}=\\sqrt{(\\Gamma-1)/(\\Gamma+1)}\\sim0.4$. We do not model the particle acceleration in the jet, but just start at the sonic point. Each jet then accelerates longitudinally along its axis due to pressure gradient and expands laterally with its initial sound speed. The bulk velocity profile, magnetic field strength, electron density and electron Lorentz factor along the jet are calculated by solving the adiabatic, relativistic Euler equation.\nIn the previous versions of the jet model we used the ratio of scattering mean free path to gyroradius of the particles in the plasma through which the shock moves (*f**s**c*), as a model parameter. However given the current lack of physical understanding of the actual shock propagation mechanism, in this paper we fixed the shock speed in the plasma (*β**s**h*) to 0.6 and used the quantity *ε**s**c* = *β**s**h*2/(ratio of scattering mean free path to gyroradius) as a model parameter instead of *f**s**c*. The parameter *ε**s**c* can be physically interpreted to be a parametrization for the shock acceleration rate. The dominant cooling processes within the jet are synchrotron and synchrotron self-Compton radiation. If the inner edge of the accretion disc is sufficiently close to the base of the jet, then the soft photons from the accretion disc, acting as seed photons for external Comptonization, can also cool the particles in the jet. The jet continuum as observed by an observer on Earth is a superposition of spectra from individual segments along the jet axis (taken to be perpendicular to the accretion disc plane), calculated by solving the radiative transfer equation, after taking into account relativistic beaming effects.\nThe irradiated disc\n-------------------\nSince the coupling between disc and jet is not well understood, we assume an independent classical thermal viscous “Shakura-Sunyaev” accretion disc, with a radial temperature profile *T*(*R*) ∝ *R*− 3/4, and modeled by the parameters $T\\_{\\rm in}$, $r\\_{\\rm in}$, the temperature and radius at the inner edge of the disc. If the outer parts of the disc are irradiated, then as discussed below, it is further possible to constrain the temperature and radius at the outer edge of the disc. Given the mounting evidence for the existence of relativistic outflows from black holes, we also explore the influence of the jet acting as a source of irradiation heating of the accretion disc.\n### Can jets influence the energetics of the accretion disc?\nWe computed the radiative jet flux incident on the accretion disc as a function of radial distance on the disc plane, taking into account special relativistic beaming effects. In order to obtain an *upper limit* on the jet-induced heating of the accretion disc, we assumed that the entire incident radiative flux from the jet *f**ν*(*R*) is thermalized upon being absorbed by the disc, thus heating the disc locally at radius *R* at a rate $H\\_{\\rm jet}(R)$, given by\n \n$$H\\_{\\rm jet}(R) = \\int\\_0^\\infty f\\_{\\nu}(R) \\mbox{ } d\\nu\n \\label{jet\\_irrad\\_eqn}$$ \nWe calculated $H\\_{\\rm jet}(R)$ at ten logarithmically spaced radii spanning from $R\\_{\\rm in}$ to $R\\_{\\rm out}$. The ratio of $H\\_{\\rm jet}(R)$ to the local viscous heating ($H\\_{\\rm visc}(R)$) is shown in Fig. [fig:jetirrad]. At large radii (*R* > 100 *R**g*), $H\\_{\\rm jet}(R)$ has a ∼ *r*− 2.4 radial profile, which is somewhat steeper than that expected from irradiation by a static corona or the inner accretion disc (for which the irradiation heating falls as $\\sim r^{-1.6{\\rm~to~}-2.0}$; see, e.g., §[sec:discirrad] and ). However the local viscous heating rate drops outward even faster ($H\\_{\\rm visc}(R) \\propto R^{-3}$), thus causing the ratio $H\\_{\\rm jet}(R)/H\\_{\\rm visc}(R)$ to slowly increase for increasing R. As the disc radius becomes small and approaches that of the jet base, the solid angle subtended by the jet increases rapidly causing the jet heating to saturate. However the viscous heating continues to rise at smaller radii and causes $H\\_{\\rm jet}(R)/H\\_{\\rm visc}(R)$ to drop at small radii. However, as evident from Fig. [fig:jetirrad], even the maximum possible jet-induced heating of the disc is about seven orders of magnitude smaller than the viscous heating. This confirms that relativistic beaming of the jet causes a very small fraction of the flux emitted by the jet to fall back on the disc, and the resulting influence of the jet on the energetics of the disc is negligible.\nLight bending effects near the black hole can enhance the flux incident on the disc, but by not more than a few tens of percent of the numbers calculated above (i.e. by assuming Minkowski metric instead of a more realistic Kerr or Schwarzschild metric).\n### An irradiation source near the compact object\nAnalysis of OIR light curves of XTE J1118+480 by suggests there might have been significant contribution of thermal flux in the optical. The near-IR flux however is reported to be significantly non-thermal in nature from studies of rapid time variability in the light curve, as well as nearly flat IR SED. As discussed in the previous paragraph as well as shown in Fig. [fig:jetirrad], the jet’s contribution in heating the disc is negligible; we have therefore added an irradiation heating term due to a somewhat more static source of irradiating X-rays near the black hole. The physical origin of this irradiating source could be the inner accretion disc near the black hole. Since we assume a radial temperature profile and do not solve the local disc structure, our model cannot discern the source of the irradiating X-rays.\nThe temperature due to irradiation heating ($T\\_{\\rm irrad}$) is usually assumed to have a power law radial dependence of the form *R*− *n*. Depending on the initial assumptions about disc structure and geometry of the irradiating source, theoretical models predict *n* to be in the range of 0.4-0.5. Given the uncertainities in de-reddening and scarcity of data points in the OIR, the exact choice of *n* is not vital for a global understanding of the broad-band SED. We used *n* = 3/7 (vertically isothermal disc with disc height *h* ∝ *r*9/7), mainly for consistency with earlier work.\nWe assume that the total disc heating is a sum of local viscous heating, and heating due to a static source of irradiation near the black hole. The effective temperature of the disc ($T\\_{\\rm eff}$) is therefore related to the viscous temperature ($T\\_{\\rm visc}$) and the irradiation temperature ($T\\_{\\rm irrad}$) as follows:\n \n$$T\\_{\\rm eff}^4(R) = T\\_{\\rm visc}^4(R) + T\\_{\\rm irrad}^4(R){\\rm,}$$ \nwhere $T\\_{\\rm visc} = T\\_{\\rm in}(R/R\\_{\\rm in})^{-3/4}$ and $T\\_{\\rm irrad}= T\\_{out}(R/R\\_{out})^{-3/7}$ with $T\\_{\\rm in/out}$, $R\\_{\\rm in/out}$ as free parameters. Other parameters, viz. masses, inclination and distance to the binary system, are taken from values published elsewhere and kept fixed during the process of fitting. The radius $R\\_{\\rm irrad}$ where irradiation heating becomes equal to viscous heating can be estimated from\n \n$$R\\_{\\rm irrad}=\\left[ (T\\_{\\rm in}/T\\_{\\rm out})\\\n R\\_{\\rm out}^{-3/7}/R\\_{\\rm in}^{-3/4}\\right]^{28/9}.$$ \nSince viscous dissipation dominates for $R < R\\_{\\rm irrad}$, the disc height at $R\\_{\\rm irrad}$ is estimated from Shakura-Sunyaev’s *α*-disc solution as\n \n$$H \\ (R\\_{\\rm irrad})=1.7\\times 10^8 \\ \\alpha^{-1/10} \\ \\dot M\\_{16}^{3/20} \n (M/{M}\\_\\odot)^{-3/8} \\ R\\_{\\rm irrad;10}^{9/8} \\ f^{3/5}\n\\label{eq:H\\_irrad}$$ \nwhere we have $\\dot M\\_{16}=\\dot M/(10^{16} \\ {\\rm g/s})$, $R\\_{\\rm irrad;10}=R\\_{\\rm irrad}/(10^{10} \\ {\\rm cm})$, $f=[1-2GM/(c^2 R\\_{\\rm irrad})]^{1/4}$ and chosen *α* = 0.05. Similarly, since *H* ∝ *R*9/7 when $R > R\\_{\\rm irrad}$, the disc height at $R\\_{\\rm out}$ is given by\n \n$$H \\ (R\\_{\\rm out}) = H(R\\_{\\rm irrad})\\times(R\\_{\\rm out} / R\\_{\\rm irrad})^{9/7}.\n\\label{eq:H\\_out}$$ \nThe solid angle subtended by the outer, irradiation dominated disc as seen from the compact object is computed from equations [eq:Hirrad] and [eq:Hout].\nThe agnjet model computes the broadband continuum from the jet and the irradiated disc but not line emission or reflection features. Therefore we used an additional Gaussian line profile to fit the iron K*α* emission complex near 6.5 keV and the convolution model reflect for the excess in the range of 10–30 keV, both features being usually attributed to reflection by the relatively cold accretion disc. The column density, taken from elsewhere published values, was fixed during the fits. For the reflect model, which accounts for reflection from neutral material in the accretion disc, we used solar abundance and tied the inclination to that of the disc (and also the jet axis). The energy of the iron line in the Gaussian model was allowed to vary between 6–7 keV and its width was fixed to 0.5 keV. The spectral coverage, along with column density ($N\\_{\\rm H}$) towards the sources, distances, and masses of the compact objects are given in Table [tab:data].\nModel fits\n----------\nSpectral fittings were performed using ISIS. Evaluation of confidence intervals for any physical model is computationally expensive, and this is true for agnjet model as well. Therefore we have used the parallelization technique described in and distributed the task of computing confidence intervals[4](#fn4) for *f* free parameters over *f*/2 or (*f* + 1)/2 nodes (depending on whether *f* is even or odd; each node is an Intel Xeon 3.4GHz processor) of a Parallel Virtual Machine running on the LISA cluster in Almere, Netherlands. This reduced the runtime to under 24 hours, a greater than 75% speedup; it also allowed us to discern finer features in parameter space by increasing the tolerance resolution by a factor of 500, while keeping the overall runtime to approximately half the serial runtime of 96 hours. While these are welcome improvements, reducing the model runtime even further would enable us to analyze new data more quickly. For this reason we are experimenting with ways of utilizing more than one processor per parameter, such as partitioning parameter space more finely, and exploring the use of OpenMP to parallelize internal loops within the model source code.\nUsing the distributed parallel computing technique described above, we determined the best-fit parameter values as well as their confidence intervals. Our confidence intervals correspond to Δ*χ*2 = 2.71 for a given parameter (which for *normal* distribution would imply a 90% single parameter confidence limit).\n### XTE J1118+480 during 2005 outburst\nOf the three data sets presented in this work, the data from the 2005 outburst of XTE J1118+480 was the faintest and had a steep power law slope in the X-rays. The flux in the HEXTE bands was extremely low for this data and while regrouping the data we found that no flux with S/N > 4.5 could be detected above 60 keV. Since the single point in the radio frequency does not constrain the length of the jet, we fixed it to 1013.1 cm which is a lower limit on the radiative length of the jet assuming that the radio data point is still on the optically thick synchrotron regime with a flat-to-slightly-inverted spectral index. While the fits require a high fraction of non-thermal electrons in the post-shock jet ( > 0.5), they are not very sensitive to the exact value of this fraction, and we fixed its value to 0.75. The data cannot be described by a non-thermal jet continuum alone and an excess in the OIR data requires additional flux which we modeled using the irradiated accretion disc model described in §[sec:irrad]. This thermal excess in the optical has also been suggested from the OIR coverage of the outburst by and.\nThe X-ray data (3–60 keV) can be well fit (*χ*2/*ν* = 88.8/55) only with a power law, however systematic residuals in the 5–7 keV range suggest that a weak iron line may be present. Adding a Gaussian line to the power law improves the fits to *χ*2/*ν* = 79.5/52, with the line center near 6.6 keV and width of 0.1 keV. Given the weak strength of the lines, we fixed the line energy as well as the width to these values, and did not use any reflection model modeling the broad band SED of XTE J1118+480.\nFits performed by allowing $r\\_{\\rm in}$ and $T\\_{\\rm in}$ of the accretion disc to vary freely results in good fits with quite small values of both $r\\_{\\rm in}$ and $T\\_{\\rm in}$. There are uncertainities in our understanding of the physical properties of the innermost regions of the disc close to ISCO, dereddening of interstellar extinction, and also uncertainities in instrumental response and calibration. These concerns make detection of a cold disc with small $r\\_{\\rm in}$, $T\\_{\\rm in}$ and its physical origin open to debate. Another issue with the XTE J1118+480 data is the jet/disc inclination. While we usually fix the inclination to a value published elsewhere, the extremely flat radio to IR spectrum of XTE J1118+480 requires a much smaller inclination than the published value of ∼ 70∘. Allowing the inclination to vary for this source results in jet inclination of ∼ 25∘.\nTherefore we tried two different models for this data set of XTE J1118+480. In the first model (Model 1; second column in Table [tab:fits]), apart from other regular parameters described above, the jet/disc inclination as well as $r\\_{\\rm in}$ and $T\\_{\\rm in}$ were allowed to vary freely. In the second, somewhat more conservative model (Model 2; third column in Table [tab:fits]), the jet inclination was fixed to 30∘, $r\\_{\\rm in}$, $T\\_{\\rm in}$ were also fixed to 30 *R**g* and 0.1 keV respectively. The best-fit values for the second model also gives similar jet parameters as the first model, although with a slightly higher jet power and steeper electron distribution index. Also the second model favors a somewhat smaller and hotter outer disc edge.\n### XTE J1118+480 during 2000 outburst\nWe used the broadband SED data of XTE J1118+480 obtained around 2000 April 18, published in and. While analyzing this data set a broad dip in the combined EUVE and Chandra spectra was noted between 0.15–2.5 keV (see e.g. the residuals in Fig. [fig:j11182000fit]) which cannot be modeled using the standardly used column density for this source ($1.3\\times 10^{22}\\ \n{\\rm cm}^{-2}$). This feature was been noted by previous works as well, and is attributed to metal absorption in a partially ionized gas,. We have therefore excluded the energy range between 0.15–2.5 keV from our broadband fitting. Results of fitting indicate that during this observation emission from the jet dominates the radio and IR regions. Contribution from the accretion disc starts becoming dominant in the optical. However, as discussed in, the EUVE fluxes are extremely sensitive to the extinction law, and the slope of the dereddened EUVE SED can range between + 2 ≥ *α* ≥ − 4 (*F**ν* ∼ *ν**α*) for the permissible range of N$\\_{\\rm H}$. Here we have used the EUVE fluxes reported by and a column depth of N$\\_{\\rm H}=1.3\\times10^{20}$ cm− 2, which results in a very steep EUVE slope. This cutoff in the EUVE data would imply a cold, truncated inner disc with with r$\\_{\\rm in}\\sim 340$ R*g* and T$\\_{\\rm in}\\sim 2.9\\times 10^5\\ {\\rm K}$. Given the low temperature of the inner disc and the consequent absence of a irradiating soft X-ray photons, it is not unexpected that we do not find any signature of irradiation in this case (i.e. T$\\_{\\rm out}$ is not constrained). As in the case of the 2005 data of this source, the X-ray emission in this model is entirely dominated by optically thin synchrotron emission from the jet. While the fits require a small nozzle (*h*0/*r*0 ∼ 0.2 − 0.6), they are not very sensitive to the exact value of this parameter, and we fixed its value to 0.4. We note that this value is abot a factor of three smaller than what is found typically in other sources, or even during the 2005 outburst of this source.\n### GX 339 − 4 during 2002 outburst\nThe X-ray data of GX 339 − 4 cannot be well fit by a power law alone. Using simple power law+ Gaussian model gives unacceptable fit to the data with reduced *χ*2 of 5.9. Convolving a power-law+gaussian with a reflection model give better fits, but still with rather a large reduced *χ*2 of 2.3. We found that the GX 339 − 4 data can be fitted well without including any thermal component, and the entire broadband SED is well described by the superposition of synchrotron and inverse Compton photons from the pre-shock region near the jet base as well as optically thick synchrotrons from post-shock regions farther from the jet base. From the radio-IR slope, the required inclination is between 45–50 degrees, but not well constrained and hence was fixed at 47 degree. The length of the jet was fixed to 1014.2 cm and fraction of non-thermal electrons fixed to 0.75 for the same reasons given for XTE J1118+480. Since the spectrum of GX 339 − 4 could be entirely dominated by jet emission, and the source was in a bright, hard X-ray state, the input jet power is much larger than that of XTE J1118+480. In fact the jet power for this observation is larger than any of the previous observations of this source. Also the acceleration region starts much farther out for the GX 339 − 4 data set.\nIf the temperature of the thermal electrons ($T\\_{\\rm e}$) at the base is left completely free, as in “Model 1” (fifth column in Table [tab:fits]), then $T\\_{\\rm e}$ for the best-fit model is comes out to be rather high, and suggests that the SED at energies > 10 keV is almost entirely due to SSC emission from the base (see top two panels in Fig. [fig:gx339fit]). The geometry of the base is rather compact and the source is in a bright hard state; therefore a strong SSC emission could imply that the photon density at the base becomes high enough for pair processes to become important. Since pair production is not included in the model, an estimate of the importance of pair processes for the best fit models were done as follows: the pair production rate ($\\dot n\\_{pp}$) was calculated from the photon field at the base of the jet using the angle averaged pair production cross section ; pair annihilation rate ($\\dot n\\_{pa}$) was calculated following. For “Model 1” of GX 339 − 4, $\\dot n\\_{pp}\\sim 10^{17}$ cm− 3*s*− 1 and $\\dot n\\_{pa}\\sim 10^{12}$ cm− 3*s*− 1, suggesting that pair processes could be important in this case. Therefore we explored the parameter space, limiting the electron temperature to be less than 5 × 1010 K and somewhat less compact base (larger r$\\_{\\rm 0}$, h$\\_{\\rm 0}$) to search for solutions which also satisfy the condition $\\dot n\\_{pp}<\\dot n\\_{pa}$ (see “Model 2”, sixth column in Table [tab:fits] and bottom panels in Fig. [fig:gx339fit]). The fit derived number density (*n*), $\\dot n\\_{pa}$ and $\\dot n\\_{pp}$ for the different models presented in Table [tab:fits], are given in Table [tab:pairs].\nIn addition to the continuum jet model, an additional Gaussian line centered at 6.3 keV and reflection from the accretion disc is needed to obtain good fits for the GX 339 − 4 data. Presence of the line and reflection suggests the presence of an accretion disc. However, as in the case for XTE J1118+480, the disc is cold and therefore not detected by the PCA.\nThe best-fit model parameters, associated confidence intervals, *χ*2 statistic and corresponding chance probabilities for both sources are shown in Table [tab:fits]. Given the simplicity of the model it is hardly unexpected that the reduced *χ*2 values obtained from the fitting are not very close to 1 in some cases. For instance, the viscous+irradiated disc model does not include details of disc atmosphere or ionization balance equations, and is therefore not able to predict the spectral breaks in the optical data. Because of its simplicity, this is not meant to be a real model of the disc, although it incorporates the dominant ongoing physical processes and can reproduce the general shape of the broadband continuum. Moreover, while fitting a data set spanning 10 decades in frequency (or energy) space, the long lever arm between radio and X-rays constrains the jet model more strongly than the overall chi-squared. Best fit models along with data and residuals are shown in Figures [fig:j11182005fit], [fig:j11182000fit] and [fig:gx339fit].\nDiscussion and Conclusions\n==========================\nWe have analyzed broad band multi-wavelength observations of the galactic black hole transient systems XTE J1118+480 and GX 339 − 4, during their outbursts in 2000 and 2005 for XTE J1118+480, and during the 2002 outburst of GX 339 − 4. The conclusions from our study are summarized below.\nThe data for XTE J1118+480 cannot be fit well with a jet continuum model alone, due to an excess in the OIR fluxes. During these observations, in OIR the source was at least 3.5 magnitude brighter than quiescence; therefore the contribution from the M1 V secondary star is small. Modeling the data from the 2000 outburst and 2005 outburst of this source we found that the outer disc was strongly irradiated during the 2005 data. It appears that the inner edge of the accretion disc was much closer to the central black hole during the observations of 2005, thus providing an ample source of soft X-ray photons to irradiate the outer disc. On the other hand, the sharp cutoff in the EUVE data and absence of any signature of irradiation from the outer disc suggests a recessed, cold accretion disc for the 2000 data. However, as discussed in §[sec:j11182000], the slope of the EUVE data depends very strongly on the assumed column depth, and a small decrease in N*H* can flatten the EUVE SED significantly. Such a flat EUVE SED would be expected from a disc with small r$\\_{\\rm in}$ and comparatively higher T$\\_{\\rm in}$.\nOur calculations in §[sec:jetirrad] show that at any point on the surface of the accretion disc, the heating of the disc by the jet is seven or more orders smaller than the viscous heating. There are two factors which largely reduce the amount of jet radiation that is incident on the disc: Firstly Doppler de-beaming - since the bulk motion of the jet is relativistic, special relativistic Doppler beaming concentrates most of the emergent radiation from the jet in the direction of its motion, i.e. away from the disc. A second, but significant factor that also contributes to the dimunition of jet flux incident on the disc is the fact that the post-shock jet synchrotron spectra peaks at increasingly longer wavelengths at increasingly larger distances from the base of the jet, thus creating fewer X-ray photons that can heat the disc. This is illustrated in Fig. [fig:jetirrad] where we show the ratio of jet-heating to viscous heating as a function of radius. Since heating by the jet alone cannot produce the OIR excess in Fig. [fig:j11182005fit] we surmised that this the OIR excess is indeed caused by the more conventional form of irradiation, i.e. irradiation by soft X-ray photons from the inner-disc. From an energetics point of view, it is easy to see that the inner accretion disc alone (see Fig. [fig:j11182005fit], at around 1 keV) radiates at least two orders or more energy than the outer region. Furthermore, it is well known that the outer edge of the accretion disc can flare-up or warp, thus making the outer parts of the disc more easily visible from the inner regions.\nFits to the 2005 data of XTE J1118+480 using the jet model+irradiated disc (described in §[sec:discirrad]) suggest a cold disc ($T\\_{\\rm in}\\sim 0.2\\ keV$) with inner radius close to the innermost stable circular orbit (6 *R**g* for a Schwarzschild black hole, and smaller if spinning). Given the uncertainities associated with using a simple *T* ∼ *R*− 3/4 model which ignores correction factors like spectral hardening, full disc structure as well as proper boundary conditions at the inner edge, the errors associated with the estimates of the inner disc parameters are most likely an underestimate. Another source of uncertainity comes from that fact that we assume the jet to be perpendicular to the disc. A misaligned jet, as seems likely for XTE J1118+480, could introduce additional error in the determination of $r\\_{\\rm in}$. Along with the uncertainities in instrument calibration, modeling and jet-disc alignment, the sharp low energy cutoff of PCA sensitivity below 3 keV makes detection of a cold disc with small $r\\_{\\rm in}$ even more difficult than a recessed cold disc. Nevertheless the numbers for $r\\_{\\rm in}$ and $T\\_{\\rm in}$ we obtained for XTE J1118+480 are similar to those obtained by several others for black hole transients in their hard state. If such a cold disc with small r$\\_{\\rm in}$ is indeed realised, then Compton scaterring of the disc photons could be strongly anisotropic, with most of the scattered emission beamed toward the disc. This could compensate partly the beaming effects due to the relativistic motion of the gas, and increase the illumination of the disc (both irradiation and reflection effects). Also in this case there could be a radiative feedback due to the local illumination of the disc by the base of the jet which is similar to the situation in standard accretion disc corona models which would act as a thermostat keeping the temperature at the base lower than what is assumed here. A somewhat more conservative model, where the temperature and radius of the inner edge of the accretion disc were fixed to 0.1 keV and 30 R$\\_{\\rm g}$ respectively (Model 2 in Table [tab:fits]) also describes the data well, showing the difficulty in detecting a cold disc with small r$\\_{\\rm in}$ from the available data.\nThe best-fit outer radii of XTE J1118+480 for both *Model 1* and *Model 2* are smaller than the circularization radius, using the latest values of orbital parameters published by. These results are therefore consistent with the assumption that accretion is occuring via Roche lobe overflow. The observation was made during the decline from outburst peak. Hence the discrepancy between the circularization radius and *R**o**u**t* inferred from fits, in the context of standard disc instability models, could be due to an inward moving cooling wave. A decreasing trend in the outer *radiative disc radius* was also observed by for the system XTE J1859+226. However the value of *R**o**u**t* in our case depends largely on the five OIR data points, all of which lie on the Rayleigh-Jeans region of the irradiated disc. Given the plethora of spectral features seen in OIR spectra of X-ray binaries in outburst, the limitation of using a few bands is obvious.\nAnalysis of the data from the 2000 outburst of XTE J1118+480 by and showed that the UV and higher energy data can be explained by an ADAF model. The flux from the ADAF model alone however falls sharply at energies below optical and underestimates IR and any lower energy data (e.g. radio), thus requiring an additional jet component. In the jet-ADAF model, the X-ray photons have a thermal Comptonization origin, whereas in the current work, the X-ray photons can originate from either synchrotron (e.g in XTE J118+480), or inverse Compton scattering, or a combination of both (e.g. in GX 339 − 4). Better simultaneous broadband coverage and future X-ray polarization studies can help to estimate the relative importance of these two emission components which dominate the X-ray spectra of X-ray binaries in hard state.\nFrom the available data, the X-ray spectrum of XTE J1118+480 during both outbursts show little or no curvature at energies higher than ∼ 10 keV. Reflection features like the iron line complex and the reflection “bump” near 20 keV are also extremely weak. Detailed analysis of the X-ray spectra taken by Chandra and RXTE, during the 2000 outburst of this source by also showed extremely weak reflection features, as would be expected for a jet+recessed disc scenario. From a phenomenological point of view, since the curvature is small, using a broken power law to describe data of X-ray binaries in hard state has been shown to work remarkably well, e.g., for GX 339 − 4 by and for Cyg X − 1 by. Residuals of a power law fit to the X-ray continuum in the 2005 data set of XTE J1118+480 show small systematic variations near 5–7 keV, which improves slightly upon addition of a Gaussian line. However, given the weakness of the line, its energy or width cannot be constrained well. Total contribution from various source of inverse Comptonization (SSC within the jet base and external Comptonization of soft thermal photons from the accretion disc) is also much smaller compared to that of GX 339 − 4. The X-ray region of the SED of XTE J1118+480 is dominated by post-shock synchrotron photons for both 2000 and 2005 outbursts. We note that the electron acceleration region seems to be in the nozzle of the jet in the case of 2005 outburst data of XTE J1118+480 while it is farther out in the jet in GX 339 − 4 as well as the 2000 outburst of XTE J1118+480. While exhaustive searching of the *χ*2 space confirms that this is required by the data, it is not clear why the acceleration region is inside the nozzle for the 2000 data of XTE J1118+480. Another unexpected result is that the base of the jet seems to be quite small (*h*0/*r*0 ∼ 0.4) for the 2000 outburst data of XTE J1118+480, whereas *h*0/*r*0 ∼ 1.5 for other outbursts of XTE J1118+480 as well as other sources (see e.g. ). Given the uncertainties in our understanding of jet formation and the acceleration region, and the simplicity of the model, here we report the best fit parameters as obtained. The particle energy distribution index (*p*) of the relativistic post-shock electrons for data from both outbursts of XTE J1118+480 is ∼ 2.5–2.6. This value is much steeper than what could be expected from an initial distribution of particles accelerated by a diffusive shock process. However it is possible that for XTE J1118+480 the observed X-ray bandpass is at a higher energy than the cooling break energy, where the particle distribution steepens from *p* to *p* + 1 due to enhanced cooling. One of the key assumptions in our model is time independence of particle distributions. However if the steep power law drop-off of the observed photon index is indeed due to enhanced cooling, or more generally a departure from the assumed equillibrium between the heating and cooling processes, a more thorough time dependent analysis of particle evolution is neccessary. We will present time dependent particle evolution in the context of jets in a forthcoming paper.\nThe jet inclination of XTE J1118+480 is still a matter of concern. The optically measured system inclination is near 70∘. However, given the flat radio-to-IR fluxes, jet inclinations higher than ∼ 30∘ would be very difficult to fit which might suggest a large misalignment between between the disc and the jet. Evidence of misalignment between disc and jet has been seen in several stellar X-ray binary systems, e.g., GRO J1655 − 40, SAX J1819 − 2525, XTE J1550 − 564, as well as in several AGNs, e.g., NGC 3079, NGC 1068 and NGC 4258. Allowing the inclination to vary for XTE J1118+480 during the fits gives a best-fit inclination of ∼ 25∘, and it is almost impossible to find statistically good fits (e.g. with *χ*2/*ν* < 2) for inclinations larger than 30∘. Previous fits by also required a similar, small inclination. In our model we assume the lateral expansion and longitudinal acceleration of the jet to be driven solely via the solutions of the relativistic Euler equation for adiabatic expansion. If this is not the case, and e.g. the jet radius and acceleration are determined by some other process(es) leading to a stronger magnetic field and/or smaller bulk velocity along the jet, that could lead to a flatter radio-IR spectrum as well.\nThe data for GX 339 − 4 does not formally require a thermal component (e.g. an accretion disc or emission from the secondary star) and the entire broad band emission from radio to X-rays is satisfactorily well fit by non-thermal photons originating from pre- and post-shock synchrotron, and synchrotron self Compton emission mainly from the jet base. Previous fits to the 3–100 keV X-ray data by also suggested that the disc contribution within this energy range is ∼ 1% of the total luminosity or less. However presence of the line near 6.5 keV and signature of reflection suggests a cold accretion disc near the black hole which cuts off exponentially at an energy significantly below the lower detection limit of the RXTE/PCA. Since the X-ray spectrum has a luminosity of ∼ 5% L$\\_{\\rm Edd}$, the disc luminosity is presumably ∼ 0.05% L$\\_{\\rm Edd}$ or smaller.\nThe data for GX 339 − 4 were taken when the source was in a bright, X-ray hard state. The jet power *N**j* is much higher than that obtained by, who studied data from previous outbursts of the same source. We also obtain a higher equipartition factor (*k*) and shallower particle distribution index. It is interesting to note that *r*0 and *z**a**c**c* for GX 339 − 4 are always found to be higher than those found for other sources. If the electron temperature at the base is left unconstrained (e.g. “Model 1” in the fifth column of Table [tab:fits]), then the density of photon field from best-fit model is high enough that pair processes may become important. In this case many photons are up-scattered above the pair production threshold, and the rate at which pairs are produced is much larger than the rate at which they are annihilated, thereby changing the density at the base of the jet. Constraining the electron temperature to be less than 5 × 1010 K, we present another fit (“Model 2”, sixth column of Table [tab:fits]) where the photon density at the base is small and pair production rate is much smaller than the pair annihilation rate. For this fit the the fractional pair annihilation rate $(\\dot n\\_{\\rm pa}/n)$ is also much smaller than unity, so that pair processes are not important for this fit.\nUsing the irradiated disc+jet model presented in this paper we have shown that in certain cases (e.g. the 2000 outburst data of XTE J1118+480) there are not enough X-ray photons from the inner regions to"},"data_id":{"kind":"string","value":"arxiv_0003266"}}},{"rowIdx":793,"cells":{"content":{"kind":"string","value":"Two decades of optical timing of the shortest-period binary star system HM Cancri\n=================================================================================\n### Accepted 2022 November 16. Received 2022 November 16; in original form 2022 September 1\n[firstpage]\nThe shortest-period binary star system known to date, RX J0806.3+1527 (HM Cancri), has now been observed in the optical for more than two decades. Although it is thought to be a double degenerate binary undergoing mass transfer, an early surprise was that its orbital frequency, *f*0, is currently increasing as the result of gravitational wave radiation. This is unusual since it was expected that the mass donor was degenerate and would expand on mass loss, leading to a decreasing *f*0. We exploit two decades of high-speed photometry to precisely quantify the trajectory of HM Cancri, allowing us to find that $\\ddot f\\_0$ is negative, where $\\ddot f\\_0~=~(-5.38\\pm2.10)\\times10^{-27}$ Hzs− 2. Coupled with our positive frequency derivative, we show that mass transfer is counteracting gravitational-wave dominated orbital decay and that HM Cnc will turn around within 2100 ± 800yrs from now. We present Hubble Space Telescope ultra-violet spectra which display Lyman-*α* absorption, indicative of the presence of hydrogen accreted from the donor star. We use these pieces of information to explore a grid of permitted donor and accretor masses with the Modules for Experiments in Stellar Astrophysics suite, finding models in good accordance with many of the observed properties for a cool and initially hydrogen-rich extremely-low-mass white dwarf ( ≈ 0.17$\\textup{M}\\_\\odot$) coupled with a high accretor mass white dwarf ( ≈ 1.0$\\textup{M}\\_\\odot$). Our measurements and models affirm that HM Cancri is still one of the brightest verification binaries for the Laser Interferometer Space Antenna spacecraft.\nbinaries: close – individual: RX J0806.3+1527, HM Cancri – stars: white dwarfs – gravitational waves\nIntroduction\n============\nThere exists a large population of double white dwarf (DWD) binary star systems in the galaxy which can form through a series of mass-transfer and common envelope events. Gravitational wave radiation drives the loss of orbital angular momentum and causes many binary orbits to decay within a Hubble time, driving them to compact (orbital period of ≈ 5–65min) configurations. Ultimately, these systems initiate mass transfer, which has a dramatic effect upon their evolution, making DWD binaries the likely progenitors of type a/.a supernovae and maybe the progenitors of R CrB stars, hot subdwarf stars and AM CVn systems. In contrast to the typical dynamics of mass transfer between non-degenerate stars, the smaller mass, larger radius white dwarf (WD) is first to overflow its Roche lobe as the donor star, with the larger mass, smaller radius WD the accretor in a DWD binary. Mass transfer from one star to a higher mass companion acts to expand the orbit, counteracting the shrinkage caused by gravitational wave losses with angular momentum transfer playing an important role in the fate of these systems.\nRX J0806.3+1527 is the shortest-period binary star system known, having a present day orbital period of 5.36min, equivalent to an orbital frequency of 3.11 mHz. There have been multiple theories suggested to explain the extremely short period of HM Cnc, namely the unipolar inductor model ; the intermediate polar model ; the AM CVn model. In brief, the unipolar inductor model is similar to the Jupiter-Io system with the binary being detached, the intermediate polar model suggests that the optical period actually represents the spins of magnetic WDs and that the binary’s orbital period is much longer, and the AM CVn model involves a compact WD binary mass-transferring helium rich material. In the case of HM Cnc, a DWD binary is required to reach the observed period for an AM CVn without a merger already taking place. The studies of and helped to reduce the options. They find clear radial-velocity variations on the optical period, difficult to explain using the intermediate polar model, and a strong indication of an accretion stream via the presence of He emission lines in the spectrum, which is not predicted by the unipolar inductor model. fail to detect any signature of metals in the X-ray spectrum, favouring a DWD AM CVn, and a DWD binary is also a natural explanation for the relative phase difference between the peak of X-ray and optical flux. Here, the peak in X-ray flux arrives slightly later than the optical, which arises from the trajectory of the accretion stream directly impacting the surface of the accretor. It would thus seem that only the DWD AM CVn theory can be used to explain all of the present observations.\nFollow-up observations of HM Cnc in the X-ray and optical domains have revealed an increase in orbital frequency, *f*, due to the loss of orbital angular momentum through gravitational wave radiation. The increase is so much so that HM Cnc will be one of the brightest “verification binaries” to optimise the performance of the Laser Interferometer Space Antenna spacecraft, given its low-frequency gravitational wave emission, strong signal strength and the precise timing measurements that have been used to quantify its orbital decay. Similarly, HM Cnc is planned to be used as the primary reference source for the TianQin spacecraft. While it took two years to detect an increase of HM Cnc’s orbital frequency, continuous timing measurements over a much longer baseline can prove insightful to determine the second derivative, $\\ddot f$, which is an indicator of the trajectory of inspiral.\nUsing X-ray data, measured the second derivative of the orbital frequency of HM Cnc, finding $\\ddot f = (-8.95\\pm1.40) \\times 10^{-27}\\,$Hz s− 2. This was the first significant detection of $\\ddot{f}$ for any AM CVn system. The combination of $\\dot f$ and $\\ddot f$ can be used to predict the timing of maximum orbital frequency of the binary, which predicts to occur in 1260 ± 200 yrs from now. In addition, was able to estimate a mass transfer rate based on the accretion luminosity of the source from X-ray observations. HM Cnc is unfortunately too dim (*G*mag = 20.9) for a distance constraint with Gaia DR3, though assuming a distance of 5 kpc and donor and accretor masses of 0.27 $\\textup{M}\\_\\odot$ and 0.55 $\\textup{M}\\_\\odot$, calculated an accretion rate $\\dot M = -1.6\\times10^{-8}\\,$$\\textup{M}\\_\\odot$ yr− 1. However, the calculated mass transfer rate is strongly dependent on the input star masses and the distance to the source. A distance measurement and a precise measurement of $\\ddot f$ are therefore extremely useful to constrain the binary’s evolution with sets of donor and companion masses.\nOur study is set out as follows. discusses our photometric and spectroscopic observations of HM Cnc and our attempts to derive a distance measurement. In we present system timing measurements and derive the orbital ephemeris. We then explore the extent to which the AM CVn channel can match our orbital ephemeris with binary evolution simulations in.\nObservations\n============\nPhotometry\n----------\nTime-series photometry was obtained using the high-speed cameras ULTRACAM, ULTRASPEC and HiPERCAM ; ULTRACAM and HiPERCAM observe simultaneously in three or five bands, respectively. Data from HiPERCAM were obtained while mounted on the 4.2m William Herschel Telescope (WHT) and the 10.4m Gran Telescopio Canarias (GTC). We observed with ULTRACAM on the 8.2m Very Large Telescope (VLT), the 3.58m ESO New Technology Telescope (NTT) and the WHT, and we observed with ULTRASPEC mounted on the the 2.4m Thai National Observatory (TNO). Exposure times were varied across the nights to reflect the seeing conditions, the aperture of the telescope and the filters used at the time. A full observing log giving dates, conditions, exposure times and the filters used for each set of observations is given in Appendix [tab:observingLog].\n [fig:LCs]\nAll of the data were bias-subtracted and flat-fielded. An additional dark current subtraction was applied to all data from ULTRACAM only as the instrument operates at a temperature high enough for there to be significant dark current. For HiPERCAM, corrections for CCD fringing were performed in the *z**s* band only using pre-obtained fringe maps for the instrument.\nWe also accessed VLT FORS1, ESO 3.6m Telescope EFOSC2 and Telescopio Nazionale Galileo (TNG) DoLoReS archival data that were presented in prior to our first observation. Extra archival data of HM Cnc were obtained from the TNG DoLoReS instrument from dates following their study. Though the study of used some of these data before, no timing solutions have been reported, so we use the archival data in our study to take advantage of the longest observational baseline possible[2](#fn2). Data obtained with the 2.56m Nordic Optical Telescope (NOT) that was presented in and with the 2.5m Isaac Newton Telescope (INT) from the RATS survey were utilised in this study too. The extracted differential photometry obtained in prior studies using data from the NOT were used, whereas raw files were handled from every other source.\nPhotometry from each instrument, besides data obtained with the NOT, was then extracted using the HiPERCAM pipeline with differential aperture photometry and the application of ’s optimal photometry algorithm when the target aperture did not need to be linked to a comparison star; otherwise regular aperture photometry was performed. To tackle poorer observing conditions, a variable aperture radius was applied, which was chosen to be a fixed multiple (1.6–1.8 × ) of the full-width at half-maximum of the stars for each individual frame. A non-variable comparison star (Gaia DR3 ID 654873149889859968) was used for all observations when possible, or another (Gaia DR3 ID 654879025405159040) if the main comparison star saturated or coincided with detector defects.\nThe observation times were corrected to the Barycentric Julian Date (BJD) Barycentric Dynamical Time (TDB) centred on the mid-point of the exposure time. Poor data (e.g. affected by clouds) were rejected by visual inspection with HiPERCAM pipeline routines. We then derived a differential atmospheric extinction term for each filter to correct for the difference in colour between the comparison star and HM Cnc. This was performed by considering the dependence of airmass on the differential flux, which was subtracted to effectively convert to a flux at zenith.\nProper Motion\n-------------\nAs of Gaia DR3, there is no measurement of the proper motion of HM Cnc, so we searched for it using our data. To do so, we added world coordinate system information to each FITS file by plate solving each exposure using the backend of Astrometry.net. We utilised a custom Gaia DR3 reference star catalogue for the local field containing stars with a right ascension (*α*) and declination (*δ*) error of less than 0.1 arcseconds. The catalogue provided the location of stars at the mid-Gaia DR3 epoch. We selected data from the *V*, *r* and *i* bands only to reduce the impact of atmospheric refraction upon the position of HM Cnc relative to the redder stars in the field, though atmospheric refraction still limits our position accuracy. A reliable coordinate calibration was possible for all frames with approximately 20 or more reference stars. Then, we converted the central (pixel) coordinate and error of the centroid for each aperture into a physical *α*/*δ* coordinate. All measurements of *α*/*δ* from each frame were then combined with a variance weighted mean, giving a measured *α*/*δ* for HM Cnc on each night.\nFrom our determined positions (as plotted in Appendix [appendix:propermotionGraphs]), we conservatively take the range of the data to find upper limits of $|\\dot\\alpha|<9.1\\,$masyr− 1 and $|\\dot\\delta|<9.2\\,$masyr− 1. Our best-fit indicates proper motions of $\\dot\\alpha=2.0\\pm1.1$ masyr− 1 and $\\dot\\delta=-3.3\\pm1.1$ masyr− 1, having a reduced *χ*2 equal to one. By taking the maximum proper motion, we can derive a loose minimum distance constraint.\nThe proper motion is inversely proportional to the distance. Using an approximate mean transverse velocity (*V**T*) of 30 kms− 1 for a WD in the galactic disc and a maximum absolute proper motion of $\\dot\\mu~=~\\sqrt{|\\dot\\alpha|^2 + |\\dot\\delta|^2 \\cos^2\\left(\\delta\\right)}~\\approx~12.7\\,$masyr− 1, we find a minimum distance of *D* = 500 pc. Our best-fit proper motion with *V**T* = 30 kms− 1 gives *D* = (1.7 ± 0.5) kpc. This result is consistent with a distance limit of *D* > 1.1 kpc if X-rays are responsible for the heating of the donor, but goes against the limit set if optical/ultra-violet irradiation is responsible, being *d* > 4.2 kpc. Our best-fitting distance measurement is consistent with the expectation of *D* ≈ 2kpc predicted by, although an absolute distance measurement is still relatively unconstrained by our assumption of a disc-like transverse velocity. Halo velocities are often larger whereby assuming *V**T* = 200 kms− 1 we obtain a lower limit of *D* = 3300pc and a best-fitting distance of *D* = (11 ± 3)kpc. We hope that future Gaia data releases will reveal a more precise proper motion and perhaps a parallax measurement, though the *G*mag = 20.9 target is at the limit of Gaia’s capabilities.\nLuminosity\n----------\nIf the mass transfer rate of HM Cnc changes as the orbit shrinks, so will the accretion luminosity. We used the ULTRACAM data using the prime and Super SDSS sets of filters to see if the brightness of HM Cnc has changed over the years. We measured HM Cnc’s magnitude relative to comparison star Gaia DR3 ID 654873149889859968 using the values of the constant offset *d* (see ) from equation ([eqn:WaveFit]). A difference in colour between HM Cnc and the much redder comparison star was accounted for by subtracting filter-dependent colour coefficients as a function of airmass from the magnitudes obtained, as in. The measured changes in magnitude are presented in Table [tab:luminosity]. Observations with the Super SDSS filters have a total observing span of 4 years, while the SDSS prime observations span 13 years in *u*ʹ and *g*ʹ, and about 7 years in *r*ʹ and *i*ʹ, hence the change in magnitude measured with the SDSS prime filters have smaller uncertainties.\nFor the most part, our measurements favour a small brightening of HM Cnc, but all measurements are within 3*σ* of no change at all, and so there has been no significant change overall.\n| Filter | Δmag (magyr− 1) | Filter | Δmag (magyr− 1) |\n| --- | --- | --- | --- |\n| u*s* | − 0.0149 ± 0.0111 | u’ | − 0.0017 ± 0.0036 |\n| g*s* | − 0.0177 ± 0.0080 | g’ | − 0.0013 ± 0.0029 |\n| i*s* | − 0.0686 ± 0.0406 | r’ | − 0.0113 ± 0.0041 |\n| | | i’ | $\\phantom{-}0.0080 \\pm 0.0106$ |\n[tab:luminosity]\nSpectroscopy\n------------\nWe utilised the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) to obtain slitless spectroscopy of HM Cnc in 2007. We observed with the High Resolution Channel (HRC) with the PR200L prism (R = 300–30) and with the Solar Blind Channel (SBC) with the PR110L prism (R = 60–10), with spectral resolution decreasing as the wavelength increases due to a very non-linear prism dispersion. All spectra were individually reduced and flux calibrated with the HST aXe software and were then stacked[3](#fn3). The resultant spectrum was de-reddened using the extinction python package following the treatment of, with A*V* = 0.10mag and R*V* = 3.1. The spectrum obtained is displayed in Fig. [fig:spectrum]. Some data at both ends of the spectra for each setup were deemed unreliable in the flux calibration due to a degraded resolution, creating a ‘winged’ effect that is typical in the ACS slitless spectroscopy setup and discussed in detail in the instrument manual. The far ultra-violet end of the impacted regions were masked in our investigation and are shown as well in Fig. [fig:spectrum].\n [fig:spectrum]\nPrior analyses of optical spectra have indirectly suggested that hydrogen is present due to blending of odd terms of the He Pickering series. Our stacked spectrum reveals a clear Lyman-*α* absorption feature in the ultra-violet, which is apparent in all spectra. This is the first direct evidence of hydrogen in the spectrum of HM Cnc and signifies that hydrogen is present in the accretion stream and thus the envelope of the donor. HM Cnc is the only known ultra-compact AM CVn to show any direct trace of hydrogen, with the possible exception of CP Eri.\nFitting to the observed spectral energy distribution allows us to obtain an improved temperature measurement of the accretor and to predict a hydrogen envelope fraction. The total mass accreted leading up to present day is expected to be approximately 0.01$\\textup{M}\\_\\odot$ with the majority of mass transferred through direct impact accretion (see ), such that the surface composition of both stars will be near identical. To constrain these properties from the spectrum, we used the Koester WD atmosphere models. These allow arbitrary atmospheric compositions including hydrogen-helium mixtures to be used in both the atmospheric structure calculation and the spectral synthesis. We performed a least-squares fit to the data including both stellar components, with model inputs being the effective temperature of the donor and accretor, the relative scaling of the donor to the accretor (i.e. *R**d*2/*R**a*2, with *R* the radius of the respective star and subscripts *d* and *a* the donor and the accretor stars) and a hydrogen-helium abundance which is identical for both stars with no metallicity. Inclusion of a scaling parameter is necessary because, with no firm distance constraint and only one spectral line in the low-resolution spectrum, it is not possible to constrain the surface gravity of either component, and so all calculations were performed at log*g* = 8. At each step in the least-squares analysis, the atmospheric structures and synthetic spectra of both components were calculated and summed together with the inclusion of the model scaling term. This combined spectrum was then convolved with a Gaussian function to match the instrumental resolution at Lyman-*α* (R = 300). Finally, the model spectrum was scaled from a radiated flux, *F*, to a flux observed at Earth, *f*obs, where\n \n$$f\\_{\\textrm{obs}} = \\frac{\\pi R\\_a^2}{D^2} \\left(F\\_a + F\\_d \\frac{R\\_{d}^2}{R\\_{a}^2} \\right)$$ \nand *π**R**a*2/*D*2 = 4.37 × 10− 26 is our determined scaling constant.\nThe derived best-fitting solution is also displayed in Fig. [fig:spectrum]. From this model, we find a number abundance ratio of log(H/He) = − 1.64 ± 0.05dex (H/He = 2.29 ± 0.26%), the hotter star (accretor) temperature to be 42 500 ± 800K and the cooler star (donor) 18 600 ± 300K. These quoted errors are purely statistical from the least-squares minimisation and do not reflect the errors in both our choice of reddening coefficients and the flux calibration when combining two instrumental setups, which are such that the cooler component remains largely unconstrained. Furthermore, it is the cooler star that contributes most significantly towards the redder wavelengths, where our spectrum is sparsely sampled and under-fits the observations. The relative contribution from the dimmer donor is *R**d*2/*R**a*2 = 2.38 ± 0.31, however the radii themselves are unknown with little constraint on the masses of the two stars. If we consider R*a* = 0.01$\\textup{R}\\_\\odot$ with our minimised scaling constant to convert from the radiated flux to the flux at Earth, we find *D* = 1.9kpc. Doubling the accretor radius would double the inferred distance.\nAlthough difficult at low resolution, other spectral lines were searched for in the stacked spectrum of all exposures and with stacked observations as a function of orbital phase. We were not able to detect any, besides the prominent Lyman-*α* absorption line.\nTiming Solutions\n================\nFitting individual observing runs\n---------------------------------\n [fig:O-C]\n[tab:HMCncsolution]\nl c c Parameter & Cubic & Quadratic \nPhase Offset, *ϕ*0 (Cycles) & 0.611 ± 0.004 & 0.611 ± 0.004 \nOrbital Frequency, *f*0 (Hz) & 0.003110224895 ± 30 pHz & 0.003110224829 ± 10 pHz \nFrequency Derivative, $\\dot f\\_0$ (Hz s− 1)& (3.538 ± 0.001) × 10− 16 & (3.538 ± 0.001) × 10− 16 \nFrequency Second Derivative, $\\ddot f\\_0$ (Hz s− 2) & ( − 5.38 ± 2.10) × 10− 27 & – \n[tab:ephemerides]\nWe fit Fourier series solutions to the extracted photometry of the form\n \n*d* + *A*cos(*ω*(*t* − *t*0)) + *B*sin(2*ω*(*t* − *t*0)) + *C*cos(2*ω*(*t* − *t*0)), \nwhere *ω* = 2*π**f* is the orbital angular frequency, *t* is the mid-exposure time of an individual observation, *t*0 is the timing solution and is the time that we seek to measure, *d* is a normalised flux offset, and *A*, *B* and *C* are amplitudes of the fundamental and first harmonic; the harmonic terms account for the non-sinusoidal variability of HM Cnc in the optical (as noticeable in Fig. [fig:LCs]). For each night of data with each filter, *d*, *A*, *B*, *C* and *t*0 were allowed to vary while *ω* was held fixed at the expected angular frequency at the time of observation from the cubic ephemeris of. Including further harmonics to equation ([eqn:WaveFit]) generated a negligible change in the timing measurement. Uncertainties on *t*0 were deduced by taking the standard deviation of 1000 bootstrapping iterations, as used by. Example Fourier series solutions and discussion on specific cases can be later found in.\nFor identical filters, if a periodic signal was difficult to detect (e.g. due to poor seeing, short observing time on target), we used the Fourier series solution of the night with the longest observation time for a given filter to search for a timing solution again. We note that there was little difference in *t*0 if we allowed the parameters of *A*, *B* and *C* to be free or if we fixed them using a solution to the combined filter dataset. This indicates that there has been no/little detectable change in the periodic signal over the full duration of observations, which may have been induced by a changing accretion spot location. We also note that there was no obvious change in the shape of the Fourier series solution about the peak X-ray flux when comparing nights of observation for identical filters spaced years apart.\nWe then searched for any filter-dependent offsets to the final timing measurements and present these computed offsets in Table [tab:offsets]. Only the *u*ʹ − *g*ʹ and *r*ʹ − *g*ʹ offsets indicate a relevant deviation from a zero-phase offset at the 1*σ* level, and so are corrected for all ULTRACAM data using the *u*ʹ and *r*ʹ filters. All other observations from data that we have obtained or from archival sources are considered to have a phase offset of zero.\nc c c c Filter & Δ*ϕ* (cycles) & Filter & Δ*ϕ* (cycles) \nu*s* − g*s* & 0.0015 ± 0.0040 & u’ − g’ & 0.0087 ± 0.0022 \nr*s* − g*s* & -0.0006 ± 0.0029 & r’ − g’ & -0.0023 ± 0.0016 \ni*s* − g*s* & 0.0019 ± 0.0047 & i’ − g’ & 0.0001 ± 0.0090 \nz*s* − g*s* & 0.0020 ± 0.0128 \n[tab:offsets]\nConstraining the ephemeris\n--------------------------\nWith the full set of filter-offset-corrected (and flickering-considered, see ) timing solutions, we proceeded by assigning each an integer cycle number, as inferred using the ephemerides of and while correcting for our own epoch, *T*0. We chose an epoch *T*0 = 55849, BJD TDB, as we found that this gave the smallest correlation between the fitting parameters and is located near the centre of the full observing span as expected. Cycle numbers from both of these ephemerides gave the same integer cycle number for all nights and inferred cycle numbers were manually verified for observations taken more recently than those in. In no case was a jump in cycle number observed, which would be easily noticeable since typical uncertainties are much less than a cycle in magnitude.\nIt is clear (see Fig. [fig:O-C]) that HM Cnc deviates by multiple cycles with respect to a linear fit that has a constant orbital frequency, so we added higher order coefficients to compute a quadratic and then a cubic fit. The set of cycle numbers were fit using\n \n$$\\phi\\left(t\\right) = \\phi\\_0 + f\\_0 \\left(t-T\\_0\\right) + \\frac{\\dot f\\_0}{2}\\left(t-T\\_0\\right)^2 + \\frac{\\ddot f\\_0}{6}\\left(t-T\\_0\\right)^3\n\\label{eqn:PhasingSol}$$ \nwith *ϕ*(*t*) the cycle number, *ϕ*0 a cycle offset at the epoch *T*0, *f*0 the frequency at the epoch *T*0, $\\dot f\\_0$ its first derivative and $\\ddot f\\_0$ its second. For a linear fit, the first two terms on the right-hand side of equation ([eqn:PhasingSol]) apply, the first three terms for a quadratic and all four for a cubic. The quadratic term corresponds to the parabola that spans 21 cycles from minimum to maximum (Fig. [fig:O-C]) which means that HM Cnc has advanced by over 80 orbital cycles relative to the period that it had upon discovery, the result of gravitational-wave driven inspiral.\nA fit to our timing measurements was performed using a least-squares analysis through the scipy lstsq solver in the scipy linear algebra package, which we also used to return 1*σ* statistical errors on all fitted parameters. In the process, the errors on observations were inflated in quadrature so that the reduced *χ*2 is equal to one, where flickering in the system largely encourages the inclusion of this procedure (as discussed in ). Our residuals for solutions with a linear and quadratic ephemeris are presented in Fig. [fig:O-C] with the ephemerides for a quadratic and cubic model given in Table [tab:HMCncsolution]. Corrections of the ephemerides due to the Shklovskii effect and galactic rotation are negligible (see Appendix [secappendix:ephemerisCorr]). The $\\dot f\\_0$ coefficient of our quadratic ephemeris is precise to a 0.03% level and is 40 times more precise than that determined by, also derived from optical data.\nTo check if the $\\ddot f\\_0$ term is statistically significant, we performed an *F*-test between the quadratic and cubic fits. Under the hypothesis that the cubic term is not a significant contribution to the ephemeris, the following *F*-ratio for *n* data points\n \n$$F\\_{\\textrm{ratio}} = \\left(\\chi^2\\_{\\textrm{quad}} - \\chi^2\\_{\\textrm{cubic}}\\right) \\left(\\frac{\\chi^2\\_{\\textrm{cubic}}}{n-4}\\right)^{-1}$$ \nis expected to have an *F*(1, *n* − 4) distribution, where the 1 in the parenthesis represents the difference in the number of degrees of freedom between the quadratic and cubic fits and *n* − 4 reflects the unique degrees of freedom in the cubic fit. We compute the *F*-ratio to be 6.08 for our *n* = 60 measurements. The cumulative distribution function of an *F*-distribution can be used to infer at what significance the null-hypothesis that a cubic term is not reflective of the data can be rejected. As such, the cubic term (and so $\\ddot f\\_0$) is significant at the 98.3% level (which is significant at 95%, or 2*σ*, but not at 99.7%, or 3*σ*). We also carried out an *F*-test on the digitised X-ray data between the quadratic and cubic solutions specified in to test the significance of $\\ddot f$ (see ). The reduced *χ*2 of a cubic fit to the X-ray data was 1.04, so we again inflated the error of all data points in quadrature to gain a reduced *χ*2 equal to one. We found $\\ddot f = -9.41\\pm1.42\\times 10^{-27}\\,$Hz s− 2 and an *F*-ratio of 44.1 for the X-ray data. From the cumulative distribution of *F*(1,172), this represents a near 100% significance. Though with sparser sampling, individual measurements for the X-ray data have higher precision since the peak flux per cycle is better defined by a sharp rise towards an intensity maximum, while the optical is more susceptible to variability. Taking the *F*-test results as evidence that $\\ddot f$ is a true characteristic of the system, all further mentions of *f*0, $\\dot f\\_0$ and $\\ddot f\\_0$ in this paper assume the cubic solution of Table [tab:ephemerides].\nOur optical measurements of $\\dot f\\_0$ and $\\ddot f\\_0$ allow us to estimate the maximum frequency of HM Cnc before a flip of sign of $\\dot f$ is observed. If $\\ddot f\\_0$ remains constant, the system frequency would be expected to evolve as\n \n$$f(t) = f\\_0 + \\dot f\\_0 (t-T\\_0) + \\frac{\\ddot f\\_0}{2} (t-T\\_0)^2\n \\label{eqn:FrequencyEvolution}$$ \nwhere a maximum frequency is reached when $t=-\\dot{f\\_0}/\\ddot{f\\_0}+T\\_0$. We would thus predict the frequency maximum to occur in approximately 2100 ± 800yrs from now. The timing of this event is consistent with the frequency maximum predicted in. As always with timing studies, a larger baseline of data would improve the precision of the ephemeris and, in particular, better constrain $\\ddot f\\_0$.\nInterestingly, our $\\ddot f\\_0$ is of opposite sign to that predicted in earlier literature, with expecting a *positive* $\\ddot f$ of magnitude 10− 28 Hz s− 2. It would be expected that $\\ddot f$ is positive for almost the entirety of inspiral until merger under purely general relativistic orbital decay, unless a re-stabilisation of the orbit due to mass transfer occurs. Consequently, it is thought that HM Cnc is in a mass transfer turn-on phase that will drive the system to smaller orbital frequencies (longer periods) following the frequency maximum.\nUnder solely gravitational-wave orbital angular momentum loss\n \n$$\\centering\n \\dot f\\_{GW} = \\frac{96}{5}~\\pi^{8/3}~\\left( \\frac{G\\mathcal{M}}{c^3}\\right)^{5/3}~f\\_{GW}^{11/3}\n \\label{eqn:fdotGR}$$ \nwith *f**G**W* the frequency of emitted gravitational waves (equal to twice the orbital frequency), *G* the gravitational constant, *c* the speed of light and M = (*m**a* *m**d*)3/5 / (*m**a* + *m**d*)1/5 the chirp mass. For our measured *f*0 and $\\dot f\\_0$, we obtain an observed chirp mass of 0.3203 ± 0.0001 $\\textup{M}\\_\\odot$. Given that the trajectory of HM Cnc is impacted by the mass accretion rate (which acts to oppose inspiral) and not from gravitational wave radiation alone, this observed chirp mass is a lower bound for the true chirp mass of the system.\nFlickering\n----------\nWe noticed variations in the timing solutions of HM Cnc between adjacent nights with ideal conditions. For data from the VLT with ULTRACAM, observations were performed for a similar duration at similar airmasses over the three nights observed. The observations had a seeing < 1.0and multiple consecutive hours spent on target, resulting in high signal-to-noise ratio (SNR) data with no clear outliers. Fig. [fig:FlickeringGraph] depicts the changes in the Fourier series solution for sets of observations that were cut to have a one hour duration, with phase-folded light curves included. There are noticeable differences between the solutions of different nights, particularly evident when comparing the phase of minimum flux. Analysis of the full set of VLT nights over an hourly timescale also indicated that there are small changes between solutions with the longest single observing period being three hours, though nightly differences were more apparent. A similar impact to the Fourier solutions and the returned timing solutions was apparent in the (simultaneously observed) u’- and r’-band VLT observations. When we compared the best-quality adjacent nights of NOT data (BJD=52645–52647) each with over 3 hours on target per night, we saw hourly light curve fluctuations at a similar level, though with larger error in individual measurements due to the smaller telescope aperture. We presume that a changing apparent light-curve morphology is a consequence of changes in the system mass transfer rate, creating a flickering effect typical of accreting binaries, while the long-term light-curve morphology is less impacted.\n [fig:FlickeringGraph]\nA changing optical periodic signal inherently limits the accuracy of all timing solutions, where, taking the mean of the VLT/ULTRACAM nights to be a reference point, the scatter in *t*0 between adjacent nights could be as much as ± 0.01–0.015 cycles for all bands. In this study, we thus combine timing solutions from all filters (phase offset corrected where applicable) to obtain a single timing measurement for the night of observations. Then, we add a fixed error in quadrature due to flickering to each night’s timing measurement, such that the reduced *χ*2 of our ephemerides is equal to one. Overall, this error adjustment was 0.014 cycles.\nAlso in support of system flickering, Fig. 7 of compares phase-folded X-ray flux profiles from the Chandra and NICER datasets. A clear difference in flux profile is noticeable after the X-ray peak flux as the flux is decreasing, and to a lesser extent at the peak X-ray flux. This shows deviation over a multiple year time-span, but it is possible that these changes occur on much shorter timescales, e.g. hours.\nWe then set out to determine whether the flickering was apparent in the X-ray timing solutions as well by comparing our data to the nearest X-ray observation. To do this, we utilised the data from Fig. 6 of by digitising and extracting the plotted data points, followed by a conversion of each to an observed time of peak X-ray flux. After, we compared our own timing solutions with the X-ray times for Chandra and NICER data separately. A phase difference between the time of X-ray peak flux and our derived solutions of *t* was compensated for by minimising the *χ*2 between our optical timing solutions and the X-ray for our cubic-fit ephemeris given in Table [tab:ephemerides], generating a relative phase difference *δ**ϕ* ≈ 0.14 cycles from the optical to X-ray. A comparison of the two sets of solutions is plotted in Fig. [fig:XrayVsOptical], where the 176 individual X-ray measurements have been combined if they coincide with half of a day of each other.\n [fig:XrayVsOptical]\nVery few optical/X-ray observations were taken within a day of each other, making a direct comparison difficult. Little scatter is seen in both datasets from the most recent observations, while the earliest show a similar scatter except for the first optical/X-ray measurements. All that we can assert from the data is that an impact on the accuracy of timing solutions at any wavelength cannot be ruled out.\nEvolutionary Modelling\n======================\nOur measured $\\ddot f\\_0$ combined with the $\\dot f\\_0$ and *f*0 is a very strong constraint upon HM Cnc’s evolution. We explored simulated binary evolution models with comparison to our observed ephemeris using the Modules for Experiments in Stellar Astrophysics (MESA) 1-D stellar evolution code, release version 15140, to investigate the AM CVn DWD configuration. It has been proposed by that the donor of HM Cnc may be an extremely low mass (ELM) WD that is not fully degenerate and has a thick hydrogen layer, which allows the period to shorten even though accretion occurs. The Lyman-*α* detection in our HST spectra supports this theory, as well as larger mass, fully-degenerate WD donors with hydrogen in their envelope. Both types of situations are included in our MESA models. Plotted in Fig. [fig:Allowedmasses] are the valid mass combinations for each star that our observed chirp mass permits, which we surveyed in the MESA models.\n [fig:Allowedmasses]\nBuilding a Donor\n----------------\nWe approximate the full binary evolution by constructing a He WD from an adaptation of the MESA test suite. The way this is performed is by evolving a pre-main sequence to a zero age main sequence star, and then by allowing fusion to take place until the core mass reaches the desired mass of a He WD minus the mass of the envelope. We then cut the evolution and gently remove the envelope of the star until only the He core resides. Next, we evolve the star to an effective temperature of 10kK and then gradually accrete pure hydrogen onto the surface, giving us full control of the total hydrogen envelope mass. What is left is a He core WD with a thin, pure hydrogen layer, which we evolve for 0.5 Gyrs while enabling diffusion, stratifying the layers of the WD to leave a realistic structure[4](#fn4).\nThe maximum hydrogen envelope mass in our models was inferred from the Z=0.0001 He WD models of, choosing models where no hydrogen flash has occurred such that the star maintains a thick hydrogen envelope[5](#fn5). For smaller or larger mass WDs than given in the range of models provided by, we linearly extrapolated their models, with the largest hydrogen envelope we used being MH = 6.5 × 10− 3 $\\textup{M}\\_\\odot$ for models of a 0.16 $\\textup{M}\\_\\odot$ donor.\nWhile the full set of models serves as a good approximation of the amount of hydrogen in the He WD’s envelope, the evolution of a compact DWD is very different to their modelling of a neutron star companion to a WD. This, as well as potential hydrogen flashes, suggests that the initial hydrogen envelope could be less than these models. With that in mind, we investigated the compatibility of different masses of hydrogen envelopes by considering our own set of He WD models, whereby a minimum donor hydrogen envelope consists of MH = 0.1 × 10− 3 $\\textup{M}\\_\\odot$. We sampled hydrogen envelope masses between these limits also, computing models for MH = 1.0 × 10− 3 $\\textup{M}\\_\\odot$ and MH = 3.0 × 10− 3 $\\textup{M}\\_\\odot$.\nBinary Modelling\n----------------\nWe modelled HM Cnc by starting the binary evolution with the fabricated donor He WD and a point mass accretor before any Roche lobe overflow has begun (typically with an initial period of 0.05 days). It is worth noting that have modelled AM CVn systems with MESA by assuming a pure He donor before any mass transfer and fully evolve two stars rather than one star and a point mass. This would be advantageous here also, however the fact that hydrogen is present in the spectrum and that the hydrogen fraction largely controls the timing of maximum orbital frequency encourages us to include the hydrogen envelope. Though evolving two stars would still be possible, computations including hydrogen to incorporate nuclear burning on the accretor’s surface are extremely computationally expensive. Our need to precisely control initial conditions, to simulate a wide grid of models and the fact that nuclear burning on the accretor’s surface is important at the onset of mass accretion but less so later in the evolution led us to proceed with the point mass accretor abstraction, as was similarly initiated by. The abstraction ignores the possible impact of thermonuclear runaways on the surface of the WD which could result in enhanced angular momentum and mass loss. Given that we know that HM Cnc has not merged and that most of the donor hydrogen envelope has been depleted, the binary has survived hydrogen-rich nova episode(s) and our models are only susceptible to complications from a helium nova episode, which may occur in the future. This makes the evolution of our models realistic leading up to *f*0, $\\dot f\\_0$ and $\\ddot f\\_0$, even though the events of the future are uncertain.\nMost importantly to the situation at hand, our dynamics revolve around the framework of the Roche lobe approximation of and mass transfer rates following the method outlined in. We evolve the orbital angular momentum, *J*orb, according to\n \n$$\\dot J\\_{\\textrm{orb}} = \\dot J\\_{\\textrm{gr}} + \\dot J\\_{\\textrm{ml}} + \\dot J\\_{\\textrm{mb}} + \\dot J\\_{\\textrm{ls}} \\label{eqn:Jorb}$$ \nwhere computed values for the rate of orbital momentum change due to gravitational wave radiation, $\\dot J\\_{\\textrm{gr}}$, stellar wind mass loss, $\\dot J\\_{\\textrm{ml}}$, magnetic breaking, $\\dot J\\_{\\textrm{mb}}$, and spin-orbit (LS) coupling, $\\dot J\\_{\\textrm{ls}}$, are handled by MESA according to the methods outlined in. Mass transfer between the two stars is considered to be perfectly efficient and we apply a tidal synchronisation following the prescription of using inbuilt MESA routines.\nWhen an accretion stream feeds into a disc around the accretor, all transferred orbital angular momentum is assumed to return to the orbit of the system; there is no contribution to equation ([eqn:Jorb]). When accretion occurs via direct impact, this condition no longer holds. To account for this, we added an extra angular momentum sink to the MESA calculation, following the prescription of \n \n$$\\dot J\\_{\\textrm{di}} = J\\_{\\textrm{orb}} \\sqrt{r\\_h~(1+q)} \\frac{\\dot M\\_d}{M\\_d}\n \\label{eq:Jdi}$$ \nwhere $\\dot J\\_{\\textrm{di}}$ is the extra loss of orbital angular momentum due to direct-impact mass transfer, *r**h* follows the prescription of equation (13) of as the equivalent radius of the orbiting accreted material and *q* = *M*d/*M*a is the mass ratio. We determined the radius of the star at which an accretion stream would pass by, resulting in disc-fed accretion, using equation (6) of. Given that we consider a point mass accretor, its radius was computed using a mass-temperature-radius relationship (MTRR) from the Montreal CO WD models, where we considered the temperature of the accretor to be 42500K from our spectrum fit (). Hence, we have direct impact accretion if the MTRR radius of the accretor is larger than the radius required for the accretion stream to pass by, or disc-fed accretion on the contrary. In our simulations, this accretor radius is a minimum since the star is likely inflated from mass transfer and partially from irradiation, such that direct impact could begin slightly earlier. For each timestep and model simulated in MESA, we computed whether matter is accreted via direct impact or not and accordingly turned on/off the inclusion of equation ([eq:Jdi]) to the computed $\\dot J\\_{\\textrm{orb}}$ in equation ([eqn:Jorb]).\nDiffusion processes were continuously modelled while evolving the binary, before and during mass transfer. The time-step integration itself was variable in the models; longest initially when there is no mass accretion and smallest when the two stars are at their closest separation. The mass transferred in each time-step was calculated implicitly. A test for a merger was analysed for every model with every time-step integration, however unstable mass transfer only occurred for the less-massive-accretor models in the range permitted in Fig. [fig:Allowedmasses].\n [fig:MESAresults]\n [fig:MESAresultsmdot]\nMESA Results\n------------\nIn the DWD configuration, the most significant variables influencing the time of maximum orbital frequency are the donor mass and the amount of hydrogen in the envelope. The mass of the donor significantly affects the duration of Roche lobe overflow, whereas a larger hydrogen fraction induces a larger donor radius and earlier mass accretion. Besides the distinction between disc/direct fed accretion, combinations of accretor and donor masses control the magnitude of $\\dot f$, where a larger accretor mass leads to a higher $\\dot f$ for a given donor mass. Furthermore, the temperature of the donor star is decisive in controlling the onset of mass transfer and the timing of a negative $\\ddot f$, since a higher temperature WD results in a larger radius.\nFor our chosen selection of hydrogen envelopes, we performed an in-depth search of the full parameter space of Fig. [fig:Allowedmasses] to determine which combinations of masses would lead to $\\dot f\\approx\\dot f\\_0$ when *f* = *f*0 at any stage of the system evolution before shortlisting to the best-agreeing candidate systems, those being where the sign of $\\ddot f$ becomes negative near *f* = *f*0. Our best-fitting models are shown in Fig. [fig:MESAresults], whereby in all cases $\\ddot f\\lessapprox0$ at *f* = *f*0.\nWe solely present the scenario where the donor star maintains an effective temperature of 5.5–6.0kK. This is at odds with the donor temperature derived in, albeit a loose constraint. We attempted to investigate the scenario where the donor effective temperature is 16kK and found that it becomes extremely difficult to maintain any hydrogen envelope for all combinations of permitted masses (Fig. [fig:Allowedmasses]) in MESA. Increasing the donor temperature requires a more massive (and with this a smaller radius) donor for the binary to reach *f*0 before $\\ddot f \\lesssim 0$ occurs. In the process, the binary reaches smaller orbital separations and we require a present-day mass transfer rate in excess of 10− 6$\\textup{M}\\_\\odot$yr− 1, with the order of magnitude of $\\ddot f$ to significantly increase over a few decades. Moreover, the total mass lost from a 16kK donor for tested models exceeded 0.015$\\textup{M}\\_\\odot$ leading up to present day, such that an extremely thick hydrogen envelope before any mass transfer begins would be required for hydrogen to be observed in the spectrum. Our models therefore indicate a preference for cooler donor stars and smaller mass ratios.\nWe conclude from Fig. [fig:MESAresults] that, if we consider our full selection of hydrogen envelope masses, the donor star has a mass in the range ≈ 0.16–0.17$\\textup{M}\\_\\odot$ to observe a negative $\\ddot f$ at *f* = *f*0. This result agrees with the expectation from, who suggest that the donor is an ELM WD. Additionally, we find that the accretor requires a minimum mass of 0.8 $\\textup{M}\\_\\odot$ to give rise to a sufficiently large $\\dot f$ that matches our ephemeris with an ELM donor. These combinations of masses combined with the modelled separation indicate that the accretion stream would be close to the direct-disc mass transfer boundary, as is particularly noticeable by the kinks from the direct-to-disc accretion transition in Fig. [fig:MESAresults] for some models. For these boundary-models in particular, direct impact occurs in the last few 10,000s of years only, although the present mass transfer mechanism for all combinations of star masses presented involve direct impact[6](#fn6). Furthermore, all models show that the binary would not have deviated far from purely general-relativistic orbital decay, meaning that the true chirp mass of the system is comparable to the observed chirp mass of 0.3203 ± 0.0001 $\\textup{M}\\_\\odot$.\nWe use the 0.169$\\textup{M}\\_\\odot$ + 1.156$\\textup{M}\\_\\odot$ and 0.183$\\textup{M}\\_\\odot$ + 0.856$\\textup{M}\\_\\odot$ models presented in Figs. [fig:MESAresults] and [fig:MESAresultsmdot] to check for consistency with the relative phase difference between the optical and X-ray peak flux. discuss how the X-ray signal arrives ≈ 0.26 cycles later than the optical. Our MESA models predict that the accretion stream directly impacts the accretor surface to produce a 0.35 and 0.30 cycle relative phase difference, respectively. This is approximately consistent with that observed, especially so as the accretor is likely to be at least partially inflated, acting to decrease the relative phase difference. For models with smaller initial hydrogen envelope masses, the relative phase differences are larger and show a worse accordance with the observations.\nOur models show a consistent hydrogen envelope fraction with the H/He = 2.29 ± 0.26% derived in. We see from Figs. [fig:MESAresults] and [fig:MESAresultsmdot] that all models have a hydrogen number abundance ratio of less than 10% when $\\ddot f = \\ddot f\\_0$, and that for the more massive donor models presented the fraction is between 1% and 5%. While it may be possible for smaller starting mass hydrogen envelope models to match the hydrogen envelope fraction from our observations, we find a best agreement with the observations for higher starting envelope masses to be consistent with the location of the direct impact spot. A comparison of the full set of models does however show that a range of initial hydrogen envelope masses can produce DWD systems that reach the orbital period of HM Cnc, such that the evolutionary channel is not exclusive to a very narrow set of initial conditions. Our models also show that the remaining hydrogen envelope of the donor will be almost entirely depleted by the time of maximum orbital frequency.\nWhen we simulated a model similar to the 0.27 $\\textup{M}\\_\\odot$ donor and 0.55 $\\textup{M}\\_\\odot$ accretor predicted by, we found that the sign of $\\ddot f$ would be *p**o**s**i**t**i**v**e* at the observed period for the full range of hydrogen envelopes. To emphasise, this would be the case for a cool donor, whereas hotter donors may be able to reproduce the observed $\\dot f\\_0$ and $\\ddot f\\_0$ at *f*0, although the aforementioned complications when modelling a 16kK donor star make this difficult to explain. As well, analysis of $\\ddot f$ indicates that it decreases much faster for a more massive donor, such that HM Cnc would spend only centuries in a turn-around phase. Our models give a mass ratio ≈ 0.15–0.21, which disagrees with the mass ratio 0.27 ≤ *q* ≤ 0.79 inferred by. We suspect that the uncertainty in the origin of light of helium emission lines and the choice of these lines to monitor the primary star makes it difficult to obtain absolute radial velocity measurements, so that the observational constraint on the mass ratio might be misleading even though a radial velocity variability is apparent.\nAs shown in Fig. [fig:MESAresultsmdot], the magnitude of $\\ddot f$ increases dramatically on approach to the maximum orbital frequency. This indicates that the timing of the frequency maximum will actually be in less than 2100 ± 800yrs and that deviations from a constant $\\ddot f$ could become evident sooner than one might expect. The duration spent between orbital periods of 6–8 minutes is significantly longer, on the timescale of 100,000–200,000 years. It hence appears fortunate that HM Cnc has been detected just before the maximum orbital frequency, which may be consequential of a strong observational bias. Though from Fig. [fig:MESAresultsmdot] we predict HM Cnc’s luminosity to increase upon approach to the maximum orbital frequency, with the mass transfer rate suspected to increase by at least 10 × , a clear observable difference is not expected in just 20 years (as suspected from ).\nThe accretor MTRR radii in our best models for a 42500K accretor are 0.0055–0.0115$\\textup{R}\\_\\odot$ for masses in the range 1.25–0.8$\\textup{M}\\_\\odot$, respectively. In, we explain that a radius constraint and a minimised flux-scaling-factor to our spectrum can be used to gain a distance measurement. Including the range of radii from our MESA models with this technique, we estimate a distance to HM Cnc of *D*=1.05–2.24kpc.\nWe follow a similar methodology to to compute the mass accretion rate with the observed X-ray luminosity. We assume that the accretion luminosity is equal to the difference between the gravitational energy at the first Lagrangian point, *ϕ**L*1, and the surface of the accretor, *ϕ**R*a, such that the accretion luminosity L$\\_{\\textrm{acc}} = \\dot M \\left(\\phi\\_{L1}-\\phi\\_{R\\_{\\textrm{a}}}\\right)$, with *ϕ**L*1 and *ϕ**R*a calculable with the prescription of. Taking donor and accretor masses of 0.17$\\textup{M}\\_\\odot$ and 1.0$\\textup{M}\\_\\odot$ with an accretor radius of 0.01$\\textup{R}\\_\\odot$ and the X-ray luminosity to be 1.8 × 1033(ΔΩ/4*π*)*D*kpc2ergs− 1, we find a mass transfer rate $\\dot M=1.84\\times10^{-10}\\times D\\_\\textrm{kpc}^2$$\\textup{M}\\_\\odot$yr− 1. If we assume a distance of 2kpc, $\\dot M=10^{-9.1}$$\\textup{M}\\_\\odot$yr− 1. As recognisable by Fig. [fig:MESAresultsmdot], this would be a clear pitfall of our models, where we would be expecting $\\dot M\\approx10^{-7.5}$$\\textup{M}\\_\\odot$yr− 1. If the distance was instead 5kpc, $\\dot M =10^{-8.3}$$\\textup{M}\\_\\odot$yr− 1, such that assuming larger distances to the source would make our models consistent. An exception is that there could be a missing luminosity component in the extreme ultra-violet that we are not sensitive to with the X-ray nor our own observations, meaning that the X-ray luminosity under-predicts the mass accretion rate. For both the 5.5–6kK and the 16kK WD donors, and with the full grid of hydrogen models, we struggle to find a $\\dot M$ consistent with 10− 9 − 10− 8$\\textup{M}\\_\\odot$yr− 1 that would generate a negative $\\ddot f\\_0$, since the higher mass accretion rates are associated with the timing of the turn-around phase.\nAs evident, there are still components of HM Cnc that remain unknown which make a precise characterisation of the system difficult. However, our models do show a good agreement with multiple observed properties, including the hydrogen fraction, the location of the impact spot and all components of the cubic ephemeris. Our models favour larger initial hydrogen envelope masses and are consistent with the theory outlined by as an origin for HM Cnc and the ELM donor conclusion of. Our models show that it is unlikely that HM Cnc had a hydrogen envelope thinner than 10− 3$\\textup{M}\\_\\odot$ before the onset of mass transfer and require thicker initial hydrogen envelopes to be consistent with the location of the accretion impact spot. Better quantification of the donor temperature and an accurate distance measurement clearly remain the key to settle the nature of HM Cnc. If *D* ≈ 2kpc, our models would need to be tweaked to include a hotter donor temperature, a more massive initial hydrogen envelope or a mixture of the two to agree with the *f*0, $\\dot f\\_0$ and $\\ddot f\\_0$ presented in this paper, since these components both permit larger mass donor stars and smaller mass accretors with an increased mass transfer rate.\nConclusions\n===========\nWe presented new timing measurements for HM Cnc following an optical observing campaign to monitor the binary over the last 20 years. We measured $\\dot f\\_0$ precise to the 0.03% level while obtaining $\\ddot f\\_0~=~(-5.38\\pm2.10)~\\times 10^{-27}$ Hzs− 2. Our ephemeris affirms that the orbital frequency is still increasing, however the negative $\\ddot f\\_0$ implies that HM Cnc may reach a maximum frequency within 2100 ± 800yrs from now.\nThen, we outlined a method whereby knowing $\\ddot f\\_0$ provides a tight condition on the present state of the system. We used MESA to explore the DWD AM CVn configuration. Since we directly witness the presence of hydrogen in our HST spectra, we surveyed multiple masses of donor hydrogen envelopes with combinations of permitted donor and accretor masses. Our best model for a cool and initially hydrogen-rich donor indicates system masses of Md ≈ 0.17$\\textup{M}\\_\\odot$ and Ma ≈ 1.0$\\textup{M}\\_\\odot$. Our model is able to reproduce the observed ephemeris, the hydrogen envelope fraction and the location of the direct impact spot, though suffers from finding a mass transfer rate higher than suggested from X-ray observations if a distance of less than 5kpc is to be assumed. Furthermore, our model indicates that, after reaching a maximum frequency, the orbital frequency will decrease as the orbital separation increases, such that the binary is unlikely to merge unless an imminent helium-powered nova event or a sub-Chandrasekhar mass type a supernova is to occur.\nOur MESA models predict that HM Cnc’s $\\dot f\\_0$ is close to that expected from the purely general relativistic orbital decay for two point masses. This means that the observed system chirp mass (0.3203 ± 0.0001 $\\textup{M}\\_\\odot$) is reflective of the true system chirp mass (0.3347$\\textup{M}\\_\\odot$ for a 0.17$\\textup{M}\\_\\odot$ + 1.0$\\textup{M}\\_\\odot$ combination) and predicts a characteristic strain of 2.67 × 10− 19/*D*kpc after a 4 year observation time with LISA. Furthermore, a 0.17 $\\textup{M}\\_\\odot$ + 1.0 $\\textup{M}\\_\\odot$ star mass pairing, an inclination of and a gravitational wave polarisation angle of generate a LISA SNR of 147 after a 4 year mission for a distance of 5kpc[7](#fn7). Even at 5kpc, HM Cnc will be one of the highest SNR binary star systems detectable by the LISA spacecraft and is still an ideal reference source for the TianQin spacecraft.\nAcknowledgements\n================\nWe thank the referee for their helpful report. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. The associated proposal for this program is #10806. Based on observations made with the Gran Telescopio Canarias (GTC), installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, on the island of La Palma. Based on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. The data presented here were obtained (in part) with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOT. This paper includes observations made in the Observatorios de Canarias del IAC with the WHT and INT operated on the island of La Palma by the IAC in the Observatorio del Roque de los Muchachos. The design and construction of HiPERCAM was funded by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) under ERC-2013-ADG Grant Agreement no. 340040 (HiPERCAM). ULTRASPEC, ULTRACAM and HiPERCAM operations, and VSD, are funded by the Science and Technology Facilities Council (grant ST/V000853/1). JM was supported by funding from a Science and Technology Facilities Council (STFC) studentship. TRM and IP were supported by STFC grant ST/T000406/1. DS acknowledges support from the STFC via grants ST/T007184/1, ST/T003103/1 and ST/T000406/1. SGP acknowledges the support of a STFC Ernest Rutherford Fellowship. For the purpose of open access, the authors has applied a creative commons attribution (CC BY) licence to any author accepted manuscript version arising. JM thanks members of the MESA mailing list and the Astrometry.net discussion forum for their helpful advice. This research made use of Astropy, NumPy and SciPy routines.\nData Availability\n=================\nTiming solutions are included in the Appendix for use. The raw photometric data and reduced HST spectra will be made available upon reasonable request to the authors. The digitised X-ray timing will be made available upon request if necessary.\nProper motion deduction\n=======================\n [fig:mylabel]\nEphemeris corrections\n=====================\nSmall corrections can be made to the obtained frequency derivatives obtained in this study. First by considering the Shklovskii effect, which can be computed with\n \n*S**h* = *V*⊥2/(*c* *D*) \nwith *c* the speed of light and *D* the distance. With our approximate transverse velocity of 30kms− 1 and distance of 1.7 ± 0.5kpc from,\n \n$$\\dot f\\_{\\textrm{Shk}} = f\\_0\\,S\\_h = (1.8\\pm0.5)\\times10^{-22}\\,\\textrm{Hz\\,s}^{-1}$$ \n \n$$\\ddot f\\_{\\textrm{Shk}}~\\approx~\\dot f\\_0\\,S\\_h = (2.0\\pm0.6)\\times10^{-35}\\,\\textrm{Hz\\,s}^{-2}$$ \nIn addition, one can account for acceleration of the system due to the galactic potential, *a*gal, as well. We calculate the following galactic corrections to $\\dot f$ and $\\ddot f$\n \n$$\\dot f\\_{\\textrm{gal}} = f\\_0\\,a\\_{\\textrm{gal}}/c = 8.3\\times10^{-22}\\,\\textrm{Hz\\,s}^{-1}$$ \n \n$$\\ddot f\\_{\\textrm{gal}} \\approx \\dot f\\_0\\,a\\_{\\textrm{gal}}/c = 9.5\\times10^{-35}\\,\\textrm{Hz\\,s}^{-2}$$ \nwhere *a*gal was computed using the MilkyWayPotential routine of gala with HM Cnc placed at a galactic coordinate 1.7kpc away from the Sun towards the north galactic pole (in Cartesian coordinates, [-8, 0, 1.7]kpc).\nThese components can then be subtracted from our measured values, hence the corrected $\\dot f$ and $\\ddot f$ are\n \n$$\\dot f\\_{\\textrm{corr}} = \\dot f\\_0 - \\dot f\\_{\\textrm{shk}} - \\dot f\\_{\\textrm{gal}}$$ \n \n$$\\ddot f\\_{\\textrm{corr}} = \\ddot f\\_0 - \\ddot f\\_{\\textrm{shk}} - \\ddot f\\_{\\textrm{gal}}$$ \nAs evident by the magnitude of each correction, all are negligible compared to the uncertainties quoted in Table [tab:HMCncsolution]. Increasing the distance to *D* = (11 ± 3)kpc for halo-like transverse velocities () results in larger corrections than those stated above. However, neither the correction of $\\dot f\\_0$ and $\\ddot f\\_0$ due to the Shklovskii effect nor galactic acceleration increase by more than 10 × and are still negligible compared to the precision of our ephemerides.\nTiming solutions and observing log\n==================================\n| Time (BJD, TDB, UTC) | ΔTime (BJD, TDB, UTC) | Telescope | Instrument | Filter |\n| --- | --- | --- | --- | --- |\n| 51634.033908 | 0.000067 | ESO 3.6m | EFOSC | B |\n| 51911.047491 | 0.000026 | TNG | DoLoReS | B |\n| 51911.103190 | 0.000052 | TNG | DoLoReS | B |\n| 52225.293041 | 0.000032 | VLT | FORS | R |\n| 52580.316805 | 0.000020 | VLT | FORS | R |\n| 52583.230663 | 0.000023 | TNG | DoLoReS | B |\n| 52619.187101 | 0.000034 | VLT | FORS | R |\n| 52636.193912 | 0.000025 | TNG | DoLoReS | V |\n| 52637.116848 | 0.000017 | TNG | DoLoReS | B |\n| 52645.084430 | 0.000015 | NOT | ALFOSC | Clear |\n| 52646.055759 | 0.000012 | NOT | ALFOSC | Clear |\n| 52647.090283 | 0.000008 | NOT | ALFOSC | Clear |\n| 52647.142372 | 0.000022 | TNG | DoLoReS | B |\n| 52675.938780 | 0.000055 | NOT | ALFOSC | Clear |\n| 52677.036455 | 0.000020 | NOT | ALFOSC | Clear |\n| 52679.035021 | 0.000050 | TNG | DoLoReS | B |\n| 52709.937606 | 0.000038 | TNG | DoLoReS | B |\n| 52710.019443 | 0.000075 | TNG | DoLoReS | V |\n| 52738.901426 | 0.000038 | TNG | DoLoReS | B |\n| 52760.905962 | 0.000021 | TNG | DoLoReS | B |\n| 52779.907568 | 0.000067 | WHT | UCAM | g’ |\n| 52780.897276 | 0.000072 | WHT | UCAM | i’ |\n| 52780.901042 | 0.000021 | WHT | UCAM | g’ |\n| 52780.901073 | 0.000051 | WHT | UCAM | u’ |\n| 52781.894725 | 0.000060 | WHT | UCAM | u’ |\n| 52781.894733 | 0.000084 | WHT | UCAM | i’ |\n| 52781.894763 | 0.000025 | WHT | UCAM | g’ |\n| 52782.892068 | 0.000062 | WHT | UCAM | u’ |\n| 52782.892226 | 0.000090 | WHT | UCAM | i’ |\n| 52782.899519 | 0.000034 | WHT | UCAM | g’ |\n| 52784.883031 | 0.000099 | WHT | UCAM | u’ |\n| 52784.883042 | 0.000034 | WHT | UCAM | g’ |\n| 52784.883089 | 0.000113 | WHT | UCAM | i’ |\n| 52955.260171 | 0.000070 | WHT | UCAM | u’ |\n| 52955.260253 | 0.000057 | WHT | UCAM | g’ |\n| 52955.260287 | 0.000058 | WHT | UCAM | i’ |\n| 52957.191561 | 0.000039 | WHT | UCAM | g’ |\n| 52957.191619 | 0.000075 | WHT | UCAM | u’ |\n| 52957.195317 | 0.000112 | WHT | UCAM | i’ |\n| 52973.238365 | 0.000018 | INT | WFC | Clear |\n| 53023.123751 | 0.000029 | NOT | ALFOSC | Clear |\n| 53024.147096 | 0.000046 | NOT | ALFOSC | Clear |\n| 53053.080944 | 0.000027 | NOT | ALFOSC | Clear |\n| 53142.904373 | 0.000035 | TNG | DoLoReS | B |\n| 53461.954123 | 0.000023 | TNG | DoLoReS | V |\n| 53701.264987 | 0.000008 | VLT | UCAM | g’ |\n| 53701.264993 | 0.000016 | VLT | UCAM | r’ |\n| 53701.265027 | 0.000023 | VLT | UCAM | u’ |\n| 53702.232700 | 0.000027 | VLT | UCAM | u’ |\n| 53702.240109 | 0.000009 | VLT | UCAM | g’ |\n| 53702.243817 | 0.000016 | VLT | UCAM | r’ |\n| 53702.288476 | 0.000020 | VLT | UCAM | u’ |\n| 53702.292164 | 0.000014 | VLT | UCAM | r’ |\n| 53702.292206 | 0.000007 | VLT | UCAM | g’ |\n| 53702.329385 | 0.000013 | VLT | UCAM | r’ |\n| 53702.329398 | 0.000008 | VLT | UCAM | g’ |\n| 53702.329419 | 0.000023 | VLT | UCAM | u’ |\n| 53702.359121 | 0.000016 | VLT | UCAM | r’ |\n| 53702.359124 | 0.000011 | VLT | UCAM | g’ |\n| 53702.359171 | 0.000034 | VLT | UCAM | u’ |\n| 53703.244843 | 0.000026 | VLT | UCAM | r’ |\n| 53703.244848 | 0.000013 | VLT | UCAM | g’ |\n| 53703.244881 | 0.000062 | VLT | UCAM | u’ |\n[tableappendix:timingSolutionsAll]\nc c c c c Time (BJD, TDB, UTC) & ΔTime (BJD, TDB, UTC) & Telescope & Instrument & Filter \n53703.267148 & 0.000016 & VLT & UCAM & r’ \n53703.267165 & 0.000009 & VLT & UCAM & g’ \n53703.267165 & 0.000024 & VLT & UCAM & u’ \n53703.296927 & 0.000013 & VLT & UCAM & r’ \n53703.296960 & 0.000021 & VLT & UCAM & u’ \n53703.300648 & 0.000008 & VLT & UCAM & g’ \n53703.334140 & 0.000016 & VLT & UCAM & r’ \n53703.334146 & 0.000008 & VLT & UCAM & g’ \n53703.334180 & 0.000032 & VLT & UCAM & u’ \n53703.356451 & 0.000013 & VLT & UCAM & r’ \n53703.356470 & 0.000008 & VLT & UCAM & g’ \n53703.360202 & 0.000024 & VLT & UCAM & u’ \n53764.997223 & 0.000025 & TNG & DoLoReS & V \n53765.983439 & 0.000018 & TNG & DoLoReS & B \n53788.136902 & 0.000030 & TNG & DoLoReS & V \n53797.015987 & 0.000019 & TNG & DoLoReS & B \n54149.909394 & 0.000040 & TNG & DoLoReS & V \n54149.957790 & 0.000044 & TNG & DoLoReS & B \n54209.968478 & 0.000032 & TNG & DoLoReS & V \n54394.235140 & 0.000012 & WHT & UCAM & g’ \n54394.235166 & 0.000023 & WHT & UCAM & r’ \n54394.235184 & 0.000029 & WHT & UCAM & u’ \n55532.296391 & 0.000031 & NTT & UCAM & u’ \n55532.300001 & 0.000040 & NTT & UCAM & i’ \n55532.303745 & 0.000011 & NTT & UCAM & g’ \n55947.917226 & 0.000026 & WHT & UCAM & r’ \n55947.920928 & 0.000013 & WHT & UCAM & g’ \n55947.920960 & 0.000027 & WHT & UCAM & u’ \n56211.250500 & 0.000048 & WHT & UCAM & u’ \n56211.250541 & 0.000027 & WHT & UCAM & g’ \n56211.250555 & 0.000049 & WHT & UCAM & r’ \n56214.168047 & 0.000047 & WHT & UCAM & u’ \n56214.168116 & 0.000019 & WHT & UCAM & g’ \n56214.168116 & 0.000034 & WHT & UCAM & r’ \n56342.883367 & 0.000113 & INT & WFC & Clear \n56658.071653 & 0.000030 & WHT & UCAM & r’ \n56658.071729 & 0.000028 & WHT & UCAM & u’ \n56658.075349 & 0.000018 & WHT & UCAM & g’ \n57037.085934 & 0.000034 & WHT & UCAM & u’ \n57037.089626 & 0.000017 & WHT & UCAM & g’ \n57040.181994 & 0.000026 & WHT & UCAM & u’ \n57040.185668 & 0.000013 & WHT & UCAM & g’ \n57040.185680 & 0.000024 & WHT & UCAM & r’ \n57700.349106 & 0.000025 & NTT & UCAM & g’ \n57700.349118 & 0.000049 & NTT & UCAM & r’ \n57700.352702 & 0.000089 & NTT & UCAM & u’ \n57702.347290 & 0.000019 & NTT & UCAM & g’ \n57702.351091 & 0.000059 & NTT & UCAM & u’ \n57702.354797 & 0.000039 & NTT & UCAM & r’ \n58044.245911 & 0.000064 & WHT & HCAM & i*s* \n58044.246005 & 0.000045 & WHT & HCAM & r*s* \n58044.246034 & 0.000021 & WHT & HCAM & g*s* \n58137.164798 & 0.000040 & NTT & UCAM & u*s* \n58137.183422 & 0.000026 & NTT & UCAM & r*s* \n58137.190855 & 0.000014 & NTT & UCAM & g*s* \n58431.326300 & 0.000015 & NTT & UCAM & g*s* \n58431.326373 & 0.000058 & NTT & UCAM & i*s* \n58431.330021 & 0.000040 & NTT & UCAM & u*s* \n58494.047163 & 0.000015 & GTC & HCAM & i*s* \n58494.047164 & 0.000008 & GTC & HCAM & r*s* \n58494.047166 & 0.000006 & GTC & HCAM & g*s* \n58494.047172 & 0.000039 & GTC & HCAM & z*s* \n58494.047174 & 0.000015 & GTC & HCAM & u*s* \n58879.239177 & 0.000051 & NTT & UCAM & u*s* \n58879.246677 & 0.000019 & NTT & UCAM & g*s* \n58879.250316 & 0.000078 & NTT & UCAM & i*s* \n58880.746361 & 0.000037 & TNO & USPEC & KG5 \n[tableappendix:timingSolutionsAllcont]\nc c c c c Time (BJD, TDB, UTC) & ΔTime (BJD, TDB, UTC) & Telescope & Instrument & Filter \n59259.711306 & 0.000032 & TNO & USPEC & KG5 \n59281.643978 & 0.000046 & TNO & USPEC & KG5 \n59526.333033 & 0.000020 & NTT & UCAM & g*s* \n59526.336784 & 0.000050 & NTT & UCAM & u*s* \n59527.319131 & 0.000044 & NTT & UCAM & u*s* \n59527.319145 & 0.000017 & NTT & UCAM & g*s* \n59527.319175 & 0.000057 & NTT & UCAM & i*s* \n59528.346290 & 0.000045 & NTT & UCAM & g*s* \n59528.346293 & 0.000075 & NTT & UCAM & u*s* \n59528.346385 & 0.000079 & NTT & UCAM & i*s* \n59642.131890 & 0.000019 & NTT & UCAM & g*s* \n59642.131937 & 0.000046 & NTT & UCAM & u*s* \n59642.132033 & 0.000102 & NTT & UCAM & i*s* \n[tableappendix:timingSolutionsAllcont2]\n[tableappendix:ObservingLog]\n| Night | MJDmid | Filters | Telescope | Instrument | Cadence (s) | Duration (min) | Comments |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 2003-05-20 | 52779.9 | u’ g’ i’ | WHT | UCAM | 20.0 | 20.3 | Seeing 1.5-2.0” |\n| 2003-05-21 | 52780.9 | u’ g’ i’ | WHT | UCAM | 9.8 | 53.2 | \\*Seeing 1.2” |\n| 2003-05-22 | 52781.9 | u’ g’ i’ | WHT | UCAM | 9.8 | 57.9 | \\*Seeing 1.0” |\n| 2003-05-23 | 52782.9 | u’ g’ i’ | WHT | UCAM | 9.8 | 62.4 | \\*Seeing 1.0” |\n| 2003-05-25 | 52784.9 | u’ g’ i’ | WHT | UCAM | 9.8 | 41.2 | \\*Seeing 1.3” |\n| 2003-11-11 | 52955.2 | u’ g’ i’ | WHT | UCAM | 10.0 | 66.1 | Seeing 1.0” |\n| 2003-11-13 | 52957.1 | u’ g’ i’ | WHT | UCAM | 10.0 | 35.5 | Seeing 1.0-1.3” |\n| 2005-11-26 | 53701.2 | u’ g’ r’ | VLT | UCAM | 3.0 | 107.5 | \\*Initially poor seeing, later 1.0-1.5” |\n| 2005-11-27 | 53702.3 | u’ g’ r’ | VLT | UCAM | 1.5 | 215.5 | \\*Seeing < 1.0” |\n| 2005-11-28 | 53703.3 | u’ g’ r’ | VLT | UCAM | 1.5 | 195.9 | \\*Seeing < 1.0” |\n| 2007-10-20 | 54394.2 | u’ g’ r’ | WHT | UCAM | 9.7 | 102.2 | Seeing 1.0–1.2” |\n| 2010-12-01 | 55532.3 | u’ g’ i’ | NTT | UCAM | 19.7 | 193.6 | Seeing 1.0-1.5”. Intermittent clouds |\n| 2012-01-21 | 55947.9 | u’ g’ r’ | WHT | UCAM | 10.0 | 113.3 | Seeing 1.0-1.5” |\n| 2012-10-10 | 56211.2 | u’ g’ r’ | WHT | UCAM | 8.0 | 32.2 | Seeing 1.0” |\n| 2012-10-13 | 56214.1 | u’ g’ r’ | WHT | UCAM | 8.5 | 49.9 | Seeing 1.4” |\n| 2013-12-31 | 56658.0 | u’ g’ r’ | WHT | UCAM | 10.0 | 57.2 | Seeing 1.5” |\n| 2015-01-14 | 57037.1 | u’ g’ i’ | WHT | UCAM | 10.0 | 66.0 | Flares in seeing; 1.0–3.0” |\n| 2015-01-17 | 57040.2 | u’ g’ r’ | WHT | UCAM | 10.0 | 107.2 | Seeing 1.0–1.4” |\n| 2016-11-07 | 57700.3 | u’ g’ r’ | NTT | UCAM | 10.0 | 35.4 | Seeing 1.5” |\n| 2016-11-09 | 57702.3 | u’ g’ r’ | NTT | UCAM | 10.0 | 56.9 | Seeing 1.2” |\n| 2018-01-18 | 58137.2 | u*s* g*s* r*s* | NTT | UCAM | 20.0 | 141.1 | Seeing 1.5” |\n| 2018-11-08 | 58431.3 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 84.7 | Seeing 1.3” |\n| 2020-01-30 | 58879.2 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 147.8 | Seeing 1.5” |\n| 2021-11-07 | 59526.3 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 74.6 | Focus issues. Good conditions. |\n| 2021-11-08 | 59527.3 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 129.7 | Seeing 1.0–1.2” |\n| 2021-11-09 | 59528.3 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 20.4 | Seeing 1.1” |\n| 2022-03-03 | 59642.1 | u*s* g*s* i*s* | NTT | UCAM | 10.0 | 71.4 | Seeing 1.2” |\n| 2017-10-17 | 58044.2 | g*s* r*s* i*s* z*s* | WHT | HCAM | 13.9 | 24.8 | Seeing 1.2” |\n| 2019-01-10 | 58494.0 | u*s* g*s* r*s* i*s* z*s* | GTC | HCAM | 10.0 | 43.3 | Seeing 0.9” |\n| 2020-02-01 | 58880.7 | KG5 | TNO | USPEC | 15.4 | 160.2 | Seeing 1.5” |\n| 2021-02-14 | 59259.7 | KG5 | TNO | USPEC | 14.2 | 96.7 | Seeing 1.0–1.4” |\n| 2021-03-08 | 59281.6 | KG5 | TNO | USPEC | 16.4 | 175.4 | Seeing 0.9” |\n[tab:observingLog]\nMESA input models\n=================\n| Donor Mass ($\\textup{M}\\_\\odot$) | Accretor Mass ($\\textup{M}\\_\\odot$) | Hydrogen Envelope (10− 3 $\\textup{M}\\_\\odot$) | Fig. [fig:MESAresults] Label |\n| --- | --- | --- | --- |\n| 0.16 | 1.20 | 0.1 | 0.156$\\textup{M}\\_\\odot$ + 1.202$\\textup{M}\\_\\odot$ |\n| 0.16 | 1.30 | 0.1 | 0.156$\\textup{M}\\_\\odot$ + 1.302$\\textup{M}\\_\\odot$ |\n| 0.17 | 1.00 | 0.1 | 0.167$\\textup{M}\\_\\odot$ + 1.001$\\textup{M}\\_\\odot$ |\n| 0.16 | 1.20 | 1.0 | 0.156$\\textup{M}\\_\\odot$ + 1.202$\\textup{M}\\_\\odot$ |\n| 0.17 | 1.00 | 1.0 | 0.166$\\textup{M}\\_\\odot$ + 1.002$\\textup{M}\\_\\odot$ |\n| 0.17 | 1.20 | 3.0 | 0.162$\\textup{M}\\_\\odot$ + 1.205$\\textup{M}\\_\\odot$ |\n| 0.17 | 1.30 | 3.0 | 0.162$\\textup{M}\\_\\odot$ + 1.305$\\textup{M}\\_\\odot$ |\n| 0.18 | 0.95 | 3.0 | 0.172$\\textup{M}\\_\\odot$ + 0.955$\\textup{M}\\_\\odot$ |\n| 0.176 | 1.25 | 5.6 | 0.163$\\textup{M}\\_\\odot$ + 1.258$\\textup{M}\\_\\odot$ |\n| 0.179 | 1.15 | 5.3 | 0.169$\\textup{M}\\_\\odot$ + 1.156$\\textup{M}\\_\\odot$ |\n| 0.195 | 0.85 | 4.9 | 0.183$\\textup{M}\\_\\odot$ + 0.856$\\textup{M}\\_\\odot$ |\n[tab:mylabel2]\n[lastpage]\n---\n1. Email: james.munday98@gmail.com[↩](#fnref1)\n2. All data presented in had calibration files and were bias-corrected and flat-fielded with custom scripts. Unfortunately in some cases either windowing setups were not specified or calibration files were not able to be recovered from the TNG archive, meaning that some data extraction was performed without bias or flat-field correction. Whenever HM Cnc overlapped an artefact, or, the image showed a noticeably non-uniform local field, the data from this night were discarded.[↩](#fnref2)\n3. It was not possible to use the typical HST “drizzle” scheme to combine spectra due to the non-linear dispersion solution. Cosmic rays were flagged by the HST ACS pipeline and were subsequently removed.[↩](#fnref3)\n4. Our WD models have Z=0.00 due to slow convergence on a solution with our crafted structure in the diffusion process. A couple of test cases of Z=0.02 were evolved as an AM CVn to ensure fair comparisons, where the impact of metallicity in the accreted material was found to be negligible.[↩](#fnref4)\n5. Though suggest that larger hydrogen envelope masses of 10− 2 $\\textup{M}\\_\\odot$ could be required for the model, we consider the non-hydrogen-flash models to have an improved maximum hydrogen envelope mass.[↩](#fnref5)\n6. We note that state the condition that direct impact can only occur in HM Cnc if 0.6 < M1 < 0.9 $\\textup{M}\\_\\odot$, however this was deduced with a zero-temperature masses-radius relationship. The masses we quote involve direct impact allowing for an accretor temperature Taccretor = 42 500 K.[↩](#fnref6)\n7. \n\nTwo decades of optical timing of the shortest-period binary star system HM Cancri\n=================================================================================\n### Accepted 2022 November 16. Received 2022 November 16; in original form 2022 September 1\n[firstpage]\nThe shortest-period binary star system known to date, RX J0806.3+1527 (HM Cancri), has now been observed in the optical for more than two decades. Although it is thought to be a double degenerate binary undergoing mass transfer, an early surprise was that its orbital frequency, *f*0, is currently increasing as the result of gravitational wave radiation. This is unusual since it was expected that the mass donor was degenerate and would expand on mass loss, leading to a decreasing *f*0. We exploit two decades of high-speed photometry to precisely quantify the trajectory of HM Cancri, allowing us to find that $\\ddot f\\_0$ is negative, where $\\ddot f\\_0~=~(-5.38\\pm2.10)\\times10^{-27}$ Hzs− 2. Coupled with our positive frequency derivative, we show that mass transfer is counteracting gravitational-wave dominated orbital decay and that HM Cnc will turn around within 2100 ± 800yrs from now. We present Hubble Space Telescope ultra-violet spectra which display Lyman-*α* absorption, indicative of the presence of hydrogen accreted from the donor star. We use these pieces of information to explore a grid of permitted donor and accretor masses with the Modules for Experiments in Stellar Astrophysics suite, finding models in good accordance with many of the observed properties for a cool and initially hydrogen-rich extremely-low-mass white dwarf ( ≈ 0.17$\\textup{M}\\_\\odot$) coupled with a high accretor mass white dwarf ( ≈ 1.0$\\textup{M}\\_\\odot$). Our measurements and models affirm that HM Cancri is still one of the brightest verification binaries for the Laser Interferometer Space Antenna spacecraft.\nbinaries: close – individual: RX J0806.3+1527, HM Cancri – stars: white dwarfs – gravitational waves\nIntroduction\n============\nThere exists a large population of double white dwarf (DWD) binary star systems in the galaxy which can form through a series of mass-transfer and common envelope events. Gravitational wave radiation drives the loss of orbital angular momentum and causes many binary orbits to decay within a Hubble time, driving them to compact (orbital period of ≈ 5–65min) configurations. Ultimately, these systems initiate mass transfer, which has a dramatic effect upon their evolution, making DWD binaries the likely progenitors of type a/.a supernovae and maybe the progenitors of R CrB stars, hot subdwarf stars and AM CVn systems. In contrast to the typical dynamics of mass transfer between non-degenerate stars, the smaller mass, larger radius white dwarf (WD) is first to overflow its Roche lobe as the donor star, with the larger mass, smaller radius WD the accretor in a DWD binary. Mass transfer from one star to a higher mass companion acts to expand the orbit, counteracting the shrinkage caused by gravitational wave losses with angular momentum transfer playing an important role in the fate of these systems.\nRX J0806.3+1527 is the shortest-period binary star system known, having a present day orbital period of 5.36min, equivalent to an orbital frequency of 3.11 mHz. There have been multiple theories suggested to explain the extremely short period of HM Cnc, namely the unipolar inductor model ; the intermediate polar model ; the AM CVn model. In brief, the unipolar inductor model is similar to the Jupiter-Io system with the binary being detached, the intermediate polar model suggests that the optical period actually represents the spins of magnetic WDs and that the binary’s orbital period is much longer, and the AM CVn model involves a compact WD binary mass-transferring helium rich material. In the case of HM Cnc, a DWD binary is required to reach the observed period for an AM CVn without a merger already taking place. The studies of and helped to reduce the options. They find clear radial-velocity variations on the optical period, difficult to explain using the intermediate polar model, and a strong indication of an accretion stream via the presence of He emission lines in the spectrum, which is not predicted by the unipolar inductor model. fail to detect any signature of metals in the X-ray spectrum, favouring a DWD AM CVn, and a DWD binary is also a natural explanation for the relative phase difference between the peak of X-ray and optical flux. Here, the peak in X-ray flux arrives slightly later than the optical, which arises from the trajectory of the accretion stream directly impacting the surface of the accretor. It would thus seem that only the DWD AM CVn theory can be used to explain all of the present observations.\nFollow-up observations of HM Cnc in the X-ray and optical domains have revealed an increase in orbital frequency, *f*, due to the loss of orbital angular momentum through gravitational wave radiation. The increase is so much so that HM Cnc will be one of the brightest “verification binaries” to optimise the performance of the Laser Interferometer Space Antenna spacecraft, given its low-frequency gravitational wave emission, strong signal strength and the precise timing measurements that have been used to quantify its orbital decay. Similarly, HM Cnc is planned to be used as the primary reference source for the TianQin spacecraft. While it took two years to detect an increase of HM Cnc’s orbital frequency, continuous timing measurements over a much longer baseline can prove insightful to determine the second derivative, $\\ddot f$, which is an indicator of the trajectory of inspiral.\nUsing X-ray data, measured the second derivative of the orbital frequency of HM Cnc, finding $\\ddot f = (-8.95\\pm1.40) \\times 10^{-27}\\,$Hz s− 2. This was the first significant detection of $\\ddot{f}$ for any AM CVn system. The combination of $\\dot f$ and $\\ddot f$ can be used to predict the timing of maximum orbital frequency of the binary, which predicts to occur in 1260 ± 200 yrs from now. In addition, was able to estimate a mass transfer rate based on the accretion luminosity of the source from X-ray observations. HM Cnc is unfortunately too dim (*G*mag = 20.9) for a distance constraint with Gaia DR3, though assuming a distance of 5 kpc and donor and accretor masses of 0.27 $\\textup{M}\\_\\odot$ and 0.55 $\\textup{M}\\_\\odot$, calculated an accretion rate $\\dot M = -1.6\\times10^{-8}\\,$$\\textup{M}\\_\\odot$ yr− 1. However, the calculated mass transfer rate is strongly dependent on the input star masses and the distance to the source. A distance measurement and a precise measurement of $\\ddot f$ are therefore extremely useful to constrain the binary’s evolution with sets of donor and companion masses.\nOur study is set out as follows. discusses our photometric and spectroscopic observations of HM Cnc and our attempts to derive a distance measurement. In we present system timing measurements and derive the orbital ephemeris. We then explore the extent to which the AM CVn channel can match our orbital ephemeris with binary evolution simulations in.\nObservations\n============\nPhotometry\n----------\nTime-series photometry was obtained using the high-speed cameras ULTRACAM, ULTRASPEC and HiPERCAM ; ULTRACAM and HiPERCAM observe simultaneously in three or five bands, respectively. Data from HiPERCAM were obtained while mounted on the 4.2m William Herschel Telescope (WHT) and the 10.4m Gran Telescopio Canarias (GTC). We observed with ULTRACAM on the 8.2m Very Large Telescope (VLT), the 3.58m ESO New Technology Telescope (NTT) and the WHT, and we observed with ULTRASPEC mounted on the the 2.4m Thai National Observatory (TNO). Exposure times were varied across the nights to reflect the seeing conditions, the aperture of the telescope and the filters used at the time. A full observing log giving dates, conditions, exposure times and the filters used for each set of observations is given in Appendix [tab:observingLog].\n [fig:LCs]\nAll of the data were bias-subtracted and flat-fielded. An additional dark current subtraction was applied to all data from ULTRACAM only as the instrument operates at a temperature high enough for there to be significant dark current. For HiPERCAM, corrections for CCD fringing were performed in the *z**s* band only using pre-obtained fringe maps for the instrument.\nWe also accessed VLT FORS1, ESO 3.6m Telescope EFOSC2 and Telescopio Nazionale Galileo (TNG) DoLoReS archival data that were presented in prior to our first observation. Extra archival data of HM Cnc were obtained from the TNG DoLoReS instrument from dates following their study. Though the study of used some of these data before, no timing solutions have been reported, so we use the archival data in our study to take advantage of the longest observational baseline possible[2](#fn2). Data obtained with the 2.56m Nordic Optical Telescope (NOT) that was presented in and with the 2.5m Isaac Newton Telescope (INT) from the RATS survey were utilised in this study too. The extracted differential photometry obtained in prior studies using data from the NOT were used, whereas raw files were handled from every other source.\nPhotometry from each instrument, besides data obtained with the NOT, was then extracted using the HiPERCAM pipeline with differential aperture photometry and the application of ’s optimal photometry algorithm when the target aperture did not need to be linked to a comparison star; otherwise regular aperture photometry was performed. To tackle poorer observing conditions, a variable aperture radius was applied, which was chosen to be a fixed multiple (1.6–1.8 × ) of the full-width at half-maximum of the stars for each individual frame. A non-variable comparison star (Gaia DR3 ID 654873149889859968) was used for all observations when possible, or another (Gaia DR3 ID 654879025405159040) if the main comparison star saturated or coincided with detector defects.\nThe observation times were corrected to the Barycentric Julian Date (BJD) Barycentric Dynamical Time (TDB) centred on the mid-point of the exposure time. Poor data (e.g. affected by clouds) were rejected by visual inspection with HiPERCAM pipeline routines. We then derived a differential atmospheric extinction term for each filter to correct for the difference in colour between the comparison star and HM Cnc. This was performed by considering the dependence of airmass on the differential flux, which was subtracted to effectively convert to a flux at zenith.\nProper Motion\n-------------\nAs of Gaia DR3, there is no measurement of the proper motion of HM Cnc, so we searched for it using our data. To do so, we added world coordinate system information to each FITS file by plate solving each exposure using the backend of Astrometry.net. We utilised a custom Gaia DR3 reference star catalogue for the local field containing stars with a right ascension (*α*) and declination (*δ*) error of less than 0.1 arcseconds. The catalogue provided the location of stars at the mid-Gaia DR3 epoch. We selected data from the *V*, *r* and *i* bands only to reduce the impact of atmospheric refraction upon the position of HM Cnc relative to the redder stars in the field, though atmospheric refraction still limits our position accuracy. A reliable coordinate calibration was possible for all frames with approximately 20 or more reference stars. Then, we converted the central (pixel) coordinate and error of the centroid for each aperture into a physical *α*/*δ* coordinate. All measurements of *α*/*δ* from each frame were then combined with a variance weighted mean, giving a measured *α*/*δ* for HM Cnc on each night.\nFrom our determined positions (as plotted in Appendix [appendix:propermotionGraphs]), we conservatively take the range of the data to find upper limits of $|\\dot\\alpha|<9.1\\,$masyr− 1 and $|\\dot\\delta|<9.2\\,$masyr− 1. Our best-fit indicates proper motions of $\\dot\\alpha=2.0\\pm1.1$ masyr− 1 and $\\dot\\delta=-3.3\\pm1.1$ masyr− 1, having a reduced *χ*2 equal to one. By taking the maximum proper motion, we can derive a loose minimum distance constraint.\nThe proper motion is inversely proportional to the distance. Using an approximate mean transverse velocity (*V**T*) of 30 kms− 1 for a WD in the galactic disc and a maximum absolute proper motion of $\\dot\\mu~=~\\sqrt{|\\dot\\alpha|^2 + |\\dot\\delta|^2 \\cos^2\\left(\\delta\\right)}~\\approx~12.7\\,$masyr− 1, we find a minimum distance of *D* = 500 pc. Our best-fit proper motion with *V**T* = 30 kms− 1 gives *D* = (1.7 ± 0.5) kpc. This result is consistent with a distance limit of *D* > 1.1 kpc if X-rays are responsible for the heating of the donor, but goes against the limit set if optical/ultra-violet irradiation is responsible, being *d* > 4.2 kpc. Our best-fitting distance measurement is consistent with the expectation of *D* ≈ 2kpc predicted by, although an absolute distance measurement is still relatively unconstrained by our assumption of a disc-like transverse velocity. Halo velocities are often larger whereby assuming *V**T* = 200 kms− 1 we obtain a lower limit of *D* = 3300pc and a best-fitting distance of *D* = (11 ± 3)kpc. We hope that future Gaia data releases will reveal a more precise proper motion and perhaps a parallax measurement, though the *G*mag = 20.9 target is at the limit of Gaia’s capabilities.\nLuminosity\n----------\nIf the mass transfer rate of HM Cnc changes as the orbit shrinks, so will the accretion luminosity. We used the ULTRACAM data using the prime and Super SDSS sets of filters to see if the brightness of HM Cnc has changed over the years. We measured HM Cnc’s magnitude relative to comparison star Gaia DR3 ID 654873149889859968 using the values of the constant offset *d* (see ) from equation ([eqn:WaveFit]). A difference in colour between HM Cnc and the much redder comparison star was accounted for by subtracting filter-dependent colour coefficients as a function of airmass from the magnitudes obtained, as in. The measured changes in magnitude are presented in Table [tab:luminosity]. Observations with the Super SDSS filters have a total observing span of 4 years, while the SDSS prime observations span 13 years in *u*ʹ and *g*ʹ, and about 7 years in *r*ʹ and *i*ʹ, hence the change in magnitude measured with the SDSS prime filters have smaller uncertainties.\nFor the most part, our measurements favour a small brightening of HM Cnc, but all measurements are within 3*σ* of no change at all, and so there has been no significant change overall.\n| Filter | Δmag (magyr− 1) | Filter | Δmag (magyr− 1) |\n| --- | --- | --- | --- |\n| u*s* | − 0.0149 ± 0.0111 | u’ | − 0.0017 ± 0.0036 |\n| g*s* | − 0.0177 ± 0.0080 | g’ | − 0.0013 ± 0.0029 |\n| i*s* | − 0.0686 ± 0.0406 | r’ | − 0.0113 ± 0.0041 |\n| | | i’ | $\\phantom{-}0.0080 \\pm 0.0106$ |\n[tab:luminosity]\nSpectroscopy\n------------\nWe utilised the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) to obtain slitless spectroscopy of HM Cnc in 2007. We observed with the High Resolution Channel (HRC) with the PR200L prism (R = 300–30) and with the Solar Blind Channel (SBC) with the PR110L prism (R = 60–10), with spectral resolution decreasing as the wavelength increases due to a very non-linear prism dispersion. All spectra were individually reduced and flux calibrated with the HST aXe software and were then stacked[3](#fn3). The resultant spectrum was de-reddened using the extinction python package following the treatment of, with A*V* = 0.10mag and R*V* = 3.1. The spectrum obtained is displayed in Fig. [fig:spectrum]. Some data at both ends of the spectra for each setup were deemed unreliable in the flux calibration due to a degraded resolution, creating a ‘winged’ effect that is typical in the ACS slitless spectroscopy setup and discussed in detail in the instrument manual. The far ultra-violet end of the impacted regions were masked in our investigation and are shown as well in Fig. [fig:spectrum].\n [fig:spectrum]\nPrior analyses of optical spectra have indirectly suggested that hydrogen is present due to blending of odd terms of the He Pickering series. Our stacked spectrum reveals a clear Lyman-*α* absorption feature in the ultra-violet, which is apparent in all spectra. This is the first direct evidence of hydrogen in the spectrum of HM Cnc and signifies that hydrogen is present in the accretion stream and thus the envelope of the donor. HM Cnc is the only known ultra-compact AM CVn to show any direct trace of hydrogen, with the possible exception of CP Eri.\nFitting to the observed spectral energy distribution allows us to obtain an improved temperature measurement of the accretor and to predict a hydrogen envelope fraction. The total mass accreted leading up to present day is expected to be approximately 0.01$\\textup{M}\\_\\odot$ with the majority of mass transferred through direct impact accretion (see ), such that the surface composition of both stars will be near identical. To constrain these properties from the spectrum, we used the Koester WD atmosphere models. These allow arbitrary atmospheric compositions including hydrogen-helium mixtures to be used in both the atmospheric structure calculation and the spectral synthesis. We performed a least-squares fit to the data including both stellar components, with model inputs being the effective temperature of the donor and accretor, the relative scaling of the donor to the accretor (i.e. *R**d*2/*R**a*2, with *R* the radius of the respective star and subscripts *d* and *a* the donor and the accretor stars) and a hydrogen-helium abundance which is identical for both stars with no metallicity. Inclusion of a scaling parameter is necessary because, with no firm distance constraint and only one spectral line in the low-resolution spectrum, it is not possible to constrain the surface gravity of either component, and so all calculations were performed at log*g* = 8. At each step in the least-squares analysis, the atmospheric structures and synthetic spectra of both components were calculated and summed together with the inclusion of the model scaling term. This combined spectrum was then convolved with a Gaussian function to match the instrumental resolution at Lyman-*α* (R = 300). Finally, the model spectrum was scaled from a radiated flux, *F*, to a flux observed at Earth, *f*obs, where\n \n$$f\\_{\\textrm{obs}} = \\frac{\\pi R\\_a^2}{D^2} \\left(F\\_a + F\\_d \\frac{R\\_{d}^2}{R\\_{a}^2} \\right)$$ \nand *π**R**a*2/*D*2 = 4.37 × 10− 26 is our determined scaling constant.\nThe derived best-fitting solution is also displayed in Fig. [fig:spectrum]. From this model, we find a number abundance ratio of log(H/He) = − 1.64 ± 0.05dex (H/He = 2.29 ± 0.26%), the hotter star (accretor) temperature to be 42 500 ± 800K and the cooler star (donor) 18 600 ± 300K. These quoted errors are purely statistical from the least-squares minimisation and do not reflect the errors in both our choice of reddening coefficients and the flux calibration when combining two instrumental setups, which are such that the cooler component remains largely unconstrained. Furthermore, it is the cooler star that contributes most significantly towards the redder wavelengths, where our spectrum is sparsely sampled and under-fits the observations. The relative contribution from the dimmer donor is *R**d*2/*R**a*2 = 2.38 ± 0.31, however the radii themselves are unknown with little constraint on the masses of the two stars. If we consider R*a* = 0.01$\\textup{R}\\_\\odot$ with our minimised scaling constant to convert from the radiated flux to the flux at Earth, we find *D* = 1.9kpc. Doubling the accretor radius would double the inferred distance.\nAlthough difficult at low resolution, other spectral lines were searched for in the stacked spectrum of all exposures and with stacked observations as a function of orbital phase. We were not able to detect any, besides the prominent Lyman-*α* absorption line.\nTiming Solutions\n================\nFitting individual observing runs\n---------------------------------\n [fig:O-C]\n[tab:HMCncsolution]\nl c c Parameter & Cubic & Quadratic \nPhase Offset, *ϕ*0 (Cycles) & 0.611 ± 0.004 & 0.611 ± 0.004 \nOrbital Frequency, *f*0 (Hz) & 0.003110224895 ± 30 pHz & 0.003110224829 ± 10 pHz \nFrequency Derivative, $\\dot f\\_0$ (Hz s− 1)& (3.538 ± 0.001) × 10− 16 & (3.538 ± 0.001) × 10− 16 \nFrequency Second Derivative, $\\ddot f\\_0$ (Hz s− 2) & ( − 5.38 ± 2.10) × 10− 27 & – \n[tab:ephemerides]\nWe fit Fourier series solutions to the extracted photometry of the form\n \n*d* + *A*cos(*ω*(*t* − *t*0)) + *B*sin(2*ω*(*t* − *t*0)) + *C*cos(2*ω*(*t* − *t*0)), \nwhere *ω* = 2*π**f* is the orbital angular frequency, *t* is the mid-exposure time of an individual observation, *t*0 is the timing solution and is the time that we seek to measure, *d* is a normalised flux offset, and *A*, *B* and *C* are amplitudes of the fundamental and first harmonic; the harmonic terms account for the non-sinusoidal variability of HM Cnc in the optical (as noticeable in Fig. [fig:LCs]). For each night of data with each filter, *d*, *A*, *B*, *C* and *t*0 were allowed to vary while *ω* was held fixed at the expected angular frequency at the time of observation from the cubic ephemeris of. Including further harmonics to equation ([eqn:WaveFit]) generated a negligible change in the timing measurement. Uncertainties on *t*0 were deduced by taking the standard deviation of 1000 bootstrapping iterations, as used by. Example Fourier series solutions and discussion on specific cases can be later found in.\nFor identical filters, if a periodic signal was difficult to detect (e.g. due to poor seeing, short observing time on target), we used the Fourier series solution of the night with the longest observation time for a given filter to search for a timing solution again. We note that there was little difference in *t*0 if we allowed the parameters of *A*, *B* and *C* to be free or if we fixed them using a solution to the combined filter dataset. This indicates that there has been no/little detectable change in the periodic signal over the full duration of observations, which may have been induced by a changing accretion spot location. We also note that there was no obvious change in the shape of the Fourier series solution about the peak X-ray flux when comparing nights of observation for identical filters spaced years apart.\nWe then searched for any filter-dependent offsets to the final timing measurements and present these computed offsets in Table [tab:offsets]. Only the *u*ʹ − *g*ʹ and *r*ʹ − *g*ʹ offsets indicate a relevant deviation from a zero-phase offset at the 1*σ* level, and so are corrected for all ULTRACAM data using the *u*ʹ and *r*ʹ filters. All other observations from data that we have obtained or from archival sources are considered to have a phase offset of zero.\nc c c c Filter & Δ*ϕ* (cycles) & Filter & Δ*ϕ* (cycles) \nu*s* − g*s* & 0.0015 ± 0.0040 & u’ − g’ & 0.0087 ± 0.0022 \nr*s* − g*s* & -0.0006 ± 0.0029 & r’ − g’ & -0.0023 ± 0.0016 \ni*s* − g*s* & 0.0019 ± 0.0047 & i’ − g’ & 0.0001 ± 0.0090 \nz*s* − g*s* & 0.0020 ± 0.0128 \n[tab:offsets]\nConstraining the ephemeris\n--------------------------\nWith the full set of filter-offset-corrected (and flickering-considered, see ) timing solutions, we proceeded by assigning each an integer cycle number, as inferred using the ephemerides of and while correcting for our own epoch, *T*0. We chose an epoch *T*0 = 55849, BJD TDB, as we found that this gave the smallest correlation between the fitting parameters and is located near the centre of the full observing span as expected. Cycle numbers from both of these ephemerides gave the same integer cycle number for all nights and inferred cycle numbers were manually verified for observations taken more recently than those in. In no case was a jump in cycle number observed, which would be easily noticeable since typical uncertainties are much less than a cycle in magnitude.\nIt is clear (see Fig. [fig:O-C]) that HM Cnc deviates by multiple cycles with respect to a linear fit that has a constant orbital frequency, so we added higher order coefficients to compute a quadratic and then a cubic fit. The set of cycle numbers were fit using\n \n$$\\phi\\left(t\\right) = \\phi\\_0 + f\\_0 \\left(t-T\\_0\\right) + \\frac{\\dot f\\_0}{2}\\left(t-T\\_0\\right)^2 + \\frac{\\ddot f\\_0}{6}\\left(t-T\\_0\\right)^3\n\\label{eqn:PhasingSol}$$ \nwith *ϕ*(*t*) the cycle number, *ϕ*0 a cycle offset at the epoch *T*0, *f*0 the frequency at the epoch *T*0, $\\dot f\\_0$ its first derivative and $\\ddot f\\_0$ its second. For a linear fit, the first two terms on the right-hand side of equation ([eqn:PhasingSol]) apply, the first three terms for a quadratic and all four for a cubic. The quadratic term corresponds to the parabola that spans 21 cycles from minimum to maximum (Fig. [fig:O-C]) which means that HM Cnc has advanced by over 80 orbital cycles relative to the period that it had upon discovery, the result of gravitational-wave driven inspiral.\nA fit to our timing measurements was performed using a least-squares analysis through the scipy lstsq solver in the scipy linear algebra package, which we also used to return 1*σ* statistical errors on all fitted parameters. In the process, the errors on observations were inflated in quadrature so that the reduced *χ*2 is equal to one, where flickering in the system largely encourages the inclusion of this procedure (as discussed in ). Our residuals for solutions with a linear and quadratic ephemeris are presented in Fig. [fig:O-C] with the ephemerides for a quadratic and cubic model given in Table [tab:HMCncsolution]. Corrections of the ephemerides due to the Shklovskii effect and galactic rotation are negligible (see Appendix [secappendix:ephemerisCorr]). The $\\dot f\\_0$ coefficient of our quadratic ephemeris is precise to a 0.03% level and is 40 times more precise than that determined by, also derived from optical data.\nTo check if the $\\ddot f\\_0$ term is statistically significant, we performed an *F*-test between the quadratic and cubic fits. Under the hypothesis that the cubic term is not a significant contribution to the ephemeris, the following *F*-ratio for *n* data points\n \n$$F\\_{\\textrm{ratio}} = \\left(\\chi^2\\_{\\textrm{quad}} - \\chi^2\\_{\\textrm{cubic}}\\right) \\left(\\frac{\\chi^2\\_{\\textrm{cubic}}}{n-4}\\right)^{-1}$$ \nis expected to have an *F*(1, *n* − 4) distribution, where the 1 in the parenthesis represents the difference in the number of degrees of freedom between the quadratic and cubic fits and *n* − 4 reflects the unique degrees of freedom in the cubic fit. We compute the *F*-ratio to be 6.08 for our *n* = 60 measurements. The cumulative distribution function of an *F*-distribution can be used to infer at what significance the null-hypothesis that a cubic term is not reflective of the data can be rejected. As such, the cubic term (and so $\\ddot f\\_0$) is significant at the 98.3% level (which is significant at 95%, or 2*σ*, but not at 99.7%, or 3*σ*). We also carried out an *F*-test on the digitised X-ray data between the quadratic and cubic solutions specified in to test the significance of $\\ddot f$ (see ). The reduced *χ*2 of a cubic fit to the X-ray data was 1.04, so we again inflated the error of all data points in quadrature to gain a reduced *χ*2 equal to one. We found $\\ddot f = -9.41\\pm1.42\\times 10^{-27}\\,$Hz s− 2 and an *F*-ratio of 44.1 for the X-ray data. From the cumulative distribution of *F*(1,172), this represents a near 100% significance. Though with sparser sampling, individual measurements for the X-ray data have higher precision since the peak flux per cycle is better defined by a sharp rise towards an intensity maximum, while the optical is more susceptible to variability. Taking the *F*-test results as evidence that $\\ddot f$ is a true characteristic of the system, all further mentions of *f*0, $\\dot f\\_0$ and $\\ddot f\\_0$ in this paper assume the cubic solution of Table [tab:ephemerides].\nOur optical measurements of $\\dot f\\_0$ and $\\ddot f\\_0$ allow us to estimate the maximum frequency of HM Cnc before a flip of sign of $\\dot f$ is observed. If $\\ddot f\\_0$ remains constant, the system frequency would be expected to evolve as\n \n$$f(t) = f\\_0 + \\dot f\\_0 (t-T\\_0) + \\frac{\\ddot f\\_0}{2} (t-T\\_0)^2\n \\label{eqn:FrequencyEvolution}$$ \nwhere a maximum frequency is reached when $t=-\\dot{f\\_0}/\\ddot{f\\_0}+T\\_0$. We would thus predict the frequency maximum to occur in approximately 2100 ± 800yrs from now. The timing of this event is consistent with the frequency maximum predicted in. As always with timing studies, a larger baseline of data would improve the precision of the ephemeris and, in particular, better constrain $\\ddot f\\_0$.\nInterestingly, our $\\ddot f\\_0$ is of opposite sign to that predicted in earlier literature, with expecting a *positive* $\\ddot f$ of magnitude 10− 28 Hz s− 2. It would be expected that $\\ddot f$ is positive for almost the entirety of inspiral until merger under purely general relativistic orbital decay, unless a re-stabilisation of the orbit due to mass transfer occurs. Consequently, it is thought that HM Cnc is in a mass transfer turn-on phase that will drive the system to smaller orbital frequencies (longer periods) following the frequency maximum.\nUnder solely gravitational-wave orbital angular momentum loss\n \n$$\\centering\n \\dot f\\_{GW} = \\frac{96}{5}~\\pi^{8/3}~\\left( \\frac{G\\mathcal{M}}{c^3}\\right)^{5/3}~f\\_{GW}^{11/3}\n \\label{eqn:fdotGR}$$ \nwith *f**G**W* the frequency of emitted gravitational waves (equal to twice the orbital frequency), *G* the gravitational constant, *c* the speed of light and M = (*m**a* *m**d*)3/5 / (*m**a* + *m**d*)1/5 the chirp mass. For our measured *f*0 and $\\dot f\\_0$, we obtain an observed chirp mass of 0.3203 ± 0.0001 $\\textup{M}\\_\\odot$. Given that the trajectory of HM Cnc is impacted by the mass accretion rate (which acts to oppose inspiral) and not from gravitational wave radiation alone, this observed chirp mass is a lower bound for the true chirp mass of the system.\nFlickering\n----------\nWe noticed variations in the timing solutions of HM Cnc between adjacent nights with ideal conditions. For data from the VLT with ULTRACAM, observations were performed for a similar duration at similar airmasses over the three nights observed. The observations had a seeing < 1.0and multiple consecutive hours spent on target, resulting in high signal-to-noise ratio (SNR) data with no clear outliers. Fig. [fig:FlickeringGraph] depicts the changes in the Fourier series solution for sets of observations that were cut to have a one hour duration, with phase-folded light curves included. There are noticeable differences between the solutions of different nights, particularly evident when comparing the phase of minimum flux. Analysis of the full set of VLT nights over an hourly timescale also indicated that there are small changes between solutions with the longest single observing period being three hours, though nightly differences were more apparent. A similar impact to the Fourier solutions and the returned timing solutions was apparent in the (simultaneously observed) u’- and r’-band VLT observations. When we compared the best-quality adjacent nights of NOT data (BJD=52645–52647) each with over 3 hours on target per night, we saw hourly light curve fluctuations at a similar level, though with larger error in individual measurements due to the smaller telescope aperture. We presume that a changing apparent light-curve morphology is a consequence of changes in the system mass transfer rate, creating a flickering effect typical of accreting binaries, while the long-term light-curve morphology is less impacted.\n [fig:FlickeringGraph]\nA changing optical periodic signal inherently limits the accuracy of all timing solutions, where, taking the mean of the VLT/ULTRACAM nights to be a reference point, the scatter in *t*0 between adjacent nights could be as much as ± 0.01–0.015 cycles for all bands. In this study, we thus combine timing solutions from all filters (phase offset corrected where applicable) to obtain a single timing measurement for the night of observations. Then, we add a fixed error in quadrature due to flickering to each night’s timing measurement, such that the reduced *χ*2 of our ephemerides is equal to one. Overall, this error adjustment was 0.014 cycles.\nAlso in support of system flickering, Fig. 7 of compares phase-folded X-ray flux profiles from the Chandra and NICER datasets. A clear difference in flux profile is noticeable after the X-ray peak flux as the flux is decreasing, and to a lesser extent at the peak X-ray flux. This shows deviation over a multiple year time-span, but it is possible that these changes occur on much shorter timescales, e.g. hours.\nWe then set out to determine whether the flickering was apparent in the X-ray timing solutions as well by comparing our data to the nearest X-ray observation. To do this, we utilised the data from Fig. 6 of by digitising and extracting the plotted data points, followed by a conversion of each to an observed time of peak X-ray flux. After, we compared our own timing solutions with the X-ray times for Chandra and NICER data separately. A phase difference between the time of X-ray peak flux and our derived solutions of *t* was compensated for by minimising the *χ*2 between our optical timing solutions and the X-ray for our cubic-fit ephemeris given in Table [tab:ephemerides], generating a relative phase difference *δ**ϕ* ≈ 0.14 cycles from the optical to X-ray. A comparison of the two sets of solutions is plotted in Fig. [fig:XrayVsOptical], where the 176 individual X-ray measurements have been combined if they coincide with half of a day of each other.\n [fig:XrayVsOptical]\nVery few optical/X-ray observations were taken within a day of each other, making a direct comparison difficult. Little scatter is seen in both datasets from the most recent observations, while the earliest show a similar scatter except for the first optical/X-ray measurements. All that we can assert from the data is that an impact on the accuracy of timing solutions at any wavelength cannot be ruled out.\nEvolutionary Modelling\n======================\nOur measured $\\ddot f\\_0$ combined with the $\\dot f\\_0$ and *f*0 is a very strong constraint upon HM Cnc’s evolution. We explored simulated binary evolution models with comparison to our observed ephemeris using the Modules for Experiments in Stellar Astrophysics (MESA) 1-D stellar evolution code, release version 15140, to investigate the AM CVn DWD configuration. It has been proposed by that the donor of HM Cnc may be an extremely low mass (ELM) WD that is not fully degenerate and has a thick hydrogen layer, which allows the period to shorten even though accretion occurs. The Lyman-*α* detection in our HST spectra supports this theory, as well as larger mass, fully-degenerate WD donors with hydrogen in their envelope. Both types of situations are included in our MESA models. Plotted in Fig. [fig:Allowedmasses] are the valid mass combinations for each star that our observed chirp mass permits, which we surveyed in the MESA models.\n [fig:Allowedmasses]\nBuilding a Donor\n----------------\nWe approximate the full binary evolution by constructing a He WD from an adaptation of the MESA test suite. The way this is performed is by evolving a pre-main sequence to a zero age main sequence star, and then by allowing fusion to take place until the core mass reaches the desired mass of a He WD minus the mass of the envelope. We then cut the evolution and gently remove the envelope of the star until only the He core resides. Next, we evolve the star to an effective temperature of 10kK and then gradually accrete pure hydrogen onto the surface, giving us full control of the total hydrogen envelope mass. What is left is a He core WD with a thin, pure hydrogen layer, which we evolve for 0.5 Gyrs while enabling diffusion, stratifying the layers of the WD to leave a realistic structure[4](#fn4).\nThe maximum hydrogen envelope mass in our models was inferred from the Z=0.0001 He WD models of, choosing models where no hydrogen flash has occurred such that the star maintains a thick hydrogen envelope[5](#fn5). For smaller or larger mass WDs than given in the range of models provided by, we linearly extrapolated their models, with the largest hydrogen envelope we used being MH = 6.5 × 10− 3 $\\textup{M}\\_\\odot$ for models of a 0.16 $\\textup{M}\\_\\odot$ donor.\nWhile the full set of models serves as a good approximation of the amount of hydrogen in the He WD’s envelope, the evolution of a compact DWD is very different to their modelling of a neutron star companion to a WD. This, as well as potential hydrogen flashes, suggests that the initial hydrogen envelope could be less than these models. With that in mind, we investigated the compatibility of different masses of hydrogen envelopes by considering our own set of He WD models, whereby a minimum donor hydrogen envelope consists of MH = 0.1 × 10− 3 $\\textup{M}\\_\\odot$. We sampled hydrogen envelope masses between these limits also, computing models for MH = 1.0 × 10− 3 $\\textup{M}\\_\\odot$ and MH = 3.0 × 10− 3 $\\textup{M}\\_\\odot$.\nBinary Modelling\n----------------\nWe modelled HM Cnc by starting the binary evolution with the fabricated donor He WD and a point mass accretor before any Roche lobe overflow has begun (typically with an initial period of 0.05 days). It is worth noting that have modelled AM CVn systems with MESA by assuming a pure He donor before any mass transfer and fully evolve two stars rather than one star and a point mass. This would be advantageous here also, however the fact that hydrogen is present in the spectrum and that the hydrogen fraction largely controls the timing of maximum orbital frequency encourages us to include the hydrogen envelope. Though evolving two stars would still be possible, computations including hydrogen to incorporate nuclear burning on the accretor’s surface are extremely computationally expensive. Our need to precisely control initial conditions, to simulate a wide grid of models and the fact that nuclear burning on the accretor’s surface is important at the onset of mass accretion but less so later in the evolution led us to proceed with the point mass accretor abstraction, as was similarly initiated by. The abstraction ignores the possible impact of thermonuclear runaways on the surface of the WD which could result in enhanced angular momentum and mass loss. Given that we know that HM Cnc has not merged and that most of the donor hydrogen envelope has been depleted, the binary has survived hydrogen-rich nova episode(s) and our models are only susceptible to complications from a helium nova episode, which may occur in the future. This makes the evolution of our models realistic leading up to *f*0, $\\dot f\\_0$ and $\\ddot f\\_0$, even though the events of the future are uncertain.\nMost importantly to the situation at hand, our dynamics revolve around the framework of the Roche lobe approximation of and mass transfer rates following the method outlined in. We evolve the orbital angular momentum, *J*orb, according to\n \n$$\\dot J\\_{\\textrm{orb}} = \\dot J\\_{\\textrm{gr}} + \\dot J\\_{\\textrm{ml}} + \\dot J\\_{\\textrm{mb}} + \\dot J\\_{\\textrm{ls}} \\label{eqn:Jorb}$$ \nwhere computed values for the rate of orbital momentum change due to gravitational wave radiation, $\\dot J\\_{\\textrm{gr}}$, stellar wind mass loss, $\\dot J\\_{\\textrm{ml}}$, magnetic breaking, $\\dot J\\_{\\textrm{mb}}$, and spin-orbit (LS) coupling, $\\dot J\\_{\\textrm{ls}}$, are handled by MESA according to the methods outlined in. Mass transfer between the two stars is considered to be perfectly efficient and we apply a tidal synchronisation following the prescription of using inbuilt MESA routines.\nWhen an accretion stream feeds into a disc around the accretor, all transferred orbital angular momentum is assumed to return to the orbit of the system; there is no contribution to equation ([eqn:Jorb]). When accretion occurs via direct impact, this condition no longer holds. To account for this, we added an extra angular momentum sink to the MESA calculation, following the prescription of \n \n$$\\dot J\\_{\\textrm{di}} = J\\_{\\textrm{orb}} \\sqrt{r\\_h~(1+q)} \\frac{\\dot M\\_d}{M\\_d}\n \\label{eq:Jdi}$$ \nwhere $\\dot J\\_{\\textrm{di}}$ is the extra loss of orbital angular momentum due to direct-impact mass transfer, *r**h* follows the prescription of equation (13) of as the equivalent radius of the orbiting accreted material and *q* = *M*d/*M*a is the mass ratio. We determined the radius of the star at which an accretion stream would pass by, resulting in disc-fed accretion, using equation (6) of. Given that we consider a point mass accretor, its radius was computed using a mass-temperature-radius relationship (MTRR) from the Montreal CO WD models, where we considered the temperature of the accretor to be 42500K from our spectrum fit (). Hence, we have direct impact accretion if the MTRR radius of the accretor is larger than the radius required for the accretion stream to pass by, or disc-fed accretion on the contrary. In our simulations, this accretor radius is a minimum since the star is likely inflated from mass transfer and partially from irradiation, such that direct impact could begin slightly earlier. For each timestep and model simulated in MESA, we computed whether matter is accreted via direct impact or not and accordingly turned on/off the inclusion of equation ([eq:Jdi]) to the computed $\\dot J\\_{\\textrm{orb}}$ in equation ([eqn:Jorb]).\nDiffusion processes were continuously modelled while evolving the binary, before and during mass transfer. The time-step integration itself was variable in the models; longest initially when there is no mass accretion and smallest when the two stars are at their closest separation. The mass transferred in each time-step was calculated implicitly. A test for a merger was analysed for every model with every time-step integration, however unstable mass transfer only occurred for the less-massive-accretor models in the range permitted in Fig. [fig:Allowedmasses].\n [fig:MESAresults]\n [fig:MESAresultsmdot]\nMESA Results\n------------\nIn the DWD configuration, the most significant variables influencing the time of maximum orbital frequency are the donor mass and the amount of hydrogen in the envelope. The mass of the donor significantly affects the duration of Roche lobe overflow, whereas a larger hydrogen fraction induces a larger donor radius and earlier mass accretion. Besides the distinction between disc/direct fed accretion, combinations of accretor and donor masses control the magnitude of $\\dot f$, where a larger accretor mass leads to a higher $\\dot f$ for a given donor mass. Furthermore, the temperature of the donor star is decisive in controlling the onset of mass transfer and the timing of a negative $\\ddot f$, since a higher temperature WD results in a larger radius.\nFor our chosen selection of hydrogen envelopes, we performed an in-depth search of the full parameter space of Fig. [fig:Allowedmasses] to determine which combinations of masses would lead to $\\dot f\\approx\\dot f\\_0$ when *f* = *f*0 at any stage of the system evolution before shortlisting to the best-agreeing candidate systems, those being where the sign of $\\ddot f$ becomes negative near *f* = *f*0. Our best-fitting models are shown in Fig. [fig:MESAresults], whereby in all cases $\\ddot f\\lessapprox0$ at *f* = *f*0.\nWe solely present the scenario where the donor star maintains an effective temperature of 5.5–6.0kK. This is at odds with the donor temperature derived in, albeit a loose constraint. We attempted to investigate the scenario where the donor effective temperature is 16kK and found that it becomes extremely difficult to maintain any hydrogen envelope for all combinations of permitted masses (Fig. [fig:Allowedmasses]) in MESA. Increasing the donor temperature requires a more massive (and with this a smaller radius) donor for the binary to reach *f*0 before $\\ddot f \\lesssim 0$ occurs. In the process, the binary reaches smaller orbital separations and we require a present-day mass transfer rate in excess of 10− 6$\\textup{M}\\_\\odot$yr− 1, with the order of magnitude of $\\ddot f$ to significantly increase over a few decades. Moreover, the total mass lost from a 16kK donor for tested models exceeded 0.015$\\textup{M}\\_\\odot$ leading up to present day, such that an extremely thick hydrogen envelope before any mass transfer begins would be required for hydrogen to be observed in the spectrum. Our models therefore indicate a preference for cooler donor stars and smaller mass ratios.\nWe conclude from Fig. [fig:MESAresults] that, if we consider our full selection of hydrogen envelope masses, the donor star has a mass in the range ≈ 0.16–0.17$\\textup{M}\\_\\odot$ to observe a negative $\\ddot f$ at *f"},"data_id":{"kind":"string","value":"arxiv_0003269"}}},{"rowIdx":794,"cells":{"content":{"kind":"string","value":"Photon bremsstrahlung from quark jet via transverse and longitudinal scatterings: single versus multiple scatterings\n====================================================================================================================\nWe study the production of jet-bremsstrahlung photons through the scattering with the constituents of a dense nuclear matter within the framework of deep-inelastic scattering off a large nucleus. Applying a gradient expansion up to the second order for the exchanged three-dimensional momentum between jet and medium, we derive the single photon bremsstrahlung spectrum with the inclusion of the contributions from the transverse broadening as well as the longitudinal drag and diffusion of the hard parton’s momentum. We also compare the medium-induced photon radiation spectra for single scattering and from the resummation of multiple scatterings. It is found that the coupling between different scatterings can give additional contribution to medium-induced photon radiation, while for small momentum exchange, the leading contribution from the drag and diffusions to the photon emission spectra remain the same for single and multiple scatterings.\nIntroduction\n============\nHard partonic jets produced from early hard collisions provide very useful tools to probe the properties of hot and dense nuclear matter produced in high-energy heavy-ion collisions, such as those at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC). During the propagation through the dense nuclear medium, the hard jet partons interact with the surrounding medium and tend to lose their energy and forward momentum before exiting the medium and fragmenting into hadrons. One of the important consequences of jet-medium interaction and parton energy loss is the suppression of high transverse momentum hadron production in relativistic heavy-ion collisions as compared to that in elementary proton-proton collisions scaled by the number of binary nucleon-nucleon collisions. This phenomenon is usually denoted as jet quenching.\nParton energy loss and jet modification originate from a combination of binary elastic collisions between the hard jet partons and the medium constituents, and medium-induced inelastic radiative/splitting processes. For light flavor jet quenching, especially the suppression of single inclusive hadron production at high transverse momentum, the medium-induced gluon radiation is usually regarded as the most important parton energy loss mechanism, while for the nuclear modification of heavy quarks and heavy flavor mesons with large finite masses, elastic processes may assume more important contribution, especially at low and intermediate transverse momentum regime. Various theoretical formalisms of jet quenching have been developed to study the collisional and radiative parts of the energy loss experienced by the hard jet partons when they traverse the dense nuclear medium (see Ref. and references therein for a more detailed comparison of the different jet quenching formalisms). Based on these theoretical formalisms, sophisticated phenomenological studies have also been performed to investigate the manifestation of jet-medium interaction in different final state observables, such as the suppression of single inclusive high hadron production at high transverse momentum regime, the nuclear modification of dihadron and photon-hadron pair correlations, etc. One of the main purposes of various jet quenching studies is to extract the values of jet transport coefficients, such as the transverse momentum broadening rate *q̂* = *d*⟨*p**T*2⟩/*d**t* and the longitudinal momentum loss rate *ê* = *d*⟨*E*⟩/*d**t*, in dense nuclear media via the systematic comparison with the available jet modification data (e.g., see Ref. ).\nCurrently, most calculations of radiative energy loss mainly focus on the gluon radiation induced by transverse momentum broadening experienced by the jet parton when propagating through the dense nuclear medium. When a jet parton scatters off the medium constituents, both transverse and longitudinal momenta are exchanged between the propagating jet and the traversed medium. The longitudinal momentum loss experienced by the propagating jet partons has been studied in many jet quenching calculations, but only in the context of evaluating purely collisional energy loss either by the leading hard parton, or by the radiated partons when studying the energy loss and the medium modification of full jet showers. Recently, the influence of longitudinal momentum exchange between the jet parton and the medium constituents on the stimulated radiation vertex has been investigate in Ref. by allowing the exchanged longitudinal momentum to be of the order of the transverse momentum.\nIn this work we study the real photon radiation from a jet parton which propagates through a dense nuclear medium and experiences both transverse and longitudinal scattering off the gluon field of the medium. This work serves as an intermediate step to evaluate medium-induced gluon emission from a hard parton interacting with the dense nuclear medium. It is also an extension of the previous work performed by one of the authors in Ref. in which the leading contribution to parton-photon double differential spectrum has been obtained in the presence of both transverse and longitudinal momentum exchanges. Here, we attempt to derive the single photon radiation spectrum induced by jet-medium interaction by integrating out the momentum of the final outgoing parton. As will be shown, in order to obtain the leading contribution from longitudinal and transverse momentum diffusion to the single photon radiation spectrum, one needs to proceed one step further in the momentum gradient expansion as compared to the previous work in Ref.. The single photon emission spectrum obtained in this work can be directly used as the input to evaluate the production of jet-medium photons in relativistic heavy-ion collisions, which are expected to provide significant contribution to the direct photon production at the intermediate transverse momentum regime. As an interesting check, we also compare the single photon bremsstrahlung spectrum from the resummation of multiple scattering scenario and that from single scattering calculation. We find that the coupling between different scatterings experienced by the hard propagating parton may give additional contribution to the medium-induced photon emission, while for small momentum exchange and only considering the leading contribution from the drag and the diffusions to the single photon emission spectrum, the results from single and multiple scattering scenarios remain the same.\nThe paper is organized as follows. In Sec. II, we provide a brief introduction to the photon bremsstrahlung process from a hard parton produced in deep-inelastic scattering (DIS) off a large nucleus at leading-twist level. In Sec. III, we investigate the induced photon radiation when a hard parton experiences a single scattering in the dense nuclear medium (at twist-four level). Applying a gradient expansion for the exchanged three-dimensional momentum between the jet parton and the nuclear medium, the contributions from the transverse momentum diffusion as well as the longitudinal momentum drag and diffusion to the single photon emission spectrum are computed. In Sec. IV, we present the computation of the single photon bremsstrahlung spectrum from a hard parton which experiences multiple scatterings in the dense nuclear medium (i.e., by resumming all-twist contributions). By resumming the number of multiple scatterings experienced by the propagating hard parton, we compute the single photon spectrum at all-twist level, and compare to the twist-four result obtained in Sec. III. Sec. V contains our summary.\nPhoton bremsstrahlung in DIS at leading twist\n=============================================\nFor completeness, in this section we present a short introduction to the photon bremsstrahlung process from a jet parton produced in semi-inclusive process of deep-inelastic scattering (DIS) off a large nucleus at leading twist. We consider the following process:\n \n$$\\begin{aligned}\ne(L\\_1) + A(P\\_A) \\to e(L\\_2) + q(l\\_q) + \\gamma(l) + X,\\end{aligned}$$ \nwhere *L*1 and *L*2 denote the momenta of the incoming and outgoing leptons, *l**q* and *l* are the momenta of the produced hard quark and the bremsstrahlung photon, and *P**A* = *A**p* is the momentum of the incoming nucleus with the atomic number *A* (i.e., each nucleon in the nucleus carries the same momentum *p*). In this work, we utilize the light-cone notation for four-momentum vectors, e.g., *p* = [*p*+, *p*−, **p**⊥], with\n \n$$\\begin{aligned}\np^+ = \\frac{1}{\\sqrt{2}}(E+p\\_z), p^- = \\frac{1}{\\sqrt{2}}(E-p\\_z).\\end{aligned}$$ \nWe note that in the Briet frame, the virtual photon *γ*\\* carries a four-momentum *q* = *L*2 − *L*1 = [ − *x**B**p*+, *q*−, **0**⊥] and the nucleus has a four-momentum *P**A* = *A**p* = *A*[*p*+, 0, **0**⊥], where *x**B* is the Bjorken variable defined as: *x**B* = *Q*2/(2*p*+*q*−), with *Q*2 = − *q*2 the invariant mass of the virtual photon.\nThe differential cross section for the lepton production from the above semi-inclusive DIS process can be written as follows:\n \n$$\\begin{aligned}\nE\\_{L\\_2} \\frac{d\\sigma}{d^3\\mathbf{L}\\_2} = \\frac{\\alpha\\_{e}}{2\\pi s} \\frac{1}{Q^4} L\\_{\\mu\\nu} {W^{\\mu\\nu}},\\end{aligned}$$ \nwhere *α**e* is the electromagnetic coupling and *s* = (*p* + *L*1)2 is the invariant mass of the lepton-nucleon collision system. The leptonic tensor *L**μ**ν* is given by:\n \n$$\\begin{aligned}\nL\\_{\\mu\\nu} = \\frac{1} {2}{\\rm Tr}[\\slashed{L}\\_1 \\gamma\\_\\mu \\slashed{L}\\_2 \\gamma\\_\\nu].\\end{aligned}$$ \nThe hadronic tensor *W**μ**ν* is defined as:\n \n$$\\begin{aligned}\nW^{\\mu\\nu} \\!\\!&=&\\!\\! \\sum\\_{X} (2\\pi)^4 \\delta^4 (q + P\\_A - P\\_X) \\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\langle A| J^\\mu(0)|X\\rangle \\langle X|J^\\nu(0)|A\\rangle.\\end{aligned}$$ \nIn the above expression, ∣*A*⟩ represents the initial state of an incoming nucleus *A*, ∣*X*⟩ denotes the final hadronic (or partonic) states, and the sum over *X* runs over all possible final states except the outgoing hard quark jet and the radiated real photon. The electromagnetic current *J**μ* = *Q**q**ψ**q**γ**μ**ψ* is for a quark of flavor *q* and the electric charge *Q**q* (in unit of the positron charge *e*). The hadronic tensor contains the detailed information about the final state interaction between the stuck quark and the traversed dense nuclear medium, and is the main focus of our current work.\n![Leading twist contribution to the hadronic tensor. ](ht_ph0 \"fig:\") [leadingtwist]\nIn Fig. [leadingtwist] we show the photon bremsstrahlung process in the semi-inclusive DIS at leading twist. In this work, we use the light-cone gauge (*A*− = 0), and other types of diagrams are power-suppressed. Fig. [leadingtwist] represents the process that a hard quark is first excited by the virtual photon from one nucleon of the nucleus, then radiates a real hard photon and exits without further interaction with the nuclear medium. One may first write down the expression of the hadronic tensor for Fig. [leadingtwist] as follows:\n \n$$\\begin{aligned}\nW\\_0^{A \\mu\\nu} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 \\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi) \\delta(l\\_q^2) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 e^{iq\\cdot y\\_0} \\int d^4z \\int d^4z' \\int \\frac{d^4 q\\_1}{(2\\pi)^4} \\int \\frac{d^4 q\\_1'}{(2\\pi)^4} \\nonumber\\\\\\!\\!&\\times&\\!\\!\ne^{-i q\\_1 \\cdot (y\\_0-z)} e^{-i q\\_1' \\cdot (z'-y\\_0')} e^{-il\\_q\\cdot (z-z')} e^{-i l\\cdot (z-z')} \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\langle A |\\bar{\\psi}(y\\_0) \\gamma^\\mu \\frac{\\slashed{q}\\_1}{q\\_1^2 - i\\epsilon} \\gamma^\\alpha \\slashed{l}\\_q \\gamma^\\beta \\frac{\\slashed{q}\\_1'}{q\\_1'^2 + i\\epsilon} \\gamma^\\nu \\psi(0)|A\\rangle \\nonumber\\\\\\!\\!&\\times&\\!\\!\nG\\_{\\alpha\\beta}(l).\\end{aligned}$$ \nHere *G**α**β*(*l*) is the sum of the photon’s polarizations:\n \n$$\\begin{aligned}\nG\\_{\\alpha\\beta}(l) = -g\\_{\\alpha \\beta} + \\frac{n\\_\\alpha l\\_\\beta + n\\_\\beta l\\_\\alpha}{n\\cdot l},\\end{aligned}$$ \nwhere *n* = [1, 0, **0**⊥] is the light-cone vector (*n* ⋅ *l* = *l*−). To simplify the above expression, we may first integrate over the photon radiation locations *z*, *z*ʹ and obtain two momentum conservation *δ* functions. Then we may perform the integration over the four-momenta *q*1, *q*1ʹ, and get the relation *q*1 = *q*1ʹ = *p*0 + *q*. By re-introducing the four-momentum *p*0 = *l**q* + *l* − *q*, the hadronic tensor can be written as follows:\n \n$$\\begin{aligned}\nW\\_0^{A \\mu\\nu} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 \\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi) \\delta(l\\_q^2)\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\int \\frac{d^4p\\_0}{(2\\pi)^4} (2\\pi)^4 \\delta^4(q + p\\_0 - l\\_q -l) \\int d^4y\\_0 e^{-ip\\_0\\cdot y\\_0} \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\langle A |\\bar{\\psi}(y\\_0) \\gamma^\\mu \\frac{\\slashed{q}\\_1}{q\\_1^2 - i\\epsilon} \\gamma^\\alpha \\slashed{l}\\_q \\gamma^\\beta \\frac{\\slashed{q}\\_1'}{q\\_1'^2 + i\\epsilon} \\gamma^\\nu \\psi(0)|A\\rangle \\hspace{-12pt}\\nonumber\\\\\\!\\!&\\times&\\!\\!\nG\\_{\\alpha\\beta}(l).\\end{aligned}$$ \nConsidering the limit of very high energy and collinear radiation, one can neglect the ⊥ -component of quark field operators and factor out the one-nucleon state from nucleus state as follows:\n \n$$\\begin{aligned}\n\\!\\!&&\\!\\! \\langle A| \\bar{\\psi}(y\\_0) \\hat{O} \\psi(0) |A\\rangle = A C\\_p^A \\langle p| \\bar{\\psi}(y\\_0) \\hat{O} \\psi(0) |p\\rangle\n\\nonumber\\\\\\!\\!&&\\!\\!\n\\approx A C\\_p^A \\langle p| \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle {\\rm Tr}[\\frac{\\gamma^-}{2} \\hat{O}],\\end{aligned}$$ \nwhere *C**p**A* denotes the probability of finding a nucleon state with a momentum *p* inside the nucleus *A*. *γ*+ and *γ*− are defined as:\n \n$$\\begin{aligned}\n\\gamma^+ = \\frac{1}{\\sqrt{2}}(\\gamma^0 + \\gamma^3), \\gamma^- = \\frac{1}{\\sqrt{2}}(\\gamma^0 - \\gamma^3).\\end{aligned}$$ \nNow we can first integrate out the four-momentum *l**q*, and then use the photon on-shell condition *δ*(*l*2) to integrate over *l*+, i.e., *l*+ = *l*⊥2/(2*l*−). The final quark’s on-shell condition gives,\n \n$$\\begin{aligned}\n(2\\pi) \\delta(l\\_q^2) = \\frac{2\\pi}{2 p^+q^-(1-y)} \\delta(x\\_0 - x\\_B - x\\_L),\\end{aligned}$$ \nwhere *y* = *l*−/*q*− is the fraction of the forward momentum carried by the radiated photon from the parent quark, and the momentum fraction *x**L* = *l*⊥2/[2*p*+*q*−*y*(1 − *y*)] is related to the photon formation time as: $\\tau\\_{\\rm form} = 1/(x\\_Lp^+)$. Using the final quark’s on-shell condition, one may now integrate out the momentum *p*0+ = *p*+*x*0. The integration over *y*0+ and **y**0 ⊥ gives a three-*δ*-function, which can be used to integrate over *p*0−, **p**0 ⊥ and sets *p*0− → 0 and **p**0 ⊥ → 0. With the above simplifications, the hadronic tensor now reads:\n \n$$\\begin{aligned}\nW\\_0^{A \\mu\\nu} \\!\\!&=&\\!\\! A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi} \\int dy \\int \\frac{d^2l\\_\\perp}{\\pi l\\_\\perp^2} \\frac{(2\\pi)f\\_q(x\\_B+x\\_L)}{8p^+(q^-)^2x\\_L} \\nonumber\\\\\\!\\!&\\times&\\!\\!\n {\\rm Tr} [ \\frac{\\gamma^-}{2} \\gamma^\\mu \\slashed{q}\\_1 \\gamma^\\alpha \\slashed{l}\\_q \\gamma^\\beta \\slashed{q}\\_1'\\gamma^\\nu ] G\\_{\\alpha\\beta}(l), \\ \\ \\ \\\\end{aligned}$$ \nwhere *f**q*(*x*) is the quark parton distribution function, with *x* the fraction of the forward momentum carried by the quark from the nucleon. It is given as:\n \n$$\\begin{aligned}\nf\\_q(x) = \\int \\frac{dy\\_0^-}{2\\pi} e^{-ixp^+y\\_0^-} \\langle p| \\bar{\\psi}(y\\_0^-) \\frac{\\gamma^+}{2} \\psi(0)|p \\rangle.\\end{aligned}$$ \nOne may carry out the trace using the commutation relations of *γ* matrices: {*γ*+, *γ*−} = 2, {*γ*±, *γ*±} = 0, and {*γ*±, *γ*⊥} = 0. The final expression for the hadronic tensor at leading twist takes the following form:\n \n$$\\begin{aligned}\n\\frac{dW\\_0^{A\\mu\\nu}}{dl\\_\\perp^2 dy} = A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2 \\pi} \\frac{P(y)}{l\\_\\perp^2} (-g\\_\\perp^{\\mu\\nu}) (2\\pi) f\\_q(x\\_B+x\\_L), \\ \\ \\ \\\\end{aligned}$$ \nwhere *P*(*y*) = [1 + (1 − *y*)2]/*y* is the quark-to-photon splitting function, and *g*⊥*μ**ν* = *g**μ**ν* − *g**μ* −*g**ν* + − *g**μ* +*g**ν* −. In the following, we investigate the effect of single and multiple scatterings experienced by the propagating hard quark on the photon bremsstrahlung process.\nPhoton bremsstrahlung from single scattering\n============================================\nIn this section, we present the computation of medium-induced single photon radiation spectrum from a hard quark undergoing a single scattering from the dense nuclear medium. This corresponds to the contribution involving twist-four parton distribution in the nucleus as compared to the leading twist contribution presented in the previous section. Fig. [11twist] shows the central-cut diagrams at the twist-four level, i.e., diagrams with a single scattering in both the amplitude and the complex conjugate. The non-central-cut diagrams at the twist-four level, i.e., two gluon insertions in the amplitude and zero insertion in the conjugate (or vice versa), are shown in Fig. [20twist]. One may write down the expressions for the hadronic tensors for all the diagrams listed in Fig. [11twist] and Fig. [20twist], and compute their contributions to the single photon radiation spectrum. Here we only provide the details for the calculation of Fig. [11twist](a). The calculations for other diagrams are analogous and their main results are provided in the Appendix.\nOur goal is to derive a closed expression for single photon radiation spectrum by a quark jet when traversing and interacting with a dense nuclear matter. For this purpose, we use the following power counting for the exchanged gluons between the hard quark and the medium constituents. We use *Q* for the hardest momentum scale and *λ* for a small dimensionless parameter, i.e., *λ**Q* denotes a softer momentum scale. If one considers a nearly on-shell projectile parton with a momentum *q* = (*q*+, *q*−, *q*⊥) ∼ (*λ*2*Q*, *Q*, 0) scattering off a nearly on-shell target parton traveling in the opposite direction with a momentum ∼ (*Q*, *λ*2*Q*, 0), then the exchanged gluons will be the standard Glauber gluons, i.e., the gluons carry momenta ∼ (*λ*2*Q*, *λ*2*Q*, *λ**Q*). If one allows the target parton not to be on shell, then the exchanged gluons can carry a longitudinal momentum component of the order of the transverse components, i.e., ∼ (*λ*2*Q*, *λ**Q*, *λ**Q*); one sometimes refers to such gluons as the longitudinal-Glauber gluons. We will investigate the effect of transverse and longitudinal momentum transfers (of the same order) on the photon radiation spectrum from a hard quark which interacts with the medium constituent via either single (this section) or multiple (next section) scatterings.\nFig. [11twist](a) shows the process that a hard virtual photon strikes a quark in the nucleus with momentum *p*0ʹ (*p*0 in the complex conjugate) at the location *y*0ʹ = 0 (*y*0 in the complex conjugate). The struck quark is then sent back to the nucleus, carrying momentum *q*1ʹ (*q*1 in the complex conjugate). During its propagating through the nucleus, the hard quark scatters off the gluon field at the location *y*1ʹ (*y*1 in the complex conjugate), and picks up a momentum *p*1ʹ (*p*1 in the complex conjugate). We denote the radiated photon’s momentum as *l* and the final outgoing quark’s momentum as *l**q*. The contribution to the hadronic tensor from Fig. [11twist](a) is given by:\n \n$$\\begin{aligned}\n\\label{bb}\nW^{A\\mu\\nu}\\_{(a)} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^2 \\frac{1}{N\\_c} {\\rm Tr}\\left[T^{a\\_1} T^{a\\_1'} \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)\\delta(l\\_q^2) \\int d^4y\\_0 e^{iq\\cdot y\\_0} \\int d^4y\\_1\\int d^4y\\_1'\\int d^4z \\int d^4z'\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n \\int \\frac{d^4q\\_1}{(2\\pi)^4} e^{-iq\\_1\\cdot (y\\_0-z)} \\int \\frac{d^4\\bar{q}\\_1}{(2\\pi)^4} e^{-i\\bar{q}\\_1\\cdot (z-y\\_1)}e^{-il\\cdot (z-z')} e^{-il\\_q\\cdot (y\\_1-y\\_1')}\\int \\frac{d^4\\bar{q}\\_1'}{(2\\pi)^4} e^{-i\\bar{q}\\_1'\\cdot (y\\_1'-z')}\\int \\frac{d^4q\\_1'}{(2\\pi)^4} e^{-iq\\_1'\\cdot (z'-y\\_0')}\n\\nonumber \\\\\\!\\!&\\times&\\!\\!\n\\langle A | \\bar{\\psi}(y\\_0) \\gamma^\\mu \\frac{\\slashed{q}\\_1}{q\\_1^2 - i\\epsilon}\\gamma^\\alpha \\frac{\\slashed{\\bar{q}}\\_1}{\\bar{q}\\_1^2 - i\\epsilon} \\slashed{A}^{a\\_1}(y\\_1)\\slashed{l}\\_q\n\\slashed{A}^{a'\\_1}(y'\\_1) \\frac{\\slashed{\\bar{q}}\\_1'}{\\bar{q}\\_1'^{2} + i \\epsilon}\n \\gamma^\\beta \\frac{ \\slashed{q}\\_1'}{{q}\\_{1}'^2 + i \\epsilon}\n \\gamma^\\nu \\psi(0) |A\\rangle \nG\\_{\\alpha\\beta}(l).\\end{aligned}$$ \n![Central-cut diagrams: a single scattering with the medium in both the amplitude and the complex conjugate. ](fig11 \"fig:\") [11twist]\n![Non-central-cut diagrams: two gluon insertions in the amplitude and zero in the complex conjugate (or vice versa). ](fig01 \"fig:\") [20twist]\nTo simplify the above expression, one may first isolate the phase factor associated with the photon insertion locations *z* and *z*ʹ: *e*− *i*(*q̄*1 + *l* − *q*1) ⋅ *z**e**i*(*q̄*1ʹ + *l* − *q*1ʹ) ⋅ *z*ʹ. After integrating out the locations *z* and *z*ʹ, one may obtain two *δ* functions to perform the integrations for the momenta *q̄*1 and *q̄*1ʹ, yielding *q̄*1 = *q*1 − *l* and *q̄*1ʹ = *q*1ʹ − *l*. Using the momentum conservation at each interaction vertex, one can write down the following relations for various momenta in Fig. [11twist](a):\n \n$$\\begin{aligned}\np\\_0=q\\_1-q, \\,\\, p\\_1=l\\_q-\\bar{q}\\_1; \\,\\,\\,\\,\np'\\_0=q'\\_1-q, \\,\\, p'\\_1=l\\_q-\\bar{q}'\\_1.\\end{aligned}$$ \nFrom the above relations, one may change the integration variables: *q*1 → *p*0 and *q*ʹ1 → *p*ʹ0. Re-introducing the four-momentum *p*1 = *q* + *p*0 − *l* − *l**q*, the phase factor for the hadronic tensor becomes: *e*− *i**p*0 ⋅ *y*0*e*− *i**p*1 ⋅ *y*1*e**i**p*ʹ1 ⋅ *y*ʹ1. In this work, we perform the calculation in the light-cone gauge (*A*− = 0) in the limit of very high energy and collinear radiation, therfore, the dominant component of the vector potential is the forward ( + )-component, which means that only ( − )-component of *γ* matrices contribute. In such limit, one may ignore the ( ⊥ )-component of the quark field operators and factor out one-nucleon state from the nucleus state as follows:\n \n$$\\begin{aligned}\n\\langle A | \\bar{\\psi}(y\\_0) \\gamma^\\mu \\hat{O} \\gamma^\\nu \\psi(0) | A \\rangle\n\\!\\!\n\\approx \\!\\! A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) | p \\rangle\n{\\rm Tr} [\\frac{\\gamma^-}{2} \\gamma^\\mu \\frac{\\gamma^+}{2} \\gamma^\\nu] {\\rm Tr} [\\frac{\\gamma^-}{2} \\langle A| \\hat{O} |A\\rangle].\\end{aligned}$$ \nWith the above simplifications, the hadronic tensor can be written as the following form:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{(a)} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^2 \\frac{1}{N\\_c} {\\rm Tr}\\left[T^{a\\_1} T^{a'\\_1}\\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)\\delta(l\\_q^2)\n\\int d^4y\\_0 \\int d^4y\\_1 \\int d^4y'\\_1e^{-ip\\_0\\cdot y\\_0} e^{-ip\\_1\\cdot y\\_1} e^{ip'\\_1\\cdot y'\\_1}\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\int \\frac{d^4 p\\_0}{(2\\pi)^4} \\int \\frac{d^4 p'\\_0}{(2\\pi)^4}\n\\int \\frac{d^4p\\_1}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - p\\_0 - p\\_1 - q)\n\\frac{1}{q\\_1^2-i\\epsilon} \\frac{1}{\\bar{q}\\_1^2-i\\epsilon}\n\\frac{1}{q\\_1'^2+i\\epsilon} \\frac{1}{\\bar{q}\\_1'^2+i\\epsilon}\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!(-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle \\langle A | A^{+a\\_1}(y\\_1) A^{+a'\\_1}(y'\\_1) |A\\rangle\n\\times {\\rm Tr}\\left[\\frac{\\gamma^-}{2} \\slashed{q}\\_1 \\gamma^\\alpha \\slashed{\\bar{q}}\\_1 \\gamma^- \\slashed{l}\\_q \\gamma^- {\\slashed{\\bar{q}}\\_1'}\n\\gamma^\\beta \\slashed{q}\\_1'\\right] \nG\\_{\\alpha\\beta}(l).\\end{aligned}$$ \nTo further simplify the above expression, we look at the quark lines before and after the photon emission point,\n \n$$\\begin{aligned}\n&& q\\_1^2 = (q+p\\_0)^2 = 2p^+q^-(1+x\\_0^-)[-x\\_B + x\\_0 - x\\_{D0}],\n\\nonumber\\\\\n&& \\bar{q}\\_1^2 = (q+p\\_0-l)^2 = 2p^+q^-(1+x\\_0^--y)[-x\\_B + x\\_0 - x\\_{C0}],\\end{aligned}$$ \nwhere we have defined the momentum fractions,\n \n$$\\begin{aligned}\n&& \\bar{x}\\_i = \\sum\\_{j=0}^i x\\_j = \\sum\\_{j=0}^i \\frac{p\\_j^+}{p^+} ; \\,\n\\bar{x}^-\\_i = \\sum\\_{j=0}^i x^-\\_j = \\sum\\_{j=0}^i \\frac{p\\_j^-}{q^-}; \\,\n\\nonumber\\\\\n&&\n\\bar{x}\\_{Di} = \\sum\\_{j=0}^i {x}\\_{Dj} = \\frac{(\\sum\\_{j=0}^i \\mathbf{p}\\_{j\\perp})^2}{2p^+q^-(1+ \\bar{x}^-\\_i)}; \\,\n\\bar{x}\\_{Ci} = \\sum\\_{j=0}^i {x}\\_{Cj} = x\\_L(1-y) + \\frac{( \\sum\\_{j=0}^i \\mathbf{p}\\_{j\\perp} - \\mathbf{l}\\_\\perp)^2}{2p^+q^-(1+\\bar{x}\\_i^--y)}.\\end{aligned}$$ \nThe momenta *q*1ʹ and *q̄*1ʹ may be treated in a similar way. In terms of the above momentum fractions, the on-shell condition for the final outgoing quark *l**q* gives,\n \n$$\\begin{aligned}\n(2\\pi)\\delta(l\\_q^2)=\\frac{1}{2p^+q^-(1+ \\bar{x}\\_1^- - y)}(2\\pi)\\delta(-x\\_B + \\bar{x}\\_1 - \\bar{x}\\_{C1}).\\end{aligned}$$ \nCombining the contributions from the denominators from all the internal quark lines and the on-shell condition for the outgoing quark, one obtains:\n \n$$\\begin{aligned}\nD\\_q \\!\\!&=&\\!\\! \\frac{C\\_q}{(2p^+q^-)^5} (2\\pi) \\delta(-x\\_B + \\bar{x}\\_1 - \\bar{x}\\_{C1})\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\frac{1}{-x\\_B+x\\_0-x\\_{D0}-i\\epsilon} \\frac{1}{-x\\_B+x\\_0-x\\_{C0}-i\\epsilon}\n\\frac{1}{-x\\_B+x\\_0'-x\\_{D0}' +i\\epsilon} \\frac{1}{-x\\_B+ x\\_0'- x\\_{C0}'+i\\epsilon}\\end{aligned}$$ \nwhere *C**q* stands for\n \n$$\\begin{aligned}\nC\\_q = \\frac{1}{1+x\\_0^-} \\frac{1}{1+x\\_0^--y} \\frac{1}{1+ x\\_0'^-}\\frac{1}{1+ x\\_0'^- -y} \\frac{1}{1+ \\bar{x}\\_1^- - y}\\end{aligned}$$ \nAs for the numerators of the quark lines (the trace part), we note the following decomposition of the photon polarization sum *G**α**β*:\n \n$$\\begin{aligned}\n G^{+-} = G^{-+} = 0; \\ G^{++} = \\frac{2l^+}{l^-}; \\ G^{\\perp+} = G^{+\\perp} = \\frac{l\\_\\perp}{l^-}; \\ G\\_{\\perp\\perp}^{\\alpha\\beta} = -g\\_\\perp^{\\alpha\\beta}.\\end{aligned}$$ \nOne may perform contraction of the trace with various components of *G**α**β*(*l*). The terms combining with the projection *G*⊥ ⊥ render:\n \n$$\\begin{aligned}\nN\\_{\\perp\\perp} \\!\\!&=&\\!\\! \\frac{2{(2q^-)}^3}{C\\_q} \\left[\\frac{\\mathbf{p}\\_{0 \\perp} \\cdot \\mathbf{p}\\_{0\\perp}'}{(1+x\\_0^-)(1+x\\_0'^-)} + \\frac{(\\mathbf{p}\\_{0 \\perp} -\\mathbf{l}\\_\\perp) \\cdot (\\mathbf{p}\\_{0\\perp}'-\\mathbf{l}\\_\\perp)}{(1+x\\_0^--y)(1+x\\_0'^--y)}\\right].\\end{aligned}$$ \nSimilarly, the terms related to the projection *G*+ + give,\n \n$$\\begin{aligned}\nN\\_{++} = \\frac{4(2q^-)^3}{C\\_q} \\frac{l\\_\\perp^2}{y^2}.\\end{aligned}$$ \nThe terms associated with the projections *G*+ ⊥ and *G*⊥ + read,\n \n$$\\begin{aligned}\nN\\_{\\perp+} + N\\_{+\\perp} = -\\frac{2(2q^-)^3}{C\\_q} \\frac{1}{y}\n\\left[\\frac{\\mathbf{p}\\_{0 \\perp} \\cdot \\mathbf{l}\\_{\\perp}}{(1+x\\_0^-)} + \\frac{\\mathbf{p}\\_{0 \\perp}' \\cdot \\mathbf{l}\\_{\\perp}}{(1+x\\_0'^-)} + \\frac{(\\mathbf{p}\\_{0 \\perp} -\\mathbf{l}\\_\\perp) \\cdot \\mathbf{l}\\_{\\perp}}{(1+x\\_0^--y)} + \\frac{(\\mathbf{p}\\_{0 \\perp}' -\\mathbf{l}\\_\\perp) \\cdot \\mathbf{l}\\_{\\perp}}{ (1+x\\_0'^--y)}\\right].\\end{aligned}$$ \nAfter the above simplifications, the hadronic tensor takes the following form,\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{(a)} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^2 \\frac{1}{N\\_c} {\\rm Tr}\\left[ T^{a\\_1} T^{a\\_1'} \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - p\\_0 - p\\_1 - q) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 \\int d^4y\\_1 \\int d^4y\\_1'\n\\int \\frac{d^3\\mathbf{p}\\_0 dx\\_0}{(2\\pi)^4}\\int \\frac{d^3\\mathbf{p}\\_1 dx\\_1}{(2\\pi)^4}\\int \\frac{d^3\\mathbf{p}\\_0' dx\\_0'}{(2\\pi)^4}\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\left(e^{-i\\mathbf{p}\\_0\\cdot \\mathbf{y}\\_0} e^{-i\\mathbf{p}\\_1\\cdot \\mathbf{y}\\_1} e^{i\\mathbf{p}\\_1'\\cdot \\mathbf{y}\\_1'}\\right)\n\\left(e^{-ix\\_0 p^+y\\_0^-} e^{-ix\\_1 p^+y\\_1^-} e^{ix\\_1' p^+ y\\_1'^-} \\right) (2\\pi)\\delta(-x\\_B + \\bar{x}\\_1 - \\bar{x}\\_{C1})\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\frac{1}{-x\\_B+x\\_0-x\\_{D0}-i\\epsilon} \\frac{1}{-x\\_B+x\\_0-x\\_{C0}-i\\epsilon} \\frac{1}{-x\\_B+x\\_0'-x\\_{D0}'+i\\epsilon} \\frac{1}{-x\\_B+x\\_0'-x\\_{C0}'+i\\epsilon}\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n(-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle\n \\langle A | A^{+a\\_1}(y\\_1) A^{+a'\\_1}(y'\\_1) |A\\rangle \\nonumber \\\\ &&\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\frac{2}{(2p^+q^-)^2}\n\\frac{1 + \\left(1 - \\frac{y}{1+{x^-\\_0}} \\right) \\left(1 - \\frac{y}{1+{x\\_0'^-}} \\right)}{y^2\\left(1 - \\frac{y}{1+{x^-\\_0}} \\right) \\left(1 - \\frac{y}{1+{x'^-\\_0}} \\right)}\n\\left(\\mathbf{l}\\_\\perp - \\frac{y}{1+{x^-\\_0}} \\mathbf{p}\\_{0\\perp} \\right)\\cdot \\left(\\mathbf{l}\\_\\perp -\\frac{y}{1+{x'^-\\_0}} \\mathbf{p}\\_{0\\perp}' \\right).\\end{aligned}$$ \nIn the above expression, we have changed the integration variables *p**i*+ → *x**i* and *p**j*ʹ+ → *x*ʹ*j*. For easy writing, the three-vector notations are utilized here for the momentum and coordinate space variables: **p** = (*p*−, **p**⊥) and **y** = (*y*+, **y**⊥); their dot product reads as: **p** ⋅ **y** = *p*−*y*+ − **p**⊥ ⋅ **y**⊥.\nNow we perform the integration over the momentum fractions *x*0, *x*1 and *x*0ʹ. The corresponding phase factors read as follows:\n \n$$\\begin{aligned}\n\\Gamma^+ =e^{-ix\\_0 p^+y\\_0^-} e^{-ix\\_1 p^+y\\_1^-} e^{i x'\\_1p^+ y\\_1'^-}=e^{-ix\\_0p^+(y\\_0^--{y'\\_1}^-)} e^{-ix\\_1p^+(y\\_1^--{y'\\_1}^-)} e^{ix\\_0' p^+(y\\_0'^-- y\\_1'^-)}\\end{aligned}$$ \nwhere we have used the overall momentum conservation *p*ʹ1 = *p*0 + *p*1 − *p*ʹ0. One may first integrate out the momentum fraction *x*1 using the on-shell condition 2*π**δ*( − *x**B* + *x̄*1 − *x̄**C*1) for the outgoing quark line, then the phase factor becomes:\n \n$$\\begin{aligned}\n\\Gamma^+ = \\left(e^{-i(x\\_B + \\bar{x}\\_{C1})p^+y\\_1^-}e^{-ix\\_0p^+(y\\_0^--y\\_1^-)}\\right) \\left(e^{i(x\\_B + \\bar{x}\\_{C1}')p^+y\\_1'^-}e^{ix'\\_0p^+(y\\_0'^--y\\_1'^-)} \\right) = \\Gamma\\_1^+ \\Gamma\\_1'^+, \\end{aligned}$$ \nwhere Γ1+ and Γ1ʹ+ denote the the parts associated with the amplitude and the complex conjugate. With the help of the contour integration technique, the integrations over the momentum fractions *x*0 and *x*ʹ0 may be performed. The *x*0 integral may be carried out by closing the contour with a counter-clockwise semi-circle in the upper half of the *x*0 complex plane:\n \n$$\\begin{aligned}\n&&\n\\int \\frac{dx\\_0}{2\\pi} \\frac{e^{-ix\\_0 p^+(y\\_0^--y\\_1^-)}}{(-x\\_B + x\\_0 - x\\_{C0} - i \\epsilon)(-x\\_B + x\\_0 - x\\_{D0} - i \\epsilon)}\n\\nonumber\\\\\n&&\n= {i\\theta(y\\_1^--y\\_0^-) e^{-ix\\_B p^+(y\\_0^--y\\_1^-)}}\n\\frac{ e^{-i x\\_{C0} p^+ (y\\_0^- - y\\_1^-)} - e^{-i x\\_{D0} p^+ (y\\_0^- - y\\_1^-)}} {x\\_{C0} - x\\_{D0}}.\\end{aligned}$$ \nTwo terms in the equation represent the fact that each one of the propagators is close to on-shell, and the *θ*-function means that the quark line propagates from *y*0− to *y*1−. We can now collect the phase factor as follows:\n \n$$\\begin{aligned}\n\\Gamma\\_1^+ \\to i\\theta(y\\_1^- - y\\_0^-) e^{-i x\\_B p^+ y\\_0^-}e^{-ix\\_{D0}p^+y\\_0^-}e^{-i x\\_{C1} p^+ y\\_1^-}\n\\frac{e^{-i\\delta x\\_{D0} p^+ y\\_0^-} - e^{-i\\delta x\\_{D0} p^+ y\\_1^-}} {\\delta x\\_{D0}}, \\ \\\\end{aligned}$$ \nwhere for convenience we have defined the momentum fraction variables *δ**x**D**i* = *x**C**i* − *x**D**i* (as well as *δ**x̄**D**i* = *x̄**C**i* − *x̄**D**i*, *δ**x**D**i*ʹ = *x**C**i*ʹ − *x**D**i*ʹ, and *δ**x̄**D**i*ʹ = *x̄**C**i*ʹ − *x̄**D**i*ʹ). The integration over *x*ʹ0 is completely analogous, except that we close the contour with a clockwise semi-circle in the upper half of the complex plane, and associated with the *θ* function is a factor of ( − *i*) instead of *i*. After the integrations over the momentum fractions *x*0, *x*1 and *x*0ʹ in the quark lines have been done, the hadronic tensor reads as follows:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{(a)} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^2 \\frac{1}{N\\_c} {\\rm Tr}\\left[ T^{a\\_1} T^{a'\\_1} \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - p\\_0 - p\\_1 - q)\n\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 \\int d^4y\\_1 \\int d^4y'\\_1\n\\int \\frac{d^3\\mathbf{p}\\_0}{(2\\pi)^3}\\int \\frac{d^3\\mathbf{p}\\_1 }{(2\\pi)^3}\\int \\frac{d^3\\mathbf{p}'\\_1}{(2\\pi)^3}\n\\left(e^{-i\\mathbf{p}\\_0\\cdot \\mathbf{y}\\_0} e^{-i\\mathbf{p}\\_1\\cdot \\mathbf{y}\\_1} e^{i\\mathbf{p}\\_1'\\cdot \\mathbf{y}\\_1'} \\right) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle \\langle A | A^{+a\\_1}(y\\_1)A^{+a'\\_1}(y'\\_1) |A\\rangle\n \\left( \\theta(y\\_1^- - y\\_0^-)\\theta(y\\_1'^- - y\\_0'^-)\\right) e^{-ix\\_Bp^+y\\_0^-}\n \\nonumber \\\\ \\!\\!&\\times&\\!\\!\n \\left(e^{-ix\\_{D0}p^+y\\_0^-} e^{-i x\\_{C1} p^+ y\\_1^-}e^{ix\\_{C1}' p^+ y\\_1'^-}\\right)\\left(e^{-i\\delta x\\_{D0} p^+ y\\_0^-} - e^{-i\\delta x\\_{D0} p^+ y\\_1^-}\\right)\n\\left( 1 - e^{i\\delta x\\_{D0}' p^+ y\\_1'^-}\\right)\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\frac{1}{\\delta x\\_{D0} \\delta x\\_{D0}'}\n\\frac{2}{(2p^+q^-)^2}\n\\frac{1 + \\left(1 - \\frac{y}{1+ x^-\\_0} \\right) \\left(1 - \\frac{y}{1+ x\\_0'^-} \\right)}{y^2\\left(1 - \\frac{y}{1+ x^-\\_0} \\right) \\left(1 - \\frac{y}{1+ x\\_0'^-} \\right)}\n\\left(\\mathbf{l}\\_\\perp - \\frac{y}{1+ x^-\\_0} \\mathbf{p}\\_{0\\perp} \\right)\\cdot \\left(\\mathbf{l}\\_\\perp -\\frac{y}{1+x\\_0'^-} \\mathbf{p}\\_{0\\perp}' \\right), \\nonumber\\end{aligned}$$ \nwhere we have changed the integral variable: *p*0ʹ → *p*1ʹ and set the location *y*0ʹ = 0. Assuming that in the limit of very high energy, the nucleus may be approximated by a weakly interacting homogenous gas of nucleons, we can simplify the matrix elements of the gluon vector potentials in the nucleus state (note that the quark operators has already been factorized out). Since a nucleon is a color singlet, one may express the expectation of field operators in the nucleus states in terms of the expectations in the nucleon states as follows:\n \n$$\\begin{aligned}\n \\langle A | A^{+a\\_1}(y\\_1) A^{+a\\_1'}(y\\_1') |A\\rangle\n= \\left(\\frac{\\rho}{2p^+}\\right) \\frac{\\delta\\_{a\\_1 a\\_1'}}{N\\_c^2 - 1} \\langle p| A^+(y\\_1) A^+(y\\_1') | p\\rangle,\\end{aligned}$$ \nwhere *ρ* is the nucleon density inside the nucleus. Also in the above expression, we have averaged the colors of gluon field operators. With the above simplification, the trace for the color matrices can be easily evaluated,\n \n$$\\begin{aligned}\n\\frac{1}{N\\_c} \\delta\\_{a\\_1 a'\\_1} {\\rm Tr}\\left[ T^{a\\_1} T^{a'\\_1} \\right] = C\\_F.\\end{aligned}$$ \nFor conveinence, we change the four-vector location variables (*y*1, *y*1ʹ) → (*Y*1, *δ**y*1),\n \n$$\\begin{aligned}\nY\\_1 = \\frac{1}{2}(y\\_1 + y\\_1'); && \\delta y\\_1 = y\\_1 - y\\_1'.\\end{aligned}$$ \nUsing the translational invariance of the correlation functions, ⟨*p*∣*A*+(*y*1)*A*+(*y*ʹ1)∣*p*⟩ ≈ ⟨*p*∣*A*+(*δ**y*1)*A*+(0)∣*p*⟩, the integration for the phase factor can be performed:\n \n$$\\begin{aligned}\n\\int d^3\\mathbf{y}\\_1 \\int d^3\\mathbf{y}\\_1' e^{-i\\mathbf{p}\\_1 \\cdot \\mathbf{y}\\_1} e^{i\\mathbf{p}\\_1'\\cdot \\mathbf{y}\\_1'}\n= (2\\pi)^3 \\delta^3(\\mathbf{p}\\_1 - \\mathbf{p}\\_1') \\int d^3\\delta \\mathbf{y}\\_1 e^{-i(\\mathbf{p}\\_1+\\mathbf{p}\\_1')\\cdot \\frac{\\delta\\mathbf{y}\\_1}{2}}.\\end{aligned}$$ \nIn the above expression, the *δ* function represents the fact that the two gluon field insertions in the amplitude and the complex cojugate carry the same momentum, **p**1ʹ = **p**1, which also implies **p**0ʹ = **p**0. The *δ* functions may be utilized to carry out the integration over the momentum **p**ʹ1. Recalling the expressions for the momentum fractions *x**D*0 and *x**C*0, one may obtain the momentum fractions *δ**x**D*0 as follows:\n \n$$\\begin{aligned}\n\\delta x\\_{D0} = x\\_{C0} - x\\_{D0}\n= \\frac{\\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+{x\\_0^-}} \\mathbf{p}\\_{0\\perp} \\right)^2}{2p^+q^-y(1-\\frac{y}{1+{x\\_0^-}})}= x\\_L.\\end{aligned}$$ \nThe expressions for *δ**x**D*0ʹ are completely analogous. We may further perform the integrations over the momentum **p**0 and the location **y**0, rendering **p**0 = 0. Using the above results, one may obtain the hard part of the matrix element (the last line of the hadronic tensor, denoted as *T*(*a*)) as:\n \n$$\\begin{aligned}\n\\label{linear\\_approximation}\nT\\_{(a)} \\!\\!&=&\\!\\! \\frac{2yP(y)}{l\\_\\perp^2}.\\end{aligned}$$ \nNow we simplify the phase factor (the second last line in the hadronic tensor). Keeping only the leading contribution, the phase factor (denoted as *S*(*a*)) can be obtained as:\n \n$$\\begin{aligned}\nS\\_{(a)} \\approx e^{-ix\\_Lp^+y\\_0^-}(1- e^{-i x\\_L p^+ (Y\\_1^-+\\frac{1}{2}\\delta y\\_1^--y\\_0^-)})( 1 - e^{ix\\_L p^+ (Y\\_1^--\\frac{1}{2}\\delta y\\_1^-)}).\\end{aligned}$$ \nWe further note that *Y*1− is the location of the photon insertion location which spans over the size of the nucleus, while *y*0− and *δ**y*1− are confined within the size of one nucleon. Thus, *y*0− and *δ**y*1− is much smaller than *Y*1− and the phase factor may be further approximated as:\n \n$$\\begin{aligned}\nS\\_{(a)} \\approx e^{-ix\\_Lp^+y\\_0^-} [2-2 \\cos(x\\_L p^+ Y\\_1^-)].\\end{aligned}$$ \nWith the above simplifications, the hadronic tensor reads as follows:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{(a)} \\!\\!&=&\\!\\! (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\n\\int dy P(y) \\int \\frac{d^2l\\_\\perp}{\\pi l\\_\\perp^2} \\int{d^3\\mathbf{l}\\_q}\n\\int dy\\_0^- e^{-i(x\\_B+x\\_L)p^+y\\_0^-} \\langle p | \\bar{\\psi}(y\\_0^-) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\int d Y\\_1^- \\int d \\delta y\\_1^- \\left(g^2 \\frac{C\\_F}{N\\_c^2-1}\\frac{\\rho}{2 p^+}\\right) \\int d^3\\delta \\mathbf{y}\\_1 \\int \\frac{d^3\\mathbf{p}\\_1}{(2\\pi)^3} e^{-i\\mathbf{p}\\_1\\cdot \\delta \\mathbf{y}\\_1} \\langle p| A^+(\\delta{y\\_1}) A^+(0) | p\\rangle\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n[2-2 \\cos(x\\_L p^+ Y\\_1^-)] \\delta^3(\\mathbf{l}+\\mathbf{l}\\_q - \\mathbf{p}\\_1 - \\mathbf{q}).\\end{aligned}$$ \nThe above expression is the contribution to the hadronic tensor from Fig.[11twist](a). The calculations for other diagrams in Fig. [11twist] and Fig. [20twist] are completely analogous; their main results are provided in the Appendix. Summing over the contributions from all central-cut diagrams in Fig. [11twist] and carrying out the integral ∫0*L*−*d**Y*1− (assuming a homogeneous medium and the transport coefficients *D**L*1, *D**L*2 and *D**T*2 are position independent), the hadronic tensor contributed from all central-cut diagrams reads as follows:\n \n$$\\begin{aligned}\n\\label{W11\\_munu}\nW^{A\\mu\\nu}\\_{(abcd)} &=& (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\n\\int dy P(y) \\int \\frac{d^2l\\_\\perp}{\\pi l\\_\\perp^2} \\int{d^3\\mathbf{l}\\_q} (2\\pi) f\\_q(x\\_B+x\\_L) \\nonumber\\\\\\!\\!&\\times&\\!\\!\nL^- \\int d\\delta y\\_1^- \\left(g^2\\frac{C\\_F}{N\\_c^2 - 1}\\frac{\\rho}{2p^+}\\right) \\int d^3\\delta \\mathbf{y}\\_1 \\int \\frac{d^3\\mathbf{p}\\_1}{(2\\pi)^3} e^{-i\\mathbf{p}\\_1\\cdot \\delta \\mathbf{y}\\_1} \\langle p | A^+(\\delta y\\_1^-, \\delta \\mathbf{y}\\_1) A^+(0) |p\\rangle\n\\nonumber \\\\ &\\times&\n\\left[1 + \\frac{y(1-y)}{1+(1-y)^2} F\\_L \\frac{{p\\_1}^-}{q^-} + y F\\_L \\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{1\\perp}}{l^2\\_\\perp} + \\frac{y}{1+(1-y)^2} \\left\\{y- (1-y)F\\_L\\right\\} \\left(\\frac{p\\_1^-}{q^-}\\right)^2 + y^2 (1 - F\\_L)\\frac{p\\_{1\\perp}^2}{l^2\\_\\perp}\n\\nonumber \\right. \\\\ &&\n\\left.\n+ 2y^2 F\\_L \\frac{(\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{1\\perp})^2}{l^4\\_\\perp}\n+ \\left\\{\\frac{y^2(1-y)}{1+(1-y)^2}(2+F\\_L) - y F\\_L\\right\\}\\frac{{p\\_1}^-}{q^-} \\cdot \\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{1\\perp}}{l^2\\_\\perp}\\right]\n\\delta^3(\\mathbf{l}+\\mathbf{l}\\_q - \\mathbf{p}\\_1 - \\mathbf{q}).\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\nF\\_L = \\frac{2\\sin(x\\_L p^+ L^-)}{x\\_L p^+ L^-}\\end{aligned}$$ \nIn the above expression, we have kept terms in the hard parts of matrix elements up to the second order in $\\frac{p\\_1^-}{q^-}$ and $\\frac{{p}\\_{1\\perp}}{{l}\\_\\perp}$ and their cross terms, to be consistent with the momentum gradient expression up to the second order as will be done in a short moment. Summing over the contributions from all non-central cut diagrams in Fig. [20twist] and carrying out the integral ∫0*L*−*d**Y*1−, the hadronic tensor contributed from all non-centra-cut diagrams read as follows:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{(efdghi)} \\!\\!&\\approx&\\!\\! (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A\\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi} \\int dy P(y) \\int \\frac{d^2l\\_{\\perp}}{\\pi l^2\\_{\\perp}}\n\\int d^3\\mathbf{l}\\_q (2\\pi) f\\_q(x\\_B+x\\_L)\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n(-L^-) \\int d \\delta y\\_1^- \\left(g^2 \\frac{C\\_F}{N^2\\_c-1} \\frac{\\rho}{2 p^+}\\right) \\langle p | A^+(\\delta y\\_1^-) A^+(0) |p\\rangle\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\delta^3(\\mathbf{l}+\\mathbf{l}\\_q - \\mathbf{q}).\\end{aligned}$$ \nTo do further simplification, we introduce the momentum gradient expansion for the hard part *H*(**p**1) (the last two lines of *W*(*a**b**c**d*)*A**μ**ν* and the last line of *W*(*e**f**d**g**h**i*)*A**μ**ν*). Assuming the momentum exchange between the hard parton and medium constituents is small, one may expand *H*(**p**1) as a series of the Taylor expansion in the three-dimensional momenta **p**1 = (*p*1−, **p**1 ⊥) around **p**1 → 0:\n \n$$\\begin{aligned}\n\\label{gradient\\_expansion\\_1}\nH = \\left[1 + p\\_1^\\alpha \\frac{\\partial}{\\partial p\\_1^\\alpha} + \\frac{1}{2} p\\_1^\\alpha p\\_1^\\beta \\frac{\\partial}{\\partial p\\_1^\\alpha} \\frac{\\partial}{\\partial p\\_1^\\beta} + \\cdots \\right] H|\\_{\\mathbf{p}\\_1 = 0}, \\ \\\\end{aligned}$$ \nwhere *α* and *β* take the values: “ − ” or `` ⊥ \". In the above expansion, we have only kept the terms up to the second order derivative in both longitudinal and transverse momenta. High-order terms are neglected and should be straightforward to include in the gradient expansion; they correspond to higher-order moments for the momentum distribution of the exchanged gluons. The zeroth order term (without momentum derivative) represents the case where the exchanged gluon carry zero momenta; it contributes to the gauge corrections to the leading twist results, and will not be considered further. One may notice that non-central-cut diagrams only contribute to the zeroth order term in the above gradient expansion, thus do not physically contribute to medium-induced single photon radiation. Now performing the integration over **p**1 by part, one may obtain:\n \n$$\\begin{aligned}\n&&\\langle p| A^+(\\delta \\mathbf{y}\\_1) A^+(0)|p \\rangle e^{-\\mathbf{p}\\_1 \\cdot \\delta\\mathbf{y}\\_1} p\\_1^\\alpha \\frac{\\partial}{\\partial p\\_1^\\alpha} = e^{-\\mathbf{p}\\_1 \\cdot \\delta\\mathbf{y}\\_1} (-i) \\langle p| \\partial^\\alpha A^+(\\delta \\mathbf{y}\\_1) A^+(0)|p \\rangle \\frac{\\partial}{\\partial p\\_1^\\alpha}, \\\\\n&&\\langle p| A^+(\\delta \\mathbf{y}\\_1) A^+(0)|p \\rangle e^{-\\mathbf{p}\\_1 \\cdot \\delta\\mathbf{y}\\_1} p\\_1^\\alpha p\\_1^\\beta \\frac{\\partial}{\\partial p\\_1^\\alpha} \\frac{\\partial}{\\partial p\\_1^\\beta} = e^{-\\mathbf{p}\\_1 \\cdot \\delta\\mathbf{y}\\_1} \\langle p| \\partial^\\alpha A^+(\\delta \\mathbf{y}\\_1) \\partial^\\beta A^+(0)|p \\rangle \\frac{\\partial}{\\partial p\\_1^\\alpha} \\frac{\\partial}{\\partial p\\_1^\\beta}.\\end{aligned}$$ \nWe note the condition **p**1 → 0 imposed by the Taylor expansion. This means that the hard part *H*(**p**1) no longer depends on **p**1, thus the integrations for **p**1 and *δ***y**1 may be directly carried out. Then the hadronic tensor may be written as follows:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu} \\!\\!&=&\\!\\! (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\n\\int dy P(y) \\int \\frac{d^2l\\_\\perp}{\\pi l\\_\\perp^2} \\int{d^3\\mathbf{l}\\_q} (2\\pi) f\\_q(x\\_B+x\\_L) \\phi(L^-, l\\_q^-, \\mathbf{l}\\_{q\\perp})\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n\\phi(L^-, l\\_q^-, \\mathbf{l}\\_{q\\perp}) \\!\\!&=&\\!\\!\nL^- \\left[ - D\\_{L1}\\frac{\\partial}{\\partial p\\_1^-} + \\frac{1}{2} D\\_{L2} \\frac{\\partial^2}{\\partial^2 p\\_1^-} + \\frac{1}{2} D\\_{T2} {\\nabla p\\_{1 \\perp}^2} \\right]\n H|\\_{\\mathbf{p}\\_1 = 0}.\\end{aligned}$$ \nHere we consider the unpolarized initial and final states and only keep the terms with non-vanishing coefficients by using the symmetry of the system. The parton transport coefficients *D**L*1, *D**L*2, *D**T*2 are defined as follows:\n \n$$\\begin{aligned}\nD\\_{L1} \\!\\!&=&\\!\\! g^2 \\frac{C\\_F}{N\\_c^2 - 1} \\int dy^- \\frac{\\rho}{2p^+} \\langle p| i\\partial^- A^+(y^-) A^+(0)|p \\rangle, \\\\\nD\\_{L2} \\!\\!&=&\\!\\! g^2 \\frac{C\\_F}{N\\_c^2 - 1} \\int dy^- \\frac{\\rho}{2p^+} \\langle p| \\partial^- A^+(y^-) \\partial^- A^+(0)|p \\rangle, \\\\\nD\\_{T2} \\!\\!&=&\\!\\! g^2 \\frac{C\\_F}{N\\_c^2 - 1} \\int dy^- \\frac{\\rho}{2p^+} \\langle p| \\partial\\_\\perp A^+(y^-) \\partial\\_\\perp A^+(0)|p \\rangle. \\ \\\\end{aligned}$$ \nUp to an overall factor, the transport coefficients *D**L*1, *D**L*2 and *D**T*2 are equivalent to *ê* (the longitudinal momentum loss rate), *ê*2 (longitudinal momentum diffusionrate), and *q̂* (the transverse momenta broadening rate), respectively. We should point out that the above definitions of parton transport coefficients are not gauge-invariant. To obtain the gauge-invariant definitions of these transport coefficients, one needs to resum all the diagrams with arbitrary numbers of soft gluon insertions, which gives the Wilson lines between the gluon field operators. One may find more discussions on gauge-invariant definition of *q̂* in Ref..\nWe now analyze the seven terms in the hard part *H* one by one:\n \n$$\\begin{aligned}\nH = H\\_0 + H\\_-^{(1)} + H\\_\\perp^{(1)} + H\\_-^{(2)} + H\\_{\\perp, 1}^{(2)} + H\\_{\\perp, 2}^{(2)} +H\\_{-\\perp}^{(2)}.\\end{aligned}$$ \nAccordingly, the distribution function *ϕ*(*L*−, *l**q*−, **l***q* ⊥) is also splitted into seven contributions:\n \n$$\\begin{aligned}\n\\phi = \\phi^{(0)} + \\phi\\_-^{(1)} + \\phi\\_\\perp^{(1)} + \\phi\\_-^{(2)} + \\phi\\_{\\perp, 1}^{(2)} + \\phi\\_{\\perp, 2}^{(2)} + \\phi\\_{-\\perp}^{(2)}.\\end{aligned}$$ \nOne can directly read off the contribution from the term *H*0:\n \n$$\\begin{aligned}\n\\phi^{(0)}\n\\!\\!&=&\\!\\!L^- \\left[ D\\_{L1}\\frac{\\partial}{\\partial l\\_q^-} + \\frac{1}{2} D\\_{L2} \\frac{\\partial^2}{\\partial^2 l\\_q^-} + \\frac{1}{2} D\\_{T2} {\\nabla\\_{l\\_{q\\perp}}^2} \\right] \\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp}),\\end{aligned}$$ \nwhere the derivative over *p*1 has been converted to the derivative over *l**q*. The distribution function *ϕ*−(1)(*L*−, *l**q*−, **l***q* ⊥) reads as:\n \n$$\\begin{aligned}\n\\phi\\_-^{(1)} \\!\\!&=&\\!\\! L^- \\left[ - D\\_{L1}\\frac{\\partial}{\\partial p\\_1^-} + \\frac{1}{2} D\\_{L2} \\frac{\\partial^2}{\\partial^2 p\\_1^-} + \\frac{1}{2} D\\_{T2} {\\nabla p\\_{1 \\perp}^2} \\right] \\left\\{\\frac{y(1-y)}{1+(1-y)^2} F\\_L \\frac{{p\\_1}^-}{q^-} \\delta^3(\\mathbf{l}+\\mathbf{l}\\_q - \\mathbf{p}\\_1 - \\mathbf{q})\\right\\}|\\_{\\mathbf{p}\\_1 = 0}\\end{aligned}$$ \nPerforming the derivative over *p*1 and then converting it to the derivative over *l**q*, one obtains:\n \n$$\\begin{aligned}\n\\phi\\_-^{(1)}\n\\!\\!&=&\\!\\!\\frac{y(1-y)}{1+(1-y)^2} F\\_L \\left(-\\frac{D\\_{L1}L^-}{q^-} - \\frac{D\\_{L2}L^-}{q^-} \\frac{\\partial}{\\partial l\\_q^-} \\right) \\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp}).\\end{aligned}$$ \nThe analysis for other five terms are completely analogous, and they read as:\n \n$$\\begin{aligned}\n\\phi\\_\\perp^{(1)} &=& y F\\_L \\left(-\\frac{ D\\_{T2}L^-}{l\\_\\perp^2} \\mathbf{l}\\_\\perp \\cdot \\nabla\\_{l\\_{q\\perp}}\\right) \\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp});\n\\nonumber\\\\\n\\phi\\_-^{(2)} &=& \\frac{y}{1+(1-y)^2} \\left\\{y - (1-y) F\\_L\\right\\} \\frac{D\\_{L2} L^-}{{{q^-}^2}}\\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp});\n\\nonumber\\\\\n\\phi\\_{\\perp,1}^{(2)} &=& 2y^2 (1 - F\\_L) \\frac{D\\_{T2}L^-}{{l\\_\\perp^2}} \\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp});\n\\nonumber\\\\\n\\phi\\_{\\perp,2}^{(2)} &=& 2y^2 F\\_L \\frac{D\\_{T2}L^-}{ l\\_\\perp^2}\\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp});\n\\nonumber\\\\\n\\phi\\_{-\\perp}^{(2)} &=& 0.\\end{aligned}$$ \nPutting seven contributions together, the hadronic tensor takes the following form:\n \n$$\\begin{aligned}\nW^{A \\mu\\nu} \\!\\!&=&\\!\\!\n(-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\n\\int dy P(y)\\int \\frac{d^2 l\\_\\perp}{\\pi l\\_\\perp^2}\\int d^3 \\mathbf{l}\\_q (2\\pi) f\\_q(x\\_B + x\\_L)\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n \\left\\{ \\left[ D\\_{L1} L^-\\frac{\\partial}{\\partial l\\_q^-} + \\frac{1}{2} D\\_{L2} L^- \\frac{\\partial^2}{\\partial^2 l\\_q^-} + \\frac{1}{2} D\\_{T2} L^- {\\nabla\\_{l\\_{q\\perp}}^2}\\right] + \\frac{y(1-y)}{1+(1-y)^2} F\\_L \\left(-\\frac{D\\_{L1} L^-}{q^-} - \\frac{D\\_{L2} L^-}{q^-} \\frac{\\partial}{\\partial l\\_q^-} \\right)\n\\nonumber\\right.\\\\\\!\\!&&\\!\\!\n\\left. + y F\\_L \\left(- D\\_{T2} L^- \\frac{\\mathbf{l}\\_\\perp \\cdot \\nabla\\_{l\\_{q\\perp}}}{l\\_\\perp^2} \\right) + \\frac{y}{1+(1-y)^2} \\left[y - (1-y) F\\_L \\right] \\frac{D\\_{L2} L^-}{{{q^-}^2}}\n+ 2y^2\\frac{D\\_{T2} L^-}{l\\_\\perp^2} \\right\\}\n\\nonumber\\\\\n\\!\\!&\\times&\\!\\!\n \\delta(l\\_q^- - q^-(1-y)) \\delta^2(\\mathbf{l}\\_{q\\perp} + \\mathbf{l}\\_{\\perp})\\end{aligned}$$ \nIn the extremely high energy and collinear limits where the exchanged momentum between the hard parton and medium is small, one can perform the integration over the three-dimensional momentum of the final outgoing quark **l***q* = (*l**q*−, **l***q* ⊥), and obtain the single-differential hadronic tensor as follows:\n \n$$\\begin{aligned}\n\\frac{dW^{A \\mu\\nu}}{dy d{l^2\\_{\\perp}}} \\!\\!&=&\\!\\! (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\\frac{P(y)}{l\\_\\perp^2} (2\\pi) f\\_q(x\\_B + x\\_L)\n\\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\left\\{ - \\frac{y(1-y)}{1+(1-y)^2} F\\_L \\frac{D\\_{L1}L^-}{q^-} + \\frac{y}{1+(1-y)^2} \\left[y - (1-y) F\\_L \\right] \\frac{D\\_{L2}L^-}{{{q^-}^2}} + 2y^2\\frac{D\\_{T2} L^-}{l\\_\\perp^2}\\right\\}\\end{aligned}$$ \nFrom the above expression, one may read off the medium-induced photon bremsstrahlung spectrum:\n \n$$\\begin{aligned}\n\\label{dNgamma\\_11}\n\\frac{dN^{\\rm med}\\_{\\gamma}}{dy d{l^2\\_{\\perp}}} \\!\\!&=&\\!\\! \\frac{\\alpha\\_e}{2\\pi} \\frac{P(y)}{l\\_\\perp^2}\n\\left\\{ - \\frac{y(1-y)}{1+(1-y)^2} F\\_L \\frac{D\\_{L1}L^-}{q^-} + \\frac{y}{1+(1-y)^2} \\left[y - (1-y) F\\_L \\right] \\frac{D\\_{L2}L^-}{{{q^-}^2}} + 2y^2\\frac{D\\_{T2} L^-}{l\\_\\perp^2}\\right\\},\\end{aligned}$$ \nwhere *y* is the fraction of the forward momentum carried by the radiated photon from the quark, and *l*⊥ is the transverse momentum of the radiated photon. One can clearly see the individual contributions from the drag and the diffusions of the longitudinal momentum and the transverse momentum diffusion to the single photon bremsstrahlung spectrum. A notable result is that unlike the transverse momentum broadening which induces additional radiation in medium, the longitudinal momentum loss (the drag) tends to suppress the medium-induced photon radiation. Another interesting observation is that the contributions from the drag and diffusions decouple for the single photon emission spectrum. This is due to the fact that we have only kept the terms up to the second order in the momentum gradient expansion. In principle, one would expect to receive the contribution from the coupling between the drag and diffusions of the hard parton’s momentum when higher order terms are included in the momentum gradient expansion.\nPhoton bremsstrahlung from multiple scatterings\n===============================================\nIn the previous section, we have computed medium-induced photon radiation spectrum from a hard quark undergoing a single scattering from the dense medium, which corresponds to the contribution involving twist-four parton distribution in the nucleus. In this section, we consider the photon bremsstrahlung process from a quark jet which experiences multiple scatterings with the medium constituents. This corresponds to higher twist contribution, i.e., containing more partonic operators in the medium. Higher twist contribution is usually suppressed by powers of the hard scale *Q*2, but a sub-class of these contributions may be length-enhanced for an extended medium. In this section, we we will isolate and resum the length-enhanced higher twist contribution to the photon bremsstrahlung process from a hard quark jet. We also compare the result from the resummation of multiple scatterings to that for single scattering.\nFig. [mntwist] shows the generic diagram for the photon bremsstrahlung process from a hard quark jet undergoing multiple scatterings in the dense nuclear medium (*m* scatterings in the amplitude and *n* scatterings in the complex conjugate). A hard virtual photon strikes a quark with momentum *p*0ʹ in the amplitude (*p*0 in the complex conjugate) in the nucleus at the location *y*0ʹ = 0 in the amplitude (*y*0 in the complex conjugate), and the struck quark is then sent back to the nucleus. During its propagation, the hard quark with momentum *q*1ʹ (*q*1 in the complex conjugate) scatters off the gluon fields in the nucleus at the locations *y**j*ʹ with 0 < *j* < *m* (*y**i* in the complex conjugate with 0 < *i* < *n*), and picks up momentum *p**j*ʹ (*p**i* in the complex conjugate) via each scattering with the medium constituents. The photon with momentum *l* is radiated from the hard quark at the location *z* between the locations *y**q* and *y**q* + 1 (at the location *z*ʹ between the locations *y**p* and *y**p* + 1 in the complex conjugate). Here the momenta of the quark lines after the scattering at the location *y*ʹ*j* are denoted as: *q*ʹ*j* + 1 with *j* ≤ *q* before the photon emission, and *q̄*ʹ*j* + 1 with *j* > *q* after the photon emission (at the location *y**i*: *q**i* + 1 with *i* ≤ *p*, and *q̄**i* + 1 with *i* > *p*, respectively, in the complex conjugate). The quark lines before and after the photon radiation vertex are denoted as *q*ʹ*q* + 1 and *q̄*ʹ*q* + 1 = *q*ʹ*q* + 1 − *l* (*q**p* + 1 and *q̄**p* + 1 = *q**p* + 1 − *l* in the complex conjugate). The final outgoing quark carries the momentum *l**q*. One may write down the hadronic tensor for Fig. [mntwist] as follows:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{nmpq} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^{n+m} \\frac{1}{N\\_c} {\\rm Tr}\\left[\\left(\\prod\\_{i=1}^{n} T^{a\\_i}\\right) \\left(\\prod\\_{j=m}^{1} T^{a'\\_j}\\right) \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)\\delta(l\\_q^2) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 e^{iq\\cdot y\\_0} \\left(\\prod\\_{i=1}^{n} \\int d^4y\\_i\\right) \\left(\\prod\\_{j=1}^{m} \\int d^4y'\\_j \\right) \\int d^4z \\int d^4z'\n\\nonumber \\\\\\!\\!&\\times&\\!\\!\n\\left(\\prod\\_{i=1}^{p} \\int \\frac{d^4q\\_i}{(2\\pi)^4} e^{-iq\\_i\\cdot (y\\_{i-1} - y\\_i)} \\right) \\left( \\int \\frac{d^4q\\_{p+1}}{(2\\pi)^4} e^{-i q\\_{p+1} \\cdot (y\\_p-z)} e^{-il\\cdot z} \\int \\frac{d^4\\bar{q}\\_{p+1}}{(2\\pi)^4} e^{-i\\bar{q}\\_{p+1}\\cdot (z-y\\_{p+1})} \\right) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\left(\\prod\\_{i=p+2}^{n} \\int \\frac{d^4\\bar{q}\\_i}{(2\\pi)^4} e^{-i\\bar{q}\\_i\\cdot (y\\_{i-1} - y\\_i)} \\right) e^{-il\\_q \\cdot (y\\_{n} - y\\_m')}\n\\left(\\prod\\_{j=q+2}^{m} \\int \\frac{d^4\\bar{q}'\\_j}{(2\\pi)^4} e^{-i\\bar{q}\\_j' \\cdot (y\\_j' - y\\_{j-1}')}\\right)\n\\nonumber \\\\ \\!\\!&\\times&\\!\\!\n\\left( \\int \\frac{d^4\\bar{q}\\_{q+1}'}{(2\\pi)^4} e^{-i \\bar{q}\\_{q+1}' \\cdot (y\\_{q+1}'-z')} e^{il\\cdot z'} \\int \\frac{d^4q\\_{q+1}'}{(2\\pi)^4} e^{-iq\\_{q+1}'\\cdot (z'-y\\_q')} \\right)\n \\left(\\prod\\_{j=1}^{q} \\int \\frac{d^4q\\_j'}{(2\\pi)^4} e^{-iq'\\_j\\cdot (y\\_j' - y\\_{j-1}')} \\right)\n\\nonumber \\\\ \\!\\!&\\times&\\!\\!\n\\langle A | \\bar{\\psi}(y\\_0) \\gamma^\\mu \\left(\\prod\\_{i=1}^{p} \\frac{\\slashed{q}\\_i}{q\\_i^2 - i\\epsilon} \\slashed{A}^{a\\_i}(y\\_i)\\right) \\frac{\\slashed{q}\\_{p+1}}{q\\_{p+1}^2 - i \\epsilon} \\gamma^\\alpha \\left(\\prod\\_{i=p+1}^{n} \\frac{\\slashed{\\bar{q}}\\_i}{\\bar{q}\\_i^2 - i\\epsilon} \\slashed{A}^{a\\_i}(y\\_i)\\right) \\slashed{l}\\_q \\nonumber\\\\&\\times&\n\\left(\\prod\\_{j=m}^{q+1} \\slashed{A}^{a\\_j'}(y\\_j') \\frac{\\slashed{\\bar{q}}\\_j'}{\\bar{q}\\_j'^2 + i \\epsilon}\\right)\n \\gamma^\\beta \\frac{\\slashed{q}\\_{q+1}'}{q\\_{q+1}'^2 + i \\epsilon}\n\\left(\\prod\\_{j=q}^{1} \\slashed{A}^{a\\_j'}(y'\\_j) \\frac{\\slashed{q}\\_j'}{q\\_j'^2 + i \\epsilon}\\right) \\gamma^\\nu \\psi(0) |A\\rangle G\\_{\\alpha\\beta}(l).\\end{aligned}$$ \n![The generic diagram for photon bremsstrahlung process from a quark undergoing multiple scatterings with the medium. ](ht_ph \"fig:\") [mntwist]\nThe simplification of the above hadronic tensor is analogous to the case of single scattering presented in the previous section. We first isolate the phase factor associated with the photon insertion locations *z* and *z*ʹ: *e*− *i*(*q̄**p* + 1 + *l* − *q**p* + 1) ⋅ *z**e**i*(*q̄*ʹ*q* + 1 + *l* − *q*ʹ*q* + 1) ⋅ *z*ʹ. The integrations over the locations *z* and *z*ʹ can be simply carried out and produce two *δ* functions, which we can be used to integrate over the momenta *q̄**p* + 1 and *q̄*ʹ*q* + 1. The remaining phase factor may be collected as: (∏*i* = 1*n**e*− *i**p**i* ⋅ *y**i*)(∏*j* = 1*m**e**i**p*ʹ*j* ⋅ *y*ʹ*j*). Using the momentum conservation at each interaction vertex, we may write down the following relations for various momenta shown in Fig. [mntwist]:\n \n$$\\begin{aligned}\n& q\\_{i+1} = q + K\\_i = q + \\sum\\_{j=0}^i p\\_j,\\,\\,\\,\\, (i \\le p+1) ;\n& \\bar{q}\\_{i+1} = q + K\\_i - l = q + \\sum\\_{j=0}^i p\\_j - l, \\,\\,\\,\\, (i \\ge p+1); \\nonumber\\\\\n& q'\\_{i+1} = q + K'\\_i = q + \\sum\\_{j=0}^i p'\\_j, \\,\\,\\,\\, (i \\le q+1);\n& \\bar{q}'\\_{i+1} = q + K'\\_i -l = q + \\sum\\_{j=0}^i p'\\_j - l, \\,\\,\\,\\, (i \\ge q+1). \\ \\ \\ \\ \\ \\\\end{aligned}$$ \nThe new variables *K**i* = ∑*j* = 0*i**p**i* and *K*ʹ*i* = ∑*j* = 0*i**p*ʹ*i* represent the total momentum exchanged between the hard quark and the dense nuclear medium. Using the above relations, one may change the integration variables *q**i* + 1 → *p**i* and *q̄*ʹ*j* + 1 → *p*ʹ*j*. We may re-introduce the *n*-th momentum *p**n* by inserting the identity:\n \n$$\\begin{aligned}\n\\int \\frac{d^4p\\_n}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - K\\_{n-1} - p\\_n - q) = 1.\\end{aligned}$$ \nNote that the *m*-th momentum *p*ʹ*m* may be obtained as: *p*ʹ*m* = *K**n* − *K*ʹ*m* − 1. As mentioned in the previous section, we will perform the calculation in the light-cone gauge (*A*− = 0) in the Breit frame in the limit of very high energy, the dominant component of the vector potential is the forward ( + )-component. Since the non-zero contribution involving *A*⊥ term appears as (*γ* ⋅ *q**i* ⊥)(*γ* ⋅ *A*⊥*a**i*), which is two-order smaller than that containing *A*+ term, we may neglect *A*⊥ component and use the approximation $\\slashed{A}= \\gamma^- A^+$, i.e, only ( − )-component survives for these *γ* matrices. Similar to the previous section, one may also factor out one-nucleon state from the nucleus state and ignore the ( ⊥ )-component of the quark field operators. With the above simplifications, the hadronic tensor now takes the following form:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{nmpq} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^{n+m} \\frac{1}{N\\_c} {\\rm Tr}\\left[\\left(\\prod\\_{i=1}^{n} T^{a\\_i}\\right) \\left(\\prod\\_{j=m}^{1} T^{a'\\_j}\\right) \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)\\delta(l\\_q^2)\n\\\\ \\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 \\left(\\prod\\_{i=1}^{n} \\int d^4y\\_i \\right) \\left(\\prod\\_{j=1}^{m} \\int d^4y'\\_j \\right)\n\\left(\\prod\\_{i=0}^{n-1} \\frac{d^4p\\_i}{(2\\pi)^4} \\right) \\left(\\prod\\_{j=0}^{m-1} \\int \\frac{d^4p'\\_j}{(2\\pi)^4} \\right)\n\\int \\frac{d^4p\\_n}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - K\\_n - q) \\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\left(\\prod\\_{i=0}^n e^{-ip\\_i\\cdot y\\_i} \\right) \\left(\\prod\\_{j=1}^m e^{ip'\\_j\\cdot y'\\_j} \\right)\n\\left(\\prod\\_{i=1}^{p+1} \\frac{1}{q\\_i^2-i\\epsilon} \\right) \\left(\\prod\\_{i=p+1}^{n} \\frac{1}{\\bar{q}\\_i^2-i\\epsilon} \\right)\n\\left(\\prod\\_{j=1}^{q+1} \\frac{1}{q\\_j'^2+i\\epsilon} \\right) \\left(\\prod\\_{j=q+1}^{m} \\frac{1}{\\bar{q}\\_j'^2+i\\epsilon} \\right) \\nonumber \\\\ \\!\\!&\\times&\\!\\!\n(-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle\n\\langle A | \\left(\\prod\\_{i=1}^{n} A^{+a\\_i}(y\\_i)\\right) \\left(\\prod\\_{j=m}^{1} A^{+a\\_j'}(y\\_j') \\right) |A\\rangle \\nonumber \\\\ \\!\\!&\\times&\\!\\!\n{\\rm Tr}\\left[\\frac{\\gamma^-}{2} \\left(\\prod\\_{i=1}^{p} {\\slashed{q}\\_i} \\gamma^- \\right) {\\slashed{q}\\_{p+1}} \\gamma^\\alpha \\left(\\prod\\_{i=p+1}^{n} {\\slashed{\\bar{q}}\\_i} \\gamma^-\n\\right) \\slashed{l}\\_q \\left(\\prod\\_{j=m}^{q+1} \\gamma^- {\\slashed{\\bar{q}}\\_j'}\\right)\n\\gamma^\\beta {\\slashed{q}\\_{q+1}'} \\left(\\prod\\_{j=q}^{1} \\gamma^- {\\slashed{q}\\_j'}\\right)\\right] G\\_{\\alpha\\beta}(l). \\nonumber \\ \\\\end{aligned}$$ \nWe now try to isolate the largest length-enhanced contribution which arises when the maximum number of quark lines are close to on-shell. For this purpose, we look at the quark lines before and after the photon emission,\n \n$$\\begin{aligned}\n&& q\\_{i+1}^2 = (q+K\\_i)^2 = 2p^+q^-(1+ \\bar{x}^-\\_i)[-x\\_B + \\bar{x}\\_i - \\bar{x}\\_{Di}],\n\\nonumber\\\\\n&& \\bar{q}\\_{i+1}^2 = (q+K\\_i-l)^2 = 2p^+q^-(1+ \\bar{x}^-\\_i-y)[-x\\_B + \\bar{x}\\_i - \\bar{x}\\_{Ci}],\\end{aligned}$$ \nwhere the following momentum fraction variables are used again for conveineince,\n \n$$\\begin{aligned}\n&& \\bar{x}\\_i = \\sum\\_{j=0}^i x\\_j \n= \\frac{K\\_i^+}{p^+}; \\,\n\\bar{x}^-\\_i = \\sum\\_{j=0}^i x^-\\_j \n= \\frac{K\\_i^-}{q^-}; \\,\n\\nonumber\\\\\n&&\n\\bar{x}\\_{Di} = \\sum\\_{j=0}^i {x}\\_{Dj} = \\frac{K\\_{i\\perp}^2}{2p^+q^-(1+ \\bar{x}^-\\_i)}; \\,\n\\bar{x}\\_{Ci} = \\sum\\_{j=0}^i {x}\\_{Cj} = x\\_L(1-y) + \\frac{( \\mathbf{K}\\_{i\\perp} - \\mathbf{l}\\_\\perp)^2}{2p^+q^-(1+\\bar{x}\\_i^--y)}.\\end{aligned}$$ \nCombining the contributions from the denominators of all the internal quark lines with the on-shell condition of the final outgoing quark line, one obtains:\n \n$$\\begin{aligned}\nD\\_q \\!\\!&=&\\!\\! \\frac{C\\_q }{(2p^+q^-)^{n+m+3}}\n\\left( \\prod\\_{i=0}^{p} \\frac{1}{-x\\_B+\\bar{x}\\_i-\\bar{x}\\_{Di}} \\right)\n\\left(\\prod\\_{i=p}^{n-1} \\frac{1}{-x\\_B+\\bar{x}\\_i-\\bar{x}\\_{Ci}} \\right)\n\\left( \\prod\\_{j=0}^{q} \\frac{1}{-x\\_B+{\\bar{x}'}\\_i-{\\bar{x}'}\\_{Di}} \\right)\n \\nonumber\\\\\n\\!\\!&\\times&\\!\\! \\left(\\prod\\_{j=q}^{m-1} \\frac{1}{-x\\_B+{\\bar{x}'}\\_i-{\\bar{x}'}\\_{Ci}} \\right)(2\\pi)\\delta(-x\\_B+\\bar{x}\\_n-\\bar{x}\\_{Cn}),\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\nC\\_q \\!\\!&=&\\!\\! \\left( \\prod\\_{i=0}^{p} \\frac{1}{1+\\bar{x}\\_i^-} \\right) \\left(\\prod\\_{i=p}^{n} \\frac{1}{1+\\bar{x}\\_i^--y} \\right)\n\\left( \\prod\\_{j=0}^{q} \\frac{1}{1+{\\bar{x}\\_j}'^-} \\right) \\left(\\prod\\_{j=q}^{m-1} \\frac{1}{1+{\\bar{x}\\_j}'^--y} \\right).\\ \\\\end{aligned}$$ \nRegarding the numerators of the quark lines, one may contract the trace terms with various components of photon polarization sum *G**α**β*(*l*). The terms contracted with the projection *G*⊥ ⊥ give,\n \n$$\\begin{aligned}\nN\\_{\\perp\\perp} = \\frac{2(2q^-)^{n+m+1}}{C\\_q}\n \\left[\\frac{\\mathbf{K}\\_{p \\perp} \\cdot \\mathbf{K}\\_{q\\perp}'}{(1+\\bar{x}\\_p^-)(1+{\\bar{x}\\_q}'^-)} + \\frac{(\\mathbf{K}\\_{p \\perp} -\\mathbf{l}\\_\\perp) \\cdot (\\mathbf{K}\\_{q\\perp}'-\\mathbf{l}\\_\\perp)}{(1+\\bar{x}\\_p^--y)(1+{\\bar{x}\\_q}'^--y)}\\right].\\end{aligned}$$ \nSimilarly, the terms associated with the projection *G*+ + produce,\n \n$$\\begin{aligned}\nN\\_{++} = \\frac{4(2q^-)^{n+m+1}}{C\\_q} \\frac{l\\_\\perp^2}{y^2}.\\end{aligned}$$ \nThe terms related to the projections *G*+ ⊥ and *G*⊥ + read,\n \n$$\\begin{aligned}\nN\\_{\\perp+} + N\\_{+\\perp} = -\\frac{2(2q^-)^{n+m+1}}{C\\_q} \\frac{1}{y}\n\\left[\\frac{\\mathbf{K}\\_{p \\perp} \\cdot \\mathbf{l}\\_{\\perp}}{1+\\bar{x}\\_p^-} + \\frac{\\mathbf{K}\\_{q \\perp}' \\cdot \\mathbf{l}\\_{\\perp}}{1+{\\bar{x}\\_q}'^-} + \\frac{(\\mathbf{K}\\_{p \\perp} -\\mathbf{l}\\_\\perp) \\cdot \\mathbf{l}\\_{\\perp}}{1+\\bar{x}\\_p^--y} + \\frac{(\\mathbf{K}\\_{q \\perp}' -\\mathbf{l}\\_\\perp) \\cdot \\mathbf{l}\\_{\\perp}}{1+{\\bar{x}\\_q}'^--y}\\right]. \\ \\\\end{aligned}$$ \nWith the above simplifications, the hadronic tensor now can be written as the following form:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{nmpq} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^{n+m} \\frac{1}{N\\_c} {\\rm Tr}\\left[\\left(\\prod\\_{i=1}^{n} T^{a\\_i}\\right) \\left(\\prod\\_{j=m}^{1} T^{a'\\_j}\\right) \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - K\\_n - q) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 \\left(\\prod\\_{i=1}^{n} \\int d^4y\\_i \\right) \\left(\\prod\\_{j=1}^{m} \\int d^4y'\\_j \\right)\n\\left(\\prod\\_{i=0}^n \\int\\frac{d^3\\mathbf{p}\\_i dx\\_i}{(2\\pi)^4} \\right) \\left(\\prod\\_{j=0}^{m-1} \\int \\frac{d^3\\mathbf{p}'\\_j dx'\\_j}{(2\\pi)^4} \\right)\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\left(\\prod\\_{i=0}^n e^{-i\\mathbf{p}\\_i\\cdot \\mathbf{y}\\_i} e^{-ix\\_ip^+y\\_i^-} \\right) \\left(\\prod\\_{j=0}^m e^{i \\mathbf{p}\\_j'\\cdot \\mathbf{y}\\_j' e^{i x\\_j' p^+ y\\_i'^-} } \\right)\n\\left( \\prod\\_{i=0}^{p} \\frac{1}{-x\\_B+\\bar{x}\\_i-\\bar{x}\\_{Di}-i\\epsilon} \\right) \\left(\\prod\\_{i=p}^{n-1} \\frac{1}{-x\\_B+\\bar{x}\\_i-\\bar{x}\\_{Ci}-i\\epsilon} \\right) \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\left( \\prod\\_{j=0}^{q} \\frac{1}{-x\\_B+{\\bar{x}'}\\_i-{\\bar{x}'}\\_{Di}+i\\epsilon} \\right) \\left(\\prod\\_{j=q}^{m-1} \\frac{1}{-x\\_B+{\\bar{x}'}\\_i-{\\bar{x}'}\\_{Ci}+i\\epsilon} \\right) (2\\pi)\\delta(-x\\_B+\\bar{x}\\_n-\\bar{x}\\_{Cn})\\nonumber\\\\\n\\!\\!&\\times&\\!\\! (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle\n \\langle A | \\left(\\prod\\_{i=1}^{n} A^{+a\\_i}(y\\_i)\\right) \\left(\\prod\\_{j=m}^{1} A^{+a'\\_j}(y'\\_j) \\right) |A\\rangle \\nonumber \\\\ &&\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\frac{2}{(2p^+q^-)^2}\n\\frac{1 + \\left(1 - \\frac{y}{1+{\\bar{x}\\_p^-}} \\right) \\left(1 - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\right)}{y^2\\left(1 - \\frac{y}{1+{\\bar{x}\\_p^-}} \\right) \\left(1 - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\right)}\n\\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+{\\bar{x}\\_p^-}} \\mathbf{K}\\_{p\\perp} \\right)\\cdot \\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\mathbf{K}\\_{q\\perp}' \\right). \\ \\\\end{aligned}$$ \nIn the above expression, we have changed the integration variables: *p**i*+ → *x**i* = *p**i*+/*p*+ and *p*ʹ*j*+ → *x*ʹ*j* = *p*ʹ*j*+/*p*+.\nWe now perform the integrations over the momentum fractions *x**i* and *x*ʹ*j*. Using the overall momentum conservation, *p**m* = ∑*i* = 0*n**p**i* − ∑*j* = 0*m* − 1*p*ʹ*j*, one may write down the phase factor as follows:\n \n$$\\begin{aligned}\n\\Gamma^+ = \\prod\\_{i=0}^n e^{-ix\\_ip^+(y\\_i^--y\\_m'^-)} \\prod\\_{j=0}^{m-1} e^{ix'\\_jp^+(y\\_j'^--y\\_m'^-)}\\!\\!,\\end{aligned}$$ \nNow we may use the on-shell condition for the outgoing quark (2*π*)*δ*( − *x**B* + *x̄**n* − *x̄**C**n*) to integrate out the momentum fraction *x**n*, and re-organize the phase factor as follows:\n \n$$\\begin{aligned}\n\\Gamma^+ = \\left(e^{-i(x\\_B+\\bar{x}\\_{Cn})p^+y\\_n^-} \\prod\\_{i=0}^{n-1} e^{-ix\\_ip^+(y\\_i^--y\\_n^-)} \\right)\n\\left(e^{i(x\\_B+\\bar{x}\\_{Cm}')p^+y\\_m'^-} \\prod\\_{j=0}^{m-1} e^{ix\\_j' p^+(y\\_j'^--y\\_m'^-)} \\right) = \\Gamma\\_n^+ \\Gamma\\_m'^+,\\end{aligned}$$ \nwhere Γ*n*+ and Γ*m*ʹ+ denote the parts associated with *x**i* (1 ≤ *i* ≤ *n*) and *x*ʹ*j* (1 ≤ *j* ≤ *m*) integrals. Using the contour integral technique, one may perform all the remaining integrations over *x**i* and *x*ʹ*j*: starting from the propagators adjacent to the cut and proceeding to those adjacent to the photon radiation point. We first look at the integration over the momentum fraction *x**n* − 1. Isolating the corresponding phase factor and closing the contour of *x**n* − 1 with a counter-clockwise semi-circle in the upper half of the complex plane, one may obtain:\n \n$$\\begin{aligned}\n\\int \\frac{dx\\_{n-1}}{2\\pi} \\frac{e^{-ix\\_{n-1}p^+(y\\_{n-1}^--y\\_n^-)}}{-x\\_B + \\bar{x}\\_{n-1} - \\bar{x}\\_{Cn-1} - i \\epsilon} = i \\theta(y\\_n^- - y\\_{n-1}^-) e^{-i(x\\_B + \\bar{x}\\_{Cn-1} - \\bar{x}\\_{n-2})p^+(y\\_{n-1}^- - y\\_n^-)},\\end{aligned}$$ \nwhere the *θ*-function means that the quark line propagates from the position *y**n* − 1− to the position *y**n*−. Combining the above result, the phase factor takes the following form:\n \n$$\\begin{aligned}\n\\Gamma\\_n^+ \\to i\\theta(y\\_n^- - y\\_{n-1}^-) e^{-i{x}\\_{Cn}p^+ y\\_n^-} e^{-i(x\\_B+\\bar{x}\\_{Cn-1})p^+y\\_{n-1}^-}\n\\left(\\prod\\_{i=0}^{n-2} e^{-ix\\_ip^+(y\\_i^--y\\_n^-)} \\right).\\end{aligned}$$ \nSimilarly, one may perform the remaining integrals for the momentum fraction *x**i* until the one for *x**p* + 1. Then the phase factor can be collected as:\n \n$$\\begin{aligned}\n\\Gamma\\_n^+ \\to \\left(\\prod\\_{i=p+2}^n i\\theta(y\\_i^- - y\\_{i-1}^-) e^{-i{x}\\_{Ci} p^+ y\\_i^-} \\right) e^{-i(x\\_B+\\bar{x}\\_{Cp+1})p^+y\\_{p+1}^-}\n \\left(\\prod\\_{i=0}^{p} e^{-ix\\_ip^+(y\\_i^--y\\_{p+1}^-)} \\right).\\end{aligned}$$ \nNow performing the contour integration for *x**p*, one may obtain:\n \n$$\\begin{aligned}\n&& \\int \\frac{dx\\_p}{2\\pi} \\frac{e^{-ix\\_p p^+(y\\_p^--y\\_{p+1}^-)}}{(-x\\_B + \\bar{x}\\_p - \\bar{x}\\_{Cp} - i \\epsilon)(-x\\_B + \\bar{x}\\_p - \\bar{x}\\_{Dp} - i \\epsilon)} \\nonumber\\\\\n&&= {i\\theta(y\\_{p+1}^--y\\_p^-) e^{-ix\\_B p^+(y\\_p^--y\\_{p+1}^-)}}\n\\frac{ e^{-i\\bar{x}\\_{Cp}p^+(y\\_p^- - y\\_{p+1}^-)} - e^{-i\\bar{x}\\_{Dp}p^+(y\\_p^- - y\\_{p+1}^-)}} {\\bar{x}\\_{Cp} - \\bar{x}\\_{Dp}} e^{i\\bar{x}\\_{p-1}p^+(y\\_p^--y\\_{p+1}^-)}. \\ \\\\end{aligned}$$ \nTwo terms in the above equation come from the fact that either of the two propagators around the photon insertion point can be used to obtain the on-shell conditions for the momentum fraction *x**p*. Collecting the above result, the phase factor now becomes:\n \n$$\\begin{aligned}\n\\Gamma\\_n^+ \\to \\left(\\prod\\_{i=p+1}^n i\\theta(y\\_i^- - y\\_{i-1}^-) e^{-i{x}\\_{Ci} p^+ y\\_i^-} \\right) e^{-i(x\\_B+\\bar{x}\\_{Dp})p^+y\\_p^-}\n\\left(\\prod\\_{i=0}^{p-1} e^{-ix\\_ip^+(y\\_i^--y\\_p^-)} \\right) \\frac{e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_p^-} - e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_{p+1}^-}} {\\delta\\bar{x}\\_{Dp}}.\\end{aligned}$$ \nThe remaining integrations over *x**i* after the photon insertion may be evaluated in a similar way. After carrying out the integration over all momentum fractions *x**i* in the complex conjugate, the phase factor may be cast into the following form:\n \n$$\\begin{aligned}\n\\Gamma\\_n^+ \\to \\left(\\prod\\_{i=p+1}^n i\\theta(y\\_i^- - y\\_{i-1}^-) e^{-i{x}\\_{Ci} p^+ y\\_i^-} \\right)\n\\left(\\prod\\_{i=1}^p i\\theta(y\\_i^- - y\\_{i-1}^-) e^{-i{x}\\_{Di} p^+ y\\_i^-} \\right)\ne^{-ix\\_Bp^+y\\_0^-} \\frac{e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_p^-} - e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_{p+1}^-}} {\\delta\\bar{x}\\_{Dp}}. \\ \\\\end{aligned}$$ \nFor the integrations over the momentum fractions *x*ʹ*j* in the amplitude, a close contour of *x**i* with a clockwise semi-circle in the upper half of the complex plane should be chosen, giving a factor of ( − *i*) instead of *i* associated with each *θ* function. After the integrations over all the quark lines have been done, the hadronic tensor may be written as:\n \n$$\\begin{aligned}\nW^{A\\mu\\nu}\\_{nmpq} \\!\\!&=&\\!\\! \\sum\\_q Q\\_q^4 e^2 g^{n+m} \\frac{1}{N\\_c} {\\rm Tr}\\left[\\left(\\prod\\_{i=1}^{n} T^{a\\_i}\\right) \\left(\\prod\\_{j=m}^{1} T^{a'\\_j}\\right) \\right]\n\\int \\frac{d^4l}{(2\\pi)^4} (2\\pi)\\delta(l^2) \\int \\frac{d^4l\\_q}{(2\\pi)^4} (2\\pi)^4 \\delta^4(l+l\\_q - K\\_n - q)\n\\\\\\!\\!&\\times&\\!\\!\n\\int d^4y\\_0 \\left(\\prod\\_{i=1}^{n} \\int d^4y\\_i \\right) \\left(\\prod\\_{j=1}^{m} \\int d^4y'\\_j \\right)\n\\left(\\prod\\_{i=0}^{n} \\frac{d^3\\mathbf{p}\\_i}{(2\\pi)^3} \\right) \\left(\\prod\\_{j=0}^{m-1} \\int \\frac{d^3\\mathbf{p}'\\_j}{(2\\pi)^3} \\right)\n \\left(\\prod\\_{i=0}^n e^{-i\\mathbf{p}\\_i\\cdot \\mathbf{y}\\_i} \\right) \\left(\\prod\\_{j=0}^m e^{i\\mathbf{p}\\_j' \\cdot \\mathbf{y}\\_j' } \\right) \\nonumber\\\\\\!\\!&\\times&\\!\\!\ne^{-ix\\_Bp^+y\\_0^-} \\left(\\prod\\_{i=1}^n i\\theta(y\\_i^- - y\\_{i-1}^-) e^{-i{x}\\_{Di} p^+ y\\_i^-} \\right)\n\\left(\\prod\\_{j=1}^m (-i)\\theta(y\\_j'^- - y\\_{j-1}'^-) e^{i{x\\_D}'\\_j p^+ y\\_j'^-}\\right) \\nonumber\\\\ \\!\\!&\\times&\\!\\!\n(-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\langle p | \\bar{\\psi}(y\\_0) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle\n \\langle A | \\left(\\prod\\_{i=1}^{n} A^{+a\\_i}(y\\_i)\\right) \\left(\\prod\\_{j=m}^{1} A^{+a'\\_j}(y'\\_j) \\right) |A\\rangle \\nonumber \\\\ \\!\\!&\\times&\\!\\!\n \\left(\\prod\\_{i=p+1}^n e^{-i\\delta{x}\\_{Di} p^+ y\\_i^-} \\right)\n \\left(\\prod\\_{j=q+1}^n e^{i\\delta{x\\_D}\\_j' p^+ y\\_j'^-} \\right)\n \\left(e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_p^-} - e^{-i\\delta\\bar{x}\\_{Dp} p^+ y\\_{p+1}^-}\\right)\n\\left(e^{i\\delta\\bar{x}\\_{Dq}' p^+ y\\_q'^-} - e^{i\\delta\\bar{x}\\_{Dq}' p^+ y\\_{q+1}'^-}\\right)\n\\nonumber\\\\ \\!\\!&\\times&\\!\\!\n\\frac{1}{\\delta\\bar{x}\\_{Dp} \\delta\\bar{x}'\\_{Dq}}\n\\frac{2}{(2p^+q^-)^2}\n\\frac{1 + \\left(1 - \\frac{y}{1+{\\bar{x}\\_p^-}} \\right) \\left(1 - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\right)}{y^2\\left(1 - \\frac{y}{1+{\\bar{x}\\_p^-}} \\right) \\left(1 - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\right)}\n\\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+{\\bar{x}\\_p^-}} \\mathbf{K}\\_{p\\perp} \\right)\\cdot \\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+{{\\bar{x}\\_q}'^-}} \\mathbf{K}\\_{q\\perp}' \\right). \\nonumber\\end{aligned}$$ \nWe now perform the resummation over different photon insertions locations (*y**p* in the complex conjugate and *y**q* in the amplitude) for given numbers of scatterings experienced by the propagating hard quark jet. In this work, we only consider the case with the same number of scatterings in both the amplitude and the complex conjugate (*n* = *m*). The diagrams with *n* ≠ *m* contribute to the gauge corrections to the terms with min(*n*, *m*) scatterings experienced by the hard quark jet. Similar to the previous section, the following assumption is invoked to simplify the matrix elements of 2*n*(*n* = *m*) gluon vector potentials in the nuclear state (note that the quark operators has already been factorized out). In the very high energy limit, nucleons are traveling in straight lines and are almost independent of each other over the time period of the interaction between the hard quark jet and the nuclear medium. Thus it is reasonable to approximate the nucleus by a weakly-interacting homogenous gas of nucleons, i.e., the expectation of gluon field operators in the nucleus states may be cast into a product of the expectations in the nucleon states. By noting that a nucleon is a color singlet, the combination of gluon (or quark) field insertions must be restricted to a color singlet, i.e., the first non-zero and the largest contribution are the terms with 2*n* gluon insertions divided into *n* singlet pairs in *n* separate nucleon states (note that the atom number *A* of the nucleus is assumed to be very large). The above statement means that the expectation of the 2*n* gluon operators in the nucleus state can be decomposed as follows:\n \n$$\\begin{aligned}\n \\langle A | \\left(\\prod\\_{i=1}^{n} A^{+a\\_i}(y\\_i)\\right) \\left(\\prod\\_{j=n}^{1} A^{+a'\\_j}(y'\\_j) \\right) |A\\rangle\n= \\left(\\frac{\\rho}{2p^+}\\right)^n \\left(\\prod\\_{i=1}^{n} \\frac{\\delta\\_{a\\_i a'\\_i}}{N\\_c^2 - 1} \\langle p| A^+(y\\_i) A^+(y'\\_i) | p\\rangle\\right),\\end{aligned}$$ \nwhere the colors of gluon field operators have been averaged. The trace for the color matrices can be easily evaluated and gives:\n \n$$\\begin{aligned}\n\\frac{1}{N\\_c(N\\_c^2 - 1)^n} {\\rm Tr}\\left[\\left(\\prod\\_{i=1}^{n} T^{a\\_i}\\!\\right) \\!\\! \\left(\\prod\\_{i=n}^1 T^{a'\\_i}\\!\\right) \\right] = \\left( \\frac{C\\_F}{N\\_c^2 - 1}\\right)^n.\\end{aligned}$$ \nNow for convenience, we change the four-vector location variables: (*y**i*, *y**i*ʹ) → (*Y**i*, *δ**y**i*),\n \n$$\\begin{aligned}\n{Y}\\_i = \\frac{1}{2}({y}\\_i + {y}'\\_i); && \\delta{y}\\_i = {y}\\_i - {y}'\\_i.\\end{aligned}$$ \nUsing the translational invariance of the correlation functions, ⟨*p*∣*A*+(*y**i*)*A*+(*y*ʹ*i*)∣*p*⟩ ≈ ⟨*p*∣*A*+(*δ**y**i*−, *δ***y***i*−)*A*+(0)∣*p*⟩, the integration for the phase factor may be performed:\n \n$$\\begin{aligned}\n\\int d^3\\mathbf{y}\\_i \\int d^3\\mathbf{y}'\\_i e^{-i\\mathbf{p}\\_i \\cdot \\mathbf{y}\\_i} e^{i\\mathbf{p}'\\_i\\cdot \\mathbf{y}'\\_i}\n= (2\\pi)^3 \\delta^3(\\mathbf{p}\\_i - \\mathbf{p}'\\_i) \\int d^3\\delta \\mathbf{y}\\_i e^{-i(\\mathbf{p}\\_i+\\mathbf{p}\\_i')\\cdot \\frac{\\delta\\mathbf{y}\\_i}{2}}.\\end{aligned}$$ \nIn the above expression, the *δ* function represents the fact that each pair of gluon field insertions in each nucleon states carry the same momentum, **p***i*ʹ = **p***i*. The *δ* functions may be utilized to carry out the integration over the momentum **p**ʹ*i*. Recalling the expressions of the momentum fraction variables *x̄**D**p* and *x̄**C**p*, one may obtain the momentum fraction *δ**x̄**D**p*:\n \n$$\\begin{aligned}\n\\delta\\bar{x}\\_{Dp} \\!\\!&=&\\!\\! \\bar{x}\\_{Cp} - \\bar{x}\\_{Dp}\n= \\frac{\\left(\\mathbf{l}\\_\\perp - \\frac{y }{1+\\bar{x}\\_p^-} \\mathbf{K}\\_{p\\perp} \\right)^2}{2p^+q^-y\\left(1-\\frac{y}{1+ \\bar{x}\\_p^-}\\right)}.\\end{aligned}$$ \nThe expression of *δ**x̄*ʹ*D**q* is completely analogous. Using the above expressions, one may obtain the hard part of the matrix element (the last line of the hadronic tensor, denoted as *T**p**q*) as follows:\n \n$$\\begin{aligned}\n\\label{linear\\_approximation}\nT\\_{pq} \\!\\!&=&\\!\\! \\frac{2yP(y)}{l\\_\\perp^2}\n\\left[1 + \\frac{y(1-y)}{1+(1-y)^2} \\frac{K\\_p^-+{K'\\_q}^-}{q^-}\n+ y \\frac{\\mathbf{l}\\_\\perp \\cdot (\\mathbf{K}\\_{p\\perp}+\\mathbf{K}'\\_{q\\perp})}{l\\_\\perp^2}\n-\\frac{y (1-y)}{1+(1-y)^2} \\frac{{K\\_p^-}^2+{K\\_q'^-}^2}{{q^-}^2}\n\\right. \\\\ \\!\\!&&\\!\\!\n+\\frac{y^2}{1+(1-y)^2}\\frac{K\\_p^-\\cdot{K'\\_q}^-}{{q^-}^2}\n-y^2\\frac{K\\_{p\\perp}^2+K\\_{q\\perp}'^2}{l\\_\\perp^2} + 2y^2\\frac{(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{p\\perp})^2+(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}'\\_{q\\perp})^2}{l\\_\\perp^4}\n+y^2\\frac{\\mathbf{K}\\_{p\\perp}\\cdot\\mathbf{K}'\\_{q\\perp}}{l\\_\\perp^2}\n\\nonumber \\\\ \\!\\!&&\\!\\!\n\\left.\n+\\left(\\frac{y^2(1-y)}{1+(1-y)^2}-y\\right)\\left(\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{p\\perp}}{l\\_\\perp^2}\\frac{K\\_p^-}{q^-}\n+\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}'\\_{q\\perp}}{l\\_\\perp^2} \\frac{{K'\\_q}^-}{q^-} \\right)\n+\\frac{y^2(1-y)}{1+(1-y)^2} \\left(\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{p\\perp}}{l\\_\\perp^2}\\frac{{K'\\_q}^-}{q^-}\n+\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}'\\_{q\\perp}}{l\\_\\perp^2} \\frac{K\\_p^-}{q^-} \\right)\\right]. \\nonumber\\end{aligned}$$ \nIn the above expression, we have kept the terms up to the second order in $\\frac{K\\_i^-}{q^-}$ and $\\frac{{K}\\_{i\\perp}}{{l}\\_\\perp}$ and their cross terms, to be consistent with the momentum gradient expansion up to the second order as will be done in a short moment. Now we simplify the phase factor (the second line of the hadronic tensor). Keeping the leading contribution, the phase factor (denoted as *S**p**q*) can be obtained as follows:\n \n$$\\begin{aligned}\nS\\_{pq} = \\left({e^{-ix\\_L p^+ y\\_p^-} - e^{-ix\\_L p^+ y\\_{p+1}^-}}\\right)\n\\left({e^{ix\\_L p^+ {y'\\_q}^-} - e^{ix\\_L p^+ {y'}\\_{q+1}^-}}\\right)\\end{aligned}$$ \nWith the above simplifications, the sum over the photon insertion locations *y**p* and *y**q*ʹ may be performed:\n \n$$\\begin{aligned}\n\\sum\\_{p=0}^{n} \\sum\\_{q=0}^n S\\_{pq} T\\_{pq} \\!\\!&=&\\!\\! \\frac{2yP(y)}{l\\_\\perp^2} e^{-ix\\_Lp^+y\\_0^-} \\left[1 + \\frac{y(1-y)}{1+(1-y)^2} \\sum\\_{k=1}^n \\frac{p\\_k^-}{q^-} F\\_k + y \\sum\\_{k=1}^n \\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{k\\perp}}{l\\_\\perp^2} F\\_k\n\\nonumber\\right. \\\\ \\!\\!&&\\!\\!\n- \\frac{y(1-y)}{1+(1-y)^2}\\sum\\_{k=1}^n\\frac{{K\\_k^-}^2-(K\\_{k-1}^-)^2}{{q^-}^2}F\\_k\n+\\frac{y^2}{1+(1-y)^2}\\sum\\_{k=1}^n\\sum\\_{p=1}^n \\frac{p\\_k^- p\\_p^-}{{q^-}^2} G\\_{kp}\n\\nonumber\\\\ \\!\\!&&\\!\\!\n-y^2\\sum\\_{k=1}^n\\frac{K\\_{k\\perp}^2-K\\_{k-1 \\perp}^2}{l\\_\\perp^2}F\\_k\n+2y^2 \\sum\\_{k=1}^n \\frac{(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{k\\perp})^2-(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{{k-1}\\perp})^2}{{l\\_{\\perp}^4}}F\\_k\n\\nonumber\\\\ \\!\\!&&\\!\\!\n+\\left(\\frac{y^2(1-y)}{1+(1-y)^2}-y\\right) \\sum\\_{k=1}^n \\left(\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{k\\perp}}{l\\_\\perp^2} \\frac{K\\_k^-}{q^-}-\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{{k-1}\\perp}}{l\\_\\perp^2}\\frac{K\\_{k-1}^-}{q^-}\\right)F\\_k\n\\nonumber\\\\ \\!\\!&&\\!\\!\n\\left.\n+y^2\\sum\\_{k=1}^n\\sum\\_{p=1}^n \\frac{\\mathbf{p}\\_{k\\perp}\\cdot \\mathbf{p}\\_{p\\perp}}{{q^-}^2} G\\_{kp}+\\frac{2y^2(1-y)}{1+(1-y)^2}\\sum\\_{k=1}^n\\sum\\_{p=1}^n\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{p\\perp}}{l\\_\\perp^2} \\frac{p\\_k^-}{q^-} G\\_{kp}\\right].\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n&& F\\_k = \n{e^{-ix\\_L p^+ (Y\\_k^-+\\frac{1}{2}\\delta y\\_k^- - y\\_0^-)} + e^{ix\\_L p^+ (Y\\_k^- - \\frac{1}{2}\\delta y\\_k^-)}};\n\\\\\n&& G\\_{kp} = \n{e^{-ix\\_L p^+ (Y\\_k^-+\\frac{1}{2}\\delta y\\_k^- - y\\_0^-)} e^{ix\\_L p^+ (Y\\_p^- - \\frac{1}{2}\\delta y\\_p^-)}}.\\end{aligned}$$ \nNote that *Y**k*−(*Y**p*−) is the location of the photon insertion point and can span over the nucleus size, while *y*0− and *δ**y**k*−(*δ**y**p*−) are confined within the nucleon size. Therefore, *y*0− and *δ**y**k*−(*δ**y**p*−) are small as compared to *Y**k*−(*Y**p*−), and can be safely dropped in the expressions of *F**k* and *G**k**p*. Then we obtain:\n \n$$\\begin{aligned}\nF\\_k \\approx 2 \\cos(x\\_L p^+ Y\\_k^-), \\, G\\_{kp} \\approx \\cos(x\\_L p^+ ({Y\\_k}^--{Y\\_p}^-)).\\end{aligned}$$ \nwhere we have exchanged the subscript *k* and *p* to symmetrize the expression of *G**k**p*. After carrying out the sum over the photon insertion locations, the hadronic tensor takes the following form:\n \n$$\\begin{aligned}\n\\label{W\\_Amunu}\nW^{A\\mu\\nu}\\_{nn} \\!\\!&=&\\!\\! \\sum\\_{p}\\sum\\_{q} W^{A\\mu\\nu}\\_{nnpq} = (-g\\_\\perp^{\\mu\\nu}) A C\\_p^A \\sum\\_q Q\\_q^4 \\frac{\\alpha\\_e}{2\\pi}\n\\int dy P(y) \\int \\frac{d^2l\\_\\perp}{\\pi l\\_\\perp^2} \\int{d^3\\mathbf{l}\\_q}\n\\int dy\\_0^- e^{-i(x\\_B+x\\_L)p^+y\\_0^-} \\langle p | \\bar{\\psi}(y\\_0^-) \\frac{\\gamma^+}{2} \\psi(0) |p\\rangle \\nonumber\\\\\\!\\!&\\times&\\!\\!\n\\frac{1}{n!} \\left(\\prod\\_{i=1}^{n} \\int dY\\_i^- \\int d\\delta y\\_i^- \\left(g^2\\frac{C\\_F}{N\\_c^2 - 1}\\frac{\\rho}{2p^+}\\right) \\int d^3\\delta \\mathbf{y}\\_i \\int \\frac{d^3\\mathbf{p}\\_i}{(2\\pi)^3} e^{-i\\mathbf{p}\\_i\\cdot \\delta \\mathbf{y}\\_i} \\langle p | A^+(\\delta y\\_i^-, \\delta \\mathbf{y}\\_i) A^+(0) |p\\rangle \\right)\n\\nonumber \\\\ \\!\\!&\\times&\\!\\!\n\\left[1 + \\frac{y(1-y)}{1+(1-y)^2} \\sum\\_{k=1}^n \\frac{p\\_k^-}{q^-} F\\_k + y \\sum\\_{k=1}^n \\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{p}\\_{k\\perp}}{l\\_\\perp^2} F\\_k - \\frac{y(1-y)}{1+(1-y)^2}\\sum\\_{k=1}^n\\frac{{K\\_k^-}^2-(K\\_{k-1}^-)^2}{{q^-}^2}F\\_k\n\\nonumber\\right. \\\\ \\!\\!&&\\!\\!\n+\\frac{y^2}{1+(1-y)^2}\\sum\\_{k=1}^n\\sum\\_{p=1}^n \\frac{p\\_k^- p\\_p^-}{{q^-}^2} G\\_{kp}-y^2\\sum\\_{k=1}^n\\frac{K\\_{k\\perp}^2-K\\_{{k-1}\\perp}^2}{l\\_\\perp^2}F\\_k+2y^2 \\sum\\_{k=1}^n \\frac{(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{k\\perp})^2-(\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{{k-1}\\perp})^2}{{l\\_{\\perp}^4}}F\\_k\n\\nonumber\\\\ \\!\\!&&\\!\\!\n+y^2\\sum\\_{k=1}^n\\sum\\_{p=1}^n \\frac{\\mathbf{p}\\_{k\\perp}\\cdot \\mathbf{p}\\_{p\\perp}}{{q^-}^2} G\\_{kp}+\\left(\\frac{y^2(1-y)}{1+(1-y)^2}-y\\right) \\sum\\_{k=1}^n \\left(\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{k\\perp}}{l\\_\\perp^2} \\frac{K\\_k^-}{q^-}-\\frac{\\mathbf{l}\\_\\perp \\cdot \\mathbf{K}\\_{{k-1}\\perp}}{l\\_\\perp^2}\\frac{K\\_{k-1}^-}{q^-}\\right)F\\_k\n\\nonumber\\\\ \\!\\"},"data_id":{"kind":"string","value":"arxiv_0003270"}}},{"rowIdx":795,"cells":{"content":{"kind":"string","value":" and assurance are likely to be affected by these weaknesses.\nExpressivity\n------------\nAn often quoted weakness of, particularly when applied to, is the “reality gap” that can exist between a naively constructed model and its corresponding real-world artefact. According to , over-reliance on simulation to test behaviour using naive and insufficiently validated models can lead to effort being applied to solving problems that do not exist in the real world. Worse, programs for robotic controllers developed in a model-based setting may fail when executed on real-world hardware, because “it is very hard to simulate the actual dynamics of the real-world” . This problem is not only true of simulation, but any form of model-based analysis, including reasoning in.\nThe fundamental problem here is that it is impossible to model the behaviour of any physical entity precisely , unless we replicate the original. Moreover, as models become more detailed, their utility decreases and they can become just as difficult to comprehend and analyse as their real-world counterparts, an observation highlighted by the famous paradox of. Nevertheless, as statistician George Box said “all models are wrong but some are useful” : we must evaluate a model not upon how “correct” it is, or how much detail it contains, but on how *informative* it is. According to , the antidote is not to abandon the use of models, but to recognise their inherent limitations and strengths, and apply them intelligently to reasoning about a specific problem. This means selecting appropriate modelling paradigms that enable specification of behaviour at a sufficiently detailed level of abstraction, and using the resulting models to guide the engineering process.\nallow and foster the use of specific abstractions to specifically inform engineers of properties critical for their assurance.\nThe effectiveness of formal models is fragile and can be significantly reduced because of uncontrollable gaps between models and their implementations.\nIntegration and Coordination\n----------------------------\nModelling notations usually employ a particular paradigm to abstract the behaviour of the real-world. For example, the state-based paradigm, employed by like Z , B , and refinement calculus , considers how the internal state of a system evolves, whilst the event-driven paradigm, employed in process calculi like CSP , CCS , and *π*-calculus , considers how behaviour may be influenced by external interactions. Consequently, individual formal methods are usually limited to considering only certain aspects or views of a system’s behaviour , which can limit their effectiveness when used in isolation. Many researchers have therefore sought to overcome this weakness by integration .\nThe 1990s saw a large number of works on semantic unification and method integration . Theoretical foundations were provided by, in his seminal work on semantic unification using the “programs-as-predicates” approach and comparative semantics . At the same time, refinement calculi were developed that would underlie the work on linking heterogeneous notations through abstraction. Also, explored the integration of state- and event-based modelling using weakest preconditions, and several other works on this topic followed . proposed a unified theory of programming that links together the three semantic styles: denotational, operational, and algebraic. These developments culminated in ’s , a general framework for integration of semantically heterogeneous notations by application of ’s approach to the formalisation of a catalogue of computational paradigms, with links between them formalised using Galois connections. This framework enables a definitive solution to the integration of states and events, along with other computational paradigms, in the Circus language family .\nAnother result of these developments was a number of seminal works on integration ., inspired by work on systematic method integration , defined a generic “meta-method” that aimed at integration of several formal and semi-formal methods using notational translations with a common predicative semantic foundation, which builds on ’s work . Meanwhile,, building on their previous work , likewise proposed the creation of hybrid with a multi-paradigm approach. Moreover, proposed that should be integrated into the V-Model of development with common semantic foundations to link the various artefacts across development steps.\nThese diverse efforts eventually led to the founding of the conference series in 1999 , with the aim of developing theoretical foundations for “combining behavioural and state-based formalisms”. For the second iteration of the conference , the scope broadened to consider all the different aspects of integration, including semantic integration, traceability, tool integration, and refinement. A few years later, a conference series was also established for , with the aim of continuing to develop unifying semantics for diverse notations within the framework.\nHowever, there is as yet no agreed and general methodology for integrating that could be applied to . Overall, integration is of particular pertinence to, since such systems are multi-layered and possess a high degree of semantic heterogeneity. As found, they “can be variously categorised as embedded, cyber-physical, real-time, hybrid, adaptive and even autonomous systems, with a typical robotic system being likely to contain all of these aspects”. When we consider, we must consider advanced computational paradigms like real-time, hybrid computation with differential equations, probability, and rigid body dynamics. This implies the use of several different modelling languages and paradigms to describe the different aspects, and therefore a variety of analysis techniques to assure properties of the overall system. Assurance of autonomous systems will certainly therefore require . on page summarises this relationship.\nraise the potential of integration and coordination of several to consistently reason about properties implemented by a combination of various technologies such as software, electronic hardware, and mechanical hardware.\nThere is currently no agreed framework for the integration of that would effectively address the needs in the domain or similar domains.\nOpportunities for Integrated Formal Methods\n===========================================\n[sec:opportunities]\nThis section continues with the *environmental part* of our analysis. Several key opportunities for the transfer of arise from ongoing assurance challenges, particularly in assurance and from looking at what other disciplines do to cope with similar challenges. In the following, we describe opportunities stemming from\n* the desire for early removal of severe errors (),\n* the desire to learn from accidents and their root causes (),\n* the desire of assurance to form a mature discipline (), and\n* the desire for adequate and dependable norms ().\nThe Desire for Early Removal of Severe Errors\n---------------------------------------------\nSummarising major challenges in automotive systems engineering in 2006, indicated that modelling languages used in practice were often not formalised and the *desired benefits could not be achieved from semi-formal languages*. Moreover, *software engineering was not well integrated* with core control and mechanical engineering processes. Domain engineers would produce software/hardware sub-systems and mechanical sub-assemblies in undesirable isolation. referred to a lack of for *overall architecture verification*.\nHas the situation changed since then? In model-centric development in embedded software practice (e.g.based on or ), drawbacks can be significant if methods and tools are not well integrated or trained personnel are missing . Likely, ’s criticism remains in contemporary automatic vehicle engineering and assurance practice. In fact, he has a recent, clearly negative, but not pessimistic answer to this question . Moreover, this view is shared by the Autonomy Assurance International Programme’s discussion of assurance barriers,[8](#fn8) that is,current challenges in assurance. These barriers (e.g.validation, verification, risk acceptance, simulation, human-robot interaction) could be addressed by formal engineering models and calculations based on such models to be used as evidence in assurance cases.\n[sec:desire-model-based] Model-based assurance uses system models to structure assurance cases and represents another opportunity for formal methods in (through-life) assurance. Assurance arguments that are purely informal can be difficult to evaluate, and may be subject to argumentation fallacies . Consequently, there have been a number of efforts to formalise and mechanise assurance cases, both at the argumentation level and the evidence level . More recently, in, the [9](#fn9) is a standardised meta-model that supports both structured argumentation and integration of evidence from diverse system models . unifies several existing argumentation notations, and also provides support for artefact traceability and terminology. It could, in the future, serve as a crucial component for in assurance.\nLeading voices from applied software engineering research periodically mention the role of as a key technology to master upcoming challenges in assuring critical software systems . A round table about the adoption of in IT security positively evaluated their overall suitability, the combination of with testing, and the achievements in automation. The panellists noticed limitations of in short-time-to-market projects and in detecting unknown vulnerabilities as well as shortcomings in training and adoption in practice.\nHowever, even for mission-critical systems, high costs from late defect removal and long defect repair cycles , as well as dangerous and fatal[10](#fn10) incidents indicate that assurance in some areas is still driven by practices failing to assist engineers in overcoming their challenges. Moreover,, an observer of a multitude of computing risks, stated that “the needs for better safety, reliability, security, privacy, and system integrity that I highlighted 24 years ago in my book, Computer-Related Risks, are still with us in one form or another today” .\nFor example,artificial intelligence software—particularly components—has been developed at a high pace and used in many non-critical applications. Recently, components are increasingly deployed in critical domains. For verification and error removal, such software has to be transparent and explainable. Preferring verifiable algorithms to heuristics, recalled the corresponding engineering principle: “We cannot trust a device unless we know how it works”. One way to follow this principle and establish transparency is to reverse engineer (i.e.,to decode) the functionality of an component even if this is not possible in general . can help extract knowledge and reverse engineer abstractions of systems to explain their behaviour. We might then ask to which extent the reverse engineered and verified functionality serves as a substitute for the original component.\nThese anecdotes make it reasonable to question current assurance practice. Seen through the eyes of assurance, they suggest that we might again be facing a dependable software engineering *crisis* similar to the one from the late 1960s .\nWe as method researchers could learn from this crisis and improve the way that can be effectively coordinated to support early error removal in practical engineering processes. We as practitioners could learn from this crisis and improve the way that we correctly engineer and certify highly automated systems such as.\nThe Desire to Learn From Accidents and Their Root Causes\n--------------------------------------------------------\nIn the title of, the word “severe” refers to the negative consequences *potentially* caused by errors we want to remove using. The more severe the potential consequences of an error, the more critical is its early removal. The usefulness of thus positively correlates with their support in the removal of critical errors. However, the estimation of severity often also requires the careful study of past field incidents .\nWe speak of *field incidents* to refer to significant operational events in the *field* (i.e.,the environment a technical system is operated) which are undesired because of their safety risks and their severe harmful consequences. Field incidents range from minor incidents to major accidents. It is important to separate the observed effect, the field incident, from its causes or, more precisely, from the *causal chains of events* leading to the observed effect. Hence, this analysis depends on the considered system perimeter . Depending on the possibilities of *observation* and the depth pursued in a, a conclusion on a possible cause can result in any combination of, for example,overall system failure, human error, adverse environmental condition, design fault, hardware fault, software fault, or specification error.[11](#fn11)\nThere are many databases about field incidents: some are comprehensive and include, others are less detailed, and some are confidential, depending on the regulations in the corresponding application domain or industry sector. Based on such databases, accident research, insurance, and consumer institutions occasionally provide brief root cause statistics along with accident statistics.[12](#fn12)\nAccident statistics allow certain predictions of the safety of systems and their operation, for examplewhether risk has been and will be acceptably low. Such statistics are also used in estimations of the amount of field testing necessary[13](#fn13) to sufficiently reduce risk .\nFor example,it is well-acknowledged that inadequately designed and implemented user interfaces are a significant contributory factor in computer-related accidents . But how and how fast have we arrived at this conclusion and how can we prevent future such incidents? Without proper empirical investigation of accident causes, such statistics are *of little use in decisions on measures for accident prevention* , particularly on improvements of engineering processes, methods (e.g.), and technologies (e.g. tools) used to build these systems. For this, we require more in-depth, possibly formal, and statistics that *relate error removal by and incident root causes*. To this extent, is a great opportunity for the investigation of effectiveness.\n reported about deficiencies of information gathering (e.g.) for databases on computer-related accidents back in the early 1990s. He noted that independent, well-known, but confidential databases might reduce under-reporting and were thus believed to improve safety culture. Reporting has not much improved as observed by. To understand the current situation, we studied a sample of 377 reports from open field incident databases (in aviation, automotive, rail, energy, and others) finding the following :\n1. in these reports were of poor quality, either because they were not going deep enough, economically or technically infeasible, or inaccessible to us.\n2. Particularly, root causes (e.g.software faults, specification errors) were rarely documented in a way that useful information about the technologies used (e.g.software) or consequences in the development process could be retrieved from the reports.\n3. Reports in some sectors (e.g.aerospace, rail, power plants, process industry) contain more in-depth than others (e.g.automotive) because of different regulations.\n4. Some sectors operate official databases (e.g.NHTSA[14](#fn14) and NTSB[15](#fn15) in the US transportation sector) and others do not (e.g.German road transportation sector).\n5. Our findings suggest that even in domains with regulated, reports in open databases tend to be less informative than reports in closed databases.\n6. The reports from the automotive industry exhibited a relatively small fraction of technology-related errors (e.g.software-related errors).\nTo validate our study and to better understand the context of our findings, we interviewed eight[16](#fn16) safety experts . One finding was that, because of an unclear separation of technologies and a lack of explicit architectural knowledge, a desirable classification of root causes is sometimes infeasible. Hence, accident analysts often close their reports with a level of detail too low to draw helpful conclusions. Additionally, one expert stated that *the hidden number of software-related or software-caused field incidents in dependable systems practice is likely much larger than the known number*. This matches our intuition but we are missing clear evidence.\n, a researcher involved in the further development of IEC 61508, demands regulations to mandate the use of systematic.[17](#fn17) In support of his view, we believe that rigorous in-depth based on can be helpful to gain clarity about actual root causes. Again, beyond this undesirable form of late error removal, RCA data is essential for the *measurement of the effectiveness* of error removal techniques, particularly.\nThe “Toyota unintended acceleration” incident exemplifies the difficulty of drawing conclusions without using powerful techniques: a first concluded that floor mats and sticky throttle pedals caused a fatal car mishap. A second carried out by NASA experts and based on *testing and automated static analysis* of the control system (i.e.,software and hardware) was not conclusive. A third [18](#fn18) based on *code reviews*—we could not find out which level of formal inspection was used—detected defects in the control software and safety architecture, demonstrated to be likely the causes of the accident .\nWe could invest in for RCA based on standardised data recording (e.g.aircraft black boxes), especially important for where human operators cannot practically perform incident response. Based on lessons from formal RCA, we could further invest in the employment of integrated formal methods in assurance and certification to prevent field incidents, major product recalls, and overly lengthy root cause investigations.\nThe Desire of Assurance to Form a Mature Discipline\n---------------------------------------------------\nIn his Turing Award acceptance speech in 1981, Tony reviewed type safety precautions in programming languages and concluded: “In any respectable branch of engineering, failure to observe such elementary precautions would have long been against the law” .\nInspired by this comparison, it can be helpful to look at other engineering disciplines such as civil, mechanical, or electrical engineering to identify transfer opportunities for. There, engineers use in many of their critical tasks. However, nowadays these methods are often hidden behind powerful software tools usable by qualified professional engineers. Although type systems, run-time bounds checking, and other variants of assertion checking have been frequently used in dependable systems practice, the overall level of adoption is still comparatively low.\nFor example,even in less critical mechanical engineering domains, vocationally trained engineers use computer-aided engineering, design, and manufacturing software. Whether for designing machine parts for serial production (i.e.,specification) or for calculations (e.g.dimensioning, force or material flow simulations) for these parts and their assembly (i.e.,for prototype verification), these engineers use tools based on canonical mathematical models.\nNowadays, drawings from computer-aided mechanical design carry at least two types of semantics: one declarative based on calculus for dimensioning (1), and one procedural for the synthesis of Computer-Numerical-Control programs for production machines processing materials to realise the drawings (2). Note that the unifying base of these two semantics is geometry, a well-studied mathematical discipline. Although higher levels of complexity demand more sophisticated analytical expertise, typically from engineers with several years of work experience, many tasks can be accomplished by less trained engineers using the corresponding tools.\nWhereas in computer-aided mechanical design both semantics seem to be used to a similar extent, in we observe that analogous semantics are rarely used even if tools are available, and less often we see (1) and (2) being consistently used. Low adoption might result from the semantics for dimensioning and production automation being usually less abstract than the semantics for verification (1) and synthesis (2) of computer programs. Accordingly, Parnas suggests a shift from correctness proof to property calculation to develop practical formal methods .\n*Patterns* have had a long history in many disciplines. In mechanical engineering, patterns are better known as *machine elements* and are particularly useful in high-reliability applications. Machine elements (and standardised forms thereof) have a stabilising impact on the outcome of an engineering project. The process of element selection and composition can take tremendous advantage not only from the *reuse* of proven design knowledge but also from the reuse of complex calculations (e.g.from gear transmissions, injection moulding tools, skeleton framings). Moreover, modern tools typically foster the use of *element libraries* and *parametric design*. Importantly, because the properties of such elements are in many cases well known, calculations for assemblies (i.e.,*compositional* verification) get relatively easy. However, the higher the required precision of these calculations, the more expensive is their computation.\nThese observations are in line with what we know from collaborations in robotics, like mechatronics, a discipline where many engineering domains have to play together well: are heavily used for the analysis of robot controllers and for various kinds of simulations and tests .\nDigital circuit engineering is a domain where such as model checking have been successfully applied decades ago. However, systematic hardware errors, such as Spectre and Meltdown, and the *unavailability of temporal specifications* of optimised operations (e.g.branch-prediction and speculative execution) discontinue the verifiability of recent computer architectures. This lack of verifiability of the assumptions (e.g.partitioning, information flow) about the execution platform complicates the verifiability of the software (e.g.an operating system) running on such a platform.[19](#fn19)\nDependability assurance has not yet successfully adopted as a vital part of their key methodologies. If seem relatively well established in other disciplines, we might also be able to successfully transfer to assurance and assurance in other domains. Beyond software design patterns, we could benefit from best practices in formal specification and development manifested in repositories of patterns. Moreover, we could aim at the further unification of established to provide common formal semantics for various domains.\nThe Desire for Adequate and Dependable Norms\n--------------------------------------------\nDependable systems practice usually includes the transition from what is called the *system specification* to an artefact called *system design*. Typically, *software and hardware specifications* are then derived from these two artefacts before delving into detailed technology choice and development. observe that safety culture in such a framework is more important than strict standards, but adequate standards and certification regimes can establish and strengthen the safety culture desirable in dependable systems practice.\nA striking finding in one of our recent discussions of dependable systems standards (e.g.IEC 61508, ISO 26262,[20](#fn20) DO-178C) is that *normative parts for specification* (i.e.,), for specification validation (i.e.,avoiding and handling requirements errors), *and for hazard and risk analysis* (particularly in early process stages) seem to be *below the state of the art* , despite several observations that significant portions of the *causes* of safety-critical software-related incidents fall into the category of *specification errors* .\nIf one identifies an error in the software specification, do causes of this error originate from an erroneous system specification? How are system, software, and hardware specifications (formally) related? Control software typically governs the behaviour of a whole machine. Hence, core components of a software specification might, modulo an input/output relation , very well govern the narrative of the overall system specification. Inspired by the issue of specification validity as highlighted in, do these standards provide guidance for checking the validity and consistency of core parts of system, software, and hardware specifications?\nThe literature provides plenty of evidence of undesired impacts of specification errors dating back as early as the investigations of and. As reported by, the 92% of computer-related field incidents caused by human-computer interaction also illustrate the gap between specifications and capabilities of humans to interact with automation. Despite these older figures, we are talking of one of the most critical parts of standards. Practitioners could expect to receive strong guidance from these parts. Moreover, the requirements to show conformance to these parts should not be vacuous.\nMany standards define *specific sets of requirements* (i.e.,for error removal and fault-tolerance) depending on the level of risk a system (or any part of it) might cause. The higher this level, the more demanding these requirements. Examples of demanding requirements include 3-4 (IEC 61508), C-D (ISO 26262), 3-4 (IEC 61508), A-B (DO-178C). Even for the highest such levels the mentioned standards only “highly recommend” but not mandate the use of.\n*Guidelines* for embedded software development such as MISRA:1994 *recommend* for 4, although MISRA:2004 no longer includes such information and instead refers back[21](#fn21) to MISRA:1994. As already mentioned, ISO 26262 as the overriding standard does not go beyond high recommendation of for D. reported in 2014 that, in the US, car manufacturers are not required to follow MISRA and that there are no other software certification requirements. This currently also applies to autonomous road vehicles.\nIn an interesting anecdote, reported on his lack of success in introducing systematic hazard (and risk) analysis methodology into normative parts of this standard . Moreover, he mentioned unsuccessful attempts to strengthen the role of in IEC 61508 and on the “broken standardisation” in assurance practice. In reaction to that, he proposed the use of evidently independent peer reviews to “dampen committee-capture by big-company bully players”.\nAdditionally, observed: “There is an expectation by the community that standards will embody the best available technology and that their presentation will allow determination of conformance to be fairly straightforward. A criticism that is seldom heard is that some standards are, in fact, technically flawed and poorly presented.” He exemplifies his critique by several issues with IEC 61508 and RTCA DO-178B and suggests to make the meaning of “conformance [or compliance] with a standard” more rigorous. Particularly, he encourages to replace *indirect* (i.e.,process-related) evidence (e.g.documentation of specification activities) in assurance cases by *direct* (i.e.,artefact-related) evidence (e.g.unsuccessful checks for presence of certain specification faults, successful checks for absence of implementation errors).[22](#fn22) With the observation in software quality control that “there is little evidence that conformance to process standards guarantees good products”, delivered a reasonable basis for ’s suggestions.\nRegarding the integration of dependability approaches and, had stated by 1993 that they “do not know how to combine formal methods assurance with metrics collected from other techniques such as fault-tolerance”. Is this still an issue? From a practical viewpoint, standards such as, for example,IEC 61508, ISO 26262, and DO-178C, provide recommendations about techniques for the reduction of both random hardware failures (e.g.by fault-tolerance techniques) and systematic hardware and software failures (e.g.by, static analysis, and testing). If can support the combined application of the recommended techniques and achieve an improvement in practice then we should really strive to demonstrate this.\nWe believe that critical fractions of strong direct evidence can be delivered through the use of. In support of ’s argument , we see a great opportunity for an assessment of how the corresponding guidelines in these standards can be extended and aligned with recent results in research.\nWe as researchers and practitioners could support the creation of adequate state-of-the-art regulations with improved guidance on specification construction and validation. That way, we could foster well-certified high-risk software in a time where dangerous autonomous machines are about to get widely deployed in our society.\nThreats to the Adoption of Integrated Formal Methods\n====================================================\n[sec:obstacles]\nThis section closes the *environmental part* of our analysis by identifying threats to transfer as well as challenges that arise from alternative or competing approaches taking the opportunities mentioned in. We also outline remedies to these threats.\nThe development of effective and their successful transfer into practice can be impeded by\n* a lack of agreement on a sound semantic base for domain-specific and cross-domain integration (),\n* missing support for widely used and established tools (),\n* a lack of interest in practical problems on the side of researchers (),\n* a lack of incentives for researchers to engage with current practice and for software practitioners to engage with recent theoretical results (),\n* a bad reputation among practitioners and applied researchers (),\n* proofs that are faulty or do not scale (),\n* the quest for soundness overriding the quest for usefulness ().\nWe discuss these threats and barriers in more detail in the following.\nDifficulties and Misconceptions of Unification\n----------------------------------------------\nAccording to , the successes and failures of semi-formal languages (e.g., ) suggest that, once wrapped in -based tools, get exposed to the *quest for a unified syntax*, one main objective of the movement in the 1990s. Rather than a unified syntax, it is more desirable to have a unified semantics and several well-defined mappings to domain-specific syntax wherever convenient (). This approach is occasionally taken up by in (). argued that one cannot achieve proper integration of methods and notations without a unifying semantics. This argument carries over to the problem of tool integration as already discussed in and revisited below. Particularly, the following challenges apply to when used in :\n1. the maintenance of a single source of information serving in the (automated) derivation of downstream artefacts (e.g.proof results, code via synthesis) ,\n2. a clear mapping between the presented to the engineer (using intuitive notation) and the semantics serving as the basis of formal verification,\n3. the embedding of a lean domain-specific formalism into a *common* data model suitable for access and manipulation by engineers through their *various* tools .\nThese challenges are complicated by irreducible unidirectionalities in automated transformations (e.g.model-to-code) limiting the desirable round-trip engineering (i.e.,the change between views of the same data).\nWe discussed as an assurance in. Likewise, *architecture description languages* (e.g.the Architecture Analysis & Design Language, EAST-ADL ) are for overall embedded system design. can be seen as one shortcut to the still ongoing efforts of arriving at a reduced version or a variant of where a semantics can be defined for the whole language .\nAt a higher level of abstraction, so-called *architecture frameworks* (cf.ISO 42010, e.g.the Department of Defense Architecture Framework) and *artefact and traceability models* have been proposed, aiming at the standardisation of specific parts of the systems and software engineering life-cycle and of the documentation and data models used there. These frameworks and models are similar to the models used in product data/life-cycle management in fields like mechanical or civil engineering.\nTo our best knowledge, no cross-disciplinary semantic unification has been undertaken yet (see ), serving as a basis for dependable systems engineering. Although many of these approaches have not been developed with the aim of formalisation and the unification of semantics, we believe that this effort has to be made when developing powerful.\nThe main threat is the lack of agreement on a sound semantic base for domain-specific and cross-domain.\nTo reduce this threat, integration and refinement-based software engineering could be better aligned with artefact models . This alignment may foster the unification of formal semantics to strengthen traceability among the artefacts and to aid in a variety of change impact analysis tasks in the engineering process .\nReluctant Integration Culture and Legacy Processes\n--------------------------------------------------\n*Tool integration* is about the integration of engineering, for example,tools for requirements specification, computer-aided software engineering, and computer-aided mechanical design. Among the wide variety of solutions to capture and track model data, the majority deals with linking or merging data models in one or another shallow way (e.g.software repositories, data exchange formats, product/engineering/application data or life-cycle management systems).\nSome tools with sustainable support are heavyweight, making it difficult to agree on lean model semantics, while others are proprietary, accompanied with interest in hiding model semantics. The surveys of,, and confirm that method and model integration have not yet been solved in,, and dependable systems practice. Moreover, frequent proposals by researchers to formalise fragments or variants of and have not yet received wide attention by practitioners and standardisation authorities.\n-based integrated development environments (e.g.using Xtext and Sirius) get close to what is suitable for -based tools. Such tools rely on a trusted representation of the formal semantics integrating the model data. For successful transfer to assurance practice, tools need to be built on a lean and open central system model .\nAn even greater barrier than loosely integrated tools are legacy *language and modelling paradigms*, an *established tool and method market* carried by *legacy stakeholders* and, possibly, a *neglected continuous improvement of education and training*.\nThe main threat is discontinuous and disintegrated education, transfer, and tool development.\nTo reduce this threat, continuous adaptation and improvement of education through teaching, of transfer through training, application, and feedback, and of tool development through regulated interface standards are necessary.\nReluctant Transfer Culture and Exaggerated Scepticism\n-----------------------------------------------------\nFinally, the vision of introducing into assurance practice might be hindered by a lack of researchers able or willing to engage with industrial assurance practice, as diagnosed by . It is certainly hard work to collect sufficient evidence for effectiveness in assurance practice because of intellectual property rights and other legal issues but also because of a lack of awareness among researchers . However, for credible method comparison experiments, recommended a sample of 20 similar projects split into two groups, 10 projects without treatment (i.e.,not using ) and 10 projects with treatment (i.e.,using ) to establish strong evidence (i.e.,evidence of level 5 or above ).\n[sec:scepticism] Exaggerated scepticism on the side of practitioners and applied researchers that has piled up over the years might be the most important barrier to cross. Early failures to meet high expectations on and transfer might have led to what can be called an “ Winter”. However, we think crossing a few other barriers first might make it easier to cope with scepticism in the assurance community and initiate an “ Spring”, at least in assurance practice. The recent successes with for certification of commercial medical devices , and the associated burgeoning field of for , are examples that could help to overcome this scepticism. Moreover, their is potential for transfer of these results to engineering, where the need for safety assured is also paramount .\nThe main threat is the reluctance of researchers to regularly engage in transfer efforts combined with an exaggerated scepticism of practitioners and with other mechanisms (e.g.lack of funding, poorly focused research evaluation, intellectual property rights) preventing both sides from engaging in an effective bidirectional transfer.\nTo reduce this threat, building awareness among researchers as well as stronger incentives (e.g.regulation) to provide continuous transfer funding for research (e.g.not relying on short-term projects like PhD theses) is needed. A good start on the academic side could then be a specific standardised repository of case studies.\nToo Many Errors in Proofs and Failure to Scale\n----------------------------------------------\nFrom the perspective of measurement, stated that “proofs of correctness sound useful, but [i] errors in the proofs themselves seem to be common failings not covered by the literature. Further, large applications may have thousands of provable algorithms, and [ii] the time required to prove them all might take many years”. For [i], the authors opposed 7% of erroneous bug repairs to up to 100% of erroneous proofs, though stating that the latter was based on an anecdote and there had been little data around. elaborated an example for [ii]: Assuming one provable algorithm per 5 function points[23](#fn23) and on average 4 proofs per day, Microsoft Windows 7 (160,000 function points) would have about 32,000 provable algorithms, taking a qualified software engineer about 36 calendar years. They highlighted that typically only around 5% of the personnel are trained to do this work, assuming that algorithms and requirements are stable during proof time.\n’s argument is based on the view of practical program verification as a largely manual activity. However, research in reasoning about programs has devised promising approaches to proof automation (e.g.bounded model checkers, interactive proof assistants). Assessing such approaches, compared rigorous but manual formal proof with mechanised or automated proof. He could not find a case where mechanised proof threw doubt upon an established mathematical theorem. observed that this underpins the robustness argument of for proofs as manual social processes rather than lengthy mechanical derivations (cf.). However, opposing ’s view, concluded that structuring of knowledge about computing ideas is the actual issue, and not the avoidance of mechanisation. Furthermore, mitigated ’s issue of the reality gap in program verification by referring to researchers’ continuous efforts in verifying hardware, programming languages, and compilers in addition to individual programs. Finally, in support of ’s concerns, highlighted the validity of specifications, manual and automated proofs rely on, as a serious issue indeed having received too little attention from researchers (cf.).\nThe main threat is the lack of qualified personnel to cope with the required amount and type of proof.\nWe believe, one angle of attacking this threat is the use of proof assistants *well-integrated* with common environments for requirements specification and software development combined with continuous research and education in the corresponding methods and tools.\nFailure to Derive Useful Tools\n------------------------------\nBeing loosely related to erroneous proofs, the *information overload through false-positive findings of errors* is a well-known problem in static program analysis. Semi-formal pattern checkers,[24](#fn24) such as PMD and FindBugs, are exposed to this threat . Additionally, FM-based verification tools, such as Terminator and ESC/Java , can be unable to correctly report all potential problems, because they are bounded and therefore unsound. While such tools can be helpful, confronting developers with many irrelevant findings (i.e.,false positives) or a high risk of critical misses (i.e.,false negatives) can lead to decreased use of -based tools.\nrelates the two information retrieval metrics, *precision* and *recall*, with two adequacy criteria of proof calculi, *soundness* and *completeness*. Precision denotes the ratio of actual (solid circle) to correct (true positive findings in the hashed area) findings, while recall measures the ratio of correct to all theoretically correct (dotted circle) findings. The ideal value for both metrics is 1, meaning that there are only true negatives and true positives. Completeness, although unachievable for richer theories, would correspond to recall and soundness would correspond to a precision of 1.\nOn the one hand, the usefulness of the calculi underlying is directly proportional only to their completeness and (traditionally) expires with a precision of less than 1. In other words, we usually try to avoid calculi that allow the derivation of false theorems. On the other hand, semi-formal pattern checkers have shown to have a wide, sometimes unacceptable, range of precision and recall of their findings. The usefulness of practical -based tools might lie somewhere in the middle between classical calculi and bug finding tools with poor precision/recall values.\nThe main threat is the struggle of academia and tool vendors to provide adequate tools, suited to adapt to novel scientific insights, to be integrated with other tools, and to be maintained in a flexible and independent manner.\nTo reduce this threat, an improvement of education and strong incentives can play an important role here (cf.).\n[dim/.style=align=flush left, tit/.style=align=left,anchor=south west, itm/.style=align=center,minimum height=3em,minimum width=3cm,fill=gray!20, lab/.style=align=center]\n(0,0) circle [radius=40pt]; (1,0) circle [radius=30pt];\n(1,0) circle [radius=30pt]; (0,0) circle [radius=40pt];\n(TP) at (2.5,1) true \npositive; (TN) at (-2.5,0) true \nnegative; (FP) at (-.8,0) false \npositive; (FN) at (2.5,-1) false \nnegative; (prcn) at (-2,1) lack of \nPrecision; (recl) at (0.7,0) Recall; (prcn) – (-0.5,0.5); (FN) – (1.5,-.5); (TP) – (1,.5);\n[t]\nXX **Method Strengths: raise the potential of …**\n* improvement of models, specification of requirements, automation of verification ()\n* early detection of systematic errors in designs ()\n* helpful abstractions to inform engineers about critical properties ()\n* integration and coordination of several to consistently reason about interrelated properties ()\n**Community Strengths: research can rely on …**\n* many transfer re-entry points from former case studies in industrial and academic labs ()\n* many assurance practitioners who perceive usefulness as positive ()\n* research designs for comparative studies of effectiveness ()\n& **Method Weaknesses: have suffered from …**\n* being difficult to learn and apply, many assurance practitioners perceive ease of use of as negative ()\n* low effectiveness of formal models because of the reality gap ()\n* fragile effectiveness and productivity in engineering ()\n**Community Weaknesses: progress has been hampered by …**\n[t]\n* no agreed framework for integration of ()\n* lack of convincing evidence of effectiveness in engineering ()\n* research ineffectively communicated in teaching/training ()\n \n**Key Opportunities for transfer and progress:**\n* The desire for early removal of erroneous behaviour and model-based assurance ()\n* The desire to learn from accidents and their root causes ()\n* The desire of assurance to be a mature discipline ()\n* The desire for adequate and dependable norms ()\nThe analysis provided in this table is an enhancement of the general analysis in in. & **Method Threats: research is threatened by …**\n* misconceptions of semantic unification in assurance ()\n* not scaling up to industry-size ()\n* faulty, tedious, or vacuous proofs ()\n* poor integration with engineering tools, processes, and education ()\n**Transfer Threats: transfer is threatened by …**\n[t]\n* a lack of roboticists’ education in ()\n* a lack of researcher engagement in transfer to practice ()\n* a lack of comprehensive access to quality data from practice ()\n \nA Vision of Integrated Formal Methods for Assurance\n===================================================\nThe following discussion applies to many domains of dependability assurance. However, the complexity of robots and autonomous systems forms a key opportunity for the progress of research and for its successful transfer. Accordingly, summarises the discussion in with an interpretation into assurance practice. Based on the strengths and opportunities described in the, we formulate our vision in terms of working hypotheses:\n1. From : Software tools for the construction of arguments and production of evidence using *can meet the challenge* of assuring safe. Computer-assisted assurance cases supported by heterogeneous formal models will increase confidence in their sufficiency, and also aid in maintenance and evolution through modularisation of arguments and evidence.\n2. From :, in particular modern verification tools, will enable *automation of the evidence gathering process*, and highlight potential problems when an assurance case changes or when an incident occurs.\n3. From : There is *no stable path to assured autonomy without the use of*. Acceptable safety will be much more likely achieved with than without them.\n4. From : The success of depends on the ability to *integrate a variety of* for different aspects of (e.g., safety-security interaction, missing human fallback, environment/world modelling, uncertain prediction/behaviour), which is not currently possible.\n5. From : Sophisticated techniques for *model integration and synchronisation are necessary* to support with. This way, will make it easier to express consistent models covering all relevant aspects, make their modelling assumptions explicit, and improve future assurance practices.\n6. From : can be beneficial in the short term. However, an important engineering principle is to be conservative and, therefore, not to change procedures unless there is compelling evidence that are effective. Such evidence can be delivered through *empirical research* and collaboration of academia and industry. That evidence *is required to re-evaluate research and foster research progress and transfer*.\n7. From : The demonstration of cost effectiveness in addition to technical effectiveness of new is necessary to justify further research.\n8. From : Norms are a lever of public interest in dependability . Current norms seem to deviate from the state of the art and may fail to guarantee product certification procedures that satisfy the public interest.\nassigns these hypotheses to the relationships between foundational and transfer-directed research by example of the domain. Overall, we believe that have great potential and can improve assurance but practitioners do not use them accordingly.\nWe could take and enhance credible measures to convince assurance practitioners of our results and effectively transfer these results. For this to happen, we have to answer further research questions.\n[dim/.style=align=flush left, tit/.style=align=left,anchor=south west, itm/.style=align=center,minimum height=3em,minimum width=2cm,fill=gray!20, lab/.style=align=center]\n(ifm) at (0,0) integrated \nFormal Methods (); (efm) at ((*i**f**m*) + ( − 8*e**m*, − 10*e**m*)) Empirical Research \nof Formal Methods; (ras) at ((*i**f**m*) + (8*e**m*, − 10*e**m*)) Assurance Practice in \nRobots and Autonomous \nSystems ();\n(ifm) edge[above,sloped,bend left] node[lab] (H1-3) have potential \nto improve (ras) (efm) edge[above] node[lab] observes (ras) (ras) edge[below,sloped] node[lab] (H4-5) pose chal- \nlenges to (ifm) (efm) edge[above,sloped,bend left] node[lab] (H6-8) examines the \neffectiveness of (ifm) ;\nEmpirical, Applied, and Foundational Research\n=============================================\nBased on the aforementioned working hypotheses, we state several objectives for foundational and transfer-directed research, formulate research questions, and show our expectations on desirable outcomes of such research.\nResearch Objectives and Tasks\n-----------------------------\nTo validate and transfer our research results, we need to\n* evaluate how assurance case construction and management for certifiable can be improved by .\n* debunk or justify arguments against the use of or -based tools in assurance.\n* foster a culture of successful research transfer to industries performing assurance.\nTaking an foundational point of view, we need\n* foundational research on the integration and unification of to tackle the complexity of .\n* a unified semantic foundation for the plethora of notations in assurance, to enable method and tool integration. There are a number of promising research directions still being investigated .\nTaking an evidence-based point of view, as already highlighted in 1993 by , we need to\n* understand the difference between the state of assurance practice and assurance research.\n* understand in which ways current assurance practices fail, and suggest effective approaches from assurance research. In this way, we can be sure that assurance practice is equipped with state-of-the-art assurance technology for defence against potential liability claims, and that assurance practitioners do not fail in fulfilling their obligations.\n* understand how results from assurance research can be validated to be sure that research follows promising directions with high potential of success in assurance practice.\nBased on that, we need to\n* set concrete directions for empirical research in assurance.\n* train researchers in applying empirical research designs in their work on rigorous assurance cases. corroborated this objective by saying that “formal methods champions need to be aware of the need to measure costs”.\n* avoid biases as found in various branches of scientific research. For example,in the social and biomedical sciences, researchers identified such biases through meta-analyses and suggested measures for bias avoidance .\n* increase the level of evidence of research to level 2 according to the hierarchy in .\n* avoid knowledge gaps about whether (a) practice is keeping up with the state of the assurance art, and (b) whether recent academic or industrial research is going in the right direction. In this way, we can be sure that we do our best to inform and serve the society.\nUsing appropriate research designs, we need to\n* invite the industry to enhance their efforts in engaging with recent research.\n* foster goal-oriented interaction (a) between assurance practitioners and researchers and (b) between researchers and assurance researchers. In this way, we can be sure to do everything to keep researchers up to date with respect to practical demands.\n* join the research and applied assurance research communities ( on page ), both vital for the progress and transfer of assurance research into assurance practice. This way, we can be sure to foster necessary knowledge transfer between these two communities.\n* summarise achievements in practical applications of for constructing assurance cases.\n* suggest improvements of curricula for assurance.\n* guide the industry in process improvement, training, and tool support.\n* guide vendors of -based assurance tools to assess and improve their tools and services.\nSome Research Questions addressing these Objectives\n---------------------------------------------------\nThe research questions below are relevant for in general. We consider these questions as crucial to be answered for assurance to address the aforementioned objectives:\n1. What is the true extent of computer-related accidents up to 2019 ? What would these figures mean for the domain?\n2. Does the use of formalism detect severe errors to a larger extent than without the use of formalism ?\n3. Does the use of formalism detect severe errors earlier than without using formalism?\n4. Why would such errors be a compelling argument for the use of?\n5. Apart from error avoidance and removal, which other benefits of in practice are evident and can be utilised for method trade-offs?\n6. Which criteria play a central role in measuring effectiveness? Particularly, how can -based tools be used at scale?\n7. How would Commercial-off-the-Shelf and System-Element-out-of-Context verification by pay off?\n8. Which hurdles need to be overcome to use in practice to the maximum benefit?\n9. How do we know when these hurdles are actually overcome?\n10. How can (from different disciplines) be used together (, unification)?\n11. How can be used to assure systems (e.g.computer vision in road vehicles) involving techniques like machine learning and deep neural networks?\n12. How can be integrated into assurance cases to support certification against international safety and security standards?\n13. How can we best combine formal and informal methods? For example,how can we deal with the issue of the validity of specifications that proofs rely on.\n14. How can we best present formal requirements, evidence, and artefacts in an assurance case?\n15. How can empirical research help in successfully demonstrating the capabilities of for rigorous and certifiable autonomy assurance?\nThis list of research questions can easily be extended by further more detailed empirical questions from the settings discussed in.\nSome Envisaged Research Outcomes\n--------------------------------\nOur vision of *rigorous assurance* implies foundational research to result in\n* novel semantic frameworks unifying best practice methods, models, and formalisms established in\n* new concepts for -based development environments\n* new computational theories to support formal modelling and verification of\n* evaluations of\n\t+ assurance tools, languages, frameworks, or platforms used in practice regarding their support of\n\t+ the integration of into modelling and programming techniques, assurance methods, and assurance processes\n\t+ languages for linking informal requirements with evidence from\n\t+ (automated) abstraction techniques used in assurance and certification\n* opinions, positions, and visions on integration and unification for rigorous practical assurance.\nOur vision of *rigorous assurance* implies applied and empirical research to result in\n* comparisons of\n\t+ projects applying in assurance practice with similar practical projects applying non- approaches\n\t+ -based (embedded software) assurance with assurance approaches in traditional engineering disciplines\n* checklists, metrics, and benchmarks (for and beyond tool performance) for\n\t+ the evaluation and comparison of -based assurance approaches (e.g.confidence level)\n\t+ relating error removal and incident root cause data (e.g.efficiency and effectiveness in removal of severe errors or in avoidance of severe accidents, cf.)\n\t+ usability and maturity assessment of (e.g.abstraction effort, proof complexity, assurance case complexity, productivity)\n\t+ the evaluation of budget cases (cf. in electronic hardware development).\n* experiences in or surveys (e.g.systematic mappings and reviews of assurance case research, interview studies with assurance practitioners, cf.) of\n\t+ transfers and applications (e.g.case studies in assurance and certification projects)\n\t+ challenges, limitations/barriers, and benefits of in assurance and certification projects,\n* research designs (e.g.for controlled field experiments) for the practical validation of in assurance and certification projects\n* opinions, positions, and visions on future research, education, and training in the use of in assurance and certification.\nSummary\n=======\nAlong the lines of, we analysed strengths, weaknesses, opportunities, and threats to determine the potential of integrated formal methods to improve the practice of dependability assurance. Emphasising robots and autonomous systems as an area in the spotlight of dependability assurance, we express our expectations of research progress and transfer. From these expectations, we derived a research and research transfer agenda with the objective of\n[(i)]\nenhancing the foundations of integrated formal methods,\ncollecting evidence on the effectiveness of integrated formal methods in practice,\nsuccessfully transferring integrated formal methods into the assurance practice, with a short-term focus on robots and autonomous systems, and\nfostering research progress, education, and training from the results of this transfer effort.\n#### Acknowledgements\nThis work is partly supported by the Deutsche Forschungsgemeinschaft (DFG) under the Grant no. 381212925 and the EPSRC projects CyPhyAssure,[25](#fn25) grant reference EP/S001190/1, and RoboCalc, grant reference EP/M025756/1. We are deeply grateful to Michael Butler, Ana Cavalcanti, John Fitzgerald, Cliff Jones, and Peter Gorm Larsen for helpful feedback on previous versions of this manuscript and for their genuine support in shaping the direction of this work.\nStudying Formal Method Practitioners: A Ballpark Figure\n=======================================================\nIn this section, we make a brief excursion to the relationship between and, closing with a rough estimate of the size of the population of users and the meaning of such an estimate for the empirical study of that population. Such studies include cross-sectional and longitudinal surveys based on questionnaires or semi-structured interviews.\n#### Formal Inspection versus Formal Methods\nencompasses a variety of techniques (e.g.peer reviews, walk-throughs) where critical process artefacts (e.g.program code) are checked (e.g.manually or using software tools) against a variety of criteria (e.g.checklists), usually by a group of independent qualified engineers. For the sake of simplicity of the following discussion, we assume that can be seen as a particularly rigorous variant of where formal specifications serve as a particular way of formulating checklists.\nWe compare the use of with the use of. According to, formal inspections are used in more than 35% of commercial defence, systems, and embedded software projects, and are estimated to be applied in less than 1% of overall commercial software engineering projects. To get an idea of this coverage data, we perform an analysis of the global embedded software market based on other global software market indicators in. We found estimates of systems and software professionals world-wide and estimates of annual US business values.[26](#fn26)\nA uniform distribution would entail roughly 37000 USD/year per person in the general software domain and 10000 USD/year per person in the embedded software domain. Clearly, geographically strongly differing salaries and part-time engagement rule out a uniform distribution, yet providing figures helpful for our purposes.\nNext, we apply the following proportions: From a worldwide population of around 18.5 million software developers in 2014 , about 19% live in the US, 10% in China, 9.8% in India, 36% Asia/Pacific region, 39 % live in Europe, the Middle East, and Africa; and 30% in the Americas.[27](#fn27) The *design to quality assurance* (i.e.,verification and test) *cost ratio* is observed to be approximately 30 : 70.[28](#fn28) About 20% of embedded software personnel are quality assurance engineers (i.e.,test, verification, or validation engineers).[29](#fn29)\n1.0p3.5cm|XXXXllX *Global market/project indicators* [Unit] & *Professional engineers / developers* [million] & *Ann. business value* [billion USD/year] & *Quality assurance personnel* [million] & *business value* [billion USD/year] & [%] & [%] & *Devices* [billion / year] \nGeneral IT hardware and devices (incl. personal computers) & & 2018: 689 & & & & & 2010: 10 \nEmbedded systems (hardware, software, connected embedded devices) in all domains & 2014: 1.2 & 2009: 882018: (161)*a* & & & (35) & & 2010: 9.8 \nIndustrial embedded systems & & & & & & & 2016: 2 \nDefence, systems, and embedded commercial software engineering & & & & & 2011: 35 & \nEmbedded software & 2014: 0.36 & 2009: 3.42018: (10) & **2010: (0.072)** & **2010: (2.38)** & & \nGeneral software (overall commercial software engineering, development) & 2014: 18.52018: 232019: 23.9 & 2013: 4072018: (515) & & & (4.6) & 2011: 1 & \nThe estimates in suggest that around 2% of the overall pure software market are allocated to the embedded pure software market. 35% coverage of formal inspection in about 13.5% of the overall software market (161/(689 + 515) = 0.134) would result in roughly 4.7% coverage of all software projects by formal inspection versus at most 1% coverage by. However, from this data we can hardly know whether rates of use get close to or beyond 10% in high-criticality systems projects.\nAssuming that in about 35% of embedded software projects the quality assurance personnel would use formal inspection and that in every fifth (1 : 4.7) of such projects formal methods would be used, the current population of regular practical users would globally amount to about 5040 ( = 72, 000 \\* 0.34 \\* 0.20) persons. Note that these numbers are rough estimates. However, we believe their order of magnitude is realistic. Given that these persons would on average earn about 100,000 USD/year each, we would speak of an annual business value of around USD 504 million.\n#### Empirical Studies of and Practitioners\nImportantly, from these data we can determine minimum sample sizes for surveys. For example,assume we want to have 95% confidence in our test results and are fine with a confidence interval of ± 7%. Then, for *regular practical users*, a population of the size of 5040 persons would require us to sample 189 independent data points (e.g.questionnaire responses). The population of *regular practical users*, 25200 ( = 72, 000 \\* .35) would imply a minimum sample size of 194.\nFor any sample of such size, any survey has to argue why the sample *represents* the population. This step depends on the possibilities given during the sampling stage. Obviously, reaching out to 189 out of 5040 persons whose locations might be largely unknown is an extremely difficult task that might only be tackled in terms of a global group effort among researchers and practitioners. Given that this challenge can be tackled, these figures determine the type of surveys required for the collection of confident evidence. Moreover, these figures limit the manual effort to perform and repeat valuable qualitative studies to a reasonable amount.\nFormal Methods in Dependable Systems Engineering\n================================================\non page depicts our research setting for this survey. There, we describe as an enhancement of and the assurance of as a special field of dependable systems engineering. Because many of the discussed issues are of a more general nature, we first prepared a analysis of in dependable systems engineering in. Then, we refined this analysis in to accommodate specifics of for.\nXX **Method Strengths:**\n[t]\n* Improvement of modelling precision, requirements clarity, verification confidence\n* High error detection effectiveness, early error removal\n**Community Strengths:**\n* Many transfer re-entry points from former case studies with industry\n* Many dependable systems practitioners perceive usefulness as positive\n& **Method Weaknesses:**\n* Difficult to learn and apply, many dependable systems practitioners perceive ease of use of as negative\n* Fragile effectiveness and productivity\n**Community Weaknesses:**\n* Lack of compelling evidence of effectiveness\n* Ineffectively communicated in teaching and training\n \n**Key Opportunities for Transfer and Research:**\n* Desire for early removal of severe errors ()\n* Desire to learn from accidents and their root causes ()\n* Desire to be a mature discipline ()\n* Desire for dependable norms ()\n& **Method Threats:**\n[t]\n* Lack of method scalability\n* Faulty, tedious, or vacuous proofs\n* Lack of user education\n* Poor tool integration, legacy tools and processes\n**Transfer Threats:**\n* Lack of researcher engagement in transfer\n* Lack of access to comprehensive high-quality data\n \n---\n1. For: Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems.[↩](#fnref1)\n2. For: Software Considerations in Airborne Systems and Equipment Certification.[↩](#fnref2)\n3. We reuse the term from the homonymous conference series .[↩](#fnref3)\n4. The SCADE Design Verifier (<http://www.esterel-technologies.com>) and the seL4 microkernel (<http://sel4.systems>) represent good although less recent examples.[↩](#fnref4)\n5. Tokeneer Project Website: <https://www.adacore.com/tokeneer>.[↩](#fnref5)\n6. Pacemaker Formal Methods Challenge: <http://www.cas.mcmaster.ca/sqrl/pacemaker.htm>.[↩](#fnref6)\n7. “Formal” refers to the formality of the procedure but does not imply the use of formal semantics or mathematical techniques.[↩](#fnref7)\n8. See <https://www.york.ac.uk/assuring-autonomy/body-of-knowledge/>.[↩](#fnref8)\n9. See <https://www.omg.org/spec/SACM/About-SACM/>.[↩](#fnref9)\n10. For example,the fatal accident involving a Tesla advanced driving assistance system, <https://www.theguardian.com/technology/2018/jun/07/tesla-fatal-crash-silicon-valley-autopilot-mode-report>.[↩](#fnref10)\n11. Specification errors are also called development failures and can be seen as flaws in the process of requirements validation.[↩](#fnref11)\n12. See for a list of accident databases from such institutions.[↩](#fnref12)\n13. For example,according to “As Low As Reasonably Practicable” or “So Far As Is Reasonably Practicable”. See <http://www.hse.gov.uk/risk/theory/alarpcheck.htm>: “Something is reasonably practicable unless its costs are grossly disproportionate to the benefits.”[↩](#fnref13)\n14. National Highway Traffic Safety Administration, <https://www.nhtsa.gov>.[↩](#fnref14)\n15. National Transportation Safety Board, <https://www.ntsb.gov>.[↩](#fnref15)\n16. These experts have experience with tool qualification (1) and safety-critical systems in aviation (3), railway (1), automotive (4), and energy systems & turbines (1). We have to keep their names confidential.[↩](#fnref16)\n17. From personal communication.[↩](#fnref17)\n18. See expert interview by embedded software journalist from EE Times in 2013 on <https://www.eetimes.com/document.asp?doc_id=1319903&page_number=1>.[↩](#fnref18)\n19. See blog post on the seL4 microkernel, <https://research.csiro.au/tsblog/crisis-security-vs-performance/>.[↩](#fnref19)\n20. For: Road Vehicles – Functional Safety.[↩](#fnref20)\n21. This might also the case for MISRA:2012 from March 2013. We are unaware of the opposite but were also unable to receive a copy of this version.[↩](#fnref21)\n22. While formal verification serves the check of absence of property violations, conventional testing can only serve as a check of presence of such violations.[↩](#fnref22)\n23. A function point is a measure of the conceptual complexity of an IT system relevant for the estimation of the amount of work required to engineer this system.[↩](#fnref23)\n24. Also called bug finding tools.[↩](#fnref24)\n25. CyPhyAssure Project: <https://www.cs.york.ac.uk/circus/CyPhyAssure/>.[↩](#fnref25)\n26. See <https://en.wikipedia.org/wiki/Software_industry> and <https://softwareengineering.stackexchange.com/questions/18959/what-proportion-of-programming-is-done-for-embedded-systems>.[↩](#fnref26)\n27. See <https://adtmag.com/Blogs/WatersWorks/2014/01/Worldwide-Developer-Count.aspx>.[↩](#fnref27)\n28. See <https://www.slideshare.net/pboulet/socdesign>.[↩](#fnref28)\n29. See <https://de.slideshare.net/vdcresearch/searching-for-the-total-size-of-the-embedded-software-engineering-market>.[↩](#fnref29)\n\nNew Opportunities for Integrated Formal Methods1\n================================================\nFormal methods have provided approaches for investigating software engineering fundamentals and also have high potential to improve current practices in dependability assurance. In this article, we summarise known strengths and weaknesses of formal methods. From the perspective of the assurance of, we highlight new opportunities for integrated formal methods and identify threats to the adoption of such methods. Based on these opportunities and threats, we develop an agenda for fundamental and empirical research on integrated formal methods and for successful transfer of validated research to assurance. Furthermore, we outline our expectations on useful outcomes of such an agenda.\n3 [sec:nomenclature]\nIntroduction\n============\nA plethora of difficulties in software practice and momentous software faults have been continuously delivering reasons to believe that a significantly more rigorous discipline of software engineering is needed . Researchers such as have collected plenty of anecdotal evidence on software-related risks substantiating this belief.\nIn, researchers have turned this belief into one of their working hypotheses and contributed formalisms, techniques, and tools to increase the rigour in engineering workflows . Examples of activities where formalisms have been brought to bear include requirements engineering , architecture design and verification, test-driven development, program synthesis, and testing, to name a few. have been an active research area for decades, serving as powerful tools for *theoretical researchers*. As a paradigm to be adopted by *practitioners* they have also been investigated by *applied researchers*. Applications of have shown their strengths but also their weaknesses .\nBased on observations contradicting the mentioned difficulties, came to ask “How did software get so reliable without proof?” He, and later, suggested that continuously improved design of programming languages and compilers, defensive programming, inspection, and testing must have effectively prevented many dangerous practices. coined the “Hoare paradox” stating that although proof was seldom used, software had shown to be surprisingly fit for purpose. However, faced with software of increasing complexity (e.g.), pondered how long ’s question will remain valid.\nIndeed, recently we can observe a plethora of difficulties with . Such systems are set to be more broadly deployed in society, thereby increasing their level of safety criticality and requiring a stringent regulatory regime. A successful method for regulatory acceptance is provided by structured assurance cases, which provide comprehensible and indefeasible safety arguments supported by evidence . However, such assurance cases—whether or not compliant with standards like IEC 61508[1](#fn1) and DO-178C[2](#fn2)—can be laborious to create, complicated to maintain and evolve, and must be rigorously checked by the evaluation process to ensure that all obligations are met and confidence in the arguments is achieved . Nevertheless, these are problems that are designed to overcome.\nIn spite of the weaknesses of current, and encouraged by their strengths, we believe that their coordinated use within established processes can reduce critical deficits observable in dependable systems engineering. state that “there is currently no general framework integrating formal methods for robotic systems”. The authors highlight the use of what are called [3](#fn3) in the construction of assurance cases and the production of evidence as a key opportunity to meet current challenges. Particularly, computer-assisted assurance techniques , supported by evidence provided by, can greatly increase confidence in the sufficiency of assurance cases, and also aid in their maintenance and evolution through automation. Moreover, the use of modern -based tools to support holistic simulation, prototyping, and verification activities, at each stage of system, hardware, and software development, can lead to systems that are demonstrably safe, secure, and trustworthy.\nContribution\n------------\nWe investigate the *potentials for the wider adoption of integrated formal methods in dependable systems engineering*, taking as a recent opportunity to foster research in such methods and to support their successful transfer and application in practice.\nWe analyse the *strengths, weaknesses, opportunities, and threats* of using formal methods in the assurance of dependable systems such as. For this analysis, we summarise experience reports on formal method transfer and application.\nAssuming that\n[(a)]\nintegration enhances and\nthe assurance of is a branch of,\nshows how we derive an agenda for fundamental and applied research.\nFrom the *strengths*, we see in recent research, and from the *opportunities* in current assurance, we argue why this domain is a key opportunity for. Particularly, we indicate how such methods can meet typical assurance challenges that are increasingly facing.\nFrom the *weaknesses*, we observe in recent research, and from the *threats* research transfer is exposed to, we derive directions of foundational and empirical research to be taken to transfer into the assurance of robots, autonomous systems, and other applications, and to use these methods to their maximum benefit.\n[dim/.style=align=flush left, tit/.style=align=left,anchor=south west, itm/.style=align=center,minimum height=3em,minimum width=3cm,fill=gray!20, lab/.style=align=center]\n(methgen) at (0,0) Formal Methods; (methspec) at ((*m**e**t**h**g**e**n*) + (0, − 5*e**m*)) integrated \nFormal Methods; (appgen) at ((*m**e**t**h**s**p**e**c*) + (18*e**m*, 0)) Dependable Systems \nEngineering; (appspec) at ((*a**p**p**g**e**n*) + (0, − 5*e**m*)) Assurance \nof ;\n(approach) at ((*m**e**t**h**g**e**n*.*n**o**r**t**h**w**e**s**t*) + (0, 0)) Approach; (research) at ((*m**e**t**h**g**e**n*.*n**o**r**t**h*) + (9*e**m*, 0)) Empirical Research; (application) at ((*r**e**s**e**a**r**c**h*.*e**a**s**t*) + (1.3, 0)) Application; (research) at ((*m**e**t**h**s**p**e**c*.*s**o**u**t**h**w**e**s**t*) + (0, − .5*e**m*)) Fundamental \nResearch;\n(methgen) edge[dotted,sloped,above] node[lab] applied and \ninvestigated in (appgen) (methspec) edge[sloped,below] node[lab] applied and \ninvestigated in (appspec) ; (methspec) edge[right] node[lab] enhances (methgen) (appspec) edge[right] node[lab] specialised core subject of (appgen) (appgen) edge node[lab] achieve wider \nadoption of (methspec) ;\nOur analysis\nelaborates on the analysis and conclusions of,\nextends their suggestions with regard to formal method experimentation and empirical evidence of effectiveness focusing on collaboration between formal method researchers and practitioners, and\ndevelops a research and research transfer road map, placing emphasis on.\nOverview\n--------\nWe provide some background including terminology () and related work () in the following. Then, we carry through an analysis of strengths, weaknesses (), opportunities (), and threats () of for assurance. Based on this analysis, we formulate our hypotheses (), pose research questions based on these hypotheses, derive a research agenda, and specify some outcomes we expect from this agenda ().\nBackground\n==========\nThis section introduces the core terminology used throughout this article, and provides a discussion of other surveys of the state of the art and a summary of similar positions and agendas.\nTerminology\n-----------\nFor the sake of clarity among readers of different provenance, we restrict the meaning of some terms we use in the following and introduce convenient abbreviations.\nWe view *robots and autonomous systems* as both dependable systems and highly automated machines capable of achieving a variety of complex tasks in support of humans. We can consider such systems by looking at four layers: the plant or process composed of the operational environment and the machine; the machine itself; the machine’s controller, and the software embedded into this controller. Based on these layers, we treat “embedded system” and “embedded software” as synonyms. Machine, controller, and software can all be distributed.\nBy *dependable systems engineering*, we refer to error-avoidance and error-detection activities in control system and embedded software development (e.g.according to the V-model). devised a comprehensive terminology and an overview of the assessment and handling of a variety of faults, errors, and failures. For critical systems, such activities are expected to be explicit (e.g.traceable, documented), to employ best practices (e.g.design patterns), and to be driven by reasonably qualified personnel (e.g.well-trained and experienced engineers or programmers).\nThe need for *dependability* often arises from the embedding of software into a cyber-physical context (i.e.,an electronic execution platform, a physical process to be controlled, and other systems or human users to interact with)., or assurance for short, encompasses the usually cross-disciplinary task of providing evidence for an assurance case (e.g.safety, security, reliability) for a system in a specific operational context .\nBy *formal methods*, we refer to the use of formal (i.e.,mathematically precise and unambiguous) modelling languages to describe system elements, such as software, hardware, and the environment, and the subjection of models written in these languages to analysis , the results of which are targeted at assurance . always require the use of both *formal syntax* and *formal semantics* (i.e.,the mapping of syntax into a mathematical structure). Semantics that allows the verification of refinement or conformance across different is said to be *unifying* . allow the coordinated application of several potentially heterogeneous, supported by interrelated layers of formal semantics .\nstand in contrast to *informal methods*, which employ artefacts without a formal syntax or semantics, such as natural language descriptions and requirements. In the gap between informal methods and there is also a variety of *semi-formal methods*, including languages like the and the, whose syntax and semantics have frequently been subject of formalisation in research .\n*-based tools* assist in the modelling and reasoning based on a. and served many opportunities for -based tools to be applied in dependable systems practice .[4](#fn4)\nWe speak of *applied or practical* to signify successful applications of in a practical context, for example,to develop embedded control software deployed in a commercial product marketed by an industrial company. We consider the use of in research projects still as *research*. *Empirical research* investigates practical, for exampleusing surveys, case studies, or controlled field experiments . We speak of *transfer* if research is transferred into practice with the aim to effectively apply and practice. We consider transfer, as discussed below, as crucial for empirical research and progress of research.\nRelated Work\n------------\nMany researchers have suggested that will, in one way or another, play a key role in mastering the difficulties discussed below and in achieving the desired guarantees (e.g.dependability, security, performance) of future critical systems .\nExpecting an increased use of to solve practical challenges in the mid 1990s, suggested integration, tool development, and continuous specialist training to foster successful transfer to practice.\nIn 2000, observed a growing number of success stories in requirements engineering. Evaluating several paradigms, he outlined weaknesses (e.g.isolation of languages, poor guidance) to be compensated and challenges to be met towards effective FM use, particularly their integration into multi-paradigm specification languages.\nAiming at the improvement of software dependability, made several key observations of recent dependability practice (e.g.lack of evidence for method effectiveness) leading to a general proposal with broad implications: rigorous dependability cases with explicit claims, the support of reuse and evolution, and the selective use of. Additionally, these authors provide a number of recommendations to tool vendors and organisations in education and research.\nAlso in the mid 2000s, spotted a decline of internet software dependability in the context of an increased level of concurrency in such systems. Their observation was backed by an earlier comparative software/hardware dependability discussion by. highlighted achievements in automation enabling an increased use of lightweight in “software engineers’ usual development environments”. Furthermore, they stressed the ability to use several in a combined manner to verify distributed (embedded) systems, avoid errors and, hence, stop the decline of software dependability.\n issued a manifesto for a “Verified Software Initiative”. Based on a consensus of strengths, weaknesses, opportunities, and threats in the software engineering community, they proposed a long-term international “research program towards the construction of error-free software systems”. This initiative aims to achieve its agenda through\n[(1)]\nnew theoretical insights into software development,\ncreation of novel automated tools, and\na collection of experiments and benchmarks.\nIn particular, the initiative is driven by a number of “grand challenges” —difficult practical verification problems that can guide future research. The experiments have broad scope, and include a smart cash card (the Mondex card), a secure entry system[5](#fn5) (Tokeneer), and a cardiac pacemaker.[6](#fn6)\nOutlining an agenda for transfer, raised the need for improved benchmarks, metrics, and infrastructure for experimental evaluation, the need for revised teaching and training curricula , and the need for research communities interested in engaging with practitioners and working on ways to scale up to large systems and to increase the usability of. The authors specified several applications with great opportunities for transfer.\nApplied researchers and practitioners interviewed by convey an optimistic picture of adoption in practice, highlighting the potentials to improve IT security, particularly in cyber-physical systems. share the view that are the most promising approach towards acceptably dependable and secure systems. The challenges they list for the security domain are similar to the challenges we perceive in assurance: integration, sound abstraction techniques, compositional guarantees, and evidence for sustainable transfer.\nWith their survey of for verification, identified difficulties of applying in the robotics domain and summarised research results and their limitations. They conclude\n[(i)]\nthat formalisation remains the most critical and most difficult task,\nthat the surveyed approaches do not provide “sufficient evidence for public trust and certification”, and\nthat would be highly desirable if the current lack of translations between the most relevant of the surveyed techniques (e.g.model checking) could be overcome.\nWe complement their observations with a further analysis of the lack of unification of and of the missing empirical evidence for the effectiveness of and. Additionally, we provide a research road map.\nStrengths and Weaknesses of Formal Methods for Assurance\n========================================================\n[sec:strengthsweaknesses]\nFollowing the guidelines for analysis by , we provide an overview of strengths and weaknesses of, regarding\n* reputation, proof culture, education, training, and use (),\n* transfer efforts (),\n* evidence of effectiveness (),\n* expressivity (), and\n* integration and coordination ().\nReputation, Proof Culture, Education, Training, and Use\n-------------------------------------------------------\nIn the guest editor’s introduction of the “50 Years of Software Engineering” IEEE Software special theme issue , the question\n> \n> “Are formal methods essential, or even useful, or are they just an intellectual exercise that gets in the way of building real-world systems?”\n> \n> \n> \ninvited us to deliberate on this topic and summarise its highlights. Applied researchers have raised the issue of *limited effectiveness and productivity* of, particularly in large practical systems with changing requirements . are known to be *difficult to apply in practice*, and *bad communication* between theorists and practitioners sustains the issue that are taught but rarely applied . In contrast, they are considered to have significant potential to cope with the toughest recent engineering problems: certifiable assurance .\nStudying the sociology of proof, identified three sources of knowledge about a system’s dependability: induction (i.e.,from observation), authority (e.g.expert opinion), and deduction (i.e.,inference from models) which is possibly the most powerful. Since the beginning of software engineering there has been a debate on the style of deductive reasoning about programs and on the usefulness of. argued that proof is a social process. Long and difficult to read computer-produced verification evidence cannot be subject to such a process and is not genuine proof. countered, albeit not as a supporter of mechanisation, to change from a personal trust-based culture of proof to the formalisation of proof steps. doubted that verification based on a model of the program can yield any knowledge about the dependability of an implementation of that program. According to, it is not the degree of formalisation making a proof convincing but the way the argument is organised. tried to arbitrate this debate between rigorous proof in ordinary mathematics and formal mechanised proof. He suggested that proof assistants have the potential to use formal methods to the maximum benefit. concluded from a discussion with Tony Hoare that formalist and empiricist perspectives, while still causing controversies between research and practice, complement each other in a fruitful way.\nNevertheless, have shown to be well-suited to *substantially improve modelling precision, requirements clarity, and verification confidence*. applications in requirements engineering such as the “Software Cost Reduction” tool set even carry the hypothesis of cost-effectiveness in its name. By the 1990s, researchers had already started to examine usefulness with the aim to respond to critical observations of practitioners . Some of these efforts culminated in empirical studies suggesting *high error detection effectiveness*, though with some controversy also caused by employed research designs.\n observe that the combination of *formal[7](#fn7) inspection*, static analysis, and formal testing has been the best approach to defect prevention with *up to 99% of accumulated defect removal efficiency*. can be seen as a rigorous and systematic form of this approach, though less often applied. In, we make a brief excursion to the relationship between and formal inspection and try to roughly estimate the population size of users.\nFrom two larger surveys, one in the early 1990s and another one in the late 2000s , we obtain a more comprehensive picture of the typical advantages of use and barriers to adoption as seen by practitioners and practical researchers. In two recent surveys, we made two, not necessarily surprising but empirically supported, observations underpinning the main findings of the former studies: many practitioners *view as promising* instruments with high potential, and would use these instruments to their maximum benefit, whether directly or through -based tools. However, the beneficial use of is still hindered by severe obstacles (e.g. are considered hard to learn, difficult to integrate in existing processes, too expensive, prone to invalid abstractions, difficult to maintain).\ncan improve modelling, the specification of requirements, and the automation of verification, fostering the early detection of systematic errors in designs. Many assurance practitioners perceive usefulness as positive.\nhave shown to be difficult to learn and apply. Many assurance practitioners perceive the ease of use of as negative. Moreover, research has been ineffectively communicated in teaching and training.\nTransfer Efforts\n----------------\ncan be effective in two ways, *ab-initio* (i.e.,before implementation) and *post-facto* (i.e.,after implementation). The ab-initio use of FMs aims at reducing late error costs through, for example,formal prototyping, step-wise refinement, formal test-driven development, crafting module assertions prior to programming, or formal contract-based development. Once an initial formalisation (e.g.invariants) is available, it is argued for families of similar systems that, from the second or third FM application onward, the benefit of having the formalisation outperforms the cumulative effort to maintain the formalisation up to an order of magnitude . This argument also addresses agile settings inasmuch as iterations or increments refer to similar systems. The post-facto use of can occur through knowledge extraction from existing artefacts and using automated tools such as, for example,formal or model-based post-facto testing tools or post-facto use of code assertion checkers. Overall, the second way of utilising is known to be more compatible with everyday software practice.\nAchievements collected by ,, and show that many researchers have been working *towards successful transfer*. Moreover, researchers experienced in particular draw positive conclusions from applications, especially in scaling through adequate tool support for continuous reasoning in agile software development . Other researchers report about progress in theorem proving of system software of industrial size and about FM-based tools for practical use . and have a history of wrapping into software tools to make access to formalisms easier and to help automating tedious tasks via and visual notations.\n*Static (program) analysis* is another branch where -based tools have been successfully practised . However, few static analysis tools are based on and many of these tools are exposed to reduced effectiveness because of *high false-positive rates*, particularly if settings are not perfectly adjusted to the corresponding project .\nFurthermore, the concolic testing technique , a post-facto, has seen multiple successes in industry . It exercises all possible execution paths of a program through systematic permutation of a sequence of branch conditions inferred by an instrumented concrete execution. It uses these symbolic execution paths and solving to obtain a series of inputs that exercise the full range of program paths. It does not depend on a predefined model of the program, but effectively infers one based on the branch conditions. It can therefore readily be used on existing program developments, and has notably been used by Samsung for verification of their flash storage platform software . Indeed, it is a belief of the authors of this latter work that post-facto methods provide greater opportunities for adoption of in industry.\nRecently, there have been several developments in the use of to assure safety requirements related to, and a growing body of literature , particularly relating to certification of commercial medical devices . There, have been shown to be successful in facilitating regulatory acceptance, for examplewith the agency in the United States. formalised the regulatory safety requirements for the user interface of a patient-controlled infusion pump, and used the PVS proof assistant to verify it., similarly, formally modelled an infusion pump interface, using the Z notation, and then used the resulting specification as an oracle to generate test cases. performed risk analysis for a new neonatal dialysis machine, formalised safety requirements, and use these requirements to analyse and verify the source code. The consensus of these authors is that can greatly reduce the defects in safety critical prior to deployment. The techniques are increasingly practically applicable through greater automation. Moreover, it is likely that these results have potential applications in other domains, such as, provided they can be transferred and scale to this greater complexity.\nThere exist many transfer re-entry points from a range of insightful case studies in industrial and academic labs. were demonstrated to be a useful basis for many static analysis and tools.\nThe number of practical (comparative) case studies using (ab-initio) or, particularly on, is still too low to draw useful and firm conclusions on effectiveness.\nEvidence of Effectiveness\n-------------------------\nWhether used as ab-initio or post-facto tools, *strong evidence* for the efficacy of in practice is *still scarce* and more anecdotal, rarely drawn from comparative studies, often primarily conducted in research labs, or not recent enough to reflect latest achievements in verification tool research . We observe that a large fraction of empirical evidence for effectiveness can be classified as level 6 or 7 according to, that is,too weak to draw effective conclusions.\n as well as, two researchers from the software engineering measurement community, support this observation. state that “there is very little empirical data on several topics that need to be well understood if proofs of correctness are to become useful tools for professional software development as opposed to academic experiments”. Moreover, the controversies about proof culture summarised in contain little data to resolve practitioners’ doubts.\n observed this lack of evidence of effectiveness in assurance argumentation. More generally, noticed insufficiently evaluated safety research. About 86% of works lack guidance to reproduce results, hence forming a barrier to the advancement of safety practice. Although their study is limited to one conference series, it indicates deficiencies in the evaluation of research. Overall, it is important to understand that the mentioned lack of evidence and successful transfer produces great opportunities for further empirical and theoretical research.\nFor (comparative) studies of effectiveness, there are several research designs and benchmark examples available from the scientific literature. In, we assess the effort and feasibility of corresponding qualitative and quantitative studies.\nhave been suffering from fragile effectiveness and productivity in dependability engineering in general. There is a lack of convincing evidence of effectiveness, particularly, of ab-initio. engineering and assurance are likely to be affected by these weaknesses.\nExpressivity\n------------\nAn often quoted weakness of, particularly when applied to, is the “reality gap” that can exist between a naively constructed model and its corresponding real-world artefact. According to , over-reliance on simulation to test behaviour using naive and insufficiently validated models can lead to effort being applied to solving problems that do not exist in the real world. Worse, programs for robotic controllers developed in a model-based setting may fail when executed on real-world hardware, because “it is very hard to simulate the actual dynamics of the real-world” . This problem is not only true of simulation, but any form of model-based analysis, including reasoning in.\nThe fundamental problem here is that it is impossible to model the behaviour of any physical entity precisely , unless we replicate the original. Moreover, as models become more detailed, their utility decreases and they can become just as difficult to comprehend and analyse as their real-world counterparts, an observation highlighted by the famous paradox of. Nevertheless, as statistician George Box said “all models are wrong but some are useful” : we must evaluate a model not upon how “correct” it is, or how much detail it contains, but on how *informative* it is. According to , the antidote is not to abandon the use of models, but to recognise their inherent limitations and strengths, and apply them intelligently to reasoning about a specific problem. This means selecting appropriate modelling paradigms that enable specification of behaviour at a sufficiently detailed level of abstraction, and using the resulting models to guide the engineering process.\nallow and foster the use of specific abstractions to specifically inform engineers of properties critical for their assurance.\nThe effectiveness of formal models is fragile and can be significantly reduced because of uncontrollable gaps between models and their implementations.\nIntegration and Coordination\n----------------------------\nModelling notations usually employ a particular paradigm to abstract the behaviour of the real-world. For example, the state-based paradigm, employed by like Z , B , and refinement calculus , considers how the internal state of a system evolves, whilst the event-driven paradigm, employed in process calculi like CSP , CCS , and *π*-calculus , considers how behaviour may be influenced by external interactions. Consequently, individual formal methods are usually limited to considering only certain aspects or views of a system’s behaviour , which can limit their effectiveness when used in isolation. Many researchers have therefore sought to overcome this weakness by integration .\nThe 1990s saw a large number of works on semantic unification and method integration . Theoretical foundations were provided by, in his seminal work on semantic unification using the “programs-as-predicates” approach and comparative semantics . At the same time, refinement calculi were developed that would underlie the work on linking heterogeneous notations through abstraction. Also, explored the integration of state- and event-based modelling using weakest preconditions, and several other works on this topic followed . proposed a unified theory of programming that links together the three semantic styles: denotational, operational, and algebraic. These developments culminated in ’s , a general framework for integration of semantically heterogeneous notations by application of ’s approach to the formalisation of a catalogue of computational paradigms, with links between them formalised using Galois connections. This framework enables a definitive solution to the integration of states and events, along with other computational paradigms, in the Circus language family .\nAnother result of these developments was a number of seminal works on integration ., inspired by work on systematic method integration , defined a generic “meta-method” that aimed at integration of several formal and semi-formal methods using notational translations with a common predicative semantic foundation, which builds on ’s work . Meanwhile,, building on their previous work , likewise proposed the creation of hybrid with a multi-paradigm approach. Moreover, proposed that should be integrated into the V-Model of development with common semantic foundations to link the various artefacts across development steps.\nThese diverse efforts eventually led to the founding of the conference series in 1999 , with the aim of developing theoretical foundations for “combining behavioural and state-based formalisms”. For the second iteration of the conference , the scope broadened to consider all the different aspects of integration, including semantic integration, traceability, tool integration, and refinement. A few years later, a conference series was also established for , with the aim of continuing to develop unifying semantics for diverse notations within the framework.\nHowever, there is as yet no agreed and general methodology for integrating that could be applied to . Overall, integration is of particular pertinence to, since such systems are multi-layered and possess a high degree of semantic heterogeneity. As found, they “can be variously categorised as embedded, cyber-physical, real-time, hybrid, adaptive and even autonomous systems, with a typical robotic system being likely to contain all of these aspects”. When we consider, we must consider advanced computational paradigms like real-time, hybrid computation with differential equations, probability, and rigid body dynamics. This implies the use of several different modelling languages and paradigms to describe the different aspects, and therefore a variety of analysis techniques to assure properties of the overall system. Assurance of autonomous systems will certainly therefore require . on page summarises this relationship.\nraise the potential of integration and coordination of several to consistently reason about properties implemented by a combination of various technologies such as software, electronic hardware, and mechanical hardware.\nThere is currently no agreed framework for the integration of that would effectively address the needs in the domain or similar domains.\nOpportunities for Integrated Formal Methods\n===========================================\n[sec:opportunities]\nThis section continues with the *environmental part* of our analysis. Several key opportunities for the transfer of arise from ongoing assurance challenges, particularly in assurance and from looking at what other disciplines do to cope with similar challenges. In the following, we describe opportunities stemming from\n* the desire for early removal of severe errors (),\n* the desire to learn from accidents and their root causes (),\n* the desire of assurance to form a mature discipline (), and\n* the desire for adequate and dependable norms ().\nThe Desire for Early Removal of Severe Errors\n---------------------------------------------\nSummarising major challenges in automotive systems engineering in 2006, indicated that modelling languages used in practice were often not formalised and the *desired benefits could not be achieved from semi-formal languages*. Moreover, *software engineering was not well integrated* with core control and mechanical engineering processes. Domain engineers would produce software/hardware sub-systems and mechanical sub-assemblies in undesirable isolation. referred to a lack of for *overall architecture verification*.\nHas the situation changed since then? In model-centric development in embedded software practice (e.g.based on or ), drawbacks can be significant if methods and tools are not well integrated or trained personnel are missing . Likely, ’s criticism remains in contemporary automatic vehicle engineering and assurance practice. In fact, he has a recent, clearly negative, but not pessimistic answer to this question . Moreover, this view is shared by the Autonomy Assurance International Programme’s discussion of assurance barriers,[8](#fn8) that is,current challenges in assurance. These barriers (e.g.validation, verification, risk acceptance, simulation, human-robot interaction) could be addressed by formal engineering models and calculations based on such models to be used as evidence in assurance cases.\n[sec:desire-model-based] Model-based assurance uses system models to structure assurance cases and represents another opportunity for formal methods in (through-life) assurance. Assurance arguments that are purely informal can be difficult to evaluate, and may be subject to argumentation fallacies . Consequently, there have been a number of efforts to formalise and mechanise assurance cases, both at the argumentation level and the evidence level . More recently, in, the [9](#fn9) is a standardised meta-model that supports both structured argumentation and integration of evidence from diverse system models . unifies several existing argumentation notations, and also provides support for artefact traceability and terminology. It could, in the future, serve as a crucial component for in assurance.\nLeading voices from applied software engineering research periodically mention the role of as a key technology to master upcoming challenges in assuring critical software systems . A round table about the adoption of in IT security positively evaluated their overall suitability, the combination of with testing, and the achievements in automation. The panellists noticed limitations of in short-time-to-market projects and in detecting unknown vulnerabilities as well as shortcomings in training and adoption in practice.\nHowever, even for mission-critical systems, high costs from late defect removal and long defect repair cycles , as well as dangerous and fatal[10](#fn10) incidents indicate that assurance in some areas is still driven by practices failing to assist engineers in overcoming their challenges. Moreover,, an observer of a multitude of computing risks, stated that “the needs for better safety, reliability, security, privacy, and system integrity that I highlighted 24 years ago in my book, Computer-Related Risks, are still with us in one form or another today” .\nFor example,artificial intelligence software—particularly components—has been developed at a high pace and used in many non-critical applications. Recently, components are increasingly deployed in critical domains. For verification and error removal, such software has to be transparent and explainable. Preferring verifiable algorithms to heuristics, recalled the corresponding engineering principle: “We cannot trust a device unless we know how it works”. One way to follow this principle and establish transparency is to reverse engineer (i.e.,to decode) the functionality of an component even if this is not possible in general . can help extract knowledge and reverse engineer abstractions of systems to explain their behaviour. We might then ask to which extent the reverse engineered and verified functionality serves as a substitute for the original component.\nThese anecdotes make it reasonable to question current assurance practice. Seen through the eyes of assurance, they suggest that we might again be facing a dependable software engineering *crisis* similar to the one from the late 1960s .\nWe as method researchers could learn from this crisis and improve the way that can be effectively coordinated to support early error removal in practical engineering processes. We as practitioners could learn from this crisis and improve the way that we correctly engineer and certify highly automated systems such as.\nThe Desire to Learn From Accidents and Their Root Causes\n--------------------------------------------------------\nIn the title of, the word “severe” refers to the negative consequences *potentially* caused by errors we want to remove using. The more severe the potential consequences of an error, the more critical is its early removal. The usefulness of thus positively correlates with their support in the removal of critical errors. However, the estimation of severity often also requires the careful study of past field incidents .\nWe speak of *field incidents* to refer to significant operational events in the *field* (i.e.,the environment a technical system is operated) which are undesired because of their safety risks and their severe harmful consequences. Field incidents range from minor incidents to major accidents. It is important to separate the observed effect, the field incident, from its causes or, more precisely, from the *causal chains of events* leading to the observed effect. Hence, this analysis depends on the considered system perimeter . Depending on the possibilities of *observation* and the depth pursued in a, a conclusion on a possible cause can result in any combination of, for example,overall system failure, human error, adverse environmental condition, design fault, hardware fault, software fault, or specification error.[11](#fn11)\nThere are many databases about field incidents: some are comprehensive and include, others are less detailed, and some are confidential, depending on the regulations in the corresponding application domain or industry sector. Based on such databases, accident research, insurance, and consumer institutions occasionally provide brief root cause statistics along with accident statistics.[12](#fn12)\nAccident statistics allow certain predictions of the safety of systems and their operation, for examplewhether risk has been and will be acceptably low. Such statistics are also used in estimations of the amount of field testing necessary[13](#fn13) to sufficiently reduce risk .\nFor example,it is well-acknowledged that inadequately designed and implemented user interfaces are a significant contributory factor in computer-related accidents . But how and how fast have we arrived at this conclusion and how can we prevent future such incidents? Without proper empirical investigation of accident causes, such statistics are *of little use in decisions on measures for accident prevention* , particularly on improvements of engineering processes, methods (e.g.), and technologies (e.g. tools) used to build these systems. For this, we require more in-depth, possibly formal, and statistics that *relate error removal by and incident root causes*. To this extent, is a great opportunity for the investigation of effectiveness.\n reported about deficiencies of information gathering (e.g.) for databases on computer-related accidents back in the early 1990s. He noted that independent, well-known, but confidential databases might reduce under-reporting and were thus believed to improve safety culture. Reporting has not much improved as observed by. To understand the current situation, we studied a sample of 377 reports from open field incident databases (in aviation, automotive, rail, energy, and others) finding the following :\n1. in these reports were of poor quality, either because they were not going deep enough, economically or technically infeasible, or inaccessible to us.\n2. Particularly, root causes (e.g.software faults, specification errors) were rarely documented in a way that useful information about the technologies used (e.g.software) or consequences in the development process could be retrieved from the reports.\n3. Reports in some sectors (e.g.aerospace, rail, power plants, process industry) contain more in-depth than others (e.g.automotive) because of different regulations.\n4. Some sectors operate official databases (e.g.NHTSA[14](#fn14) and NTSB[15](#fn15) in the US transportation sector) and others do not (e.g.German road transportation sector).\n5. Our findings suggest that even in domains with regulated, reports in open databases tend to be less informative than reports in closed databases.\n6. The reports from the automotive industry exhibited a relatively small fraction of technology-related errors (e.g.software-related errors).\nTo validate our study and to better understand the context of our findings, we interviewed eight[16](#fn16) safety experts . One finding was that, because of an unclear separation of technologies and a lack of explicit architectural knowledge, a desirable classification of root causes is sometimes infeasible. Hence, accident analysts often close their reports with a level of detail too low to draw helpful conclusions. Additionally, one expert stated that *the hidden number of software-related or software-caused field incidents in dependable systems practice is likely much larger than the known number*. This matches our intuition but we are missing clear evidence.\n, a researcher involved in the further development of IEC 61508, demands regulations to mandate the use of systematic.[17](#fn17) In support of his view, we believe that rigorous in-depth based on can be helpful to gain clarity about actual root causes. Again, beyond this undesirable form of late error removal, RCA data is essential for the *measurement of the effectiveness* of error removal techniques, particularly.\nThe “Toyota unintended acceleration” incident exemplifies the difficulty of drawing conclusions without using powerful techniques: a first concluded that floor mats and sticky throttle pedals caused a fatal car mishap. A second carried out by NASA experts and based on *testing and automated static analysis* of the control system (i.e.,software and hardware) was not conclusive. A third [18](#fn18) based on *code reviews*—we could not find out which level of formal inspection was used—detected defects in the control software and safety architecture, demonstrated to be likely the causes of the accident .\nWe could invest in for RCA based on standardised data recording (e.g.aircraft black boxes), especially important for where human operators cannot practically perform incident response. Based on lessons from formal RCA, we could further invest in the employment of integrated formal methods in assurance and certification to prevent field incidents, major product recalls, and overly lengthy root cause investigations.\nThe Desire of Assurance to Form a Mature Discipline\n---------------------------------------------------\nIn his Turing Award acceptance speech in 1981, Tony reviewed type safety precautions in programming languages and concluded: “In any respectable branch of engineering, failure to observe such elementary precautions would have long been against the law” .\nInspired by this comparison, it can be helpful to look at other engineering disciplines such as civil, mechanical, or electrical engineering to identify transfer opportunities for. There, engineers use in many of their critical tasks. However, nowadays these methods are often hidden behind powerful software tools usable by qualified professional engineers. Although type systems, run-time bounds checking, and other variants of assertion checking have been frequently used in dependable systems practice, the overall level of adoption is still comparatively low.\nFor example,even in less critical mechanical engineering domains, vocationally trained engineers use computer-aided engineering, design, and manufacturing software. Whether for designing machine parts for serial production (i.e.,specification) or for calculations (e.g.dimensioning, force or material flow simulations) for these parts and their assembly (i.e.,for prototype verification), these engineers use tools based on canonical mathematical models.\nNowadays, drawings from computer-aided mechanical design carry at least two types of semantics: one declarative based on calculus for dimensioning (1), and one procedural for the synthesis of Computer-Numerical-Control programs for production machines processing materials to realise the drawings (2). Note that the unifying base of these two semantics is geometry, a well-studied mathematical discipline. Although higher levels of complexity demand more sophisticated analytical expertise, typically from engineers with several years of work experience, many tasks can be accomplished by less trained engineers using the corresponding tools.\nWhereas in computer-aided mechanical design both semantics seem to be used to a similar extent, in we observe that analogous semantics are rarely used even if tools are available, and less often we see (1) and (2) being consistently used. Low adoption might result from the semantics for dimensioning and production automation being usually less abstract than the semantics for verification (1) and synthesis (2) of computer programs. Accordingly, Parnas suggests a shift from correctness proof to property calculation to develop practical formal methods .\n*Patterns* have had a long history in many disciplines. In mechanical engineering, patterns are better known as *machine elements* and are particularly useful in high-reliability applications. Machine elements (and standardised forms thereof) have a stabilising impact on the outcome of an engineering project. The process of element selection and composition can take tremendous advantage not only from the *reuse* of proven design knowledge but also from the reuse of complex calculations (e.g.from gear transmissions, injection moulding tools, skeleton framings). Moreover, modern tools typically foster the use of *element libraries* and *parametric design*. Importantly, because the properties of such elements are in many cases well known, calculations for assemblies (i.e.,*compositional* verification) get relatively easy. However, the higher the required precision of these calculations, the more expensive is their computation.\nThese observations are in line with what we know from collaborations in robotics, like mechatronics, a discipline where many engineering domains have to play together well: are heavily used for the analysis of robot controllers and for various kinds of simulations and tests .\nDigital circuit engineering is a domain where such as model checking have been successfully applied decades ago. However, systematic hardware errors, such as Spectre and Meltdown, and the *unavailability of temporal specifications* of optimised operations (e.g.branch-prediction and speculative execution) discontinue the verifiability of recent computer architectures. This lack of verifiability of the assumptions (e.g.partitioning, information flow) about the execution platform complicates the verifiability of the software (e.g.an operating system) running on such a platform.[19](#fn19)\nDependability assurance has not yet successfully adopted as a vital part of their key methodologies. If seem relatively well established in other disciplines, we might also be able to successfully transfer to assurance and assurance in other domains. Beyond software design patterns, we could benefit from best practices in formal specification and development manifested in repositories of patterns. Moreover, we could aim at the further unification of established to provide common formal semantics for various domains.\nThe Desire for Adequate and Dependable Norms\n--------------------------------------------\nDependable systems practice usually includes the transition from what is called the *system specification* to an artefact called *system design*. Typically, *software and hardware specifications* are then derived from these two artefacts before delving into detailed technology choice and development. observe that safety culture in such a framework is more important than strict standards, but adequate standards and certification regimes can establish and strengthen the safety culture desirable in dependable systems practice.\nA striking finding in one of our recent discussions of dependable systems standards (e.g.IEC 61508, ISO 26262,[20](#fn20) DO-178C) is that *normative parts for specification* (i.e.,), for specification validation (i.e.,avoiding and handling requirements errors), *and for hazard and risk analysis* (particularly in early process stages) seem to be *below the state of the art* , despite several observations that significant portions of the *causes* of safety-critical software-related incidents fall into the category of *specification errors* .\nIf one identifies an error in the software specification, do causes of this error originate from an erroneous system specification? How are system, software, and hardware specifications (formally) related? Control software typically governs the behaviour of a whole machine. Hence, core components of a software specification might, modulo an input/output relation , very well govern the narrative of the overall system specification. Inspired by the issue of specification validity as highlighted in, do these standards provide guidance for checking the validity and consistency of core parts of system, software, and hardware specifications?\nThe literature provides plenty of evidence of undesired impacts of specification errors dating back as early as the investigations of and. As reported by, the 92% of computer-related field incidents caused by human-computer interaction also illustrate the gap between specifications and capabilities of humans to interact with automation. Despite these older figures, we are talking of one of the most critical parts of standards. Practitioners could expect to receive strong guidance from these parts. Moreover, the requirements to show conformance to these parts should not be vacuous.\nMany standards define *specific sets of requirements* (i.e.,for error removal and fault-tolerance) depending on the level of risk a system (or any part of it) might cause. The higher this level, the more demanding these requirements. Examples of demanding requirements include 3-4 (IEC 61508), C-D (ISO 26262), 3-4 (IEC 61508), A-B (DO-178C). Even for the highest such levels the mentioned standards only “highly recommend” but not mandate the use of.\n*Guidelines* for embedded software development such as MISRA:1994 *recommend* for 4, although MISRA:2004 no longer includes such information and instead refers back[21](#fn21) to MISRA:1994. As already mentioned, ISO 26262 as the overriding standard does not go beyond high recommendation of for D. reported in 2014 that, in the US, car manufacturers are not required to follow MISRA and that there are no other software certification requirements. This currently also applies to autonomous road vehicles.\nIn an interesting anecdote, reported on his lack of success in introducing systematic hazard (and risk) analysis methodology into normative parts of this standard . Moreover, he mentioned unsuccessful attempts to strengthen the role of in IEC 61508 and on the “broken standardisation” in assurance practice. In reaction to that, he proposed the use of evidently independent peer reviews to “dampen committee-capture by big-company bully players”.\nAdditionally, observed: “There is an expectation by the community that standards will embody the best available technology and that their presentation will allow determination of conformance to be fairly straightforward. A criticism that is seldom heard is that some standards are, in fact, technically flawed and poorly presented.” He exemplifies his critique by several issues with IEC 61508 and RTCA DO-178B and suggests to make the meaning of “conformance [or compliance] with a standard” more rigorous. Particularly, he encourages to replace *indirect* (i.e.,process-related) evidence (e.g.documentation of specification activities) in assurance cases by *direct* (i.e.,artefact-related) evidence (e.g.unsuccessful checks for presence of certain specification faults, successful checks for absence of implementation errors).[22](#fn22) With the observation in software quality control that “there is little evidence that conformance to process standards guarantees good products”, delivered a reasonable basis for ’s suggestions.\nRegarding the integration of dependability approaches and, had stated by 1993 that they “do not know how to combine formal methods assurance with metrics collected from other techniques such as fault-tolerance”. Is this still an issue? From a practical viewpoint, standards such as, for example,IEC 61508, ISO 26262, and DO-178C, provide recommendations about techniques for the reduction of both random hardware failures (e.g.by fault-tolerance techniques) and systematic hardware and software failures (e.g.by, static analysis, and testing). If can support the combined application of the recommended techniques and achieve an improvement in practice then we should really strive to demonstrate this.\nWe believe that critical fractions of strong direct evidence can be delivered through the use of. In support of ’s argument , we see a great opportunity for an assessment of how the corresponding guidelines in these standards can be extended and aligned with recent results in research.\nWe as researchers and practitioners could support the creation of adequate state-of-the-art regulations with improved guidance on specification construction and validation. That way, we could foster well-certified high-risk software in a time where dangerous autonomous machines are about to get widely deployed in our society.\nThreats to the Adoption of Integrated Formal Methods\n====================================================\n[sec:obstacles]\nThis section closes the *environmental part* of our analysis by identifying threats to transfer as well as challenges that arise from alternative or competing approaches taking the opportunities mentioned in. We also outline remedies to these threats.\nThe development of effective and their successful transfer into practice can be impeded by\n* a lack of agreement on a sound semantic base for domain-specific and cross-domain integration (),\n* missing support for widely used and established tools (),\n* a lack of interest in practical problems on the side of researchers (),\n* a lack of incentives for researchers to engage with current practice and for software practitioners to engage with recent theoretical results (),\n* a bad reputation among practitioners and applied researchers (),\n* proofs that are faulty or do not scale (),\n* the quest for soundness overriding the quest for usefulness ().\nWe discuss these threats and barriers in more detail in the following.\nDifficulties and Misconceptions of Unification\n----------------------------------------------\nAccording to , the successes and failures of semi-formal languages (e.g., ) suggest that, once wrapped in -based tools, get exposed to the *quest for a unified syntax*, one main objective of the movement in the 1990s. Rather than a unified syntax, it is more desirable to have a unified semantics and several well-defined mappings to domain-specific syntax wherever convenient (). This approach is occasionally taken up by in (). argued that one cannot achieve proper integration of methods and notations without a unifying semantics. This argument carries over to the problem of tool integration as already discussed in and revisited below. Particularly, the following challenges apply to when used in :\n1. the maintenance of a single source of information serving in the (automated) derivation of downstream artefacts (e.g.proof results, code via synthesis) ,\n2. a clear mapping between the presented to the engineer (using intuitive notation) and the semantics serving as the basis of formal verification,\n3. the embedding of a lean domain-specific formalism into a *common* data model suitable for access and manipulation by engineers through their *various* tools .\nThese challenges are complicated by irreducible unidirectionalities in automated transformations (e.g.model-to-code) limiting the desirable round-trip engineering (i.e.,the change between views of the same data).\nWe discussed as an assurance in. Likewise, *architecture description languages* (e.g.the Architecture Analysis & Design Language, EAST-ADL ) are for overall embedded system design. can be seen as one shortcut to the still ongoing efforts of arriving at a reduced version or a variant of where a semantics can be defined for the whole language .\nAt a higher level of abstraction, so-called *architecture frameworks* (cf.ISO 42010, e.g.the Department of Defense Architecture Framework) and *artefact and traceability models* have been proposed, aiming at the standardisation of specific parts of the systems and software engineering life-cycle and of the documentation and data models used there. These frameworks and models are similar to the models used in product data/life-cycle management in fields like mechanical or civil engineering.\nTo our best knowledge, no cross-disciplinary semantic unification has been undertaken yet (see ), serving as a basis for dependable systems engineering. Although many of these approaches have not been developed with the aim of formalisation and the unification of semantics, we believe that this effort has to be made when developing powerful.\nThe main threat is the lack of agreement on a sound semantic base for domain-specific and cross-domain.\nTo reduce this threat, integration and refinement-based software engineering could be better aligned with artefact models . This alignment may foster the unification of formal semantics to strengthen traceability among the artefacts and to aid in a variety of change impact analysis tasks in the engineering process .\nReluctant Integration Culture and Legacy Processes\n--------------------------------------------------\n*Tool integration* is about the integration of engineering, for example,tools for requirements specification, computer-aided software engineering, and computer-aided mechanical design. Among the wide variety of solutions to capture and track model data, the majority deals with linking or merging data models in one or another shallow way (e.g.software repositories, data exchange formats, product/engineering/application data or life-cycle management systems).\nSome tools with sustainable support are heavyweight, making it difficult to agree on lean model semantics, while others are proprietary, accompanied with interest in hiding model semantics. The surveys of,, and confirm that method and model integration have not yet been solved in,, and dependable systems practice. Moreover, frequent proposals by researchers to formalise fragments or variants of and have not yet received wide attention by practitioners and standardisation authorities.\n-based integrated development environments (e.g.using Xtext and Sirius) get close to what is suitable for -based tools. Such tools rely on a trusted representation of the formal semantics integrating the model data. For successful transfer to assurance practice, tools need to be built on a lean and open central system model .\nAn even greater barrier than loosely integrated tools are legacy *language and modelling paradigms*, an *established tool and method market* carried by *legacy stakeholders* and, possibly, a *neglected continuous improvement of education and training*.\nThe main threat is discontinuous and disintegrated education, transfer, and tool development.\nTo reduce this threat, continuous adaptation and improvement of education through teaching, of transfer through training, application, and feedback, and of tool development through regulated interface standards are necessary.\nReluctant Transfer Culture and Exaggerated Scepticism\n-----------------------------------------------------\nFinally, the vision of introducing into assurance practice might be hindered by a lack of researchers able or willing to engage with industrial assurance practice, as diagnosed by . It is certainly hard work to collect sufficient evidence for effectiveness in assurance practice because of intellectual property rights and other legal issues but also because of a lack of awareness among researchers . However, for credible method comparison experiments, recommended a sample of 20 similar projects split into two groups, 10 projects without treatment (i.e.,not using ) and 10 projects with treatment (i.e.,using ) to establish strong evidence (i.e.,evidence of level 5 or above ).\n[sec:scepticism] Exaggerated scepticism on the side of practitioners and applied researchers that has piled up over the years might be the most important barrier to cross. Early failures to meet high expectations on and transfer might have led to what can be called an “ Winter”. However, we think crossing a few other barriers first might make it easier to cope with scepticism in the assurance community and initiate an “ Spring”, at least in assurance practice. The recent successes with for certification of commercial medical devices , and the associated burgeoning field of for , are examples that could help to overcome this scepticism. Moreover, their is potential for transfer of these results to engineering, where the need for safety assured is also paramount .\nThe main threat is the reluctance of researchers to regularly engage in transfer efforts combined with an exaggerated scepticism of practitioners and with other mechanisms (e.g.lack of funding, poorly focused research evaluation, intellectual property rights) preventing both sides from engaging in an effective bidirectional transfer.\nTo reduce this threat, building awareness among researchers as well as stronger incentives (e.g.regulation) to provide continuous transfer funding for research (e.g.not relying on short-term projects like PhD theses) is needed. A good start on the academic side could then be a specific standardised repository of case studies.\nToo Many Errors in Proofs and Failure to Scale\n----------------------------------------------\nFrom the perspective of measurement, stated that “proofs of correctness sound useful, but [i] errors in the proofs themselves seem to be common failings not covered by the literature. Further, large applications may have thousands of provable algorithms, and [ii] the time required to prove them all might take many years”. For [i], the authors opposed 7% of erroneous bug repairs to up to 100% of erroneous proofs, though stating that the latter was based on an anecdote and there had been little data around. elaborated an example for [ii]: Assuming one provable algorithm per 5 function points[23](#fn23) and on average 4 proofs per day, Microsoft Windows 7 (160,000 function points) would have about 32,000 provable algorithms, taking a qualified software engineer about 36 calendar years. They highlighted that typically only around 5% of the personnel are trained to do this work, assuming that algorithms and requirements are stable during proof time.\n’s argument is based on the view of practical program verification as a largely manual activity. However, research in reasoning about programs has devised promising approaches to proof automation (e.g.bounded model checkers, interactive proof assistants). Assessing such approaches, compared rigorous but manual formal proof with mechanised or automated proof. He could not find a case where mechanised proof threw doubt upon an established mathematical theorem. observed that this underpins the robustness argument of for proofs as manual social processes rather than lengthy mechanical derivations (cf.). However, opposing ’s view, concluded that structuring of knowledge about computing ideas is the actual issue, and not the avoidance of mechanisation. Furthermore, mitigated ’s issue of the reality gap in program verification by referring to researchers’ continuous efforts in verifying hardware, programming languages, and compilers in addition to individual programs. Finally, in support of ’s concerns, highlighted the validity of specifications, manual and automated proofs rely on, as a serious issue indeed having received too little attention from researchers (cf.).\nThe main threat is the lack of qualified personnel to cope with the required amount and type of proof.\nWe believe, one angle of attacking this threat is the use of proof assistants *well-integrated* with common environments for requirements specification and software development combined with continuous research and education in the corresponding methods and tools.\nFailure to Derive Useful Tools\n------------------------------\nBeing loosely related to erroneous proofs, the *information overload through false-positive findings of errors* is a well-known problem in static program analysis. Semi-formal pattern checkers,[24](#fn24) such as PMD and FindBugs, are exposed to this threat . Additionally, FM-based verification tools, such as Terminator and ESC/Java , can be unable to correctly report all potential problems, because they are bounded and therefore unsound. While such tools can be helpful, confronting developers with many irrelevant findings (i.e.,false positives) or a high risk of critical misses (i.e.,false negatives) can lead to decreased use of -based tools.\nrelates the two information retrieval metrics, *precision* and *recall*, with two adequacy criteria of proof calculi, *soundness* and *completeness*. Precision denotes the ratio of actual (solid circle) to correct (true positive findings in the hashed area) findings, while recall measures the ratio of correct to all theoretically correct (dotted circle) findings. The ideal value for both metrics is 1, meaning that there are only true negatives and true positives. Completeness, although unachievable for richer theories, would correspond to recall and soundness would correspond to a precision of 1.\nOn the one hand, the usefulness of the calculi underlying is directly proportional only to their completeness and (traditionally) expires with a precision of less than 1. In other words, we usually try to avoid calculi that allow the derivation of false theorems. On the other hand, semi-formal pattern checkers have shown to have a wide, sometimes unacceptable, range of precision and recall of their findings. The usefulness of practical -based tools might lie somewhere in the middle between classical calculi and bug finding tools with poor precision/recall values.\nThe main threat is the struggle of academia and tool vendors to provide adequate tools, suited to adapt to novel scientific insights, to be integrated with other tools, and to be maintained in a flexible and independent manner.\nTo reduce this threat, an improvement of education and strong incentives can play an important role here (cf.).\n[dim/.style=align=flush left, tit/.style=align=left,anchor=south west, itm/.style=align=center,minimum height=3em,minimum width=3cm,fill=gray!20, lab/.style=align=center]\n(0,0) circle [radius=40pt]; (1,0) circle [radius=30pt];\n(1,0) circle [radius=30pt]; (0,0) circle [radius=40pt];\n(TP) at (2.5,1) true \npositive; (TN) at (-2.5,0) true \nnegative; (FP) at (-.8,0) false \npositive; (FN) at (2.5,-1) false \nnegative; (prcn) at (-2,1) lack of \nPrecision; (recl) at (0.7,0) Recall; (prcn) – (-0.5,0.5); (FN) – (1.5,-.5); (TP) – (1,.5);\n[t]\nXX **Method Strengths: raise the potential of …**\n* improvement of models, specification of requirements, automation of verification ()\n* early detection of systematic errors in designs ()\n* helpful abstractions to inform engineers about critical properties ()\n* integration and coordination of several to consistently reason about interrelated properties ()\n**Community Strengths: research can rely on …**\n* many transfer re-entry points from former case studies in industrial and academic labs ()\n* many assurance practitioners who perceive usefulness as positive ()\n* research designs for comparative studies of effectiveness ()\n& **Method Weaknesses: have suffered from …**\n* being difficult to learn and apply, many assurance practitioners perceive ease of use of as negative ()\n* low effectiveness of formal models because of the reality gap ()\n* fragile effectiveness and productivity in engineering ()\n**Community Weaknesses: progress has been hampered by …**\n[t]\n* no agreed framework for integration of ()\n* lack of convincing evidence of effectiveness in engineering ()\n* research ineffectively communicated in teaching/training ()\n \n**Key Opportunities for transfer and progress:**\n* The desire for early removal of erroneous behaviour and model-based assurance ()\n* The desire to learn from accidents and their root causes ()\n* The desire of assurance to be a mature discipline ()\n* The desire for adequate and dependable norms ()\nThe analysis provided in this table is an enhancement of the general analysis in in. & **Method Threats: research is threatened by …**\n* misconceptions of semantic unification in assurance ()\n* not scaling up to industry-size ()\n* faulty, tedious, or vacuous proofs ()\n* poor integration with engineering tools, processes, and education ()\n**Transfer Threats: transfer is threatened by …**\n[t]\n* a lack of roboticists’ education in ()\n* a lack of researcher engagement in transfer to practice ()\n* a lack of comprehensive access to quality data from practice ()\n \nA Vision of Integrated Formal Methods for Assurance\n===================================================\nThe following discussion applies to many domains of dependability assurance. However, the complexity of robots and autonomous systems forms a key opportunity for the progress of research and for its successful transfer. Accordingly, summarises the discussion in with an interpretation into assurance practice. Based on the strengths and opportunities described in the, we formulate our vision in terms of working hypotheses:\n1. From : Software tools for the construction of arguments and production of evidence using *can meet the challenge* of assuring safe. Computer-assisted assurance cases supported by heterogeneous formal models will increase confidence in their sufficiency, and also aid in maintenance and evolution through modularisation of arguments and evidence.\n2. From :, in particular modern verification tools, will enable *automation of the evidence gathering process*, and highlight potential problems when an assurance case changes or when an incident occurs.\n3. From : There is *no stable path to assured autonomy without the use of*. Acceptable safety will be much more likely achieved with than without them.\n4. From : The success of depends on the ability to *integrate a variety of* for different aspects of (e.g., safety-security interaction, missing human fallback, environment/world modelling, uncertain prediction/behaviour), which is not currently possible.\n5. From : Sophisticated techniques for *model integration and synchronisation are necessary* to support with. This way, will make it easier to express consistent models covering all relevant aspects, make their modelling assumptions explicit, and improve future assurance practices.\n6. From : can be beneficial in the short term. However, an important engineering principle is to be conservative and, therefore, not to change procedures unless there is compelling evidence that are effective. Such evidence can be delivered through *empirical research* and collaboration of academia and industry. That evidence *is required to re-evaluate research and foster research progress and transfer*.\n7. From : The demonstration of cost effectiveness in addition to technical effectiveness of new is necessary to justify further research.\n8. From : Norms are a lever of public interest in dependability . Current norms seem to deviate from the state of the art and may fail to guarantee product certification procedures that satisfy the public interest.\nassigns these hypotheses to the relationships between foundational and transfer-directed research by example of the domain. Overall, we believe that have great potential and can improve assurance but practitioners do not use them accordingly.\nWe could take and enhance credible measures to convince assurance practitioners of our results and effectively transfer these results. For this to happen, we have to answer further research questions.\n[dim/.style=align=flush left, tit/.style=align=left,anchor=south west, itm/.style=align=center,minimum height=3em,minimum width=2cm,fill=gray!20, lab/.style=align=center]\n(ifm) at (0,0) integrated \nFormal Methods (); (efm) at ((*i**f**m*) + ( − 8*e**m*, − 10*e**m*)) Empirical Research \nof Formal Methods; (ras) at ((*i**f**m*) + (8*e**m*, − 10*e**m*)) Assurance Practice in \nRobots and Autonomous \nSystems ();\n(ifm) edge[above,sloped,bend left] node[lab] (H1-3) have potential \nto improve (ras) (efm) edge[above] node[lab] observes (ras) (ras) edge[below,sloped] node[lab] (H4-5) pose chal- \nlenges to (ifm) (efm) edge[above,sloped,bend left] node[lab] (H6-8) examines the \neffectiveness of (ifm) ;\nEmpirical, Applied, and Foundational Research\n=============================================\nBased on the aforementioned working hypotheses, we state several objectives for foundational and transfer-directed research, formulate research questions, and show our expectations on desirable outcomes of such research.\nResearch Objectives and Tasks\n-----------------------------\nTo validate and transfer our research results, we need to\n* evaluate how assurance case construction and management for certifiable can be improved by .\n* debunk or justify arguments against the use of or -based tools in assurance.\n* foster a culture of successful research transfer to industries performing assurance.\nTaking an foundational point of view, we need\n* foundational research on the integration and unification of to tackle the complexity of .\n* a unified semantic foundation for the plethora of notations in assurance, to enable method and tool integration. There are a number of promising research directions still being investigated .\nTaking an evidence-based point of view, as already highlighted in 1993 by , we need to\n* understand the difference between the state of assurance practice and assurance research.\n* understand in which ways current assurance practices fail, and suggest effective approaches from assurance research. In this way, we can be sure that assurance practice is equipped with state-of-the-art assurance technology for defence against potential liability claims, and that assurance practitioners do not fail in fulfilling their obligations.\n* understand how results from assurance research can be validated to be sure that research follows promising directions with high potential of success in assurance practice.\nBased on that, we need to\n* set concrete directions for empirical research in assurance.\n* train researchers in applying empirical research designs in their work on rigorous assurance cases. corroborated this objective by saying that “formal methods champions need to be aware of the need to measure costs”.\n* avoid biases as found in various branches of scientific research. For example,in the social and biomedical sciences, researchers identified such biases through meta-analyses and suggested measures for bias avoidance .\n* increase the level of evidence of research to level 2 according to the hierarchy in .\n* avoid knowledge gaps about whether (a) practice is keeping up with the state of the assurance art, and (b) whether recent academic or industrial research is going in the right direction. In this way, we can be sure that we do our best to inform and serve the society.\nUsing appropriate research designs, we need to\n* invite the industry to enhance their efforts in engaging with recent research.\n* foster goal-oriented interaction (a) between assurance practitioners and researchers and (b) between researchers and assurance researchers. In this way, we can be sure to do everything to keep researchers up to date with respect to practical demands.\n* join the research and applied assurance research communities ( on page ), both vital for the progress and transfer of assurance research into assurance practice. This way, we can be sure to foster necessary knowledge transfer between these two communities.\n* summarise achievements in practical applications of for constructing assurance cases.\n* suggest improvements of curricula for assurance.\n* guide the industry in process improvement, training, and tool support.\n* guide vendors of -based assurance tools to assess and improve their tools and services.\nSome Research Questions addressing these Objectives\n---------------------------------------------------\nThe research questions below are relevant for in general. We consider these questions as crucial to be answered for assurance to address the aforementioned objectives:\n1. What is the true extent of computer-related accidents up to 2019 ? What would these figures mean for the domain?\n2. Does the use of formalism detect severe errors to a larger extent than without the use of formalism ?\n3. Does the use of formalism detect severe errors earlier than without using formalism?\n4. Why would such errors be a compelling argument for the use of?\n5. Apart from error avoidance and removal, which other benefits of in practice are evident and can be utilised for method trade-offs?\n6. Which criteria play a central role in measuring effectiveness? Particularly, how can -based tools be used at scale?\n7. How would Commercial-off-the-Shelf and System-Element-out-of-Context verification by pay off?\n8. Which hurdles need to be overcome to use in practice to the maximum benefit?\n9. How do we know when these hurdles are actually overcome?\n10. How can (from different disciplines) be used together (, unification)?\n11. How can be used to assure systems (e.g.computer vision in road vehicles) involving techniques like machine learning and deep neural networks?\n12. How can be integrated into assurance cases to support certification against international safety and security standards?\n13. How can we best combine formal and informal methods? For example,how can we deal with the issue of the validity of specifications that proofs rely on.\n14. How can we best present formal requirements, evidence, and artefacts in an assurance case?\n15. How can empirical research help in successfully demonstrating the capabilities of for rigorous and certifiable autonomy assurance?\nThis list of research questions can easily be extended by further more detailed empirical questions from the settings discussed in.\nSome Envisaged Research Outcomes\n--------------------------------\nOur vision of *rigorous assurance* implies foundational research to result in\n* novel semantic frameworks unifying best practice methods, models, and formalisms established in\n* new concepts for -based development environments\n* new computational theories to support formal modelling and verification of\n* evaluations of\n\t+ assurance tools, languages, frameworks, or platforms used in practice regarding their support of\n\t+ the integration of into modelling and programming techniques, assurance methods, and assurance processes\n\t+ languages for linking informal requirements with evidence from\n\t+ (automated) abstraction techniques used in assurance and certification\n* opinions, positions, and visions on integration and unification for rigorous practical assurance.\nOur vision of *rigorous assurance* implies applied and empirical research to result in\n* comparisons of\n\t+ projects applying in assurance practice with similar practical projects applying non- approaches\n\t+ -based (embedded software) assurance with assurance approaches in traditional engineering disciplines\n* checklists, metrics, and benchmarks (for and beyond tool performance) for\n\t+ the evaluation and comparison of -based assurance approaches (e.g.confidence level)\n\t+ relating error removal and incident root cause data (e.g.efficiency and effectiveness in removal of severe errors or in avoidance of severe accidents, cf.)\n\t+ usability and maturity assessment of (e.g.abstraction effort, proof complexity, assurance case complexity, productivity)\n\t+ the evaluation of budget cases (cf. in electronic hardware development).\n* experiences in or surveys (e.g.systematic mappings and reviews of assurance case research, interview studies with assurance practitioners, cf.) of\n\t+ transfers and applications (e.g.case studies in assurance and certification projects)\n\t+ challenges, limitations/barriers, and benefits of in assurance and certification projects,\n* research designs (e.g.for controlled field experiments) for the practical validation of in assurance and certification projects\n* opinions, positions, and visions on future research, education, and training in the use of in assurance and certification.\nSummary\n=======\nAlong the lines of, we analysed strengths, weaknesses, opportunities, and threats to determine the potential of integrated formal methods to improve the practice of dependability assurance. Emphasising robots and autonomous systems as an area in the spotlight of dependability assurance, we express our expectations of research progress and transfer. From these expectations, we derived a research and research transfer agenda with the objective of\n[(i)]\nenhancing the foundations of integrated formal methods,\ncollecting evidence on the effectiveness of integrated formal methods in practice,\nsuccessfully transferring integrated formal methods into the assurance practice, with a short-term focus on robots and autonomous systems, and\nfostering research progress, education, and training from the results of this transfer effort.\n#### Acknowledgements\nThis work is partly supported by the Deutsche Forschungsgemeinschaft (DFG) under the Grant no. 381212925 and the EPSRC projects CyPhyAssure,[25](#fn25) grant reference EP/S001190/1, and RoboCalc, grant reference EP/M025756/1. We are deeply grateful to Michael Butler, Ana Cavalcanti, John Fitzgerald, Cliff Jones, and Peter Gorm Larsen for helpful feedback on previous versions of this manuscript and for their genuine support in shaping the direction of this work.\nStudying Formal Method Practitioners: A Ballpark Figure\n=======================================================\nIn this section, we make a brief excursion to the relationship between and, closing with a rough estimate of the size of the population of users and the meaning of such an estimate for the empirical study of that population. Such studies include cross-sectional and longitudinal surveys based on questionnaires or semi-structured interviews.\n#### Formal Inspection versus Formal Methods\nencompasses a variety of techniques (e.g.peer reviews, walk-throughs) where critical process artefacts (e.g.program code) are checked (e.g.manually or using software tools) against a variety of criteria (e.g.checklists), usually by a group of independent qualified engineers. For the sake of simplicity of the following discussion, we assume that can be seen as a particularly rigorous variant of where formal specifications serve as a particular way of formulating checklists.\nWe compare the use of with the use of. According to, formal inspections are used in more than 35% of commercial defence, systems, and embedded software projects, and are estimated to be applied in less than 1% of overall commercial software engineering projects. To get an idea of this coverage data, we perform an analysis of the global embedded software market based on other global software market indicators in. We found estimates of systems and software professionals world-wide and estimates of annual US business values.[26](#fn26)\nA uniform distribution would entail roughly 37000 USD/year per person in the general software domain and 10000 USD/year per person in the embedded software domain. Clearly, geographically strongly differing salaries and part-time engagement rule out a uniform distribution, yet providing figures helpful for our purposes.\nNext, we apply the following proportions: From a worldwide population of around 18.5 million software developers in 2014 , about 19% live in the US, 10% in China, 9.8% in India, 36% Asia/Pacific region, 39 % live in Europe, the Middle East, and Africa; and 30% in the Americas.[27](#fn27) The *design to quality assurance* (i.e.,verification and test) *cost ratio* is observed to be approximately 30 : 70.[28](#fn28) About 20% of embedded software personnel are quality assurance engineers (i.e.,test, verification, or validation engineers).[29](#fn29)\n1.0p3.5cm|XXXXllX *Global market/project indicators* [Unit] & *Professional engineers / developers* [million] & *Ann. business value* [billion USD/year] & *Quality assurance personnel* [million] & *business value* [billion USD/year] & [%] & [%] & *Devices* [billion / year] \nGeneral IT hardware and devices (incl. personal computers) & & 2018: 689 & & & & & 2010: 10 \nEmbedded systems (hardware, software, connected embedded devices) in all domains & 2014: 1.2 & 2009: 882018: (161)*a* & & & (35) & & 2010: 9.8 \nIndustrial embedded systems & & & & & & & 2016: 2 \nDefence, systems, and embedded commercial software engineering & & & & & 2011: 35 & \nEmbedded software & 2014: 0.36 & 2009: 3.42018: (10) & **2010: (0.072)** & **2010: (2.38)** & & \nGeneral software (overall commercial software engineering, development) & 2014: 18.52018: 232019: 23.9 & 2013: 4072018: (515) & & & (4.6) & 2011: 1 & \nThe estimates in suggest that around 2% of the overall pure software market are allocated to the embedded pure software market. 35% coverage of formal inspection in about 13.5% of the overall software market (161/(689 + 515) = 0.134) would result in roughly 4.7% coverage of all software projects by formal inspection versus at most 1% coverage by. However, from this data we can hardly know whether rates of use get close to or beyond 10% in high-criticality systems projects.\nAssuming that in about 35% of embedded software projects the quality assurance personnel would use formal inspection and that in every fifth (1 : 4.7) of such projects formal methods would be used, the current population of regular practical users would globally amount to about 5040 ( = 72, 000 \\* 0.34 \\* 0.20) persons. Note that these numbers are rough estimates. However, we believe their order of magnitude is realistic. Given that these persons would on average earn about 100,000 USD/year each, we would speak of an annual business value of around USD 504 million.\n#### Empirical Studies of and Practitioners\nImportantly, from these data we can determine minimum sample sizes for surveys. For example,assume we want to have 95% confidence in our test results and are fine with a confidence interval of ± 7%. Then, for *regular practical users*, a population of the size of 5040 persons would require us to sample 189 independent data points (e.g.questionnaire responses). The population of *regular practical users*, 25200 ( = 72, 000 \\* .35) would imply a minimum sample size of 194.\nFor any sample of such size, any survey has to argue why the sample *represents* the population. This step depends on the possibilities given during the sampling stage. Obviously, reaching out to 189 out of 5040 persons whose locations might be largely unknown is an extremely difficult task that might only be tackled in terms of a global group effort among researchers and practitioners. Given that this challenge can be tackled, these figures determine the type of surveys required for the collection of confident evidence. Moreover, these figures limit the manual effort to perform and repeat valuable qualitative studies to a reasonable amount.\nFormal Methods in Dependable Systems Engineering\n================================================\non page depicts our research setting for this survey. There, we describe as an enhancement of and the assurance of as a special field of dependable systems engineering. Because many of the discussed issues are of a more general nature, we first prepared a analysis of in dependable systems engineering in. Then, we refined this analysis in to accommodate specifics of for.\nXX **Method Strengths:**\n[t]\n* Improvement of modelling precision, requirements clarity, verification confidence\n* High error detection effectiveness, early error removal\n**Community Strengths:**\n* Many transfer re-entry points from former case studies with industry\n* Many dependable systems practitioners perceive usefulness as positive\n& **Method Weaknesses:**\n* Difficult to learn and apply, many dependable systems practitioners perceive ease of use of as negative\n* Fragile effectiveness and productivity\n**Community Weaknesses:**\n* Lack of compelling evidence of effectiveness\n* Ineffectively communicated in teaching and training\n \n**Key Opportunities for Transfer and Research:**\n* Desire for early removal of severe errors ()\n* Desire to learn from accidents and their root causes ()\n* Desire to be a mature discipline ()\n* Desire for dependable norms ()\n& **Method Threats:**\n[t]\n* Lack of method scalability\n* Faulty, tedious, or vacuous proofs\n* Lack of user education\n* Poor tool integration, legacy tools and processes\n**Transfer Threats:**\n* Lack of researcher engagement in transfer\n* Lack of access to comprehensive high-quality data\n \n---\n1. For: Functional Safety of Electrical/Electronic/Programmable Electronic Safety-related Systems.[↩](#fnref1)\n2. For: Software Considerations in Airborne Systems and Equipment Certification.[↩](#fnref2)\n3. We reuse the term from the homonymous conference series .[↩](#fnref3)\n4. The SCADE Design Verifier (<http://www.esterel-technologies.com>) and the seL4 microkernel (<http://sel4.systems>) represent good although less recent examples.[↩](#fnref4)\n5. Tokeneer Project Website: <https://www.adacore.com/tokeneer>.[↩](#fnref5)\n6. Pacemaker Formal Methods Challenge: <http://www.cas.mcmaster.ca/sqrl/pacemaker.htm>.[↩](#fnref6)\n7. “Formal” refers to the formality of the procedure but does not imply the use of formal semantics or mathematical techniques.[↩](#fnref7)\n8. See <https://www.york.ac.uk/assuring-autonomy/body-of-knowledge/>.[↩](#fnref8)\n9. See <https://www.omg.org/spec/SACM/About-SACM/>.[↩](#fnref9)\n10. For example,the fatal accident involving a Tesla advanced driving assistance system, <https://www.theguardian.com/technology/2018/jun/07/tesla-fatal-crash-silicon-valley-autopilot-mode-report>.[↩](#fnref10)\n11. Specification errors are also called development failures and can be seen as flaws in the process of requirements validation.[↩](#fnref11)\n12. See for a list of accident databases from such institutions.[↩](#fnref12)\n13. For example,according to “As Low As Reasonably Practicable” or “So Far As Is Reasonably Practicable”. See <http://www.hse.gov.uk/risk/theory/alarpcheck.htm>: “Something is reasonably practicable unless its costs are grossly disproportionate to the benefits.”[↩](#fnref13)\n14. National Highway Traffic Safety Administration, <https://www.nhtsa.gov>.[↩](#fnref14)\n15. National Transportation Safety Board, <https://www.ntsb.gov>.[↩](#fnref15)\n16. These experts have experience with tool qualification (1) and safety-critical systems in aviation (3), railway (1), automotive (4), and energy systems & turbines (1). We have to keep their names confidential.[↩](#fnref16)\n17. From personal communication.[↩](#fnref17)\n18. See expert interview by embedded software journalist from EE Times in 2013 on <https://www.eetimes.com/document.asp?doc_id=1319903&page_number=1>.[↩](#fnref18)\n19. See blog post on the seL4 microkernel, <https://research.csiro.au/tsblog/crisis-security-vs-performance/>.[↩](#fnref19)\n20. For: Road Vehicles – Functional Safety.[↩](#fnref20)\n21. This might also the case for MISRA:2012 from March 2013. We are unaware of the opposite but were also unable to receive a copy of this version.[↩](#fnref21)\n22. While formal verification serves the check of absence of property violations, conventional testing can only serve as a check of presence of such violations.[↩](#fnref22)\n23. A function point is a measure of the conceptual complexity of an IT system relevant for the estimation of the amount of work required to engineer this system.[↩](#fnref23)\n24. Also called bug finding tools.[↩](#fnref24)\n25. CyPhyAssure Project: <https://www.cs.york.ac.uk/circus/CyPhyAssure/>.[↩](#fnref25)\n26. See <https://en.wikipedia.org/wiki/Software_industry> and <https://softwareengineering.stackexchange.com/questions/18959/what-proportion-of-programming-is-done-for-embedded-systems>.[↩](#fnref26)\n27. See <https://adtmag.com/Blogs/WatersWorks/2014/01/Worldwide-Developer-Count.aspx>.[↩](#fnref27)\n28. See <https://www.slideshare.net/pboulet/socdesign>.[↩](#fnref28)\n29. See <https://de.slideshare.net/vdcresearch/searching-for-the-total-size-of-the-embedded-software-engineering-market>.[↩](#fnref29)\n"},"data_id":{"kind":"string","value":"arxiv_0003277"}}},{"rowIdx":796,"cells":{"content":{"kind":"string","value":"**Operator space tensor norms**\n===============================\n[1](#fn1)\nnamedefsubjclassname@2020 Mathematics Subject Classification\nThe use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. The aim of this manuscript is to kick off the development of a *systematic* theory of tensor products and tensor norms for operator spaces and its interplay with their associated mapping ideals. Based on the theory of tensor products in Banach spaces, we provide the corresponding natural definitions in the operator space framework. The theory is not a mere translation of what is known in the classical setting and new insights, techniques, ideas or hypotheses are required in many cases. As a consequence, notable differences in the theory appear when compared to the classical one.\nIntroduction\n============\nIn the thirties von Neumann in collaboration with Murray began a “mathematical quantization program”, aiming to formulate an operator version of integration theory. They were successful, with the key idea being to replace function spaces by \\* -algebras of bounded operators on Hilbert spaces. This was the seed that started a long road of “noncommutative analysis” which lead to what is now known as the theory of operator spaces. Along this road and prior to the formalization of this theory, many concepts which are now typical in this framework (e.g., completely bounded mappings) were introduced in the context of the *C*\\*-algebras.\nAn operator space is a Banach space $\\textbf X$ with an extra “matricial norm structure”: in addition to the usual norm on $\\textbf X$, we have norms on all the spaces $M\\_n(\\textbf X)$ of *n* × *n* matrices with entries from $\\textbf X$ (often called the matrix levels), where these norms must satisfy certain consistency requirements. The connection with operator algebras comes from the fact that when $\\textbf X$ is a subspace of an algebra B(*H*) of bounded linear operators on a Hilbert space *H*, there is a canonical norm on $M\\_n(\\textbf X)$ coming from the identification between *M**n*(B(*H*)) and B(*H**n*). The natural morphisms in this category are no longer just bounded maps, but the *completely bounded ones*: they are required to be uniformly bounded on all the matrix levels. Operator spaces are thus a quantized or noncommutative version of Banach spaces, giving rise to a theory that not only is mathematically attractive but it is also naturally well-positioned to have applications to quantum physics. The systematic study of operator spaces begun with Ruan’s work, and was deepened mainly by Effros and Ruan, Blecher and Paulsen, Pisier and Junge (see the monographs and the references therein). Operator spaces have often provided an appropriate framework to develop many areas of non-commutative analysis.\nIt is quite natural to investigate to what extent the classical theory of Banach spaces can be translated to the noncommutative context, and this has been extensively studied throughout the years. Though some properties do carry over, many do not and these differences are one of the reasons making the new theory so interesting and rich. And even for those properties that do admit a generalization, the proofs are often different and require the development of new tools. Along these lines, the main goal of this monograph is to take the first step towards the systematic development of a complete theory of tensor products and tensor norms in the operator space setting and its connections with the parallel theory of their associated mapping ideals.\nIt should be noted that both ideals of operators and tensor norms have inspired important developments in the operator space setting. Noncommutative versions of nuclear, integral, summing, and other ideals of operators have played significant roles in, e.g.,, whereas various specific tensor norms for operator spaces (most notably the Haagerup one) have also played a significant role (see for a good summary).\nBlecher and Paulsen have showed that part of elementary and basic theory of tensor norms of Banach spaces carries over to operator spaces, initiating a “tensor norm program” for operator spaces further developed in. However, many of the abstract tools of operators ideals and tensor norms in the sense of do not appear to have been treated yet. Moreover, while it is clear to specialists that there are obstacles making it impossible to fully translate the classical tensor product theory into the operator space setting, it has not been highlighted exactly which portions do translate and which ones do not. Hence, we will focus on seeing to what extent the classical theory can be transferred. In many situations, this is not the case and additional assumptions about the structure of the operator spaces or the tensor norms involved are really necessary. A good example is local reflexivity: since all Banach spaces have this property but not all operator spaces do, it is well known that operator space versions of classical results will often need local reflexivity as an additional assumption. However, when it comes to tensor norms it turns out that in some cases we can still get operator space versions of some classical results without requiring local reflexivity, as long as the tensor norm satisfies an extra condition.\nAlthough the theory of operator spaces has its own peculiarities, the classical theory will be our reference and roadmap to obtain and develop non-commutative versions of the theory of tensor norms. Therefore, let us now recall briefly summarize the history of the classical theory’s development. Tensor products burst into the world of functional analysis in the late thirties with the works of Murray, von Neumann and Schatten, but it was Grothendieck who first truly revealed the richness of properties for tensor products. His famous article *“Résumé de la théorie métrique des produits tensoriels topologiques”* established the basis of what we know today as ‘local theory’ — that is, the study of Banach spaces in terms of their finite-dimensional subspaces — and showed the relevance of the use of tensor products in the theory of normed spaces. All this helped him to establish a very fruitful theory of duality.\nWhile nowadays the *“Résumé”* is considered foundational work in the area, this masterpiece remained mostly unnoticed for many years. It was not until the end of the sixties when it started to be better valued by the scientific community. Lindenstrauss and Pełczyński presented some important connections and applications to the theory of absolutely *p*-summing operators, translating results written in terms of certain tensor norms by Grothendieck into properties of operator ideals. On the other hand, almost simultaneously, a general theory of operator ideals on the class of Banach spaces was developed by Pietsch (and his school in Jena) but without the use — and the language — of tensor norms. Pietsch’s book *“Operator Ideals”*, which appeared at the end of the seventies, led this theory to become one of the central themes of study in Banach space theory.\nAt the time functional analysts generally chose the language of operator ideals over that of tensor products, so the former received more attention. The theory of tensor products in Banach spaces became stronger and more ubiquitous during the eighties with the seminal work of Pisier, which showed the usefulness of these tools.\nWith time, the theory of tensor norms became an interesting field in its own right. Defant and Floret, in their famous monograph *“Tensor Norms and Operator Ideals”*, made it clear that the theory of tensor products and the theory of operator ideals are two sides of the same coin. Their work had a tremendous legacy: nowadays authors use indistinctly both languages. This same approach has also been developed in various other books on tensor products of Banach spaces.\nAs stated in, “both theories, the theory of tensor norms and of norm operator ideals, are more easily understood and also richer if one works with both simultaneously”. With this in mind, certain problems are more easily handled using either the *categorical perspective* due to Pietsch, or Grothendieck’s *cycle of ideas on tensor products*. With the present work, it is our intent to bring the benefits of the aforementioned parallel perspectives to the operator space setting. As mentioned above both the approaches of ideals and tensor norms have been repeatedly used in the operator space literature, but there has not been a methodical exploration of the connections between the two. It is worth mentioning that this work does not pretend to be an extensive monograph like. Our purpose is much more modest: initiate a program that, over time, can be enriched with the contributions of the operator space community. Thus, numerous topics covered in have been left out. Some expected results that do not appear in the literature can be found in this monograph, and also some new insights that do not have a counterpart or differ from the theory of Banach spaces.\nThe manuscript is organized as follows. Section [Basic definitions] is devoted to background material. We set some notation, give basic definitions that will be needed throughout the manuscript (in particular what we mean by an operator space tensor norm, or o.s. tensor norm), and present some known examples of operator space tensor norms. Also, we introduce a new family of Haagerup-style tensor norms which are examples of *λ*-tensor products (developed in ). Additionally, we emphasize the main relevant differences between the operator space and the Banach space frameworks which have an impact in the study of tensor norms in both settings.\nIn Section [hulls] we deal with the notions of finitely and cofinitely generated o.s. tensor norms. The former essentially means that the tensor norm can be approximated using finite-dimensional subspaces, while the latter means we can approximate it using quotients over subspaces of cofinite dimension. We also define the finite and cofinite hull of a given o.s. tensor norm.\nIn Section [fivebasic] we present the “Five Basic Lemmas” (see Section 13 in ) for the operator space setting. These are the *Approximation Lemma*, the *Extension Lemma*, the *Embedding Lemma*, the *Density Lemma* and the $\\mathcal L\\_p$-*Technique Lemma*. To prove some of these results, unlike the classical case, we additionally need certain strong hypotheses such as local reflexivity. However, in the case of *λ*-o.s. tensor norms this extra assumption is often not necessary.\nIn Section [dual section] we deal with dual tensor norms and develop an operator space version of the *Duality Theorem*. We also introduce right and left-accessible tensor norms and prove that any *λ*-o.s. tensor norm is accessible. Further, in order to prove some duality relations we define the notion of local accessibility, a weaker version of accessibility which involves locally reflexive operator spaces.\nIn Section [complete bap] we discuss completely metric and completely bounded approximation properties and relate these properties with the weak density of a multiple of *B**E*ʹ ⊗ min*F* in *B*CB(*E*, *F*). Also, we consider a weak completely bounded approximation property which provides an accurate characterization in terms of the duality between the proj and the min o.s. tensor norms.\nIn Section [mapping ideals] we discuss the notion of mapping ideal (defined in ) and present some examples already studied in the literature. To maintain the core of the Banach space concept and allow the natural relationship with tensor norms, we have to slightly strengthen the definition of mapping ideal from and add an extra condition. Also, we introduce some typical procedures: injective/surjective hulls of a given mapping ideal, the dual mapping ideal, and present some properties.\nSections [represent sec] and [minimal sec] deal with maximal and minimal mapping ideals and their *Representation theorems*. As the names indicate, maximal and minimal ideals associated to a given tensor norms are the largest and smallest with respect to the inclusion which coincides over finite dimensional spaces. The representation theorem for maximal mapping ideals shows how the ideal can be seen as a dual of a tensor product. Again, the hypothesis of local reflexivity is needed here in order to obtain a full characterization. As a consequence, we prove an *Embedding Theorem* (which gives a natural completely isometric inclusion from a tensor product into the mapping ideal) and show how dual and adjoint ideals behave in terms of their associated tensor norms. For minimal ideals, the representation theorem provides a natural completely quotient mapping from a tensor product onto the mapping ideal. Therefore, in a sense, both theorems (for maximal and minimal ideals) relate a mapping ideal with a tensor product.\nIn Section [capsulas inyectivas], we give the definitions of right and left completely injective and completely projective hulls of an o.s. tensor norm. We examine some of their interesting properties and see when they are finitely generated. Also, we define (both sides) completely injective and completely projective hulls showing that in both cases left and right procedures commute.\nWe establish in Section [mapping accessibility sec] that, unlike what happens in the Banach space setting, accessibility is a required property of a tensor norm in order to obtain the usual relations between duality and injective/projective hulls. Accessibility is also needed to associate a left (or right) injective hull of a norm to the surjective (respectively, injective) hull of a mapping ideal. Based on the classical theory, we also provide a definition of accessibility for mapping ideals. As expected, right-accessible finitely-generated tensor norms are associated with right-accessible mapping ideals but, surprisingly, the left version of this result does not hold. We see that a left-accessible o.s. tensor norm might have an associated mapping ideal which is not left-accessible (although we show in Proposition [prop-equivalences-accessibility] that, in this case, the mapping ideal does have a weaker version of left-accessibility).\nSection [normasnaturales sec] deals with the structure of *natural* norms, in the sense of Grothendieck. We prove that some results from the Banach space theory about equivalences or inequalities between injective and projective hulls remain valid in this new context but the whole picture is undoubtedly substantially different. In the classical theory Grothendieck’s inequality (which for instance implies that there is not possible for a tensor norm to be both injective and projective) has a strong impact in the description of natural tensor norms. The existence of an injective and projective Haagerup tensor norm shows us that the situation in the operator space setting could not be at all a translation from the classical case. We use the term *proj family* for the set of norms obtained from the norm proj after taking left or right injective/projective hulls finitely many times. Analogously, we define the *min family*. Unlike the classical setting, these families are not intertwined: each member of the proj family dominates any member of the min family. We present incomplete pictures of both of these families, proving some dominations and non-equivalences and leaving a large list of open questions about them. On the other hand, we completely describe the list of all natural norms that come from applying to min or proj two-sided hull operations (injective or projective). Precisely, it consists of six o.s. tensor norms. Again, this differs from the Banach space case where there are actually four (see Section [twosided]).\nBasic definitions\n=================\nWe only assume familiarity with the basic theory of operator spaces, although we recall some elementary definitions. The books and are excellent references on this topic. Our notation follows closely that from. For technical reasons, unlike the general literature in the subject, we will not assume that our operator spaces are complete. Thus, to avoid confusion, we choose to call them *normed operator spaces* (when completeness is not assumed) and *Banach operator spaces* (when completeness is required). The letters *E*, *F* and *G* will always denote normed operator spaces, that is, normed vector spaces with an additional structure at the matricial level that we now explain.\nAs usual, *M**n*(*E*) will stand for the set of *n* × *n*-matrices of elements in *E* and for *a* ∈ *M**n*, *m* (*n* × *m*-matrices of complex scalars), ∥*a*∥ denotes its norm as an operator from ℓ2*m* to ℓ2*n*.\nGiven *x* = (*x**i*, *j*) ∈ *M**n*(*E*) and *y* = (*y**k*, *l*) ∈ *M**m*(*E*), we consider *x* ⊕ *y* ∈ *M**n* + *m*(*E*) to be the matrix $\nx\\oplus y=\\left(\n \\begin{array}{cc}\n (x\\_{i,j}) & 0 \\\\\n 0 & (y\\_{k,l}) \\\\\n \\end{array}\n \\right).\n$\n*E* is a *normed operator space* if, for each *n*, there is a norm ∥ ⋅ ∥*n* on *M**n*(*E*) satisfying:\n1. $\\|x\\oplus y\\|\\_{M\\_{n+m}(E)}=\\max\\big\\{\\|x\\|\\_{M\\_n(E)},\\|y\\|\\_{M\\_m(E)}\\big\\}$, for all *x* ∈ *M**n*(*E*) and *y* ∈ *M**m*(*E*).\n2. ∥*a**x**b*∥*M**n*(*E*) ≤ ∥*a*∥ ⋅ ∥*x*∥*M**m*(*E*) ⋅ ∥*b*∥, for all *x* ∈ *M**m*(*E*), *a* ∈ *M**n*, *m* and *b* ∈ *M**m*, *n*.\nIf *H* is a Hilbert space and $E\\subset \\mathcal B(H)$ is a subspace, there is a natural norm in *M**n*(*E*) given through the identification $M\\_n(\\mathcal B(H))=\\mathcal B(H^n)$, which endows *E* with a normed operator space structure. On the other hand, by Ruan’s representation theorem for any operator space *E* there exist a Hilbert space *H* and an inclusion $E\\subset\\mathcal B(H)$ which is an isometry for all the matrix levels.\nEvery linear mapping *T* : *E* → *F* induces, for each *n* ∈ N, a linear mapping *T**n* : *M**n*(*E*) → *M**n*(*F*) (called the *amplification of *T**) given by \n*T**n*(*x*) = (*T*(*x**i**j*)), for all *x* = (*x**i**j*) ∈ *M**n*(*E*). \nThe *completely bounded norm* of *T* is defined by \n$$\\|T\\|\\_{\\operatorname{cb}}=\\sup\\_{n\\in\\mathbb{N}}\\|T\\_n\\|=\\sup\\_{n\\in\\mathbb{N}} \\big\\{\\|T\\_n(x)\\|\\_{M\\_n(F)}:\\, \\|x\\|\\_{M\\_n(E)}\\le 1\\big\\}.$$ \nWe say that *T* is *completely bounded* if ∥*T*∥cb is finite and we denote by CB(*E*, *F*) the space of completely bounded mappings from *E* to *F*. This is an operator space with the structure given by the identification *M**n*(CB(*E*, *F*)) = CB(*E*, *M**n*(*F*)).\nWe say that *T* is a *complete isometry* if each *T**n* : *M**n*(*E*) → *M**n*(*F*) is an isometry and we write *E* = *F* to indicate that *E* and *F* are completely isometrically isomorphic.\nThe operator space structure of a dual space *E*ʹ comes from the identification *M**n*(*E*ʹ) = CB(*E*, *M**n*). The norm of a matrix in *M**n*(*E*ʹ) can also be computed through the *matrix pairing* ⟨⟨ ⋅ , ⋅ ⟩⟩ : *M**n*(*E*) × *M**n*(*E*ʹ) → *M**n*2, where ⟨⟨(*x**i**j*), (*φ**k**l*)⟩⟩ = (*φ**k**l*(*x**i**j*)). Precisely, \n$$\\|(\\varphi\\_{kl})\\|\\_{M\\_n(E')}=\\sup\\big\\{\\|\\langle\\langle (x\\_{ij}),(\\varphi\\_{kl})\\rangle\\rangle\\|\\_{M\\_{n^2}}: \\|(x\\_{ij})\\|\\_{M\\_n(E)}\\le 1\\big\\}.$$ \nGiven a bilinear mapping *ϕ* : *E* × *F* → *G*, we denote by *ϕ**n* : *M**n*(*E*) × *M**n*(*F*) → *M**n*2(*G*) the mapping defined, for each *n* ∈ N, as follows: \n*ϕ**n*(*v*, *w*) = (*ϕ*(*v**i**j*, *w**k**l*)), for all *v* = (*v**i**j*) ∈ *M**n*(*E*), *w* = (*w**k**l*) ∈ *M**n*(*F*). \nWe say that *ϕ* is *jointly completely bounded* when \n∥*ϕ*∥jcb ≡ sup*n* ∈ N∥*ϕ**n*∥ < ∞. \n The space JCB(*E* × *F*, *G*) of all jointly completely bounded bilinear mappings from *E* × *F* to *G* has an operator space structure given by the identification \n*M**n*(JCB(*E* × *F*, *G*)) = JCB(*E* × *F*, *M**n*(*G*)). \nAnother natural associate to *ϕ* is the bilinear mapping *ϕ*(*n*) : *M**n*(*E*) × *M**n*(*F*) → *M**n*(*G*) related with the matrix product and given by \n*ϕ*(*n*)(*v*, *w*) = (∑*k* = 1*n**ϕ*(*v**i**k*, *w**k**l*)), for all *v* = (*v**i**j*) ∈ *M**n*(*E*), *w* = (*w**k**l*) ∈ *M**n*(*F*). \nThe bilinear mapping *ϕ* is *multiplicatively bounded* if \n∥*ϕ*∥mb = sup*n* ∈ N∥*ϕ*(*n*)∥ < ∞. \nAgain, for the space MB(*E* × *F*, *G*) of all multiplicatively bounded bilinear mappings from *E* × *F* to *G* an operator space structure is provided via the identification \n*M**n*(MB(*E* × *F*, *G*)) = MB(*E* × *F*, *M**n*(*G*)). \nDefinition of o.s. tensor norm\n------------------------------\nWe begin by recalling the definitions of the three most common tensor norms in operator space theory: the minimal, projective, and Haagerup tensor norms.\nThe *operator space injective tensor norm* (also known as *minimal tensor norm*) of *u* ∈ *M**n*(*E* ⊗ *F*) is defined as\n \nmin(*u*) = sup{∥(*f* ⊗ *g*)*n*(*u*)∥ : *f* ∈ *M**p*(*E*ʹ), *g* ∈ *M**q*(*F*ʹ), ‖*f*‖ ≤ 1, ‖*g*‖ ≤ 1} \nThe *operator space projective tensor norm* of *u* ∈ *M**n*(*E* ⊗ *F*) is defined as\n \nproj(*u*) = inf{‖*a*‖ ⋅ ‖*x*‖ ⋅ ‖*y*‖ ⋅ ‖*b*‖}, \nwhere the infimum is taken over all the representations of *u* = *a*(*x* ⊗ *y*)*b* with *x* ∈ *M**p*(*E*), *y* ∈ *M**q*(*F*), *a* ∈ *M**n*, *p* ⋅ *q*, *b* ∈ *M**p* ⋅ *q*, *n* for some *p*, *q* ∈ N.\nThe *Haagerup tensor norm* of *u* ∈ *M**n*(*E* ⊗ *F*) is defined as\n \n*h*(*u*) = inf{‖*x*‖ ⋅ ‖*y*‖ : *u* = *x* ⊙ *y*, *x* ∈ *M**n*, *r*(*E*), *y* ∈ *M**r*, *n*(*F*), *r* ∈ N}, \nwhere ⊙ denotes the standard matrix product. Precisely, \n*x* ⊙ *y* = (∑*k* = 1*r**x**i**k* ⊗ *y**k**j*)*i*, *j*. \nThe three aforementioned well-known tensor norms can be framed into the following concept.\nAn *operator space cross-norm* *α*, on the class ONORM of all normed operator spaces, is an assignment of a normed operator space *E* ⊗ *α**F* to each pair (*E*, *F*) of normed operator spaces, in such a way that *E* ⊗ *α**F* is the algebraic tensor product *E* ⊗ *F* together with a matricial norm structure on *E* ⊗ *F*, that we write as *α**n* or ∥ ⋅ ∥*α**n*, and such that\n \n*α**n**m*(*x* ⊗ *y*) = ∥*x*∥*M**n*(*E*) ⋅ ∥*y*∥*M**m*(*F*)for every *x* ∈ *M**n*(*E*), *y* ∈ *M**m*(*F*). \nThis implies that the identity map *E* ⊗ proj*F* → *E* ⊗ *α**F* is completely contractive. If in addition the identity map *E* ⊗ *α**F* → *E* ⊗ min*F* is also completely contractive, we say that *α* is *reasonable*. Note that if both mappings *E* ⊗ proj*F* → *E* ⊗ *α**F* and *E* ⊗ *α**F* → *E* ⊗ min*F* are completely contractive then we obviously obtain.\nMoreover, an operator space cross-norm *α* is called *uniform* if additionally it satisfies the *complete metric mapping property*: if *S* ∈ CB(*E*1, *E*2) and *T* ∈ CB(*F*1, *F*2), then \n∥*S* ⊗ *T*: *E*1 ⊗ *α**F*1 → *E*2 ⊗ *α**F*2∥cb ≤ ∥*S*∥cb∥*T*∥cb. \nAn operator space cross-norm on ONORM that is both reasonable and uniform will be called an *operator space tensor norm*, or *o.s. tensor norm* for simplicity.\nAn operator tensor norm *α* is an assignment to each *E*, *F* ∈ ONORM of a matricial norm in *E* ⊗ *F* such that the following two conditions are satisfied:\n1. *α* is reasonable: The mappings *E* ⊗ proj*F* → *E* ⊗ *α**F* → *E* ⊗ min*F* are completely contractive.\n2. *α* is uniform: If *S* ∈ CB(*E*1, *E*2) and *T* ∈ CB(*F*1, *F*2), then \n∥*S* ⊗ *T*: *E*1 ⊗ *α**F*1 → *E*2 ⊗ *α**F*2∥cb ≤ ∥*S*∥cb∥*T*∥cb.\nA degree of caution is required when consulting different works dealing with operator space tensor products, since the term “tensor norm” is not always taken to have the exact same meaning. We are using the same definition as, which is slightly different than the one from ; we refer to for an explicit comparison. It is clear that analogous definitions for operator space tensor norms can also be made on a subclass of operator spaces, for example the class OFIN of all finite-dimensional operator spaces, or the class of dual operator spaces.\nBasic properties of tensor norms\n--------------------------------\nAn o.s. tensor norm *α* is *symmetric* if *E* ⊗ *α**F* and *F* ⊗ *α**E* are canonically completely isometric (via the transposition map). A norm *α* is *associative* if (*E* ⊗ *α**F*) ⊗ *α**G* and *E* ⊗ *α*(*F* ⊗ *α**G*) are canonically completely isometric (via the identity map). It is known that min and proj are both symmetric and associative, while the Haagerup o.s. tensor norm *h* is associative but not symmetric.\nFor operator space tensor norms *α* and *β* and a constant *c*, we write “*α* ≤ *c**β* on *E* ⊗ *F*” to indicate that the identity map *E* ⊗ *β**F* → *E* ⊗ *α**F* has cb-norm at most *c*. If no reference to spaces is made, we mean that the inequality holds for any pair of normed operator spaces.\nThe Schatten spaces and their vector-valued versions\n----------------------------------------------------\nRecall that for a Hilbert space *H* the Schatten class *S**p*(*H*) is defined for 1 ≤ *p* < ∞ as the space of all compact operators *T* on *H* such that $\\operatorname{tr}\\big( |T|^p\\big) < \\infty$ equipped with the norm ${ \\left\\|T\\right\\| }\\_{S\\_p} = \\big( \\operatorname{tr}\\big( |T|^p \\big) \\big)^{1/p}$; in the case *p* = ∞, we denote by *S*∞(*H*) the space of all compact operators on *H* (also denoted by $\\mathcal K(H)$) endowed with the operator norm. To endow the spaces *S**p*(*H*) with an operator space structure, we follow Pisier’s approach : since *S*∞(*H*) is a *C*\\*-algebra it has a canonical operator space structure, by the duality *S*1(*H*)ʹ = *B*(*H*) we also get a natural operator space structure on *S*1(*H*), and by complex interpolation for operator spaces we get an operator space structure for each of the intermediate *S**p*(*H*) spaces. In the special case *H* = ℓ2, we simply write *S**p* instead of *S**p*(ℓ2). For an operator space *E* and 1 ≤ *p* ≤ ∞ we can similarly obtain an operator space which is an *E*-valued version of *S**p* and will be denoted by *S**p*[*E*]: we define *S*∞[*E*] = *S*∞ ⊗ min*E* and *S*1[*E*] = *S*1 ⊗ proj*E*, and once again in the case 1 < *p* < ∞ we define *S**p*[*E*] via complex interpolation between *S*∞[*E*] and *S*1[*E*].\nReplacing *S*1 by the space *S*1*n* of *n* × *n* matrices with the trace norm, and *S*∞ by the space *M**n*, the same procedure analogously yields operator spaces *S**p**n* and *S**p**n*[*E*]. We refer the reader to for a detailed study of these spaces.\nDirect sums\n-----------\n Given a family {*E**γ*}*γ* ∈ Γ of operator spaces, its ℓ∞ direct sum which we will denote by ℓ∞({*E**γ* : *γ* ∈ Γ}) is the direct sum *E* = ⨁*γ* ∈ Γ*E**γ* with the operator space structure given as follows: for *u* = (*u**γ*)*γ* ∈ Γ ∈ *M**n*(*E*), \n$$\\|u\\|\\_{\\ell\\_\\infty( \\{E\\_\\gamma \\;:\\; \\gamma \\in \\Gamma\\}) } = \\sup \\big\\{ \\|u\\_\\gamma\\|\\_{M\\_n(E\\_\\gamma)} \\;:\\; \\gamma \\in \\Gamma \\}.$$ \n When we have only two summands, we will often use the notation *E*1 ⊕ ∞*E*2 instead.\nThe ℓ1-sum of {*E**γ*}*γ* ∈ Γ will be similarly denoted by ℓ1({*E**γ* : *γ* ∈ Γ}); for simplicity, below we only give the full details in the case *E*1 ⊕ 1*E*2 where we have two summands. For *u* ∈ *M**n*(*E*1 ⊕ *E*2), \n$$\\|u\\|\\_{M\\_n(E\\_1 \\oplus\\_1 E\\_2)} = \\sup\\big\\{ { \\left\\|(T\\_1 \\oplus T\\_2)\\_n(u)\\right\\| }\\_{M\\_n(\\mathcal{B}(H))} \\;:\\; { \\left\\|T\\_j : E\\_j \\to \\mathcal{B}(H)\\right\\| }\\_{\\operatorname{cb}} \\le 1 \\text{ for } j=1,2\\big\\}$$ \n where *T*1 ⊕ *T*2 : *E*1 ⊕ *E*2 → B(*H*) is the mapping (*T*1 ⊕ *T*2)(*x*, *y*) = *x* + *y*.\nInjections and projections\n--------------------------\nA linear map *q* : *E* → *F* between normed operator spaces is called a *complete quotient* or *complete metric surjection* or *complete projection* if it is onto and the associated map from *E*/ker(*q*) to *F* is a completely isometric isomorphism. In, it is proved that a linear map *q* : *E* → *F* is a complete quotient if and only if its adjoint *q*ʹ : *F*ʹ → *E*ʹ is a completely isometric embedding. Note that if a linear map *q* : *E* → *F* between normed operator spaces is a complete quotient, then for every *x* ∈ *M**n*(*F*) we have \n∥*x*∥*M**n*(*F*) = inf{∥*y*∥*M**n*(*E*) : *y* ∈ *M**n*(*E*), *q**n*(*y*) = *x*}. \nA completely isometric embedding *T* : *E* → *F* will be called a *complete injection* for short.\nAn o.s. tensor norm *α* is called *completely left projective* (resp. *completely left injective*) if for any normed operator space *G* and any complete projection (resp. complete injection) *T* : *E* → *F*, the map *T* ⊗ *i**d**G* : *E* ⊗ *α**G* → *F* ⊗ *α**G* is a complete projection (resp. complete injection) as well. Completely right projective and completely right injective operator space tensor norms are defined analogously, and an o.s. tensor norm will be called completely projective (resp. completely injective) when it is both completely left and right projective (resp. injective).\nA normed operator space *E* is said to be *completely injective* if whenever *F* ⊆ *G* are operator spaces and *T* : *F* → *E* is a completely bounded linear map, there exists an extension $\\widetilde{T} : G \\to E$ with ${ \\left\\|T\\right\\| }\\_{\\operatorname{cb}} = \\|\\widetilde{T}\\|\\_{\\operatorname{cb}}$. By the Arveson extension theorem, B(*H*) is completely injective for any Hilbert space *H*. Let us now observe that any completely injective normed operator space *E* has to be complete. Indeed, consider the diagram \n$$\\xymatrix{\n\\widetilde{E} \\ar[dr]^T &\\\\\nE \\ar[u] \\ar[r]\\_{id\\_E} & E}$$ \n where the vertical arrow is simply the inclusion of *E* in its completion $\\widetilde{E}$, and the operator *T* is given by the injectivity of *E*. If we take a Cauchy sequence (*v**n*)*n* in *E*, and we take its limit $v \\in \\widetilde{E}$, then *T*(*v*) ∈ *E* is the limit of (*v**n*)*n* since \n$$T(v) = T\\Big( \\lim\\_{n\\to\\infty} v\\_n \\Big) = \\lim\\_{n\\to\\infty} T(v\\_n) = \\lim\\_{n\\to\\infty} v\\_n.$$ \nA normed operator space *E* is said to be *completely projective* if for any completely bounded linear map *T* : *E* → *G*/*F* into a quotient space and any *ɛ* > 0 there exists a lifting $\\widetilde{T} : E \\to G$ of *T* with $\\|\\widetilde{T}\\|\\_{\\operatorname{cb}} \\le (1+\\varepsilon) { \\left\\|T\\right\\| }\\_{\\operatorname{cb}}$. The basic examples of projective operator spaces are the spaces *S*1*n* for any *n* ∈ N, and we will also use the fact that an ℓ1-sum of projective operator spaces is again projective.\nEvery Banach operator space *E* can be seen as a quotient of a completely projective space. Indeed, there is a set *I* and a family (*n**i*)*i* ∈ *I* ⊂ N such that *E* is the quotient of ℓ1({*S*1*n**i* : *i* ∈ *I*}) (see e.g., ). We denote the latter space by *Z**E* and $q\\_E : Z\\_E \\twoheadrightarrow E$ the corresponding complete quotient mapping. The fact that *Z**E* is completely projective can be tracked in.\nSee for more on completely injective and completely projective operator spaces.\nExamples of o.s. tensor norms\n-----------------------------\nIn addition to the three fundamental o.s. tensor norms defined above (injective, projective, Haagerup), a number of other examples have appeared in the literature and we list some of them below. The first three are discussed in, where it is shown that they satisfy the complete metric mapping property. To conclude that they are o.s. tensor norms, we then just need to check that they are between min and proj.\n### The nuclear tensor product\n*E* ⊗ nuc*F* is defined as the quotient of *E* ⊗ proj*F* by the kernel of the canonical identity map *E* ⊗ proj*F* → *E* ⊗ min*F*. From the definition it is clear that min ≤ nuc ≤ proj, so nuc is reasonable.\n### The extended Haagerup tensor product\n*E* ⊗ eh*F* is the space of maps *u* : *E*ʹ × *F*ʹ → C which are normal (i.e. weak\\* continuous in each variable) and multiplicatively bounded. If we denote the space of such maps by MB*σ*(*E*ʹ × *F*ʹ, C), the matrix norms on *E* ⊗ eh*F* are given by the identification ${M\\_n\\big( \\operatorname{MB}^\\sigma(E'\\times F', {{\\mathbb{C}}}) \\big)} = \\operatorname{MB}^\\sigma(E'\\times F', M\\_n)$. It then follows from that min ≤ eh.\nThe operator space structure on *E* ⊗ eh*F* can be described in the same way as but replacing *r* ∈ N with an arbitrary set, hence the *extended* in the name. This in particular shows eh ≤ *h*, and thus eh ≤ proj. The extended Haagerup tensor product is associative, and completely injective but not completely projective.\n### The normal Haagerup tensor product\nThis is only defined for dual operator spaces, by *E*ʹ ⊗ σh*F*ʹ = (*E* ⊗ eh*F*)ʹ. Since the identity *E* ⊗ proj*F* → *E* ⊗ eh*F* is a complete contraction, dualizing yields a contraction \n*E*ʹ ⊗ σh*F*ʹ = (*E* ⊗ eh*F*)ʹ → (*E* ⊗ proj*F*)ʹ = CB(*E*, *F*ʹ), \n which implies that the identity *E*ʹ ⊗ σh*F*ʹ → *E*ʹ ⊗ min*F*ʹ is a complete contraction. The fact that σh ≤ proj is a complete contraction, i.e. the identity *E*ʹ ⊗ proj*F*ʹ → (*E* ⊗ eh*F*)ʹ is a complete contraction, follows easily from and the description of *E* ⊗ eh*F* as normal multiplicatively bounded mappings (see [subsubsection-extended-Haagerup]).\nRecall from above that the extended Haagerup tensor product is not completely projective, that is, there exist complete projections *π**j* : *E**j* → *F**j*, *j* = 1, 2 such that *π*1 ⊗ *π*2 : *E*1 ⊗ eh*E*2 → *F*1 ⊗ eh*F*2 is not a complete projection. By dualizing, we have complete injections *π**j*ʹ : *F**j*ʹ → *E**j*ʹ, *j* = 1, 2 such that *π*1ʹ ⊗ *π*2ʹ : *F*1ʹ ⊗ σh*F*2ʹ → *E*1ʹ ⊗ σh*E*2ʹ is not a complete injection, so the normal Haagerup tensor product is not completely injective.\n### The symmetrized Haagerup tensor norms\nGiven normed operator spaces *E* and *F* and *u* ∈ *M**n*(*E* ⊗ *F*), we define \n$$(h \\cap h^t)\\_n(u;E,F) = \\max\\big\\{ h\\_n(u; E,F), h^t\\_n(u; E,F) \\big\\}.$$ \n It is easy to see that *h* ∩ *h**t* is a symmetric o.s. tensor norm, and from the definition it is obvious that there is a completely isometric embedding \n$$E \\otimes\\_{ h \\cap h^t} F \\hookrightarrow (E \\otimes\\_h F) \\oplus\\_\\infty (E \\otimes\\_{h\\_t} F)$$ \n given by *u* ↦ (*u*, *u*). Since *h* and *h**t* are completely injective, it is clear that *h* ∩ *h**t* is completely injective as well. Notice that if *α* is a symmetric o.s. tensor norm such that *h* ≤ *α*, by transposing we have *h**t* ≤ *α* and therefore *h* ∩ *h**t* ≤ *α*. Thus, we call *h* ∩ *h**t* the minimal symmetrized Haagerup tensor norm. This tensor norm has appeared in.\nOn the other hand, for normed operator spaces *E* and *F* and *u* ∈ *M**n*(*E* ⊗ *F*), we define \n$$(h+h^t)\\_n(u;E,F) = \\inf\\big\\{ \\|(v,w)\\|\\_{M\\_n\\big( (E \\otimes\\_h F)\\oplus\\_1(E \\otimes\\_{h^t} F) \\big)} \\;:\\; u = v+w \\big\\}.$$ \n That is, by definition the mapping \n*q* : (*E* ⊗ *h**F*) ⊕ 1(*E* ⊗ *h**t**E*) → *E* ⊗ *h* + *h**t**F* \n given by *q*(*u*, *v*) = *u* + *v* is a complete quotient. Once again, it is easy to check that this defines a symmetric o.s. tensor norm which is completely projective because so is *h*. The o.s. tensor norm *h* + *h**t* was introduced in, were it was denoted by *μ* and it was shown that it is neither associative nor completely injective. We will call *h* + *h**t* the maximal symmetrized Haagerup tensor norm because of the following property. Suppose that *β* is a symmetric tensor norm with *β* ≤ *h*. Then *β* ≤ *h**t*, so that the formal identity maps \n*E* ⊗ *h**F* → *E* ⊗ *β**F*, *E* ⊗ *h**t**F* → *E* ⊗ *β**F* \n are complete contractions. Therefore, the map (*v*, *w*) ↦ *v* + *w* is a complete contraction \n(*E* ⊗ *h**F*) ⊕ 1(*E* ⊗ *h**t**E*) → *E* ⊗ *β**F* \n and by the standard properties of quotients, the identity map *E* ⊗ *h* + *h**t**F* → *E* ⊗ *β**F* is a complete contraction as well, that is, *β* ≤ *h* + *h**t*.\n### The Chevet-Saphar tensor products\nInspired by the definition of the Banach space case, for 1 ≤ *p* ≤ ∞, the right and left *p*-Chevet-Saphar o.s. tensor norms, *d**p* and *g**p*, are defined in in following way: given normed operator spaces *E* and *F*, for *u* ∈ *E* ⊗ *F*,\n \n$$\\begin{aligned}\n\\label{dp-os}\nd\\_p (u) = \\inf \\left\\{\\Vert (x\\_{ij}) \\Vert\\_{S^n\\_{p'}\\otimes\\_{\\min} E} \\; \\Vert(y\\_{ij})\\Vert\\_{S^n\\_p[F]} : u = \\sum\\_{i,j=1}^n x\\_{ij} \\otimes y\\_{ij} \\right\\}, \\\\\ng\\_p (u) = \\inf \\left\\{ \\Vert(x\\_{ij})\\Vert\\_{S^n\\_p[E]} \\; \\Vert (y\\_{ij}) \\Vert\\_{S^n\\_{p'}\\otimes\\_{\\min} F} : u = \\sum\\_{i,j=1}^n x\\_{ij} \\otimes y\\_{ij} \\right\\}\\end{aligned}$$ \nwhere the infimum runs over all the possible ways in which the tensor *u* can be written as a finite sum as above.\nNote that the above expressions only define norms on *E* ⊗ *F*, the full operator space structure of the tensor products $E \\widehat \\otimes\\_{d\\_p } F$ and $E \\widehat \\otimes\\_{g\\_p } F$ are given by the following quotient mappings (see )\n \n$$\\begin{aligned}\n q\\_{d\\_p} : (S\\_{p'} \\widehat{\\otimes}\\_{\\min} E) \\widehat{\\otimes}\\_{\\operatorname{proj}} S\\_p[F] \\to E \\widehat{\\otimes}\\_{d \\_p} F, \\\\\n q\\_{g\\_p} : S\\_p[E] \\widehat{\\otimes}\\_{\\operatorname{proj}} (S\\_{p'} \\widehat{\\otimes}\\_{\\min} F) \\to E \\widehat{\\otimes}\\_{g \\_p} F,\n \\end{aligned}$$ \nwhere $\\widehat \\otimes $ means the completion of the corresponding tensor product. In it is shown that both *d**p* and *g**p* are o.s. tensor norms.\n### *λ*-tensor products\nFor each *k* ∈ N, let \n*B**λ**k* : *M**k* × *M**k* → *M**τ*(*k*) \n (where *τ*(*k*) ∈ N is a natural number only depending on *k*) be a bilinear map; we will denote the sequence (*B**λ**k*)*k* by *λ*. Two basic examples to keep in mind are the tensor product ⊗ (i.e. when *B**λ**k*(*x*, *y*) = *x* ⊗ *y* for all *k*) and the matrix product ⊙ (i.e. when *B**λ**k*(*x*, *y*) = *x* ⊙ *y* for all *k*). We point out that the notation above is slightly different from that of, in order to keep consistency in the notation of the present work. For *u* ∈ *M**k*(*E* ⊗ *F*), define\n \n*λ**k*(*u*) = inf{∥*a*∥∥*v*1∥∥*v*2∥∥*b*∥} \nwhere the infimum is taken over arbitrary decompositions *u* = *a* ⊗ *B**λ**j*(*v*1, *v*2)*b* with *a* ∈ *M**k*, *τ*(*j*), *b* ∈ *M**τ*(*j*), *k*, *v*1 ∈ *M**j*(*E*), *v*2 ∈ *M**j*(*F*), where ⊗ *B**λ**j* : *M**j*(*E*) × *M**j*(*F*) → *M**τ*(*j*)(*E* ⊗ *F*) is the bilinear map given by (*a*1 ⊗ *x*, *a*2 ⊗ *y*) ↦ *B**λ**j*(*a*1, *a*2) ⊗ *x* ⊗ *y*. Observe that the case *λ* = ⊗ corresponds to the projective tensor norm, and *λ* = ⊙ yields the Haagerup tensor norm.\nIn order to guarantee that the norms defined above give an o.s. tensor norm, we will need for *λ* to satisfy certain technical conditions. First, some notation. We let *ɛ**i*, *j* : = *ɛ**i*, *j*[*k*, *l*] ∈ *M**k*, *l* denote the matrix which is 1 in the (*i*, *j*)-th entry and zero elsewhere, *ɛ**i*, *j*[*k*] : = *ɛ**i*, *j*[*k*, *k*], *ɛ**i* : = *ɛ**i*, *i*, *ɛ**i*[*k*] : = *ɛ**i*, *i*[*k*, *k*] and *ɛ**i*, *j*[*k*, *l*] = 0 if $(i,j) \\notin \\{1,\\ldots,k\\} \\times \\{1,\\ldots,l\\}$. As usual we will denote by *I**n* the identity *n* × *n*-matrix.\n[ and ] For any operator spaces *E* and *F* the sequence *λ**k*( ⋅ ) defined above gives an operator space structure on *E* ⊗ *F*, whenever *λ* satisfies the following conditions:\n* For all *k* ∈ N there exist *p* ∈ N and matrices *S* ∈ *M**k*, *τ*(*p*), *T* ∈ *M**τ*(*p*), *k*, *a*1, ⋯, *a**k* ∈ *M**p* such that for all *j*1, *j*2 ∈ {1, ⋯, *k*}: \n$$S {B^\\lambda}\\_p(a\\_{j\\_1},a\\_{j\\_2}) T= \\left\\{\n \\begin{array}{ll}\n \\varepsilon\\_j^{[k]} & \\qquad \\text{if}\\;\\; j\\_1=j\\_2=j, \\\\\n 0 & \\qquad \\text{otherwise.}\n \\end{array}\n \\right.$$\n* For all *r*, *s* ∈ N there exist matrices *P* ∈ *M**τ*(*r*) + *τ*(*s*), *τ*(*r* + *s*), with ∥*P*∥ ≤ 1 such that for all (*i**k*, *j**k*) ∈ {1, ⋯, *r*}2 ∪ {*r* + 1, ⋯, *r* + *s*}2 with *k* = 1, 2:\n \n$$\\begin{aligned}\n\\begin{split}\n& P {B^\\lambda}\\_{r+s}( \\varepsilon\\_{i\\_1, j\\_1}^{[r+s]}, \\varepsilon\\_{i\\_2,j\\_2 }^{[r+s]})P^\\* \\\\\n&= \\mathrm{diag}\\Big( {B^\\lambda}\\_r (\\varepsilon\\_{i\\_1,j\\_1}^{[r]}, \\varepsilon\\_{i\\_2,j\\_2}^{[r]}), {B^\\lambda}\\_{s}(\\varepsilon\\_{i\\_1-r,j\\_1-r}^{[s]}, \\varepsilon\\_{i\\_2-r,j\\_2-r}^{[s]})\\Big).\\notag\n\\end{split}\\end{aligned}$$\n* *B**λ*1(1, 1) = 1 and sup*k* ∈ N∥*B**λ**k*∥ < ∞.\nIn this case, we denote by *E* ⊗ *λ**F* the corresponding operator space. Moreover, the complete metric mapping property is satisfied.\nNote that, in an abuse of notation, we are using the symbol *λ* to denote both the sequence of bilinear maps *M**k* × *M**k* → *M**τ*(*k*) and the operator space structure induced on the tensor product.\nJust as for the projective and Haagerup tensor products, the dual of a *λ*-tensor product can be identified with a certain space of bilinear forms. For any bilinear map *ϕ* : *E* × *F* → *W* where *E*, *F*, *W* are operator spaces, define the bilinear maps *ϕ**B**λ**k* : *M**k*(*E*) × *M**k*(*F*) → *M**τ*(*k*)(*W*) given on elementary tensors by \n$$\\phi\\_{{B^\\lambda}\\_k}\\big( a\\_1 \\otimes v\\_1, a\\_2 \\otimes v\\_2 \\big) = {B^\\lambda}\\_k(a\\_1,a\\_2) \\otimes \\phi(v\\_1,v\\_2), \\qquad a\\_1, a\\_2 \\in M\\_k, v\\_1 \\in E, v\\_2 \\in F.$$ \n We also define \n∥*ϕ*∥cb, *λ* = sup*k* ∈ N{∥*ϕ**B**λ**k*(*x*, *y*)∥*M**τ*(*k*)(*W*) : ∥*x*∥*M**k*(*E*) ≤ 1, ∥*y*∥*M**k*(*F*) ≤ 1} \n and \nCB*λ*(*E* × *F*; *W*) = {*ϕ* : *E* × *F* → *W*bilinear : ∥*ϕ*∥cb, *λ* < ∞}. \n This space is an operator space with the identification \n$$M\\_k\\big(\\operatorname{CB}\\_\\lambda(E \\times F; W)\\big) = \\operatorname{CB}\\_\\lambda\\big(E \\times F; M\\_k(W)\\big).$$ \n Note that for the cases *λ* = ⊗ (projective tensor product) and *λ* = ⊙ (Haagerup tensor product) we recover the usual dual spaces CB⊗ = JCB and CB⊙ = MB.\n[][duality-lambda-tensor] If *λ* satisfies (E1), (E2), and (E3), then the natural identification yields a canonical complete isometry \nCB(*E* ⊗ *λ**F*, *W*) = CB*λ*(*E* × *F*; *W*) \n so in particular \n(*E* ⊗ *λ**F*)ʹ = CB*λ*(*E* × *F*; C) = : CB*λ*(*E* × *F*). \nLet us denote by ⊙ *t* : *M**k* × *M**k* → *M**k* the transposition of the usual matrix product, that is, *A* ⊙ *t**B* = *B* ⊙ *A* for *A*, *B* ∈ *M**k*. It is clear that taking *λ* = ⊙ *t*, the associated operator space structure on *E* ⊗ *F* is precisely *E* ⊗ *h**t**F*, the transposition of the Haagerup tensor product. We now introduce a new family of Haagerup-style o.s. tensor products which are *λ*-tensor products, and which can be understood as a sort of “interpolation” between *h* and *h**t*. For any *θ* ∈ [0, 1] and *k* ∈ N, we let ⊙ *k**θ* : *M**k* × *M**k* → *M**k* be (1 − *θ*) ⊙ + *θ* ⊙ *t*, that is, ⊙ *k**θ*(*A*, *B*) = (1 − *θ*)*A**B* + *θ**B**A* for *A*, *B* ∈ *M**k*. It is clear that each ⊙ *k**θ* is bilinear, and let us now show that they all induce operator space structures on the tensor product.\n[htheta-satisfies-Es] For each *θ* ∈ [0, 1], ⊙ *θ* satisfies (E1), (E2) and (E3).\nClearly 1 ⊙ 1 = 1 ⊙ *t*1 = 1, and ∥ ⊙ *k*∥ = ∥ ⊙ *k**t*∥ = 1 (since ∥*A* ⊙ *B*∥ ≤ ∥*A*∥∥*B*∥ for *A*, *B* ∈ *M**k*). Taking a convex combination yields (E3) for ⊙ *θ*.\nAs in the proof of, (E1) is satisfied for ⊙ by taking *p* = *k*, *S* = *T* = *I**k*, and *a**j**i* = *ɛ**j**i*[*k*]. Since all the matrices involved are diagonal all the products commute, and it follows that the same choices yield (E1) for ⊙ *t*. Taking convex combinations, the same choices once again yield (E1) for ⊙ *θ*. The exact same type of argument works for (E2): this is proved for ⊙ in using *P* = *I**r* + *s*.\n[htheta-is-cross] For each *θ* ∈ [0, 1], ⊙ *θ* is a cross-norm.\nLet *E*, *F* be operator spaces, *x* ∈ *M**m*(*E*), *y* ∈ *M**n*(*F*). It is well-known that *x* ⊗ *y* = (*x* ⊗ *I**n*) ⊙ (*I**m* ⊗ *y*), see e.g.. The exact same calculation shows *x* ⊗ *y* = (*x* ⊗ *I**n*) ⊙ *t*(*I**m* ⊗ *y*), and therefore *x* ⊗ *y* = ⊙ *m**n**θ*(*x* ⊗ *I**n*, *I**m* ⊗ *y*). Thus, by the definition of ⊙ *m**n**θ*, \n ⊙ *m**n**θ*(*x* ⊗ *y*) ≤ ∥*x* ⊗ *I**n*∥*M**m**n*(*E*)∥*I**m* ⊗ *y*∥*M**m**n*(*F*) = ∥*x*∥*M**m*(*E*) ⋅ ∥*y*∥*M**n*(*F*). \nIn order to conclude that ⊙ *θ* is an o.s. tensor norm, the only missing ingredient is to check that min ≤ ⊙ *θ*. We prove a more general result.\n[min-less-than-lambda] Suppose that *λ* satisfies (E1), (E2), (E3), and ∥*B**λ**k*∥jcb ≤ 1 for each *k* ∈ N. Then the identity mapping *E* ⊗ *λ**F* → *E* ⊗ min*F* is a complete contraction.\nBy, it suffices to show that for every *U* ∈ *M**k*(*E* ⊗ *F*), *ϕ* ∈ *M**m*(*E*ʹ), *ψ* ∈ *M**n*(*F*ʹ) we have \n∥⟨⟨*ϕ* ⊗ *ψ*, *U*⟩⟩∥*M**m**n**k* ≤ ∥*ϕ*∥*M**m*(*E*ʹ)∥*ψ*∥*M**n*(*F*ʹ)*λ**k*(*U*). \n Now, by the identification (*E* ⊗ *λ**F*)ʹ = CB*λ*(*E* × *F*) we already know that \n∥⟨⟨*ϕ* ⊗ *ψ*, *U*⟩⟩∥*M**m**n**k* ≤ ∥*ϕ* ⊗ *ψ*∥*M**m**n*(CB*λ*(*E* × *F*))*λ**k*(*U*). \n Recalling that $M\\_{mn}\\big(\\operatorname{CB}\\_\\lambda(E \\times F)\\big) = \\operatorname{CB}\\_\\lambda\\big(E \\times F; M\\_{mn}\\big)$, all we need is to show that \n∥*φ*∥*λ*, cb ≤ ∥*ϕ*∥*M**m*(*E*ʹ)∥*ψ*∥*M**n*(*F*ʹ) \n where *φ* = *ϕ* ⊗ *ψ* : *E* × *F* → *M**m**n*; note that here we are interpreting *ϕ* ∈ *M**m*(*E*ʹ) = CB(*E*, *M**m*) and *ψ* ∈ *M**n*(*F*ʹ) = CB(*F*, *M**n*).\nTo calculate the norm of *φ* in CB*λ*(*E* × *F*; *M**m**n*), take *u* ∈ *M**k*(*E*) and *v* ∈ *M**k*(*F*) with ∥*u*∥*M**k*(*E*), ∥*v*∥*M**k*(*F*) ≤ 1. Represent *u* = ∑*i**a**i* ⊗ *u**i*, *v* = ∑*j**b**j* ⊗ *v**j* where *a**i*, *b**j* ∈ *M**k*, *u**i* ∈ *E*, *v**j* ∈ *F*. Then, by the definition of *φ**B**λ**k*,\n \n$$\\begin{gathered}\n \\varphi\\_{{B^\\lambda}\\_k}(u,v) = \\sum\\_{i,j} {B^\\lambda}\\_k(a\\_i,b\\_j) \\otimes \\varphi(u\\_i,v\\_j) = \\sum\\_{i,j} {B^\\lambda}\\_k(a\\_i,b\\_j) \\otimes \\phi(u\\_i) \\otimes \\psi(v\\_j) \\\\\n = \\otimes\\_{{B^\\lambda}\\_k} \\Big(\\sum\\_i a\\_i \\otimes \\phi(u\\_i), \\sum\\_j b\\_j \\otimes \\psi(v\\_j) \\Big) \n = \\otimes\\_{{B^\\lambda}\\_k} \\big( (I\\_k \\otimes \\phi)u, (I\\_k\\otimes \\phi)v \\big).\\end{gathered}$$ \nNote that the assumption ∥*B**λ**k*∥jcb ≤ 1 precisely means that for any *A* ∈ *M**k*(*M**m*) and *B* ∈ *M**k*(*M**n*) we have \n∥ ⊗ *B**λ**k*(*A*, *B*)∥*M**τ*(*k*)*m**n* ≤ ∥*A*∥*M**k*(*M**m*)∥*B*∥*M**k*(*M**n*), \n so\n \n$$\\begin{gathered}\n{ \\left\\|\\varphi\\_{{B^\\lambda}\\_k}(u,v)\\right\\| } \\le { \\left\\|(I\\_k \\otimes \\phi)u\\right\\| }\\_{M\\_k(M\\_m)} { \\left\\|(I\\_k\\otimes \\phi)v\\right\\| }\\_{M\\_k(M\\_n)} \\\\\n\\le { \\left\\|I\\_k \\otimes \\phi\\right\\| } { \\left\\|u\\right\\| } { \\left\\|I\\_k\\otimes \\phi\\right\\| } { \\left\\|v\\right\\| } \\le { \\left\\|\\phi\\right\\| } { \\left\\|\\phi\\right\\| } \\end{gathered}$$ \nwhich yields the desired conclusion.\nFor each *θ* ∈ [0, 1], ⊙ *θ* is an o.s. tensor norm.\nThis is an immediate consequence of Propositions [htheta-satisfies-Es], [htheta-is-cross] and [min-less-than-lambda], where for the latter we use the fact that ∥ ⊙ ∥jcb ≤ 1 and therefore ∥ ⊙ *k**θ*∥jcb ≤ 1.\nThroughout the rest of this paper, whenever we consider any *λ*-tensor norm we will always be assuming that it satisfies (E1), (E2), (E3) and additionally it is an o.s. tensor norm. To emphasize this point, we will call them **λ*-o.s. tensor norms*.\nWe have previously pointed out that our notion of uniformity in the definition of o.s. tensor norm is weaker than that of. This has the advantage of allowing the ⊙ *θ* to be covered by the theory: the only assumptions we are aware of which imply that a *λ*-tensor norm satisfies the uniformity condition in are conditions (W1) and (W2) in, but ⊙ *θ* only satisfies (W2) in the extreme cases *θ* = 0, 1.\nSome usual notation\n-------------------\nFor a normed operator space *E*, we denote by OFIN(*E*) (resp. OCOFIN(*E*)) the set of all finite-dimensional (resp. finite-codimensional) subspaces of *E*. Given *L* ∈ OCOFIN(*E*), let *q**L**E*: *E* → *E*/*L* be the canonical projection and given a subspace *F* of *E* let *i**F**E*: *F* → *E* be the canonical injection. To avoid an overload of notation, when the inclusion is the typical complete isometry for operator spaces *E* ⊂ B(*H*) we just denote the canonical injection by $i\\_E:E\\to \\mathcal B(H)$. Also, following the usual Banach space notation, for a normed operator space *E*, *κ**E*: *E* → *E*ʺ denotes the canonical injection into the bidual.\nGiven vector spaces *E* and *F*, we identify a linear map *T* : *E* → *F*ʹ with a bilinear map *β**T* : *E* × *F* → C via *β**T*(*x*, *y*) = (*T**x*)(*y*); note that *β**T* can also be identified with an element of (*E* ⊗ *F*)ʹ.\nA normed operator space *E* is an OS*p*, *C* space if there is a family (*F**i*)*i* ∈ *I* of finite-dimensional subspaces of *E* whose union is dense in *E* and such that for every index *i* there is a natural number *n**i* such that *d*cb(*S**p**n**i*, *F**i*) ≤ *C* where *d*cb stands for the completely bounded Banach-Mazur distance, i.e., \n$$d\\_{\\operatorname{cb}}(E, F):= \\inf\\big\\{\\Vert T \\Vert\\_{cb} \\Vert T^{-1} \\Vert\\_{cb} : T \\in \\operatorname{CB}(E,F) \\mbox{ is a complete isomorphism} \\big\\}.$$ \n If *E* is an OS*p*, *C*ʹ space for every *C*ʹ > *C*, we say that *E* is an OS*p*, *C* + space.\nIf *W* is a Banach space, its maximal operator space structure Max(*W*) is defined by, for *A* ∈ *M**n*(*W*) \n$${ \\left\\|A\\right\\| }\\_{M\\_n(\\operatorname{Max}(W))} = \\big\\{ { \\left\\| u\\_n(A) \\right\\| }\\_{M\\_n(B(H))} \\;:\\; { \\left\\| u : W \\to B(H) \\right\\| } \\le 1 \\big\\}.$$ \n See for more details.\nOperator approximation property\n-------------------------------\nA normed operator space *E* is said to have the *operator approximation property* (OAP) if there exists a net of finite rank mappings *T**η* ∈ CB(*E*, *E*) such that the net *i**d*K ⊗ *T**η* converges pointwise to the identity in *K*∞(*E*) = K ⊗ min*E*, where K denotes the compact operators on ℓ2. Similarly, *E* is said to have the **C*-completely bounded approximation property* (*C*-CBAP) if there exists a net of finite rank mappings *T**η* ∈ CB(*E*, *E*) such that ∥*T**η*∥cb ≤ *C* and ∥*T**η**x* − *x*∥ → 0 for every *x* ∈ *E*. Note that by, we also have ∥(*T**η*)*n**x* − *x*∥ → 0 for every *x* ∈ *M**n*(*E*). In the particular case *C* = 1, we say that *E* has the *completely metric approximation property* (CMAP).\nRelevant differences with the Banach space setting\n--------------------------------------------------\nIt is well-known that the theory of Banach spaces cannot be painlessly transferred to the operator space framework, and the theory of tensor norms is no exception. Let us point out three important differences in this context.\n### Local reflexivity\nA normed operator space *E* is said to be *locally reflexive* if for each finite-dimensional operator space *F*, any complete contraction *φ* : *F* → *E*ʺ may be approximated in the point-weak\\* topology by a net of complete contractions *φ**η* : *F* → *E*. The class of all locally reflexive normed operator spaces is denoted by OLOC.\nSince local reflexivity is relevant to prove many properties related to Banach space tensor norms, it is often an additional hypothesis in the operator space versions of those results. Moreover, in some cases this is known to be necessary: is an example of an identity involving tensor norms whose validity turns out to characterize local reflexivity. Nevertheless, there are some particular situations where we can omit this assumption (for instance, for *λ*-o.s. tensor norms, see Lemma [extension-lemma] and Lemma [embedding-lemma]).\nWe point out, for future reference, that given a complete operator space *E*, the completely projective space *Z**E* introduced in Section [injections-and-projections] is in fact locally reflexive. Indeed, its dual ℓ∞({*M**n**i* : *i* ∈ *I*}) is clearly a *C*\\*-algebra and thus a von Neumann algebra, and preduals of von Neumann algebras are locally reflexive (see e.g., ).\n### Exactness\nEvery finite-dimensional Banach space embeds (1 + *ɛ*)-isomorphically into a space of the form ℓ∞*n*, but the analogous property in the operator space setting does not hold in general: an operator space *E* is said to be **C*-exact* if for any *ɛ* > 0 every finite-dimensional subspace of *E* is (*C* + *ɛ*)-completely isomorphic to a subspace of an *M**n* space. When an operator space is 1-exact we simply say that it is exact.\nRemark [rmk: bad news] gives an unsurprising example where this causes a significant difference between the classical tensor norm theory and the one for operator spaces.\nAnother useful property (for Banach tensor products arguments) which is not transferable to operator spaces due to exactness issues is the one proved in about finite dimensional pieces of a quotient:\nThe following statement is false:\n> \n> For every complete quotient between Banach operator spaces $q : E \\twoheadrightarrow F$, every *F*0 ∈ OFIN(*F*), and every *ɛ* > 0, there exists *E*0 ∈ OFIN(*E*) such that *q*(*E*0) = *F*0 and for every *y* ∈ *M**n*(*F*0) there exists *x* ∈ *M**n*(*E*0) such that *q**n*(*x*) = *y* and ∥*x*∥*M**n*(*E*0) ≤ (1 + *ɛ*)∥*y*∥*M**n*(*F*0).\n> \n> \n> \nFirst let us remark that if *E*0 ∈ OFIN(*E*) satisfies the conditions in the statement, then it is clear that *F*0 is (1 + *ɛ*)-completely isomorphic to $E\\_0/\\ker({{ \\left.\\kern-\\nulldelimiterspace q \\right|\\_{E\\_0} }})$ via the canonical map $E\\_0/\\ker({{ \\left.\\kern-\\nulldelimiterspace q \\right|\\_{E\\_0} }}) \\to F\\_0$ induced by ${{ \\left.\\kern-\\nulldelimiterspace q \\right|\\_{E\\_0} }}$.\nLet *F*0 be a finite-dimensional operator space. By, there exists a complete quotient $q : S\\_1 \\twoheadrightarrow F\\_0$. Given *ɛ* > 0, suppose that there exists *E*0 ∈ OFIN(*S*1) satisfying the conditions in the statement.\nSince *E*0 ∈ OFIN(*S*1), we can find *G*0 ∈ OFIN(*S*1) such that *E*0 ⊆ *G*0 and *G*0 is (1 + *ɛ*)-completely isomorphic to *S*1*N* for some *N* ∈ N using perturbation arguments as in. First, take a basis of *E*0 and approximate it by finitely supported matrices. This yields a subspace *E*1 ⊂ *S*1 which is completely isomorphic to *E*0 and naturally sits inside a subspace *G*1 ⊂ *S*1 that is completely isometric to *S*1*N* for some *N*. Perturbing *G*1 to “put *E*1 back onto *E*0” yields the desired *G*0.\nBecause *G*0 ⊇ *E*0, it is clear that *G*0 also satisfies the conditions in the statement. This gives that *F* is (1 + *ɛ*)2-completely isomorphic to a quotient of *S*1*N*, and therefore *F*ʹ is (1 + *ɛ*)2-completely isomorphic to a subspace of *M**N*.\nBut this is, in general, false: from, for any *n* ≥ 2 any complete isomorphism from Max(ℓ1*n*) onto a subspace of an *M**N* space has constant at least $\\frac{n}{2\\sqrt{n-1}}$.\nAs one would expect, this idea of approximating quotients using finite-dimensional pieces is useful when investigating projectivity of Banach space tensor norms. For example, it is used in to show that a tensor norm is projective when its dual tensor norm is injective. Not only does this argument not work in the operator space setting, but the corresponding result is in fact not true: see Remark [consecuencias min] below, where we also point out that a related duality result for completely injective and completely projective hulls claimed without proof in does not hold.\n### Approximation properties of completely injective spaces\nEach Banach space *E* isometrically embeds into ℓ∞(*B**E*ʹ), which is an injective space in the Banach space setting. The fact that ℓ∞(*B**E*ʹ) has the approximation property plays a relevant role in many tensor product results in the Banach space framework, especially those related with the Approximation Lemma and completely injective tensor norms. In contrast, for an operator space *E* the standard complete isometry into a completely injective space is the one into $\\mathcal B(H)$ for some Hilbert space *H*. The lack of the approximation property in $\\mathcal B(H)$ is another important dissimilarity between both frameworks.\nFinite and cofinite hulls\n=========================\nGiven an o.s. tensor norm *α* on OFIN, we can use the following two procedures to extend it to the class of all operator spaces.\n[def-finite-and-cofinite-hulls] Given operator spaces *E* and *F*, and *u* ∈ *M**n*(*E* ⊗ *F*), let\n \n$$\\begin{aligned}\n\\overrightarrow{\\alpha}\\_n(u ; E, F) &= \\inf\\left\\{\\alpha\\_n(u; E\\_0,F\\_0) \\colon E\\_0 \\in \\operatorname{OFIN}(E), F\\_0 \\in\\operatorname{OFIN}(F), u \\in M\\_n(E\\_0 \\otimes F\\_0)\\right\\}\\end{aligned}$$ \nand\n \n$$\\begin{aligned}\n\\overleftarrow{\\alpha}\\_n(u ; E, F) = \\sup\\left\\{\\alpha\\_n\\big( (q\\_K^E \\otimes q\\_L^F)\\_n(u); E/K,F/L\\big) \\colon K \\in \\operatorname{OCOFIN}(E), L \\in\\operatorname{OCOFIN}(F) \\right\\}\\end{aligned}$$ \nAn o.s. tensor norm *α* on ONORM is called *finitely-generated* if *α* = *α⃗*, and *cofinitely-generated* if *α* = *α⃖*.\nClearly, if *α* ≤ *c**β* it follows that *α⃖* ≤ *c**β⃖* and *α⃗* ≤ *c**β⃗*. Since min and *h* are completely injective, $\\min = \\overrightarrow{\\min}$ and *h* = *h⃗*. Observe also that the description of the projective norm as an infimum gives $\\operatorname{proj}= \\overrightarrow{\\operatorname{proj}}$.\nFor the same reason, any *λ*-o.s. tensor norm is finitely generated.\n[prop-order-of-hulls] Let *α* be an o.s. tensor norm on OFIN. Then the *finite hull* *α⃗* of *α* and the *cofinite hull* *α⃖* of *α* are o.s. tensor norms on ONORM with \nmin ≤ *α⃖* ≤ *α⃗* ≤ proj, *α⃖*|OFIN = *α⃗*|OFIN = *α*. \n If *α* is defined on ONORM (and not just on OFIN) then \n*α⃖* ≤ *α* ≤ *α⃗*. \nThe complete metric mapping property of *α* yields *α⃖* ≤ *α* ≤ *α⃗*; the remarks before the statement of the Proposition then show min ≤ *α⃖* ≤ *α⃗* ≤ proj. Since it is obvious that *α* agrees with *α⃖* and *α⃗* on finite-dimensional spaces, the only thing left to prove is that both hulls are in fact o.s. tensor norms. But we have already proved that they are between min and proj, so we just have to check the complete metric mapping property.\nTo that effect, let *S* ∈ CB(*E*1, *E*2), *T* ∈ CB(*F*1, *F*2) and *u* ∈ *M**n*(*E*1 ⊗ *F*1). Then, using the complete metric mapping property of *α*,\n \n$$\\begin{gathered}\n\\overrightarrow{\\alpha}\\_n\\big( (S\\otimes T)\\_n u ; E\\_2, F\\_2\\big) \\\\\n= \n \\begin{aligned}[t] \n\\inf\\Big\\{\\alpha\\_n\\big((S\\otimes T)\\_n u; E^2\\_0,F^2\\_0\\big) \\colon E^2\\_0 \\in \\operatorname{OFIN}(E\\_2), F^2\\_0 \\in\\operatorname{OFIN}(F\\_2), (S\\otimes T)\\_nu \\in M\\_n(E^2\\_0 \\otimes F^2\\_0)\\Big\\} \n\\end{aligned}\n\\\\\n\\le \n \\begin{aligned}[t] \n \\inf\\Big\\{\\alpha\\_n\\big((S\\otimes T)\\_nu&; SE^1\\_0,TF^1\\_0\\big) \\colon \\\\ &E^1\\_0 \\in \\operatorname{OFIN}(E\\_1), F^1\\_0 \\in\\operatorname{OFIN}(F\\_1), u \\in M\\_n(E^1\\_0 \\otimes F^1\\_0)\\Big\\}\n\\end{aligned}\n\\\\\n\\le { \\left\\|S\\right\\| }\\_{\\operatorname{cb}}{ \\left\\|T\\right\\| }\\_{\\operatorname{cb}} \n \\begin{aligned}[t] \n\\inf\\Big\\{\\alpha\\_n\\big(u; &E^1\\_0,F^1\\_0\\big) \\colon \\\\&E^1\\_0 \\in \\operatorname{OFIN}(E\\_1), F^1\\_0 \\in\\operatorname{OFIN}(F\\_1), u \\in M\\_n(E^1\\_0 \\otimes F^1\\_0)\\Big\\} \n\\end{aligned}\n\\\\\n= { \\left\\|S\\right\\| }\\_{\\operatorname{cb}}{ \\left\\|T\\right\\| }\\_{\\operatorname{cb}} \\overrightarrow{\\alpha}\\_n\\big( u ; E\\_1, F\\_1\\big),\\end{gathered}$$ \nwhich shows the complete metric mapping property for *α⃗*.\nNow, consider *K*02 ∈ OCOFIN(*E*2) and *L*02 ∈ OCOFIN(*F*2). Observe that *S*− 1*K*02 ∈ OCOFIN(*E*1) and *T*− 1*L*02 ∈ OCOFIN(*F*1). Moreover, by the basic properties of quotients there exist maps *S**K*02 : *E*1/*S*− 1*K*02 → *E*2/*K*02 and *T**L*02 : *F*1/*T*− 1*L*02 → *F*2/*L*02 making the following diagrams commutative \n$$\\xymatrix{\n E\\_1 \\ar[d]\\_{ q^{E\\_1}\\_{S^{-1}K^2\\_0} } \\ar[r]^{S} & E\\_2 \\ar[d]^{ q^{E\\_2}\\_{K^2\\_0} }\\\\\n E\\_1/S^{-1}K^2\\_0 \\ar[r]\\_{S\\_{K^2\\_0}} & E\\_2/K^2\\_0\n }\n \\qquad \\qquad\n \\xymatrix{\n F\\_1 \\ar[d]\\_{ q^{F\\_1}\\_{T^{-1}L^2\\_0} } \\ar[r]^{T} & F\\_2 \\ar[d]^{ q^{F\\_2}\\_{L^2\\_0} }\\\\\n F\\_1/T^{-1}L^2\\_0 \\ar[r]\\_{T\\_{L^2\\_0}} & F\\_2/L^2\\_0\n }$$ \n and moreover we have that ∥*S**K*02∥cb = ∥*q**K*02*E*2*S*∥cb ≤ ∥*S*∥cb and ∥*T**L*02∥cb = ∥*q**L*02*F*2*T*∥cb ≤ ∥*T*∥cb. Therefore, once again using the complete metric mapping property of *α*,\n \n$$\\begin{gathered}\n \\overleftarrow{\\alpha}\\_n\\big( (S\\otimes T)\\_n u ; E\\_2, F\\_2\\big) \\\\\n =\n \\begin{aligned}[t]\n \\sup\\Big\\{\\alpha\\_n\\big( \\big(q^{E\\_2}\\_{K^2\\_0} \\otimes q^{F\\_2}\\_{L^2\\_0}\\big)\\_n (S\\otimes T)\\_n (u) &; E\\_2/K^2\\_0,F\\_2/L^2\\_0\\big) \\colon \\\\ \n &K^2\\_0 \\in \\operatorname{OCOFIN}(E\\_2), L^2\\_0 \\in\\operatorname{OCOFIN}(F\\_2) \\Big\\}\n \\end{aligned}\n \\\\\n = \n \\begin{aligned}[t] \n \\sup\\Big\\{\\alpha\\_n\\big( (S\\_{K^2\\_0}\\otimes T\\_{L^2\\_0})\\_n \\big(q^{E\\_1}\\_{S^{-1}K^2\\_0} \\otimes &q^{F\\_1}\\_{T^{-1}L^2\\_0} \\big)\\_n (u); E\\_2/K^2\\_0,F\\_2/L^2\\_0\\big) \\\\\n &\\colon K^2\\_0 \\in \\operatorname{OCOFIN}(E\\_2), L^2\\_0 \\in\\operatorname{OCOFIN}(F\\_2) \\Big\\} \n \\end{aligned}\n \\\\\n \\le { \\left\\|S\\right\\| }\\_{\\operatorname{cb}}{ \\left\\|T\\right\\| }\\_{\\operatorname{cb}} \n \\begin{aligned}[t] \n \\sup\\Big\\{\\alpha\\_n\\big( \\big(q^{E\\_1}\\_{K^1\\_0} \\otimes q^{F\\_1}\\_{L^1\\_0} \\big)\\_n &(u); E\\_1/K^1\\_0,F\\_1/L^1\\_0\\big) \\colon\\\\ \n &K^1\\_0 \\in \\operatorname{OCOFIN}(E\\_1), L^1\\_0 \\in\\operatorname{OCOFIN}(F\\_1) \\Big\\}\n \\end{aligned}\n \\\\\n = { \\left\\|S\\right\\| }\\_{\\operatorname{cb}}{ \\left\\|T\\right\\| }\\_{\\operatorname{cb}} \\overleftarrow{\\alpha}\\_n\\big( u ; E\\_1, F\\_1\\big),\\end{gathered}$$ \nwhich shows the complete metric mapping property for *α⃖*.\nAs a consequence of the previous proposition, since we already observed that $\\min = \\overrightarrow{\\min}$ it follows that $\\min = \\overrightarrow{\\min}= \\overleftarrow{\\min}$.\nWe will see also in Remark [h cofin] that the Haagerup o.s. tensor norm satisfies *h* = *h⃗* = *h⃖*.\nNext we show that, just as in the classical theory, the projective norm is not cofinitely-generated.\n[proj no es cofinitamente generada]\n1. Let *W* be a Banach space and *F* a normed operator space. Then ${\\operatorname{Max}(W) \\otimes\\_{\\overleftarrow{\\operatorname{proj}}} F}$ and *W* ⊗ *π⃖**F* are isometrically isomorphic as Banach spaces.\n2. $\\overleftarrow{\\operatorname{proj}} \\neq \\operatorname{proj}$.\n(*a*) Let *u* ∈ *W* ⊗ *F*. From the definitions,\n \n$$\\begin{gathered}\n\\overleftarrow{\\operatorname{proj}}(u ; \\operatorname{Max}(W), F) = \\sup\\big\\{\\operatorname{proj}\\big( (q\\_K^W \\otimes q\\_L^F)(u); \\operatorname{Max}(W)/K,F/L\\big) \\colon \\\\ K \\in \\operatorname{OCOFIN}(\\operatorname{Max}(W)), L \\in\\operatorname{OCOFIN}(F) \\big\\}\\end{gathered}$$ \nand\n \n$$\\begin{aligned}\n\\overleftarrow{\\pi}(u ; W, F) =\n\\sup\\left\\{\\pi\\big( (q\\_K^W \\otimes q\\_L^F)(u); W/K,F/L\\big) \\colon K \\in \\operatorname{COFIN}(W), L \\in \\operatorname{COFIN}(F) \\right\\}.\\end{aligned}$$ \nFrom we have that Max(*W*/*K*) = Max(*W*)/*K*. Therefore, it follows from that \n$$\\operatorname{proj}\\big( (q\\_K^W \\otimes q\\_L^F)(u); \\operatorname{Max}(W)/K,F/L\\big) = \\pi\\big( (q\\_K^W \\otimes q\\_L^F)(u); W/K,F/L\\big)$$ \n and thus $\\overleftarrow{\\operatorname{proj}}(u ; \\operatorname{Max}(W), F) = \\overleftarrow{\\pi}(u ; W, F)$.\n(*b*) Since *π⃖* ≠ *π*, there exist Banach spaces *W* and *V* such that *W* ⊗ *π**V* and *W* ⊗ *π⃖**V* are different; it then follows from part (*a*) and that ${\\operatorname{Max}(W) \\otimes\\_{\\overleftarrow{\\operatorname{proj}}} \\operatorname{Max}(V)}$ and Max(*W*) ⊗ projMax(*V*) are different (even just as Banach spaces).\nThe right-finite hull is presented in Definition [def:right-finite hull]. The left-finite hull can be defined analogously.\n[prop-dp-and-gp-finitely-generated] For 1 ≤ *p* ≤ ∞, the o.s. tensor norms *d**p* and *g**p* are finitely generated.\nBy symmetry, it suffices to prove it for *d**p*. Observe that the result will follow immediately if we can prove that for any *u* ∈ *M**n*(*E* ⊗ *F*) we have\n \n(*d**p*)*n*(*u*; *E*, *F*) = inf{∥*w*∥*M**n*((*S**p*ʹ*k* ⊗ min*E*) ⊗ proj*S**p**k*[*F*]) : *k* ∈ N, (*q**p*, *k*)*n**w* = *u*} \nwhere *q**p*, *k* : (*S**p*ʹ*k* ⊗ min*E*) ⊗ proj*S**p**k*[*F*] → *E* ⊗ *F* is the tensor contraction. Indeed, let *ɛ* > 0 be given and choose *k* ∈ N, *w* ∈ *M**n*((*S**p*ʹ*k* ⊗ min*E*) ⊗ proj*S**p**k*[*F*]) such that (*q**p*, *k*)*n**w* = *u* and ∥*w*∥*M**n*((*S**p*ʹ*k* ⊗ min*E*) ⊗ proj*S**p**k*[*F*]) ≤ (1 + *ɛ*)(*d**p*)*n*(*u*; *E*, *F*). Since proj is finitely generated, we can 1 + *ɛ* approximate the norm of *w* by passing to finite-dimensional subspaces of *S**p*ʹ*k* ⊗ min*E* and *S**p**k*[*F*]. Without loss of generality, by enlarging the subspaces if necessary, we can assume that the aforementioned finite-dimensional subspaces are of the form *S**p*ʹ*k* ⊗ min*E*0 and *S**p**k*[*F*0] for some *E*0 ∈ OFIN(*E*), *F*0 ∈ OFIN(*F*), and also that *u* ∈ *M**n*(*E*0 ⊗ *F*0). It now follows that \n(*d**p*)*n*(*u*; *E*0, *F*0) ≤ (1 + *ɛ*)2(*d**p*)*n*(*u*; *E*, *F*), \n showing that *d**p* is finitely generated.\nLet us now prove. Recall that the tensor contraction ${q}^p : \\big( S\\_{p'}\\widehat{\\otimes}\\_{\\min} E \\big) \\widehat{\\otimes}\\_{\\operatorname{proj}} S\\_p[F] \\to E \\widehat{\\otimes}\\_{d\\_p } F$ is a complete 1-quotient. By dualizing, there is a completely isometric embedding $(E \\otimes\\_{d\\_p } F)' \\hookrightarrow \\operatorname{CB}(S\\_{p'}\\widehat{\\otimes}\\_{\\min} E, S\\_{p'}[F'])$. Since the union of the spaces *S**p*ʹ*k* ⊗ min*E* is dense in $S\\_{p'}\\widehat{\\otimes}\\_{\\min} E$, norms in $\\operatorname{CB}(S\\_{p'}\\widehat{\\otimes}\\_{\\min} E, S\\_{p'}[F'])$ (and more generally, norms of matrices over $\\operatorname{CB}(S\\_{p'}\\widehat{\\otimes}\\_{\\min} E, S\\_{p'}[F'])$) can be calculated by taking the supremum over the restrictions to CB(*S**p*ʹ*k* ⊗ min*E*, *S**p*ʹ*k*[*F*ʹ]), as *k* ranges over N. Thus, the norm of *u* in *M**n*(*E* ⊗ *d**p**F*) can be calculated as the supremum of the norms of all pairings of *u* against a *φ* which is a matrix of norm at most one in $\\operatorname{CB}(S\\_{p'}\\widehat{\\otimes}\\_{\\min} E, S\\_{p'}[F'])$, and by the above we can consider instead *φ* to be a matrix of norm at most one in CB(*S**p*ʹ*k* ⊗ min*E*, *S**p*ʹ*k*[*F*ʹ]), but this is exactly the same as the right-hand side of.\n[prop-symmetrized-Haagerup-finitely-generated] The symmetrized Haagerup o.s. tensor norms *h* ∩ *h**t* and *h* + *h**t* are finitely generated.\nSince *h* and *h**t* are completely injective, it is clear that so is *h* ∩ *h**t* and therefore the latter is finitely generated.\nConsider operator spaces *E* and *F*, *u* ∈ *M**n*(*E* ⊗ *F*), and *ɛ* > 0. Let *v*, *w* ∈ *M**n*(*E* ⊗ *F*) such that *u* = *v* + *w* and \n$$\\|(v,w)\\|\\_{M\\_n\\big( (E \\otimes\\_h F)\\oplus\\_1(E \\otimes\\_{h^t} F) \\big)} < (h+h^t)\\_n(u; E,F) + \\varepsilon.$$ \n Let *E*0 ∈ OFIN(*E*) and *F*0 ∈ OFIN(*F*) such that *v*, *w* ∈ *M**n*(*E*0 ⊗ *F*0). Since *h* and *h**t* are both completely injective, and so are ℓ1-sums (this is clear from the description of ℓ1-sums in Section [direct-sums] and the injectivity of B(*H*)), we have that \n$$\\|(v,w)\\|\\_{M\\_n\\big( (E\\_0 \\otimes\\_h F\\_0)\\oplus\\_1(E\\_0 \\otimes\\_{h^t} F\\_0) \\big)} = \\|(v,w)\\|\\_{M\\_n\\big( (E \\otimes\\_h F)\\oplus\\_1(E \\otimes\\_{h^t} F) \\big)}.$$ \n Since $(h+h^t)\\_n(u; E\\_0, F\\_0) \\le \\|(v,w)\\|\\_{M\\_n\\big( (E\\_0 \\otimes\\_h F\\_0)\\oplus\\_1(E\\_0 \\otimes\\_{h^t} F\\_0) \\big)}$, combining the inequalities gives (*h* + *h**t*)*n*(*u*; *E*0, *F*0) < (*h* + *h**t*)*n*(*u*; *E*, *F*) + *ɛ* which implies that *h* + *h**t* is finitely generated.\nThe five basic lemmas\n=====================\nIn the theory of tensor products of normed spaces, “The Five Basic Lemmas” (see Section 13 in Defant and Floret’s book ) are rather simple results which turn out to be “basic for the understanding and use of tensor norms”. Namely, they are the *Approximation Lemma*, the *Extension Lemma*, the *Embedding Lemma*, the *Density Lemma* and the *Local Technique Lemma*. We present here the analogous results for the operator space setting and also exhibit some applications as example of their potential. Our presentation follows the lines of. Although the proofs are similar to the Banach space case, the operator space nature of our tensor products introduces some difficulties, as we can see, for example, in our version of the Extension Lemma [extension-lemma], whose proof involves additional hypothesis of local reflexivity.\nThe approximation lemma\n-----------------------\n[approximation-lemma] Let *α* and *β* be o.s. tensor norms (on ONORM), *E* and *F* normed operator spaces, *c* ≥ 1 and \n*α* ≤ *c**β* on *E* ⊗ *N* \n for cofinally many *N* ∈ OFIN(*F*). If *F* has the completely bounded approximation property with constant *C* ≥ 1, then \n*α* ≤ *C**c**β*on *E* ⊗ *F*. \nFix *n* ∈ N and *z* ∈ *M**n*(*E* ⊗ *F*), and take *ɛ* > 0. Since *F* has the *C*-completely bounded approximation property, there is a net (*T**η*)*η* of finite rank operators in CB(*F*, *F*) with cb-norm bounded by *C* such that ∥(*T**η*)*n**x* − *x*∥ → 0 for every *x* ∈ *M**n*(*E*). Therefore (*i**d**E* ⊗ *T**η*)*n*(*z*) converges to *z* in the projective norm, and thus also in the *α* norm. Hence we have *α**n*(*z* − (*i**d**E* ⊗ *T**η*)*n*(*z*); *E*, *F*) < *ɛ* for some *η* large enough. If we take a subspace *N* containing *T**η*(*E*) satisfying the hypothesis of the lemma, by the complete metric mapping property of the o.s. tensor norm *β* we have\n \n$$\\begin{aligned}\n \\alpha\\_n(z;E,F) & \\leq \\alpha\\_n(z-(id\\_E \\otimes T\\_\\eta)\\_n(z) ; E,F ) + \\alpha\\_n((id\\_E \\otimes T\\_\\eta)\\_n(z) ; E,F ) \\\\\n & \\leq \\varepsilon + \\alpha\\_n((id\\_E \\otimes T\\_\\eta)\\_n(z) ; E,N ) \\\\\n & \\leq \\varepsilon + c \\beta\\_n((id\\_E \\otimes T\\_\\eta)\\_n(z) ; E,N ) \\\\\n & \\leq \\varepsilon + c { \\left\\|T\\_\\eta: E \\to N \\right\\| }\\_{\\operatorname{cb}} \\beta\\_n(z; E,F) \\\\\n & \\leq \\varepsilon + C c \\beta\\_n(z; E,F).\n \\end{aligned}$$ \nSince this holds for every *ɛ* > 0, we have *α**n*(*z*; *E*, *F*) ≤ *C**c**β**n*(*z*; *E*, *F*).\nBy Proposition [prop-order-of-hulls] and the previous lemma we obtain the following.\n[cor-cbap-implies-equivalence-of-hulls] Let *α* be an o.s. tensor norm (on OFIN), *E* and *F* normed operator spaces with the completely bounded approximation property with constants *C**E* and *C**F*, respectively. Then \n*α⃖* ≤ *α⃗* ≤ *C**E**C**F**α⃖* on *E* ⊗ *F*. \n In particular, if *E* and *F* both have the completely metric approximation property then *α⃖* = *α⃗* on *E* ⊗ *F*.\nThe extension lemma\n-------------------\nEvery *φ* ∈ (*E* ⊗ proj*F*)ʹ = CB(*E*, *F*ʹ) has a canonical extension *φ*∧ ∈ (*E* ⊗ proj*F*ʺ)ʹ = CB(*E*, *F*ʺ) which satisfies for every *x* ∈ *E* and *y*ʺ ∈ *F*ʺ the relation \n⟨*φ*∧, *x* ⊗ *y*ʺ⟩ = ⟨*L**φ**x*, *y*ʺ⟩*F*ʹ, *F*ʺ, \n where *L**φ* ∈ CB(*E*, *F*ʹ) is the linear map associated with *φ*. The extension lemma deals with this situation for more general o.s. tensor norms. Unlike in the Banach space case, local reflexivity does not come for free and we are forced to require it. The item (*a*) of the following statement previously appeared in. We include the proof here for the reader’s benefit. Also, the proof of item (*b*) is inspired by ) which presents the same argument just for the norm proj.\n[extension-lemma] (Right extension lemma) Let *α* be a o.s. tensor norm (on ONORM), *E* and *F* be normed operator spaces and let *φ* ∈ (*E* ⊗ proj*F*)ʹ. Then \n*φ* ∈ (*E* ⊗ *α**F*)ʹif and only if *φ*∧ ∈ (*E* ⊗ *α**F*ʺ)ʹ \n and the map *φ* ↦ *φ*∧ is a complete isometry from (*E* ⊗ *α**F*)ʹ to (*E* ⊗ *α**F*ʺ)ʹ in the following cases:\n1. *α* is finitely-generated and *F* is locally reflexive.\n2. *α* is a *λ*-o.s. tensor norm.\nThe complete metric mapping property implies that $id\\_E\\otimes \\kappa\\_F:E \\otimes\\_\\alpha F \\hookrightarrow E \\otimes\\_\\alpha F''$ is a complete contraction, and hence so is the restriction map (*i**d**E* ⊗ *κ**F*)ʹ : (*E* ⊗ *α**F*ʺ)ʹ → (*E* ⊗ *α**F*)ʹ. This shows that for any $(\\varphi\\_{ij}) \\in M\\_m\\big( (E\\otimes\\_{\\operatorname{proj}} F)' \\big)$, \n∥(*φ**i**j*)∥*M**m*((*E* ⊗ *α**F*)ʹ) ≤ ∥(*φ**i**j*∧)∥*M**m*((*E* ⊗ *α**F*ʺ)ʹ). \n(*a*) Now consider *n* ∈ N, *z*0 ∈ *M**n*(*E* ⊗ *F*ʺ), *M* ∈ OFIN(*E*) and *N* ∈ OFIN(*F*ʺ) such that *z*0 ∈ *M**n*(*M* ⊗ *N*). Since *F* is locally reflexive, by given *ɛ* > 0 there exists *R* ∈ CB(*N*, *F*) with ∥*R*∥cb ≤ 1 + *ɛ* such that for every *y*ʺ ∈ *N*, *x* ∈ *M* and 1 ≤ *i*, *j* ≤ *m* \n⟨*y*ʺ, *L**φ**i**j**x*⟩*F*ʺ, *F*ʹ = ⟨*R**y*ʺ, *L**φ**i**j**x*⟩*F*, *F*ʹ. \n This means that \n⟨*φ**i**j*∧, *x* ⊗ *y*ʺ⟩ = ⟨*φ**i**j*, (*i**d**E* ⊗ *R*)(*x* ⊗ *y*ʺ)⟩, \n thus \n⟨⟨(*φ**i**j*∧), *z*0⟩⟩ = ⟨⟨(*φ**i**j*), (*i**d**E* ⊗ *R*)*z*0⟩⟩ \n and hence\n \n$$\\begin{aligned}\n { \\left\\| {{\\langle\\langle (\\varphi\\_{ij}^\\wedge), z\\_0 \\rangle\\rangle}} \\right\\| }\\_{M\\_{mn}} &\\le { \\left\\| (\\varphi\\_{ij}) \\right\\| }\\_{ M\\_m( (E\\otimes\\_{\\alpha} F)' ) } \\alpha\\_n((id\\_E \\otimes R)z\\_0; E, F) \\\\\n &\\le\n { \\left\\| (\\varphi\\_{ij}) \\right\\| }\\_{ M\\_m( (E\\otimes\\_{\\alpha} F)' ) }\n { \\left\\|id\\_E \\otimes R\\right\\| }\\_{\\operatorname{cb}} \\alpha\\_n(z\\_0 ; E, N) \\\\\n &\\le\n { \\left\\| (\\varphi\\_{ij}) \\right\\| }\\_{ M\\_m( (E\\otimes\\_{\\alpha} F)' ) }\n { \\left\\|R\\right\\| }\\_{\\operatorname{cb}} \\alpha\\_n(z\\_0 ; E, N) \\\\\n &\\le\n { \\left\\| (\\varphi\\_{ij}) \\right\\| }\\_{ M\\_m( (E\\otimes\\_{\\alpha} F)' ) }\n (1+\\varepsilon) \\alpha\\_n(z\\_0 ; E, N)\n \\end{aligned}$$ \nwhich implies the result since *α* is finitely-generated.\n(*b*) Recall that by Theorem [duality-lambda-tensor] we can identify (*E* ⊗ *λ**F*)ʹ = CB*λ*(*E* × *F*), so all we need to show is that the map *φ* ↦ *φ*∧ gives a completely isometric embedding CB*λ*(*E* × *F*) → CB*λ*(*E* × *F*ʺ). Notice that at the beginning of the proof it has already been shown that this is a complete contraction. That we get a complete isometry is then clear from the definition of ∥ ⋅ ∥cb, *λ* and the fact that the unit ball of *M**k*(*F*) is weak\\*-dense in that of *M**k*(*F*ʺ) = *M**k*(*F*)ʺ by Goldstine’s theorem.\nIn a similar way a left extension for each *φ* ∈ (*E* ⊗ proj*F*)ʹ is defined: let ∧*φ* ∈ (*E*ʺ ⊗ *α**F*)ʹ given by, for *x*ʺ ∈ *E*ʺ and *y* ∈ *F*, \n⟨∧*φ*, *x*ʺ ⊗ *y*⟩ = ⟨*x*ʺ, *R**φ**y*⟩*E*ʺ, *E*ʹ, \n where *R**φ* ∈ CB(*F*, *E*ʹ) is the mapping *R**φ*(*y*)(*x*) = ⟨*φ*, *x* ⊗ *y*⟩.\n[left-extension-lemma] (Left extension lemma) Let *α* be a o.s. tensor norm (on ONORM), *E* and *F* be normed operator spaces and let *φ* ∈ (*E* ⊗ proj*F*)ʹ. Then \n*φ* ∈ (*E* ⊗ *α**F*)ʹif and only if ∧*φ* ∈ (*E*ʺ ⊗ *α**F*)ʹ. \n and the map *φ* ↦ ∧*φ* is a complete isometry from (*E* ⊗ *α**F*)ʹ to (*E*ʺ ⊗ *α**F*)ʹ in the following cases:\n1. *α* is finitely-generated and *E* is locally reflexive.\n2. *α* is a *λ*-o.s. tensor norm.\nNow, both procedures can be applied to a given *φ* ∈ (*E* ⊗ *α**F*)ʹ to obtain two (possibly different) extensions to (*E*ʺ ⊗ *α**F*ʺ)ʹ.\n[cor-extension] Let *α* be a o.s. tensor norm (on ONORM), *E* and *F* be normed operator spaces and let *φ* ∈ (*E* ⊗ proj*F*)ʹ. Then the following maps from (*E* ⊗ *α**F*)ʹ to (*E*ʺ ⊗ *α**F*ʺ)ʹ are complete isometries: \n*φ* ↦ ∧(*φ*∧) and *φ* ↦ (∧*φ*)∧ \n in the following cases:\n1. *α* is finitely-generated and *E* and *F* are locally reflexive.\n2. *α* is a *λ*-o.s. tensor norm.\nAs we have already mentioned, the fact that we do not need local reflexivity in the extension lemmas if the o.s. tensor norm involved is proj was proved in. In the same article this fact is also shown for the Haagerup o.s. tensor norm *h* using that it satisfies the following property: for any operator spaces *E* and *F* and *φ* ∈ (*E* ⊗ *h**F*)ʹ, there is a unique extension $\\widetilde\\varphi \\in (E'' \\otimes\\_h F'')'$ which is separately *w*\\*-continuous, see.\nA word can be said about the equality/non-equality of the two extensions. Repeating the arguments of we obtain the analogous result in our setting. That is, given *φ* ∈ (*E* ⊗ proj*F*)ʹ, we have ∧(*φ*∧) = (∧*φ*)∧ if and only if *L**φ* : *E* → *F*ʹ is weakly compact. This is also equivalent to the existence of an extension of *φ* to (*E*ʺ ⊗ proj*F*ʺ)ʹ that is separately *w*\\*-continuous. Note that, by the comment above, if *φ* ∈ (*E* ⊗ *h**F*)ʹ then *L**φ* is weakly compact.\nNote that we have the separating duality pair $\\big\\langle E \\otimes F'', (E\\otimes\\_{\\alpha}F)' \\big\\rangle$ given by \n⟨*z*, *φ*⟩ = *φ*∧(*z*). \nUsing the bipolar theorem it is easy to see that, in the Banach space realm, the unit ball *B**E* ⊗ *α**F* is *σ*(*E* ⊗ *F*ʺ, (*E* ⊗ *α**F*)ʹ)-dense in the unit ball *B**E* ⊗ *α**F*ʺ, whenever *α* is a finitely generated tensor norm. In other words, given *z* ∈ *B**E* ⊗ *α**F*ʺ there is a net (*u**η*)*η* ⊂ *B**E* ⊗ *α**F* such that for every *φ* ∈ (*E* ⊗ *α**F*)ʹ, \n*φ*(*u**η*) → *φ*∧(*z*). \nIn our setting, we can consider for each *n* ∈ N, the following duality pair:\n \n$$\\big\\langle M\\_n(E \\otimes F''),S\\_1^n[(E\\otimes\\_{\\alpha}F)'] \\big\\rangle$$ \n defined as \n⟨(*z**i**j*), (*φ**i**j*)⟩ = ∑*i*, *j**φ**i**j*∧(*z**i**j*). \nIt is easy to see that this dual pairing system is separating. Moreover, we have the following density result.\nLet *α* be a o.s. tensor norm (on ONORM), *E* and *F* be normed operator spaces.\nFor every *n* ∈ N, the unit ball *B**M**n*(*E* ⊗ *α**F*) is *σ*(*M**n*(*E* ⊗ *F*ʺ), *S*1*n*[(*E* ⊗ *α**F*)ʹ])-dense in the unit ball *B**M**n*(*E* ⊗ *α**F*ʺ) in the following cases:\n1. *α* is finitely-generated and *F* is locally reflexive.\n2. *α* is a *λ*-o.s. tensor norm.\nSince (*M**n*(*E* ⊗ *α**F*))ʹ is *S*1*n*[(*E* ⊗ *α**F*)ʹ] isometrically, it is clear that the polar (*B**M**n*(*E* ⊗ *α**F*))∘ (in this pairing system) is *B**S*1*n*[(*E* ⊗ *α**F*)ʹ]. By the bipolar theorem it is enough to prove that *B**M**n*(*E* ⊗ *α**F*ʺ) ⊂ (*B**S*1*n*[(*E* ⊗ *α**F*)ʹ])∘. Indeed, given (*z**i**j*) ∈ *B**M**n*(*E* ⊗ *α**F*ʺ) and (*φ**i**j*) ∈ *B**S*1*n*[(*E* ⊗ *α**F*)ʹ]; by the Extension Lemma [extension-lemma] along with we know that (*φ**i**j*∧) ∈ *B**S*1*n*[(*E* ⊗ *α**F*ʺ)ʹ]. Using again (*M**n*(*E* ⊗ *α**F*ʺ))ʹ = *S*1*n*[(*E* ⊗ *α**F*ʺ)ʹ] isometrically, we derive that that \n∣⟨(*z**i**j*), (*φ**i**j*)⟩∣ = ∣∑*i*, *j**φ**i**j*∧(*z**i**j*)∣ ≤ 1. \n This concludes the proof.\nThe embedding lemma\n-------------------\nOperator space tensor norms generally do not respect subspaces, but the embedding into the bidual is respected under certain conditions.\n[embedding-lemma] Let *α* be an o.s. tensor norm (on ONORM). The mapping \n*i**d**E* ⊗ *κ**F* : *E* ⊗ *α**F* → *E* ⊗ *α**F*ʺ \n is a complete isometry in the following cases:\n1. *α* is finitely-generated and *F* is locally reflexive.\n2. *α* is cofinitely-generated.\n3. *α* is a *λ*-o.s. tensor norm.\nLet *n* ∈ N and *z* ∈ *M**n*(*E* ⊗ *α**F*). By the complete metric mapping property \n*i**d**E* ⊗ *κ**F* : *E* ⊗ *α**F* → *E* ⊗ *α**F*ʺ \n is always a complete contraction, so \n*α**n*(*z*; *E*, *F*ʺ) ≤ *α**n*(*z*; *E*, *F*). \n (*a*) By the extension lemma,\n \n$$\\begin{aligned}\n \\alpha\\_n(z ; E, F) &= \\sup \\big\\{ { \\left\\|{{\\langle\\langle z, \\Phi \\rangle\\rangle}}\\right\\| }\\_{M\\_{nm}} \\; : \\; \\Phi \\in M\\_m((E \\otimes\\_\\alpha F )'), { \\left\\|\\Phi\\right\\| }\\_{M\\_m( (E \\otimes\\_\\alpha F )')} \\le 1 \\big\\} \\\\\n &= \\sup \\big\\{ { \\left\\|{{\\langle\\langle z, \\Phi^\\wedge \\rangle\\rangle}}\\right\\| }\\_{M\\_{nm}} \\; : \\; \\Phi \\in M\\_m((E \\otimes\\_\\alpha F )'), { \\left\\|\\Phi\\right\\| }\\_{M\\_m( (E \\otimes\\_\\alpha F )')} \\le 1 \\big\\} \\\\\n &\\le \\sup \\big\\{ { \\left\\|{{\\langle\\langle z, \\Psi \\rangle\\rangle}}\\right\\| }\\_{M\\_{nm}} \\; : \\; \\Psi \\in M\\_m((E \\otimes\\_\\alpha F'' )'), { \\left\\|\\Psi\\right\\| }\\_{M\\_m( (E \\otimes\\_\\alpha F'' )')} \\le 1 \\big\\} \\\\\n &= \\alpha\\_n(z ; E,F''),\n \\end{aligned}$$ \nwhich gives the reverse inequality.\n(*b*) Let *L* ∈ OCOFIN(*F*), then *L*00 (formed in *F*ʺ) is in OCOFIN(*F*ʺ) and the map \n*κ**F*/*L* : *F*/*L* → (*F*/*L*)ʺ = *F*ʺ/*L*00 \n is completely isometric and surjective; moreover, *q**L*00*F*ʺ ∘ *κ**F* = *κ**F*/*L* ∘ *q**L**F*. Therefore,\n \n$$\\begin{aligned}\n \\alpha\\_n\\big((id\\_E \\otimes q\\_L^F)\\_n(z); E,F/L\\big)\n &= \\alpha\\_n\\big((id\\_E \\otimes (\\kappa\\_{F/L} \\circ q^F\\_L) )\\_n(z); E,(F/L)''\\big) \\\\\n &= \\alpha\\_n\\big((id\\_E \\otimes q^{F''}\\_{L^{00}}) \\circ (id\\_E \\otimes \\kappa\\_F) )\\_n(z); E, F''/L^{00} \\big) \\\\\n &\\le \\overleftarrow{\\alpha}\\_n( (id\\_E \\otimes \\kappa\\_F) )\\_n(z); E, F'' ) \\\\\n &= \\overleftarrow{\\alpha}\\_n( z; E, F'' ) = \\alpha\\_n(z ; E, F'')\n \\end{aligned}$$ \nThe same argument with *E*/*K* instead of *E* allows us to conclude *α**n*(*z*; *E*, *F*) ≤ *α**n*(*z*; *E*, *F*ʺ).\n(*c*) The same proof as in the case (*a*) works here because a *λ*-o.s. tensor norm is finitely-generated and the hypothesis of local reflexivity of *F* is not needed for the extension lemma with this norm.\nWe point out that part (c) above is a generalization of, where the spaces are assumed to be *C*\\*-algebras and the embedding is only shown to be an isometry instead of a complete isometry.\nNote that the completion of an operator space *F* has the same bidual as *F*. Therefore, as a consequence of the previous result we have:\n[Cor:extender al completado] Let *α* be an o.s. tensor norm (on ONORM) and let $\\widetilde F$ the completion of a normed operator space *F*. The mapping \n$$id\\_E \\otimes \\iota\\_F : E \\otimes\\_\\alpha F \\to E \\otimes\\_\\alpha \\widetilde F,$$ \n where $\\iota\\_F : F \\to \\widetilde F$ is the canonical inclusion, is a complete isometry with dense range in the following cases:\n1. *α* is finitely-generated and *F* is locally reflexive.\n2. *α* is cofinitely-generated.\n3. *α* is a *λ*-o.s. tensor norm.\nThe density lemma\n-----------------\nWe now state the so-called density lemma, which asserts that a given completely bounded mapping defined on the tensor product of certain normed operator spaces can be extended to a completely bounded mapping defined on the tensor product of the completions of the spaces. As usual, we need a local reflexivity hypothesis for the finitely-generated case. The proof easily follows from Corollary [Cor:extender al completado].\nLet *α* be an o.s. tensor norm (on ONORM), *E*, *F* and *G* normed operator spaces, *E*0 and *F*0 dense subspaces of *E* and *F* respectively. Suppose *T* ∈ *C**B*(*E* ⊗ proj*F*; *G*) such that *T*∣*E*0 ⊗ *α**F*0 ∈ *C**B*(*E*0 ⊗ *α**F*0; *G*). Then *T* ∈ *C**B*(*E* ⊗ *α**F*; *G*) and \n‖*T*‖*C**B*(*E*0 ⊗ *α**F*0; *G*) = ‖*T*‖*C**B*(*E* ⊗ *α**F*; *G*), \n in the following cases:\n1. *α* is finitely-generated and *F* is locally reflexive.\n2. *α* is cofinitely-generated.\n3. *α* is a *λ*-o.s. tensor norm.\nThe OS*p*-local technique lemma\n-------------------------------\nIn the Banach space setting the $\\mathscr L\\_p$-local technique lemma allow us to transfer results from tensor products of ℓ*p**n* spaces to general $\\mathscr L\\_p$-spaces (which, loosely speaking, are space whose finite dimensional subspaces “uniformly look” like ℓ*p**n*’s). In the operator space setting, one of the possible extension of the $\\mathscr L\\_p$-spaces are the OS*p*-spaces (see Section [usual notation] for the definition). For this extension, we have the following local technique lemma.\n[local-technique-lemma] Let *α* and *β* be o.s. tensor norms, *c* ≥ 0 and *E* a normed operator space such that for all *k* ∈ N, \n*α* ≤ *c**β* on *E* ⊗ *S**p**k*. \n Then *α* ≤ *C**c**β⃗* on *E* ⊗ *F* for every OS*p*, *C*-space *F*.\nFor *N* ∈ OFIN(*F*), by we can take a factorization \n$$\\xymatrix{\n N \\ar[rr]^{id\\_N} \\ar[dr]\\_{R} & &F\\\\\n & S\\_p^k \\ar[ru]\\_{S} &\n }$$ \n with ∥*R*∥cb∥*S*∥cb ≤ *C*(1 + *ɛ*). Let *M* : = *S*(*S**p**k*). Then for every *z* ∈ *M**n*(*E* ⊗ *N*),\n \n$$\\begin{gathered}\n \\alpha\\_n(z ; E, M) = \\alpha\\_n( (id\\_E \\otimes (S \\circ R))\\_n z; E, M )\n \\le { \\left\\|S\\right\\| }\\_{\\operatorname{cb}} \\alpha\\_n( (id\\_E \\otimes R)\\_nz ; E, S\\_p^k ) \\\\\n \\le { \\left\\|S\\right\\| }\\_{\\operatorname{cb}} c \\beta\\_n( (id\\_E \\otimes R)\\_n z ; E, S\\_p^k )\n \\le { \\left\\|S\\right\\| }\\_{\\operatorname{cb}} { \\left\\|R\\right\\| }\\_{\\operatorname{cb}} c \\beta\\_n(z ; E, N).\n \\end{gathered}$$ \nThe statement follows.\nNote that more generally the proof above applies whenever *F* has the *γ**p*-AP, thanks to. Also note that the same argument yields versions of the local technique lemma for other operator space versions of the L*p* spaces, e.g. the OL*p* spaces from.\nDual o.s. tensor norms\n======================\nIn a nutshell, the dual o.s. tensor norm *α*ʹ for the o.s. tensor norm *α* is the one that makes \n*E* ⊗ *α*ʹ*F* = (*E*ʹ ⊗ *α**F*ʹ)ʹ \n into a complete isometry for any *E*, *F* ∈ OFIN. More explicitly, when *E*, *F* ∈ OFIN for *z* ∈ *M**n*(*E* ⊗ *F*) we define \n$$\\alpha'\\_n(z ; E,F) := \\sup \\big\\{ { \\left\\| {{\\langle\\langle u, z \\rangle\\rangle}} \\right\\| }\\_{M\\_{mn}} \\mid \\alpha\\_m(u ; E', F') \\le 1 \\big\\}.$$ \nLet us check that, for example, the minimal and projective o.s. tensor norms are in duality with each other. Consider finite-dimensional operator spaces *E* and *F*. The operator space structure on *E* ⊗ min*F* is, by definition, the one induced by the embedding *E* ⊗ min*F* → CB(*E*ʹ, *"},"data_id":{"kind":"string","value":"arxiv_0003280"}}},{"rowIdx":797,"cells":{"content":{"kind":"string","value":"The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regionsI. Free waves\n===================================================================================================================================================\nIn particular, tidal inertial waves may strongly interact with zonal flows at the so-called corotation resonances, where the wave’s Doppler-shifted frequency cancels out. The energy dissipation at such resonances could deeply modify the orbital and spin evolutions of tidally interacting systems. In this context, we aim to provide a deep physical understanding of the dynamics of tidal waves at corotation resonances, in the presence of differential rotation profiles that are typical of low-mass stars and giant planets. In this work, we have developed an analytical local model of an inclined shearing box describing a small patch of the differentially rotating convective zone of a star or a planet. We investigate the propagation and the transmission of free inertial waves at corotation, and more generally at critical levels, which are singularities in the governing wave differential equation. Through the construction of an invariant called the wave action flux, we identify different regimes of wave transmission at critical levels, which are confirmed with a one-dimensional three-layer numerical model. We find that inertial waves can be either fully transmitted, strongly damped, or even amplified after crossing a critical level. The occurrence of these regimes depends on the assumed profile of differential rotation, on the nature as well as the latitude of the critical level, and on wave parameters such as the inertial frequency and the longitudinal and vertical wavenumbers. Waves can thus either deposit their action flux to the fluid when damped at critical levels, or they can extract action flux to the fluid when amplified at critical levels. Both situations could lead to significant angular momentum exchange between the tidally interacting bodies. \nIntroduction\n============\nTidal interactions are known to drive the late evolution of short-period planetary systems, like Hot-Jupiters orbiting around their host star, and in our solar system the satellites around Jupiter and Saturn. In particular, the dissipation of tides in the convective envelope of low-mass host stars and giant planets can modify the spin of the tidally pertubed body, the orbital period and the spin-orbit angle of the perturber. Inertial waves, which are driven by tidal forcing and restored by the Coriolis acceleration, are an important source of tidal dissipation in stellar and planetary convective zones, where the action of turbulent motions on tidal flows is most often modelled as an effective frictional force or a viscous force with an effective viscosity that is much larger than the molecular viscosity. For coplanar and circular systems, inertial waves are excited so long as the companion orbits beyond half its corotation radius (the orbit where the host’s rotation frequency is equal to the mean motion). Low-mass stars from K to F spectral type and giant gaseous planets both harbour a convective envelope surrounding a radiative and a solid (or diluted) core, respectively. In these objects, inertial waves then propagate in a spherical shell and do not form regular normal modes of oscillation as in spherical. In contrast, they can focus on limit cycles also called attractors of characteristics that are confined within the convective envelope. With a non-zero viscosity, attractors take the form of shear layers where the tidal wave’s energy and angular momentum can be deposited by viscous dissipation. Besides, viscous dissipation across shear layers can be more important as viscosity is weaker, as demonstrated notably by, and. In that respect, tidal dissipation of inertial waves can compete with the dissipation of gravito-inertial waves in the radiative core or be greater by several orders of magnitude than the dissipation of equilibrium tidal flows in the convective zone. The dissipation of tidally-forced waves can have a great impact on the orbital and rotational evolution of the system. Moreover, the dissipation of the stellar dynamical and equilibrium tides varies significantly along the evolution of the star, and is highly dependent on stellar parameters like the mass, the angular velocity, and the metallicity of stars. This makes desirable the inclusion of all stellar processes on tidal interaction, in particular differential rotation.\nThe frequency-averaged tidal dissipation is often used to quantify the response of a body subject to tidal perturbations. Yet, the dissipation of a tidally-forced inertial wave is strongly correlated with the presence of an attractor at a specific eigenfrequency of the spherical shell. Tidal dissipation at a given frequency may then alter differently each orbital and spin elements of the two-body systems as postulated for instance by to explain the survival of hot-Jupiters with completely damped spin-orbit angle, and revisited by with an improved treatment of dynamical tides in the convective region. In addition in the context of Jupiter and Saturn moon systems, and also investigated the dependence in frequency of tidal dissipation to explain rapid outward migration of the moons, through resonant locking of tidally-forced internal modes in the giant gaseous planets. This concept could for example explain the high dissipation observed in Saturn as derived from astrometric measurements at the frequency of Rhea, and at the frequency of Titan.\nFurthermore, the fact that all layers in a star or a planet do not rotate at the same speed, i.e. differential rotation, is rarely taken into account in the determination of tidal dissipation. Yet, differential rotation seems ubiquitous in low-mass stars and giant gaseous planets. The Sun’s surface is rotating in ∼ 25 days at the equator versus ∼ 35 days near the poles, and a latitude-dependent rotational gradient has also been observed in the Sun’s convective envelope thanks to helioseismology. Through asteroseismology, latitudinal shears have been found to be comparable to that of the Sun for Sun analogs, and can be even larger for solar-like stars. Essentially, differential rotation in low-mass stars depends on the effective temperature, and seems to be more important as the convective envelope is thinner. Solar-like and anti-solar-like (with faster poles and slower equator) rotation profiles are expected for G and K-type stars based on 3D numerical simulations, while cylindrical rotation profile is expected for fast rotators. Regarding giant gaseous planets in our solar system, the extent of zonal winds, which are visible on their surface as running lengthwise bands, has been recently constrained by the probes Cassini and Juno. They extend to $3000\\,\\kilo\\meter$ depth for Jupiter, while they penetrate down to 9000 km in Saturn. Thus, the outermost molecular convective envelopes are the seat of cylindrical differential rotation.\nThe study of the impact of differential rotation on the propagation and dissipation properties of inertial modes of oscillation began with the work of. They examined the impact of either a shellular (radial) or a cylindrical rotation profile on free inertial waves in an incompressible background, by means of a Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) linear analysis for an inviscid fluid and by solving the linearised hydrodynamics equations for a viscous fluid via a spectral code. Their linear analysis highlighted major differences compared to the case of solid-body rotation. Two regimes of propagation have been found, in which inertial modes of oscillation can develop along curved paths of characteristic in the entire convective shell (which the authors named D modes), or in a restricted region of the convective shell, encompassed between a turning surface and one of the shell’s boundaries (DT modes). Compared to solid-body rotation, the frequency range of propagation of inertial modes is broader. also pointed out strong dissipation of wave energy at corotation resonances where the Doppler-shifted wave frequency vanishes within the fluid. All these new properties have been retrieved by, who in turn examined a conical (latitudinal) rotation profile, which is typical of low-mass (F- to K-type) stars. They also confirmed the existence of unstable inertial modes (i.e., modes with positive growth rate) at corotation resonances, which were found only for shellular rotation in. Tidal forcing of inertial waves with conical rotation has been introduced by within a linear numerical exploration, which also underlined the strong dissipation of inertial waves at corotation resonances, particularly at low viscosities. also studied tidally-forced inertial waves, but through non-linear numerical simulations. Differential rotation was triggered in their simulations by tidal waves depositing energy and angular momentum in an initially uniformly rotating spherical shell. In some cases, they observed hydrodynamical shear instabilities when the Ekman number (the ratio between the viscous and Coriolis accelerations) is sufficiently small.\nUnderstanding how inertial waves interact with corotation resonances is thus a key issue in quantifying tidal dissipation, especially since waves may deeply interact with the background flow at this particular location, which in turn may alter the background flow. In binary systems and for late-type stars, have shown that the angular momentum transported by gravity waves and exchanged at corotation can lead to the successive synchronisation of the layers, from the base to the top of the radiative envelope. More generally, a body of work in various domains from astrophysical disks to geophysical fluid dynamics has tried to understand the properties of wave propagation and dissipation around corotation, and more generally at all special locations in fluids that correspond to singularities in the linear wave propagation equation. We will refer to them as critical levels in the following, or to critical layers in the case of a viscous medium. This distinction is analogous to that between shear layers and attractors of characteristics that are kind of singularities for the governing equation of inertial waves in a spherical shell. The aforementioned singularities can act very differently, with either severe absorption at the critical level, or no attenuation if the wave propagates in a peculiar direction. In other cases, a critical level may even give rise to wave amplification under certain conditions related to the first and second derivatives of the mean flow velocity. These studies have in common the use of an invariant quantity (the Reynolds stress or the wave action for rotating or magnetic flows) as a diagnostic tool to interpret the role of the critical level in terms of energy transmission and to quantify exchanges between the wave and the mean flow.\nIn light of these various studies, it is necessary to consider carefully corotation in differentially rotating convective zones. A local model can notably allow us a detailed understanding of physical processes at critical levels. While the propagation through a critical level of gravito-inertial waves in stratified shear flows and of Rossby waves in baroclinic and barotropic flows has been largely studied in the past decades, the behaviour of inertial waves in a latitudinal sheared flow with critical levels has been poorly investigated so far. This is why we develop in this work a local Cartesian shearing box model to understand the complex interplay between tidal waves and zonal flows near critical levels. The concept of a shearing box has been introduced by to investigate the interactions between and convective motions. In our model, we focus on latitudinal differential rotation of the mean flow, varying the box orientation to model either cylindrical or conical rotation. The behaviour of free inertial waves in this framework is then examined near critical levels using both analytical and numerical approaches.\nThis paper is organised as follows. In Sect. [sec2], we describe the local shear model with its main assumptions and the system of governing equations. In Sect. [sec3], we establish a second-order ordinary differential equation (ODE) for the latitudinal perturbed velocity, and we derive the propagation properties of inertial waves for an inviscid fluid. This is solved near each critical level for both conical and cylindrical rotation profiles, and we interpret energy flux exchanges between the waves and the mean flow. We use in Sect. [secnum] a three-layer numerical model to test our analytical predictions at critical levels. Viscosity is included and non-linear mean flow profiles are also used. Astrophysical applications are discussed in Sect. [astrodi] with implications for low-mass stars hosting close exoplanets and giant gaseous planets in our solar system. In sect. [conclu], we summarise the main results of the paper, and discuss some perspectives and caveats.\nLocal Cartesian model including differential rotation\n=====================================================\n [box]\nPresentation of the model\n-------------------------\nThe local model takes the form of an inclined sheared box, centred at a point *C* of a convective shell, as illustrated in Fig. [box]. The inclined box model has already been used by to characterise analytically the properties of tidal gravito-inertial waves in the presence of viscous and thermal diffusion in stably stratified or convective regions, and by in layered semi-convective regions in giant planets interiors. The local coordinate system (*x*, *y*, *z*) corresponds to the local azimuthal, latitudinal and radial direction of global spherical coordinates, respectively, as presented in Table [tabcorres]. The mean flow velocity ${{\\bm{U}}}$ is directed along the local azimuthal axis ${{{\\bm{e}}}}\\_x$ (we neglect possible meridional flows) and differential rotation is embodied by a latitudinal shear ∂*y**U*. As the box is tilted by an angle *θ*0 relative to the rotation axis, the rotation vector in the local coordinate system is\n \n$$2{{\\bm{\\Omega}}}\\_0=(0,2\\Omega\\_0\\sin\\theta\\_0,2\\Omega\\_0\\cos\\theta\\_0)=2\\Omega\\_0\\,(0,{\\widetilde{f}},f),$$ \nwhere Ω0 is the rotation frequency of the star at the pole, ${\\widetilde{f}}$ and *f* are the normalised horizontal and vertical Coriolis components, respectively. Note that the inclusion of both these components means that we go beyond the traditional *f*-plane approximation. Furthermore, we make several hypotheses to model wave propagation in a latitudinal shear flow. The buoyancy acceleration is kept in the fluid equations for the background flow. the vector ${{\\bm{g}}}$ is supposed to be uniform and constant in the whole box. This requires that the typical length of the box *L* *L* ≪ *H**p*, where *H**p* = − *p*(d*z*/d*p*) is the vertical pressure scale height, with *p* the pressure. We can assume this because tidally excited waves are expected to have small-scale structures. Moreover, the dimensions of the box are chosen to be small compared to the depth of the convective envelope so as to remove curvature effects.\nMean flow profile\n-----------------\nIn global spherical geometry, the mean flow based on a conical rotation profile Ω(*θ*) is written :\n \n$${{\\bm{u}}}=\\boldsymbol\\Omega\\times{{\\bm{r}}}=r\\sin\\theta\\Omega(\\theta){{\\bm{e}}}\\_\\varphi,\n\\label{mean\\_flow\\_g}$$ \nwhere ${{\\bm{e}}}\\_\\varphi$ is the azimuthal unit vector, *r* and *θ* are the radius and colatitude, respectively. We introduce ${{\\bm{u}}}\\_0=r\\sin\\theta\\Omega\\_0{{\\bm{e}}}\\_\\varphi$ the mean flow at a point *M* inside the box (see Fig. [box]) without differential rotation, where we remind that Ω0 is the spin frequency at the pole. We shall also use the shear contrast *δ*Ω = Ω(*θ*) − Ω0, i.e. the difference between the angular frequency at colatitude *θ* and at the pole. The shear contrast is positive for the Sun since the equator rotates faster than the pole, and negative for anti-solar-like rotating stars. Using the notations of Fig. [box], the centre *C* of the box is located at a distance *r*0sin*θ*0 from the rotation axis. Accordingly, the of the point *M* in the local frame is\n \n*y* = *r*sin(*θ*0 − *θ*). \nIt should be noted that the radial coordinate *r* of the point *M* in spherical geometry can be written as *r* = *r*0 + *z*. Nevertheless, we neglect vertical displacements in the expression of the local shear, because we are interested in how the (one-dimensional) horizontal shear affects the wave dynamics while a lot of studies on differential rotation in stars have focused on the vertical shear. Since *y*/*r* and so *θ*0 − *θ* are small, we have written in Table [tabcorres] the correspondences in terms of mean flows and shears between the two geometries.\nc|cc Geometry &Local Cartesian &Global Spherical \n---\nBasis &$({{\\bm{e}}}\\_x,{{\\bm{e}}}\\_y, {{\\bm{e}}}\\_z)$ & $({{\\bm{e}}}\\_\\varphi,-{{\\bm{e}}}\\_\\theta, {{\\bm{e}}}\\_r)$ \n---\n| |\n| --- |\n| Conical coordinate |\n& *y*/*r*0 & *θ*0 − *θ* \n---\nMean flow & ${{\\bm{{U}}}}$ & ${{\\bm{u}}}-{{\\bm{u}}}\\_0$ \n---\nConical shear & ∂*y**U* & − ∂*θ*(sin*θ**δ*Ω) \n[tabcorres]\nAs an example, the shear contrast from solid-body rotation used by was:\n \n*δ*Ω(*θ*) = Ω0*χ*sin2*θ*, \nwhere *χ* is the magnitude of the shear between the equator and the pole. Performing a second-order Taylor expansion around a fixed colatitude *θ*0, such that *θ* = *θ*0 − *y*/*r*0 and at a specified depth *r*0 inside the convective region, the local mean flow *U* can be recast as\n \n$$\\begin{aligned}\nu-u\\_0& =r\\_0\\Omega\\_0\\chi\\sin^3\\theta\\\\\n&\\simeq \\Omega\\_0\\chi\\left[r\\_0\\sin^3\\theta\\_0-3\\cos\\theta\\_0\\sin^2\\theta\\_0 y\\textcolor{white}{\\frac{3}{2}}\\right.\\\\\n&\\hspace{1cm}\\left.+\\frac{y^2}{r\\_0}\\left(3\\cos\\theta^2\\_0\\sin\\theta\\_0-\\frac{3}{2}\\sin^3\\theta\\_0\\right)+O(y^3)\\right].\n\\end{aligned}\n\\label{U\\_conic}$$ \nWe point out that the Taylor expansion must be pushed further at the pole *θ*0 = 0 (and at the pole *θ*0 = *π* with an opposite sign):\n \n$$u-u\\_0\\simeq-\\frac{\\Omega\\_0\\chi}{r\\_0^2}y^3+O(y^4).\n\\label{cubicpro}$$ \nAccordingly, we can approximate a conical shear as a linear mean flow at the first order when the box is tilted. We recall that conical shear has been observed in the solar convective zone and is expected in slowly and in moderately rotating solar-like stars. When the box is at the pole, *y* becomes the distance from the rotation axis (called hereafter axial distance). Thus, the mean flow mimics a cylindrical differential rotation that can be modelled using a cubic profile in *y* given Eq. ([cubicpro]). This rotation profile is found in Jupiter and Saturn, as well as in rapidly rotating stars as demonstrated for instance by and.\nSystem of equations\n-------------------\nTo derive the system of governing equations for tidal waves in the local reference frame, we made several hypotheses. Stratification terms, which usually drive the propagation of internal gravity waves, have been kept for clarity sake and will be carefully kept or removed after applying the Boussinesq approximation and setting the equations for inertial waves. Moreover, we assume that the action of turbulence can be modelled as a Rayleigh friction term in the momentum equation with an effective *σ**f*. This of the fluid equations compared to the usual modelling of turbulence as an effective viscous force. The momentum, continuity and thermodynamic equations for tidal waves in a differentially rotating Cartesian framework thus are:\n \n$$\\begin{aligned}\n{\\frac{\\mathrm{d}{{\\bm{u}}}}{\\mathrm{d}t}}+2{{\\bm{\\Omega}}}\\_0\\times{{\\bm{u}}}&= -\\frac{{{{\\bm{\\nabla}}}}p}{\\rho}+{{\\bm{g}}}-{\\sigma\\_{f}}{{\\bm{u}}}+{{\\bm{f}}} \\label{eq:1},\\\\\n\\frac{\\partial\\rho}{\\partial t}+{{{\\bm{\\nabla}}}}\\cdot(\\rho{{\\bm{u}}}) &=0\\label{eq:2},\\\\\n{\\frac{\\mathrm{d}\\rho}{\\mathrm{d}t}} -\\frac{1}{c\\_\\mathrm{s}^2}{\\frac{\\mathrm{d}p}{\\mathrm{d}t}}&=0\\label{eq:3},\\end{aligned}$$ \nwhere ${{\\bm{u}}}$, *p*, *ρ*, and ${{\\bm{f}}}$ denote the velocity, pressure, density and volumetric tidal forcing, respectively. We have also introduced *c*s the sound speed and is the total derivative operator.\nAll variables are then linearised at first order: zero-order terms correspond to background equilibrium quantities while first-order terms represent the leading perturbation. The local velocity, density and pressure are therefore written:\n \n$$\\left\\{\\begin{aligned}\n{{\\bm{u}}}&={U}(y)\\,{{\\bm{e}}}\\_x + {{\\bm{u}}}'\\\\\\\n\\rho&={\\rho\\_0}+\\epsilon\\rho'\\\\\np&=p\\_0+\\epsilon p'\n\\end{aligned}\\right.,$$ \nwhere ${{\\bm{u}}}=(u,v,w)$ in the local Cartesian basis. We have introduced the dimensionless parameter *ε*\n \n$$\\epsilon=\\frac{(2\\Omega\\_0)^2 L}{g},$$ \nwhere we have used 1/(2Ω0) a characteristic time scale and *L* a characteristic length scale of the mean flow. These notations are based on those of, and adapted to our model. In the following, we will work with dimensionless variables using the above scaling, including $2\\Omega\\_{{\\textcolor{black}{0}}} L$ to scale velocity and *ρ*T*g**L* to scale pressure, with *ρ*T the reference density. The dimensionless momentum equation of the mean flow is:\n \n$$\\epsilon\\,{{{\\bm{n}}}}\\times{{\\bm{{U}}}}=-\\frac{{{{\\bm{\\nabla}}}}p\\_0}{{\\rho\\_0}}-{{\\bm{e}}}\\_z,\n\\label{eq0}$$ \nwith ${{{\\bm{n}}}}$ the unit vector parallel to the rotation axis. Projecting Eq. ([eq0]) into Cartesian coordinates, one can derive:\n \n$$\\left\\{\\begin{aligned}\n\\partial\\_xp\\_0&=0,\\\\\n\\partial\\_yp\\_0&=-\\epsilon{\\rho\\_0}f{U},\\\\\n\\partial\\_zp\\_0&=-{\\rho\\_0}+\\epsilon{\\rho\\_0}{\\widetilde{f}}{U}.\n\\end{aligned}\\right.$$ \nAt the leading order in *ε*, one can recognise the hydrostatic balance, and at the first-order the geostrophic balance. We underline that tending *ε* to zero is similar to assuming the Boussinesq approximation. Indeed, all density variations are neglected, except the ones involved in the buoyancy force. The dimensionless Brunt-Väisälä frequency is\n \n$$N^2=\\epsilon^{-1}\\left(F\\frac{\\partial\\_zp\\_0}{{\\rho\\_0}}-\\partial\\_z\\ln{\\rho\\_0}\\right),$$ \nwhere we have introduced the dimensionless number *F* = *g**L*/*c*s2, which is small when filtering acoustic waves. Consequently, the curl of Eq. ([eq0]) gives\n \n$$\\left\\{\\begin{aligned}\n&\\partial\\_x{\\rho\\_0}=0, \\\\\n&\\partial\\_y\\ln{\\rho\\_0}=\\epsilon\\left({\\widetilde{f}}\\partial\\_y{U}-fUF\\right),\\\\\n&\\partial\\_z\\ln{\\rho\\_0}=\\epsilon\\left(F{\\widetilde{f}}U-N^2\\right)-F,\n\\end{aligned}\\right.\n\\label{rhorel}$$ \nwhere we neglect the second-order terms in *ε*.\nNow, we make several assumptions to treat the propagation of inertial waves. As the convective motions are essentially adiabatic, the convective zone can be assumed neutrally stratified to a first approximation. Hence, the Brunt-Väisälä frequency *N* is cancelled out in the third density relationship Eq. ([rhorel]). Moreover, we make the Boussinesq approximation, which means that we terms in *ε* and *F* in the final set of perturbed equations. Thus, the dimensionless linearised momentum, continuity, and thermodynamic equations are finally:\n \n$$\\begin{aligned}\n{\\frac{\\mathrm{d}{{\\bm{u}}}'}{\\mathrm{d}t}}+v\\partial\\_y{{\\bm{U}}}+{{{\\bm{n}}}}\\times{{\\bm{u}}}'&= -\\frac{{{{\\bm{\\nabla}}}}p'}{{\\rho\\_0}}-\\frac{\\rho'}{\\rho\\_0}{{\\bm{e}}}\\_z-{\\sigma\\_{f}}{{\\bm{u}}}'+{{\\bm{f}}} \\label{pert:1},\\\\\n{{{\\bm{\\nabla}}}}\\cdot{{\\bm{u}}}' &=0,\\label{pert:2}\\\\\n{\\frac{\\mathrm{d}\\rho'}{\\mathrm{d}t}}&=-\\rho\\_0v{\\widetilde{f}}\\partial\\_y{U}{\\textcolor{black}{,}} \\label{rhop}\\end{aligned}$$ \n. We emphasise that, although vertical stratification has been filtered in the limit *N* goes to zero, an horizontal stratification term remains in Eq. ([rhop]). As a result, we consider the inertial waves propagating in the inclined shear box where the mean flow is maintained by the thermal-wind balance.\nEquilibrium state of the background flow\n----------------------------------------\nIt is noteworthy to discuss the choice of keeping buoyancy forces in the zero-order momentum equation. Without gravitational forces, the momentum equation for mean dimensional variables is written as a geostrophic balance:\n \n$$2{\\rho\\_0}{{\\bm{\\Omega}}}\\_{{\\textcolor{black}{0}}}\\times{{\\bm{U}}}=-{{{\\bm{\\nabla}}}}p\\_0.\n\\label{stab}$$ \nThis balance satisfies the Taylor-Proudman theorem, namely the geostrophic flow is independent of the coordinate parallel to the rotation axis. When taking the x-axis (the only non-zero) projection of the curl of this equation, one gets the following relationship:\n \n$$(2{{\\bm{\\Omega}}}\\_{{\\textcolor{black}{0}}}\\cdot{{{\\bm{\\nabla}}}})({\\rho\\_0}{U})=0.\n\\label{relstab}$$ \nWithout vertical stratification embodied by the Brunt-Väisälä frequency, nor latitudinal stratification, so for an incompressible fluid, the equilibrium of mean flow is not ensured. An alternative to conserve the equilibrium without stratification would be to consider a *z*-dependence of the mean flow. Such other possibility is not considered in this paper since we are mainly interested in latitudinal mean flow profiles. Furthermore, in addition to maintaining differential rotation, the latitudinal stratification can allow to construct an invariant that is useful for studying energy transfer at critical levels: the wave action flux. This will further discussed in Sect. [sec3]. Lastly, since ${\\widetilde{f}}=0$ at the poles, the latitudinal stratification term will not appear in the perturbed fluid equations (as we can see from Eq. ([rhop])).\nDynamics of inertial waves at critical levels: analytical predictions\n=====================================================================\nIn this section, we investigate analytically the behaviour of inertial waves at critical levels in a non-dissipative fluid at various colatitudes. For this purpose, we consider perturbations *q* in the normal mode\n \n*q*(*x*, *y*, *z*, *t*) = *q*(*y*)exp*i*(*k**x**x* + *k**z**z* − *ω**t*) + c.c. \nwith *ω* the complex inertial frequency, *k**x* and *k**z* the real streamwise and vertical wavenumbers, respectively, and c.c. the complex conjugate.\n equation\n---------\nUsing the modal form ([q]) for *ρ*, *p* and ${{\\bm{u}}}$, we solve the set of hydrodynamic equations, Eqs. ([pert:1]) to ([rhop]), for the latitudinal velocity *v*. Considering free inertial waves (i.e. without forcing terms), the set of perturbation equations can be recast into a single second-order ODE for *v*:\n \n*A**v*ʺ + *B**v*ʹ + *C**v* = 0, \nwhere the prime now denotes the derivative according to *y*, and *A*, *B*, and *C* are the coefficients that can be simplified without friction as follows:\n \n$$\\begin{aligned}\nA=&\\sigma^2-{\\widetilde{f}}^2,\\\\\nB=&-2k\\_x{\\widetilde{f}}^2\\frac{{U}'}{\\sigma}-2ik\\_z{\\widetilde{f}}f,\\\\\nC=&-k\\_\\perp^2\\sigma^2+k\\_z^2f(f-{U}')-2ik\\_xk\\_z{\\widetilde{f}}f\\frac{U'}{\\sigma}\\\\\n&-2k\\_x^2{\\widetilde{f}}^2\\frac{{U}'^2}{\\sigma^2}+\\frac{k\\_x{U}''}{\\sigma}\\left(\\sigma^2-{\\widetilde{f}}^2\\right),\n\\label{coefs}\n\\end{aligned}$$ \nwhere $k\\_{\\perp}=\\sqrt{k\\_x^2+k\\_z^2}$ is the absolute wavenumber in the direction perpendicular to the *y*-direction. We refer the reader to the Appendix [derPc] for the detailed ODE derivation with and tidal source terms. Eq. ([Pc]) becomes singular when *A* = 0 or *σ* = 0, and these singular points are called critical levels. The critical level where the Doppler-shifted frequency equals to zero (i.e. *σ* = 0) can be met when the mean flow matches the local phase velocity, and is also known as corotation resonance. When the Coriolis acceleration is not taken into account, as to treat internal gravity waves, the corotation resonance is the unique critical level. At colatitudes other than the poles, the critical levels come in three flavours, the corotation *σ* = 0 and two other critical levels that are defined, in our model, by $\\sigma=\\pm{\\widetilde{f}}$ (where we remind that ${\\widetilde{f}}$ in the latitudinal component of the rotation vector). These critical levels were similarly reported for vertical shear flows as in the studies of and for vertical and inclined rotation vectors, respectively. In these work, the Doppler-shifted frequency at critical levels other than the corotation resonance equals to ± 2Ω*v* where Ω*v* is the vertical component of the rotation vector.\nPropagation properties\n----------------------\n### Dispersion relation, group and phase velocities\nThe 3D dispersion relation is a fourth-order equation in Doppler-shifted frequency when injecting wave-like solutions in the three directions *x*, *y*, and *z* in Eq. ([Pc]). In order to understand the main properties of waves at the critical level, we make the short-wavelength approximation as in and in the. This involves keeping only the second-order derivatives in the *y* and *z* directions and it reduces the relation dispersion to a second-order equation when injecting plane wave-like solutions. In the local meridional plane, the equation reduces to :\n \n$$\\left(\\sigma^2-{\\widetilde{f}}^2\\right)\\partial\\_{y,y}v-2{\\widetilde{f}}f\\partial\\_{y,z}v+\\left[\\sigma^2+f\\left({U}'-f\\right)\\right]\\partial\\_{z,z}v=0,$$ \nMoreover, we set *v* ∝ exp − *i*(*k**z**z* − *k**x**x*) so as to write the wave dispersion relation for the Doppler-shifted frequency *σ*:\n \n$$\\sigma^2=\\frac{1}{||{{\\bm{k}}}||^2}\\left[\\left({{{\\bm{n}}}}\\cdot{{\\bm{k}}}\\right)^2-k\\_z^2f{U}'\\right],$$ \nwhere $||{{\\bm{k}}}||=k\\_y^2+k\\_z^2$ is the norm of the wave vector in the plane (e.g. for fixed *k**x*) like in. Compared to solid-body rotation, an additional term (*k**z*2*f**U*ʹ) is present, which accounts for the latitudinal shear., we therefore introduce\n \n$$\\gamma=\\sqrt{({{{\\bm{n}}}}\\cdot{{\\bm{k}}})^2-k\\_z^2f{U'}}.$$ \nWe can then explicit the phase velocity in the plane:\n \n$${{\\bm{v}}}\\_\\phi=\\frac{\\sigma}{{||{{\\bm{k}}}||}^2}{{{\\bm{k}}}}=\\pm \\frac{\\gamma}{||{{\\bm{k}}}||^3}{{{\\bm{k}}}}.\n\\label{vp}$$ \nIn the same way, we can derive the expression for the group velocity in the plane:\n \n$${{\\bm{v}}}\\_\\mathrm{g}={{\\bm{\\nabla}}}\\_{{{\\bm{k}}}}\\sigma=\\pm\\frac{\\gamma}{||{{\\bm{k}}}||^3}\\left\\{-{{\\bm{k}}}+\\left(\\frac{||{{\\bm{k}}}||}{\\gamma}\\right)^2\\left[{{{{\\bm{n}}}}}({{{\\bm{n}}}}\\cdot{{\\bm{k}}})-k\\_zf{U'}{{\\bm{e}}}\\_z\\right]\\right\\}.\n\\label{vg}$$ \nNote that without differential rotation, the group velocity reduces to its well-known expression for solid-body rotation :\n \n$${{{\\bm{v}}}\\_\\mathrm{g}}=\\pm \\frac{{{{\\bm{k}}}}\\wedge({{{{\\bm{n}}}}\\wedge{{\\bm{k}}}})}{||{{{\\bm{k}}}}||^3}.$$ \nMoreover, as in solid-body rotation, the group velocity (Eq. [vg]) and the phase velocity (Eq. [vp]) lie in perpendicular planes: ${{\\bm{v}}}\\_\\mathrm{g}\\cdot{{\\bm{v}}}\\_\\phi=0$.\nWhen the box is located at the North pole (*θ*0 = 0 in Fig. [box]), by setting *κ* = 1 − *U*ʹ we recover\n \n$$\\left\\{\n\\begin{aligned}\n\\sigma^2&=\\frac{k\\_z^2}{||{{\\bm{k}}}||^2}\\kappa^2\\\\\n{{\\bm{v}}}\\_\\phi&=\\pm\\kappa\\frac{k\\_z{{\\bm{k}}}}{||{{\\bm{k}}}||^3}\\\\\n{{\\bm{v}}}\\_\\mathrm{g}&=\\pm\\kappa\\frac{k\\_y}{||{{\\bm{k}}}||^3}(-k\\_z{{\\bm{e}}}\\_y+k\\_y{{\\bm{e}}}\\_z)\n\\end{aligned}\\right.$$ \nas in or, where *κ* can be identified to their epicyclic frequency and *k**y* corresponds to the cylindrical component of the wavenumber (*k**s*).\n### Phase and group velocity at singularities\nWe derive in this section the conditions required to meet singularities in terms of wavenumbers and shear, and examine what the implications are for the phase and group velocities. When the box is inclined, for *σ* → 0 we must have:\n* *γ* → 0, meaning that $\\bm v\\_\\phi\\rightarrow\\bm0$ while $\\ |\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_y|\\rightarrow\\infty$ and $|\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_z|\\rightarrow\\infty$.\n found similar results by studying the propagation of free inertial waves in a global frame with conical shear, namely when their parameter B (which is homogeneous to a frequency and equivalent to our *γ* parameter) goes to zero, the group velocity goes to infinity while the phase velocity cancels out. According to their work, an inertial wave may propagate across the corotation. \nNow to get $\\sigma\\rightarrow\\pm{\\widetilde{f}}$, we either need:\n* ∣*k**y*∣ → ∞ at fixed *k**z*, which implies $\\bm v\\_\\phi \\rightarrow\\bm0$ and $\\bm v\\_\\mathrm{g}\\rightarrow\\bm0$ and means that inertial waves cannot get through the critical level,\n* ∣*k**z*∣ → 0 at fixed *k**y*, which gives $\\bm v\\_\\phi\\cdot\\bm e\\_z \\rightarrow0$ and $\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_y\\rightarrow0$ while $|\\bm v\\_\\phi\\cdot\\bm e\\_y|\\rightarrow{\\widetilde{f}}/k\\_y$, and $|\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_z|\\rightarrow f/k\\_y$: the wave may then cross the critical level with some preferential direction.\nAgain, these conditions share some similarities with those observed for corotation in a global spherical geometry. The first aforementioned possibility (first item above) is analogous to the global phase and group velocities tending to zero when *k**s* → ∞, with *k**s* the axial wavenumber in cylindrical coordinates. This makes sense since the axial distance is *s* = *r*sin*θ*, and *y* ∼ *r*0(*θ*0 − *θ*) here. However, the second aforementioned condition (second item above) is slightly different from both these previous works, in that ∣*k**z*∣ → 0 at fixed *k**s*, with *k**z* the global vertical wavenumber, i.e. along the rotation axis unlike our local vertical wavenumber *k**z* along the spherical radial coordinate.\nWe point out that the singularities at $\\sigma=\\pm{\\widetilde{f}}$ arise in our model because the rotation vector is inclined with respect to the local vertical axis of the box. In the global model of, three conditions for a wave to meet the corotation exist, and these conditions are actually quite similar to the three above conditions for waves in our model to interact either with the corotation *σ* = 0 or the other critical levels at $\\sigma=\\pm{\\widetilde{f}}$. Hence, the local critical levels at $\\sigma=\\pm{\\widetilde{f}}$ behave partly like the corotation in the global framework, as if we partially broke the degeneracy in the local framework of the origin of the corotation found in the global framework.\nWhen the box is at the North pole, the conditions to meet corotation are similar but lead to different relationships for the phase and group velocities:\n* *κ* → 0 (i.e. *U*ʹ → 1) meaning that $\\bm v\\_\\phi\\rightarrow\\bm0$, and $\\bm v\\_\\mathrm{g}\\rightarrow{{\\bm{0}}}$: the wave is totally absorbed at corotation.\n* ∣*k**y*∣ → ∞ at fixed *k**z*, which implies $\\bm v\\_\\phi \\rightarrow\\bm0$ and $\\bm v\\_\\mathrm{g}\\rightarrow\\bm0$: same conclusion as in the previous case and analogous case as when the box is tilted,\n* ∣*k**z*∣ → 0 at fixed *k**y*, which gives ${{\\bm{v}}}\\_\\phi\\rightarrow{{\\bm{0}}}$, $\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_y\\rightarrow0$ while $|\\bm v\\_\\mathrm{g}\\cdot\\bm e\\_z|\\rightarrow \\kappa/k\\_y$: the wave energy does not cross the corotation in the latitudinal direction.\nAt the north pole, we actually have a perfect match with the conditions given by when using a cylindrical rotation profile for the mean flow.\n### Energetical aspects\nIn this section, we examine the energetic balance associated inertial waves in our inclined shear box model, without assuming the short-wavelength approximation. This energetic balance does not include potential energy because of the adiabaticity of the convective region but two additional terms appear compared to the solid-body rotation case, coming from the differential rotation. We denote by *η* and *ζ* the displacements along the vertical and latitudinal directions, respectively. Considering that *ω* = *k**x**c*, where *c* is the longitudinal phase velocity, we can use the first-order definition\n \n$$\\begin{aligned}\nv&={\\frac{\\mathrm{d}\\zeta}{\\mathrm{d}t}}=({U}-c)\\zeta\\_x,\\\\\nw&={\\frac{\\mathrm{d}\\eta}{\\mathrm{d}t}}=({U}-c)\\eta\\_x.\n\\end{aligned}$$ \nIt first allows us to express the perturbed density from Eq. ([rhop]) as $\\rho=-\\rho\\_0\\zeta{\\widetilde{f}}{U}'$. Then, by multiplying the momentum equation ([pert:1]) by ${\\rho\\_0}{{\\bm{u}}}$, we can get the energy balance equation:\n \n$$(U-c)\\partial\\_xe\\_\\mathrm{k}+{{{\\bm{\\nabla}}}\\cdot}(p{{\\bm{u}}})={\\rho\\_0}\\left[-v{U'}u+\\zeta{\\widetilde{f}}{U}' w-{\\sigma\\_{f}}{{\\bm{u}}}^2+{{\\bm{f}}}\\cdot{{\\bm{u}}}\\right]\n\\label{balance\\_e},$$ \nwhere $e\\_\\mathrm{k}={\\rho\\_0}\\bm u^2/2$ is the kinetic energy density, and $p{\\bm u}$ the so-called acoustic flux. We now integrate the above energy balance equation over *x* and *z*, and over one wave period, as the perturbed quantities have a wave-like form in these directions. Further assuming that the box is *δ* thick in the *y*-direction, the energetic balance yields:\n \n$$\\mathcal{P}\\_\\mathrm{ext}=\\mathcal{P\\_\\mathrm{shear}}+D\\_\\mathrm{visc}+\\mathcal{P}\\_\\mathrm{f},\n\\label{power}$$ \nwhere we have introduced from left to right the power of the external pressures at the boundaries − *δ*/2 and *δ*/2 on the perturbed latitudinal flow, the work of the shear, the viscous dissipation, and the forcing power, which read respectively:\n \n$$\\begin{aligned}\n\\mathrm{P}\\_\\mathrm{ext}&=\\left[\\textcolor{white}{{\\widetilde{f}}}\\overline{pv}\\textcolor{white}{{\\widetilde{f}}}\\right]\\_{-\\delta/2}^{\\delta/2},\\\\\n\\mathcal{P\\_\\mathrm{shear}}&=-\\int\\_{-\\delta/2}^{\\delta/2}{\\rho\\_0}{U'}\\left(\\overline{uv}-{\\widetilde{f}}\\,\\overline{w\\zeta}\\right)\\,\\mathrm{d}y,\\\\\nD\\_\\mathrm{visc}&=-\\int\\_{-\\delta/2}^{\\delta/2}{\\rho\\_0}{\\sigma\\_{f}}\\overline{{{\\bm{u}}}^2}\\,\\mathrm{d}y,\\\\ \n\\mathcal{P}\\_\\mathrm{f}&=\\int\\_{-\\delta/2}^{\\delta/2}{\\rho\\_0}\\overline{{{\\bm{f}}}\\cdot{{\\bm{u}}}}\\,\\mathrm{d}y,\n\\end{aligned}$$ \nwhere the bar represents the average in the (*z*, *x*) plane over one period. Note that the energy density and the acoustic flux in the *x* − and *z* − directions drop out in Eq. ([power]) when integrating, because of the wave periodicity in those directions. The quantity Pshear can also be seen as the power transferred from the mean flow to the perturbation (or conversely) by the Reynolds stress:\n \n$$\\tau=-{\\rho\\_0}(\\overline{uv}-{\\widetilde{f}}\\,\\overline{w\\zeta})=-{\\rho\\_0}\\overline{v\\left(u+{\\widetilde{f}}\\eta\\right)},\n\\label{Re}$$ \nwhere we have used partial integration and the periodicity of perturbations in the *x* and *z* directions. At the pole, ${\\widetilde{f}}=0$ so we recover the definition of the Reynolds stress in, who studied the stability of a 2D stratified *y*-sheared flow, i.e. $\\tau=-{\\rho\\_0}\\overline{uv}$. This quantity can also be called the latitudinal flux of horizontal (in the sense (*z*, *x*) plane) momentum in reference to the vertical flux of horizontal momentum in stratified *z*-sheared flows. Moreover, we emphasise that the latitudinal flux of energy $\\overline{pv}$ is not conserved even in the inviscid free-wave problem. This is due to the presence of the shear, as already stated for example by, who studied stratified vertically sheared flows. They underline that when the mean flow varies with height, the kinetic energy of the mean motion can be converted into wave energy. Without and forcing, the *y*-derivative of the latitudinal flux is:\n \n$${\\frac{\\mathrm{d}}{\\mathrm{d}y}}\\overline{pv}=-{\\rho\\_0}{U'}\\overline{v\\left(u+{\\widetilde{f}}\\eta\\right)}.\n\\label{flux}$$ \nUsing the same method as, we multiply the *x*-projection of the inviscid force-free momentum equation by *ζ*:\n \n$$-p\\_x\\zeta=\\rho\\_0({U}-c)\\zeta u\\_x+\\rho\\_0({U}'-f)v\\zeta+{\\rho\\_0}{\\widetilde{f}}w\\zeta.$$ \nBy multiplying by (*U* − *c*), the latitudinal flux of energy can thus be written as:\n \n$$\\overline{pv}=-{\\rho\\_0}({U}-c)\\overline{v\\left(u+{\\widetilde{f}}\\eta\\right)}.$$ \nBy, and by equalising with Eq. ([flux]), one can obtain:\n \n$$-{\\rho\\_0}(U-c)\\frac{\\mathrm{d}}{\\mathrm{d}y}\\left\\{\\overline{v\\left(u+{\\widetilde{f}}\\eta\\right)}\\right\\}=0,\n\\label{rel}$$ \nthat is $(U-c)\\cfrac{\\mathrm{d}\\tau}{\\mathrm{d} y}=0$ with *τ* the Reynolds stress. Eq. ([rel]) is naturally satisfied at corotation, where *U* − *c* = 0, or if the Reynolds stress is uniform. have shown that the Reynolds stress is discontinuous at a critical level, highlighting exchanges between wave energy and the mean flow. Compared to the analysis of for 3D stratified shear flows, Eq. ([rel]) is not vectorial, because our base flow is unidirectional.\n### Polarisation relations\nFor the forthcoming analysis, it is useful to derive expressions of the perturbed projected velocities and the perturbed reduced pressure Π = *p*/*ρ*0, namely the polarisation relations (see Appendix [derPi] for more details). In the inviscid free-wave problem, these perturbed quantities can be written in terms of the latitudinal velocity, its derivative, and the shear:\n \n$$\\begin{aligned}\n\\Pi&=\\frac{1}{\\sigma k\\_\\perp^2}\\left\\{iAv'+\\left[ik\\_x\\frac{{U}'}{\\sigma}A+f(k\\_z{\\widetilde{f}}-ik\\_x\\sigma)\\right]v\\right\\},\\\\\nw&=\\frac{1}{\\sigma k\\_\\perp^2}\\left\\{\\left(i\\sigma k\\_z-k\\_x{\\widetilde{f}}\\right)v'+k\\_x\\left[ik\\_z({U}'-f)-k\\_x{\\widetilde{f}}\\frac{{U}'}{\\sigma}\\right]v\\right\\},\\\\\nu&=\\frac{1}{\\sigma k\\_\\perp^2}\\left\\{\\left(i\\sigma k\\_x+k\\_z{\\widetilde{f}}\\right)v'+k\\_z\\left[ik\\_z(f-{U}')+k\\_x{\\widetilde{f}}\\frac{{U}'}{\\sigma}\\right]v\\right\\}.\n\\end{aligned}\n\\label{pol}$$ \nWithout shear and at *θ*0 = 0, we recover the polarisation relations in the solid-body rotation case.\n### Conservation of the wave action flux\nWhile the latitudinal flux of energy is not conserved in the whole domain, there is a conserved quantity, called the wave action flux as introduced in ’s paper:\n \n$${\\mathcal{A}}=\\frac{\\overline{pv}}{\\sigma}=\\frac{{\\rho\\_0}}{2}\\Re\\left\\{\\frac{\\Pi v^\\*}{\\sigma}\\right\\},\n\\label{A}$$ \nwhich is the latitudinal flux averaged over vertical and longitudinal wavelengths divided by the Doppler-shifted frequency. A general treatment for the derivation of the wave action as a conserved quantity can be similarly found in. The wave action flux is related to the Reynolds stress *τ* as A = − *τ*/*k**x*. By using the expression for the perturbed reduced pressure Π derived in the previous section, the wave action flux now reads:\n \n$${\\mathcal{A}}=\\frac{1}{2}{\\rho\\_0}\\Re\\left\\{\\frac{iAv\\_yv^\\*+f{\\widetilde{f}}k\\_z|v|^2}{\\sigma^2k\\_\\perp^2} \\right\\}.\n\\label{wA}$$ \nUnlike the latitudinal flux of energy, but similarly to the Reynolds stress, this wave action flux is conserved along the latitudinal direction. One can demonstrate that Aʹ = 0 in the whole domain except at critical levels, by using the expression for the reduced pressure in Eqs. ([pol]) and the ([Pc]). Several works have shown that a properly defined (i.e., conserved) angular momentum transport parameter can be found in *z*-sheared mean flows without rotation, with rotation under the traditional approximation, and with rotation under the non-traditional approximation. Verifying the conservation in the whole domain except at critical levels is really important because it brings to the fore energy transfers due to the critical levels. We specify that A is a measure of wave energy through a surface (in the (*z*, *x*) plane) since $\\overline{pv}$ is the energy density transported by the group velocity *V*g in the latitudinal direction. It should be underlined that the wave action flux has been defined in the inviscid limit and is not conserved when the is taken into account.\nInertial waves at critical levels when the box is tilted\n--------------------------------------------------------\nIn this section, we analytically investigate waves passing through the various critical levels in the tilted box. We examine the behaviour of the waves around the corotation *σ* = 0 and the critical levels $\\sigma=\\pm{\\widetilde{f}}$ when the box is tilted (for the corotation when the box is at the pole, see Section [polarcase]).\n### Critical levels at $\\sigma=\\pm{\\widetilde{f}}$\n[ht] [atnudeg]\nIn this subsection, we treat both singularities $\\sigma=\\pm{\\widetilde{f}}$ simultaneously. Although Eq. ([Pc]) does not have analytical solutions in general, it is still possible to study the behaviour of an inertial wave close to the critical levels defined by $\\sigma=\\pm{\\widetilde{f}}$ by approximating the ODE through its first-order Taylor expansion in the vicinity of these singularities, and then by applying the Frobenius method. We introduce *y*±, the location of the related critical level $\\sigma=\\pm{\\widetilde{f}}$. For a linear mean flow profile *U* = Λ*y*, with Λ a constant, *y*± are given by:\n \n$$y\\_\\pm=\\frac{\\omega\\mp{\\widetilde{f}}}{k\\_x\\Lambda}.$$ \nWithout any assumption on the mean flow profile, the first-order Taylor expansion of the ODE ([Pc]) near *y*± is:\n \n$$v''+\\frac{{\\mathcal{B}}}{y-y\\_\\pm} v'=0,\n\\label{PcDL\\_ft}$$ \nwith\n \n$$\\begin{aligned}\n{\\mathcal{B}}=&1\\pm i\\frac{k\\_zf}{k\\_x{U}'\\_{{\\textcolor{black}{\\pm}}}},\\\\\n\\end{aligned}$$ \nwhere the symbol ± refers to the regular singularities *y*+ and *y*−, respectively The Frobenius method consists in injecting the power function (*y* − *y*±)*λ* in Eq. ([PcDLft]), with *λ* a constant to be determined. The corresponding indicial equation is then:\n \n*λ*(*λ* − 1) + B*λ* = 0, \nwith solutions:\n \n$$\\lambda\\_\\pm=\\left\\{0,\\ \\mp i\\frac{k\\_zf}{k\\_x{U}\\_{{\\textcolor{black}{\\pm}}}'}\\right\\}.$$ \nTherefore, the two independent solutions of Eq. ([PcDLft]) can be written as follows:\n \n*v*1, ±(*y*) = ∑*n* = 0+ ∞*a**n*(*y* − *y*±)*n* and *v*2, ±(*y*) = ∑*n* = 0+ ∞*b**n*(*y* − *y*±)*n* + *λ*±, \nwhere *a**n* and *b**n* are complex constants. Both solutions are valid in the vicinity of the critical level around which they are built, up to the next singularity if it exists. The coefficients *a*0 and *b*0 are unconstrained and depend on boundary conditions, unlike the other factors that can be determined by injecting solutions ([sol]) into the\nNear the critical points *y*±, the total solution *v*± is well approximated by the lowest orders of *v*1, ± and *v*2, ±:\n \n*v*±(*y*) ≃ *a*0 + *b*0(*y* − *y*±)*λ*±. \nOwing to the existence of a branch point at *y*± (since *λ*± is complex), reconnecting solutions on either part of the critical levels is not straightforward. This requires both physical and mathematical arguments. In order to remove degeneracy of the path from positive to negative *y* − *y*± (i.e. choose either e+ *i**π* or e− *i**π*), we make use of a complex inertial frequency *ω* = *ω*R + *i**ω*I, assuming the radiation condition *ω*I > 0. This condition ensures a non-growing wave toward infinity. The Taylor expansion of the base flow at first-order in *y* − *y*± gives\n \n$$y-y\\_\\pm=\\frac{{U}(y)-{U}\\_{{\\textcolor{black}{\\pm}}}}{{U}'\\_{{\\textcolor{black}{\\pm}}}},$$ \nand by definition, we have\n \n$${U}\\_{{\\textcolor{black}{\\pm}}}=\\frac{\\omega\\mp{\\widetilde{f}}}{k\\_x}.$$ \nConsequently, the solution below the critical level is unambiguous in terms of the above solution coefficients, and depends on\n \n$$\\operatorname{sign}[\\Im\\{y-y\\_\\pm\\}]=-\\operatorname{sign}\\left[k\\_x{U}'\\_{{\\textcolor{black}{\\pm}}}\\right].\n\\label{branch}$$ \nIn other words, when taking *y* − *y*± to decrease from positive to negative values, its complex argument changes continuously from 0 to $-\\operatorname{sign}\\left[k\\_x{U}'\\_{{\\textcolor{black}{\\pm}}}\\right]\\pi$. Thus, the appropriate path for determining the branch of (*y* − *y*±)*λ*± passes under (above) *y*± as long as $k\\_x{U}'\\_{{\\textcolor{black}{\\pm}}}>0$ ($k\\_x{U}'\\_{{\\textcolor{black}{\\pm}}}<0$). Therefore, the solution on both sides of the critical level *y*± is:\n \n$$v\\_{\\pm}(y)\\simeq\\left\\{\\begin{aligned}\n&a\\_0+b\\_0|y-y\\_\\pm|^{\\mp i k\\_z f/k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}}}&\\text{ for }y>y\\_\\pm\\\\\n&a\\_0+b\\_0|y-y\\_\\pm|^{\\mp i k\\_z f/k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}}}\\mathrm{exp}(\\mp \\pi k\\_z f/|k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}} |)&\\text{ for }y<y\\_\\pm\n\\end{aligned}\\right..$$ \nThe remaining issue is now to know in which direction the wave is propagating. The second part of the solution can be assimilated to a wave-like solution with the varying latitudinal wavenumber $\\mp (k\\_z f/k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}}) \\log|y-y\\_{\\pm}|$. Moreover, according to Eq. ([wA]), the wave action flux on either side of *y*± is:\n \n$${\\mathcal{A}}=\n\\frac{{\\rho\\_0}k\\_zf}{2{\\widetilde{f}}k\\_\\perp^2}\\left\\{\\begin{aligned}\n&|a\\_0|^2-|b\\_0|^2&\\text{ for }y>y\\_\\pm\\\\\n&|a\\_0|^2-|b\\_0|^2\\mathrm{exp}(\\mp2\\pi k\\_z f/|k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}}|)&\\text{ for }y<y\\_\\pm\n\\end{aligned}\\right..\n\\label{atnu}$$ \nThe group velocity gives the direction towards which the energy is transported, recalling that ${V\\_\\mathrm{g}}{\\mathcal{E}}=\\overline{pv}$ with *V*g the group velocity and E the local energy density. By consequence, sign(*V*g) = sign(A*σ*) = ∓ sign(*f**k**z*) for the solution featuring the coefficient *b*0. If *k**z**f* is positive, this wave transports energy downward (upward) across the critical level $\\sigma={\\widetilde{f}}$ ($\\sigma=-{\\widetilde{f}}$). If *k**z**f* is negative, the wave transports energy upward (downward) across the critical level $\\sigma={\\widetilde{f}}$ ($\\sigma=-{\\widetilde{f}}$). In all cases, the action flux of the wave with the amplitude ∣*b*0∣ will be transmitted (in the direction given by the sign of *k**z**f* and the critical level *y*+ or *y*−) by a factor *T**θ*0 where\n \n$$T\\_{\\theta\\_0}=\\exp(-2\\pi\\frac{|k\\_zf|}{|k\\_x{U'}\\_{{\\textcolor{black}{\\pm}}}|})=\\exp(-2\\pi\\frac{|\\alpha\\_k\\cos\\theta\\_0|}{|{R\\_\\mathrm{o}}|}),$$ \nwith *α**k* = *k**z*/*k**x* and ${R\\_\\mathrm{o}}={U}'\\_{{\\textcolor{black}{\\pm}}}$, after passing through the critical level. Such wave will always be attenuated since *T**θ*0 ≤ 1. The transmission factor $T\\_{\\theta\\_0=10\\degree}$ and $T\\_{\\theta\\_0=80\\degree}$ are displayed in Fig. [atnudeg] in terms of the absolute value of the shear Rossby number ∣*R*o∣ and the ratio of wave numbers *α**k*. The lower the amplitude of the Rossby number and the lower the inclination, the more likely the wave is to be strongly attenuated at any *α**k*. We remind that a low Rossby number refers either to fast rotating stars or to low differential rotation. At the equator, one should note that. Results are the same for *θ*0 + *k**π*/2 with *k* ∈ {0, 1, 2, 3}, and for negative Rossby numbers. However, it has to be emphasised that the cases where the inclination satisfies *θ*0 = *k**π* with *k* ∈ {0, 1} are not well described by the attenuated factor *T**θ*0 and require a specific treatment, as discussed in Sect. [polarcase].\nIt is important to note that, with fixed parameters {*k**z*, *θ*0, *R*o}, the attenuation of the wave action flux is specific to a single direction of wave propagation, i.e. the solution featuring the coefficient *b*0. The solution of coefficient *a*0 is not affected by the attenuation. It is the so-called valve effect introduced by in the context of hydromagnetic waves in a rotating fluid. It was also evidenced by, and further discussed in for magneto-gravito-inertial waves in an inviscid and compressible z-sheared fluid.\n### Inertial wave crossing corotation\nWe perform the same analysis as in the previous section to treat the corotation point *y*0 where *σ* = 0 (i.e. $U({y\\_0}) {\\textcolor{black}{=U\\_0}}=\\omega/k\\_{x}$). Taylor expansion of is: where $U\\_{{\\textcolor{black}{0}}}'$ and $U\\_{{\\textcolor{black}{0}}}''$ are evaluated at the critical level *y*0. The singularity at the corotation is a regular singularity and we can use again the Frobenius method. The indicial equation has solutions *λ* = {2, 1}. Since the difference between the two of the exponent is an integer, one expects a second independent solution *v*2 of Eq. ([Pcorot])\nLike in the works of and, it is tempting to investigate the asymptotic behaviour of a wave when *y* → ∞, in order to better constrain the propagation of waves through one or multiple critical levels. Nevertheless, the term − *σ*2*k*⊥2 in the ODE (Eq. ([Pc])), which can not be overlooked like in the aforementioned studies since we do not have vertical stratification, makes the singularity *y* = ∞ an essential (or irregular) singularity and the Frobenius method can not be applied. This term also prevents us to apply an analysis like the WKBJ approximation because even far from critical levels the coefficients *C* of the ODE (in Eq. ([Pc])) still have a strong dependence on the latitudinal coordinate when the box is tilted.\nInertial waves when the box is at the poles\n-------------------------------------------\nWhen the box is located at the North or the South pole, ${\\widetilde{f}}=0$ and the (Eq. ([Pc])) is greatly simplified. For *θ*0 = 0, the dimensionless equation becomes indeed\n \n$$v''+\\left[\\frac{k\\_z^2(1-{U'})}{\\sigma^2}+\\frac{k\\_x{U}''}{\\sigma}-k\\_\\perp^2\\right]v=0.\n\\label{Pcpole}$$ \nAt the South pole (i.e. *θ*0 = *π*), the term 1 − *U*ʹ in Eq. ([Pcpole]) is replaced by 1 + *U*ʹ. Note that this equation is reminiscent of the differential equation for Rossby waves in the *β*-plane, i.e. $2{{\\bm{\\Omega}}} =(0,{\\widetilde{f}},f)$ and constant d*f*/d*y* = *β*, with *f* the Coriolis parameter. However, we can not make a direct comparison at corotation, because the singularity in the equations for Rossby waves and inertial waves is not of the same order. We have a second-order pole around the corotation while only first-order poles are found in the aforementioned studies. In fact, Eq. ([Pcpole]) is similar to the wave equation in stratified z-sheared flows.\nIn our polar configuration the y-coordinate is now the axial distance, and it means that the mean flow has a cylindrical profile. Such a rotation profile is expected in giant planets such as Jupiter and Saturn as a natural outcome of the Proudman-Taylor theorem for fast-rotating bodies. The propagation and dissipation of inertial modes of oscillations in the presence of critical levels for this kind of mean flow have been investigated by in a spherical shell.\n### Analytical solutions with constant shear\nAnalytical solutions of the ODE Eq. ([Pcpole]) are difficult to find for general profiles of the mean flow, e.g. a quadratic mean-flow profile. A linear mean-flow profile, on the other hand, has analytic solutions, that is why we use in this section such a profile, i.e. *U* = *R*o*y*, with *R*o the shear Rossby number which is taken constant here. Eq. ([Pcpole]) then becomes:\n \n$$v''+\\left[\\frac{\\alpha\\_k^2(1-{R\\_\\mathrm{o}})/{R\\_\\mathrm{o}}^2}{(y-{y\\_0})^2}-k\\_\\perp^2\\right]v=0,\n\\label{pole\\_cstUp}$$ \nwhere *y*0 = *ω*/(*k**x**R*o) and *α**k* = *k**z*/*k**x* the vertical to longitudinal wave number ratio. When the box is located at the South pole, the left-hand term in the bracket is *α**k*2(1 + *R*o)/*R*o2 in the numerator. This equation takes the form of Whittaker’s equation and solutions can be written in terms of the Whittaker functions *M*:\n \n$$v(y)=AM\\_{0,\\,\\mu\\,}(\\tilde y)+BM\\_{0,\\,-\\mu\\,}(\\tilde y),$$ \nwith $\\tilde y=2k\\_\\perp(y-{y\\_0})$,\n \n$$\\mu=\\sqrt{\\frac{1}{4}-\\frac{{\\alpha\\_k}^2(1-{R\\_\\mathrm{o}})}{{R\\_\\mathrm{o}}^2}},$$ \nand *A* and *B* are complex constants given by boundary conditions. The Whittaker function *M*0, − *μ* allows quite straightforward analytic continuation:\n \n$$M\\_{0,\\,\\mu\\,}(\\operatorname{e}^{-i\\pi}\\tilde y)=-i\\operatorname{e}^{-i\\pi\\mu}M\\_{0,\\,\\mu\\,}(\\tilde y).$$ \nBy consequence, the solution below the critical point *y* = *y*0 is:\n \n$$v\\_\\mathrm{W}(y)=-i\\varsigma A\\operatorname{e}^{- i\\varsigma\\pi\\mu}M\\_{0,\\,\\mu\\,}(-\\tilde y)- i\\varsigma B\\operatorname{e}^{i\\varsigma\\pi\\mu}M\\_{0,\\,-\\mu\\,}(-\\tilde y).$$ \nAlthough the Whittaker functions do not feature precisely as wave-like forms, we can already have a good idea of the attenuation factor thanks to analytic continuation as will be shown in the following section.\nIt is important to point out that *μ* can be real or complex depending on the value of\n \n*R**θ*0 = 0, *π* = *α**k*2(1 ∓ *R*o)/*R*o2, \nwhich we will simply denote by *R* in the following. This can drastically change the behaviour of a wave passing through the corotation. A necessary, but not sufficient condition to find an instability is that *R* < 1/4 as we will demonstrate in Sect. [inst]. This condition is similar to the Miles-Howard theorem for stratified *z*-sheared flow. In these studies, the prerequisite for instability is that Ri < 1/4 where Ri is the Richardson number, i.e. the squared ratio of the Brunt-Väisälä frequency and the vertical shear. In our model, unlike cases where the box is tilted, a WKBJ analysis can be performed for a linear mean flow, mainly provided that ∣*R*∣ ≫ 1/4, in line with the condition of stability derived in the coming sections, and detailed in Appendix [wkbj].\nThese various situations regarding the value of *R**θ*0 = 0, *π* at the North and South poles are illustrated in Fig. [O2]. We stress the particular case where *R*o = 1 (*R*o = − 1) at the North (South) pole, and where the differential equation and its solutions take a quite simple form:\n \n*v*ʺ − *k*⊥2*v* = 0, with *v* ∝ e± *k*⊥*y*. \nSolutions are then fully evanescent for such shears. One can notice that Eq. ([eva]) is the same far from corotation, for any mean flow.\nFinally, it is clear from Fig. [O2] that wave propagation is the same at the North or South pole provided a Rossby number of opposite sign. As a result, only the equations at the North pole will be treated in the following, and the word “pole” now refers to the North pole.\n [O2]\n### Frobenius method at the pole\nThough analytic solutions are known, it is still useful to determine Frobenius solutions near corotation for two main reasons. First, these solutions can be derived for any mean flow profile near the corotation. Close to corotation, the mean flow is approximated by a Taylor expansion at the first-order ${U}=U'\\_{{\\textcolor{black}{0}}}(y-{y\\_0})$. Secondly, Frobenius solutions may feature wave-like forms, which is helpful for physical interpretation. Therefore, Eq. ([Pcpole]) can be written near corotation:\n \n$$v''+\\frac{\\alpha\\_k^2(1-{R\\_\\mathrm{o}})/{R\\_\\mathrm{o}}^2}{(y-{y\\_0})^2}\nv=0,\n\\label{Fropoleeq}$$ \nwhere ${R\\_\\mathrm{o}}={U'}\\_{{\\textcolor{black}{0}}}$ and ${R\\_\\mathrm{o}}'={U'}'\\_{{\\textcolor{black}{0}}}$. The indicial equation gives:\n \n$$\\lambda=\\frac{1}{2}\\pm\\mu.$$ \nIn the two next subsections, we examine both cases where *μ* is imaginary or real.\n### Theoretical stable regime (*R* > 1/4)\nWe address here the case where *R* > 1/4. The same analysis as in Sect. [Fro] can be carried out to determine how a wave behaves upon crossing the corotation. The solutions of the indicial equation can be recast as\n \n$$\\lambda=\\frac{1}{2}\\pm i|\\mu|.$$ \nThe first-order solutions to Eq. ([Fropoleeq]) in the vicinity of *y*0 are :\n \n$$\\left\\{\\begin{aligned}\n&a\\_0(y-{y\\_0})^{\\frac{1}{2}+i|\\mu|}+b\\_0(y-{y\\_0})^{\\frac{1}{2}-i|\\mu|},&y>{y\\_0}\\\\\n&-i\\varsigma\\left[a\\_0({y\\_0}-y)^{\\frac{1}{2}+i|\\mu|}e^{\\varsigma\\pi|\\mu|}+b\\_0({y\\_0}-y)^{\\frac{1}{2}-i|\\mu|}e^{-\\varsigma\\pi|\\mu|}\\right],&y<{y\\_0}\\end{aligned}\\right.\n\\label{solpol}$$ \nfor *a*0, *b*0 ∈ C. One can recover the same form of Frobenius solutions as in, who examined radially stratified mean flows in spherical geometry. As ${\\widetilde{f}}=0$, the wave action flux Eq. ([wA]) reduces to\n \n$${\\mathcal{A}}=-\\frac{{\\rho\\_0}}{2k\\_\\perp^2}\\Im{v\\_yv^\\*},\n\\label{wApol}$$ \nthat is, injecting the solutions on both sides of the critical level:\n \n$$\\mathcal{A}=-\\frac{{\\rho\\_0}}{2k\\_\\perp^2}|\\mu|\\left\\{\n\\begin{aligned}\n&|a\\_0|^2-|b\\_0|^2,&y>{y\\_0}\\\\\n&-|a\\_0|^2\\operatorname{e}^{2\\varsigma\\pi|\\mu|}+|b\\_0|^2\\operatorname{e}^{-2\\varsigma\\pi|\\mu|},&y<{y\\_0}\\end{aligned}\\right..\n\\label{Apol\\_st}$$ \nThis formulation is quite similar to the expression of the Reynolds stress (*τ*) in vertically stratified mean flows, which can be found in in Cartesian geometry. We recall indeed that *τ* = − *k**x*A. Moreover, given Eq. ([Re]), the Reynolds stress in our model reads $\\tau=-{\\rho\\_0}\\overline{uv}$. Using the polarisation relations for *u*, we recover the wave action flux in Eq. ([wApol]).\n![Transmission rate T_\\mathrm{F} of the wave action flux across the corotation when the box is located at the pole for {R}>1/4. It is displayed against the Rossby number {R_\\mathrm{o}} and the ratio of the horizontal wave numbers \\alpha_k=k_z/k_x. The forbidden region where {R}<1/4 is painted in white, and the dark red cone corresponds to values of T_\\mathrm{F} lower than 10^{-9}.](figures/colormap_atnu_deg0 \"fig:\") [atnuO2]\nThe pre-factor *i* in the solutions ([solpol]) below the corotation does not affect the energy flow and simply indicates that the wave undergoes a phase shift of *π*/2 through the critical level. Above the critical level, the normalised Doppler-shifted frequency satisfies $\\operatorname{sign}(\\sigma)=-\\varsigma$ as for the corotation in the inclined case. The sign is reversed below the critical level. Thus, the first solution of main amplitude ∣*a*0∣ carries its latitudinal flux of energy upward (downward) for $\\varsigma=+1$ ($\\varsigma=-1$), while the second solution transfers its energy in the opposite direction in the various cases. Therefore, the energy flux of an upward or downward wave is always attenuated by a factor\n \n*T*F = exp{ − 2*π*∣*μ*∣}. \nThis attenuation factor is shown in Fig. [atnuO2] versus *R*o and *α**k*. We observe that the wave is largely absorbed at the critical level and thus deposits most, if not all its energy for most couples (*α**k*, *R*o).\n### Possible unstable regime (*R* < 1/4)\nWe now deal with the case where *R* < 1/4, i.e. *μ* is real. Contrary to the situation where *R* > 1/4, we can no longer assimilate solutions to wave-like functions. The exponential form of solutions for *R* < 1/4 near the critical level reads\n \n$$v(y)=\\left\\{\\begin{aligned}\n&a\\_0(y-{y\\_0})^{\\frac{1}{2}+|\\mu|}+b\\_0(y-{y\\_0})^{\\frac{1}{2}-|\\mu|}, &y>{y\\_0}\\\\\n&-i\\varsigma\\left[\\operatorname{e}^{-i\\varsigma\\pi|\\mu|}a\\_0(y-{y\\_0})^{\\frac{1}{2}+|\\mu|}\\right.\\\\\n&\\left.~~~~~~~+\\operatorname{e}^{i\\varsigma\\pi|\\mu|}b\\_0(y-{y\\_0})^{\\frac{1}{2}-|\\mu|}\\right], &y<{y\\_0}\\end{aligned}\\right.\n\\label{solpol\\_inst}$$ \nand makes this region fully evanescent. Furthermore, the associated wave action flux is\n \n$$\\mathcal{A}=-\\frac{{\\rho\\_0}}{k\\_\\perp^2}\\mu\\left\\{\n\\begin{aligned}\n&\\Im{a\\_0b\\_0^\\*},&y>{y\\_0}\\\\\n&-\\Im{a\\_0b\\_0^\\*\\operatorname{e}^{-2i\\varsigma\\pi\\mu}},&y<{y\\_0}\\end{aligned}\\right..\n\\label{Apol\\_inst}$$ \n[ht!]\nc|c|| c|c\n---\nBox & critical level & attenuation & amplification \n---\n& $\\sigma={\\widetilde{f}}$ & yes for waves ↓ & no \n---\n& $ \\sigma=-{\\widetilde{f}}$ & yes for waves ↑ & no \n---\n& *σ* = 0 & no & no \n---\n& *σ* = 0 &\n| |\n| --- |\n| yes if *R* < 1/4 |\n| yes if $\\left\\{\\begin{aligned}R>1/4 \\\\\\varsigma{\\mathcal{A}}^+>0 \\end{aligned}\\right.$ |\n& yes if $\\left\\{\\begin{aligned}R>1/4 \\\\\\varsigma{\\mathcal{A}}^+<0 \\end{aligned}\\right.$ \n---\n& & & \n---\n& *σ* = 0 & no & no \n[recapana]\nWithout knowing the direction of the wave or the energy flux since $\\operatorname{sign}({V\\_\\mathrm{g}})=\\varsigma\\operatorname{sign}\\left(\\Im{a\\_0b\\_0^\\*}\\right)$, it is difficult to assess the impact of the critical level on wave propagation, i.e. whether it will attenuate waves or on the contrary amplify them.\n found a way to investigate the behaviour of internal gravity waves passing through a critical level in a regime similar to ours (Ri < 1/4) where solutions are fully evanescent. Their work has been taken up by spherical geometry. The method is to determine the reflection and transmission coefficients in a three-zone model. The evanescent region where the Richardson number Ri < 1/4 and where the critical level is located (zone II), is sandwiched between two propagating wave layers (zones I and III). Using a linear mean flow profile so as to establish solutions inside zone II, and both used continuity relations of the perturbed vertical (or radial) velocity and its derivatives at the interfaces between zones in order to get the transmission and reflection coefficients. The critical level is located in the middle of zone II of width *δ*. By consequence, the reflection and transmission coefficients depend, in their works, on the shear and more precisely the Richardson number, and on the width *δ*. They both found that, depending on the Richardson number and *δ*, the reflection and transmission coefficients can be greater than one, meaning that the wave can be over-reflected and/or over-transmitted. However, this result is conditioned by the geometry of the model. As shown by in his review and references therein, models with one or even two layers with evanescent and eventually a wave-like region, do not allow such phenomena. A first region that allows the wave propagation is mandatory and is combined with a “sink” that pulls the wave to cross the critical level. According to and, the nature of the sink for wave flux can be either another propagative region or an evanescent region, as in zone II, subject to friction processes. Given this peculiar geometry, instabilities can occur under boundary conditions that allow the wave to return successively to the critical level. Many studies have tried to relate over-reflection and shear instability for a specific wave geometry.\nIn the present study, we do not investigate further shear instability by doing, for instance, a temporal analysis to estimate the waves’ growth rate. On the contrary, we give arguments, such as *R* < 1/4, of necessary but not sufficient condition to find instabilities. It is important to note that *R* is constant in the whole domain for a linear mean flow profile, and thus one is stuck with either a propagative (stable) or an evanescent regime. Therefore, finding an adequate geometry to allow over-reflection and over-transmission requires at least that the Rossby number is not the same in the whole domain, by using for instance a non-linear mean flow profile. Furthermore, in the particular case where *R* = 0, (i.e. *R*o = 1 or *α**k* = 0 in Eq. ([Pcpole]) when the wave with *k**z* = 0 propagates in the (*x*, *y*)-plane), a necessary condition for instability is given by the Inflection Point Theorem. This theorem is particularly used to study barotropic instabilities for Rossby waves. In other words, a necessary condition to have unstable modes for *R*o = 1 is that *U*ʺ cancels out in the domain of wave propagation. \nWe summarise in Table [recapana] the main analytical results of Sects. [Fro] and [polarcase] about wave and wave action flux transmission, either when the box is relative to the rotation axis, at the North pole.\nA three-zone numerical model\n============================\nIn order to test the analytical predictions of the previous section, we have built up a three-zone numerical model to simulate waves passing through critical levels. A similar model has been used, for instance, by to explore the behaviour of internal gravity waves passing through critical levels in a fluid with rotation and vertical shear. In our model, we solve the two first-order ODEs satisfied by *v* and Π, the combination of which led to the equation ([Pc]). By imposing boundary conditions such that waves satisfy the dispersion relations (see also Appendix [solIIII]), we examine the dynamics of inertial waves propagating in the shear region. Also, whenever possible, we calculate analytically the wave transmission and reflection coefficients as the wave-like solution crosses the shear region.\n![Mean flow profiles used in the three-zone numerical model against y. In the no-shear regions I (y<0) and III (y > 1), the mean flow is uniform and set respectively to {U}=0 and {U}=\\Lambda. The sheared region II (gray-shaded) can have a linear, square, or cubic mean flow profile.](figures/mean_flows \"fig:\") [meanflow]\nDescription of the model\n------------------------\nThe mean flow profile that is used in the three-zone model is illustrated in Fig. [meanflow]. The zone with shear (zone II) is surrounded by two no-shear regions, one with no mean flow (zone I), and one with a uniform mean flow (zone III). In the whole domain, the mean flow profile that we adopt is expressed as\n \n$$U(y)=\\left\\{\\begin{aligned}\n&0~~~&\\mathrm{for}&~y<0: & \\textit{zone I},&\\\\\n&\\Lambda y^n~~~&\\mathrm{for}&~0\\leq y\\leq1:&\\textit{zone II},&\\\\\n&\\Lambda~~~&\\mathrm{for}&~y>1:& \\textit{zone III}.&\\\\\n\\end{aligned}\\right.\n\\label{U\\_pro}$$ \nwhere *U*(*y*) is continuous at each interface, and *n* is an integer: *n* = 1 for a linear shear flow, *n* = 2 for a square shear flow or *n* = 3 for a cubic shear flow (see also Fig. [meanflow]). In zone I, we assume that there is an incident wave that enters the shear zone as well as a wave that is reflected at the interface between zones I and II or in zone II, i.e.:\n \n*v*(*y*) = *A**I*exp(*i**k**I**y*) + *A**R*exp(*i**k**R**y*), \nwhere *A**I* and *A**R* are the amplitudes, *k**I* and *k**R* are the wavenumbers of the incident and reflected waves, respectively. We further impose as boundary condition in zone III a transmitted wave that propagates towards positive *y*-values:\n \n*v*(*y*) = *A**T*exp(*i**k**T**y*), \nwhere *A**T* and *k**T* are the amplitude and wavenumber of the transmitted wave, respectively. We impose *A**T* = 1 without loss of generality and compute the remaining amplitudes *A**I* and *A**R*. More details on the solutions and the dispersion relations of the waves in zones I and III can be found in Appendix [solIIII]. We have ensured that the transmitted wave carries energy upwards, by deriving the wave action flux in zones I and III (see Appendix [wAIIII]).\nWe impose the continuity of the latitudinal velocity and reduced pressure at the interfaces (at *y* = 0 and *y* = 1). By doing so, the wave action flux is continuous at both interfaces. Thus, in the absence of critical points, the wave action flux is conserved in the whole domain, namely AT = AI-R where AT is the wave action flux of the transmitted wave and AI-R = AI − AR with AI and AR for the incident and reflected wave action fluxes, respectively.\nTo solve the ODE in the three zones and in particular near singularities, we have used MATLAB’s solver ode15s, which is suitable for solving stiff differential equations. To avoid strict singularities at *σ*2 = *f̃*2, we have added a small friction *σ**f* = 10− 8 in our set of units. Given the boundary conditions, the numerical solver deals with two first-order ODEs for *v* and Π, which take the form\n \n$$\\left\\{\n\\begin{aligned}\nF\\_{1,v}v+F\\_{1,p}\\Pi=Av',\\\\\nF\\_{2,v}v+F\\_{2,p}\\Pi=A\\Pi',\n\\end{aligned}\\right.\n\\label{ODEcode}$$ \nwhere\n \n$$\\begin{aligned}\nF\\_{1,v}&={\\widetilde{f}}\\frac{{U'}}{\\sigma}\\left(k\\_x{\\widetilde{f}}+ik\\_zs\\right)-({U'}-f)\\left(k\\_xs+ik\\_z{\\widetilde{f}}\\right),\\\\\nF\\_{1,p}&=-ik\\_\\perp^2s,\\\\\nF\\_{2,v}&=is\\left[s^2+f({U'}-f)\\right]-i{\\widetilde{f}}^2\\left(s+f\\frac{{U'}}{\\sigma}\\right),\\\\\nF\\_{2,p}&=ik\\_zf{\\widetilde{f}}-k\\_xfs,\\\\\n\\end{aligned}$$ \nand where we recall that *s* = *σ* + *i**σ**f* is the modified Doppler-shifted frequency due to Rayleigh friction. While *A**T* = 1 is imposed by the boundary condition, we compute *A**I* and *A**R* by comparing numerical solutions of the system Eq. ([ODEcode]) at *y* = 0 with the definition of velocity in zone I (Eq. ([IRwaves])) and its associated reduced pressure (see Eq. ([Pivis])).\nNumerical exploration at the pole for a constant shear\n------------------------------------------------------\n### Reflection and transmission coefficients\n[ht] [RTinst]\n[ht] [RTst]\nIn most cases, for any inclination of the box and any mean flow profile, there is no analytical solutions in zone II. Nevertheless, we have shown in Sect. [anapole] that, when the box is at the pole and for a linear mean shear flow, solutions can be found in terms of Whittaker functions. In this section, we will find reflection and transmission coefficients similarly as in and, though there are a few differences.. Moreover, our boundary conditions are different, as detailed in Sect. [inst], and the thickness of our shear region is fixed to one in scaled units while and leave the thickness *δ* as a control parameter. We also check the existence of a critical level in the shear zone and the frequency range that delineates the regimes with and without the critical level.\nWe consider that the perturbed reduced pressure Π and velocity *v* are continuous at the interfaces *y* = 0 and *y* = 1. In the presence of the critical level *y*0 in zone II, we have a set of four analytical solutions whose values at *y* = 0 and *y* = 1 allow us to determine the reflection and transmission coefficients. The solutions to equation in zones I, II (below and after the critical level) and III are:\n \n$$v=\\left\\{\\begin{aligned}\n&A\\_\\mathrm{I}\\operatorname{e}^{ik\\_\\mathrm{I}y}+A\\_\\mathrm{R}\\operatorname{e}^{ik\\_\\mathrm{R}y},\\ &\\text{for }y<0\\\\\n&-i\\left[A\\operatorname{e}^{-i\\pi\\mu}M\\_{0,\\,\\mu\\,}(-\\tilde y)+B\\operatorname{e}^{i\\pi\\mu}M\\_{0,\\,-\\mu\\,}(-\\tilde y)\\right],\\ &\\text{for }0<y<{y\\_0}\\\\\n&AM\\_{0,\\,\\mu\\,}(\\tilde y)+BM\\_{0,\\,-\\mu\\,}(\\tilde y),\\ &\\text{for }{y\\_0}<y<1\\\\\n&A\\_\\mathrm{T}\\operatorname{e}^{ik\\_\\mathrm{T}y},\\ &\\text{ for }y>1\n\\end{aligned}\\right.$$ \nwhere *A* and *B* are complex coefficients which we will express below. We remind *A**T* = 1 in our numerical model. In the shear region (zone II), the reduced pressure perturbation is given by Eqs. ([pol]), which, at the pole, can be recast as\n \n$$\\Pi=\\frac{i}{k\\_\\perp^2}\\left[\\sigma v\\_y+k\\_x({R\\_\\mathrm{o}}-1)v \\right]\n\\label{Pipole},$$ \nwith *R*o = Λ. In the regions with no shear (zones I and III), Π takes a simpler expression with *R*o = 0, and *σ* = *ω* in zone I and *σ* = *ω* − *k**x**U*(1) = *ω* − *k**x**R*o in zone III (noted *σ*3 in the following). Note that while the reduced pressure is kept continuous to conserve the wave action flux across the interfaces, the first derivative of the latitudinal velocity *v*ʹ is not necessarily continuous at the interfaces. To find the transmission and reflection coefficients, we solve the system of equations that consist of matching conditions at interfaces as follows:\n*v* is~continuous~at *y* = 0 : \n*A*I + *A*R = *v*W(0);\nΠ is~continuous~at *y* = 0 : \n(*i**k*I*ω* − *k**x*)*A*I + (*i**k*R*ω* − *k**x*)*A*R = *ω**v*ʹW(0) + *k**x*(*R*o − 1)*v*W(0), \n*v* is~continuous~at *y* = 1 : \ne*i**k*T = *v*W(1), \nΠ is~continuous~at *y* = 1 : \n(*i**k*T*σ*3 − *k**x*)e*i**k*T = *σ*3 *v*ʹW(1) + *k**x*(*R*o − 1)*v*W(1), \nwith the Whittaker functions *v*W. At the interfaces below and above the critical level *y*0 (i.e. *y* = 0 and *y* = 1), we have\n \n$$\\begin{aligned}\nv\\_\\mathrm{W}(0)&=-i\\left[A\\operatorname{e}^{-i\\pi\\mu}M\\_{0,\\,\\mu\\,}(2k\\_\\perp{y\\_0})+B\\operatorname{e}^{i\\pi\\mu}M\\_{0,\\,-\\mu\\,}(2k\\_\\perp{y\\_0})\\right],\\\\\nv\\_\\mathrm{W}(1)&=AM\\_{0,\\,\\mu\\,}\\left(2k\\_\\perp(1-{y\\_0})\\right)+BM\\_{0,\\,-\\mu\\,}\\left(2k\\_\\perp(1-{y\\_0})\\right).\n\\end{aligned}\n\\label{whit\\_01}$$ \nPlease note that the first derivative of the Whittaker functions can be computed either numerically or analytically via the relationships in.\nThe equations in the above continuity relationships 1. to 4. are independent two by two (1. and 2., 3. and 4.), and *A* and *B* can be found first:\n \n$$\\begin{aligned}\n A = &\\operatorname{e}^{ik\\_\\mathrm{T}}\\frac{{\\sigma\\_3}(M'\\_{-\\mu,1}-ik\\_\\mathrm{T}M\\_{-\\mu,1})+M\\_{-\\mu,1}k\\_x{R\\_\\mathrm{o}}}{{\\sigma\\_3}(M\\_{\\mu,1}M'\\_{-\\mu,1}-M'\\_{\\mu,1}M\\_{-\\mu,1})},\\\\\n B = &\\operatorname{e}^{ik\\_\\mathrm{T}}\\frac{{\\sigma\\_3}(M'\\_{\\mu,1}-ik\\_\\mathrm{T}M\\_{\\mu,1})+M\\_{\\mu,1}k\\_x{R\\_\\mathrm{o}}}{{\\sigma\\_3}(M'\\_{\\mu,1}M\\_{\\mu,1}-M\\_{\\mu,1}M'\\_{-\\mu,1})}.\n\\end{aligned}$$ \nThe amplitude of the incident and reflected waves can be written, in terms of *A* and *B*, as:\n \n$$\\begin{aligned}\n A\\_\\mathrm{I}=& \\frac{A\\operatorname{e}^{-i\\mu\\pi}X\\_\\mu(k\\_\\mathrm{R})+B\\operatorname{e}^{i\\mu\\pi}X\\_{-\\mu}(k\\_\\mathrm{R})}{\\omega(k\\_\\mathrm{R}-k\\_\\mathrm{I})},\\\\\nA\\_\\mathrm{R}=& \\frac{A\\operatorname{e}^{-i\\mu\\pi}X\\_\\mu(k\\_\\mathrm{I})+B\\operatorname{e}^{i\\mu\\pi}X\\_{-\\mu}(k\\_\\mathrm{I})}{\\omega(k\\_\\mathrm{I}-k\\_\\mathrm{R})},\n\\end{aligned}\n\\label{AIR}$$ \nwhere\n \n$$\\begin{aligned}\nX\\_{\\mu}(k)=\\omega(M'\\_{\\mu,0}-ikM\\_{\\mu,0})+k\\_x{R\\_\\mathrm{o}}M\\_{\\mu,0}.\\\\\n\\end{aligned}$$ \nThe transmission and reflection coefficients are then:\n \n$$|T|=\\frac{|A\\_{\\mathrm{T}}|}{|A\\_{\\mathrm{I}}|}=\\frac{1}{|A\\_\\mathrm{I}|}\\ \\text{ and }\\ |R|=\\frac{|A\\_\\mathrm{R}|}{|A\\_\\mathrm{I}|}.$$ \nWe emphasise that these factors depend notably on the location of the critical level and on the inertial frequency, which was not the case in and.\n[ht!] [wAIRT]\nWe display in Figs. [RTinst] and [RTst] the transmission and reflection coefficients as a function of the Rossby number *R*o and the normalised inertial frequency *ω* in the regimes *R* < 1/4 and *R* > 1/4, respectively. We choose *k**x* = *k**z* = 0.1 in these plots and *R* = 1/4 gives ${R\\_\\mathrm{o}}=-2\\pm\\sqrt{8}\\approx \\{-4.8,0.8\\}$, which delineates the two regimes as we can see in Fig. [O2]. Areas that are hatched do not possess the corotation point *σ* = 0. In addition, the wavenumber *k*T was chosen with a positive sign in these regions to maintain an upward propagating wave (see Appendices [solIIII] and [wAIIII]). Areas that are not hatched feature a critical level, according to the table of Appendix [wAIIII]. In the case where *R* < 1/4, over-reflection and over-transmission are both possible (see Fig. [RTinst]). One should notice that for *R*o > 1, we always have by definition of *R*\n \n$$\\frac{R}{(y-{y\\_0})^2}-k\\_\\perp^2<0,$$ \nregardless of *y*, which makes the solutions of Eq. ([polecstUp]) tend towards pure exponential functions, i.e. without any imaginary part. Also, we do not see any over-reflection, nor over-transmission in the hatched areas where there is no corotation point. This highlights the essential role of the critical level in inducing over-reflection or over-transmission of inertial waves crossing the shear region in this regime.\nThe regime where *R* > 1/4 (in Fig. [RTst]) is more delicate to analyse. According to our discussion in Sect. [stabpole], we expect a strong attenuation of the wave and of the wave action flux as shown in Fig. [atnuO2]. From this figure and for *α**k* = 0.1, the damping is very strong for low positive Rossby numbers. This tendency is also found for both transmission and reflection coefficients. Nevertheless, one can also observe an unexpected regime of over-transmission near *R*o = 0.8 and low frequency *ω*. Still, we must not forget that solutions in this regime, even near the critical level (see Eq. ([solpol])), are not rigorously equivalent to wave-like functions. In particular, the amplification term (*y* − *y*0)1/2 that can be found at the first-order in the Frobenius solutions becomes more prominent as the thickness of the shear zone is larger. This is especially true for the transmission coefficient. Assuming that Eq. ([solpol]) holds throughout zone II and corresponds to upward and downward waves, the transmission coefficient is modulated by ∣1 − *y*0∣1/2/∣0 − *y*0∣1/2, the amplitude ratio between the transmitted and incident waves. This term can be greater than one in the shear region. In particular, it is always greater than one when *y*0 < 0, i.e. no critical level in the regime *R* > 1/4 (hatched areas in Fig. [RTst]). On the contrary, this ratio is not present for the reflection coefficient since ∣*R*∣ is function of the incident and reflected waves evaluated at *y* = 0. Though this hand-waving explanation does not formally demonstrate the origin of this amplification, it stresses the important role of the shear-region thickness and more generally of the geometry of the model.\nIn order to clarify whether the amplification is due to the geometry or the critical level, we need to investigate how the wave action flux changes before and after the critical level. The wave action flux is indeed the relevant quantity to investigate energy flux exchanges at a critical level.\n### Wave action fluxes below and above the shear region\nSince *v* and Π are continuous at the interfaces between the shear and no-shear regions, the wave action flux is preserved and continuous in all three zones in the absence of and critical levels. However, it is discontinuous at the corotation point as demonstrated in Sects. [stabpole] and [inst]. Given the amplitude of the incident and reflected waves (Eq. ([AIR]), we can calculate the ratio of the wave action flux below and after the corotation (see Appendix D.2 for the detailed calculation):\n \n$$\\frac{{\\mathcal{A}}\\_\\mathrm{T}}{{\\mathcal{A}}\\_\\mathrm{I-R}}=\\pm\\frac{\\omega^2\\sqrt{{\\sigma\\_3}^2\\left[k\\_z^2f^2+k\\_\\perp^2\\left({\\widetilde{f}}^2-{\\sigma\\_3}^{2}\\right)\\right]}}{{\\sigma\\_3}^2\\sqrt{\\omega^2\\left[k\\_z^2f^2+k\\_\\perp^2\\left({\\widetilde{f}}^2-\\omega^{2}\\right)\\right]}\\left(\\left|A\\_\\mathrm{I}\\right|^{2}-\\left|A\\_\\mathrm{R}\\right|^{2}\\right)}.\n\\label{rwA}$$ \nThe signs + or − can be chosen in regards to the wave action flux of the transmitted wave that can be positive or negative depending on the presence of the critical level, while the energy flux is always positive in order to have an upward propagating wave in zone III (see Appendix [wAIIII] for a more detailed discussion). This wave action flux ratio is displayed in Fig. [wAIRT] in the two regimes $R\\lessgtr1/4$. As expected, this ratio is equal to one when no critical level is present (hatched areas). Unlike in the previous section, the regime where *R* > 1/4 has no longer amplification areas, ∣AT/AI-R∣ < 1 everywhere. This supports the idea that the critical level has nothing to do with the amplification phenomenon observed in the left panel of Fig. [RTst]. As already observed in Fig. [atnuO2], the damping due to the critical level is strong except for Rossby numbers close to the threshold between the two regimes. Moreover, AT/AI-R < 0 means that ∣*A*I∣2 > ∣*A*R∣2 since the minus sign is taken in Eq. ([rwA]). Therefore, no over-reflection due to the critical level is expected in this regime. The other regime (*R* < 1/4, right panel) features areas where the wave is over-reflected for ∣*A*I∣2 < ∣*A*R∣2 (i.e. when AT/AI-R > 0) and areas where the wave is over-transmitted for ∣AT/AI-R∣ > 1. For the first inequality (∣*A*I∣2 < ∣*A*R∣2), the threshold between under and over reflection (around *R*o ≈ 0.9) is the same than for the reflection factor (in the right panel of Fig. [RTinst]). For the second one (∣AT/AI-R∣ > 1), the comparison with the transmission factor (in the left panel of Fig. [RTinst]) is more questionable. Still, these two points suggest that the critical level can induce the over-reflection and over-transmission phenomena in the regime where *R* < 1/4.\n### Numerical solutions\n[ht] [vApole]\nIn the previous sections, we have examined how the shear parameter and the inertial wave frequency impact the reflection and transmission coefficients as well as the wave action flux. We now study particular cases of wave propagation through the critical level for fixed sets of parameters in both regimes $R\\lessgtr1/4$. To do that, we have numerically solved Eqs. ([ODEcode]) for the three-layer model described in Sect. [numMod], for *θ*0 = 0 and a linear shear flow (*n* = 1 in zone II). We have selected three pairs of values for the inertial frequency and the shear, two in the regime *R* > 1/4 and one in the regime *R* < 1/4. These values are marked by crosses in Figs. [RTinst], [RTst] and [wAIRT]. In each case, the latitudinal velocity and the wave action flux have been calculated through the three zones successively and plotted in Fig. [vApole]. The numerical solution, which has been computed by imposing the boundary condition *A*T = 1 and the continuous interfacial conditions for *v* and Π at *y* = {0, 1}, is the sum of incident and reflected waves in zone I, and equal to a transmitted wave in zone III. The expressions for the incident, reflected and transmitted waves are given by Eqs. ([IRwaves]) and ([Twave]) (see also Appendix [solIIII]). In the shear region (zone II) of the upper panel of Fig. [vApole], the Whittaker solution has been added and it matches perfectly with the numerical solution below and above the critical level in each case. Moreover, Frobenius approximations for the latitudinal velocity (Eqs. ([solpol]) when *R* > 1/4, and ([solpolinst]) when *R* < 1/4) and for the wave action flux (Eqs. ([Apolst]) when *R* > 1/4, ([Apolinst]) otherwise) have also been included. The coefficients *a*0 and *b*0 have been determined by matching the numerical solution and its derivative to the Frobenius approximation for the velocity close to the critical level. For both the latitudinal velocity and the wave action flux, this first-order approximation gives satisfactory agreement with numerical solutions, although a slight deviation (regarding velocity) from the numerical solution can be observed as one moves away from the critical level. In particular, it should be mentioned that for the far left panels, *a*0(*y* − *y**c*)1/2 + *i*∣*μ*∣ corresponding to an upward wave is sufficient to fit correctly the numerical solution, meaning that the counter-propagating wave in zone I is reflected at *y* = 0. However, for the middle and right-hand side panels, it is not clear whether the first-order Frobenius solutions can be reconnected to the incident and reflected waves at *y* = 0.\nWe now examine attenuation or amplification phenomena in each column of panels in Fig. [vApole], from left to right. In the left panels, for which *R* > 1/4, the latitudinal velocity is strongly attenuated at the critical levels, and so is the wave flux action as we can expect from the left panel of Fig. [wAIRT]. While the transmitted wave is totally absorbed, the reflected wave remains, which is consistent with analytical values of the transmission and reflection coefficients in Fig. [RTst] (see white crosses). In the middle panels, where we also have *R* > 1/4, the wave is over-transmitted but not over-reflected, which is also consistent with Fig. [RTst] (see black crosses). In view of the wave action flux, the amplification of the transmitted wave does not seem to be related to the critical level because this quantity is greatly reduced after the critical level (see also the white cross in the left panel of Fig. [wAIRT]). The third column of panels now refers to the regime where *R* < 1/4. The wave is over-reflected by a factor of ∼ 1.5 and over-transmitted by a factor of ∼ 2 in concordance with the reflection and transmission coefficients plotted in Fig. [RTinst]. The wave action flux is negative and ∣AI-R∣ < ∣AT∣ by a factor of 3 as observed in Fig. [wAIRT]. These three case studies reinforce the idea of in the case of stratified z-sheared flows that the wave energy can be lost to the mean flow, or on the contrary that the wave can take energy from the mean flow.\n[ht] [vA10deg]\nNumerical exploration at constant shear when the box is inclined\n----------------------------------------------------------------\nWe now investigate wave propagation through the different critical levels when the box is inclined with respect to the rotation axis. We still assume that the shear region (zone II) has a linear shear flow profile (*n* = 1). In contrast to the polar configuration, we do not have analytical solutions to the ordinary differential equation. Instead of going for an extensive numerical investigation of the parameter space, we will rather focus on the dynamics of inertial waves in our three-layer model as they cross the critical levels $\\sigma=\\pm{\\widetilde{f}}$ and *σ* = 0. Our results are presented in Fig. [vA10deg] for a box inclination of $10\\degree$ relative to the rotation axis, a shear fixed to *R*o = 0.3 and wave numbers set to *k**x* = *k**z* = 1. The value of the frequency *ω* determines the existence and the nature of the critical level as detailed in the table of Appendix [wAIIII]. Like in Fig. [vApole], we plot in each column of Fig. [vA10deg] the latitudinal velocity and the wave action flux. From left to right, we illustrate our results for the critical levels $\\sigma=+{\\widetilde{f}}$, *σ* = 0, and $\\sigma=\\{0,-{\\widetilde{f}}\\}$ (i.e., there are two critical levels in the rightmost panels). One can notice that the first-order Frobenius approximation is not in good agreement anymore with the numerical solution in the entire shear region, though it remains a reasonable approximation close enough to a critical level. Unlike the polar case, the discrepancy far outside the critical levels is due to the linear approximation that the governing ODE takes around critical levels (see Eq. ([PcDLft]) and ([Pcorot])).\nIn the left panels of Fig. [vA10deg], the reflected and transmitted waves are strongly attenuated at the critical level *y* = *y*+. Part of the wave energy is laid down to the mean flow, as corroborated by the drop in the wave action flux. In the middle panels, we do not see any discontinuity at the corotation *y* = *y*0, which is in line with the theoretical analysis in Sect. [corotinclined] for a constant shear. However, the wave is over-reflected and over-transmitted, possibly due to the polynomial form of the solutions in the Frobenius series around *y* = *y*0. In the right panels, the wave encounters successively critical levels at *y* = *y*0 and at *y* = *y*−. Although the wave going through the shear region is not attenuated at the corotation *y* = *y*0, it is completely absorbed at the second critical level *y* = *y*− where the wave action flux drops to zero. This is consistent with the transmission coefficient in the left panel of Fig. [atnudeg]. The latitudinal velocities displayed in the top-left and top-right panels of Fig. [vA10deg] support the concept of a valve effect. Indeed, according to our analysis in Sect. [ccft] with the Frobenius method and given the shear and wave numbers of Fig. [vA10deg], the attenuation is strong for a downward wave meeting the critical level *y*+ (first panel) while the attenuation is strong for an upward wave that meets the critical level *y*− (third panel). Before and after the critical levels *y*− and *y*+, respectively, we observe fast oscillations of shorter and shorter period close to the critical level as already evidenced by. The analysis to determine how the wave is reflected in the shear zone can hardly be taken any further, because Frobenius solutions are not fully separable into upward and downward waves.\nWe emphasise that the behaviour of the wave at corotation *y* = *y*0 when the box is at the pole stands out clearly different from the case when the box is inclined for a linear mean flow profile. This is particularly true in terms of the absolute value of the wave action flux that is subject to rise and drop in the polar configuration, whereas it remains conserved when the box is inclined. In this inclined case, the only way for a wave to be attenuated without is that it meets critical levels $\\sigma=\\pm{\\widetilde{f}}$. Depending on the critical level encountered, an upward or downward wave will not be attenuated in the same way as known by the valve effect. In addition, we no longer observe amplification due to the critical level but still “geometric” amplification (for instance in the middle panels where we can observe over-transmission and over-reflection), that can be explained by the exponential form of the Frobenius series.\nNumerical results with a non-constant shear\n-------------------------------------------\n[ht!]\n c|c ||c c c | c c c | c c c && & & \n---\n& & & & & & & & & & \n---\n&*R*o = 0.3 & 1.3 10− 4 & 9.1 10− 3 & 1.5 10− 2 & 1.3 10− 1 & 2.1 10− 2 & 7.4 10− 2 & − 3.2 10− 8 & − 1.6 10− 4 & − 1.1 10− 2 \n---\n& *R*o = 0.8 & 7.4 10− 1 & 5.7 10− 3 & 1.7 10− 4 & 1.4 & 9.1 10− 5 & 4.0 10− 6 & − 1.1 10− 1 & − 2.1 10− 10 & − 4.1 10− 13 \n---\n&*R*o = 1.8 & 1.5 & 6.4 10− 1 & 7.0 10− 1 & 1.8 & 1.1 & 9.7 10− 1 & 2.9 & − 1.9 & − 1.8 \n---\n&*y*+ & 8.2 10− 2 & 2.2 10− 2 & 2.4 10− 2 & 1.0 10− 1 & 8.9 10− 2 & 8.3 10− 2 & 3.5 10− 1 & 2.7 10− 1 & 2.4 10− 1 \n---\n& *y*0 & 5.3 & 4.9 & 4.7 & 4.9 & 4.4 & 4.3 & 1.0 & 1.0 & 1.0 \n---\n&*y*0 and *y*− & 5.0 & 6.2 & 6.1 & 2.3 10− 4 & 4.0 10− 2 & 2.2 10− 1 & 1.1 10− 9 & 2.0 10− 5 & 6.7 10− 4\n[tabRTA]\nThe choice of a linear mean flow profile allows the resolution of the ordinary differential equation at the pole and a simpler implementation of the Frobenius method. Nevertheless, the correspondence between a global and a local mean flows as presented in Sect. [meanflowsect] involves higher-order terms than a simple linear dependence. Therefore, it is important to examine the effect of different mean flow profiles with non-zero *U*ʺ. Now assuming *n* > 1 for the shear flow profile in zone II (see Eq. [Upro]), the Frobenius method in the inclined and polar cases still holds provided that\n \n$$\\left\\{\\begin{aligned}\n&{y\\_0}=\\sqrt[n]{\\frac{\\omega}{k\\_x\\Lambda}},\\ \\text{ and }\\ y\\_\\pm=\\sqrt[n]{\\frac{\\omega\\mp{\\widetilde{f}}}{k\\_x\\Lambda}}\\\\\n&{R\\_\\mathrm{o}}=\\Lambda n(y\\_{0,\\pm})^{n-1}.\n\\end{aligned}\\right.$$ \nMoreover, we specify that the critical level in *y* < 0 for even values of *n* is not examined given our numerical set up where the shear region is located in the range *y* ∈ [0, 1]. The conditions to have critical levels inside the shear region (zone II) are the same as those for a constant shear (see Appendix [wAIIII]).\nWe show in Table [tabRTA] numerical values of the reflection and transmission coefficients along with the ratio of the wave action flux below and after the critical level in the six parameter sets illustrated in Figs. [vApole] and [vA10deg] for linear, square and cubic mean flow profiles. For all cases, *R*, *T* and AT/AI-R change quite significantly between *n* = 1, 2 and 3. At the pole, the (“geometric”) over-transmission found for *R*o = 0.8 disappears for *n* = 2 and *n* = 3, where the reflected and transmitted waves are strongly attenuated. Similarly, in the case where *R*o = 1.8, the over-transmission and over-reflection disappear when *n* = 3, whereas ∣AT/AI-R∣ > 1 which entails that the transmitted wave is taken wave action flux from the mean flow just after the critical level. We find that the Frobenius method does not give consistent results for the case where *R*o = 1.8 and *n* = {2, 3}. When the box is tilted and for the critical level *y*+, the reflected wave is more attenuated than the transmitted one, which is consistent with stronger attenuation of the counter-propagating wave discussed in the previous section. At the corotation and when the box is inclined (fifth row of Table [tabRTA]),. We speculate that the over-transmission and over-reflection are due to the polynomial form of the solutions in the shear region.. In the last case (sixth row), where two critical levels co-exist, the transmitted wave is less attenuated at *y* = *y*− from *n* = 1 to *n* = 3 but it remains more attenuated than the reflected wave as discussed in the previous section. Again, no jump is found in the wave action flux at the corotation despite analytical predictions.\nThe same analysis was carried out for different amplitudes of the Rayleigh friction force, up to *σ**f* = 10− 2. Of course, the wave action flux is no longer constant, but we observe that all three parameters of Table [tabRTA] change very little compared to the case where *σ**f* = 10− 8. This result is consistent with in the context of gravity waves and vertical shear. We comment that while a low is mandatory in the numerical code to solve the ODE at singularities *y*±, it is not the case at the corotation *y* = *y*0.\nAstrophysical discussion\n========================\n![Absolute values of the shear contrast normalised by the rotation at the pole against the age of stars of K and G spectral types. Solid lines feature 0.3<R_\\mathrm{of}<0.9, dashed lines R_\\mathrm{of}\\lesssim0.3 and transparent lines R_\\mathrm{of}>1.](figures/Dom_age_medn \"fig:\") [Dom]\nLatitudinal differential rotation in stars\n------------------------------------------\nIn stars, latitudinal differential rotation is often characterised by the difference in rotation frequency between the equator and the pole, that is the quantity ΔΩ = Ωeq − Ω0, where Ωeq is the rotation frequency at the equator. We will now refer to ΔΩ as the shear contrast. Different regimes are distinguished according to the value of ΔΩ: anti-solar-like rotation for ΔΩ < 0, cylindrical rotation for ∣ΔΩ∣ ≪ 1, and solar-like rotation rotation for (not-too-low) positive ΔΩ (e.g. ΔΩ/Ω0 ≃ 0.3 for the Sun). Several works based on three-dimensional numerical simulations have explored the range of physical parameters leading to each aforementioned regime in stars and in giant planets In particular, have derived a criterion based on mixing length theory and calibrated with 3D simulations that determines the rotation profile of a solar-like star. This criterion is based on the fluid Rossby number *R*of, defined as\n \n*R*of = *R*of,\\odot Ω\\*− 0.82*M*\\*1.53, \nwhere *R*of,\\odot = 0.89 is the solar fluid Rossby number, and Ω\\* and *M*\\* are the mean rotation and the mass of the star respectively, normalised with their solar value. highlighted the following three regimes:\n* *R*of > 1 for anti-solar-like rotation,\n* 0.3 < *R*of < 0.9 for solar-like rotation,\n* $R\\_{\\mathrm{of}}\\lesssim0.3$ for cylindrical rotation.\nFurthermore, they introduced the shear contrast ΔΩS at the colatitude $30\\degree$ as\n \n$$\\Delta\\Omega\\_\\mathrm{S}=\\Omega\\_\\mathrm{eq}-\\Omega(\\theta=30\\degree),$$ \nas the rotation frequency is often ill-defined at low colatitudes in 3D numerical simulations in spherical geometry. From their 3D simulations, obtained the following scaling:\n \nΔΩS = ΔΩS, ⊙ *M*\\*0.73Ω0.66, \nwhere $\\Delta\\Omega\\_{\\mathrm{S},\\odot}\\simeq565\\times{\\,10^{-9}}\\second^{-1}$ is the solar value of ΔΩS calculated from. Using the mean flow profile (Eq. ([law])), ΔΩS can be related to our shear contrast ΔΩ via\n \n$$\\frac{\\Delta\\Omega}{\\Omega\\_0}=\\chi=\\frac{4}{3}\\frac{\\Delta\\Omega\\_\\mathrm{S}}{\\Omega\\_0}.\n \\label{deltaOmega}$$ \nWe show in Fig. [Dom] the quantity ΔΩ/Ω0, expressed in Eq. ([deltaOmega]), versus the age of solar-like stars for K to G spectral types. To compute this quantity, we have used grids of the 1D stellar evolution code STAREVOL. In light of Fig. [Dom], the stars in the pre-main sequence ($\\mathrm{age}\\lesssim 100\\,\\mega\\mathrm{yr}$) exhibit cylindrical rotation as they are fast rotating. During the main sequence, stars mostly feature solar-like rotation, while anti-solar-like rotation is observed at the end of the main sequence from 0.8 to 1.1 *M*⊙. According to Fig. [Dom], a limit on the absolute value of the normalised shear contrast can be set to ∣ΔΩ∣/Ω0 < 0.5. However, as already stressed by, the latitudinal shear inferred by asteroseismology can be much larger than predicted by numerical simulations. This can actually be inferred by comparing the shear factors in their work with ours given in Fig. [Dom]. Moreover, according to their study, cylindrical and anti-solar differential rotation are hardly unambiguously detectable. Finally, one should recall that ’s scaling laws given in Eqs. ([B1]) and ([B2]) are derived for K and G spectral type stars only. \nSince we now know values of the shear contrast, we can calculate the “shear” Rossby number *R*o = *U*ʹ/(2Ω), given by the following relationship:\n \n$${R\\_\\mathrm{o}}=-\\frac{3}{2}\\cos\\theta\\_0\\sin^2\\theta\\_0\\,\\chi,$$ \nwhich has been derived from Eq. ([Uconic]) by keeping only zero-order terms in *y*. Taking *χ* ≃ 0.3 as a representative value of the shear contrast for main sequence G and K stars, we find that the Rossby number is maximal when $\\theta\\_0\\simeq55\\degree$, its maximum value being *R*o ≃ − 0.17. In particular, *R*o ≃ − 0.013 for $\\theta\\_0=10\\degree$ and *R*o ≃ − 0.076 for $\\theta\\_0=80\\degree$. These values are useful to interpret wave flux action transmission at critical levels $\\sigma=\\pm{\\widetilde{f}}$, considering Fig. [atnudeg]. A downward (upward) propagating wave through $\\sigma={\\widetilde{f}}$ ($\\sigma=-{\\widetilde{f}}$) is\n* totally absorbed provided that $k\\_z\\gtrsim0.1 k\\_x$ ($\\alpha\\_k\\gtrsim0.1$) at $\\theta\\_0=10\\degree$, and that $k\\_z\\gtrsim k\\_x$ ($\\alpha\\_k\\gtrsim1$) at $\\theta\\_0=80\\degree$,\n* strongly attenuated for *k**z* ∼ 0.1*k**x* (*α**k* ∼ 0.1) at $\\theta\\_0=80\\degree$,\n* fully transmitted given that *k**z* ≪ 10− 1*k**x* (*α**k* ≪ 10− 1) for both inclinations.\nThese results also hold for anti-solar-like differential rotation, since the transmission factor *T**θ*0 is function of ∣*R*o∣. For larger values of ∣*R*o∣, i.e. for larger values of the shear contrast, waves are less damped at critical levels $\\sigma=\\pm{\\widetilde{f}}$ at a given *α**k* = *k**z*/*k**x*. The connection between this ratio of the vertical and azimuthal wavenumbers in the local model and an equivalent ratio of global wavenumbers in the spherical geometry is not straightforward. A first hint can be to state that *k**z* ∼ *k**r*, where *k**r* is the wavenumber in the global radial direction, while *k**x* ∼ *m*/(*r*0sin*θ*0), where *m* is the azimuthal order of the considered mode of the tidal potential. Then, we have *α**k* ≡ *k**r**r*0sin*θ*0/*m* ≡ 2*π* *r*0/*λ**r* × sin*θ*0/*m*, by introducing *λ**r* the radial wavelength of the tidal wave. In the case where *r*0 > *λ**r**m*, we should thus be in the regime where the tidal wave is attenuated.\nCylindrical differential rotation in Jupiter and Saturn\n-------------------------------------------------------\n [JuSa]\n[t!]\nc|cc|cc|c| c\n---\n& & & & \n---\n& *l* & *m* & *m* & *l*Θ & & \n---\nasynchronous & 2 & 2 & 2 & & & absorption \n---\ninclined &2 & 1 & 1 & 2 & 2 & total absorption \n---\neccentric & 2 & 0 & 0 & 2 & → ∞ & total absorption\n[lm]\nMathis et al. (in prep.) have developed an equatorial model to examine inertial wave properties in the outer convective layers of giant gaseous planets such as Saturn and Jupiter, which are subject to cylindrical differential rotation. In their model (built in cylindrical coordinates), they derived a Schrödinger-like differential equation for $\\Psi=\\sqrt{\\rho}r^2v\\_r$ under the anelastic approximation, where *ρ* is the density, *r* the axial distance coordinate, and *v**r* the axial velocity. For free inertial waves, their second-order differential equation is:\n \n$${\\frac{\\mathrm{d}^2\\Psi}{\\mathrm{d}r^2}}+\\left(\\frac{\\kappa\\_r^2l\\_\\Theta^2}{\\hat\\sigma^2r^2}-\\frac{l\\_\\Theta^2+m^2}{r^2}\\right)\\Psi=0,\n\\label{equa}$$ \nwhere *l*Θ and *m* denote the equatorial and azimuthal wavenumbers, respectively, $\\hat\\sigma=\\omega+m\\Omega(r)$ is the (linear) Doppler-shifted frequency, and *κ**r* the “axial” epicyclic frequency defined as $\\kappa\\_r^2=4\\Omega^2+2\\Omega r{\\frac{\\mathrm{d}\\Omega}{\\mathrm{d}r}}$. For cylindrical differential equation, the corotation resonance $\\hat\\sigma=0$ results in critical cylinders characterised by a critical axial distance *r* = *r*c. The Taylor expansion of Eq. ([equa]) at first order around *r*c gives:\n \n$${\\frac{\\mathrm{d}^2\\Psi}{\\mathrm{d}r^2}}+\\left(\\frac{ l\\_\\Theta^2(1+{R\\_\\mathrm{o}})}{m^2{R\\_\\mathrm{o}}^2(r-{r\\_{\\mathrm{c}}})^2}-\\frac{l\\_\\Theta^2+m^2}{{r\\_{\\mathrm{c}}}^2}\\right)\\Psi=0,\n\\label{equa\\_exp}$$ \nby setting ${R\\_\\mathrm{o}}={r\\_{\\mathrm{c}}}\\left.{\\frac{\\mathrm{d}\\Omega}{\\mathrm{d}r}}\\right|\\_{r={r\\_{\\mathrm{c}}}}/(2\\Omega\\_\\mathrm{c})$ the local Rossby number in cylindrical coordinates. By writing *α**k*2 = *l*Θ2/*m*2, Eq. ([equaexp]) becomes very similar to our Eq. ([Pc]) when the box is located at the South pole and for a constant shear (i.e. with ${\\widetilde{f}}=0$, *f* = − 1 and *U*ʹʹ = 0). Note that, when the local shear box model is located at the equator, the latitudinal coordinate *y* is directed along the (vertical) rotation axis, whereas when the box is at the poles, *y* is directly the axial distance. That is why a polar configuration of the box best reproduces the “equatorial” model of Mathis et al. (in prep.). Moreover, the convention of a plus sign in the Doppler-shifted frequency explains why Eq. ([equaexp]) is analogous to our equation when the box is at the South pole rather than at the North pole.\nIn Fig. [JuSa], we show Jupiter’s and Saturn’s local Rossby number from Mathis et al. (in prep.). Cylindrical differential rotation extends in the outer layer of the convective envelope of both planets, in agreement with Juno and Cassini Grand Finale observations. According to Fig. [O2], where we notably plotted *R* = *α**k*2(1 + *R*o)/*R*o2 at the South pole (dark red and purple areas), we can assess the role of the critical level for wave transmission across the corotation, in terms of the wavenumber ratio *α**k*. In Fig. [JuSa], the Rossby number ${R\\_\\mathrm{o}}\\lesssim0.27$ for Saturn and ${R\\_\\mathrm{o}}\\lesssim0.07$ for Jupiter. Given this range of values, two regimes can be evidenced for waves and wave action fluxes through the corotation:\n* waves are strongly attenuated for $l\\_\\Theta\\gtrsim m$ (see also Fig. [atnuO2] for *R*o = − ∣*R*o∣ as the transmission factor in this figure is plotted for the North pole).\n* waves can be over-reflected and over-transmitted for *l*Θ ≪ *m*, and can potentially lead to instabilities given specific boundary conditions.\nTo give an idea of the values these wavenumbers can take, we have listed in Table [lm] three typical orbital states, where, in order, the asynchronous, eccentricity and obliquity tides are supposed to be dominant. These states are described by the “spherical” quadrupolar components of the dominant terms in the tidal potential, i.e. the degree *l* and the order *m* of the spherical harmonics. The analogy with the equatorial model is then made to get *m*, (Mathis et al. in prep.)..\nConclusion and perspectives\n===========================\nThe present study was motivated by the works of and, who showed that differential rotation can strongly affect the propagation and dissipation properties of (tidal) linear inertial waves. They considered different rotation profiles typical of stellar and planetary interiors and they pointed out that tidal waves can deeply interact with zonal flows at corotation resonances, leading to intense wave energy dissipation, alongside possible instabilities. In this paper, we have investigated the transmission of free inertial waves with latitudinal stratification and differential rotation, at the corotation resonance (characterised by a zero Doppler-shifted wave frequency) and more broadly at critical levels (any singularities of the governing second-order wave propagation equation in the inviscid limit). For this purpose, we have built a new local Cartesian box model with horizontal shear, modelling a small patch of the convective zone of a low-mass star or a giant planet. By considering the inclination of the local reference frame relative to the rotation axis, we have examined the effect on wave propagation through a critical level of a conical rotation profile at a general colatitude when the box is tilted, or of a cylindrical rotation profile when the box is at the North or South poles. These rotation profiles are inspired by those observed or expected in the Sun, low-mass stars, and giant gaseous planet in our solar system. Three critical levels can be identified when the box is inclined relative to the rotation axis: the corotation resonance and two others critical levels that arise from the inclination between the gravity and the rotation vectors, and which are defined by a Doppler-shifted frequency equal to plus or minus the latitudinal component of the rotation frequency. When the box is at the poles, critical levels are restricted to the corotation.\nIn order to diagnose the behaviour of a wave passing through a critical level for both aforementioned rotation profiles, we made use of an invariant called the wave action flux that is independent of the latitudinal coordinate in a non-dissipative fluid flow. This invariant was used when the “directional” flux of angular momentum (here latitudinal) can not be constructed easily from the mean perturbed velocity, as it is the case for example in for Rossby waves in plane-parallel shear flows. The wave action flux has already been used in vertically-stratified shear flows in the presence of rotation or magnetic fields to interpret the role of critical levels. Using the condition that this invariant is discontinuous at critical levels, we have demonstrated in Sect. [sec3] that waves can be either fully transmitted, damped or even amplified after passing through critical levels as a result of wave action flux exchanges. These different regimes of wave transmission are found both with conical and cylindrical rotation profiles; they depend on the critical level encountered, on the wave properties (e.g. the propagation direction, wavenumbers) and on the profile of the mean flow. Table [recapana] summarises the main analytical results.\nWe have then confronted our analytical results with a three-layer numerical model that comprises a shear zone in which the critical level is located and two surrounding shear-free zones that allow incident, reflected and transmitted waves. A difference with the analytical model is the introduction of small dissipative force under the form of a Rayleigh friction to avoid strict singularities. It does not seem to affect the results, since analytical and numerical results match well when using a and by varying the. This conclusion is also shared by the work of who studied corotation resonances for gravity waves propagating in stratified and vertically shear flows.\nBased on the analytical results, we have discussed in Sect. [astrodi] possible applications to stellar and planetary interiors. We have estimated the rate of differential rotation in solar-like stars by the shear contrast (the rotation difference between the pole and the equator) and in giant gaseous planets through the local Rossby number (the ratio between the shear and the rotation frequency in cylindrical coordinates). We find that for K and G-type stars along their lifetime, and for Jupiter and Saturn at the present time, a regime where inertial waves are strongly damped is largely preferred in the convective envelope of these objects. Similar conclusions were found by for internal gravity waves through critical levels in the core of solar-like stars. \nIt is interesting to discuss different regimes of the wave transmission in terms of angular momentum transfer for cases of strong damping and wave amplification. First, we have to underline that the theoretical analysis presented in Section 3 (using the Frobenius method) does not adequately characterise wave (over-)reflection as similarly observed in the numerical section [secnum]. What we can access is the wave action flux on either side of the critical level. The analysis on changes in the wave action flux across the critical level allows us to understand whether energy is deposited to or extracted from the mean flow, in line with the work carried out by. In the presence of a locally conical differential rotation, we have demonstrated that a valve effect can be found for critical levels other than the corotation, analogous to the results of and for hydromagnetic and gravito-inertial waves with a vertical shear. For these peculiar critical levels, waves can be attenuated when going one direction, mainly featured by the sign of the rotation components in the box, or fully transmitted when going the other direction. For cylindrical differential rotation, we have found a criterion analogous to the Miles-Howard theorem for stratified shear flows, which for inertial waves as in this work can be formulated as:\n \n$$R=\\frac{k\\_z}{k\\_x}\\frac{1-{R\\_\\mathrm{o}}}{{R\\_\\mathrm{o}}^2}\n\\left\\}\\begin{aligned}\n&>1/4 \\text{: wave attenuation;}\\\\\n&<1/4 \\text{: possible wave over- }\\left|\\begin{tabular}{l} reflection, \\\\ transmission. \\end{tabular}\\right.\n\\end{aligned}\\right.\n\\label{resum}$$ \nThe above criterion depends on the shear Rossby number *R*o = *U*ʹ/(2Ω) and the vertical (*k**z*) and longitudinal (*k**x*) wavenumbers. We stress that is very different from the Rayleigh’s inflection point theorem for Rossby waves, which are a subclass of inertial waves when neglecting the vertical or the radial perturbed velocity. The Miles-Howard criterion allows to disentangle between critical levels where strong wave attenuation is expected for Ri > 1/4 (where Ri is the Richardson number), and those where over-reflection and over-transmission can lead to potential shear instabilities for Ri < 1/4. warns, however, that over-reflection and over-transmission are a necessary but not sufficient condition for shear instability. Such amplifications leading to the instability require peculiar conditions in a three-layer model, where the shear zone that features the critical level is surrounded by a region of incoming propagating waves, and a “sink” zone to force waves to cross the evanescent shear zone. Special boundary conditions are necessary for the wave to return successively to the critical level and induce wave amplitude growth. Recent studies have revisited instabilities in stratified shear flows by studying multiple counter-propagating waves that can interact to grow in amplitude with time (with conditions as phase-locking). A parallel is drawn between over-reflection mechanisms and interacting counter-propagating waves by to describe baroclinic instabilities for Rossby waves.\nContrary to what our results predict, did not observe any instabilities of inertial waves when using cylindrical differential rotation. Several reasons can be put forward to explain this discrepancy, such as boundary conditions (as discussed in the previous paragraph), or the values of the shear and horizontal wavenumbers, since they may not be in the regime allowing instabilities according to the criterion Eq. ([resum]), which further needs to be adapted to their global cylindrical geometry. We stress that when exploring different power-laws for the mean flow profiles, over-reflection was not retrieved for a non-linear mean flow in cylindrical differential rotation. With conical differential rotation, did observe instabilities, but only for sufficiently low viscosities, whereas our study showed little dependence on the, and rather highlights possible over-transmission for non-linear flows"},"data_id":{"kind":"string","value":"arxiv_0003282"}}},{"rowIdx":798,"cells":{"content":{"kind":"string","value":"see §3.1) \n2) a ray-tracing algorithm based on the pre-calculation of a lower limit for the RFED (see §3.2 and §3.3) \n3) an iterative procedure for the optimization of the angular density for the rays departing from each emitting cell (§3.3). \nFurthermore, parallelized versions of the code have been written for its use on shared-memory machines and computer clusters. \nIn order to verify the code accuracy, we performed comparisons with the results provided by other codes. Specifically, we used our code to calculate solutions for a spherical dusty shell illuminated by a central point source and for an axis symmetric galaxy model. For the first configuration, we considered as benchmark a set of solutions calculated by using the DUSTY 1D code (Ivezic & Elitzur 1997). We showed that the equilibrium dust temperature radial profiles and the outgoing flux spectra derived by our 3D calculations agree within a few percent with the benchmark solutions for models with low radial optical depths (*τ*1 = 1, 2) but present larger discrepancies for more optically thick models (*τ*1 = 5, 10). The residual discrepancies, especially for the models with higher optical depths, are most probably due to the lack of dust self-heating in our code and the lower spatial resolution of the 3D calculations compared to the 1D ones. Since the geometry of the source emission/opacity of star/dust shell does not reproduce that for which our algorithm was developed, namely that of an extended distribution of stellar emission and dust, we also used a second benchmark. Thus, we considered the 2D calculations by Popescu et al. (2011) for the RFED distribution within their galaxy model. We calculated the contribution to the RFED distribution due to an old stellar disc and a young stellar disc separately and we compared the results for radial and vertical RFED profiles derived for a set of reference radii and vertical distances. We found a general good agreement between the 3D and 2D calculations within a few percent in most of the cases. At least part of the residual discrepancy can be accounted by the relatively low spatial resolution of the grid used in the 3D calculation, which is not sufficient to properly resolve the thin disc component of the galaxy model of Popescu et al. (2011). We showed an example of a 3D application of the code by performing RT for a spiral galaxy model where in one case the emissivity of the young stellar population and associated dust opacity are distributed in logarithmic spiral arms and in another case are distributed in exponential discs. We found that the edge-on images, the attenuation as a function of inclination and the azimuthally average RFED profiles on the galaxy plane are approximately the same for the two models. This suggests that the spatially integrated SEDs of spirals can be well described by 2D models.\nThe tests we performed have shown that, in the conditions where dust self-heating is negligible and the 3D spatial resolution is high enough to resolve emission and opacity distributions, our code can be used to calculate accurate solutions for the RFED. This characteristic is particularly important for the calculation of stochastically heated dust emission, which requires both the overall intensity and the colour of the radiation field to be calculated in an accurate way. In a future work we will show applications of the code for the calculation of infrared emission. This will be performed using our 3D RT code coupled with the dust emission code used by Popescu et al. (2011), which self-consistently calculates the stochastic emission from small grains and PAH molecules. In this way, it will be possible to use our code to obtain both integrated SEDs and images in the mid- and fir-infrared for galaxies with arbitrary geometries. In addition, an important step will be to further optimize the code in order to make it possible to run on grids containing millions of cells in a reasonably short time. This will allow us to improve further the accuracy of the calculation for RFED within multi-scale structures spanning at least three orders of magnitude, such as from ≈ 10pc to ≈ 10kpc in the case of a galaxy ISM. \nAcknowledgements\n================\nWe acknowledge support from the UK Science and Technology Facilities Council (STFC; grant ST/J001341/1). GN thanks S. Dalla, K. Foyle, E. Kafexhiu, T. Laitinen, J. Steinacker for useful suggestions and/or discussions. CCP thanks the Max Planck Institute für Kernphysik for support during a sabbatical, when this work was completed.\nAbel, T., Wandelt, B. D. 2002, MNRAS, 330, 53 Baes, M., Verstappen, J., De Looze, I., Fritz, J., Saftly, W. et al. 2011, ApJS, 196, 22 Bendo, G. J., Boselli, A., Dariush, A., Pohlen, M., Roussel, H. et al. 2012, MNRAS, 419, 1833 Bianchi, S., Ferrara, A., Giovanardi, C. 1996, ApJ, 465, 127 Bianchi, S. 2008, A&A, 490, 461 Bisbas, T. G., Bell, T. A., Viti, S., Yates, J., Barlow, M. 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The actual version of the code does not consider the dust absorption/scattering of light emitted by dust at other positions (so called “dust self-heating”). This effect can be neglected in case the galaxies are optically thin at infrared wavelengths[↩](#fnref2)\n3. Throughout the text by “volume emissivity” we will always mean the luminosity per unit volume per unit solid angle and per unit wavelength interval of the stellar radiation at each position. To not be confused with the emissivity coefficient used to characterise the emission properties of e.g. gas or dust.[↩](#fnref3)\n4. For example, the latter condition for a single grain stochastically heated can be expressed as: \n∫*Q**λ*, *a**b**s*∫*B**λ*(*T*)*P*(*T*)*d**T**d**λ* = (*c*/4*π*)∫*Q**λ*, *a**b**s**U**λ**d**λ* \n where *Q**λ*, *a**b**s* is the grain absorption efficiency at wavelength *λ*, *B**λ*(*T*) is the Planck function calculated for dust temperature *T* and *P*(*T*) is the probability for the dust grain to have a temperature equal to *T*. Numerical methods, such as the one presented in Guhathakurta & Draine (1989), allow us to derive *P*(*T*) and therefore the dust emission spectra, once the absorption efficiency *Q**λ*, *a**b**s* and the RFED *U**λ* are known..[↩](#fnref4)\n5. This approach can be redundant because the rays originating from different cells can often go through almost the same paths. On the other side, it is convenient because it easily allows the implementation of the acceleration techniques presented in this work.[↩](#fnref5)\n6. Note that, as shown in the right-hand panel of Fig.[encrit], there could be emitting cells which are part of a large emitting cell distribution, whose individual RFED contributions *δ**U**λ* might be lower than the assumed energy density threshold *f**U**U**λ*, *L**L*, but the cumulative contribution is not negligible. Overlooking the presence of this kind of cells can be avoided by choosing an appropriate value for the constant *f**U*. However, in cases such as that of an extended distribution of uniform volume emissivity within an optically thin system, the value of *f**U* to be chosen, in order to reach an accurate solution, could be extremely low. In those cases, there might be no substantial advantage in terms of speed by using the presented algorithm, since the majority of emitting cells contribute significantly to the RFED at each position in the entire volume.[↩](#fnref6)\n7. We use a grid refinement factor equal to 3, since it has the advantage of preserving the position of cell centres after each cell subdivisions. For technical reasons, this facilitates the inclusion in the grid of central emitting point sources such those in the solutions described in the next section.[↩](#fnref7)\n8. Actually the numerical implementation requires to assume a small threshold value higher than zero, such that *e*− *τ**λ* ≠ 1 when the exponential function is evaluated.[↩](#fnref8)\n9. The SCATT\\_EN array can be very large: dimension = number of cells × $4\\pi/\\Omega\\_{\\rm{HP,INT}}$.[↩](#fnref9)\n10. see upper-left panel of Fig.[hpixsphere] for definition of “HEALPix main sector”[↩](#fnref10)\n11. We note that the models are truncated at the radius *R* = 24kpc. The unit luminosities in Table E2 of P11 refer to the luminosities within the truncation radius.[↩](#fnref11)\n12. The choice of *f**U* and $N\\_{\\rm rays}$ is such to guarantee global energy balance within a few percent accuracy (see §4).[↩](#fnref12)\n\nDART-Ray: a 3D ray-tracing radiative transfer code for calculating the propagation of light in dusty galaxies\n=============================================================================================================\n[firstpage]\nWe present DART-Ray, a new ray-tracing 3D dust radiative transfer (RT) code designed specifically to calculate radiation field energy density (RFED) distributions within dusty galaxy models with arbitrary geometries. In this paper we introduce the basic algorithm implemented in DART-Ray which is based on a pre-calculation of a lower limit for the RFED distribution. This pre-calculation allows us to estimate the extent of regions around the radiation sources within which these sources contribute significantly to the RFED. In this way, ray-tracing calculations can be restricted to take place only within these regions, thus substantially reducing the computational time compared to a complete ray-tracing RT calculation. Anisotropic scattering is included in the code and handled in a similar fashion. Furthermore, the code utilizes a Cartesian adaptive spatial grid and an iterative method has been implemented to optimize the angular densities of the rays originated from each emitting cell. In order to verify the accuracy of the RT calculations performed by DART-Ray, we present results of comparisons with solutions obtained using the DUSTY 1D RT code for a dust shell illuminated by a central point source and existing 2D RT calculations of disc galaxies with diffusely distributed stellar emission and dust opacity. Finally, we show the application of the code on a spiral galaxy model with logarithmic spiral arms in order to measure the effect of the spiral pattern on the attenuation and RFED.\nIntroduction\n============\nInterstellar dust is of primary importance in determining the spectral energy distribution (SED) of the radiation escaping from galaxies at wavelengths ranging from the ultraviolet (UV) to the submillimetre (submm) and radio. Dust attenuates and redistributes the light, originating mainly from stars and, if present, from an active galactic nucleus (AGN), by either absorbing or scattering photons. The absorbed luminosity is then re-emitted in the infrared regime. The dust may be situated in complex geometries with respect to these sources, affecting the observed structure of the galaxy at each wavelength as well as its integrated SED. Modeling the propagation of light within real galaxies is thus a challenging task. Nevertheless, it is essential to do such modeling, if physical quantities of interest, such as the distribution and properties of the stellar populations and the interstellar medium (ISM) as traced by dust and the interstellar radiation fields, are to be derived from multiwavelength images and SEDs. \nTaking advantage of the approximate cylindrical symmetry of galaxies, 2D dust radiative transfer (RT) models, such as the one presented by Popescu et al. (2011), already contain the main ingredients needed to predict integrated galaxy SEDs, average profiles, dust emission and attenuation for the case of normal star-forming disc galaxies. However, there are a number of reasons why 3D dust RT codes are desirable. First, spiral galaxies, although well modeled with 2D codes, show the presence of multiple and irregular features such as spiral structures, bars, warps and local clumpiness of the ISM. Also, galaxies may host a central AGN whose polar axis may not be aligned with that of the galaxy. For mergers or post-merger galaxies there is clearly no fundamental symmetry of the distribution of stars and dust. Finally, solutions for the distribution of stars and ISM provided by numerical simulations of forming and evolving galaxies generally require processing with a 3D RT code in order to predict the appearance in different bands. \nThe main challenge in realizing 3D solutions of the dust RT problem is the computational expense. The stationary 3D dust RT equation is a non-local non-linear equation: non-local in space (photons propagate within the entire domain), direction (due to scattering, absorption/re-emission) and wavelength (absorption/re-emission). Even using a relatively coarse resolution in each of the six fundamental variables, namely the three spatial coordinates, the two angles specifying the radiation direction and the wavelength, solving the 3D dust RT problem require an impressive amount of both memory and computational speed, at the limits of the capabilities of current computers. \nPossibly the quickest way to calculate an image of a galaxy in direct and scattered light in a particular direction is by using Monte Carlo (MC) methods (including modern acceleration techniques, see Steinacker et al. 2013). There is a rich history of applications of MC codes to dust RT problems, starting with the pioneering works of e.g. Mattila (1970), Roark, Roark & Collins (1974), Witt & Stephens (1974) and Witt (1977). In the following decades the MC RT technique was further developed by many authors such as e.g. Witt, Thronson & Capuano (1992), Fischer, Henning & Yorke (1994), Bianchi, Ferrara & Giovanardi (1996), Witt & Gordon (1996) and Dullemond & Turolla (2000). Nowadays, this method can be considered as the mainstream approach to 3D dust RT calculations (see e.g. Gordon et al. 2001, Ercolano et al. 2005, Jonsson 2006, Bianchi 2008, Chakrabarti & Whitney 2009, Baes et al. 2011, Robitaille 2011, but also see table 1 of Steinacker et al. 2013 for a recent list of published 3D dust RT codes). The MC approach to dust RT consists of a simulation of the propagation of photons within a discretized spatial domain, based on a probabilistic determination of the location of emission of the photons, their initial propagation direction, the position where an interaction event (absorption or scattering) occurs and the new propagation direction after a scattering event. Thus, the MC technique mimics closely the actual processes occurring in nature which shape the appearance of galaxies in UV/optical light. However, since it is based on a probabilistic approach to determine the photon propagation directions, an RT MC calculation does not necessarily determine the radiation field energy density (RFED) accurately in the entire volume of the calculation. The reason is that regions which have a low probability of being illuminated are reached by only few photons unless the total number of photons in the RT run is substantially increased. Nonetheless, in the case of disc galaxies, accurate calculation of radiation field intensities throughout the entire volume is needed, in particular for the calculation of dust emission. Indeed, far-infrared/submm observations of spiral galaxies show that most of the dust emission luminosity is emitted longwards of 100*μ*m (see e.g. Sodroski et al. 1997, Odenwald et al. 1998, Popescu et al. 2002, Popescu & Tuffs 2002, Dale et al. 2007, 2012, Bendo et al. 2012) through grains situated in the diffuse ISM which are generally located at very considerable distances from the stars heating the dust.\nAnother method to solve the RT problem in galaxies, alternative to the mainstream MC approach, is by using a ray-tracing algorithm. This method consists in the calculation of the variation of the radiation specific intensity along a finite set of directions, usually referred to as “rays”. Ray-tracing algorithms can be specifically designed to calculate radiation field intensities throughout the entire volume considered in the RT calculation. Also, it should be pointed out that MC codes already make large use of ray-tracing operations (see Steinacker et al. 2013). It is thus interesting to pursue in the developing of pure ray-tracing 3D RT codes, which can be sufficiently efficient for the modelling of galaxies with 3D arbitrary geometries, if appropriate acceleration techniques are implemented. Similar to MC codes, ray-tracing dust RT codes have had a rich history in astrophysics (see e.g. Hummer & Rybicki 1971, Rowan-Robinson 1980, Efstathiou & Rowan-Robinson 1990, Siebenmorgen et al. 1992, Semionov & Vansevicius 2005, 2006). Application to analysis of galaxies started with the 2D code of Kylafis & Bahcall (1987). Although originally implemented only for the calculation of optical images (see also Xilouris et al. 1997, 1998,1999), this algorithm was later adapted by Popescu et al. (2000) for the calculation of radiation fields and was coupled with a dust emission model (including stochastic heating of grains) to predict the full mid-infrared (MIR)/FIR/submm SED of spiral galaxies (see also Misiriotis et al. 2001, Popescu et al. 2011). Thus far, extensions of the ray-tracing technique to 3D have been implemented but are specifically designed for solving the RT problem for star forming clouds (e.g. Steinacker et al. 2003, Kuiper et al. 2010), heated by few dominant discrete sources, rather than for very extended distributions of emission and dust as encountered in galaxies. \nIn this paper, we present DART-Ray[1](#fn1), a new ray-tracing algorithm which is optimized for the solution of the 3D dust RT problem for galaxies with arbitrary geometries and moderate optical depth at optical/UV wavelengths[2](#fn2). The main challenge faced by this model is the construction of an efficient algorithm for the placing of rays throughout the volume of the galaxy. In fact, a complete ray-tracing calculation between all the cells, used to discretise a model, is not a viable option, since it is by far too computationally expensive even for relatively coarse spatial resolution. Our algorithm circumvents the problem by performing an appropriate pre-calculation, whose goal is to provide a lower limit to the RFED distribution throughout the model. In this way, the ray angular density needed in the actual RT calculation can be dynamically adjusted such that the ray contributions to the local RFED are calculated only within the fraction of the volume where these contributions are not going to be negligible. \nFurthermore, the code we developed can be coupled with any dust emission model. Applications of the 3D code for calculation of infrared emission from stochastically heated dust grains of various sizes and composition, including heating of Polycyclic Aromatic Hydrocarbons molecules, utilizes the dust emission model from Popescu et al. (2011), and will be given in a future paper.\nThe paper is structured as follows. In §2 we provide some background information and motivation behind our particular ray-tracing solution strategy. In §3 we give a technical description of our code. In §4 we provide some notes on implementation and performance of the code. In §5 we compare solutions provided by our code with those calculated by the 1D code DUSTY and the 2D RT calculations performed by Popescu et al. (2011). In §6 we show the application of the code on a galaxy model including logarithmic spiral arms. A summary closes the paper. A list of definitions for the terms and expressions used throughout the paper can be found in Table [tabterms].\n|p4cm|p10cm| Term & Definition \n$A\\_{\\rm{EM}}$ & Projected area of an emitting cell \n$A\\_{\\rm{INT}}$ & Projected area of an intersected cell \n*f**L* & Input parameter needed to set the threshold value for Δ*L**λ* below which scattering iterations are stopped \n*f**U* & Input parameter needed to set the threshold value for *δ**U**λ* below which rays are stopped \n*g**λ* & Henyey-Greenstein scattering phase function parameter \n*I**λ* & Radiation specific intensity \n < *I**λ* > & Average of *I**λ* over the path crossed by a ray within an intersected cell \n*I**λ*, *i* & Radiation specific intensity of a ray before crossing a cell *i* \n$I\\_{\\lambda,\\rm{esc}}$ & Radiation specific intensity of a ray once it has reached the model border \n$I\\_{\\lambda,\\rm{ext}} $ & Amount of radiation specific intensity of a ray extincted within an intersected cell \n$\\delta I\\_{\\lambda,\\rm{sca}}(\\theta,\\phi)$ & Ray contribution to the specific intensity of the radiation scattered into direction (*θ*, *ϕ*), see Fig.[isca] \n*j**λ* & Volume emissivity, that is, the luminosity emitted at a certain position per unit volume, unit solid angle and unit wavelength interval (also *j**λ*, *c* when refereed to the average volume emissivity within a cell) \n*κ**λ* & Dust extinction coefficient per unit dust mass (also $\\kappa\\_{\\lambda,\\rm{abs}}$ or $\\kappa\\_{\\lambda,\\rm{sca}}$ when referred to the extinction coefficients due to dust absorption or scattering) \n*L**λ* & Total stellar luminosity density of the model \nΔ*L**λ* & Amount of luminosity still to be processed during scattering iterations \n$L\\_{\\lambda,\\rm{ray}}$ & Luminosity associated with a ray beam \n$N\\_{\\rm{rays}}$ & Input parameter specifying the minimum number of rays crossing a cell within the fully sampled region \nΦ*λ* & Scattering phase function \n*ρ* & Dust mass density (also *ρ**c* when referred to the average dust mass density within a cell) \nΔ*r* & Ray path within an intersected cell \nSCATT\\_EN(*θ*, *ϕ*) & Array storing the luminosity of the radiation scattered by dusty cells into a finite set of solid angles \n*τ**λ* & Optical depth \n*U**λ* & Radiation field energy density (also referred to as RFED) \n*δ**U**λ* & Contribution to the local RFED carried by a ray (also $U\\_{\\lambda,\\rm{INT}}$ when referred to the contribution to an intersected cell RFED) \n$U\\_{\\lambda,\\rm{LL}}$ & Lower limit to the RFED \n$U\\_{\\rm{TEMP}}$ & Temporary array used to store RFED contributions from an emitting cell throughout the model \n$U\\_{\\lambda, \\rm{FINAL}}$ & Array storing the RFED distribution which is output by the code \n$V\\_{\\rm{INT}}$ & Intersected cell volume \n*ω**λ* & Albedo \n$\\Omega\\_{\\rm{HP,EM}}$ & Solid angle associated with the HEALPix spherical pixels used to define the directions of rays from an emitting cell \n$\\Omega\\_{\\rm{HP,INT}}$ & Solid angle associated with the HEALPix spherical pixels used to define the directions of the radiation scattered by an intersected cell \n$\\Omega\\_{\\rm{HP,MS}}$ & Solid angle associated with an HEALPix main sector (see Fig.[hpixsphere]) \n$\\Omega\\_{\\rm{INT}}$ & Solid angle subtended by projected area of the intersected cell $A\\_{\\rm{INT}}$ (see Fig.[isca] \n“Dusty cell” & A cell where the average value of dust density is higher than zero \n“Dust self-heating” & The dust absorption of radiation emitted by dust at other positions (not included in this version of the code) \n“Emitting cell” & A cell where the average value of the stellar light or scattered light volume emissivity is higher than zero \n“Escaping radiation” & The direct or scattered stellar radiation propagating outside the borders of the volume considered in the RT calculation \n“Full sampling” (of a region) & The process of launching enough rays from a source, such that all the cells within a region are intersected by multiple rays \n“Intersected cell” & A cell intersected by a ray \n“Leaf cell” & A cell of the 3D adaptive grid which is not further subdivided \n“Lost luminosity” & Amount of stellar luminosity not considered in the RT calculation (to be kept low to guarantee approximate energy balance, see §4) \nMC & Monte Carlo \nRFED & Radiation Field Energy Density \nRT & Radiative Transfer \n“Source influence volume” & The fraction of model volume within which a source contributes significantly to the RFED \n“Volume emissivity” & see *j**λ* \n[tabterms]\nBackground and Motivations\n==========================\nIn this section we describe the basic characteristics of the time-independent 3D dust RT equation, briefly introduce the ray-tracing approach used in our code and provide the main motivations behind our solution strategy. Finally, we describe the main steps of the new algorithm we present in this work. A much more technical description of our code can be found in §3.\nDust continuum 3D Radiative Transfer: Ray-tracing and Solution Strategy\n-----------------------------------------------------------------------\nGiven an input distribution of stellar luminosity and dust mass, solving the RT problem requires in principle the resolution of the following equation for the specific intensity *I*(*λ*, **x,n**), which represents the luminosity per unit area, solid angle and wavelength interval propagating at point **x** into the direction **n**:\n \n$$\\mathbf{n}\\nabla\\_{\\mathbf x} I\\_\\lambda(\\mathbf{x,n}) = -k\\_\\lambda(\\mathbf{x})\\rho(\\mathbf{x})\\left[I\\_\\lambda(\\mathbf{x,n}) \\\\\n - \\omega\\_\\lambda\\int\\_\\Omega{\\Phi\\_\\lambda(\\mathbf{n,n'}) I\\_\\lambda(\\mathbf{x,n'})d\\Omega'}\\right]+j\\_\\lambda(\\mathbf{x}) \n\\label{rf\\_equ}$$ \nwhere *k**λ*(**x**) is the total extinction coefficient per unit mass of dust (including both absorption and scattering), *ρ*(**x**) is the dust mass density, *ω**λ* is the albedo, defined such that *ω**λ* × *k**λ*(**x**) gives the fraction of extinction due to scattering, Φ*λ*(**n,n'**) is the scattering phase function, which gives the probability for radiation coming from direction **n'** to be scattered into direction **n**, and *j**λ*(**x**) is the distribution of stellar volume emissivity[3](#fn3). The first term on the right-hand side of Eq.[rfequ] acts to reduce the radiation specific intensity *I*(*λ*, **x,n**) by a quantity that is proportional to the radiation intensity itself. The second term instead acts as a source term and gives a positive contribution to the radiation intensity by adding the light coming from all directions to point **x** and then scattered into the direction **n**. The third term gives a positive contribution to the propagating radiation specific intensity, which is due to stellar emission and it is assumed to be isotropic at each position **x**. Since in 3D RT there are six independent variables, namely wavelength, three spatial coordinates and two angular directions, the solution vector for *I**λ*(**x,n**) can be extremely large, making the problem very challenging also from the point of view of memory requirements, apart from the difficulty of solving the integro-differential equation itself in three dimensions. Note also that we did not include a term that represents the re-emission by dust, important at infrared wavelengths. The resolution algorithm presented in this work is designed only to handle the propagation of direct and scattered light from stellar populations and does not consider the self-heating of dust. \nInstead of seeking to obtain a solution for *I**λ*(**x,n**), the main aim of this work is to construct an algorithm optimized to derive the radiation field energy density *U**λ* (hereafter also refereed to as RFED) at each position **x**, in order to be able to calculate successively the dust emission spectra assuming local energy balance between dust radiative heating and emission[4](#fn4)\nIn terms of *I**λ*(**x,n**), one can express *U**λ* as:\n \n$$U\\_\\lambda(\\mathbf{x})=\\frac{\\int{I\\_\\lambda(\\mathbf{x,n})d\\Omega}}{c}\n\\label{en\\_equ}$$ \nIn order to calculate *U**λ*(**x**) at a specific point **x**, one can simply sum up the contributions *δ**U**λ* provided by the radiation coming from all the emitting sources to the value of *U**λ*(**x**) at that position. A numerical method to calculate *U**λ*(**x**) at any position can be implemented by considering “rays” originating from each emitting source and propagating throughout the whole volume considered in the calculation. Along each ray path one follows the variation of the radiation intensity and one can thus calculate the *δ**U**λ* contributions at a finite set of positions (this solution technique is among those known as “ray-tracing” methods). More specifically, a solution algorithm one could use to derive the distribution of *U**λ*(**x**) for a single wavelength *λ*, given an input 3D distribution of stellar luminosity and dust mass, is the following.\nFirst, the entire model is subdivided in an adaptive grid of cubic cells and to each cell one assigns the average values for both the dust density and stellar volume emissivity within the cell volume. A cell for which the average value of stellar volume emissivity is higher than zero can be treated approximately as a discrete radiation source. In the following, we will refer to this kind of cell as an “emitting cell”. Similarly, cells with average dust density higher than zero will be referred to as “dusty cells”.\nThen, one performs ray-tracing for a large set of directions originating from the centres of the emitting cells. That is, one follows the variation of the specific intensity *I**λ*(**x,n**) from the cell centres along rays corresponding to each direction **n**. While following a ray, one considers the increase of radiation intensity *I**λ*(**x,n**) due to the stellar volume emissivity in the cell originating the ray but not in the intersected cells, where only the decrease of intensity due to dust absorption and scattering is considered. This allows us to calculate separately the contributions *δ**U**λ* provided only by the emitting cell originating the ray to the final value of *U**λ*(**x**) in all the cells intersected by the same ray[5](#fn5). When a ray intersects a cell *i* different from the original cell, the new value of the specific intensity will then be:\n \n*I**λ*, *i* + 1 = *I**λ*, *i**e*− *k**ρ**c*, *i*Δ*r* \nwhere *ρ**c* is the cell dust density and Δ*r* is the *crossing path*. For each ray-cell intersection one calculates an appropriate average of the value of *I**λ* within the ray crossing path and, thus, the contribution *δ**U**λ* to the local value of *U**λ*(**x**) by using a discrete version of Eq. [enequ]. When all the rays from one emitting cell have been processed, the ray-tracing is performed from another emitting cell and so on until all the emitting cells have been considered. Scattered radiation, whose intensity in a finite number of directions has also been stored locally for each ray-cell intersection, is then processed in a similar fashion.\nIf one creates a grid sufficiently fine in resolution and launches a sufficiently large number of rays, the calculated cell RFED values are very close to the exact values of *U**λ*(**x**) at the cell centres and can be used to calculate the dust heating. Furthermore, the described procedure is extremely flexible and capable to handle completely arbitrary 3D distributions of stellar luminosity and dust mass. Unfortunately, the implementation of the procedure in the simple form described above is too computationally expensive (scaling approximately as *N*5/3, with *N* being the total number of cells, in the case of uniform spatial sampling), considering also that the calculation has to be performed in an iterative way for the scattered light and for different wavelengths. Nonetheless, for well mixed emitters–absorbers geometries, such as in the case of galaxy dust–stellar distributions, the full ray-tracing calculation for rays propagating throughout the entire model from each emitting cell does not have to be necessarily performed to obtain a reasonably accurate solution for *U**λ*(**x**).\nIn fact, each radiation source in a general RT problem not necessarily contributes significantly to *U**λ* at each position within the volume considered for the RT calculation but often only within a fraction of it, which we call the “source influence volume”. In the ray-tracing method described above, if one knew in advance the extent of the influence volume for each emitting cell, it would be possible to reduce the number of calculations by simply performing ray-tracing only within this volume. Unfortunately, it is not possible to know this information a priori, since the final values of *U**λ* at each position are available only *after* performing the RT calculation. However, one can estimate in a conservative way the extent of the influence volume for a given emitting cell. The basic idea is to identify a point at some distance from the emitting cell where it can be proved that the contribution *δ**U**λ* carried by a ray is negligible compared to the local final value of *U**λ*. If this is the case, this can imply that the emitting cell will not contribute significantly to *U**λ* beyond that position, which is already outside the influence volume of the emitting cell [see Fig.[encrit] for simple examples where this criteria will work (left-hand panel) or can fail (middle and right-hand panels) if not properly applied].\nIn practice, one can implement this method in the following way. First, one calculates a lower limit $U\\_{\\lambda,\\rm{LL}}$ for *U**λ* within the entire volume by performing ray-tracing from each emitting cell only until a certain arbitrary distance (see Fig.[ferayprop]). Once the lower limit for *U**λ* is calculated, one can start the RT calculation again from the beginning but this time is able to check if a contribution *δ**U**λ*, carried by a ray to a particular cell, is going to be significant or not. In fact, if for a certain intersected cell *δ**U**λ* is negligible compared to the local value of the lower limit $U\\_{\\lambda,\\rm{LL}}$, then it will be negligible also compared to the final value of *U**λ* at that position. The actual implementation of this method consists in checking at each cell intersection if *δ**U**λ* < *f**U* × *U**λ*, *L**L* with *f**U* equal to a very small number to be chosen appropriately (see §4 for discussion on this point). When this condition is realized and if the chosen value of *f**U* is sufficiently small, then the contributions *δ**U**λ* will also be negligible for all the cells beyond that position in the same ray direction.\nThis implies that, once *δ**U**λ* < *f**U* × *U**λ*, *L**L* at a certain distance from an emitting cell along a ray-path, one can stop the ray-tracing calculation for that particular ray at that position. In fact, in that case, the intersected cell is already outside the influence volume of the emitting cell originating the ray (see Fig.[figstep1]). In this way, the total amount of calculations to be performed can be substantially reduced, making it feasible to use a modified version of the above ray-tracing algorithm to infer the *U**λ* distribution within complex dust/stars structures as those observed in galaxies. In the following subsection, we provide a simplified description of the RT algorithm we have developed based on this approach. \n [encrit]\n [ferayprop]\n [figstep1]\nBasic Description of the DART-Ray Algorithm\n-------------------------------------------\nIn the following we provide a simple description of the three steps performed by the algorithm which implements the solution strategy outlined above to calculate the RFED distribution *U**λ* at a single wavelength *λ*. This description assumes that a grid of cells subdividing the entire model has already been created. A complete technical description of the code will be given in Section §3. \n*First Step: Calculation of a Lower Limit for *U**λ** \nThe first step consists of ray-tracing from each emitting cell adopting a ray angular density such that *all* the cells within a certain radius or a certain optical depth from each emitting cell are intersected by multiple rays, as shown in Fig.[ferayprop] (hereafter, we will refer to the process of intersecting all cells in a certain region with multiple rays as “fully sampling” that region). The specific values for the limit radius and optical depth can be specified in the input. They should be large enough to let the rays cover a relatively large fraction of the entire model but small enough to avoid a too long computational time in this step. The inferred contributions *δ**U**λ* to the value of *U**λ* in each cell crossed by rays are accumulated into an array $U\\_{\\lambda,\\rm{LL}}$ (LL stays as “lower limit”). As said before, the inferred RFED distribution $U\\_{\\lambda,\\rm{LL}}$ represents only a lower limit for the final value *U**λ* since there could be other emitting cells, at distances larger than those adopted limits, whose contribution to *U**λ* at each particular point has not been considered yet. In addition, scattered light contributions have also been neglected at this point. \n*Second Step: Processing of Source Direct Light* \nIn the second step the ray-tracing procedure is repeated again from the beginning but this time the rays fully sample the regions around each emitting cell, until the ray contribution *δ**U**λ* for an intersected cell become smaller than a very small fraction of the lower limit $U\\_{\\lambda,\\rm{LL}}$, that is until *δ**U**λ* < *f**U* × *U**λ*, *L**L* with *f**U* being a very small number (see §4 for more details about how to choose an appropriate value for *f**U*). When this condition is realized, it means that the position reached by the ray is outside the emitting cell influence volume (see Fig.[figstep1]) and the final value of *U**λ* in the crossed cell is contributed mainly by emitting cells different from the emitting cell originating the ray[6](#fn6). Therefore, to the purpose of calculating the RFED distribution *U**λ*, there is no reason to proceed with a full sampling of the cells beyond the limit determined in this way. Apart from storing the values of the RFED contributions *δ**U**λ*, after each ray crosses a cell, the scattered energy information are also stored for that cell. That is, the luminosity scattered within a discrete set of solid angles is stored for each cell containing dust. These values will be used in the next step. Before going to the third step, the value of $U\\_{\\lambda,\\rm{LL}}$ is updated with the current estimation for *U**λ*. \n*Third Step: Scattering Iterations* \nAfter the direct light has been processed in the second step, a third step is started where there is a series of iterations to process the scattered radiation. Each cell which originates scattered light is treated as an emitting cell exactly in the same way as in step 2. Ray-tracing is performed from each dusty cell by fully sampling all the volume surrounding the cell until the contribution *δ**U**λ* is negligible compared to a small fraction of $U\\_{\\lambda,\\rm{LL}}$, the lower limit for *U**λ* updated with the new value of *U**λ* found in step 2. Again, scattering information are stored as well after each ray crossing and this higher order scattered radiation intensity is processed in successive iterations. These scattering iterations continue until the vast majority of the luminosity of the system has been either absorbed by dust or has escaped outside the model. That is, until Δ*L**λ* < *f**L**L**λ*, where *L**λ* is the total stellar luminosity emitted within the model, Δ*L**λ* is the amount of luminosity still to be processed (that is, not absorbed or escaped yet) and *f**L* is a parameter to be set in the input. \nThe disadvantage of this procedure is that many of the calculations performed in the first step are repeated once again in the second step. The advantage is that the number of calculations avoided can be very high, thus reducing significantly the total calculation time for those geometries where the influence volumes of the emitting cells are only a small fraction of the total volume. Further characteristics of the code, not mentioned in the simplified algorithm above, include the adaptive and directional-dependent angular density of the rays (see §3.2) and the methods implemented to calculate the specific intensity of the radiation escaping outside the model in a finite set of directions (see Section §3.4).\nThe Ray-Tracing 3D Code: Technical Description\n==============================================\nIn this section we provide an extensive description of the code we have developed. The code consists of two main programs performing the adaptive grid creation and the RT calculation respectively. In the following subsections we describe the adaptive grid creation (§3.1), the basic ray-tracing routine (§3.2) and each of the three steps of the RT algorithm in detail (§3.3). Finally, we describe the methods implemented to derive the escaping radiation specific intensity (§3.4).\nAdaptive Grid Creation\n----------------------\nGiven an input spatial distribution of dust mass and stellar luminosity (either defined by analytical formulae or in a tabulated form), an adaptive grid is created in a way such that the spatial resolution is higher in regions where the radiation field intensity is expected to vary in a more rapid way, such as those where the density of dust is higher. A parent cell of cubic shape, enclosing the entire model, is subdivided in 3x3x3=27 child cubic cells of equal size[7](#fn7). Then, the average dust density and stellar volume emissivity are calculated within the volume of each newly created cell, together with an estimation of their variation within each cell. Further cell subdivision proceeds for those child cells which do not satisfy user-defined criteria and those cells become parents of even smaller child cells. After that, the estimation of the cell dust density/stellar volume emissivity and the cell subdivision procedure are performed for the new cells and so on. In this way, a tree of cells is constructed iteratively until the chosen criteria are satisfied for all the smallest cells within each original parent cell. Also, in order to obtain a smooth variation of the grid resolution, further cell subdivision is performed to avoid differences in cell subdivision level higher than one between neighbour cells. The cells which have not been further subdivided, after the entire grid creation has been completed, are called the leaf cells.\nThe criteria for cell subdivision should be chosen in order to obtain both numerical accuracy and an adequate coverage of the RFED distribution within the model. In order to achieve a good numerical accuracy, it is important to have leaf cells with small total optical depths and with small gradients of dust density and stellar volume emissivity within the cells. For example, typical conditions for a cell to be a leaf cell could be:\n \n*τ**λ* < < 1 \n \n$$\\frac{\\Delta\\rho}{\\rho\\_c} < 0.5 \n \\label{delta\\_rho}$$ \nwhere *τ**λ* = *k**λ**ρ**c**l**c* is the total cell optical depth, Δ*ρ* is an estimate of the dust density variation within a cell and *ρ**c* is the average cell dust density. An equivalent condition, as the one expressed by Eq. [deltarho], has to be fulfilled by the cell stellar volume emissivity *j**λ*, *c*. These conditions allow us to use the simple expression given by Eq. [inewequ] to calculate the variation of the specific intensity within a cell to a good degree of accuracy. However, note that an additional constraint is the maximum allowed number of subdivision levels NLVL\\_MAX, which has to be chosen in the input as well. The value of this parameter should be such to guarantee that the input cell parameter requirements, such as those above, are valid for all or at least the vast majority of cells in the model and, at the same time, avoid to create models with too many cells. It is desirable to keep the number of cells as low as possible, since it is one of the main parameters affecting the total computational time. After creating the grid, the program creates a table where the user can read the maximum and average cell parameters and, thus, quickly verify to which degree the input cell requirements have been fulfilled.\nAll the information which define the grid is printed on a file that can be read from the RT program. This includes: \n– cell id number \n– position of cell centre in Cartesian coordinates \n– cell child number : which is equal to “-1” if the cell is a leaf cell or to the id number of the first child cell created during the cell subdivision \n– cell index number : binary code expressing the position of the cell within the cell tree \n– cell size *l**c* \n– cell dust density *ρ**c* \n– cell stellar volume emissivity *j**λ*, *c* \nThe basic Ray-Tracing Routine\n-----------------------------\nIn this subsection we describe the basic ray-tracing calculation for rays departing from an emitting cell and propagating throughout the model. This is the core routine used by the RT algorithm described in the next subsection. Given an emitting cell, rays are casted into multiple directions defined by using the HEALPix sphere pixelation scheme (Gorski et al. 2005, see also e.g. Abel & Wandelt 2002 and Bisbas et al. 2012 for other applications in RT codes). In this scheme a sphere is divided in 12 main sectors of equal size, which can be further subdivided in smaller spherical pixels (see Fig.[hpixsphere]). By assuming that an HEALPix sphere is centred on the emitting cell centre, the lines connecting the cell centre to the centre of the HEALPix spherical pixels define directions along which one can follow the variation of the specific intensity within the model. The use of the HEALPix routines PIX2ANG\\_NEST (translating pixel index numbers into spherical coordinates) and its inverse ANG2PIX\\_NEST is a convenient way to handle sets of rays with an adaptive angular density (see below).\n [hpixsphere]\nFor each ray, the following approximations are implemented: \n1) the calculation is performed as if all the cell luminosity propagates through solid angles defined by the pixels of an HEALPix sphere centred on the cell centre; \n2) since one can only follow the exact variation of *I**λ* along the finite set of HEALPix directions, it is assumed that the evolution of *I**λ* along all the directions included within the total solid angle associated with a ray (determined by the adopted HEALPix scheme angular resolution) is exactly the same as for the main central ray direction (see Fig.[beamprop]). \nThe above approximations imply that the total luminosity density associated with a ray and flowing through an HEALPix solid angle $\\Omega\\_{\\rm{HP,EM}}$ at any distance *r* from the centre of the emitting cell is given by:\n \n$$L\\_{\\lambda,\\rm{ray}}(r)=I\\_\\lambda(r)\\Omega\\_{\\rm{HP,EM}}A\\_{EM}\n \\label{lum\\_ray\\_beam}$$ \nwhere $A\\_{\\rm{EM}}$ is the projected area of the emitting cell (assumed to be equal to the emitting cell size squared). \nThe variation of *I**λ*, when the associated ray crosses a cell, is calculated using the following expressions. The variation due to the crossed optical depth *τ**λ* = *k**λ**ρ**c*Δ*r* and cell volume emissivity *j**λ*, *c* within the cell originating the ray is equal to:\n \n$$I\\_{\\lambda,0}=\\frac{j\\_{\\lambda,c}\\Delta r}{\\tau\\_\\lambda}(1-e^{-\\tau\\_\\lambda})$$ \nif *τ**λ* > 0 (cell containing dust)[8](#fn8). If *τ**λ* = 0, we assumed:\n \n*I**λ*, 0 = *j**λ*, *c*Δ*r* \nwhich is consistent with the previous expression when *τ**λ* → 0. During the scattering iterations, the scattered light is considered for the calculation of the cell volume emissivity, as it will be shown later. Instead, the new value of *I**λ* after the crossing of a ray through a cell different from the original one is simply given by:\n \n*I**λ*, *i* + 1 = *I**λ*, *i**e*− *τ**λ* \nApart from calculating a new value for *I**λ*, when a ray crosses a cell, the code determines the contribution of the ray to the intersected cell RFED and the amount of energy scattered by the dust in the intersected cell. However, these contributions can be accurately calculated only if a sufficiently high ray angular density is used. In fact, depending on the cell sizes and the adopted HEALPix angular resolution, at any distance from the emitting cell, the ray beam as a whole (not just the main ray direction) can intersect either a single cell or a group of cells at the same time. At distances large enough from the emitting cell, the beam, originally intersecting only one cell at a time, will begin intersecting more cells simultaneously (see Fig.[beamprop]). As said before, the code traces the evolution of *I**λ* only along the main direction of the ray beam and it is assumed that for all the directions within the beam the *I**λ* variation is exactly the same. However, far away from the emitting cell, the beam propagates through larger physical volumes and the previous approximation can become very inaccurate. Actually, in order to obtain a precise calculation of the RFED contributed by an emitting cell $C\\_{\\rm{EM}}$ to a certain cell $C\\_{\\rm{INT}}$, it is desirable that several rays originating from $C\\_{\\rm{EM}}$ are intersecting $C\\_{\\rm{INT}}$. In this way, the calculation of RFED is more accurate because of the better estimation of the average *I**λ* within the intersected cell.\n [beamprop]\nIf one defines $A\\_{\\rm{INT}}$, the projected area of the intersected cell (approximated by the cell size squared), and $\\Omega\\_{\\rm{INT}}=\\frac{A\\_{\\rm{INT}}}{r^2}$, the solid angle subtended by the projected area of the intersected cell and with origin in the centre of the emitting cell (see Fig.[isca]), in order to have more rays from $C\\_{\\rm{EM}}$ crossing $C\\_{\\rm{INT}}$ one requires that:\n \n$$\\Omega\\_{\\rm{HP,EM}} < \\frac{\\Omega\\_{\\rm{INT}}}{N\\_{\\rm{rays}}}\n\\label{beam\\_cond}$$ \nwhere $N\\_{\\rm{rays}}$ is an input parameter equal to the minimum number of rays which should cross $C\\_{\\rm{INT}}$. If this condition is fulfilled by all the intersected cells within a certain region around an emitting cell, we say that the rays are “fully sampling” that region, using the same terminology already introduced in §2.2.\nOnce the above condition is fulfilled for an intersected cell, the code defines $\\Omega=\\Omega\\_{\\rm{HP,EM}}$ and the contribution of a ray to the intersected cell RFED $\\delta U\\_{\\lambda,\\rm{INT}}$ is given by\n \n$$\\delta U\\_{\\lambda,\\rm{INT}}=\\frac{<I\\_{\\lambda}>A\\_{\\rm{EM}}\\Omega\\frac{\\Delta r}{c}}{V\\_{\\rm{INT}}}\n\\label{en\\_dens\\_contr}$$ \nwhere *c* is the speed of light, $\\frac{\\Delta r}{c}$ is equal to the time needed by the light to cross the intersected cell, $V\\_{\\rm{INT}}$ is the volume of the intersected cell and the average value of *I**λ* along the crossing path is equal to:\n \n$$<I\\_{\\lambda}> = \\frac{I\\_{\\lambda,i}(1-e^{-\\tau\\_\\lambda})}{\\tau\\_\\lambda}$$ \nif *τ**λ* > 0 and\n \n < *I**λ* > = *I**λ*, *i* \nif *τ**λ* = 0.\nThe ray contribution to the specific intensity scattered by the intersected cell $\\delta I\\_{\\lambda,\\rm{sca}}(\\theta,\\phi)$ is (see Fig.[isca]):\n \n$$\\delta I\\_{\\lambda,\\rm{sca}}(\\theta,\\phi)=\n\\frac{I\\_{\\lambda,\\rm{ext}}w\\_{\\rm{\\lambda,\\rm{sca}}}A\\_{\\rm{EM}}\\Omega \\Phi(\\theta,\\phi)}{A\\_{\\rm{INT}}\\Omega\\_{\\rm{HP,INT}}}\n\\label{scatt\\_contr}$$ \nwhere $\\omega\\_{\\rm{\\lambda}}=\\frac{k\\_{\\lambda,\\rm{sca}}}{k\\_{\\lambda,\\rm{ext}}}$, $\\Omega\\_{\\rm{HP,INT}}$ is the solid angle determined by the HEALPix angular resolution adopted to store the scattered radiation intensity in the intersected cell, Φ*H**G*(*θ*, *ϕ*) is a term representing the integration of the Henyey-Greenstein scattering phase function over the solid angle $\\Omega\\_{\\rm{HP,INT}}$:\n \n$$\\Phi\\_{HG}(\\theta,\\phi)=\\frac{\\Omega\\_{\\rm{HP,INT}}}{4\\pi}\\frac{1-g\\_\\lambda^2}{[1+g\\_\\lambda^2-2g\\_\\lambda\\cos (\\theta)]^{3/2}]}\n\\label{HG\\_phase}$$ \nand $I\\_{\\lambda,\\rm{ext}}$ is the amount of ray specific intensity absorbed or scattered by the cell, which is given by:\n \n*I**λ*, *e**x**t* = *I**λ*, *i*(1 − *e*− *τ**λ*) \nThe scattered luminosity values (numerator of Eq. [scattcontr]) are accumulated for each cell on an array SCATT\\_EN(*θ*, *ϕ*) for a certain set of HEALPix directions, whose angular density can be specified in the input according to memory availability [9](#fn9). Of course, a higher numerical accuracy will be achieved if the scattered intensity values are stored with a higher angular resolution.\nIn case the condition expressed by Eq. [beamcond] is not fulfilled, all the equations above are still used but with $\\Omega=\\Omega\\_{\\rm{HP,EM}}$ if $\\Omega\\_{\\rm{HP,EM}} < \\Omega\\_{\\rm{INT}}$ and $\\Omega=\\Omega\\_{\\rm{INT}}$ if $\\Omega\\_{\\rm{HP,EM}} > \\Omega\\_{\\rm{INT}}$. That is, we consider only the beam luminosity passing through the intersected cell. As it will be explained later, condition [beamcond] is not required to be fulfilled beyond the regions where full sampling is desired. That means, beyond a certain distance from an emitting cell there could be cells which are not intersected by any ray and therefore none of the above quantities is calculated by the code for those cells.\nThe formulae given above are different when the crossed cell coincides with the emitting cell originating the rays. In this case one has also to take into account the internal volume emissivity of the cell. Therefore, the average value of *I**λ* needed to calculate the ray contribution to the local RFED by using Eq. [endenscontr] is given by:\n \n$$<I\\_{\\lambda}>=\\frac{j\\_{\\lambda,c}\\Delta r}{\\tau\\_\\lambda^2}(e^{-\\tau\\_\\lambda}-\\tau\\_\\lambda-1)$$ \nif *τ**λ* > 0 and\n \n$$<I\\_{\\lambda}>=\\frac{j\\_{\\lambda,c}\\Delta r}{2}$$ \nif *τ**λ* = 0. Instead the value of $I\\_{\\lambda,\\rm{ext}}$ in Eq. [scattcontr] is given by:\n \n$$I\\_{\\lambda,\\rm{ext}}=\\frac{j\\_{\\lambda,c}\\Delta r\\left(e^{-\\tau\\_\\lambda}+\\tau\\_\\lambda-1\\right)}{\\tau\\_\\lambda}$$ \nand all the quantities with subscript $\\rm{INT}$ in Eq. [endenscontr] and [scattcontr] refer to the emitting cell in this case. During scattering iterations, the term *j**λ*, *c*Δ*r* in the previous equations is substituted by $I\\_{\\lambda,\\rm{sca}}(\\theta, \\phi)$, the specific intensity of the total scattered radiation accumulated locally on each cell (see Eq.[scattcontr]).\n [isca]\nThe radiative transfer algorithm\n--------------------------------\n### Step 1: Calculation of a Lower Limit for the RFED\nIn the first step of the RT algorithm, ray-tracing is performed from each emitting cell by fully sampling all the cells until the rays cross a limit optical depth or distance specified in the input (see Fig.[ferayprop]). In this way the contributions from each emitting cell to the RFED in the areas around those cells are summed up to obtain a lower limit $U\\_{\\lambda,\\rm{LL}}$ for the RFED distribution in the entire model. Full sampling of a region of cells centred on an emitting cell requires a sufficient amount of rays to be launched from the emitting cell. However, the ray angular density necessary to achieve full sampling can vary depending on the angular direction. In order to optimize the number of rays, our code has the additional peculiarity that the angular distribution of rays originating from emitting cells is not homogeneous. Specifically, the ray angular density is optimized within an initial HEALPix sector by the following iterative procedure (see flow diagram in Fig.[flowdiagfe]).\nA first ray-tracing attempt is performed with an initial HEALPix angular resolution, e.g. the calculation is performed only for the direction passing by the centre of an HEALpix main sector[10](#fn10). The beam defined by the chosen HEALPix pixel will intersect the cells progressively more distant from the emitting cell. As said in the previous subsection, in order to have an accurate estimation of the radiation field contribution *δ**U**λ* carried by the beam to an intersected cell, it is necessary that the entire beam is passing through the cell, together with at least several other adjacent beams. Therefore, when an intersected cell is found, the code checks if the condition expressed by Eq.[beamcond] is fulfilled. When this happens, the contribution to the RFED is stored in a temporary array $U\\_{\\rm{TEMP}}$ and the ray-tracing continues to the next intersected cell in the same way. When the above condition is not realized, the $U\\_{\\rm{TEMP}}$ array is initialized and the radial ray-tracing re-starts from the beginning but with a higher HEALpix angular resolution. That is, more rays are launched within the initial HEALpix sector. In this way, smaller beams are generated until condition ([beamcond]) is always fulfilled for all the intersected cells within the input-defined distance or optical depth crossed by each ray. When this is realized, the ray-tracing can be performed without interruptions for all the rays within the chosen initial HEALPix sector. After the last ray within the initial HEALPix sector has been processed, the values of RFED stored in $U\\_{\\rm{TEMP}}$ are added to the array *U**L**L* and then $U\\_{\\rm{TEMP}}$ is initialized. This iterative procedure is performed for all the initial HEALpix sectors, covering the entire sphere, and for each emitting cell.\n### Step 2: Processing Direct Radiation\nIn the second step, the ray-tracing is started from the beginning again following the same iterative procedure explained in the previous subsection but with some differences (see flow diagram in Fig.[flowdiagrt]). As before, the angular density of the rays is increased until the intersected cells, where the contribution to the local RFED needs to be accurately calculated, are crossed by multiple rays. However, during this step the increasing of ray angular density is stopped if the contribution by a ray to an intersected cell RFED is a very small fraction of the value stored at that position in $U\\_{\\lambda,\\rm{LL}}$. That is, when:\n \n*δ**U**λ* < *f**U* × *U**λ*, *L**L* \nwith *f**U* defined in the input (see §4). When this condition is fulfilled, it means that the emitting cell, originating the ray, is not contributing substantially to the RFED of the intersected cell under consideration and to all the other cells beyond that in the direction of the ray. In addition, at variance with step one, ray-tracing is not necessarily limited in the region where full sampling is required, but optionally rays can keep propagating until the model border (this option will be referred to as “ray mode 2”). Similarly as before, the values of the RFED contributions are first stored in $U\\_{\\rm{TEMP}}$ and then added to the array $U\\_{\\lambda,\\rm{FINAL}}$, after the ray-tracing from all the directions within the initial HEALPix sector is completed. After this procedure has been performed for all the emitting cells, $U\\_{\\lambda,\\rm{FINAL}}$ contains all the contributions from direct radiation but still lacks the contribution from scattered light, which will be calculated in the next step. However, during this step, the scattered radiation energy information is stored after each cell crossing in the array SCATT\\_EN(*θ*, *ϕ*). As explained in section §3.2, this array contains the scattered radiation luminosity in a finite number of HEALPix beams, whose angular density is defined by the user according to memory availability. When a ray crosses a cell, the fraction of scattered luminosity is angularly distributed according to the Henyey-Greenstein phase function (see Eq.[HGphase]).\n### Step 3: Processing Scattered Radiation\nThe last step consists of the processing of the scattered radiation. Since the scattered radiation itself can be scattered multiple times, this step requires several iterations. However, the procedure for the ray-tracing is completely the same as the one in the previous step once the scattered luminosity values, stored in the SCATT\\_EN array, are considered to calculate the volume emissivity from each cell. The only difference is that, while in the previous steps the emission from each cell was isotropic, in this step the ray luminosities can depend on the angular direction. The SCATT\\_EN values from each cell are on turn processed and initialized. Using the same optimisation procedure described above, ray-tracing from each cell is performed to calculate the contribution to $U\\_{\\lambda,\\rm{FINAL}}$ and to the SCATT\\_EN arrays of the intersected cells. After all the emitting cells have been processed once, a check on the remaining luminosity Δ*L**λ* stored in the SCATT\\_EN array is performed. If this luminosity is higher than a very small fraction *f**L* of the total luminosity *L**λ* emitted by the entire model, that is, when Δ*L**λ* > *f**L**L**λ*, a new scattering iteration is started. If not, scattering iterations are stopped and the output of the entire calculation is printed on a file. The output includes both the RFED distribution and the escaping radiation specific intensity in several directions, calculated as described in the next section.\nCalculations of the Escaping Radiation Specific Intensity\n---------------------------------------------------------\nAlthough the algorithm we developed is optimized for the calculation of RFED distributions, our code can be used to derive the specific intensity of the radiation emitted or scattered by each cell and escaping outside the model volume. The code can calculate either averages for the radiation propagating within large solid angles or the specific intensity for the radiation propagating into single directions defined in the input. As it has been shown before, our code optimizes the angular density of the rays departing by each emitting cell in order to obtain a full sampling of cells where the ray RFED contribution is important. Beyond the region fully sampled, if specified in the input (so-called “ray mode 2”), rays can keep propagating throughout the model until the model border (although they can miss a progressively higher fraction of cells). When a ray arrives to the model border, it is possible to calculate the specific intensity of the escaping radiation, generated by the emitting cell in the ray direction:\n \n$$I\\_{\\lambda,\\rm{esc}}=I\\_{\\lambda,o} e^{-\\tau\\_\\lambda}\n \\label{i\\_esc\\_formula}$$ \nwhere *τ**λ* is the total optical depth crossed by the ray from the emitting cell to the model border. The calculation of $I\\_{\\lambda,\\rm{esc}}$ can be done for all the rays belonging to the sets of rays within an HEALPix main sector, within which the ray angular density has been optimized as described in §3.3. It is then straightforward to calculate the following average:\n \n$$<I\\_{\\lambda,\\rm{esc}}>=\\frac{\\sum I\\_{\\lambda,esc,i}\\Omega\\_{HP,EM,i}}{\\Omega\\_{\\rm{HP,MS}}}$$ \nwhere $\\Omega\\_{\\rm{HP,MS}}=4\\pi/12$ is the solid angle of an HEALPix main sector and the sum is performed for all rays passing within that solid angle. The average value $<I\\_{\\lambda,\\rm{esc}}>$ derived in this way can be used to measure the escaping luminosity within a HEALPix main sector using Eq. [lumraybeam] with *I**λ*(*r*) = < *I**λ*, *e**s**c* > . Note that this estimate assumes that the missed cells in the volumes beyond the fully sampled region have attenuation properties similar to those of the intersected cells at the same distance from the emitting cell.\nThe estimate of the escaping luminosity within an entire beam is useful to calculate the total amount of stellar luminosity escaping from the system. However, one would also like to store the escaping radiation specific intensity in a set of directions, which is what it would actually be observed on astronomical maps. To do this, we store the values of escaping radiation specific intensity from each cell, calculated using formula ([iescformula]), for a user defined set of directions. Since far away from the model the rays reaching the observer are all parallel, one can consider the escaping radiation specific intensity coming from each emitting cell along parallel directions in order to create visual maps of the model as seen from different view angles (see an example in Fig. [spiralmodelcomparison]).\n [flowdiagfe]\n [flowdiagrt]\nNotes on Approximations, Implementation and Performance\n=======================================================\n3D dust RT requires extremely high computational resources if one wants to obtain numerically accurate solutions for systems where both radiation sources and dust are distributed over different spatial scales. Practically, this implies that 3D dust RT codes need to be developed trying to balance the need for numerical accuracy with approximations reducing the total computational time to acceptable amounts. \nWhile implementing the algorithm described in the previous section, we made use of the following approximations: \n \n1) the dust density value is assumed to be constant within each cell and, thus, within each cell crossing path. This allows us to use the simple exponential expression given by Eq.[inewequ] to calculate the new value of the specific intensity of a ray after a cell crossing. However, the accuracy of the results is then affected by the spatial resolution of the grid. A more precise method would consist in using adaptive steps along a ray and calculating new values for the dust density at each position within the crossing path (see e.g. Steinacker et al. 2003). However, these extra calculations would increase substantially the total calculation time. \n \n2) the angular sampling of scattered light in each cell is performed using a finite number of solid angles covering the entire sphere (see §3 for more details). That is, the contribution to the scattered light, calculated after a cell intersection, is stored in a number of directions corresponding to the dimension of the local SCATT\\_EN array, defined in §3 (typically 48 or 96 directions are used in the calculation presented below). This means, that some of the information on the angular distribution of the scattered light is neglected because of the relatively coarse sampling on the sphere. \n \n3) the increase of angular density for the rays departing from each emitting cell is stopped when the condition expressed by Eq.[encontrcond] for the ray contribution to the local RFED is realized. This is one of the characteristic features of our RT algorithm and probably the main one which allows us to reduce substantially the computational time. As mentioned in §3.3 and §3.4, the code we developed gives the possibility of either simply stopping the rays at the locations where the above condition is realized (ray mode 1) or only avoiding the further increase of ray angular density beyond those points but still continuing the ray-tracing until the model border (ray mode 2). In both cases, a fraction of the luminosity carried by the beam associated with the rays is not considered in the calculation of the RFED distribution. Controlling the actual amount of this “lost luminosity”, not processed during the RT calculation, is fundamental to assure that approximate global energy balance is reached between the luminosity emitted by the stars and the sum of the luminosities absorbed and escaping from the system. In order to check this, we set up a counter of “lost luminosity” which sum up the ray luminosity which is not processed during the calculation. Contributions to this sum are provided by the entire beam luminosity at the locations where condition [encontrcond] is realized. The input parameters directly affecting the amount of “lost luminosity” are *f**U* and $N\\_{\\rm rays}$, which have to be chosen carefully such to guarantee energy balance. Practically, one first finds an appropriate combination of *f**U* and $N\\_{\\rm rays}$ for a typical model such to obtain a low fraction of “lost luminosity” at the end of the calculation. After that, it is usually enough to use the same combination for these parameters for similar models and check that approximate energy balance has been achieved after each calculation. For all the calculations presented in the next section, we checked that, for the assumed combinations of those input parameters, the total lost luminosity is always less than 1-2% of the total stellar luminosity. \nThe algorithm has been implemented in a Fortran 90 code, which is a typical language used in high-performance computing (HPC). Parallelization of the code has been implemented using the application programming interface OpenMP, which allows parallel computing on shared-memory machines. We opted for OpenMP parallelization since it required only a relatively small number of extra lines within the existing code. Furthermore, given the additive nature of the RT problem, it has been possible to develop simple programs to distribute the calculations for different sets of emitting cells among different nodes in a computer cluster. Within each node the calculation can be performed in parallel mode without the need of communication between nodes, except when arrays sums are needed at the end of each step of the RT algorithm or between scattering iterations. Extra routines have been written to perform these sums and this has been sufficient to perform RT calculations on computer clusters without the need of more elaborated MPI parallelization. \nThe test runs presented in the next subsection have been performed using the computing facilities at the University of Central Lancashire in Preston (in particular the local HPC cluster) and at the Max Planck Institute für Kernphysik in Heidelberg. The computational times and memory required vary depending on the geometry of the model, the spatial resolution adopted and the number of CPUs used. For example, a typical single wavelength calculation for a disc galaxy model (see §5.2) with a grid containing of the order of 105 cells can take about 2 days by using 64 CPUs with 2.57 GHz clock rate (and by using a *f**U* parameter low enough to achieve energy balance within less than a percent). In terms of memory, this calculation requires few gigabytes of RAM memory (exact amount varies during the calculation because of the temporary “allocatable” arrays vastly used by our code). The mentioned calculation time seems to be rather longer compared to those needed by MC codes where acceleration techniques are well developed and which are able to handle multiwavelength photon packages simultaneously (see e.g. Baes et al. 2011, Jonsson 2006). However, one should notice that, at variance with ray-tracing codes, MC codes are not typically used to obtain accurate calculations of the monochromatic RFED distribution. Applications of MC methods focus on obtaining a good calculation for the escaping multiwavelength spectra. In order to predict the dust emission, this requires only the dust temperature in each cell to converge and not necessarily each single value of the RFED at each wavelength.\nComparisons with other codes\n============================\nIn the following we describe and show the results of the comparisons with two codes: the 1D code DUSTY (§5.1) and the 2D code used in Popescu et al. (2011) (§5.2)\nComparison with DUSTY code\n--------------------------\nWe performed a series of tests by comparing the results provided by our code with those obtained using the latest version of the RT code DUSTY (Ivezic & Elitzur 1997). Specifically, we considered the geometric configurations of the benchmark solutions of Ivezic et al. (1997, I97), with parameters equivalent or similar to those adopted in that paper. The DUSTY code can be used to solve the dust RT problem for a geometry consisting of a single central radiation source illuminating a spherically symmetric shell of dust with an arbitrary dust density radial profile. In this case, one can reduce the 3D RT equation to a 1D equation for the mean radiation field intensity by taking advantage of the spherical symmetry of this configuration. Furthermore, the DUSTY code utilizes several scaling properties of the RT problem. These are used to calculate sets of solutions which do not depend on the absolute values of the luminosity of the central source and of the dust density and opacity but only on their spatial variation or wavelength dependence. Thus, given a dust radial density profile in the shell and the wavelength dependence of the dust optical properties, this specific RT problem can be completely defined once the following parameters are specified: \n- the effective temperature $T\\_{\\rm source}$ of the central source, which we assumed it emits as a black body \n- the dust temperature *T*1 at the inner radius of the shell \n- the total radial optical depth *τ* \n- the outer to inner radius ratio *Y* = *r*2/*r*1. \nAs in I97, we performed RT calculations for two types of radial profiles of the dust density distribution: \n1) a constant density profile *ρ*(*r*) = *ρ**o*; \n2) a power-law density profile $\\rho(r)=\\rho\\_o\\left(\\frac{r}{r\\_1}\\right) ^{-2}$ \nfor *r* in the range [*r*1, *r*2] and *ρ*(*r*) = 0 for any other value of *r*. \nThe dust opacity coefficients per unit dust mass are defined as follows: \n \n$$q\\_{\\lambda,abs}=q\\_{\\lambda,\\rm{sca}}=1\n\\label{q1}$$ \nfor *λ* < 1*μ**m* and\n \n$$q\\_{\\lambda,abs}=\\frac{1}{\\lambda} \\; ; \\; q\\_{\\lambda,\\rm{sca}}=\\frac{1}{\\lambda^4} \n\\label{q2}$$ \nfor *λ* > 1*μ**m*. \nScattering is considered isotropic in the DUSTY code, which corresponds to assuming *g**λ* = 0 in the scattering Henyey-Greenstein phase function used by our code. \nFor both forms of dust density profiles specified above, we performed two series of tests where we compared the results for the dust temperature radial profile and the outgoing radiation spectra. First, we created a grid of models by varying the inner radius dust temperature *T*1 and keeping all the other parameters fixed to the same values. For a fixed amount of dust mass, a higher value of *T*1 implies a higher dust luminosity which can be self-absorbed by dust. Since self-absorption is not included in our code yet, at least some of the discrepancies evidenced by this test can be due to this effect. We will refer to this test in the following as “the dust temperature” test. In a second series of tests, we have varied the value of *τ* while maintaining all the other parameters constant. By increasing the value of *τ*, the source luminosity is absorbed more efficiently and dust heating due to self-absorption could become progressively more dominant, especially at larger radii. We will refer to this test as “the optical depth” test. \nThe specific parameters used in the “dust temperature” test are the following: Y=1000, $T\\_{\\rm source}=2500$ K, optical depth at 1*μ*m *τ*1 = 1 and *T*1 = 200, 400, 800K. In the “optical depth” test we used these parameters: Y=1000, $T\\_{\\rm source}=2500$ K, *τ*1 = 2, 5, 10 and *T*1 = 200K. \nThe first step performed by our code is the creation of a spatial grid sampling the entire model. To do this we require the absolute values of the central source luminosity and the physical distances corresponding to the inner and outer radius of the shell *r*1 and *r*2, as these quantities are not explicitly specified in the input parameters of the DUSTY code. We derived these quantities from the standard output of the DUSTY code, which assumes that the bolometric central source luminosity is equal to $10^4~\\rm{L}\\_\\odot$. Also, we obtained the absolute scaling of the dust density distribution *ρ**o* by using the following formulae:\n \n$$\\rho\\_o=\\frac{\\tau\\_1}{q\\_{1,\\rm{ext}}(R\\_2-R\\_1)}$$ \nfor the case of the constant dust density radial profile and\n \n$$\\rho\\_o=\\frac{\\tau\\_1}{q\\_{1,\\rm{ext}}\\frac{R\\_1}{R\\_2}(R\\_2-R\\_1)}$$ \nfor the case of the power law dust density radial profile. In the previous formulae $q\\_{1,\\rm{ext}}=q\\_{1,abs}+q\\_{1,\\rm{sca}}$ and the opacity coefficients are all evaluated at 1*μ**m*. We used the coefficients at this wavelength since they are the highest. This implies that the cell optical depths of the grid so created are the same or smaller at other wavelengths. For simplicity we used the same grid for all the wavelengths. \nWhile creating the grid, we assigned average density values to each leaf cell containing dust. In the constant density profile case we simply assumed $<\\rho\\_{\\rm dust}(r)>=\\rho\\_o$. In the power law case, we used the following expression for the cell dust density, corresponding to the cell density average along a radial direction:\n \n$$<\\rho\\_{\\rm dust}(r)>=\\rho\\_o R\\_1^2[\\frac{1}{r\\_c-\\Delta r/2}-\\frac{1}{r\\_c+\\Delta r/2}]/\\Delta r$$ \nwhere *r**c* is the radius corresponding to the cell centre and Δ*r* is equal to the cell size. In the cases of the cells at the inner or outer border of the shell, we calculated averages by integrating the density only in the part of the cell containing dust and then dividing by the cell size. \nWe imposed the following conditions in the input of the grid creation program: 1) a cell has to be subdivided in smaller cells if the cell optical depth exceeds a small fraction of the total radial optical depth *τ*1, typically a factor of 0.01-0.03; 2) the minimum cell subdivision level is 3; 3) the maximum allowed cell subdivision level is equal to 7 and 8 for the constant and power law dust density profile, respectively; 4) in the case of the power law density profile, we also added the constraint that the maximum cell optical depth gradient is Δ*τ*1*μ**m*/*τ*1*μ**m* = 0.4. These conditions have been chosen in order to achieve a solution with good numerical accuracy but also to avoid to create too many cells (that is, less than ≈ 106 cells), thus reducing the amount of calculation time needed. In fact, note that the amount of cell subdivisions is limited by condition (3). Thus, in some parts of the model, a cell subdivision required by conditions (1) or (4) might not be performed because it would conflict with condition (3). The maximum and average values of the cell optical depths in each model are shown in Tables [deltatautdup] and [deltatautauup]. \n| p | $\\Delta\\tau\\_{1,\\rm{max}}$ | < Δ*τ*1 > | *f**U* | $N\\_{\\rm{rays}}$ | *f**L* | N$\\_{\\rm cells}$ |\n| --- | --- | --- | --- | --- | --- | --- |\n| 0 | 0.029 | 0.023 | 0.0001 | 16 | 0.001 | 302481 |\n| 2 | 0.23 | 0.006 | 0.001 | 16 | 0.001 | 214326 |\n[deltatautdup]\n| p | *τ*1 | $\\Delta\\tau\\_{1,\\rm{max}}$ | < Δ*τ*1 > | *f**U* | $N\\_{\\rm{rays}}$ | *f**L* | N$\\_{\\rm cells}$ |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 0 | 2 | 0.06 | 0.046 | 0.0001 | 16 | 0.001 | 302481 |\n| 0 | 5 | 0.15 | 0.11 | 0.001 | 4 | 0.001 | 302481 |\n| 0 | 10 | 0.29 | 0.23 | 0.001 | 4 | 0.001 | 302481 |\n| 2 | 2 | 0.45 | 0.013 | 0.001 | 16 | 0.001 | 214326 |\n| 2 | 5 | 1.13 | 0.032 | 0.001 | 16 | 0.001 | 214326 |\n| 2 | 10 | 2.2 | 0.064 | 0.001 | 16 | 0.001 | 214326 |\n[deltatautauup]\nOnce the grid has been created, before starting the RT calculation, one has to assign values to the following parameters (see §3): 1) *f**U*, the relative energy density contribution threshold above which a ray contribution to a crossed cell RFED is considered non negligible; 2) $N\\_{\\rm{rays}}$, the minimum number of rays which has to cross a cell when the ray contributions are found not negligible; 3) *f**L*, the escaping luminosity threshold parameter, used to determine when the scattering iterations have to stop. The adopted values are also shown in Tables [deltatautdup] and [deltatautauup].\nFor each model, we performed the calculation at the following wavelengths: 0.443, 1.05, 1.1, 1.15, 1.2, 1.259, 1.3, 1.8, 2.2, 5.0, 10, 20*μ*m. We calculated only one point for *λ* < 1*μ**m*, that is at *λ* = 0.443*μ**m*, because the opacity coefficients used by I97 are the same in that wavelength range (see Eq. [q1] and [q2]) and the inferred RFED scales only with the luminosity of the central source at the different wavelengths. For *λ* > 1*μ**m*, the chosen wavelength steps are smaller between 1 and 2 *μ**m*. Since *T**s**o**u**r**c**e* = 2500*K*, the emission from the central source peaks in that wavelength region while the dust opacity is still relatively high. As a consequence, a consistent fraction of source luminosity is absorbed or scattered within the system at those wavelengths. Therefore, a good sampling in that wavelength region is desirable to obtain a more accurate solution for the dust temperature. At longer wavelengths, a finer sampling is not as important since both the opacity and the radiation intensity from the central source decrease rapidly. \nAfter the RT calculation has been performed at all the wavelengths specified above, we obtained a grid of radiation field energy density spectra at each cell position. Then, we calculated the energy density values in the entire wavelength range 0.01 − 1*μ**m* by scaling the value inferred at *λ* = 0.443*μ**m* according to the central source luminosity at different wavelengths. This is possible because for *λ* < 1*μ**m* the dust opacity is assumed to be constant. In the wavelength range 1 − 100*μ**m* we simply interpolated the inferred values within that range. Then, in a way consistent with the DUSTY code calculation, we derived the equilibrium dust temperature *T**d* at each position such that:\n \n$$\\int{q\\_{\\lambda,\\rm{abs}}B\\_{\\lambda}(T\\_d)d\\lambda}=\\frac{c}{4\\pi}\\int{q\\_{\\lambda,\\rm{abs}}U\\_{\\lambda}d\\lambda}\n \\label{loc\\_en\\_bal\\_dusty}$$ \nWe also calculated the spectra of the outgoing radiation flux at the outer radius of the shell, a quantity given in the output of the DUSTY code, as follows. First, we derived the total escaping luminosity $L\\_{\\lambda,\\rm{esc}}$ by taking advantage of the cell average escaping brightness $<I\\_{\\lambda,\\rm{esc}}>$ within each HEALPix main direction, as provided by our code (see §3.4). In fact, $L\\_{\\lambda,\\rm{esc}}$ can be expressed as:\n \n$$L\\_{\\lambda,\\rm{esc}}=\\sum\\_i <I\\_{\\lambda,\\rm{esc}}>\\_i \\Omega\\_{\\rm{HP,MS}}^iA\\_{\\rm{EM}}^i$$ \nwhere the sum is performed for all the leaf cells and all the HEALPix main sectors. The outgoing flux can then be derived by simply dividing $L\\_{\\lambda,\\rm{esc}}$ by 4*π**R*22. In order to obtain the values of $L\\_{\\lambda,\\rm{esc}}$ for a larger set of wavelengths, we used the same procedure already applied for the scaling and interpolation of the inferred RFED (see above). We also calculated the contribution to the outgoing flux due to dust emission, which we derived assuming that the dust in each layer of the shell emits accordingly to the dust temperature radial profile we derived. Finally, we summed up both the central source and dust emission contributions to obtain the total outgoing radiation spectra. \nIn Fig.[tdups2500p0]-[tauups2500flux] we show the comparison of the dust temperature radial profiles and outgoing radiation spectra we inferred with our code with those obtained by the DUSTY code. Fig.[tdups2500p0] and [tdups2500p0flux] shows the results for the constant density profile for the “dust temperature” test. As shown in Table [sigmaivezicdust temp], the average difference between the DUSTY code dust temperature profile and the one generated by our code is about 1 − 2% for all the values of *T*1. The average difference between the inferred outgoing spectra is about 2 − 3% for the points actually calculated by our code (triangles in the figure, hereafter referred to as “calculated fluxes”. In this estimate of the discrepancy we did not consider the near-infrared points which are dominated by dust emission). The discrepancy increases to 5 − 7% if one consider the entire UV-to-IR SED we derived as explained above. The highest discrepancies are observed in the MIR region. An excess of flux in the MIR range is expected since we did not take into account dust self-attenuation in our code and dust opacity is still relatively high at MIR wavelengths. For the power law density profile, the discrepancy between the inferred temperature profiles/SEDs increases while going to higher values of *T*1, as shown in Fig.[tdups2500] and [tdups2500flux]. Specifically, the average difference between the inferred dust temperature profiles is equal to 1.6%, 2.3% and 3.4% for *T*1 equal to 200, 400 and 800 K respectively. We note that the discrepancy is due to a systematic underestimation of the dust temperature by the 3D code, which is also expected because we did not include the extra-heating due to self-absorption. For the outgoing radiation spectra the average difference is in the range 2 − 4% for the calculated fluxes and about 5 − 13% for the global SEDs, with the highest discrepancies again in the MIR range. \nThe results of these tests show that there is only a rather small difference for the dust temperature radial profile inferred by the two codes for models at fixed optical depth *τ*1 = 1 and with dust temperature *T*1 varying between 200 and 800 K. However, the discrepancy is higher for the outgoing spectra. \nFig.[tauups2500p0]-[tauups2500flux] show the results for the “optical depth” tests for the constant and power law dust density profiles. Average discrepancies are tabulated in Table [sigmaivezicoptdepth]. In the case of the constant density profile (see Figs.[tauups2500p0]-[tauups2500p0flux]), all the three models (*T*1 = 200 and *τ*1 = 2, 5 and 10) present an average difference for the dust temperature profiles of order of 2%. The outgoing radiation spectra differ on average by 5 − 17% for the calculated fluxes and 9 − 29% for the total SEDs. For the power law profile, the average differences in the temperature profiles are 3.2%, 6.5% and 15% going from *τ*1 = 2 to *τ*1 = 10. Instead the differences for the outgoing radiation spectra are within 6 − 16% for the calculated fluxes and 8 − 30% for the total inferred SEDs. The discrepancy increases systematically with the optical depth of the model considered. \nTo summarize, from the comparison with the DUSTY code we obtained the following results. For the dust temperature test, we found that: \n- the dust temperature radial profile and the calculated fluxes agree within few percent for both the constant and power law dust density profile \n- the average discrepancy for the total SEDs is of order of 5 − 10%, with the highest discrepancies in the MIR region. \nFor the optical depth test, we found that: \n- the dust temperature radial profiles agree within few percent for the constant dust density case, while for the power law case the discrepancy increases with the optical depth of the model (up to 15% for *τ*1 = 10) \n- The discrepancy for both the calculated fluxes and the total SEDs increases with the model optical depth in a similar way for both the dust density profiles. For the highest optical depth *τ*1 = 10, it is of order of 15% for the calculated fluxes and 30% for the total SEDs. As before, the highest discrepancies when comparing the total SEDs are found in the MIR region. \nAlthough the results provided by the two codes seem to be quite consistent for models with optical depths *τ*1 = 1 and 2, there is still some residual discrepancy for more optically thick models. A certainly important cause of the observed discrepancy is that, as already pointed out before, dust self-heating needs to be included in the code in order to predict accurate dust temperature profiles and output spectra, especially in the MIR region. However, another source of error is also the relatively low resolution of the 3D calculation compared to the 1D one. As explained before, the 3D grid spatial resolution is higher in regions with higher dust density but it has been limited to keep the total number of cells in the range 105 − 106. Thus, in the grids used in the calculations some regions have relatively high optical depths (see maximum cell optical depths in Table [deltatautauup]). An increased spatial resolution would have been beneficial to improve the accuracy of the solution but at the expense of a much longer computational time. This problem is more important for models with higher optical depths and might explain the residual discrepancy for the calculated fluxes in the UV-optical regime. \nBecause of the lack of dust self-heating in our code and the RT geometry assumed in this test, the solutions provided by the DUSTY code do not provide ideal benchmarks to test our code. In particular, the geometry of the emission source/opacity of a star/dust shell does not resemble that of a galaxy, which is the class of object for which we developed our algorithm. For these reasons, we decided to compare solutions for a galaxy type geometry of stars and dust, using the 2D calculations of Popescu et al. (2011). The results of this comparison are shown in the next subsection.\n| p | *T*1 | $<\\frac{\\Delta T}{T}>$ | $<\\frac{\\Delta\\lambda F\\_\\lambda}{\\lambda F\\_\\lambda}>\\_{\\rm{CALC}}$ | $<\\frac{\\Delta\\lambda F\\_\\lambda}{\\lambda F\\_\\lambda}>\\_{\\rm{SED}}$ |\n| --- | --- | --- | --- | --- |\n| 0 | 200 K | 0.018 | 0.025 | 0.07 |\n| 0 | 400 K | 0.017 | 0.035 | 0.05 |\n| 0 | 800 K | 0.01 | 0.03 | 0.068 |\n| 2 | 200 K | 0.016 | 0.024 | 0.052 |\n| 2 | 400 K | 0.023 | 0.03 | 0.063 |\n| 2 | 800 K | 0.034 | 0.043 | 0.13 |\n[sigmaivezicdust temp]\n| p | *τ*1 | $<\\frac{\\Delta T}{T}>$ | $<\\frac{\\Delta\\lambda F\\_\\lambda}{\\lambda F\\_\\lambda}>\\_{\\rm{CALC}}$ | $<\\frac{\\Delta\\lambda F\\_\\lambda}{\\lambda F\\_\\lambda}>\\_{\\rm{SED}}$ |\n| --- | --- | --- | --- | --- |\n| 0 | 2 | 0.019 | 0.052 | 0.087 |\n| 0 | 5 | 0.022 | 0.14 | 0.2 |\n| 0 | 10 | 0.021 | 0.17 | 0.29 |\n| 2 | 2 | 0.032 | 0.064 | 0.08 |\n| 2 | 5 | 0.065 | 0.14 | 0.15 |\n| 2 | 10 | 0.15 | 0.16 | 0.3 |\n[sigmaivezicoptdepth]\n [tdups2500p0]\n [tdups2500p0flux]\n [tdups2500]\n [tdups2500flux]\n [tauups2500p0]\n [tauups2500p0flux]\n [tauups2500]\n [tauups2500flux]\nComparison with 2D calculations of Popescu et al. (2011)\n--------------------------------------------------------\nMost of the dust RT solutions considered as benchmarks in the literature (e.g. Ivezic et al. 1997, Pascucci et al. 2004) are designed to test RT codes in cases resembling star forming clouds or proto-planetary discs. Those systems can be well approximated by a central luminous source illuminating a spherical dust distribution or a dusty disc. Although able to handle completely arbitrary geometries, our code has been designed for the purpose of solving the RT problem in a galaxy type geometry, which consists of an extended distribution of sources illuminating a dust distribution. For this reason, we decided to perform a comparison with the 2D RT calculations presented by Popescu et al. (2011, P11), which assume a disc galaxy geometry. The accuracy of the solutions for the radiation fields from P11 have been tested in Popescu & Tuffs 2013 against analytic solutions. For the cases where the analytic solution is an exact solution, the accuracy of the 2D code has been proven to be better than 1%.\nP11 used a modified version of the Kylafis & Bahcall (1987) 2D ray-tracing code to calculate radiation fields within a galaxy model comprising of three stellar components (a bulge, a disc and a thin disc) and two dust discs (called “thick dust disc” and “thin dust disc”). Both the stellar volume emissivity and dust density for the disc type components are described by a double exponential distribution:\n \n$$f(R,z)=f(0,0)\\exp^{-\\frac{R}{h\\_{d,s}}-\\frac{|z|}{z\\_{d,s}}}\n \\label{double\\_exp}$$ \nwhere *h**d*, *s* and *z**d*, *s* are the scale-length and scale-height of the disc components. As described in P11, the stellar discs (referred to as “disc” for the old stellar component and “thin disc” for the young stellar component) are characterized by the geometrical parameters scale-length and -height, as reported in Table E1 of that paper. They are also described by two parameters *o**l**d* and *S**F**R*. The *S**F**R* is a parameter defining the luminosity of the young stellar population (the thin disc) and *o**l**d* is a parameter defining the luminosity of the old stellar population (the disc). The spectral luminosity densities corresponding to the unit values of these parameters, *S**F**R* = 1 *M*⊙/*y**r* and *o**l**d* = 1 are given in Table E2 of P11[11](#fn11). The parameter values for the stellar discs in the B-band and the dust discs are also shown in Table [tableP11parameters], since they are used in the RT calculations we present below. The scaling of the dust density double exponential distributions is determined by the face-on central optical depth in the B-band *τ**B**f*. For the galaxy model of P11, a fixed ratio equal to 0.387 is assumed for the face-on central optical depths of the thick dust disc and thin dust disc (see Eqn.10 in P11). The assumed opacity and scattering coefficients are those for *R**V*=3.1 from Weingartner & Draine (2001), revised by Draine & Li (2007). Note that, unlike the comparison with the DUSTY code, the scattering is considered anisotropic in this case. \nFor the comparison between the solutions provided by the two codes, we used the library of radiation fields presented in Popescu & Tuffs (2013, PT13). From the library we obtained the contribution of the old stellar disc and young stellar disc to the radiation field energy density distribution separately, thus allowing a more careful check on the accuracy of the 3D calculation. \nAs for the comparison with the DUSTY code, the first step is to create the grids used in the calculations. We input in the grid creation algorithm (see §3.1) the functional shapes for the stellar and dust discs and impose the following criteria for the cell subdivision: 1) a maximum value for cell optical depth; 2) a maximum value for the cell luminosity; 3) a maximum cell subdivision level (typically about 6-7); 4) a subdivision level equal to the maximum subdivision level for the cells in the region close to the galaxy centre (∣*z*∣ < 1000 and *R* < 500). By using condition 4, we required a higher spatial resolution in the galaxy central regions. Those regions are the ones where we expect the radiation field to vary more rapidly. An alternative way to obtain an increase of the spatial resolution in those regions would be by lowering further the threshold values for e.g. the cell optical depth required for the cell subdivision. However, this usually results in a substantial increase of the total number of cells even for relatively small changes for the required thresholds. For simplicity we decided to impose a condition simply based on the distance from the axis origin.\nFor each leaf cell we assigned for both the dust density and stellar volume emissivity the values averaged over the cell volume. That is, we numerically integrated the double exponential functions *f*(*R*, *z*) describing the stellar and dust distribution within the cell volume *V**c* and then divided by *V**c*. That is:\n \n$$<f\\_c>=\\frac{\\int\\_{V\\_c}{f(R,z)dxdydz}}{V\\_c}$$ \nAs mentioned above, we performed 3D calculations for the models including either the old stellar disc or the young stellar disc. The details and results of these tests are described in the following two subsections.\n### Calculations for the old stellar disc\nWe considered the old stellar disc of the P11 galaxy model for *o**l**d* = 1 and we calculated the RFED distribution in the B-band for the following models (maximum and average cell optical depth and luminosity in Table [tableP11testgrid]. For all the models we used *f**U* = 10− 7, $N\\_{\\rm{rays}}=2$ and *f**L* = 10− 3)[12](#fn12): \n \n1) *τ**B**f* = 0 (optically thin model) \n2) *τ**B**f* = 1, only the thick dust disc \n3) *τ**B**f* = 1, both the thick and thin dust discs \nModel 1) has been calculated to check the accuracy of the 3D code when dust is not included. Fig.[mdtau0B] shows a comparison of the RFED radial and vertical profiles obtained by this calculation (diamonds symbols) and by the 2D code used by P11 (continuous line, same convention hereafter). For the comparison we selected the vertical profiles at *R* = 0, 5, 10, 18kpc and radial profiles at *z* = 0, 400, 1000, 2000pc (note that while the plots for *R* = 0 and *z* = 0pc show the exact values obtained from the 3D calculation, the other plots show values obtained through interpolation within the 3D grid). As shown in Table [sigmapopescu11], the average discrepancy for the vertical profiles is of order of 1-2% for *R* = 0, 5, 10kpc and about 5% for *R* = 18kpc. For the radial profiles the discrepancies are between 2-5%, although most of the disagreement is found at large radial distances. \nModel 2) includes only the thick dust disc but not the thin dust disc. We calculated this model because both the old stellar disc and the thick dust disc can be well resolved by an adaptive grid containing of order of 105 cells. Resolving the thin dust disc properly requires an order of magnitude more cells and this is avoided in this calculation to reduce the calculation time (but see model 3 below). Because the thin dust disc is not present, all the opacity is assigned to the thick dust disc (at variance with the original P11 galaxy model). For this test we show three sets of results. Fig.[mdtau1Bnotdustdir] shows the profile of the radiation field including only the direct light from the stellar distribution. Fig.[mdtau1Bnotdustsca1] includes the direct light and the first order scattered light. This calculation has been performed since the 2D code of P11 is based on an algorithm which explicitly calculates only the first order scattered light, assuming that the ratio between the specific intensity of successive scattering orders is constant (see Kylafis & Bahcall 1987 for more details). Thus, it is important to verify the agreement at this stage before comparing the final solution including all order scattering, shown in Fig.[mdtau1Bnotdust]. The calculations including only direct light agree within 1-4% for both vertical and radial profiles. The agreement for the solution including also the first order scattered light is within about 0.5-2% for the vertical profiles and within 1-3% for the radial profiles. Finally, the comparison for the solution including all order scattered light shows an agreement within 1-2% for the vertical profiles and 2-3% for the radial profiles. \nFor model 3) the calculations includes both dust discs, although the thin dust disc is not well resolved especially at large radii (cell sizes on the galaxy plane varying between 65 and 592pc, while the scaleheight of the thin dust disc is about 91 pc). Similarly as before, Fig. [mdtau1Bdir], [mdtau1Bsca1] and [mdtau1B] show the solutions for the RFED profiles including the direct light, the direct light plus the first order scattered light and the final values including all scattered light respectively. The comparison for the direct light shows an agreement within about 2-6% while the calculations including direct light and first order scattered light agree within 1-4%. For the last calculation including all the stellar and scattered light contributions to the RFED, the average discrepancies are within 1-4% for the vertical profiles, about 7% for the radial profile at *z* = 0pc and about 3-4% for the radial profiles at *z* = 400, 1000, 2000pc. \nThe 3D calculations performed for the old stellar disc show a good agreement with the 2D solutions, with the residual discrepancies being plausibly due to the resolution of the 3D grid which does not resolve properly the thin dust disc in the central regions and both the discs at large radii.\n| | |\n| --- | --- |\n| $h\\_s^{\\rm disc}$ | 5670 pc |\n| $z\\_s^{\\rm disc}$ | 419.58 pc |\n| $h\\_s^{\\rm tdisc}$ | 5670 pc |\n| $z\\_s^{\\rm tdisc}$ | 90.72 pc |\n| $h\\_d^{\\rm disc}$ | 7972.02 pc |\n| $z\\_d^{\\rm disc}$ | 272.16 pc |\n| $h\\_d^{\\rm tdisc}$ | 5670 pc |\n| $z\\_d^{\\rm tdisc}$ | 90.72 pc |\n| $L\\_\\nu^{\\rm disc}$ | 4.771 × 1021W/Hz |\n| $L\\_\\nu^{\\rm tdisc}$ | 2.271 × 1021W/Hz |\n[tableP11parameters]\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| model | $\\Delta\\tau\\_{B,\\rm{max}}$ | < Δ*τ**B* > | $\\Delta L\\_{B,\\rm{max}}$ | < Δ*L**B* > | N$\\_{\\rm cells}$ |\n| | | | 1017 W/Hz | 1017 W/Hz | |\n| OLD (1,3) | 0.4 | 0.047 | 4.76 | 0.46 | 109620 |\n| OLD (2) | 0.28 | 0.048 | 4.76 | 0.50 | 97200 |\n| YOUNG (1,2) | 0.4 | 0.059 | 2.27 | 0.25 | 95040 |\n[tableP11testgrid]\n### Calculations for the young stellar disc\n3D calculations to obtain the RFED distribution due to the thin stellar disc are more challenging because, as previously stated, in order to properly resolve the young stellar and thin dust discs, one should in principle create a grid containing of order of 106 cells. We preferred to use a grid of about 105 cells which underresolve the thin discs but it allows shorter computational times. The downside is that the 3D calculations will be less accurate numerically. \nFor this set of tests we considered a young stellar disc with *S**F**R* = 1 *M*⊙/*y**r* and we calculated the RFED in the B-band for the following models (as before, we used *f**U* = 10− 7, $N\\_{\\rm{rays}}=2$ and *f**L* = 10− 3): \n \n1) *τ**B* = 0 (optically thin case) \n2) *τ**B* = 1, both dust discs \n \nFig.[mtdtau0B] shows the results for model 1). The calculated vertical profiles from the 2D and 3D codes agree within 1-4% for *R* = 0, 5, 10kpc while the average discrepancy is about 10% for *R* = 18kpc (see Table [sigmapopescu11]). The discrepancies tend to be higher for the radial profiles, that is, about 5-10% with most of the discrepancy found at large radii. However, in this case the discrepancy seems not to be due only to the coarse 3D grid resolution at large radii. In fact, the 2D calculation show an artificial feature on the *R* = 18kpc plot for low values of *z*. This feature is due to inaccuracies in the 2D calculation. \nFig.[mtdtau1B] shows the calculations for model 2). The average discrepancies are of order of 3-6% for the vertical profiles and about 5-7% for the radial profiles. Note that the “steps” appearing in the radial profile for *Z* = 0 are located at the position where the adaptive grid changes cell size. They appear also in the old stellar disc solutions but in a less evident way. \nThe 3D calculations for the young stellar disc provide solutions which still present a fairly good agreement, although there is a higher discrepancy compared to the old stellar disc calculations. As said before, this is most probably mainly due to the low resolution of the 3D grid, unable to properly resolve the stellar and dust components in this test.\ncccccccccc Model & Type & \n& & *R*1 & *R*2 & *R*3 & *R*4 & *z*1 & *z*2 & *z*3 & *z*4 \nOLD (1) & NODUST & 0.009 & 0.02 & 0.02 & 0.048 & 0.017 & 0.046 & 0.036 & 0.036 \nOLD (2) & DIR & 0.014 & 0.018 & 0.019 & 0.037 & 0.038 & 0.03 & 0.016 & 0.028 \nOLD (2) & DIR+SCA1 & 0.019 & 0.008 & 0.006 & 0.022 & 0.014 & 0.029 & 0.017 & 0.024 \nOLD (2) & ALL & 0.016 & 0.015 & 0.015 & 0.013 & 0.02 & 0.022 & 0.034 & 0.032 \nOLD (3) & DIR & 0.03 & 0.04 & 0.054 & 0.068 & 0.0079 & 0.06 & 0.034 & 0.027 \nOLD (3) & DIR+SCA1 & 0.014 & 0.01 & 0.02 & 0.034 & 0.027 & 0.036 & 0.021 & 0.023 \nOLD (3) & ALL & 0.036 & 0.013 & 0.01 & 0.025 & 0.073 & 0.027 & 0.035 & 0.035 \nYOUNG (1) & NODUST & 0.0078 & 0.02 & 0.043 & 0.095 & 0.055 & 0.097 & 0.066 & 0.027 \nYOUNG (2) & ALL & 0.052 & 0.061 & 0.046 & 0.032 & 0.069 & 0.065 & 0.054 & 0.048 \n[sigmapopescu11]\n [mdtau0B]\n [mdtau1Bnotdustdir]\n [mdtau1Bnotdustsca1]\n [mdtau1Bnotdust]\n [mdtau1Bdir]\n [mdtau1Bsca1]\n [mdtau1B]\n [mtdtau0B]\n [mtdtau1B]\nAn RT 3D application: including spiral arms in galaxy models\n============================================================\nFor most practical applications to large statistical samples of galaxies (e.g. Driver et al. 2007, 2008, 2012, Gunawardhana et al. 2011, Silva et al. 2011, Grootes et al. 2013), RT modelling of the observed spatially integrated direct and dust-reradiated starlight necessarily (in the absence of detailed images) adopts 2D axisymmetric approximation of the distribution of stars and dust. However, the distribution of light at UV and short optical wavelengths from young massive stars is well known in real galaxies to be biased towards a spiral pattern of enhanced dust density, rather than the smooth exponential disc function typically assumed by these models. The question therefore arises whether this effect introduces any systematic bias into 2D model predictions of dust attenuation of integrated starlight and averaged RFED in spiral galaxies. To evaluate this bias, we have performed a RT calculation for a galaxy model including logarithmic spiral arms. We considered a typical model galaxy from P11, consisting of a disc with *o**l**d* = 1 and a thin stellar disc with *S**F**R* = 1*M*⊙/*y**r*, and two dust discs with *τ**B**f* = 1 (see Sect. 5.2 for a description of the model parameters of P11). We modified this model using the same procedure as that adopted in P11 for the inclusion of circular spiral arms. Thus, we considered the same double exponential distribution for the thick stellar and dust disc but we redistributed the thin disc stellar luminosity and dust mass within spiral arms. As shown in Schechtman-Rook et al. (2012), the implementation of logarithmic spiral arms can be done by multiplying a logarithmic spiral disc perturbation *ξ* to the double exponential formula describing the stellar volume emissivity and dust density (see Eqn. [doubleexp]). We adopted the expression for *ξ* in their formula 10 (two spiral arms):\n \n$$\\xi=\\left[1-w + \\prod\\_{n=2,n+2}^N \\frac{n}{n-1} \\sin^N\\left(\\frac{\\ln(\\sqrt{x^2+y^2})}{\\tan(p)}-\\tan^{-1}(\\frac{y}{x})+\\frac{\\pi}{4} \\right) \\right]$$ \nwith *w* the fraction of stellar light or dust within the spiral arms, *p* the pitch angle determining how tightly the spirals turn around each other and the exponent *N* which regulates the relative size of the arm and interarm regions. For these parameters, we adopted the values *w* = 0.9, *p* = 10 ∘ and *N* = 10. Then, we performed an RT calculation for a galaxy model (including both old and young stellar discs) in the B-band with face-on central optical depth *τ**B**f* = 1 and disc parameter values as in Table [tableP11parameters].\nFig.[spiralmodelcomparison] shows the comparison for the output surface brightness images at different inclinations between a pure double exponential model (upper row) and for the model including spirals (lower low). The images show the different morphology of the stellar emission for the face-on and low-inclination images. However, the edge-on images are remarkably similar for the two models. We also made a comparison for the total attenuation as a function of galaxy inclination for the two models, which is shown in Fig.[plotattmodel]. The attenuation curves are quite close to each other, within 0.02 dex, showing that the spiral pattern does not affect much the total attenuation of the galaxy for the adopted parameters.\nFinally, we compared the RFED profiles in the galaxy plane. Fig.[ploturadprofmodel] shows the profiles for the pure double exponential model (squares), a cut along the x-axis of the model including spiral arms (blue line) and its azimuthally averaged RFED profile. Interestingly enough, although the RFED along the x-axis shows the variation due to the spiral arms, the azimuthally averaged profile is very close to the profile for the model without spiral arms.\nAlthough P11 have already shown that the spatially integrated dust and PAH emission SED of a typical spiral galaxy does not depend on whether the young stellar population and associated dust is distributed in a circular spiral arm or in a disc, here we show for the first time that it is at the level of the radiation fields that heat the dust that the distributions in the spirals start to resemble the disc distributions on the average.\nThe results on the global attenuation, images and radiation fields, all suggest that double exponential models can be a quite good representation for spiral disc galaxies, and that the spatially integrated SEDs of spirals can be accounted by 2D models. Although this is in qualitative agreement with previous works (Misiriotis et al. 2000, Semionov et al. 2006, Popescu et al. 2011), a more extensive study is needed to see how the different parameters, e.g. the face-on optical depth, affect the attenuation curve and the radiation fields in models with and without spiral arms.\n [spiralmodelcomparison]\n [plotattmodel]\n [ploturadprofmodel]\nSummary and Outlook\n===================\nIn this paper we present a new ray-tracing dust radiation transfer algorithm which is able to handle arbitrary 3D geometries and it is specifically designed to calculate accurate RFED within galaxy models. The main optimization characteristics of this algorithm are the following: \n1) an adaptive 3D grid (see §3.1) \n2) a ray-tracing algorithm based on the pre-calculation of a lower limit for the RFED (see §3.2 and §3.3) \n3) an iterative procedure for the optimization of the angular density for the rays departing from each emitting cell (§3.3). \nFurthermore, parallelized versions of the code have been written for its use on shared-memory machines and computer clusters. \nIn order to verify the code accuracy, we performed comparisons with the results provided by other codes. Specifically, we used our code to calculate solutions for a spherical dusty shell illuminated by a central point source and for an axis symmetric galaxy model. For the first configuration, we considered as benchmark a set of solutions calculated by using the DUSTY 1D code (Ivezic & Elitzur 1997). We showed that the equilibrium dust temperature radial profiles and the outgoing flux spectra derived by our 3D calculations agree within a few percent with the benchmark solutions for models with low radial optical depths (*τ*1 = 1, 2) but present larger discrepancies for more optically thick models (*τ*1 = 5, 10). The residual discrepancies, especially for the models with higher optical depths, are most probably due to the lack of dust self-heating in our code and the lower spatial resolution of the 3D calculations compared to the 1D ones. Since the geometry of the source emission/opacity of star/dust shell does not reproduce that for which our algorithm was developed, namely that of an extended distribution of stellar emission and dust, we also used a second benchmark. Thus, we considered the 2D calculations by Popescu et al. (2011) for the RFED distribution within their galaxy model. We calculated the contribution to the RFED distribution due to an old stellar disc and a young stellar disc separately and we compared the results for radial and vertical RFED profiles derived for a set of reference radii and vertical distances. We found a general good agreement between the 3D and 2D calculations within a few percent in most of the cases. At least part of the residual discrepancy can be accounted by the relatively low spatial resolution of the grid used in the 3D calculation, which is not sufficient to properly resolve the thin disc component of the galaxy model of Popescu et al. (2011). We showed an example of a 3D application of the code by performing RT for a spiral galaxy model where in one case the emissivity of the young stellar population and associated dust opacity are distributed in logarithmic spiral arms and in another case are distributed in exponential discs. We found that the edge-on images, the attenuation as a function of inclination and the azimuthally average RFED profiles on the galaxy plane are approximately the same for the two models. This suggests that the spatially integrated SEDs of spirals can be well described by 2D models.\nThe tests we performed have shown that, in the conditions where dust self-heating is negligible and the 3D spatial resolution is high enough to resolve emission and opacity distributions, our code can be used to calculate accurate solutions for the RFED. This characteristic is particularly important for the calculation of stochastically heated dust emission, which requires both the overall intensity and the colour of the radiation field to be calculated in an accurate way. In a future work we will show applications of the code for the calculation of infrared emission. This will be performed using our 3D RT code coupled with the dust emission code used by Popescu et al. (2011), which self-consistently calculates the stochastic emission from small grains and PAH molecules. In this way, it will be possible to use our code to obtain both integrated SEDs and images in the mid- and fir-infrared for galaxies with arbitrary geometries. In addition, an important step will be to further optimize the code in order to make it possible to run on grids containing millions of cells in a reasonably short time. This will allow us to improve further the accuracy of the calculation for RFED within multi-scale structures spanning at least three orders of magnitude, such as from ≈ 10pc to ≈ 10kpc in the case of a galaxy ISM. \nAcknowledgements\n================\nWe acknowledge support from the UK Science and Technology Facilities Council (STFC; grant ST/J001341/1). GN thanks S. Dalla, K. Foyle, E. Kafexhiu, T. Laitinen, J. Steinacker for useful suggestions and/or discussions. 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V., Papamastorakis, J. 1999, A&A, 344, 868 Xilouris, E. M., Kylafis, N. D., Papamastorakis, J., Paleologou, E. V., Haerendel, G. 1997, A&A, 325, 135 Weingartner, J. C., Draine, B. T. 2001, ApJ, 548, 296 Witt, A. N. 1977, ApJS, 35, 1 Witt, A. N., Gordon, K. D. 1996, ApJ, 463, 681 Witt, A. N., Stephens, T. C. 1974, AJ, 79, 948 Witt, A. N., Thronson, H. A., Jr., Capuano, J. M., Jr. 1992, ApJ, 393, 611\n---\n1. The name of the code can be seen as the acronym for “Dust Adaptive Radiative Transfer Ray-tracing”[↩](#fnref1)\n2. The actual version of the code does not consider the dust absorption/scattering of light emitted by dust at other positions (so called “dust self-heating”). This effect can be neglected in case the galaxies are optically thin at infrared wavelengths[↩](#fnref2)\n3. Throughout the text by “volume emissivity” we will always mean the luminosity per unit volume per unit solid angle and per unit wavelength interval of the stellar radiation at each position. To not be confused with the emissivity coefficient used to characterise the emission properties of e.g. gas or dust.[↩](#fnref3)\n4. For example, the latter condition for a single grain stochastically heated can be expressed as: \n∫*Q**λ*, *a**b**s*∫*B**λ*(*T*)*P*(*T*)*d**T**d**λ* = (*c*/4*π*)∫*Q**λ*, *a**b**s**U**λ**d**λ* \n where *Q**λ*, *a**b**s* is the grain absorption efficiency at wavelength *λ*, *B**λ*(*T*) is the Planck function calculated for dust temperature *T* and *P*(*T*) is the probability for the dust grain to have a temperature equal to *T*. Numerical methods, such as the one presented in Guhathakurta & Draine (1989), allow us to derive *P*(*T*) and therefore the dust emission spectra, once the absorption efficiency *Q**λ*, *a**b**s* and the RFED *U**λ* are known..[↩](#fnref4)\n5. This approach can be redundant because the rays originating from different cells can often go through almost the same paths. On the other side, it is convenient because it easily allows the implementation of the acceleration techniques presented in this work.[↩](#fnref5)\n6. Note that, as shown in the right-hand panel of Fig.[encrit], there could be emitting cells which are part of a large emitting cell distribution, whose individual RFED contributions *δ**U**λ* might be lower than the assumed energy density threshold *f**U**U**λ*, *L**L*, but the cumulative contribution is not negligible. Overlooking the presence of this kind of cells can be avoided by choosing an appropriate value for the constant *f**U*. However, in cases such as that of an extended distribution of uniform volume emissivity within an optically thin system, the value of *f**U* to be chosen, in order to reach an accurate solution, could be extremely low. In those cases, there might be no substantial advantage in terms of speed by using the presented algorithm, since the majority of emitting cells contribute significantly to the RFED at each position in the entire volume.[↩](#fnref6)\n7. We use a grid refinement factor equal to 3, since it has the advantage of preserving the position of cell centres after each cell subdivisions. For technical reasons, this facilitates the inclusion in the grid of central emitting point sources such those in the solutions described in the next section.[↩](#fnref7)\n8. Actually the numerical implementation requires to assume a small threshold value higher than zero, such that *e*− *τ**λ* ≠ 1 when the exponential function is evaluated.[↩](#fnref8)\n9. The SCATT\\_EN array can be very large: dimension = number of cells × $4\\pi/\\Omega\\_{\\rm{HP,INT}}$.[↩](#fnref9)\n10. see upper-left panel of Fig.[hpixsphere] for definition of “HEALPix main sector”[↩](#fnref10)\n11. We note that the models are truncated at the radius *R* = 24kpc. The unit luminosities in Table E2 of P11 refer to the luminosities within the truncation radius.[↩](#fnref11)\n12. The choice of *f**U* and $N\\_{\\rm rays}$ is such to guarantee global energy balance within a few percent accuracy (see §4).[↩](#fnref12)\n"},"data_id":{"kind":"string","value":"arxiv_0003287"}}},{"rowIdx":799,"cells":{"content":{"kind":"string","value":"4 for both channels and different values of *η*, and show the result in panel (b) of Fig. [fig:triangularresults]. As is clear, in the close vicinity of *J*2/*J*1 = 1/8 the binding energy is positive in both channels, but is larger for the “chiral” *l* = 3 channel. At some value $J\\_2/J\\_1 \\gtrsim 1/8$, which is marked by a black squares in Fig. [fig:triangularresults](b) and increases with *η*, the “nematic” *l* = 2 channel becomes dominant, implying a transition from a chiral instability towards a Potts-nematic instability. Note that the curves exhibit a cusp (i.e., discontinuous derivative) at *J*2/*J*1 = 1/8, which is due to the aforementioned abrupt change of the global minimum.\n[fig:nonzeroCoM]![The value of the interaction parameter \\eta = J_i^z / J_i required to create an exciton with the binding energy \\varepsilon_{{\\bf q}}= 0.05 J_1 for different values of the center-of-mass momentum {{\\bf K}}, taking values along the \\Gamma-K-M-\\Gamma contour in the Brillouin zone (horizontal axis). The minimal \\eta corresponds to the {{\\bf K}}= 0 case (\\Gamma point), which makes it the leading instability channel and justifies considering zero center-of-mass momentum excitons in the rest of the paper. ](NonzeroCoM \"fig:\")\nTo further understand the formation of nematic two-magnon bound states, we follow the weak coupling approach described in Sec. [ssec:wc] and compute the magnon density of states *ν**M*△ and effective coupling constant *λ*△ by expanding around the *M* points. The density of states is given by [DOStriangular] M= and the effective coupling constant is found as = 8(J1z + J2z). [lambdatriangular] These two parameters enter the expression for the binding energy in Eq. and have been used to compute the black dashed curves in Fig. [fig:triangularresults](a). (The prefactor coming from the cutoff was fitted.) The density of states, which is shown in panel (d) of Fig. [fig:triangularresults], is singular and diverges at *J*2 = *J*1. This is caused by the fact that for *J*2 > *J*1 the minima of $J\\_{{\\bf q}}$ move away from *M* towards Γ, making the *M* points saddle points rather than minima. In contrast, *ν**M*△ is regular at *J*2/*J*1 = 1/8, because *M* remains a local minimum. Since the effective coupling remains finite and does not change significantly in the interval 1/8 < *J*2/*J*1 < 1, the main dependence of the binding energy *ɛ**b* on *J*2/*J*1 ratio comes from *J*2/*J*1 dependence of *ν**M*, which is illustrated by Figs. [fig:triangularresults](c) and (d).\n![ (a-1) Zigzag (blue) and striped (yellow) phases on the honeycomb lattice for antiferromagnetic nearest neighbor coupling (J_1 < 0 case). Effective Potts-nematic coupling constant is positive only at J_2 > J_1/2, so vestigial state can emerge only in the part of the zigzag domain (shown in darker blue). Lines demonstrate contours at which perturbative theory predicts the width of the intermediate phase to be (D_c^* - D_c) / J_1 \\sim 10^{-1} (solid), \\sim 10^{-2} (dashed), and \\sim 10^{-3} (dot-dashed). We assumed J_i^z = J_i when plotting this picture. (a-2) Similar plot for J_1 > 0 case. In that case vestigial phase can appear for all points inside the zigzag and striped magnetic domains. (b) (c) The core real-space ({\\boldsymbol{\\rho} }) structure of the Potts-nematic exciton on the honeycomb lattice in the perturbative regime for a) zigzag phase; b) striped phase. The spin-up magnon resides on the first sublattice, while the spin-down magnon is located at a site, shifted by {\\boldsymbol{\\rho} }. The solution is doubly degenerate in each case. Slow exponential decay is not shown. ](honeycomb_results \"fig:\") [fig:honeycombresults]\nAs explained in Section [sec:schroedingerformagnonpairs], excitonic solution in the Potts-nematic channel correspond to *l* = ± 2 values of the angular momentum and thus has a twofold degeneracy, which we observe both in numerical and weak coupling approaches. In the latter, it is straightforward to obtain an explicit expression for the wavefunctions in the real space, which up to a constant factor reads: 1(2)( ) = e-r / l, [eq:wftriangular] Here ${\\boldsymbol{\\rho} }$ is the vector connecting the two lattice sites occupied by magnons constituting the exciton and *ω* = *e*2*π**i*/3 is the cubic root of unity. The factor *e*− *r*/*l**ɛ*, where *l**ɛ* ∼ *ɛ*− 1/2 ≫ 1 appears in the next orders of the perturbation theory and makes the wavefunction normalizable. Eq. represents the fact that the solution is a linear combination of exponentials $e^{i {{\\bf M}}\\_i \\cdot{\\boldsymbol{\\rho} }}$ with the wavevectors ${{\\bf M}}\\_i$ pointing to the minima of the single-magnon dispersion, so that the exciton is formed form the most low-lying excitations. The three exponentials are combined in a function with *l* = ± 2 in the same way as the nematic order parameter is constituted from the spin densities at ${{\\bf M}}\\_i$ points. Taking then real (imaginary) part of *l* = 2 wavefunction [Eq. ] gives the wavefunctions with the symmetry of *d**x**y* (*d**x*2 − *y*2) orbitals. The third linearly independent combination of exponentials $e^{i {{\\bf M}}\\_1\\cdot {\\boldsymbol{\\rho} }} + e^{i {{\\bf M}}\\_2 \\cdot{\\boldsymbol{\\rho} }} + e^{i {{\\bf M}}\\_3\\cdot {\\boldsymbol{\\rho} }}$ provides the rotation-symmetric *l* = 0 wavefunction, which contradicts assumptions made in and thus should be discarded.\nWe show the bound state wavefunctions graphically in Fig. [fig:triangularwavefunction] ignoring the slow decaying factor. Note that the radial structure is not completely trivial: one can see that the angular harmonics are not aligned on the neighboring shells but are rotated with respect to each other. That makes $\\psi({\\boldsymbol{\\rho} })$ obey the approximate translational symmetry ${\\boldsymbol{\\rho} }\\rightarrow {\\boldsymbol{\\rho} }+ 2 {\\boldsymbol{\\delta} }\\_{1(2)}$ with doubled period reflecting the fact the $2 {{\\bf M}}\\_i \\sim 0$ in the reciprocal space. (The translation symmetry is not exact due to the decaying factor *e*− *ρ*/*l**ɛ*.) Note also that the excitonic wavefunction $\\psi({\\boldsymbol{\\rho} })$ nullifies when the two magnons reside on the third-nearest sites. In accordance with this, we found that introducing third-nearest neighbor interaction *J*3 does not change the value of the effective coupling.\nFinally, to make sure that excitons with zero center-of-mass momentum ${{\\bf K}}$ represent the leading instability, we run a numerical calculation for the case of a non-zero ${{\\bf K}}$, which still boils down to solving with $\\mathcal M$ given by Eq. . In this place we opted for fixing the binding energy at the value of *ɛ* = 0.05*J*1 and evaluating the minimal interaction strength *η* required to create such bound state. The result is presented in Fig. [fig:nonzeroCoM], where we allowed the center-of-mass momentum *K* to take values across a path in the Brillouin zone, connecting the points Γ − *K* − *M* − Γ. Indeed, we observed that ${{\\bf K}}= 0$ requires the minimal amount of interaction for the bound state formation, which justifies considering ${{\\bf K}}=0$ in the rest of the paper. Interestingly, we found that ${{\\bf K}}= {{\\bf M}}\\_i$ choice is also favorable for exciton formation, however, this instability remains subleading.\nHoneycomb lattice magnets [ssec:honeycomb]\n------------------------------------------\nNext, we examine Potts-nematic two-magnon bound state formation on the honeycomb lattice. The motivation for considering the honeycomb lattice is twofold. First, it has the same symmetry group as the triangular lattice, but contrary to the latter it has sublattice structure, thus providing a generalization of the analysis presented in Sec. [sec:theory] to non-Bravais lattices. Second, a variety of experimentally available candidate materials are realizations of the honeycomb lattice model which makes the honeycomb lattice highly relevant from an experimental perspective. The connection to experimental compounds will be discussed in more detail below in Sec. [sec:expts].\nAs mentioned in Sec. [sec:Z3breaking], as far as the exchange interactions included in Eq. are concerned, we consider a frustrated *J*1–*J*2–*J*3 honeycomb model with up to third-nearest neighbor couplings. The classical phase diagram of the isotropic *J*1–*J*2–*J*3 Heisenberg model is known and notably includes two magnetic phases with broken threefold rotation symmetry. The corresponding regions in the phase diagram are defined by property that the minima of $J\\_{{\\bf q}}$ are located at the *M* points. Note that since $J\\_{{\\bf q}}$ is a matrix in the space of the two sublattices, two distinct solutions exist, which are related by the relative orientation of the spins on the sublattices, i.e., either aligned or anti-aligned. The single-*Q* collinear orderings with *M*-point wave vectors that are selected by fluctuations are schematically shown in Fig. [fig:honeycombresults], and are generally referred to as stripe (sublattice-even) and zigzag (sublattice-odd) orders. In our analysis we restrict to exchange couplings (*J*1, *J*2, *J*3) which favor zigzag or stripe order on the classical level.\nTo account for the two sublattices in the system, we extend the general theory presented in Sec. [sec:schroedingerformagnonpairs] to the case of a non-Bravais lattice (see Appendix [app:gen]). The main occurring difference is the appearance of projectors to the Bloch bands in the expression for the effective matrix elements. We present solutions of the modified equations obtained by numerical Brillouine zone integration complemented by the band summation in Fig. [fig:honeycombcomparison]. For that plot, we picked values of the spin couplings inside the zigzag domain on the Fig. [fig:honeycombresults] (a) (*J*1 < 0, *J*2/∣*J*1∣ = 1, *J*3/∣*J*1∣ = 0.3) and varied the effective interaction strength *η* = *η**i* = *J**i**z*/*J**i*. We observed that bound state with the angular momentum *l* = ± 2 has the largest binding energy and its dependence on *η* is captured well by the perturbation theory with a fitted prefactor (dashed line). As in the case of triangular lattice, that means that proliferation of excitations of this kind constitutes the leading instability, when *D* is lowered and that marks the transition to the Potts-nematic phase.\nThe weak coupling theory for a non-Bravais lattice in general follows the previous derivation given in Section [ssec:wc]. Again, the main contribution stems from the most low-lying excitations, which now also means that it is sufficient to consider the lower band. We expand around the minima of the latter (located at ${{\\bf M}}\\_i$ points) and, after some algebra, get expressions for the effective coupling constant in the Potts-nematic channel [lambdahoneycomb] = 8 ( J1z + 2 J2z ), which appears to be the same for both zigzag and stripe *M*-orderings. The density of low-lying states at *M* points equals: [doshoneycomb] M =, where *J*13 = *J*12 + *J*1*J*3 − 4*J*32 and *β* = sign (*J*1 − 3*J*3). Eqs. and should be plugged into the general perturbative formula for the binding energy, which gives a result consistent with numerical computation (Fig. [fig:honeycombcomparison]).\nPerturbative wavefunction of *l* = ± 2 excitions are again doubly degenerate and appear to be a linear combination of the exponentials $e^{i {{\\bf M}}\\_i {{\\bf r}}}$ as in the case of triangular lattice, but bear a sublattice structure, which slightly differs between the zigzag and stripe phase. We present it graphically in Fig. [fig:honeycombresults] (b) for underlying zigzag ordering an Fig. [fig:honeycombresults] (c) for the stripe one, choosing the *d**x**y* and *d**x*2 − *y*2 basis and using the ${\\boldsymbol{\\rho} }$ plane, where ${\\boldsymbol{\\rho} }$ is the distance between the magnons in the pair. We assumed that the spin-up magnon resides on the first sublattice; the opposite case leads to similar pictures. The figure depicts the core of the exciton wavefunction with omitted slow decay factor *e*− *ρ*/*l**ɛ*.\nWith obtained analytical approximation, we explore exciton formation for all striped phases in the phase diagram of Fig. [fig:honeycombresults] (a). The fist important observation is that for the bound state to emerge, the effective coupling constant needs to be positive. It turns out that at *J*1 < 0 and inside the zigzag domain (*J*3 > 0), $\\lambda\\_{\\hexagon} > 0$ only when *J*2/*J*1 > 1/2, which defines a subregion potent to develop a vestigial phase (shown by darker tone and solid frame). Contrary to that, for the stripe phase (*J*3 < 0) and for both phases at *J*1 > 0, the effective coupling is positive everywhere inside the magnetic domain, so that in the latter cases magnetic ordering should always be preempted by a vestigial phase.\nWe now turn to the discussion of how prominent should be the effect of exciton formation across the honeycomb lattice phase diagram. As suggested by Eq. and common knowledge, enhancement of the density of states at minima of the dispersion is beneficial for the formation of the bound state. Inspection of implies that such enhancement happens in the vicinity of the right boundaries of zigzag and stripe domains. That can be explained by the fact that at these lines the system undergoes phase transition into the spiral ordering, characterized by a non-commensurate value of ordering wavevector ${{\\bf Q}}$. After the transition, the minima of the coupling matrix $J\\_{{{\\bf q}}}$ eigenvalues and, consequently, the minima of the single-magnon dispersion on top of paramagnet drift away from the ${{\\bf M}}$-points, turning the ${{\\bf M}}$-point itself into a saddle point rather than the minimum. Before the transition but near the boundary the effect is manifested by appearing of the soft direction at each ${{\\bf M}}$-point, leading to the increase of the single-magnon density of states. This phenomenon is completely analogous to the one, which happens on the triangular lattice in the vicinity of *J*2/*J*1 = 1 point as described in Section [ssec:triangular]. To further illustrate this statement and make it quantitative, we plot contours, at which our perturbative theory predicts the binding energy (and hence the width of the Potts-nematic phase) to be of the order 10− 1 (solid line), 10− 2 (dashed line), and 10− 3 (dot-dashed line) in the units of ∣*J*1∣ [Fig. [fig:honeycombresults] (a)]. As can be seen in the picture, the most perceptible effect appears near the aforementioned boundary of the domains.\n![ The width of the Potts-nematic phase (D_c^* - D_c)/J_1 obtained from exciton binding energy as a function of interaction strength \\eta = J_i^z / J_i for the honeycomb lattice model. Solid line shows numerical solution, while the dashed one is for the analytical perturbative result, valid at \\eta \\ll 1. The prefactor in the latter was fitted. ](Honeycomb_comparison \"fig:\") [fig:honeycombcomparison]\nFCC lattice magnets [ssec:fcc]\n------------------------------\n![ Binding energy of the Potts-nematic excitons on the face-centered cubic (FCC) lattice as a function of interaction strength parameter \\eta = J_i^z / J_i. Note that the bound state exists only at \\eta > \\eta_c due to the peculiarity of a three-dimensional problem. Dashed lines demonstrate the asymptotics obtained by fitting the data with Eq. .](FCC_lattice \"fig:\") [fig:fcclattice]\nIn the final part of this section we turn to an application of our theory to a model in three dimensions: the face-centric cubic (FCC) lattice. Our goal is to demonstrate that the analysis developed in Sec. [sec:theory] is not limited to two dimensions, but extends to three dimensions as well. The three-dimensional case is different in one important way, however. Since it is well known that bound states do not exist for arbitrarily shallow potential in three dimensions, making it a threshold phenomenon, we expect the two-magnon bound states to occur only for finite interaction between the magnons. In particular, this means that Eq. does not apply to the three-dimensional case. Below we describe how the analysis of Sec. [ssec:wc] can be adapted to the case of the FCC lattice.\nWe consider a *J*1-*J*2 model on the FCC lattice with antiferromagnetic *J*1 and small ferromagnetic *J*2. For a simple nearest neighbor FCC antiferromagnet (*J*2 = 0), the Fourier transformed exchange coupling $J\\_{{{\\bf q}}}$ has minima on lines in momentum space. The lines of minima reflects the frustration of the FFC lattice and are defined by the constraint (*q**x*, *q**y*) = (2*π*, 0) with arbitrary *q**z* (all symmetry equivalent constraints also define minima). Introducing ferromagnetic coupling between the next-nearest neighbors (*J*2 < 0) lifts that degeneracy and results in minima of $J\\_{{{\\bf q}}}$ at the three wave vectors ${{\\bf Q}}\\_1= (2\\pi,0,0)$, ${{\\bf Q}}\\_2=(0,2\\pi,0)$, and ${{\\bf Q}}\\_3=(0,0,2\\pi)$. These wave vectors satisfy Eq. . The fact that there are three equivalent minima points implies that $\\mathds{Z}\\_3$ symmetry is broken in the ordered phase with the ordering wavevector provided by one of these minima. Such ordering is formed by stripes, which now can take one of the three directions parallel to the principal axes in the three-dimensional space. We thus expect that in this case also one might expect intermediate Potts-nematic state at a range of single-ion anisotropy parameter *D* between the ordered and paramagnetic phases.\nWe start with the demonstration of the solution for the binding energy *ɛ**b*, obtained by numerical integration in. It is shown in Fig. [fig:fcclattice], where *ɛ**b* is plotted against the interaction strength *η* = *η**i* = *J**i**z*/*J**i* for a number of *J*2/*J*1 values. As predicted above, we observe that the bound state is formed only at *η* > *η**c*, which is the main distinction with respect to the two-dimensional case. At $\\eta \\gtrsim \\eta\\_c$, the binding energy has approximately quadratic dependence on *η* − *η**c*. A more accurate formula contains a logarithmic correction and is discussed below. We plot it in dashed in Fig. [fig:fcclattice] and observe that it captures well the numerical dependence at small *η* − *η**c* (with fitted parameters).\nHaving identified the threshold nature of the bound state formation, we studied the dependence of the critical interaction *η**c* on the spin-spin couplings (Fig. [fig:criticalzeta]). The resulting dependence is monotonous and drops to zero in a sharp nonanalytical way at *J*2 → 0. This behavior is explained by the fact that at *J*2 = 0 the single-magnon dispersion reaches its minimum on a set of lines, which effectively renders the problem two-dimensional. From a quantitative perspective, it is interesting that for isotropic Heisenberg model with *η* = *J**i**z*/*J**i* = 1, the bound states exist only at $|J\\_2| / J\\_1 \\lesssim 0.1$.\n![Critical values of the interaction parameter \\eta_c on the face-centered cubic (FCC) lattice as a function of spin couplings ratio |J_2| / J_1. Formation of Potts-nematic excitons and, hence, the vestigial phase occurs only at \\eta = J_i^z / J_i > \\eta_c due to the corresponding property of the three-dimensional potential well problem. ](Critical_Zeta \"fig:\") [fig:criticalzeta]\nWe proceed with the weak coupling theory for the three-dimensional case. Naive attempt at taking the integral in using expansion in the vicinity of ${{\\bf Q}}\\_i$ points leads to a linearly divergent integral due to increased phase volume in three dimensions. The common resolution would be to subtract from the integrand of its value at *ɛ**b* = 0 (to be taken numerically over the whole Brillouin zone) and then proceed with the regularized integral. In the regime of small binding energies *ɛ**b* that leads to the following implicit expression for *ɛ**b*(*η*): -1 = c-1 - B. [3Dsolution] where *B* is a constant determined by the density of states in the vicinity of *M*-points. The result indeed signifies that the bound state exists only at *η* > *η**c*. However, we found that Eq. is suitable to describe the asymptotics of the exact solution only at very small values of *ɛ**b*, which correspond to a much smaller scale than the one in Fig. [fig:fcclattice]. This observation is explained by the following considerations. As discussed in the beginning of this subsection, at ∣*J*2∣ → 0 the dispersion acquires a very soft direction near the minimum, which eventually transforms into a line of degenerate minima. This process leads to the increase of the single-magnon density of states, which, as discussed in previous sections, enhances the effect. Given the smallness of relevant *J*2 (in Fig. [fig:fcclattice] $|J\\_2| / J\\_1 \\lesssim 0.1$) that means that at moderate *ɛ**b* the value of is not captured well by the quadratic expansion. Instead, one can expect logarithmical corrections to due to the nascent crossover to 2D. Indeed, we find that a corrected formula -1 = c-1 - B [3Dsolutionwithlog] captures the numerical dependence much better (see dashed lines in Fig. [fig:fcclattice]).\nTo conclude, our consideration predicts that the effect of exciton formation and appearance of the vestigial phase is weaker in three dimensions for nonspecific values of the spin-spin interactions. However, in the vicinity of special points in the phase diagram (*J*2 = 0 in our case), the effect becomes more pronounced due to the enhanced density of states of relevant magnons.\nConnection to experiment [sec:expts]\n====================================\nThe analysis presented in the previous section indicates that the instability towards a Potts-nematic phase is rather general and does not depend on the specific realization of the proposed model on any specific lattice. This generality motivates the important question whether material realizations of Potts-nematic quantum magnets exist among experimentally known material systems. In pursuit of this question, in this section we highlight and discuss a number of promising candidate materials to which our theory applies or relates.\nFor our theory to directly apply, candidate materials should satisfy three requirements. First, the interactions between the spin must be such that magnetic order with broken $\\mathds{Z}\\_3$ crystal symmetry is favored. This implies a tacit constraint on the symmetry of the crystal structure and limits the search for materials to hexagonal, trigonal, tetrahedral, or octahedral systems. In practice, the search should be focused on materials or families of materials in which rotation symmetry broken magnetic order is observed, which is clearest signature of the nature of the exchange couplings. Examples will be discussed below. The second requirement is that the candidate materials exhibit significant easy-plane single-ion anisotropy, which implies that spin-orbit coupling is important. In our model, the single-ion anisotropy controls the (quantum phase) transition between the paramagnet and the ordered state, and therefore would be responsible for driving a magnetically ordered state into a putative Potts-nematic phase. Closely related to this second requirement is the third requirement: the quantum spins must be *S* = 1 degrees of freedom, such that in the limit of large single-ion anisotropy the ground state is a perfect quantum paramagnet product state. In principle other integer-spin systems may be considered, such as *S* = 2, since these also admit a quantum paramagnetic ground state.\nGiven these requirements, one of the more promising material classes is the family of transition-metal phosphorous trichalcogenides (*M*P*X*3). These are layered van der Waals materials in which the transition metal *M* = (Fe, Ni, Mn, Co) sites are magnetic and form a honeycomb lattice within each layer. The *X* are occupied by chalcogen atoms, typically sulfur (S) or selenium (Se). The *M*P*X*3 materials have attracted much attention recently as a versatile platform for exploring intrinsic two-dimensional magnetism in few-layer or monolayer systems. While some members of this class show antiferromagnetic Neél order, others were found to realize antiferromagnetic zigzag order—precisely the type of order discussed in Sec. [ssec:honeycomb]. Furthermore, strong easy-axis or easy-plane behavior was observed, such as Ising-like easy-axis behavior in FePS3 and XY-like easy-plane behavior in NiPS3 . To explain the observed magnetic orderings across the family of *M*P*X*3 compounds a *J*1–*J*2–*J*3 honeycomb lattice spin model with single-ion anisotropy was proposed, which, quite remarkably, is identical to the honeycomb lattice model studied in this work. It is for this reason that the trichalcogenide magnets offer a particularly compelling venue for Potts-nematicity associated with rotation symmetry broken magnetic order.\nWithin the family of *M*P*X*3 materials NiPS3 deserves special attention. Not only do the ordered moments of the observed zigzag phase lie in the plane , suggesting easy-plane anisotropy with *D* > 0, but the Ni2 + ions also give rise to *S* = 1 spins on the honeycomb sites. This implies that NiPS3 provides a realization of the model studied in this work.\nIt is important to point out one caveat concerning the bulk *M*P*X*3 materials, in particular concerning the stacking of the constituent layers in the bulk structures. In the sulfur-based compounds *M*PS3 with *M* = (Fe,Ni,Mn) the layers are stacked in monoclinic fashion, giving rise to space group *C*2/*m*. This means that the threefold rotation symmetry of the constituent honeycomb layers is already broken in the bulk structure, thus precluding the spontaneous breaking of rotation symmetry by the magnetic interactions. The coupling between the layers is believed to be very weak, however, and may be weak enough to prevent a selection of the nematic director by the monoclinic stacking direction. Indeed, encouraging evidence for this has recently been reported by linear and non-linear optical measurements, which have found three different magnetic domains below the Neél temperature within one homogeneous structural domain.\nA second material of interest is the iron-intercalated transition metal dichalcogenide (TMD) Fe*x*NbS2 with *x* = 1/3, which is a member of the class of intercalated TMDs *M**x**T**A*2 . Here *M* is a transition metal, *T*=(Ta, Nb), and *A*=(S, Se). The magnetic Fe atoms are intercalated between the adjacent van der Waals layers of 2*H*-NbS2 and form a triangular lattice with $\\sqrt{3}\\times\\sqrt{3}$ periodicity. The two neighboring triangular lattice layers of Fe atoms that make up the unit cell (and are located at *c* = 1/4 and *c* = 3/4) are shifted with respect to each other, effectively forming a honeycomb structure. The resulting space group of the intercalated structure is *P*6322. Importantly, both optical birefringence and detailed neutron crystal diffraction measurements have reported observation of rotational symmetry broken stripe order. Here, the two triangular lattice layers each form the ordering pattern shown in Fig. [fig:triangular] and together form the honeycomb lattice stripy order shown in Fig. [fig:honeycombresults]. As such, the observed magnetic order falls in the class of orderings considered in our work. (Note that Ref. reported great sensitivity of the ordering pattern to small changes in the intercalation parameter *x* close to the commensurate value 1/3.) Furthermore, the Fe2 + ions give rise to *S* = 2 spins. To understand the observed magnetic orderings, recent theory works proposed a Heisenberg spin with extended exchange interactions and single-ion anisotropy , essentially equivalent to our honeycomb lattice model. An open question is the nature of the single-ion anisotropy, however, with initial estimates suggesting it might be negative, thus implying an easy-axis anisotropy .\nWe conclude this section with a general remark. To obtain the nematic phase at low temperatures one needs a suitable tuning knob to weaken and eventually destroy the magnetic order. This can be provided by varying strain and pressure in the system. An alternative possibility is the formation of Potts-nematic phase at the thermal phase transition as function of temperature, rather than the quantum phase transition studied here. A theory of thermal transition to the Potts-nematic phase will be reported elsewhere.\nDiscussion and Conclusion\n=========================\nThe central result of this paper is the demonstration that a magnetically disordered Potts-nematic phase can exist as an intermediate phase at the quantum phase transition from a quantum paramagnet to a $\\mathds{Z}\\_3$ threefold rotation symmetry broken antiferromagnet. Since such a phase is magnetically disordered, spin-rotation symmetry and time-reversal symmetry are preserved, yet a $\\mathds{Z}\\_3$ crystal rotation symmetry is broken, thus giving rising to nematic order. The relevant order parameter for the nematic phase is expressed in terms of bilinears of the primary magnetic degrees of freedom, i.e., the spin variables.\nOur conclusion is inferred from the analysis of two-magnon excitations in the paramagnetic state of an *S* = 1 quantum spin model, in which the transition between the paramagnet and the magnetic state is regulated by the value of easy-plane single-ion anisotropy. With both numerical and analytical methods we established the existence of two-magnon bound states—“excitons”—with the angular momentum *l* = 2. That makes the two-particle gap close slightly before the single-particle gap as the transition is approached, leading to the proliferation of excitons. The latter can be described by a Jastrow ansatz. We derived a general equation for the excitonic bound state in terms of lattice harmonics [Eqs. and ] and derived its analytical solution for the weak-coupling regime. The latter expresses the binding energy and, hence, the width of the vestigial phase, as a function of magnon density of states at the dispersion minima (in the paramagnetic state) and the effective interaction strength, which is a linear combination of the spin-spin coupling constants. Analytical perturbation theory is corroborated by a detailed numerical analysis for the cases of triangular, honeycomb, and face-centered cubic lattices (see Figs. [fig:triangularresults]–[fig:criticalzeta]).\nIt is important to describe the relation between the character of the vestigial phase determined by the two-magnon wavefunction and the properties of the single-magnon dispersion above the paramagnetic phase. In the present paper we assumed that the dispersion has three degenerate minima in the Brillouin zone (*Q*1, *Q*2, and *Q*3 momenta, satisfying 2*Q**i* ∼ 0), which are related to each other by an in-plane rotation for the cases of triangular and honeycomb lattice. Under these assumptions, perturbative expression for the effective coupling constants in different angular momenta channels (*l*) nullifies for *l* = 1 and *l* = 3 leaving only Potts-nematic channel with *l* = 2 and unphysical symmetric wavefunction with *l* = 0. Similarly, we checked that in the setup of Ref. , in which the dispersion had two inequivalent minima, the only physical channel arising in the perturbation theory is the chiral one (*l* = 3). This relation provides a universal scheme allowing to determine the nature of the vestigial phase given the single-particle spectrum. In the numerical calculation we observed that the effective couplings in the ‘wrong’ angular momentum channels are nonzero, but they are small and thus irrelevant deep inside the domain with a given magnetic ordering [see Fig. [fig:triangularresults](b)].\nWhile in the present work we focused on the specific case of an *S* = 1 spin model, our analysis straightforwardly generalizes to any integer spin system. Indeed, the ground state at large values of the single-ion anisotropy in that case is still a perfect paramagnet, given by a product of *S**z* = 0 states on each site. A spin flip to a state with *S**z* = *s* on any site costs an energy *D**s*2 and therefore, the energetically lowest excitations in this case still correspond to *s* = ± 1. It means that for the purpose of determining the instability of the paramagnet marked by a gap closure, one can restrict the consideration to these two branches only, which makes all further analysis the same as in the *S* = 1 scenario. In contrast, we expect that half-integer models behave qualitatively different, since they do not admit a paramagnetic product state.\nThe prediction of the formation of Potts-nematic phase at low temperatures in the vicinity of the quantum phase transition poses a natural question: does there exist a similar vestigial phase at the thermal phase transition between a striped antiferromagnet and the symmetry high-temperature phase? This question will be addressed in a separate paper.\nWe conclude by emphasizing that there exist a number of different materials which provide promising venues for experimental observation of Potts-nematicity in quantum magnets. As discussed in Sec. [sec:expts], given its currently established magnetic properties, the *M*P*X*3 family of transition-metal trichalcogenides is of particular interest. Within this class of materials NiPS3 is most promising, as it is likely to provide a realization of the model considered in this work. Another possible candidate material is the Fe-intercalated dichalcogenide NbS2.\nAcknowledgements\n================\nWe gratefully acknowledge useful discussions with R. M. Fernandes, A. V. Chubukov, and L. Wu. This research was supported by the National Science Foundation Award No. DMR-2144352.\nGeneralization to non-Bravais lattices [app:gen]\n================================================\nHere we describe the generalization of the two-magnon problem to the case of a non-Bravais lattice, which requires a proper account for the sublattice degree of freedom. We start we the expression for the *X**X**Z* Hamiltonian of the form\n \n$$\\begin{gathered}\n \\mathcal{H} = \\frac{1}{2} \\sum\\_{{{\\bf r}}, {\\boldsymbol{\\delta} }, \\alpha, \\beta} (J\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha\\beta} (S\\_{{{\\bf r}}\\alpha}^{x}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{x} + S\\_{{{\\bf r}}\\alpha}^{y}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{y}) \\\\\n+ \\frac{1}{2} \\sum\\_{{{\\bf r}}, {\\boldsymbol{\\delta} }, \\alpha, \\beta} (J^z\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha\\beta} S\\_{{{\\bf r}}\\alpha}^{z}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{z},\\end{gathered}$$ \nwhere ${{\\bf r}}$ and ${{\\bf r}}+{\\boldsymbol{\\delta} }$ are Bravais lattice vectors and *α*, *β* label the spins within the unit cell, i.e., the sublattice degrees of freedom. Two-magnon wavefunction $\\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}' \\beta}$ also carries sublattice indices. For the clarity of the narration, we will consider zero center-of-mass momentum excitons (generalization to the opposite case is straightforward). Then, the Fourier transform of the wavefunction reads **r**, ; **r**’ = **q** (**q**) ei**q**(**r**+ - **r**’ - ), where ${\\boldsymbol{\\ell} }\\_{\\alpha}$ determines the position of *α*-sublattice site inside a unit cell.\nWe proceed with projecting the Schroedinger equation to the subspace of two opposite-spin magnons as was done in the main part [Eq. ]. It is convenient to interpret $\\psi\\_{\\alpha \\beta} ({{\\bf q}})$ as a matrix and write the variational Schroedinger equation in the matrix form: [Schroedingermultiband] (E - 2D) (**q**) - **q** (**q**) - (**q**) **q**= []. Here $\\hat{J}\\_{{{\\bf q}}}$ is the Fourier transform of the interaction matrix [Jqmatrixdefinition] (J**q**) = (J) ei**q**( + - ), which reduces to the single-magnon energy $\\varepsilon({{\\bf q}})$ in the Bravais-lattice case [see paragraph under ]. Int[*ψ*] is the interaction term appearing due to magnon-magnon interaction, provided by the *S**z**S**z* part of the spin coupling. It has the following matrix elements: ([]) = - (Jz) B,, ei**q**( + - ), where $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }} = \\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}+ {\\boldsymbol{\\delta} }, \\beta}$ is the real-space wavefunction amplitude at two magnons residing on *α* and *β* sublattices in the unit cells separated by the Bravais lattice vector ${\\boldsymbol{\\delta} }$. That is a correct definition because $\\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}+ {\\boldsymbol{\\delta} }, \\beta}$ does not depend on ${{\\bf r}}$ in the case of zero center-of-mass momentum. Importantly, contains only the elements $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$ which are multiplied by nonzero spin-spin couplings $(J\\_{{\\boldsymbol{\\delta} }}^z)\\_{\\alpha\\beta}$. Thus, for a model with finite-range interaction, there is a finite number of $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$ components. For instance, on a *J*1 − *J*2 − *J*3 model on a honeycomb lattice for each of the two choices for *α*, there are three nearest neighbors, six next-nearest, and three third-nearest ones, constituting 12 elements for one *α* and 24 total components of $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$. To proceed, we need to express the latter via the momentum-space wavefunction: [Bdefinition] B,, = e-i (), where we introduced the following notation for brevity: = + -.\nNext step is to diagonalize the l.h.s. of by expressing $\\psi\\_{\\alpha \\beta} ({{\\bf q}})$ via the eigenstates $e\\_{a}({{\\bf q}})$ of $\\hat{J}\\_{{{\\bf q}}}$ matrix: [psithrougheigenvectors] (**q**) = ab cab (**q**) ea(**q**) ebT(-**q**), **q** ea(**q**) = **q**(a) ea(**q**), where *a* enumerates the Bloch bands of magnons. Then, takes the form [Scroedingermultibanddiagonalized] ab cab ea(**q**) ebT(-**q**) = [], Assuming that $e\\_{a}({{\\bf q}})$ constitute an orthonormal basis for each ${{\\bf q}}$ and multiplying by $e\\_{a'}^{\\dagger}({{\\bf q}})$ from the left and by $e\\_{b'}({{\\bf q}})$ from the right, we get *c**a**b*: [ccoefs] cab(**q**) = -,, . We then substitute into and further into to find the self-consistency equation [equationforBnonBravais] B,, = ’, ’, ’,, ; ’, ’, ’ B’, ’, ’, where [Mmultiband],, ; ’, ’, ’ = **q**,a,b, where $P^{(a)} ({{\\bf q}}) = e\\_{{{\\bf q}}}^{(a)} [e\\_{{{\\bf q}}}^{(a)}]^{\\dagger}$ is the projector to the *a*th band acting in the sublattice space, and *E* = 0 value was taken since we are looking for the instability, which arises when the excitons become gapless.\nCondition for to have solution is given by the same equation detM − 1 = 0 as in the main text [Eq. ]. However, expression for $\\mathcal M$ is slightly more involved and includes projectors to magnon bands and summation over the latter. Nevertheless, for an isotropic model one can still block diagonalize $\\mathcal M$ with the help of lattice harmonics and proceed with the numerical computation. In Section [ssec:honeycomb] we demonstrate how that works for the case of the honeycomb lattice.\nWeak coupling theory for a non-Bravais lattice is regulated by the parameters $(\\eta\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha \\beta} = (J\\_{{\\boldsymbol{\\delta} }}^z)\\_{\\alpha \\beta} / (J\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha \\beta}$, which determine interaction between the single-magnon excitations. In an isotropic model the couplings depend only on the relative distance $\\tilde{{\\boldsymbol{\\delta} }}$ between the sites: (J) = J(| + - |). For simplicity, we would also assume that *η* is the same for all couplings so that the theory has just one perturbation parameter. At small *η*, which corresponds to a shallow well problem, common knowledge is to expect the exciton binding energy to be small, while its size to be large in real space and small in reciprocal space. That allows to consider only the low-lying excitations in the summation in, which boils down to considering only the lower band and expanding the dispersions $\\varepsilon\\_{\\bm{q}}^{(a)}$ in the vicinity of their minima, which we denote as ${{\\bf Q}}\\_i$. Moreover, we will evaluate the numerator of strictly at ${{\\bf Q}}\\_i$ points. Thus, we rewrite the summation in the vicinity of each minimum in terms of $\\bm{p} = {{\\bf q}}- \\bm{Q}\\_i$. [Mmultibandlowest],, ; ’, ’, ’ = i, where $P\\_{\\alpha \\beta}^{{{\\bf Q}}\\_i} = P\\_{\\alpha \\beta}^{(\\text{low})} ({{\\bf Q}}\\_i)$ are projectors onto the lowest magnon band, binding energy $\\varepsilon\\_b = 2D + \\varepsilon\\_{{{\\bf Q}}\\_i} + \\varepsilon\\_{-{{\\bf Q}}\\_i}$, and (Ai)mn =. qmqn |**Q**i is the expansion of the magnon dispersion in the vicinity of the global minima ${{\\bf Q}}\\_i$. Eq. has the same general form as the analogous equation for a Bravais lattice. For a quadratic magnon band minima one can proceed with integrating over $\\xi = {{\\bf p}}^T A\\_i {{\\bf p}}$ as was done in. The corresponding density of states $\\nu\\_{{{\\bf Q}}\\_i}$ should contain only the contribution of the lowest band at ${{\\bf Q}}\\_i$ point. Such integration leads to the same functional dependence as in with some value of *λ* and density of states to be understood as explained above. For the case of the honeycomb lattice, which has dispersion minima at M-points, these quantities are given by and. We discuss the result of the weak-coupling approach for this case in detail in Section [ssec:honeycomb].\n---\n1. J. Strockoz and D. S. Antonenko contributed equally to this work.[↩](#fnref1)\n2. If one considers ansatz, in which $\\ket{\\Omega}$ mixes with the two-magnon states, the resulting wavefunction will be dominated by the $\\ket{\\Omega}$ component as it is lower in energy by at least 2*D* compared to the two-magnon states. The solution can be interpreted as the ground state, corrected by the interaction with the two-magnon states.[↩](#fnref2)\n3. The six nearest neighbor and next-nearest neighbor vectors are related by sixfold rotation; some are shown in Fig. [fig:triangular].[↩](#fnref3)\n\nExcitonic instability towards a Potts-nematic quantum paramagnet\n================================================================\n[1](#fn1)\nMagnetic frustration can lead to peculiar magnetic orderings that break a discrete symmetry of the lattice in addition to the fundamental magnetic symmetries (i.e., spin rotation invariance and time-reversal symmetry). In this work, we focus on frustrated quantum magnets and study the nature of the quantum phase transition between a paramagnet and a magnetically ordered state with broken threefold ($\\mathds{Z}\\_3$) crystal rotation symmetry. We predict the transition to happen in two stages, giving rise to an intermediate nematic phase in which rotation symmetry is broken but the system remains magnetically disordered. The nematic transition is described by the three-state Potts model. This prediction is based on an analysis of bound states formed from two-magnon excitations in the paramagnet, which become gapless while single-magnon excitations remain gapped. By considering three different lattice models, we demonstrate a generic instability towards two-magnon bound state formation in the Potts-nematic nematic channel. We present both numerical results and a general analytical perturbative formula for the bound state binding energy similar to BCS theory. We further discuss a number of different materials that realize key features of the model considered, and thus provide promising venues for possible experimental observation.\nIntroduction [sec:introduction]\n===============================\nThe study of ordering phenomena is one of the central pillars of condensed matter physics. Landau’s theory of phase transitions provides a general framework for understanding the structure of ordered phases and the nature of the transitions between different phases. The order parameter is a fundamental concept in the study of phase transitions and characterizes the ordered phase in terms of its symmetry. In the simplest and most common cases, the ordering transition is associated with the breaking of a fundamental symmetry, such as translation symmetry in the case of crystalline order; charge conservation in the case of superconductivity; and time-reversal combined with spin-rotation symmetry in the case of magnetic order. Ordering transitions with a much richer structure arise when, in addition to the fundamental symmetry, one or more secondary symmetries are broken in the ordered state, such as spatial rotation or inversion symmetry. This occurs, for instance, in unconventional non-*s*-wave superconductors and superfluids, where the internal structure of the Cooper pairs (i.e., their spin and orbital angular momentum) can lead to spontaneous spatial anisotropy associated with broken rotation symmetry . Similar phenomena can occur in magnets when the exchange interactions are (strongly) frustrated, which prohibits the formation of simple ferromagnetic or antiferromagnetic Néel order. Whereas the ferromagnet and antiferromagnet do not break the symmetries of the crystal lattice, the more complicated magnetic configurations that arise as a result of frustration generally do break one or more spatial symmetries.\nIn this paper, we study a particular class of such frustrated magnets: magnets which spontaneously break a discrete $\\mathds{Z}\\_3$ (i.e., threefold) crystal rotation symmetry in the ordered state. Such phases naturally arise in simple Heisenberg spin models with frustrated exchange interactions, for instance in two-dimensional (or layered quasi-two-dimensional) magnets with triangular or honeycomb lattice geometry. The phase diagrams of triangular and honeycomb lattice spin models with further neighbor exchange couplings are known to include collinear single-*Q* magnetic states, characterized by staggered ferromagnetic stripes (see Fig. [fig:triangular]), which are stabilized by the “order by disorder” mechanism. Since a single ordering wave vector is spontaneously selected from a set of three wave vectors related by crystal rotation symmetry, the latter symmetry is spontaneously broken in these single-*Q* states. This may be compared to Heisenberg antiferromagnets with tetragonal symmetry, such as frustrated square lattice magnets, which can support single-*Q* stripe order described by either one of two wave vectors (*π*, 0) and (0, *π*). Such single-*Q* phases break a $\\mathds{Z}\\_2$ Ising symmetry corresponding to the two possible orientations of the stripes, which in two dimensions implies a finite temperature Ising transition to an Ising nematic phase with no magnetic order. In three dimensions the Ising and magnetic transitions may still occur at different temperatures, in which case a so-called vestigial Ising nematic phase arises . The emergence of Ising nematicity is well-known example of the rich ordering phenomenology that can occur in complex magnets with multiple broken symmetries .\nIn contrast to the Ising case, the breaking of a discrete $\\mathds{Z}\\_3$ symmetry is governed by the three-state Potts model and thus provides a manifestly different—and much less studied—window into the nature of magnetic ordering in unconventional complex magnets . The purpose of this paper is to explore the nature of the phase transition in magnets that break an additional $\\mathds{Z}\\_3$ symmetry, such as a threefold crystal rotation, and thus belong to the three-state Potts universality class. As far as the thermal phase transition is concerned, general arguments suggest that, similar to the Ising case, a finite temperate transition to a Potts-nematic phase is possible .\nRather than the thermal phase transition, our goal here is to study the nature of the zero temperature quantum phase transition between a quantum paramagnet and a magnetically ordered phase which breaks a discrete $\\mathds{Z}\\_3$ crystal rotation symmetry. The question we specifically seek to answer is whether an intermediate Potts-nematic phase, in which the system remains paramagnetic but breaks the $\\mathds{Z}\\_3$ rotation symmetry, can exist.\nTo address this question, we consider a minimal XXZ model of *S* = 1 quantum spins with single-ion anisotropy. In this minimal model, defined in Eq., the single-ion anisotropy *D* controls the transition between the quantum paramagnet realized for sufficiently large *D* and the magnetically ordered state. The latter is controlled by the frustrated exchange interactions, which are chosen such that, at the classical level, the energy is minimized by a set of degenerate ordering wave vectors related by threefold rotation symmetry. Following the approach of Ref. , we start from the quantum paramagnet and study its instability towards the formation of two-magnon bound states as *D* is lowered, which may occur due to an attractive interaction between single-magnon excitations. If the energy gap of two-magnon bound states closes at a value of *D* larger than the value at which the single-magnon gap closes, this indicates an instability towards bound state formation. Furthermore, if the internal structure of the bound state solution, defined by the relative angular momentum of the magnons, breaks the $\\mathds{Z}\\_3$ rotation symmetry, this suggests that proliferation of two-magnon excitations gives rise to an intermediate Potts-nematic quantum paramagnetic phase. Such analysis is similar in spirit to the classic Cooper problem in the context of superconductivity , which addresses the instability of the Fermi sea to the formation of two-particle bound states (Cooper pairs).\nWe apply this analysis to three different models of frustrated quantum magnets: the triangular lattice, the honeycomb lattice, and the three-dimensional face-centered cubic (FCC) lattice. In all three cases we focus on a part of the phase diagram where rotation symmetry broken single-*Q* magnetic order is favored. We obtain a matrix Schrödinger equation for the two-magnon bound states, show that it decouples in channels of distinct symmetry quantum numbers, and solve it numerically. We demonstrate that when the exchange interactions favor magnetic order broken rotation symmetry, generically two-magnon bound states have largest (positive) binding energy in the nematic channel, suggesting a generic instability towards a Potts-nematic phase in this class of quantum magnets. The numerical analysis is supplemented by a perturbative analytical approach, is independent of the underlying lattice structure and in two dimensions yields an expression for the width of the intermediate Potts-nematic phase similar to BCS theory. In particular, as expressed in Eq. , the binding energy, which is proportional to the width of the nematic phase, depends on the single-magnon density of states and the effective coupling constant in the nematic channel.\n![Triangular lattice model. (a) Triangular lattice Heisenberg model with exchange couplings J_1 between nearest neighbors (exemplified by red arrows) and J_2 between next-nearest neighbor (blue arrow). When both J_{1,2}>0 and 1/8 < J_2/J_1 <1, the minima of the Fourier-transformed exchange coupling matrix J_{{\\bf q}} are located at the three M points in the Brillouine zone with the wave vectors Q_{1,2,3}, indicated by blue dots on the right. The generating lattice vectors {{\\bf a}}_{1,2} are indicated in red. (b) Three possible realizations of triangular lattice single-Q ordering corresponding to the three M-point wave vectors. Black (white) solid dots represent classical moments pointing out of (into) the page. ](triangular \"fig:\") [fig:triangular]\nTo make a connection with possible candidate materials, we identified a number of compounds that can host the proposed Potts-nematic vestigial phase under certain conditions. We point to two specific material classes, which we propose as a prospective platform to test our predictions. One of the two groups is the family of transition metal trichalcogenides (M*P*X3), which have attracted much attention recently. The other is iron-intercalated transition metal dichalcogenide Fe*x*NbS2, which was recently reported to host a $\\mathds{Z}\\_3$-breaking magnetism, revealed in optical measurements.\nThe paper is organized as follows. In the next Section [sec:Z3breaking] we define our model and give general introduction to the physics of $\\mathds{Z}\\_3$-symmetry breaking magnets. We proceed with the Section [sec:theory], in which we develop the main formalism used in our work and derive its perturbative limit. We then apply this machinery to the specific cases of triangular, honeycomb, and face-centered cubic lattices and present the results in Section [sec:results]. Finally, we propose concrete experimental implementations in Section [sec:expts] and conclude with a discussion (Section [sec:discussion]). The details of the derivation for a non-Bravais lattice are relegated to Appendix [app:gen]. Throughout the paper we consider zero temperature and work in units of the lattice constant (i.e., *a* = 1).\nRotation symmetry breaking in frustrated antiferromagnets [sec:Z3breaking]\n==========================================================================\nThis work is focused on frustrated Heisenberg antiferromagnets described by the general Hamiltonian H = 12 ij Jij **S**i **S**j, [eq:HHB] where *J**i**j* = *J**j**i* are the exchange coupling constants. When the spins $\\{ {{\\bf S}}\\_i \\}$ are considered as classical vectors, the exchange couplings describe interactions between pairs of spins which either favor alignment (*J**i**j* < 0) or anti-alignment (*J**i**j* > 0). Frustration arises when the interactions between different pairs of spins compete and the energetic requirement of perfect alignment or anti-alignment cannot be simultaneously satisfied for all pairs of spins. Frustration may originate from the geometry of the lattice, as is the case for the nearest neighbor triangular lattice antiferromagnet, or from the presence of multiple competing exchange constants, such as in *J*1–*J*2–*J*3 models.\nA standard approach to determining the classical spin configuration with the lowest energy is the Luttinger-Tisza method . It relies on expanding the spin variables in Fourier modes as ${{\\bf S}}\\_{{{\\bf r}}\\alpha} = \\sum\\_{{{\\bf q}}} e^{i {{\\bf q}}\\cdot ({{\\bf r}}+{\\boldsymbol{\\ell} }\\_\\alpha)} {{\\bf S}}\\_{{{\\bf q}}\\alpha}$, where ${\\boldsymbol{\\ell} }\\_\\alpha$ denotes the position of sublattice site *α* with respect to the lattice vector ${{\\bf r}}$, and writing the spin Hamiltonian in terms of the Fourier modes as $H = (N/2) \\sum\\_{{{\\bf q}}} ( J\\_{{{\\bf q}}} )\\_{\\alpha\\beta} {{\\bf S}}^\\*\\_{{{\\bf q}}\\alpha} \\cdot {{\\bf S}}\\_{{{\\bf q}}\\beta}$. Here $ J\\_{{{\\bf q}}}$ is the Fourier transform of the exchange couplings, which in the case of non-Bravais lattices is matrix in sublattice space. Minimization of the classical energy then requires computing the eigenvalues of $J\\_{{{\\bf q}}}$ for all ${{\\bf q}}$ and finding the minimum of lowest eigenvalue branch with respect to ${{\\bf q}}$. The minimum occurs at some specific wave vector ${{\\bf Q}}$ or at a set of symmetry related (i.e., “degenerate”) wave vectors $\\{{{\\bf Q}}\\_i\\}$. The latter case is a consequence and signature of frustration. The eigenmodes corresponding to the minima of $J\\_{{\\bf q}}$ are generally referred to as the optimal Luttinger-Tisza modes and can be used to construct spin states that minimize the magnetic energy. This construction must be subject to the fixed-length constraint, i.e., $|{{\\bf S}}\\_i|^2=S^2$, however, and this implies that, in the case of non-Bravais lattices, it may not be possible to construct ground states from optimal Luttinger-Tisza modes only. In this case, the Luttinger-Tisza method only provides a lower bound the classical energy. Note that in the case of primitive Bravais lattices, however, it is always possible to construct ground states from the optimal Luttinger-Tisza modes.\nIn this work we focus on a particular class of frustrated magnetic systems for which the minima of the eigenvalues of $J\\_{{\\bf q}}$ occur at a set of three symmetry-related wave vectors $\\{{{\\bf Q}}\\_i\\}$ (with *i* = 1, 2, 3) in some part of the phase diagram. The degenerate wave vectors are related by crystal rotation symmetry and satisfy the two additive relations 2 **Q**i 0, **Q**1+**Q**2+**Q**3 0, [eq:Q-relation] Note that the first of these relations implies that the Fourier modes are real. Prominent examples of Heisenberg magnets which exhibit such phases in the phase diagram are the triangular *J*1–*J*2 model , the honeycomb lattice *J*1–*J*2–*J*3 model , and the three-dimensional face-centered cubic (FCC) lattice *J*1–*J*2 model . These three models will be the subject of the remainder of this paper and will be discussed in more detail below. In the case of the triangular and honeycomb lattices, the wave vectors ${{\\bf Q}}\\_i$ correspond to the high symmetry *M* points of the Brillouin zone shown in Fig. [fig:triangular](a), which are clearly related by threefold rotation. The triangular lattice provides the simplest example of the magnetism of interest. The minima of $J\\_{{\\bf q}}$ are located at the three *M* points when *J*1 and *J*2 are both positive (and hence frustrated) and 1/8 < *J*2/*J*1 < 1.\nTo understand the nature of the classical ground state manifold when the optimal Luttinger-Tisza modes derive from the wave vectors $\\{{{\\bf Q}}\\_i\\}$, consider first the simple case of a Bravais lattice. The triangular and FCC lattices are examples. Up to a global spin rotation, the most general parametrization of the Fourier modes ${{\\bf S}}\\_{{{\\bf Q}}\\_i}$ consistent with the fixed length requirement takes the form **S****Q**1= mx **x****S****Q**2= my **y**, **S****Q**3= mz **z**, [eq:Fourier-S] where *S* is the spin length and $\\vec m=(m\\_x,m\\_y,m\\_z)\\equiv (\\sin\\theta\\cos\\phi, \\sin\\theta\\sin\\phi, \\cos\\theta) $ is a unit vector defined in terms of the angles (*θ*, *ϕ*). The angles (*θ*, *ϕ*) parametrize a family of degenerate classical ground states, which can be represented as points on the Bloch sphere. Generically, such points correspond to triple-*Q* magnetic states, but there are special lines and points on the Bloch sphere correspond to double-*Q* and single-*Q* states. In particular, states which correspond to $\\vec m = (1,0,0)$ and its equivalents define single-*Q* configurations.\nA natural and convenient way to resolve the degeneracy of the classical ground states is to consider composite order parameters built from the primary magnetic Fourier modes. The distinct symmetry properties of the composite orders unambiguously expose the symmetries of magnetic configurations and can thus be used to sharply distinguish classical ground states. In the present context, we define the nematic composite order parameter *n* and the chiral order parameter *χ* as n &=& |**S****Q**3|2 + |**S****Q**1|2 + 2|**S****Q**2|2, [eq:n-def] \n&=& **S****Q**1**S****Q**2 **S****Q**3. [eq:chi-def] Here *ω* = *e*2*π**i*/3 is a cubic root of unity. It is clear from these definitions that both *n* and *χ* transform trivially under translations, but *n* is time-reversal even whereas *χ* is time-reversal odd. Most importantly, since nonzero *n* must originate from an unequal contribution of the three wave vectors, it signals the breaking of crystal rotation symmetry in the magnetic state. Substituting the Fourier mode representation into and one finds *n* and *c**h**i* in terms of $\\vec m$ as n & ~& mz2+ mx2 + 2my2, \n&~& mxmymz. This shows that *n* is zero when *m**x*2 = *m**y*2 = *m**z*2, and that finite *χ* requires all components of $\\vec m$ to be nonzero (i.e., noncoplanar order). Note in particular that single-*Q* ordered states are associated with ⟨*n*⟩ ≠ 0.\nSince any energy functional of *n* contains a cubic Potts-anisotropy ∝ *n*3 + (*n*\\*)3, thus giving rise to three distinct nematic ground states, *n* describes Pott-nematic order . As a result, *n* is uniquely associated with the spontaneous breaking of threefold crystal rotation symmetry in the magnetically ordered phase.\nThe continuous degeneracy of the classical ground states parametrized by $\\vec m$ does not survive the effect of fluctuations. This is due to the well known “order-by-disorder” mechanism, which tends to favor collinear spin alignment . For instance, whereas the classical energy is independent of the angles (*θ*, *ϕ*), the energy associated with the quantum zero-point motion of the magnon excitations does depend on the orientation of $\\vec m$, and is lowest for the single-*Q* states such as $\\vec m = (1,0,0)$ .\nInstabilities of a quantum paramagnet [sec:theory]\n==================================================\nIn this section we turn to the central part of this work. As outlined in the introduction, our goal is to determine the instability of a quantum paramagnet towards the formation of nematic magnon bound states. To achieve this, we start from an XXZ model for a system of *S* = 1 spins, introduced in Ref. , which is defined by the Hamiltonian\n \n$$\\begin{gathered}\n\\label{eq:H\\_XXZ}\n\\mathcal{H} = \\frac{1}{2} \\sum\\_{{{\\bf r}}, {\\boldsymbol{\\delta} }} J\\_{{\\boldsymbol{\\delta} }} (S\\_{{{\\bf r}}}^{x}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} }}^{x} + S\\_{{{\\bf r}}}^{y}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} }}^{y} + \\eta\\_{{\\boldsymbol{\\delta} }} S\\_{{{\\bf r}}}^{z}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} }}^{z}) \\\\\n+ D \\sum\\_{{{\\bf r}}}(S\\_{{{\\bf r}}}^{z})^{2}.\\end{gathered}$$ \nThe Hamiltonian is a sum of two terms. The first term is of the general form introduced in and describes exchange couplings between the spins. Specifically, pairs of spins separated by a distance ${{\\bf r}}-{{\\bf r}}' = {\\boldsymbol{\\delta} }$ interact with coupling constant $J\\_{{\\boldsymbol{\\delta} }}$. Note that compared to here we have also introduced an XXZ anisotropy given by $\\eta\\_{{\\boldsymbol{\\delta} }} \\equiv J^z\\_{{\\boldsymbol{\\delta} }}/ J\\_{{\\boldsymbol{\\delta} }}$, which denotes the ratio of the exchange couplings in the *z* direction and the *x**y* plane. This is useful, since the parameter $\\eta\\_{{\\boldsymbol{\\delta} }}$ can be interpreted as controlling the interactions between magnons. As far as the magnetic ground state is concerned, there is no difference with. The second term in describes a single-ion anisotropy of strength *D* > 0.\nFor simplicity, in what follows we focus the discussion on the case of primitive Bravais lattices and leave the straightforward generalization to non-Bravais lattices to Appendix [app:gen], the results of which will be used when applying the analysis to the honeycomb lattice. Since we are interested in the quantum behavior of this model we consider the system at *T* = 0.\nAs argued in Ref. , in the limit where *D* is much larger than the exchange couplings $J\\_{{\\boldsymbol{\\delta} }}$, the ground state is a quantum paramagnet with all spins in the $\\ket{0}\\equiv\\ket{S=1,0}$ state. We denote the paramagnetic ground state as $\\ket{\\Omega} = \\otimes\\_j \\ket{0\\_j}$. Instead, when *D* → 0 one expects a magnetically ordered state, the nature of which is determined by the exchange couplings $J\\_{{\\boldsymbol{\\delta} }}$. Crucially, in what follows we assume that the exchange couplings take values such that $J\\_{{\\bf q}}$ has minima at three degenerate wave vectors $\\{{{\\bf Q}}\\_i\\}$, giving rise to a degenerate magnetic ground manifold at the classical level. This is precisely the part of the phase diagram considered in the previous section. In this case, *D* describes the transition from a quantum paramagnet to a magnetically ordered state which spontaneously breaks a discrete $\\mathds{Z}\\_3$ rotation symmetry. Our goal is to examine this transition and establish whether or not the paramagnetic ground state $\\ket{\\Omega}$ is unstable towards the formation of *nonmagnetic* two-magnon bound states in the vicinity of the magnetic transition.\nSchrödinger equation for magnon pairs [ssec:schroedinger]\n---------------------------------------------------------\nTo address this question, we follow the approach of Ref. and consider a general two-magnon state $\\ket{\\psi}$ defined as |=**r****r**’**r****r**’ |**r**, **r**’, |**r**, **r**’=S**r**+S**r**’-|, [eq:psi-def] where $\\psi\\_{{{\\bf r}}{{\\bf r}}'}$ is the wave function, which is a function of the positions of the two magnons. By projecting Schrödinger equation $\\mathcal H \\ket{\\psi} = E\\ket{\\psi}$ into the two subspace of two-magnon states, we find the equation\n \n$$\\begin{gathered}\n \\label{Scroedinger\\_coordinate\\_space}\n (E - 2D)\\psi\\_{{{\\bf r}}{{\\bf r}}'} = \\\\\n \\sum\\_{{\\boldsymbol{\\delta} }} J\\_{{\\boldsymbol{\\delta} }}(\\psi\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} }, {{\\bf r}}'} + \\psi\\_{{{\\bf r}}, {{\\bf r}}' + {\\boldsymbol{\\delta} }}) - J\\_{{\\boldsymbol{\\delta} }}^z \\, \\psi\\_{{{\\bf r}}{{\\bf r}}'} \\delta\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} }, {{\\bf r}}' },\\end{gathered}$$ \nwhere $J\\_{{\\boldsymbol{\\delta} }}^z \\equiv \\eta\\_{{\\boldsymbol{\\delta} }}J\\_{{\\boldsymbol{\\delta} }}$. Note that, in accordance with, the two-magnon wavefunction is understood to vanish when ${{\\bf r}}'={{\\bf r}}$ (i.e., $\\psi\\_{{{\\bf r}}{{\\bf r}}}=0$) [2](#fn2). The first key observation is that the second term on the right hand side, which is proportional to $J\\_{{\\boldsymbol{\\delta} }}^z$, generically describes an attractive interaction between the magnons, whereas the first term represents a kinetic term for the two magnons. To proceed, we expand the wave function $\\psi\\_{{{\\bf r}}{{\\bf r}}'}$ in Fourier modes as [WignercoordinatesFourier] **r****r**’=**K**, **q**ei**K**(**r**+ **r**’)/2ei**q**(**r**- **r**’) **q**(**K**), where ${{\\bf K}}$ is the momentum conjugate to the center of mass coordinate and ${{\\bf q}}$ is conjugate to the relative coordinate ${{\\bf r}}- {{\\bf r}}'$. Substituting the Fourier mode expansion into the Schrödinger equation yields the equation [integralequation] **q**(**K**) = - Jz ei **q** B, where [magnonpairenergy] E**q**(**K**) = 2 J ei **K** /2 **q**, can be viewed as a kinetic energy of the two magnons, and $B\\_{{\\boldsymbol{\\delta} }} $, defined as B = **p** e -i **p** **p**(**K**), [Bdef] corresponds to the real-space wavefunction as a function of the relative coordinate ${{\\bf r}}- {{\\bf r}}' = {\\boldsymbol{\\delta} }$. Note that in the case of zero center of mass momentum one has $\\mathcal E\\_{{\\bf q}}({{\\bf K}}=0) \\equiv \\mathcal E\\_{{\\bf q}}= 2\\varepsilon\\_{{\\bf q}}$, where $\\varepsilon\\_{{\\bf q}}= \\sum\\_{{\\boldsymbol{\\delta} }} J\\_{{\\boldsymbol{\\delta} }} \\cos {{\\bf q}}\\cdot {\\boldsymbol{\\delta} }$ is the single-magnon dispersion. To solve Eq. , we divide by the kernel $E - 2 D - \\mathcal E\\_{{\\bf q}}({{\\bf K}}) $ and use the definition of $B\\_{{\\boldsymbol{\\delta} }}$ to obtain a matrix equation given by B = ’ ’ B ’, [matrixequation] where the matrix $\\mathcal{M}\\_{{\\boldsymbol{\\delta} }{\\boldsymbol{\\delta} }'}$ is defined as [Mdefinition] ’ = **q** Solutions to the Schrödinger equation are then determined by the condition (-) = 0. [detzero] In our case, we seek solutions corresponding to zero energy (*E* = 0), which means that the energy for creating two-magnon excitations vanishes. The general strategy for solving after setting *E* = 0 is to diagonalize M and determine when the eigenvalues of M − 1 vanish. This is done as a function of *D* and the first eigenvalue which vanishes as *D* decreases corresponds to the solution of interest.\nInspection of the structure of M reveals that it is possible to bring it in block diagonal form by making use of the symmetry properties of the underlying lattice and the center of mass momentum ${{\\bf K}}$. To illustrate how group theory machinery can be employed to block diagonalize M, it is useful to consider the example of the triangular lattice *J*1–*J*2 model discussed in the previous section.\nThe triangular lattice has six nearest neighbor vectors ${{\\bf a}}\\_j$ and six next-nearest neighbor vectors ${{\\bf a}}'\\_j$ [3](#fn3), which we label *j* = 0, …, 5. As a result, ${\\boldsymbol{\\delta} },{\\boldsymbol{\\delta} }'$ in and take values in this set of twelve vectors. Let us furthermore consider the case ${{\\bf K}}=0$. The matrix $\\mathcal M$ is then block diagonalized by a unitary matrix $\\mathcal U$ of the form U =\nU & \n&U\n, where the matrix elements of *U* are Ujl = ei 2l j /6/. Here *U* transforms from the basis of (next)-nearest neighbor vectors to a basis of angular momentum labeled by *l* = − 2, …, 3. Since the triangular lattice (as well as the center of mass momentum ${{\\bf K}}=\\Gamma$) has full hexagonal symmetry, different angular momentum channels cannot mix, implying that $\\mathcal U^\\dagger \\mathcal M \\mathcal U $ is block diagonal. The blocks therefore correspond to distinct angular momentum channels, which we denote M*l* and are given by [Mblock] l= **q**\nJz1 | fl**q**|2 & Jz2 fl \\***q**gl**q** \nJz1 gl \\***q**fl**q**&Jz2 | gl**q**|2\n. Here $f^l\\_{{\\bf q}}$ and $g^l\\_{{\\bf q}}$ are the nearest neighbor and next-nearest neighbor symmetry-adapted lattice harmonics of the triangular lattice and are defined as fl**q**= j Ujl ei **q****a**j, gl**q**= j Ujl ei **q****a**’j. [harmonics] It is worth noting that the matrix in can be decomposed as\nJz1 | fl**q**|2 & Jz2 fl \\***q**gl**q** \nJz1 gl \\***q**fl**q**&Jz2 | gl**q**|2\n=\nfl \\***q** \ngl \\***q**\nJz1 & 0 \n0 &Jz2\nfl**q**& gl**q**\n, which leads to further simplifications within a weak coupling approach, see Sec. [ssec:wc]. We further point out that the *l* = 0 angular momentum channel is special and should be excluded, since it leads to a two-magnon wavefunction $\\psi\\_{{{\\bf r}}{{\\bf r}}'}$ that is nonzero when ${{\\bf r}}'={{\\bf r}}$, thus contradicting the assumptions that underlie.\nEquation now reduces to a set of separate equations in each angular momentum channel given by Det(M*l* − 1) = 0. These equations can be solved—generally numerically—to obtain the critical value *D**c*\\* at which magnon pairs can form with zero energy. Each angular momentum channel will yield a different value of *D**c*\\*. The critical values *D**c*\\* are then compared to *D**c*, the value of *D* at which the gap to single magnon excitations closes, which is obtained by setting $2D\\_c + \\mathcal E\\_{{{\\bf Q}}\\_i}=2( D\\_c + \\varepsilon\\_{{{\\bf Q}}\\_i})=0$. It thus follows that *D**c* is determined by the minimum of the magnon dispersion $ \\varepsilon\\_{{{\\bf Q}}\\_i}$. (Note that $\\varepsilon\\_{{{\\bf Q}}\\_i}$ is negative.) When *D**c*\\* > *D**c* for one or more *l*, two-magnon bound states can form with positive binding energy b = 2(D\\*c -Dc). [eq:epsb-def] The angular momentum channel with the largest value *D**c*\\* then determines the structure of excitons, much like the structure of Cooper pairs is determined by the relative angular momentum of the constituent electron pairs.\nIt is worth noting that symmetries generally imply degeneracies of angular momentum channels. For instance, in the case of the triangular lattice, *l* = ± 2 (as well as *l* = ± 1) are degenerate and form a single irreducible channel.\nThis detailed example of the triangular lattice serves to illustrate the general symmetry-based method for solving. In particular, the matrix $\\mathcal M$ can always be block diagonalized such that each block M*i* corresponds to distinct symmetry quantum numbers (i.e., representation of the lattice symmetry group) and matrix elements within each block are products of symmetry-adapted lattice harmonics. So, Eq. reduces to a set of decoupled equations in each symmetry channel of the form Det(M*i* − 1) = 0. Their solutions determine whether two-magnon bound states can form and if so, in what symmetry channel they form. This general method applies to finite center-of-mass momentum ${{\\bf K}}$ as well, with the modification that the relevant symmetry group is the little group of ${{\\bf K}}$.\nWeak coupling [ssec:wc]\n-----------------------\nFurther analysis of Eq. is possible when the interaction between the magnons is small. It follows from Eq. that the interaction between magnons originates from the *S**z**S**z* coupling and is parametrized by $\\eta\\_{{\\boldsymbol{\\delta} }} = J\\_{{\\boldsymbol{\\delta} }}^z / J\\_{{\\boldsymbol{\\delta} }}$, which may be treated as a perturbative parameter. Similar to Ref. , this assumption will allow us to derive a compact expression for the exciton binding energy, which is shown to depend on: (i) an effective coupling constant and (ii) magnon density of states.\nIn Section [sec:schroedingerformagnonpairs], we used to express the two-magnon problem to Schrödinger equation in terms of center-of-mass and relative coordinates. Under this transformation, the magnon interaction translates into quantum well potential for the relative coordinate ${{\\bf r}}- {{\\bf r}}'\\ = {\\boldsymbol{\\delta} }$. As a result, the weak coupling limit is equivalent to the shallow well approximation in quantum mechanics. This is why we expect the bound state binding energy *ɛ**b* to be small compared to the exchange couplings $J\\_{{\\boldsymbol{\\delta} }}$, while the size of the two-magnon bound state (i.e., the “exciton”) should be large on the lattice scale. Consequently, the extent of the exciton in reciprocal space will be small, which allows us to expand the dispersion in the vicinity of its minima.\n![Two-magnon exciton bound states in the triangular lattice model. (a) Binding energy \\varepsilon_b defined in Eq. as a function of \\eta = J^z_{1,2}/J_{1,2} for different values of J_2/J_1. Solid (and colored) curves correspond to numerical solutions of Eq. for l=2. Dashed (and black) curves correspond to the binding energy computed using Eq. (with prefactor fitted), showing excellent agreement for small \\eta. (b) The binding energies as a function of J_2/J_1 computed numerically from Eq. for the l=2 (solid curves) and l=3 (dashed curves) angular momentum channels. Black squares indicate where the nematic l=2 channel becomes dominant. (c) The binding energies as a function of J_2 / J_1 in the interval 1/8 < J_2 / J_1 < 1 (bounded by vertical dashed lines). (d) Single-magnon density of states \\nu_M (in units of J^{-1}_1) for magnon excitations on top of the paramagnetic phase evaluated at the M points. \\nu_M is given by Eq. and explains the general behavior of \\varepsilon_b shown in (c). ](triangular_results \"fig:\") [fig:triangularresults]\nTo illustrate the weak coupling analysis, we will again consider the case of triangular lattice, and relegate the generalization to non-Bravais lattices to Appendix [app:gen]. Following the scheme outlined above, we assume that the summand in is concentrated in the vicinities of the wave vectors ${{\\bf Q}}\\_i$, the locations of the minima of. Assuming further that ${{\\bf K}}=0$, we expand $\\mathcal{E}\\_{{\\bf q}}$ in the denominator up to the second order in ${{\\bf p}}= {{\\bf q}}- {{\\bf Q}}\\_i$ and evaluate the lattice harmonics in the numerator at ${{\\bf Q}}\\_i$. We also use the fact that the summand is invariant under rotation of ${{\\bf q}}$ by 2*π*/3, which allows to consider just one out of three ${{\\bf Q}}\\_i$ points; the contributions from each wave vector must be equal. Then the matrix structure of is determined by a single ${{\\bf q}}$ point and has rank one. Evidently, for fixed angular momentum *l* it has just one nonzero eigenvalue corresponding to the eigenvector $( f^{l}\\_{{{\\bf Q}}\\_i}, g^{l}\\_{{{\\bf Q}}\\_i})^T$. That leads to the following equation, where the summation was replaced by integration in the vicinity of ${{\\bf Q}}\\_i$: [finaleqexpand] l = 1. Here *V*BZ is the area of the Brillouin zone, ${{\\bf p}}$ is a small momentum reckoned from ${{\\bf Q}}\\_i$, and $(A\\_i)\\_{mn} = \\partial^2 \\mathcal E\\_{{\\bf q}}/ \\partial q\\_m\\partial q\\_n |\\_{{{\\bf Q}}\\_i} $. The effective coupling constants *λ**l* and the exciton binding energy *ɛ**b* are defined as [lambdageneral] l = 3 ( Jz1 | fl**Q**i |2+ Jz2 | gl**Q**i |2 ), b = 2D + E**Q**i. Integral is most easily performed by switching to the integration over single-magnon energy $\\xi = {{\\bf p}}^T A\\_i{{\\bf p}}/ 2$: [finaleqlog] **Q**i l d = = 1, where *ω**c* is a cutoff on the order of the exchange energies *J*1, 2 and $\\nu\\_{{{\\bf Q}}\\_i}$ is magnon density of states at one of the three minima. The latter is finite in two dimensions at a quadratic dispersion minimum and can be evaluated by *A**i* diagonalization followed by a linear coordinate transform for the integration variables *p**x*, *p**y*. If *ρ*1 and *ρ*2 are the eigenvalues of *A**i* (obviously, they are the same for each ${{\\bf Q}}\\_i$), then $\\nu\\_{{{\\bf Q}}\\_i} \\propto 1/\\sqrt{\\rho\\_1\\rho\\_2}$. Note that flattening of the minima enhances the density of states and increases the binding energy. The value of the latter follows from : [bindingenergy] b = c (- ), which is a key result of this section. The expression for the binding energy closely resembles the one for the Cooper problem of two attracting electrons above the two-dimensional Fermi surface. That is expected as both problems are in a weak coupling regime and effectively two spacial dimensions.\nThe energy of the two-magnon state is given by the familiar expression for two-particle bound states, i.e., $E = 2(D + \\varepsilon\\_{{{\\bf Q}}\\_i}) - \\varepsilon\\_b$, which implies that zero energy solutions exist for a critical value $D^\\*\\_c = -\\varepsilon\\_{{{\\bf Q}}\\_i} + \\varepsilon\\_b/2 = D\\_c+ \\varepsilon\\_b/2$, reproducing Eq. . Since the binding energy is positive for channels with nonzero (attractive) coupling constant, one has *D**c*\\* > *D**c*, demonstrating that the quantum paramagnet is unstable towards the formation of non-magnetic two-magnon excitons. The internal structure of the excitons, as defined by their symmetry quantum numbers, directly follows from comparing the coupling constants *λ**l*: the angular momentum channel with the largest coupling gives rise to the largest value of *D**c*\\* and determines the leading instability. As explained in the previous Section, zero angular momentum channel should be eliminated.\nWe expect the result $\\eqref{binding\\_energy}$ to be general for two-dimensional materials with three equivalent minima of the magnon dispersion. Details of the model in each case are encoded into magnon density of states and effective coupling constants *λ**l*. For non-Bravais lattices, Eq. should be modified by introducing projectors to the lower magnon band as explained in Appendix [app:gen]. We also comment on generalization to three spacial dimensions in the next Section.\nPotts-nematic magnon bound states [sec:results]\n===============================================\nIn this section, which forms the heart of the paper, we present a detailed application of the theory developed in the previous section to three particular lattice models that exhibit rotation symmetry broken magnetism. Specifically, we consider the triangular and honeycomb lattice in two dimensions, and the FCC lattice in three dimensions. We present both the outcome of the exact solution of the two-particle problem obtained by numerical integration in Eq. and the prediction of the analytical perturbation theory given by. For the case of the FCC lattice we show how weak coupling approach, which in two dimensions gives rise to, still yields useful approximations to the numerically exact result.\nTriangular lattice magnets [ssec:triangular]\n--------------------------------------------\nWe begin by considering the triangular lattice model briefly introduced in Sec. [sec:Z3breaking] (see also Fig. [fig:triangular]). This model includes antiferromagnetic exchange couplings between nearest neighbors, given by *J*1 > 0, and next-nearest neighbors, given by *J*2 > 0, which define the XXZ part of. Since both couplings are antiferromagnetic the interactions are frustrated. We specifically focus on the part of the phase diagram where 1/8 < *J*2/*J*1 < 1, since in this regime $J\\_{{{\\bf q}}}$ has minima at three commensurate *M*-point wave vectors (i.e., the centers of the edges of the Brillouin zone), thus implying a magnetically ordered state of collinear stripes at small *D* [shown in Fig. [fig:triangular](b)]. Furthermore, since the dispersion of the single-magnon excitations in the quantum paramagnetic state is proportional to $J\\_{{{\\bf q}}}$ (i.e., $\\varepsilon\\_{{{\\bf q}}}\\sim J\\_{{\\bf q}}$, see Sec. [sec:schroedingerformagnonpairs]), its minima are also located at the *M* points.\n![Real space structure of the two-magnon bound state wave function on the triangular lattice in the l=2 angular momentum channel. Shown are the wave functions \\psi_{1,2}({\\boldsymbol{\\rho} }) given in Eq. , where {\\boldsymbol{\\rho} }= {{\\bf r}}-{{\\bf r}}' is the distance between the magnons. Doubly degenerate solution for the exciton wavefunction on the triangular lattice in the weak coupling regime, shown in real space [\\psi({\\boldsymbol{\\rho} }) = (1/2\\pi) \\int d {{\\bf q}}\\, \\psi_{{\\bf q}}e^{i {{\\bf q}}\\cdot {\\boldsymbol{\\rho} }}, where {\\boldsymbol{\\rho} } is the vector connecting the two magnons]. The values slowly decay in space due to the finite exciton size (not shown). ](wf_triangular \"fig:\") [fig:triangularwavefunction]\nConsider first the numerical solutions of Eq. . These are obtained straightforwardly by numerical evaluation of the sum and the key results are presented in Fig. [fig:triangularresults]. Most importantly, we find that in the regime of interest, i.e., $1/8 \\lesssim J\\_2 / J\\_1 < 1$, the binding energy for two-magnon bound states is positive and largest in the *l* = 2 angular momentum channel, which is the doubly degenerate (Potts-)nematic channel. In Fig. [fig:triangularresults](a) we show the binding energy *ɛ**b* obtained from solving for *l* = 2 as a function of *η*, which controls the strength of the attractive magnon interaction. For convenience here we have taken *η*1, 2 = *η*, where *η**i* = *J**i**z*/*J**i*, i.e., the ratio between *J**z* and *J* is equal for nearest and next-nearest neighbor couplings. Different (solid and colored) curves correspond to different values of *J*2/*J*1, showing that the binding energy increases as *J*2/*J*1 decreases. For comparison, the dashed black curves show the binding energy calculated using Eq. , which was derived under the assumption that *η* is small. We see that the BCS-like formula is in excellent agreement with the numerically exact result in this regime.\nIn panels (b) and (c) of Fig. [fig:triangularresults] we show the binding energy, or equivalently *D**c*\\* − *D**c*, as a function of the strength of *J*2 relative to *J*1 for various values of *η*. As expected, panel (c) shows that the binding energy increases for increasing *η*. Furthermore, we find that the binding energy is a convex function of *J*2/*J*1 and increases towards to boundaries of the interval 1/8 < *J*2/*J*1 < 1. As discussed below, this behavior can be explained by, in particular the density of states *ν**M* at the *M* points.\nThe exchange coupling ratio *J*2/*J*1 = 1/8 is of special importance since the minima of $J\\_{{\\bf q}}$ change at this value, thus implying that the classical ground state changes. In particular, for *J*2/*J*1 < 1/8 the minima of $J\\_{{\\bf q}}$ are located at the *K* points, giving rise to the well-known coplanar 120∘ spiral state on the classical level. Note that in the vicinity of *J*2/*J*1 = 1/8 both the *M* points and the *K* are local minima of $J\\_{{\\bf q}}$; it is the global minimum which changes at *J*2/*J*1 = 1/8.\nIn the limit *J*2/*J*1 = 0, Ref. showed that two-magnon bound states can form in the (nondegenerate) *l* = 3 channel, which is the channel naturally associated with the *K* points. Excitons with *l* = 3 break an Ising $\\mathds{Z}\\_2$ symmetry, which corresponds to the chirality of the 120∘ classical order. To examine the behavior in the vicinity of *J*2/*J*1 = 1/8 we compute the binding energy in the interval 0 < *J*2/*J*1 < 1/4 for both channels and different values of *η*, and show the result in panel (b) of Fig. [fig:triangularresults]. As is clear, in the close vicinity of *J*2/*J*1 = 1/8 the binding energy is positive in both channels, but is larger for the “chiral” *l* = 3 channel. At some value $J\\_2/J\\_1 \\gtrsim 1/8$, which is marked by a black squares in Fig. [fig:triangularresults](b) and increases with *η*, the “nematic” *l* = 2 channel becomes dominant, implying a transition from a chiral instability towards a Potts-nematic instability. Note that the curves exhibit a cusp (i.e., discontinuous derivative) at *J*2/*J*1 = 1/8, which is due to the aforementioned abrupt change of the global minimum.\n[fig:nonzeroCoM]![The value of the interaction parameter \\eta = J_i^z / J_i required to create an exciton with the binding energy \\varepsilon_{{\\bf q}}= 0.05 J_1 for different values of the center-of-mass momentum {{\\bf K}}, taking values along the \\Gamma-K-M-\\Gamma contour in the Brillouin zone (horizontal axis). The minimal \\eta corresponds to the {{\\bf K}}= 0 case (\\Gamma point), which makes it the leading instability channel and justifies considering zero center-of-mass momentum excitons in the rest of the paper. ](NonzeroCoM \"fig:\")\nTo further understand the formation of nematic two-magnon bound states, we follow the weak coupling approach described in Sec. [ssec:wc] and compute the magnon density of states *ν**M*△ and effective coupling constant *λ*△ by expanding around the *M* points. The density of states is given by [DOStriangular] M= and the effective coupling constant is found as = 8(J1z + J2z). [lambdatriangular] These two parameters enter the expression for the binding energy in Eq. and have been used to compute the black dashed curves in Fig. [fig:triangularresults](a). (The prefactor coming from the cutoff was fitted.) The density of states, which is shown in panel (d) of Fig. [fig:triangularresults], is singular and diverges at *J*2 = *J*1. This is caused by the fact that for *J*2 > *J*1 the minima of $J\\_{{\\bf q}}$ move away from *M* towards Γ, making the *M* points saddle points rather than minima. In contrast, *ν**M*△ is regular at *J*2/*J*1 = 1/8, because *M* remains a local minimum. Since the effective coupling remains finite and does not change significantly in the interval 1/8 < *J*2/*J*1 < 1, the main dependence of the binding energy *ɛ**b* on *J*2/*J*1 ratio comes from *J*2/*J*1 dependence of *ν**M*, which is illustrated by Figs. [fig:triangularresults](c) and (d).\n![ (a-1) Zigzag (blue) and striped (yellow) phases on the honeycomb lattice for antiferromagnetic nearest neighbor coupling (J_1 < 0 case). Effective Potts-nematic coupling constant is positive only at J_2 > J_1/2, so vestigial state can emerge only in the part of the zigzag domain (shown in darker blue). Lines demonstrate contours at which perturbative theory predicts the width of the intermediate phase to be (D_c^* - D_c) / J_1 \\sim 10^{-1} (solid), \\sim 10^{-2} (dashed), and \\sim 10^{-3} (dot-dashed). We assumed J_i^z = J_i when plotting this picture. (a-2) Similar plot for J_1 > 0 case. In that case vestigial phase can appear for all points inside the zigzag and striped magnetic domains. (b) (c) The core real-space ({\\boldsymbol{\\rho} }) structure of the Potts-nematic exciton on the honeycomb lattice in the perturbative regime for a) zigzag phase; b) striped phase. The spin-up magnon resides on the first sublattice, while the spin-down magnon is located at a site, shifted by {\\boldsymbol{\\rho} }. The solution is doubly degenerate in each case. Slow exponential decay is not shown. ](honeycomb_results \"fig:\") [fig:honeycombresults]\nAs explained in Section [sec:schroedingerformagnonpairs], excitonic solution in the Potts-nematic channel correspond to *l* = ± 2 values of the angular momentum and thus has a twofold degeneracy, which we observe both in numerical and weak coupling approaches. In the latter, it is straightforward to obtain an explicit expression for the wavefunctions in the real space, which up to a constant factor reads: 1(2)( ) = e-r / l, [eq:wftriangular] Here ${\\boldsymbol{\\rho} }$ is the vector connecting the two lattice sites occupied by magnons constituting the exciton and *ω* = *e*2*π**i*/3 is the cubic root of unity. The factor *e*− *r*/*l**ɛ*, where *l**ɛ* ∼ *ɛ*− 1/2 ≫ 1 appears in the next orders of the perturbation theory and makes the wavefunction normalizable. Eq. represents the fact that the solution is a linear combination of exponentials $e^{i {{\\bf M}}\\_i \\cdot{\\boldsymbol{\\rho} }}$ with the wavevectors ${{\\bf M}}\\_i$ pointing to the minima of the single-magnon dispersion, so that the exciton is formed form the most low-lying excitations. The three exponentials are combined in a function with *l* = ± 2 in the same way as the nematic order parameter is constituted from the spin densities at ${{\\bf M}}\\_i$ points. Taking then real (imaginary) part of *l* = 2 wavefunction [Eq. ] gives the wavefunctions with the symmetry of *d**x**y* (*d**x*2 − *y*2) orbitals. The third linearly independent combination of exponentials $e^{i {{\\bf M}}\\_1\\cdot {\\boldsymbol{\\rho} }} + e^{i {{\\bf M}}\\_2 \\cdot{\\boldsymbol{\\rho} }} + e^{i {{\\bf M}}\\_3\\cdot {\\boldsymbol{\\rho} }}$ provides the rotation-symmetric *l* = 0 wavefunction, which contradicts assumptions made in and thus should be discarded.\nWe show the bound state wavefunctions graphically in Fig. [fig:triangularwavefunction] ignoring the slow decaying factor. Note that the radial structure is not completely trivial: one can see that the angular harmonics are not aligned on the neighboring shells but are rotated with respect to each other. That makes $\\psi({\\boldsymbol{\\rho} })$ obey the approximate translational symmetry ${\\boldsymbol{\\rho} }\\rightarrow {\\boldsymbol{\\rho} }+ 2 {\\boldsymbol{\\delta} }\\_{1(2)}$ with doubled period reflecting the fact the $2 {{\\bf M}}\\_i \\sim 0$ in the reciprocal space. (The translation symmetry is not exact due to the decaying factor *e*− *ρ*/*l**ɛ*.) Note also that the excitonic wavefunction $\\psi({\\boldsymbol{\\rho} })$ nullifies when the two magnons reside on the third-nearest sites. In accordance with this, we found that introducing third-nearest neighbor interaction *J*3 does not change the value of the effective coupling.\nFinally, to make sure that excitons with zero center-of-mass momentum ${{\\bf K}}$ represent the leading instability, we run a numerical calculation for the case of a non-zero ${{\\bf K}}$, which still boils down to solving with $\\mathcal M$ given by Eq. . In this place we opted for fixing the binding energy at the value of *ɛ* = 0.05*J*1 and evaluating the minimal interaction strength *η* required to create such bound state. The result is presented in Fig. [fig:nonzeroCoM], where we allowed the center-of-mass momentum *K* to take values across a path in the Brillouin zone, connecting the points Γ − *K* − *M* − Γ. Indeed, we observed that ${{\\bf K}}= 0$ requires the minimal amount of interaction for the bound state formation, which justifies considering ${{\\bf K}}=0$ in the rest of the paper. Interestingly, we found that ${{\\bf K}}= {{\\bf M}}\\_i$ choice is also favorable for exciton formation, however, this instability remains subleading.\nHoneycomb lattice magnets [ssec:honeycomb]\n------------------------------------------\nNext, we examine Potts-nematic two-magnon bound state formation on the honeycomb lattice. The motivation for considering the honeycomb lattice is twofold. First, it has the same symmetry group as the triangular lattice, but contrary to the latter it has sublattice structure, thus providing a generalization of the analysis presented in Sec. [sec:theory] to non-Bravais lattices. Second, a variety of experimentally available candidate materials are realizations of the honeycomb lattice model which makes the honeycomb lattice highly relevant from an experimental perspective. The connection to experimental compounds will be discussed in more detail below in Sec. [sec:expts].\nAs mentioned in Sec. [sec:Z3breaking], as far as the exchange interactions included in Eq. are concerned, we consider a frustrated *J*1–*J*2–*J*3 honeycomb model with up to third-nearest neighbor couplings. The classical phase diagram of the isotropic *J*1–*J*2–*J*3 Heisenberg model is known and notably includes two magnetic phases with broken threefold rotation symmetry. The corresponding regions in the phase diagram are defined by property that the minima of $J\\_{{\\bf q}}$ are located at the *M* points. Note that since $J\\_{{\\bf q}}$ is a matrix in the space of the two sublattices, two distinct solutions exist, which are related by the relative orientation of the spins on the sublattices, i.e., either aligned or anti-aligned. The single-*Q* collinear orderings with *M*-point wave vectors that are selected by fluctuations are schematically shown in Fig. [fig:honeycombresults], and are generally referred to as stripe (sublattice-even) and zigzag (sublattice-odd) orders. In our analysis we restrict to exchange couplings (*J*1, *J*2, *J*3) which favor zigzag or stripe order on the classical level.\nTo account for the two sublattices in the system, we extend the general theory presented in Sec. [sec:schroedingerformagnonpairs] to the case of a non-Bravais lattice (see Appendix [app:gen]). The main occurring difference is the appearance of projectors to the Bloch bands in the expression for the effective matrix elements. We present solutions of the modified equations obtained by numerical Brillouine zone integration complemented by the band summation in Fig. [fig:honeycombcomparison]. For that plot, we picked values of the spin couplings inside the zigzag domain on the Fig. [fig:honeycombresults] (a) (*J*1 < 0, *J*2/∣*J*1∣ = 1, *J*3/∣*J*1∣ = 0.3) and varied the effective interaction strength *η* = *η**i* = *J**i**z*/*J**i*. We observed that bound state with the angular momentum *l* = ± 2 has the largest binding energy and its dependence on *η* is captured well by the perturbation theory with a fitted prefactor (dashed line). As in the case of triangular lattice, that means that proliferation of excitations of this kind constitutes the leading instability, when *D* is lowered and that marks the transition to the Potts-nematic phase.\nThe weak coupling theory for a non-Bravais lattice in general follows the previous derivation given in Section [ssec:wc]. Again, the main contribution stems from the most low-lying excitations, which now also means that it is sufficient to consider the lower band. We expand around the minima of the latter (located at ${{\\bf M}}\\_i$ points) and, after some algebra, get expressions for the effective coupling constant in the Potts-nematic channel [lambdahoneycomb] = 8 ( J1z + 2 J2z ), which appears to be the same for both zigzag and stripe *M*-orderings. The density of low-lying states at *M* points equals: [doshoneycomb] M =, where *J*13 = *J*12 + *J*1*J*3 − 4*J*32 and *β* = sign (*J*1 − 3*J*3). Eqs. and should be plugged into the general perturbative formula for the binding energy, which gives a result consistent with numerical computation (Fig. [fig:honeycombcomparison]).\nPerturbative wavefunction of *l* = ± 2 excitions are again doubly degenerate and appear to be a linear combination of the exponentials $e^{i {{\\bf M}}\\_i {{\\bf r}}}$ as in the case of triangular lattice, but bear a sublattice structure, which slightly differs between the zigzag and stripe phase. We present it graphically in Fig. [fig:honeycombresults] (b) for underlying zigzag ordering an Fig. [fig:honeycombresults] (c) for the stripe one, choosing the *d**x**y* and *d**x*2 − *y*2 basis and using the ${\\boldsymbol{\\rho} }$ plane, where ${\\boldsymbol{\\rho} }$ is the distance between the magnons in the pair. We assumed that the spin-up magnon resides on the first sublattice; the opposite case leads to similar pictures. The figure depicts the core of the exciton wavefunction with omitted slow decay factor *e*− *ρ*/*l**ɛ*.\nWith obtained analytical approximation, we explore exciton formation for all striped phases in the phase diagram of Fig. [fig:honeycombresults] (a). The fist important observation is that for the bound state to emerge, the effective coupling constant needs to be positive. It turns out that at *J*1 < 0 and inside the zigzag domain (*J*3 > 0), $\\lambda\\_{\\hexagon} > 0$ only when *J*2/*J*1 > 1/2, which defines a subregion potent to develop a vestigial phase (shown by darker tone and solid frame). Contrary to that, for the stripe phase (*J*3 < 0) and for both phases at *J*1 > 0, the effective coupling is positive everywhere inside the magnetic domain, so that in the latter cases magnetic ordering should always be preempted by a vestigial phase.\nWe now turn to the discussion of how prominent should be the effect of exciton formation across the honeycomb lattice phase diagram. As suggested by Eq. and common knowledge, enhancement of the density of states at minima of the dispersion is beneficial for the formation of the bound state. Inspection of implies that such enhancement happens in the vicinity of the right boundaries of zigzag and stripe domains. That can be explained by the fact that at these lines the system undergoes phase transition into the spiral ordering, characterized by a non-commensurate value of ordering wavevector ${{\\bf Q}}$. After the transition, the minima of the coupling matrix $J\\_{{{\\bf q}}}$ eigenvalues and, consequently, the minima of the single-magnon dispersion on top of paramagnet drift away from the ${{\\bf M}}$-points, turning the ${{\\bf M}}$-point itself into a saddle point rather than the minimum. Before the transition but near the boundary the effect is manifested by appearing of the soft direction at each ${{\\bf M}}$-point, leading to the increase of the single-magnon density of states. This phenomenon is completely analogous to the one, which happens on the triangular lattice in the vicinity of *J*2/*J*1 = 1 point as described in Section [ssec:triangular]. To further illustrate this statement and make it quantitative, we plot contours, at which our perturbative theory predicts the binding energy (and hence the width of the Potts-nematic phase) to be of the order 10− 1 (solid line), 10− 2 (dashed line), and 10− 3 (dot-dashed line) in the units of ∣*J*1∣ [Fig. [fig:honeycombresults] (a)]. As can be seen in the picture, the most perceptible effect appears near the aforementioned boundary of the domains.\n![ The width of the Potts-nematic phase (D_c^* - D_c)/J_1 obtained from exciton binding energy as a function of interaction strength \\eta = J_i^z / J_i for the honeycomb lattice model. Solid line shows numerical solution, while the dashed one is for the analytical perturbative result, valid at \\eta \\ll 1. The prefactor in the latter was fitted. ](Honeycomb_comparison \"fig:\") [fig:honeycombcomparison]\nFCC lattice magnets [ssec:fcc]\n------------------------------\n![ Binding energy of the Potts-nematic excitons on the face-centered cubic (FCC) lattice as a function of interaction strength parameter \\eta = J_i^z / J_i. Note that the bound state exists only at \\eta > \\eta_c due to the peculiarity of a three-dimensional problem. Dashed lines demonstrate the asymptotics obtained by fitting the data with Eq. .](FCC_lattice \"fig:\") [fig:fcclattice]\nIn the final part of this section we turn to an application of our theory to a model in three dimensions: the face-centric cubic (FCC) lattice. Our goal is to demonstrate that the analysis developed in Sec. [sec:theory] is not limited to two dimensions, but extends to three dimensions as well. The three-dimensional case is different in one important way, however. Since it is well known that bound states do not exist for arbitrarily shallow potential in three dimensions, making it a threshold phenomenon, we expect the two-magnon bound states to occur only for finite interaction between the magnons. In particular, this means that Eq. does not apply to the three-dimensional case. Below we describe how the analysis of Sec. [ssec:wc] can be adapted to the case of the FCC lattice.\nWe consider a *J*1-*J*2 model on the FCC lattice with antiferromagnetic *J*1 and small ferromagnetic *J*2. For a simple nearest neighbor FCC antiferromagnet (*J*2 = 0), the Fourier transformed exchange coupling $J\\_{{{\\bf q}}}$ has minima on lines in momentum space. The lines of minima reflects the frustration of the FFC lattice and are defined by the constraint (*q**x*, *q**y*) = (2*π*, 0) with arbitrary *q**z* (all symmetry equivalent constraints also define minima). Introducing ferromagnetic coupling between the next-nearest neighbors (*J*2 < 0) lifts that degeneracy and results in minima of $J\\_{{{\\bf q}}}$ at the three wave vectors ${{\\bf Q}}\\_1= (2\\pi,0,0)$, ${{\\bf Q}}\\_2=(0,2\\pi,0)$, and ${{\\bf Q}}\\_3=(0,0,2\\pi)$. These wave vectors satisfy Eq. . The fact that there are three equivalent minima points implies that $\\mathds{Z}\\_3$ symmetry is broken in the ordered phase with the ordering wavevector provided by one of these minima. Such ordering is formed by stripes, which now can take one of the three directions parallel to the principal axes in the three-dimensional space. We thus expect that in this case also one might expect intermediate Potts-nematic state at a range of single-ion anisotropy parameter *D* between the ordered and paramagnetic phases.\nWe start with the demonstration of the solution for the binding energy *ɛ**b*, obtained by numerical integration in. It is shown in Fig. [fig:fcclattice], where *ɛ**b* is plotted against the interaction strength *η* = *η**i* = *J**i**z*/*J**i* for a number of *J*2/*J*1 values. As predicted above, we observe that the bound state is formed only at *η* > *η**c*, which is the main distinction with respect to the two-dimensional case. At $\\eta \\gtrsim \\eta\\_c$, the binding energy has approximately quadratic dependence on *η* − *η**c*. A more accurate formula contains a logarithmic correction and is discussed below. We plot it in dashed in Fig. [fig:fcclattice] and observe that it captures well the numerical dependence at small *η* − *η**c* (with fitted parameters).\nHaving identified the threshold nature of the bound state formation, we studied the dependence of the critical interaction *η**c* on the spin-spin couplings (Fig. [fig:criticalzeta]). The resulting dependence is monotonous and drops to zero in a sharp nonanalytical way at *J*2 → 0. This behavior is explained by the fact that at *J*2 = 0 the single-magnon dispersion reaches its minimum on a set of lines, which effectively renders the problem two-dimensional. From a quantitative perspective, it is interesting that for isotropic Heisenberg model with *η* = *J**i**z*/*J**i* = 1, the bound states exist only at $|J\\_2| / J\\_1 \\lesssim 0.1$.\n![Critical values of the interaction parameter \\eta_c on the face-centered cubic (FCC) lattice as a function of spin couplings ratio |J_2| / J_1. Formation of Potts-nematic excitons and, hence, the vestigial phase occurs only at \\eta = J_i^z / J_i > \\eta_c due to the corresponding property of the three-dimensional potential well problem. ](Critical_Zeta \"fig:\") [fig:criticalzeta]\nWe proceed with the weak coupling theory for the three-dimensional case. Naive attempt at taking the integral in using expansion in the vicinity of ${{\\bf Q}}\\_i$ points leads to a linearly divergent integral due to increased phase volume in three dimensions. The common resolution would be to subtract from the integrand of its value at *ɛ**b* = 0 (to be taken numerically over the whole Brillouin zone) and then proceed with the regularized integral. In the regime of small binding energies *ɛ**b* that leads to the following implicit expression for *ɛ**b*(*η*): -1 = c-1 - B. [3Dsolution] where *B* is a constant determined by the density of states in the vicinity of *M*-points. The result indeed signifies that the bound state exists only at *η* > *η**c*. However, we found that Eq. is suitable to describe the asymptotics of the exact solution only at very small values of *ɛ**b*, which correspond to a much smaller scale than the one in Fig. [fig:fcclattice]. This observation is explained by the following considerations. As discussed in the beginning of this subsection, at ∣*J*2∣ → 0 the dispersion acquires a very soft direction near the minimum, which eventually transforms into a line of degenerate minima. This process leads to the increase of the single-magnon density of states, which, as discussed in previous sections, enhances the effect. Given the smallness of relevant *J*2 (in Fig. [fig:fcclattice] $|J\\_2| / J\\_1 \\lesssim 0.1$) that means that at moderate *ɛ**b* the value of is not captured well by the quadratic expansion. Instead, one can expect logarithmical corrections to due to the nascent crossover to 2D. Indeed, we find that a corrected formula -1 = c-1 - B [3Dsolutionwithlog] captures the numerical dependence much better (see dashed lines in Fig. [fig:fcclattice]).\nTo conclude, our consideration predicts that the effect of exciton formation and appearance of the vestigial phase is weaker in three dimensions for nonspecific values of the spin-spin interactions. However, in the vicinity of special points in the phase diagram (*J*2 = 0 in our case), the effect becomes more pronounced due to the enhanced density of states of relevant magnons.\nConnection to experiment [sec:expts]\n====================================\nThe analysis presented in the previous section indicates that the instability towards a Potts-nematic phase is rather general and does not depend on the specific realization of the proposed model on any specific lattice. This generality motivates the important question whether material realizations of Potts-nematic quantum magnets exist among experimentally known material systems. In pursuit of this question, in this section we highlight and discuss a number of promising candidate materials to which our theory applies or relates.\nFor our theory to directly apply, candidate materials should satisfy three requirements. First, the interactions between the spin must be such that magnetic order with broken $\\mathds{Z}\\_3$ crystal symmetry is favored. This implies a tacit constraint on the symmetry of the crystal structure and limits the search for materials to hexagonal, trigonal, tetrahedral, or octahedral systems. In practice, the search should be focused on materials or families of materials in which rotation symmetry broken magnetic order is observed, which is clearest signature of the nature of the exchange couplings. Examples will be discussed below. The second requirement is that the candidate materials exhibit significant easy-plane single-ion anisotropy, which implies that spin-orbit coupling is important. In our model, the single-ion anisotropy controls the (quantum phase) transition between the paramagnet and the ordered state, and therefore would be responsible for driving a magnetically ordered state into a putative Potts-nematic phase. Closely related to this second requirement is the third requirement: the quantum spins must be *S* = 1 degrees of freedom, such that in the limit of large single-ion anisotropy the ground state is a perfect quantum paramagnet product state. In principle other integer-spin systems may be considered, such as *S* = 2, since these also admit a quantum paramagnetic ground state.\nGiven these requirements, one of the more promising material classes is the family of transition-metal phosphorous trichalcogenides (*M*P*X*3). These are layered van der Waals materials in which the transition metal *M* = (Fe, Ni, Mn, Co) sites are magnetic and form a honeycomb lattice within each layer. The *X* are occupied by chalcogen atoms, typically sulfur (S) or selenium (Se). The *M*P*X*3 materials have attracted much attention recently as a versatile platform for exploring intrinsic two-dimensional magnetism in few-layer or monolayer systems. While some members of this class show antiferromagnetic Neél order, others were found to realize antiferromagnetic zigzag order—precisely the type of order discussed in Sec. [ssec:honeycomb]. Furthermore, strong easy-axis or easy-plane behavior was observed, such as Ising-like easy-axis behavior in FePS3 and XY-like easy-plane behavior in NiPS3 . To explain the observed magnetic orderings across the family of *M*P*X*3 compounds a *J*1–*J*2–*J*3 honeycomb lattice spin model with single-ion anisotropy was proposed, which, quite remarkably, is identical to the honeycomb lattice model studied in this work. It is for this reason that the trichalcogenide magnets offer a particularly compelling venue for Potts-nematicity associated with rotation symmetry broken magnetic order.\nWithin the family of *M*P*X*3 materials NiPS3 deserves special attention. Not only do the ordered moments of the observed zigzag phase lie in the plane , suggesting easy-plane anisotropy with *D* > 0, but the Ni2 + ions also give rise to *S* = 1 spins on the honeycomb sites. This implies that NiPS3 provides a realization of the model studied in this work.\nIt is important to point out one caveat concerning the bulk *M*P*X*3 materials, in particular concerning the stacking of the constituent layers in the bulk structures. In the sulfur-based compounds *M*PS3 with *M* = (Fe,Ni,Mn) the layers are stacked in monoclinic fashion, giving rise to space group *C*2/*m*. This means that the threefold rotation symmetry of the constituent honeycomb layers is already broken in the bulk structure, thus precluding the spontaneous breaking of rotation symmetry by the magnetic interactions. The coupling between the layers is believed to be very weak, however, and may be weak enough to prevent a selection of the nematic director by the monoclinic stacking direction. Indeed, encouraging evidence for this has recently been reported by linear and non-linear optical measurements, which have found three different magnetic domains below the Neél temperature within one homogeneous structural domain.\nA second material of interest is the iron-intercalated transition metal dichalcogenide (TMD) Fe*x*NbS2 with *x* = 1/3, which is a member of the class of intercalated TMDs *M**x**T**A*2 . Here *M* is a transition metal, *T*=(Ta, Nb), and *A*=(S, Se). The magnetic Fe atoms are intercalated between the adjacent van der Waals layers of 2*H*-NbS2 and form a triangular lattice with $\\sqrt{3}\\times\\sqrt{3}$ periodicity. The two neighboring triangular lattice layers of Fe atoms that make up the unit cell (and are located at *c* = 1/4 and *c* = 3/4) are shifted with respect to each other, effectively forming a honeycomb structure. The resulting space group of the intercalated structure is *P*6322. Importantly, both optical birefringence and detailed neutron crystal diffraction measurements have reported observation of rotational symmetry broken stripe order. Here, the two triangular lattice layers each form the ordering pattern shown in Fig. [fig:triangular] and together form the honeycomb lattice stripy order shown in Fig. [fig:honeycombresults]. As such, the observed magnetic order falls in the class of orderings considered in our work. (Note that Ref. reported great sensitivity of the ordering pattern to small changes in the intercalation parameter *x* close to the commensurate value 1/3.) Furthermore, the Fe2 + ions give rise to *S* = 2 spins. To understand the observed magnetic orderings, recent theory works proposed a Heisenberg spin with extended exchange interactions and single-ion anisotropy , essentially equivalent to our honeycomb lattice model. An open question is the nature of the single-ion anisotropy, however, with initial estimates suggesting it might be negative, thus implying an easy-axis anisotropy .\nWe conclude this section with a general remark. To obtain the nematic phase at low temperatures one needs a suitable tuning knob to weaken and eventually destroy the magnetic order. This can be provided by varying strain and pressure in the system. An alternative possibility is the formation of Potts-nematic phase at the thermal phase transition as function of temperature, rather than the quantum phase transition studied here. A theory of thermal transition to the Potts-nematic phase will be reported elsewhere.\nDiscussion and Conclusion\n=========================\nThe central result of this paper is the demonstration that a magnetically disordered Potts-nematic phase can exist as an intermediate phase at the quantum phase transition from a quantum paramagnet to a $\\mathds{Z}\\_3$ threefold rotation symmetry broken antiferromagnet. Since such a phase is magnetically disordered, spin-rotation symmetry and time-reversal symmetry are preserved, yet a $\\mathds{Z}\\_3$ crystal rotation symmetry is broken, thus giving rising to nematic order. The relevant order parameter for the nematic phase is expressed in terms of bilinears of the primary magnetic degrees of freedom, i.e., the spin variables.\nOur conclusion is inferred from the analysis of two-magnon excitations in the paramagnetic state of an *S* = 1 quantum spin model, in which the transition between the paramagnet and the magnetic state is regulated by the value of easy-plane single-ion anisotropy. With both numerical and analytical methods we established the existence of two-magnon bound states—“excitons”—with the angular momentum *l* = 2. That makes the two-particle gap close slightly before the single-particle gap as the transition is approached, leading to the proliferation of excitons. The latter can be described by a Jastrow ansatz. We derived a general equation for the excitonic bound state in terms of lattice harmonics [Eqs. and ] and derived its analytical solution for the weak-coupling regime. The latter expresses the binding energy and, hence, the width of the vestigial phase, as a function of magnon density of states at the dispersion minima (in the paramagnetic state) and the effective interaction strength, which is a linear combination of the spin-spin coupling constants. Analytical perturbation theory is corroborated by a detailed numerical analysis for the cases of triangular, honeycomb, and face-centered cubic lattices (see Figs. [fig:triangularresults]–[fig:criticalzeta]).\nIt is important to describe the relation between the character of the vestigial phase determined by the two-magnon wavefunction and the properties of the single-magnon dispersion above the paramagnetic phase. In the present paper we assumed that the dispersion has three degenerate minima in the Brillouin zone (*Q*1, *Q*2, and *Q*3 momenta, satisfying 2*Q**i* ∼ 0), which are related to each other by an in-plane rotation for the cases of triangular and honeycomb lattice. Under these assumptions, perturbative expression for the effective coupling constants in different angular momenta channels (*l*) nullifies for *l* = 1 and *l* = 3 leaving only Potts-nematic channel with *l* = 2 and unphysical symmetric wavefunction with *l* = 0. Similarly, we checked that in the setup of Ref. , in which the dispersion had two inequivalent minima, the only physical channel arising in the perturbation theory is the chiral one (*l* = 3). This relation provides a universal scheme allowing to determine the nature of the vestigial phase given the single-particle spectrum. In the numerical calculation we observed that the effective couplings in the ‘wrong’ angular momentum channels are nonzero, but they are small and thus irrelevant deep inside the domain with a given magnetic ordering [see Fig. [fig:triangularresults](b)].\nWhile in the present work we focused on the specific case of an *S* = 1 spin model, our analysis straightforwardly generalizes to any integer spin system. Indeed, the ground state at large values of the single-ion anisotropy in that case is still a perfect paramagnet, given by a product of *S**z* = 0 states on each site. A spin flip to a state with *S**z* = *s* on any site costs an energy *D**s*2 and therefore, the energetically lowest excitations in this case still correspond to *s* = ± 1. It means that for the purpose of determining the instability of the paramagnet marked by a gap closure, one can restrict the consideration to these two branches only, which makes all further analysis the same as in the *S* = 1 scenario. In contrast, we expect that half-integer models behave qualitatively different, since they do not admit a paramagnetic product state.\nThe prediction of the formation of Potts-nematic phase at low temperatures in the vicinity of the quantum phase transition poses a natural question: does there exist a similar vestigial phase at the thermal phase transition between a striped antiferromagnet and the symmetry high-temperature phase? This question will be addressed in a separate paper.\nWe conclude by emphasizing that there exist a number of different materials which provide promising venues for experimental observation of Potts-nematicity in quantum magnets. As discussed in Sec. [sec:expts], given its currently established magnetic properties, the *M*P*X*3 family of transition-metal trichalcogenides is of particular interest. Within this class of materials NiPS3 is most promising, as it is likely to provide a realization of the model considered in this work. Another possible candidate material is the Fe-intercalated dichalcogenide NbS2.\nAcknowledgements\n================\nWe gratefully acknowledge useful discussions with R. M. Fernandes, A. V. Chubukov, and L. Wu. This research was supported by the National Science Foundation Award No. DMR-2144352.\nGeneralization to non-Bravais lattices [app:gen]\n================================================\nHere we describe the generalization of the two-magnon problem to the case of a non-Bravais lattice, which requires a proper account for the sublattice degree of freedom. We start we the expression for the *X**X**Z* Hamiltonian of the form\n \n$$\\begin{gathered}\n \\mathcal{H} = \\frac{1}{2} \\sum\\_{{{\\bf r}}, {\\boldsymbol{\\delta} }, \\alpha, \\beta} (J\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha\\beta} (S\\_{{{\\bf r}}\\alpha}^{x}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{x} + S\\_{{{\\bf r}}\\alpha}^{y}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{y}) \\\\\n+ \\frac{1}{2} \\sum\\_{{{\\bf r}}, {\\boldsymbol{\\delta} }, \\alpha, \\beta} (J^z\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha\\beta} S\\_{{{\\bf r}}\\alpha}^{z}S\\_{{{\\bf r}}+ {\\boldsymbol{\\delta} },\\beta}^{z},\\end{gathered}$$ \nwhere ${{\\bf r}}$ and ${{\\bf r}}+{\\boldsymbol{\\delta} }$ are Bravais lattice vectors and *α*, *β* label the spins within the unit cell, i.e., the sublattice degrees of freedom. Two-magnon wavefunction $\\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}' \\beta}$ also carries sublattice indices. For the clarity of the narration, we will consider zero center-of-mass momentum excitons (generalization to the opposite case is straightforward). Then, the Fourier transform of the wavefunction reads **r**, ; **r**’ = **q** (**q**) ei**q**(**r**+ - **r**’ - ), where ${\\boldsymbol{\\ell} }\\_{\\alpha}$ determines the position of *α*-sublattice site inside a unit cell.\nWe proceed with projecting the Schroedinger equation to the subspace of two opposite-spin magnons as was done in the main part [Eq. ]. It is convenient to interpret $\\psi\\_{\\alpha \\beta} ({{\\bf q}})$ as a matrix and write the variational Schroedinger equation in the matrix form: [Schroedingermultiband] (E - 2D) (**q**) - **q** (**q**) - (**q**) **q**= []. Here $\\hat{J}\\_{{{\\bf q}}}$ is the Fourier transform of the interaction matrix [Jqmatrixdefinition] (J**q**) = (J) ei**q**( + - ), which reduces to the single-magnon energy $\\varepsilon({{\\bf q}})$ in the Bravais-lattice case [see paragraph under ]. Int[*ψ*] is the interaction term appearing due to magnon-magnon interaction, provided by the *S**z**S**z* part of the spin coupling. It has the following matrix elements: ([]) = - (Jz) B,, ei**q**( + - ), where $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }} = \\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}+ {\\boldsymbol{\\delta} }, \\beta}$ is the real-space wavefunction amplitude at two magnons residing on *α* and *β* sublattices in the unit cells separated by the Bravais lattice vector ${\\boldsymbol{\\delta} }$. That is a correct definition because $\\psi\\_{{{\\bf r}}, \\alpha; {{\\bf r}}+ {\\boldsymbol{\\delta} }, \\beta}$ does not depend on ${{\\bf r}}$ in the case of zero center-of-mass momentum. Importantly, contains only the elements $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$ which are multiplied by nonzero spin-spin couplings $(J\\_{{\\boldsymbol{\\delta} }}^z)\\_{\\alpha\\beta}$. Thus, for a model with finite-range interaction, there is a finite number of $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$ components. For instance, on a *J*1 − *J*2 − *J*3 model on a honeycomb lattice for each of the two choices for *α*, there are three nearest neighbors, six next-nearest, and three third-nearest ones, constituting 12 elements for one *α* and 24 total components of $B\\_{\\alpha, \\beta, {\\boldsymbol{\\delta} }}$. To proceed, we need to express the latter via the momentum-space wavefunction: [Bdefinition] B,, = e-i (), where we introduced the following notation for brevity: = + -.\nNext step is to diagonalize the l.h.s. of by expressing $\\psi\\_{\\alpha \\beta} ({{\\bf q}})$ via the eigenstates $e\\_{a}({{\\bf q}})$ of $\\hat{J}\\_{{{\\bf q}}}$ matrix: [psithrougheigenvectors] (**q**) = ab cab (**q**) ea(**q**) ebT(-**q**), **q** ea(**q**) = **q**(a) ea(**q**), where *a* enumerates the Bloch bands of magnons. Then, takes the form [Scroedingermultibanddiagonalized] ab cab ea(**q**) ebT(-**q**) = [], Assuming that $e\\_{a}({{\\bf q}})$ constitute an orthonormal basis for each ${{\\bf q}}$ and multiplying by $e\\_{a'}^{\\dagger}({{\\bf q}})$ from the left and by $e\\_{b'}({{\\bf q}})$ from the right, we get *c**a**b*: [ccoefs] cab(**q**) = -,, . We then substitute into and further into to find the self-consistency equation [equationforBnonBravais] B,, = ’, ’, ’,, ; ’, ’, ’ B’, ’, ’, where [Mmultiband],, ; ’, ’, ’ = **q**,a,b, where $P^{(a)} ({{\\bf q}}) = e\\_{{{\\bf q}}}^{(a)} [e\\_{{{\\bf q}}}^{(a)}]^{\\dagger}$ is the projector to the *a*th band acting in the sublattice space, and *E* = 0 value was taken since we are looking for the instability, which arises when the excitons become gapless.\nCondition for to have solution is given by the same equation detM − 1 = 0 as in the main text [Eq. ]. However, expression for $\\mathcal M$ is slightly more involved and includes projectors to magnon bands and summation over the latter. Nevertheless, for an isotropic model one can still block diagonalize $\\mathcal M$ with the help of lattice harmonics and proceed with the numerical computation. In Section [ssec:honeycomb] we demonstrate how that works for the case of the honeycomb lattice.\nWeak coupling theory for a non-Bravais lattice is regulated by the parameters $(\\eta\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha \\beta} = (J\\_{{\\boldsymbol{\\delta} }}^z)\\_{\\alpha \\beta} / (J\\_{{\\boldsymbol{\\delta} }})\\_{\\alpha \\beta}$, which determine interaction between the single-magnon excitations. In an isotropic model the couplings depend only on the relative distance $\\tilde{{\\boldsymbol{\\delta} }}$ between the sites: (J) = J(| + - |). For simplicity, we would also assume that *η* is the same for all couplings so that the theory has just one perturbation parameter. At small *η*, which corresponds to a shallow well problem, common knowledge is to expect the exciton binding energy to be small, while its size to be large in real space and small in reciprocal space. That allows to consider only the low-lying excitations in the summation in, which boils down to considering only the lower band and expanding the dispersions $\\varepsilon\\_{\\bm{q}}^{(a)}$ in the vicinity of their minima, which we denote as ${{\\bf Q}}\\_i$. Moreover, we will evaluate the numerator of strictly at ${{\\bf Q}}\\_i$ points. Thus, we rewrite the summation in the vicinity of each minimum in terms of $\\bm{p} = {{\\bf q}}- \\bm{Q}\\_i$. [Mmultibandlowest],, ; ’, ’, ’ = i, where $P\\_{\\alpha \\beta}^{{{\\bf Q}}\\_i} = P\\_{\\alpha \\beta}^{(\\text{low})} ({{\\bf Q}}\\_i)$ are projectors onto the lowest magnon band, binding energy $\\varepsilon\\_b = 2D + \\varepsilon\\_{{{\\bf Q}}\\_i} + \\varepsilon\\_{-{{\\bf Q}}\\_i}$, and (Ai)mn =. qmqn |**Q**i is the expansion of the magnon dispersion in the vicinity of the global minima ${{\\bf Q}}\\_i$. Eq. has the same general form as the analogous equation for a Bravais lattice. For a quadratic magnon band minima one can proceed with integrating over $\\xi = {{\\bf p}}^T A\\_i {{\\bf p}}$ as was done in. The corresponding density of states $\\nu\\_{{{\\bf Q}}\\_i}$ should contain only the contribution of the lowest band at ${{\\bf Q}}\\_i$ point. Such integration leads to the same functional dependence as in with some value of *λ* and density of states to be understood as explained above. For the case of the honeycomb lattice, which has dispersion minima at M-points, these quantities are given by and. We discuss the result of the weak-coupling approach for this case in detail in Section [ssec:honeycomb].\n---\n1. J. Strockoz and D. S. Antonenko contributed equally to this work.[↩](#fnref1)\n2. If one considers ansatz, in which $\\ket{\\Omega}$ mixes with the two-magnon states, the resulting wavefunction will be dominated by the $\\ket{\\Omega}$ component as it is lower in energy by at least 2*D* compared to the two-magnon states. The solution can be interpreted as the ground state, corrected by the interaction with the two-magnon states.[↩](#fnref2)\n3. The six nearest neighbor and next-nearest neighbor vectors are related by sixfold rotation; some are shown in Fig. [fig:triangular].[↩](#fnref3)\n"},"data_id":{"kind":"string","value":"arxiv_0003295"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":7,"numItemsPerPage":100,"numTotalItems":46503,"offset":700,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6Im1sR1pFYktIVHRVSGRhX1RRdEczQ0N1S3k5ME9Qd25DNHY5elRjM3FVYzgifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4MzgxNjcyOSwianRpIjoiYmNkOTdlNTctMmQ0OS00YmUwLTg0MTctMzMyMDI4ZWNmYmJiIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODM4MjAzMjksImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.yGEh6U2uJwqbHmIgDHBx7awohWOmd3gpo0tq1fkQ_SwwboZSqA8fUO2f8-atfal3RZT4laG18u3IT7ZF7vKbBg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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Stochastic processes with competing reinforcements
==================================================
[1](#fn1)
[2](#fn2)
We introduce a simple but powerful strategy to study processes driven by two or more reinforcement mechanisms in competition. We apply our method to two types of models: to non conservative zero ran... | arxiv_0002856 |
A Non-Deterministic Multiset Query Language
===========================================
We develop a multiset query and update language executable in a term rewriting system. Its most remarkable feature, besides non-standard approach to quantification and introduction of fresh values, is non-determinism — a query resul... | arxiv_0002857 |
↦ *x* ⊗ 1, *y* ↦ *y* ⊗ 1.
Then, by Corollary [cor:W-aut], the map *W* allows to define unital homomorphism of algebras $\Delta:\Bbbk\_q[x,y] \to \Bbbk\_q[x,y]\otimes \Bbbk\_q[x,y]$, given on generators by
Δ(*x*) = 1 ⊗ *x* + *x* ⊗ *y*, Δ(*y*) = *y* ⊗ *y*.
Such a Δ is coassociative
$$[(\Delta \otimes \math... | arxiv_0002860 |
Beryllium abundances in stars with planets:
===========================================
Extending the sample[1](#fn1)
=============================
### Received November 2009; accepted
We present here an extension of previous beryllium abundance studies. The complete sample consists of 70 stars hosting planets and 30 s... | arxiv_0002862 |
Preparing contributions to CERN reports (school, workshop, and conference proceedings)
======================================================================================
This document explains instructions for authors who want to prepare electronic copy to be published as a CERN Report.
Introduction
============
Au... | arxiv_0002865 |
The longevity of the oldest open clusters:
==========================================
Structural parameters of NGC188, NGC2420, NGC2425, NGC2682, NGC6791, NGC6819
============================================================================
We investigate the spatial distribution and structural parameters of six open cl... | arxiv_0002870 |
Information dynamics and the arrow of time
==========================================
Why does time appear to pass irreversibly? To investigate, we introduce a class of partitioned cellular automata (PCAs) whose cellwise evolution is based on the chaotic baker’s map. After imposing a suitable initial condition and rest... | arxiv_0002872 |
Ergodic and Mixing Properties of the Boussinesq Equations with a Degenerate Random Forcing[1](#fn1)
===================================================================================================
-4mm
-1mm
-1mm
-1mm
-1mm
We establish the existence, uniqueness and attraction properties of an ergodic invariant measur... | arxiv_0002878 |
Sgoldstino inflation
====================
Introduction
============
Scalar fields are abundant in supersymmetric theories. They all couple to each other with at least gravitational strength interactions. Planck suppressed couplings are generically unimportant when describing processes at low energies, but such a decoup... | arxiv_0002883 |
Differential gene co-expression networks viaBayesian biclustering models
========================================================================
Identifying latent structure in large data matrices is essential for exploring biological processes. Here, we consider recovering gene co-expression networks from gene expres... | arxiv_0002884 |
Differential graded cell 2-representations
==========================================
This article develops a theory of cell combinatorics and cell 2-representations for differential graded 2-categories. We introduce two types of partial preorders, called the strong and weak preorder, and analyse and compare them. The ... | arxiv_0002885 |
to identify representative subclusters.
In summary, the dendrogram illustrated in Figure [figclustering] suggests that curriculum learning works should be first divided by the underlying learning paradigm: supervised learning, reinforcement learning, and self-paced learning. At the second level, the scientific works t... | arxiv_0002887 |
Learning Equilibria in Matching Markets from Bandit Feedback
============================================================
Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale a... | arxiv_0002888 |
i.e. it lacks large-scale dust and molecular concentrations. On the basis of photometry, high-resolution echelle spectroscopy, and 21cm HI spectrum survey data, they concluded that NGC602 formed at the periphery of a low-density HI cloud, under conditions that are marginal for star formation.
The other regions listed ... | arxiv_0002897 |
Near-critical spanning forests and renormalization
==================================================
We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deleti... | arxiv_0002898 |
, say, Alice is interested in exponential discounting, then she can estimate a preference without the need to impose this assumption on the data. If Bob is indeed discounting exponentially, then the preference estimates are guaranteed to converge to a preference that follows exponential discounting. And if the preferen... | arxiv_0002900 |
formation and evolution is to understand the relationship between galaxies and their host dark matter halos. In ΛCDM models galaxies grow hierarchically, and we expect to see the signatures of such growth in satellite galaxies, which trace the accreted dark-matter sub-halos. The distribution of satellite galaxies can ... | arxiv_0002902 |
Singular holomorphic foliations by curves I: Integrability of holonomy cocycle in dimension 2
=============================================================================================
We study the holonomy cocycle $\mathcal H$ of a holomorphic foliation F by Riemann surfaces defined on a compact complex projective ... | arxiv_0002906 |
On the nonexistence of -continuous path and tree-decompositions
===============================================================
Bojanczyk and Pilipczuk showed in their celebrated article *Definability equals recognizability for graphs of bounded treewidth* (LICS 2016) that monadic second-order logic can define tree-dec... | arxiv_0002910 |
VI-modules in non-describing characteristic, part I
===================================================
Let VI be the category of finite dimensional F*q*-vector spaces whose morphisms are injective linear maps, and let **k** be a noetherian ring. We study the category of functors from VI to **k**-modules in the case wh... | arxiv_0002913 |
obtained for LoTSS-DR1 (where both *σ**R**A* and *σ**D**E**C* were $\sim0.2\arcsec$)
We also explored the positional uncertainties as a function of S/N and found that, as expected, they increase gradually below approximately an S/N of 20 and reach a maximum of 0.5$\arcsec$ at a S/N of 5 - we characterise this behaviou... | arxiv_0002916 |
& 0.51
SVISS-SN54 & CC & 0.812& 0.77
SVISS-SN261& CC & 0.734& 0.37
SVISS-SN55 & CC & 0.995& 0.83
SVISS-SN31 & CC & 0.999& 0.12
SVISS-SN56 & CC & 0.930& 0.57
SVISS-SN357& CC & 1.000& 1.40
SVISS-SN24 & CC & 0.643& 0.81
For the sample of supernovae considered in this paper we used the pipeline described a... | arxiv_0002918 |
$$
In particular, if *n* is prime and congruent to 1 mod 3, then BB*n* = *p*(*n*).
There are 1203397779055806181762759 fullerenes with 10000 carbon atoms. This trails the leading term $\frac{809}{2^{15}3^{13}5^2}\sigma\_9(5000)$ by about 0.73%.
The final answer is a linear combination of Eisenstein series because of ... | arxiv_0002920 |
**Multiple-stopping time Sequential Detection for Energy Efficient Mining in Blockchain-Enabled IoT**
=====================================================================================================
What are the optimal times for an Internet of Things (IoT) device to act as a blockchain miner? The aim is to minimi... | arxiv_0002925 |
**Do halos that form early, have high concentration, are part of a pair, or contain a central galaxy potential host more pronounced planes of satellite galaxies?**
============================================================================================================================================================... | arxiv_0002926 |
When cold, dense quarks in 1+1 and 3+1 dimensions are not a Fermi liquid
========================================================================
=1 =7
We analyze the behavior of quarks coupled to a *S**U*(*N**c*) gauge theory in 1+1 dimensions. In the limit of strong coupling, the model reduces to a Wess-Zumino-Noviko... | arxiv_0002927 |
An Analytical, Statistical Approximate Solution for Dissipative and non-DissipativeBinary-Single Stellar Encounters
===================================================================================================================
We present a statistical approximate solution of the bound, non-hierarchical three-body ... | arxiv_0002929 |
Flavored ABJM theory on the sphere and holographic *F*-functions
================================================================
Introduction
============
UV-complete quantum field theories may be constructed as conformal field theories (CFTs) deformed by a relevant operator, which drives a renormalization group (RG) ... | arxiv_0002933 |
Velocity-Aware Handover Management in Two-Tier Cellular Networks
================================================================
While network densification is considered an important solution to cater the ever-increasing capacity demand, its effect on the handover (HO) rate is overlooked. In dense 5G networks, HO del... | arxiv_0002940 |
Reflected and Doubly RBSDEs with Irregular Obstacles and a Large Set of Stopping Strategies
===========================================================================================
We introduce a new formulation of reflected BSDEs and doubly reflected BSDEs associated with irregular obstacles. In the first part of t... | arxiv_0002950 |
Centaurus Lupus being one magnitude fainter than their USco counterparts according to theoretical models. They could also have escaped from the nearby *ρ* Oph star-forming region ( ≤ 1 Myr; 140 pc) where the occurence of disks is expected to be higher than in Upper Sco.
Three young accreting sources
-----------------... | arxiv_0002952 |
An Exact Consistent Tangent Stiffness Matrix for a Second Gradient Model for Porous Plastic Solids: Derivation and Assessment.
===============================================================================================================================
It is well known that the use of a consistent tangent stiffness ... | arxiv_0002956 |
Color-Magnitude Relations of Active and Non-Active Galaxies in the *Chandra* Deep Fields: High-Redshift Constraints and Stellar-Mass Selection Effects
======================================================================================================================================================
Å
We extend color... | arxiv_0002959 |
ʹ, *W*(*T*) ≤ *W*(*T*ʹ), with *T*ʹ be a spanning tree of *G* and *W*(*S*) = ∑*e* ∈ *S**w*(*e*) be the cost of the subgraph *S*).
The Algorithm
=============
In this section, we describe our self-stabilizing algorithm for the MST problem. We call this algorithm. In the next section, we shall prove the correctness of thi... | arxiv_0002965 |
Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching
==========================================================================
Introduction and sketch of the idea
===================================
In this paper, we connect the semi-classical analysis of asymptotically free quantum field theory ... | arxiv_0002973 |
Proper Plasma Analysis Practice (PPAP),an Integrated Procedure of the Extinction Correction and Plasma Diagnostics:a Demo with an HST/WFC3 Image Set of NGC6720
===============================================================================================================================================================
... | arxiv_0002976 |
ALMA spectroscopic survey in the Hubble Ultra Deep Field: CO luminosity functions and the evolution of the cosmic density of molecular gas
===========================================================================================================================================
In this paper we use ASPECS, the ALMA Sp... | arxiv_0002977 |
ON DEFORMATIONS OF CLASSICAL MECHANICS DUE TO PLANK-SCALE PHYSICS
=================================================================
Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This m... | arxiv_0002985 |
+ 1.193 | + 0.258 | + 0.921 |
| G4300 | + 1.513 | + 1.876 | + 1.482 | + 1.765 |
| HgA | − 4.383 | − 3.974 | + 3.414 | − 2.845 |
| HgF | − 2.630 | − 2.116 | + 0.999 | − 1.857 |
| Fe4383 | + 1.976 | + 3.753 | − 4.099 | + 1.361 |
| Ca4455 | + 0.582 | + 1.224 | + 0.062 | + 0.733 |
| Fe4531 | + 0.865 ... | arxiv_0003004 |
Higher Categories and Topological Quantum Field Theories
========================================================
We construct a state-sum type invariant of smooth closed oriented 4-manifolds out of a *G*-crossed braided spherical fusion category (*G*-BSFC) for *G* a finite group. The construction can be extended to ob... | arxiv_0003005 |
IFT-UAM/CSIC-18-038
arXiv:1805.05748
On Missing Bianchi Identities in Cohomology Formulation
Xin Gao†[1](#fn1), Pramod Shukla‡ ◇[2](#fn2), Rui Sun$^\spadesuit$[3](#fn3)
†Dipartimento di Fisica, Universita di Roma ``Tor Vergata“, and
I.N.F.N. Sezione di Roma ``Tor Vergata”,
Via della Ricerca Scientifica, 00133 Roma... | arxiv_0003009 |
inceton.edu`. Yingying Li, Department of Information Systems, Business Statistics and Operations Management, Hong Kong University of Science and Technology, Hong Kong. E-mail: `yyli@ust.hk`. Ke Yu, Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08540. E-mail:`kyu@Prince... | arxiv_0003010 |
K})^{1/2}$ . The basic ingredient for this relation is the thermal conductivity in the envelope, where ions are in the liquid phase and the conductivity is dominated by electrons . The presence of light elements and accretion are also important.
Old NSs, at the verge of photon domination, can decisively constrain QCD ... | arxiv_0003013 |
and Xe comparison arcs were obtained to calibrate the wavelength scale. The slit width was 15 and its position on the sky is shown in Fig. [fig:saltslit]. The position of the slit was chosen in this way, to be able to obtain the spectra of WR 48a and the WC8 star D2-3 (2MASS J13125770-6240599) simultaneously. As one c... | arxiv_0003015 |
be communicative. Each (meta)society is regarded as appearing and behaving independently of the others. The adopted metasocietal distribution function is
$$\begin{gathered}
\label{sec:MetaDistClassic}
{\zgDist (\zTuple | \gTuple)} = \fDirac(\zPosition - \hPositionMetaOrigin) \fDirac(\zSize) {\gPDF{\zDuration, \zFC,... | arxiv_0003016 |
Active Ising Models of Flocking: A Field-Theoretic Approach
===========================================================
Introduction
============
Flocking, in which a group of self-propelled particles align and move in the same direction, is displayed by a wide variety of biological and soft-matter systems. The alignme... | arxiv_0003028 |
Galois groups of uplifted de Sitter vacua
=========================================
Introduction
============
The Swampland program is a research program in string theory that aims to identify which effective quantum field theories can be consistently embedded into a theory of quantum gravity, such as string theory, an... | arxiv_0003033 |
0,4cm}m{0,4cm}m{0,4cm}m{0,4cm}m{0,4cm}m{0,4cm}m{0,4cm}}
10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 8 & 1 & 8 & 6 \\
0 & 4 & 9 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 8 & 10 & 0 & 0 & 0 & 0 & 0 \\
0 & 10 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 8 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 8 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 8 ... | arxiv_0003041 |
Artificial-Noise-Aided Secure MIMO Wireless Communications via Intelligent Reflecting Surface
=============================================================================================
This paper considers a MIMO secure wireless communication system aided by the physical layer security technique of sending artificia... | arxiv_0003046 |
Binary Neutron Star Mergers
===========================
We review the current status of studies of the coalescence of binary neutron star systems. We begin with a discussion of the formation channels of merging binaries and we discuss the most recent theoretical predictions for merger rates. Next, we turn to the quasi-... | arxiv_0003058 |
The faint end of the Centaurus A satellite luminosity function
==============================================================
The Panoramic Imaging Survey of Centaurus and Sculptor (PISCeS) is constructing a wide-field map of the resolved stellar populations in the extended halos of these two nearby, prominent galaxies... | arxiv_0003060 |
Superconducting and Pseudogap Transition Temperatures in High-Tc Cuprates and the *T**c* Dependence on Pressure
===============================================================================================================
We derive universal analytic expressions for the critical temperatures of the superconducting (S... | arxiv_0003061 |
)(0.22,0.12)2(1,0)0.22 (26.94,53.95)(0.21,0.14)2(1,0)0.21 (26.56,53.62)(0.13,0.11)3(1,0)0.13 (26.21,53.26)(0.12,0.12)3(0,1)0.12 (25.9,52.86)(0.1,0.13)3(0,1)0.13 (25.63,52.43)(0.13,0.21)2(0,1)0.21 (25.41,51.97)(0.11,0.23)2(0,1)0.23 (25.23,51.5)(0.18,0.47)1(0,1)0.47 (25.1,51.01)(0.13,0.49)1(0,1)0.49 (25.03,50.51)(0.08,0.... | arxiv_0003075 |
Local Helioseismology: Three Dimensional Imaging of the Solar Interior
======================================================================
Solar oscillations, convection zone dynamics, solar magnetism, sunspots, solar cycle
The Sun supports a rich spectrum of internal waves that are continuously excited by turbulent... | arxiv_0003079 |
Two-resonator circuit QED: Dissipative Theory
=============================================
[1](#fn1)
We present a theoretical treatment for the dissipative two-resonator circuit quantum electrodynamics setup referred to as quantum switch. There, switchable coupling between two superconducting resonators is mediated by... | arxiv_0003082 |
Rational Models for Inflation-Linked Derivatives
================================================
We construct models for the pricing and risk management of inflation-linked derivatives. The models are rational in the sense that linear payoffs written on the consumer price index have prices that are rational functions ... | arxiv_0003084 |
\rm bol})\le 10.4$). The galaxies selected in IR and not detected in UV span a very wide range of bolometric luminosity and hence of SFRs. The average trends found at z=0 are also reported in Fig. [lIRUV-lbol]. The samples at z=0 and z=0.7 cover the same range of luminosity. The slight difference found in the distribut... | arxiv_0003085 |
Two-Loop QCD Corrections to the Higgs plus three-parton amplitudes with Top Mass Correction
===========================================================================================
Introduction
============
Scattering amplitudes play an indispensable role in particle physics. They act as a bridge between theory and ... | arxiv_0003086 |
the lower panels of Fig. [fig:examplebigclasses] and Fig. [fig:exampleminorclasses], with the history always belonging to the galaxy for which the radial density profiles are shown in the upper panels of the same figures, and the intensity maps are shown in the according panels of Fig. [fig:showplots]. The redshifts o... | arxiv_0003097 |
On the characterisation of the underlying event
===============================================
The measurement of the underlying event (UE) and its separation from hard interactions in hadron-collider events is a conceptually and practically challenging task. We develop a simple, mostly analytical toy model for the UE... | arxiv_0003105 |
≈ 0.01 magnitudes for *z* = 0.1, but can be as as large as *σ**m* ≈ 0.05 magnitudes for *z* = 0.3.
### Background Estimation
To fully specify our filters, we also need to describe our background model. We assume the background galaxy density is constant in space, so that $b({{\mathbf{x}}})=2\pi R \bar \Sigma\_g(m\_i,c)... | arxiv_0003107 |
order book model
=====================================================================================================
We study a microscopic limit order book model, in which the order dynamics depend on the current best bid and ask price and the current volume density functions, simultaneously, and derive its macrosc... | arxiv_0003114 |
Testing Screened Modified Gravity
=================================
Long range scalar fields with a coupling to matter appear to violate known bounds on gravitation in the solar system and the laboratory. This is evaded thanks to screening mechanisms. In this short review, we shall present the various screening mechani... | arxiv_0003116 |
Activity cycles in members of young loose stellar associations
==============================================================
### Received ; accepted
In the present paper we searched for activity cycles in a sample of 90 young solar-like stars with ages between 4 and 95 Myr with the aim to investigate the properties of... | arxiv_0003122 |
In this paper we present with *algebraic trees* a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch point map which assigns to each triple of points their branch point. We give ... | arxiv_0003127 |
Secular dynamics of hierarchical multiple systems composed of nested binaries, with an arbitrary number of bodies and arbitrary hierarchical structure. First applications to multiplanet and multistar systems
================================================================================================================... | arxiv_0003136 |
Table [Basetransactions] (Page ). For *minsupp* = *4* and *minbond* = *0.20*, the representation RFCCP of the FCP set is equal to: {( (*C*, 4, 4), (*BE*, 4, 4)}.
The next theorem proves that the proposed RFCCP representation is a condensed exact representation of the FCP set of frequent correlated patterns.
The repres... | arxiv_0003138 |
\_2 k^2}{N^2} + O(k N^{-2}) +O(k^3 N^{-3}).$$
Fortunately, one-step method *θ*(1) could achieve a lower upper bound of MSE with only one additional step. we will show the following in Section [asym]:
$$\mathbb{E} [ \| \theta^{(1)} - \theta\_0 \|^2 ] \le \frac{C\_1}{N} + O(N^{-2}) + O(k^4 N^{-4}).$$
Main Results... | arxiv_0003140 |
Minimal elastic networks
========================
We consider planar networks of three curves that meet at two junctions with prescribed equal angles, minimizing a combination of the elastic energy and the length functional. We prove existence and regularity of minimizers, and we show some properties of the minimal con... | arxiv_0003142 |
Modified toric code models with flux attachment from Hopf algebra gauge theory
==============================================================================
Kitaev’s toric code is constructed using a finite gauge group from gauge theory. Such gauge theories can be generalized with the gauge group generalized to any fi... | arxiv_0003147 |
*d**P*(*u*, *c*\*) finds the most preferred candidate of voter *u*. In this way, *d**P*(*u*, *c*\*) can be 0 if *u* votes for *c*\*, even if the distance between *u* and *c*\* is not 0. Hence the complexity of *d**P*(*u*, *c*\*) is *O*(*m*), where *m* is the number of candidates. As explained above, most of the experi... | arxiv_0003158 |
From the hyperbolic 24-cell to the cuboctahedron
================================================
### May 2008
=1
[section] [thm]Lemma [thm]Corollary [thm]Main Lemma [thm]Proposition [thm]Conjecture [thm]Remark
[1] *Proof of #1:*
[section]
We describe a family of 4-dimensional hyperbolic orbifolds, constructed by defor... | arxiv_0003160 |
Estimating Average Treatment Effects with a Double-Index Propensity Score
=========================================================================
[firstpage]
We consider estimating average treatment effects (ATE) of a binary treatment in observational data when data-driven variable selection is needed to select relev... | arxiv_0003167 |
Creation of Ballot Sequences in a Periodic Cellular Automaton
=============================================================
Introduction
============
It is well-known that the soliton equations provide infinite dimensional completely integrable systems in the sense that they possess infinitely many first integrals. The... | arxiv_0003173 |
The role of gas accretion in galaxy evolution is still a matter of debate. The presence of inflows of metal-poor gas that trigger star formation bursts of low metallicity has been proposed as an explanation for the local anti-correlation between star formation rate (SFR) and gas-phase metallicity ($\rm Z\_g$) found in ... | arxiv_0003175 |
3pt);; (3.45,2.65) node *b*; (4,0) – +(-1,0); (4,0) – +(0,1); (4,0) – +(-1,1) – +(-1,0); (4,0) – +(-1,1) – + (0,1); ;
[scale=.5] (0,-1) (5,4); (1,0) rectangle (3,1); (3,1) rectangle (4,3); (1,3) rectangle (3,4); (1,1) – (4,1); (1,3) – (4,3); (1,4) – (4,4); (1,0) – (1,4); (3,0) – (3,4); (3.5,3.5) node $\varnothing$; (2... | arxiv_0003185 |
is also the highest *a**a* threshold that does not lead to a splitting of some large *I**n**t*(*a**p*) events into separate, smaller *I**n**t*(*a**a*) events).
A GPD ML fit to the extreme *I**n**t*(*a**a*) distribution was obtained by the same EVT method as before, giving *ξ* ∼ − 0.28 ± 0.6 and *σ* ∼ 4100 ± 3000 for ... | arxiv_0003186 |
At the borderline of shape coexistence: Mo and Ru
=================================================
Background
Even-even isotopes of Mo (*Z* = 42) and Ru (*Z* = 44) are nuclei close to the subshell closure at *Z* = 40, where shape coexistence plays a significant role. As a result, their spectroscopic properties are exp... | arxiv_0003203 |
Out-of-equilibrium steady states of a locally driven lossy qubit array
======================================================================
We find a rich variety of counterintuitive features in the steady states of a qubit array coupled to a dissipative source and sink at two arbitrary sites, using a master equation... | arxiv_0003206 |
}
= 1.96 \times 10^{-3} M\_\odot,\end{aligned}$$
where the inner region (*D* ≤ 5*R*S) contributes only ∼ 10% ( = 2.40 × 10− 4*M*⊙). The total ejecta mass in Eq. (16) is only a few percent of our assumed (initial) BH-torus mass ( ∼ 0.1*M*⊙; Eq. (11)). This may justify our assumption of steady-state neutrino-driven ou... | arxiv_0003211 |
*E**a**l**l* + *E*0,
where, we consider the exchange interaction *J*12 between spins at site 1 and 2. $\textbf{h}\_1= \sum\_{i\neq1,2} J\_{1i}\textbf{S}\_{i}, \textbf{h}\_2= \sum\_{i\neq1,2} J\_{2i}\textbf{S}\_{i}, E\_{\rm all}= \sum\_{i\neq1,2} J\_{ij}\textbf{S}\_{i}.\textbf{S}\_{j}$, and *E*0 contains all other no... | arxiv_0003221 |
Uplink HARQ for Cloud RAN via Separation of Control and Data Planes
===================================================================
The implementation of uplink HARQ in a Cloud- Radio Access Network RAN (C-RAN) architecture is constrained by the two-way latency on the fronthaul links connecting the Remote Radio Hea... | arxiv_0003226 |
Bit threads in higher-curvature gravity
=======================================
Introduction
============
Holographic entanglement entropy (HEE) has had a profound impact on our understanding of quantum gravity, by directly connecting quantum information and geometry. According to the Ryu-Takayanagi (RT) formula, the e... | arxiv_0003229 |
shift=(c0)] 120:0.61) ;
; ;
(b1) at ([shift=(b)] -80:1.5) ; (b2) at ([shift=(b)] 140:1.5) ;
(b0-60) at ([shift=(b0)] 60:0.61) ; (b0-240) at ([shift=(b0)] 240:0.61) ;
(b1-20) at ([shift=(b1)] 20:0.61) ; (b1-200) at ([shift=(b1)] 200:0.61) ;
(b0-20) at ([shift=(b0)] 20:0.61) ; (b0-200) at ([shift=(b0)] 200:0.61) ;
(b0-30... | arxiv_0003231 |
Piecewise Strong Convexity of Neural Networks
=============================================
We study the loss surface of a feed-forward neural network with ReLU non-linearities, regularized with weight decay. We show that the regularized loss function is piecewise strongly convex on an important open set which contains... | arxiv_0003236 |
Compactness of self-shrinkers in $\R^3$ with fixed genus
========================================================
We prove the compactness of self-shrinkers in $\mathbb R^3$ with bounded entropy and fixed genus. As a corollary, we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus... | arxiv_0003240 |
KCL-MTH-13-02
KIAS P-13017
Phases, Flops and F-theory:
*S**U*(5) Gauge Theories
1 *School of Physics, Korea Institute for Advanced Study,
Seoul 130-722, Korea*
hayashi kias.re.kr
2 *Department of Mathematics, King’s College, London
The Strand, London WC2R 2LS, England*
We consider F-theory and M-the... | arxiv_0003241 |
cont2]
[cont3]
[ACAcont]
[alloutflows]
---
1. Here, these sizes are analogous to diameters of the sources.[↩](#fnref1)
2. The protostellar systems may show different dust temperatures of the envelope+disk system based on the stellar luminosity. If these sources also have an extended but colder envelope, the mass of the... | arxiv_0003246 |
-enhanced population approximately matched a [C/Fe] of -0.3 dex and a [N/Fe] of 1.4 dex. The same sets of abundances were then found to match S(3839) and CH band measurements of main sequence stars by. These bands were then used to make abundance measurements for individual stars by, which also agreed with those previo... | arxiv_0003250 |
Consistent Right-Invariant Fixed-Lag Smootherwith Application to Visual Inertial SLAM
=====================================================================================
State estimation problems without absolute position measurements routinely arise in navigation of unmanned aerial vehicles, autonomous ground vehicl... | arxiv_0003252 |
Effective Electrostatic Interactions in Colloid-Nanoparticle Mixtures
=====================================================================
Interparticle interactions and bulk properties of colloidal suspensions can be substantially modified by addition of nanoparticles. Extreme asymmetries in size and charge between c... | arxiv_0003253 |
) and 0.3928 / 41.96% (Exp.4: class-imbalanced), respectively.
Thirdly, we can also observe that the heterogeneous problem existing between source and target databases raises the level of difficulty of the CDMER tasks. It is known that the samples in SMIC (NIR) are recorded by a near-infrared camera, whose image-qualit... | arxiv_0003263 |
Constraining Jet/Disk Geometry and Radiative Processes in Stellar Black Holes XTE J1118+480 and GX 339 − 4
==========================================================================================================
### Accepted 2009 April 8. Received in original form 2008 June 9
[firstpage]
We present results from model... | arxiv_0003266 |
Two decades of optical timing of the shortest-period binary star system HM Cancri
=================================================================================
### Accepted 2022 November 16. Received 2022 November 16; in original form 2022 September 1
[firstpage]
The shortest-period binary star system known to date... | arxiv_0003269 |
Photon bremsstrahlung from quark jet via transverse and longitudinal scatterings: single versus multiple scatterings
====================================================================================================================
We study the production of jet-bremsstrahlung photons through the scattering with the ... | arxiv_0003270 |
and assurance are likely to be affected by these weaknesses.
Expressivity
------------
An often quoted weakness of, particularly when applied to, is the “reality gap” that can exist between a naively constructed model and its corresponding real-world artefact. According to , over-reliance on simulation to test behavi... | arxiv_0003277 |
**Operator space tensor norms**
===============================
[1](#fn1)
namedefsubjclassname@2020 Mathematics Subject Classification
The use of a tensor product perspective has enriched functional analysis and other important areas of mathematics and physics. The context of operator spaces is clearly no exception. Th... | arxiv_0003280 |
The complex interplay between tidal inertial waves and zonal flows in differentially rotating stellar and planetary convective regionsI. Free waves
===================================================================================================================================================
In particular, tidal ine... | arxiv_0003282 |
see §3.1)
2) a ray-tracing algorithm based on the pre-calculation of a lower limit for the RFED (see §3.2 and §3.3)
3) an iterative procedure for the optimization of the angular density for the rays departing from each emitting cell (§3.3).
Furthermore, parallelized versions of the code have been written for its ... | arxiv_0003287 |
4 for both channels and different values of *η*, and show the result in panel (b) of Fig. [fig:triangularresults]. As is clear, in the close vicinity of *J*2/*J*1 = 1/8 the binding energy is positive in both channels, but is larger for the “chiral” *l* = 3 channel. At some value $J\_2/J\_1 \gtrsim 1/8$, which is marked... | arxiv_0003295 |
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