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When [MATH] and [MATH] one uses [EQUATION] to obtain the contribution [EQUATION] When [MATH] and [MATH] , the matrix element [EQUATION] |
yields [EQUATION] Next one considers [MATH] but [MATH] , with matrix element [EQUATION] to find [EQUATION] where we have defined the average |
[EQUATION] Finally the case [MATH] [MATH] [EQUATION] provides the last contribution [EQUATION] Summing these contributions one finds the variance of [MATH] |
[EQUATION] Recall that the average value of [MATH] is [MATH] , and [MATH] is not constant, so [MATH] and therefore the [MATH] contribution does not vanish. |
What about the unanchored [MATH] ? Recall that the [MATH] term in [MATH] is canceled by a term in [MATH] , and so could the [MATH] contribution to the unanchored [MATH] vanish? The [MATH] term enters at [MATH] so it may seem promising. The trouble is the first term in ( 4.18 ). It is identical to the [MATH] term consid... |
5.2 The Variance of [MATH] at [MATH] Once the anchor is included, one arrives at our master formula for [MATH] in Eq. ( 4.23 ). Here all terms that could potentially give [MATH] contributions on dimensional grounds are gone. The constant term does not contribute to the variance and so we will drop it. We will also shif... |
[EQUATION] The variance is just [EQUATION] We are interested in the [MATH] contribution, which arises entirely from the Gaussian correlations, corresponding to disconnected pairs of 2 point functions. When more than one pairing is available, the sum over pairings may increase the diagram by a factor of 2 or 3 however t... |
We begin with easiest, corresponding to [EQUATION] There are only two distinct values of [MATH] and [MATH] . As [MATH] and [MATH] , this implies that [MATH] and [MATH] . Since both points are degenerate, this means that also [MATH] and [MATH] . Now in this case and in all cases that follow, the matrix element is entire... |
[EQUATION] where we have defined [EQUATION] We next consider the diagram [EQUATION] Here again there are only two values of [MATH] and [MATH] and so again [MATH] and [MATH] . One of these is a double point. If it is [MATH] then [MATH] , but if it is [MATH] then [MATH] . These two cases give equal contributions, and so ... |
[EQUATION] The first line gives [MATH] . To simplify the second line, we can use the binned version of the Bethe equation ( 4.15 ) to sum over [MATH] and [MATH] , leaving |
[EQUATION] where we have again used the fact that the average value of [MATH] is [MATH] The third diagram is [EQUATION] Now there are three columns, and so there are inequivalent pairings of [MATH] and [MATH] . One may have [MATH] [MATH] [MATH] or [MATH] . The first two give equal contributions, as there is a symmetry ... |
[EQUATION] While in the second it is [EQUATION] This explains the factor of two difference. The signs are different because in the second case one exchanges one pair of [MATH] . Both [MATH] and also the zeroed form of [MATH] are antisymmetric with respect to this interchange. |
Summarizing, we only need to consider the first of the four possibilities, and the contribution of the other diagrams will give a weight factor of [MATH] . This is |
[EQUATION] Note that this is equal to our first expression for [MATH] in Eq. ( 5.19 ) except for the [MATH] integral which is multiplied by a factor of [MATH] and the matrix element which is multiplied by [MATH] . Therefore |
[EQUATION] The next diagram is three by three [EQUATION] Again, corresponding to the three columns, there are three possible values of the [MATH] and [MATH] , yielding the same four pairings as above. The integration factors are the same and so again the weights are [MATH] [MATH] [MATH] and [MATH] and so it will suffic... |
[EQUATION] The first line yields [MATH] . As in the case of [MATH] , the [MATH] and [MATH] may be summed in the last line using the binned Bethe equation, leaving |
[EQUATION] Comparing with Eq. ( 5.23 ) we see that [EQUATION] The final diagram is [EQUATION] Again there are three columns and so the same three values of [MATH] and [MATH] , with the same weights and so we need only consider the first case [MATH] . Unlike the case of [MATH] , now [MATH] implies that [MATH] . Thus we ... |
[EQUATION] The first row is identical to that of [MATH] in Eq. ( 5.2 ). What about the second row? If one exchanges [MATH] with [MATH] then the [MATH] terms look the same, but with their indices reversed. Transposing the indices gives a minus sign in each summand. However [MATH] is also antisymmetric under the exchange... |
[EQUATION] Adding all of these terms together we find that the variance of the anchored [MATH] , at [MATH] , is [EQUATION] Is this zero? We numerically integrated the continuum expressions for [MATH] and [MATH] in Eq. ( 2.40 ) to obtain |
[EQUATION] and so [EQUATION] Is this compatible with zero? It is nearly twice the best fit Gaussian variance found at [MATH] in Fig. , but this is not obviously a sign of incompatibility as the [MATH] term could easily drive it down, with a coefficient of order unity. |
Conclusions Our goal is to devise a method to calculate, to arbitrary accuracy, the ground state and first excited state wave functionals of the [MATH] nonlinear sigma model. We would like to study the behavior of these wave functionals acting on a fixed-time configuration which circumnavigates the target space at each... |
This model is equivalent to a high spin Heisenberg XXX spin chain, for which the states are in principle known, but in a rather unwieldy form which would be difficult to map to the sigma model. Therefore one needs a prescription to calculate the spin chain matrix elements which is sufficiently simple so that it can be ... |
We begin, for sanity’s sake, with spin [MATH] . To cast our problem in a way which is close to continuum field theory, we collected the lattice sites into bins. We believe that it is the bins, and not individual pairs of sites, which will eventually correspond to points in the continuum field theory. We then average aw... |
Thus the validity of our binning approach is, for the time being, taken as a postulate. Once we are able to calculate the matrix elements, we may be able to use them to calculate [MATH] -point functions. These are known, and so we can in principle test the consistency of the postulate. Even if the postulate is true, we... |
Assuming this binning postulate, we found that standard combinatorial arguments in terms of partitions describe the behavior of the bins. Thus instead of complexities which are polynomial in [MATH] , the chain length [MATH] essentially disappears from the problem. This combinatorial approach partially fixes the behavio... |
Our strategy is to encode the information about a matrix element in a single function, [MATH] , which is the density of phases [MATH] in the CBA. The Fourier transform of [MATH] gives a matrix element. Such an approach would be possible even without binning, but we use the combinatorics of the binning to calculate the ... |
Our initial hope was that [MATH] would be a Gaussian, and so this would be straightforward. However it turned it that the variance was of order [MATH] . In the Gaussian approximation this would lead to matrix elements of order [MATH] , which is inconsistent with the fact that there are only [MATH] states. Our next hope... |
To fix this, we modified [MATH] by introducing an anchor which leaves the matrix elements invariant. This anchor has a number of nice properties. First, using the binning approximation we were able to show that the variance of the anchored [MATH] is only [MATH] . Numerically we were able to show, at [MATH] , that the a... |
In general we focused our attention on a single matrix element, that relating the classical and quantum ground states [MATH] . The quantum ground state enters rather superficially in the last step, when one performs a numerical integral, and so it is likely that the generalization to other quantum states is not difficu... |
What about the Gaussian approximation? If indeed [MATH] is a Gaussian, then matrix elements of [MATH] are only obtained if the variance of our anchored [MATH] is [MATH] . The anchor eliminates the [MATH] part and we have calculated here the [MATH] contribution. We found that the [MATH] coefficient is quite small and in... |
If the variance contains a term of [MATH] , then that term will dominate the variance at large [MATH] , which is the limit of interest. But the question is whether it will dominate the matrix elements. If it does, then the matrix elements will be of order [MATH] and so cannot be normalized and we will arrive at an inco... |
Consider the following three distributions [MATH] . The first is a Gaussian with variance that scales as [MATH] at large [MATH] . The second is the weighted sum of two Gaussians with [MATH] -independent weights, one with a variance of [MATH] and the other with a variance of [MATH] . The third, which generalizes the sec... |
[EQUATION] In the first case, the matrix elements will be [MATH] and so we will have an inconsistency. In the second, at large [MATH] the broader Gaussian simply ceases to contribute to the matrix elements, and so the matrix elements are of [MATH] as desired, determined entirely from the thin Gaussian. These first two ... |
So which is the case at hand? Having only calculated the variance, it is too early to say. The calculation of higher moments can distinguish these cases, although at any finite moment, assumptions about the form of [MATH] will be necessary to determine the potential. In other words, a concrete statement of the absence ... |
In fact, it is possible for us to go beyond simply calculating moments. Eq. ( 4.4 ) is the entire joint probability density function (PDF) for [MATH] [MATH] [MATH] and [MATH] . It can be put in a useful form with the expansion ( 4.3 ) and [MATH] can be found by imposing that [MATH] as was done in Subsec. 4.3 . Summing ... |
Summarizing, we appear to be well along the way to calculating the matrix element of the classical and quantum ground states of the [MATH] model, and the other matrix elements appear to be similar. It is possible in principle that the anchor that we have adapted does not render [MATH] sufficiently close to a Gaussian f... |
And if this all works, how do we get to higher spin? After all, there is no CBA in these cases ? The algebraic Bethe Ansatz provides a much more complicated construction of these states. However on the bright side they are still constructed from [MATH] commuting copies of the creation operators [MATH] , and so there is... |
Acknowledgement JE is supported by the CAS Key Research Program of Frontier Sciences grant QYZDY-SSW-SLH006 and the NSFC MianShang grants 11875296 and 11675223. JE also thanks the Recruitment Program of High-end Foreign Experts for support. |
Appendix A The Map Between the Spin Chain and Sigma Model The [MATH] nonlinear sigma model and the antiferromagnetic XXX spin chain at spin [MATH] are equivalent in the limit [MATH] . This was shown classically by Haldane in Refs. |
, where it was seem that classically finite [MATH] corresponds to a finite coupling of the sigma model. At the quantum level, the sigma model coupling runs and so there is no such dimensionless free parameter. Nonetheless the exact quantum correspondence in the infinite [MATH] limit was shown in Ref. |
. We will review that argument, following the presentation in Ref. In Sec. we introduced the spin [MATH] antiferromagnetic XXX spin chain. The general spin [MATH] spin chain, introduced in Ref. |
, is similar. In this case, the Hilbert space at each lattice site is [MATH] , and the [MATH] Lie algebra, with generators [MATH] at each lattice site [MATH] acts on this Hilbert space in the [MATH] -dimensional representation. The Hamiltonian must include higher order couplings of neighboring sites if one demands inte... |
Define the following combinations of operators [EQUATION] The intuition for the connection to the [MATH] sigma model is as follows. In the classical ground state, neighboring spins are antialigned and so [MATH] . Classically one may replace the [MATH] with their eigenvalues and so conclude that [MATH] and so |
[EQUATION] which in the large [MATH] limit tends to unity. More nontrivially, in the large [MATH] limit this antialignment holds even quantum mechanically, in the sense that the energy required to get a finite fractional difference between the eigenvalues of the spin operators at adjacent sites becomes infinite. Thus i... |
Now for a more rigorous description of the equivalence with the [MATH] model. The algebra satisfied by these new operators is easily calculated from that of [MATH] to be |
[EQUATION] We may recognize the first two of these as the canonical commutation relations of the discretized [MATH] model with coupling [MATH] if [MATH] are the coordinates at the [MATH] th lattice point and [MATH] is the canonical momentum |
[EQUATION] However the third relation in ( A.3 ) agrees with the commutation relations of the canonically quantized sigma model only in the limit [MATH] , where it vanishes. Therefore the finite [MATH] spin chain corresponds to a noncommutative deformation of the [MATH] sigma model. |
# Source: arxiv 1806.07874 # Title: Optimizing momentum resolution with a new fitting method for silicon-strip detectors # Sections: all # Downloaded: 2026-03-03T05:16:55.480622+00:00 |
Optimizing momentum resolution with a new fitting method for silicon-strip detectors Abstract A new fitting method is explored for momentum reconstruction. The tracker model reproduces a set of silicon micro-strip detectors in a constant magnetic field. The new fitting method gives substantial increases of momentum res... |
[MATH] -algorithm (the best one) and the two-strip center of gravity. The gain obtained in momentum resolution is measured as the virtual magnetic field and the virtual signal-to-noise ratio required by the two standard fits to reach an overlap with the best of two new methods. For the low noise side, the virtual magne... |
keywords: Performance of High Energy Physics Detectors, Pattern recognition, cluster finding, calibration and fitting methods, Si microstrip and pad detectors, Analysis and statistical methods |
Introduction The momenta of the charged particles are fundamental pieces of information for the event reconstructions in high energy particle physics. Very complex instruments (trackers) have been developed for this task where the key element is a uniform (or near to) magnetic field. In a homogeneous magnetic field a f... |
(to mention the most recent ones of a very large set) are addressed to the first three effects. The handling of the full non linearity of the particle path is clearly described in ref. |
This work is addressed to an accurate study of the statistical effects in the positioning algorithms of minimum ionizing particle (MIP) and how to reduce at minimum their influence in the momentum reconstruction. For this task, realistic probability density functions (PDFs; probability distributions in the physicist ja... |
where exact analytical forms were demonstrated for its systematic error. Due to its (apparent) simplicity the COG is one of the most used positioning algorithms: [MATH] , where |
[MATH] and [MATH] are the signals and the positions used. Even if a single name is used for the COG, slight different version are in use. The principal differences are in number of signal used, each selection implies different analytical properties and different systematic errors. The differences of each COG-version ar... |
and the mathematical approaches explored there for the COG study are very complex, but these complexities are typical of the fight against the systematic errors. For example, Gauss demonstrated what he called |
"Theorema Egregium" for the systematic errors of his topographic maps, and the recommendations on the systematic error suppression are always preliminary statements in his papers on least squares |
With heteroscedasticity as an essential assumption, ref deviates largely from the standard expositions of the least squares, those simplifications (often presented as Gauss-Markov theorem) surely generate many suboptimal fits. |
An interesting achievement, to suppress the COG systematic error, was reported in ref. i.e. the [MATH] -algorithm. A preliminary assumption of the method is the uniform "illumination" of the strips by MIP signals from parallel incidence. In this case the non uniform distribution of the COG , given by systematic error, ... |
is limited to symmetric signal distributions collected by the two (adjacent) leading strips of a cluster. It is evident that for inclined tracks or in presence of a magnetic field the symmetry, if present at orthogonal incidence, is surely destroyed. To overcame these limitations, the analytic methods developed in ref. |
was essential. It was used in ref. to generalize the [MATH] -algorithm beyond the two strip limitation and to extract the required corrections for non symmetric configurations. The set of generalizations will be indicated as [MATH] -algorithm, where the index |
[MATH] is connected to the number of strips contained in the COG algorithm from which the [MATH] algorithm is derived. Here, we will use always the COG and the [MATH] -algorithm. At our orthogonal incidence, the signals collected by the two leading strips suffice for the reconstructions. |
The fit of straight tracks was the first problem we faced with this new fitting method, even if the straight tracks are not the principal aim in high energy physics. The results obtained were described in ref. |
The limitation to straight tracks was mainly due to the complexity of the method we applied for the first time. Hence, working with two parameters, the debugging and testing of the application can be followed on a surface. Furthermore, the data, elaborated in our approach, were collected in a CERN test beam |
in the absence of a magnetic field. The detectors, we used, were few samples of double-sided silicon microstrip detector as those composing the PAMELA tracker |
The results of ref. showed a drastic improvement of the fitted track parameters respect to the results of the least squares methods. We observed excellent reconstructions even in presence of very noisy hits, often called outliers , that generally produce bad fits. This achievement is almost natural for the heavy tails ... |
We have to recall that the perception of a rough handling of the hit PDFs is well present in literature, and Kalman-filter modifications are often studied to accept extended deviations from a pure Gaussian model. These extensions use linear combinations of different Gaussians |
, the number of them is limited to avoid the intractability of the equations. The unknown parameters are selected from an optimization of the fit results. In late sense, our schematic approximations could be reconnected to those extensions. In fact, to speed the convergence of the maximum likelihood search, we calculat... |
The confidence gained in ref. with straight tracks allow us to deal with more complex and appealing tasks, i.e. the reconstruction of tracks in a magnetic field and the measure of their momenta. To face this extension with the minimum modifications of our previous work, and saving a backward compatibility, we will util... |
thus this reference will be often quoted as a repository of details not reported here. Now those data are adapted to the PAMELA geometry and to a constant magnetic field of [MATH] near to the average field of the PAMELA tracker. This relatively low magnetic field introduces small modifications of the parameters obtaine... |
[MATH] . Around this incidence angle, the average signal distribution of a MIP is practically identical to that at orthogonal incidence in the absence of a magnetic field. Thus, without further notice, we will assume a rotation of the detectors of their Lorentz angle respect to our MIP direction. These assumptions allo... |
The simulations on this side, as bending side, can give a glimpse of our approach for other type of trackers even for the (small angle stereo) double-sided detectors of the ALICE |
experiment. For these reasons, this side will be indicated as normal noisy side. In section 2, few details of this new fitting method will be recalled with a derivation of the general probability distribution. Sections 3 and 4 are devoted to the momentum PDF for the each side of this two sided detectors. In section 5 t... |
Details of the method The possibility of different statistical properties of each hit is based on well known properties of the MIP hitting a silicon strip (or gas discharge) detectors, largely reported in literature: a) the charge released in each hit has a Landau distribution with different signal-to-noise ratio. b) T... |
for gas ionization chambers) c) Each strip has its own noise. Thus the assumption of homoscedasticity (on an entire detector layer) is inconsistent for the previous well known reasons, which are common to a large class of detectors. To handle these random effects a complex mathematical infrastructure is required, it mu... |
supporting a general aspect of this border effect. The floating strip side is almost always much better, the vertical scale of the right plot is around three times that of the left plot. |
The demonstration of different PDF for groups of hits, or as in our case a different PDF for each hit, poses serious problems to the use the standard least squares as a fitting tools. A frequently used assumption to derive the least squares (or linear regression as it is often called) properties is the identity of the ... |
where heteroscedasticity is assumed from the very beginning. 2.1 A short derivation of the PDF for the COG algorithm The incidence angle of our MIPs imposes the use of the minimum number of signal strips to reduce the noise, hence, as stated above, only two signal strips will be used. In ref. |
we indicated the principal steps required to obtain the PDF for the COG (two-strip COG), those steps followed the standard method described in the books about the theory of probability. At first one has to obtain the cumulative distribution with integrations on complex surfaces, after, the derivative of the cumulative ... |
[MATH] [MATH] , and [MATH] , respectively the signal of the right strip, central strip (with the maximum signal) and left strip. If [MATH] |
the COG is [MATH] , if [MATH] it is [MATH] . Thus: [EQUATION] where [MATH] is the PDF to have the signals [MATH] from the strips [MATH] . The signals [MATH] are at their final elaboration stage (pedestals, common noise, etc.) and ready to be used for the hit-position reconstruction. The function [MATH] is the Heaviside... |
[MATH] for [MATH] [MATH] for [MATH] , and [MATH] is the Dirac [MATH] -function. The normalization of [MATH] can be easily verified by an integration on [MATH] Equation can be further elaborated with the following transformations. Splitting the sum of eq. in two independent integrals and transforming the variables [MATH... |
[MATH] , the jacobian of the transformation is one and the integrals in [MATH] and [MATH] can be performed with the rule: [EQUATION] |
Applying eq. to eq. and using the limitations of the two [MATH] -functions, eq. becomes: [EQUATION] This form underlines very well the similarity with the Cauchy PDF; in the limit of [MATH] the [MATH] -part of the [MATH] arguments are |
[MATH] and [MATH] for large [MATH] 2.2 The probability [MATH] for small [MATH] The probability [MATH] can handle a strict correlation among its arguments, we release this strict correlation with the weakest one: the mean values of the strip signals [MATH] are correlated, but the fluctuations around [MATH] are independe... |
(the strip additive noise is well reproduced by a gaussian in the absence of MIP signals). To simplify, the constants [MATH] are the noiseless signals released by a MIP with impact point [MATH] |
[EQUATION] Even with the gaussian functions, the integrals of eq. have no analytical expressions and effective approximations must be constructed. We will not report our final forms that are very long, instead we will illustrate a limiting case which gives a simple approximation and eliminates a disturbing singularity ... |
[MATH] converge to two Dirac [MATH] -functions. Hence, for small [MATH] (or better for [MATH] ), the integrals of eq. can be expressed as: |
[EQUATION] Equation is correct for [MATH] , but it is useful beyond this limit, as far as the pole for [MATH] is irrelevant, and contains many ingredients of more complex expressions. |
An example of eq. can be seen in the right side of figure The essential elements are the two maxima (gaussian-like) centered in the possible noiseless two-strip COG: [MATH] and [MATH] The widths of the two maxima are modulated by the signal-to-noise ratio [MATH] and [MATH] , the [MATH] |
factor is relevant at the borders of the [MATH] -range, i.e. [MATH] . Each maximum has a scaling factor [MATH] or [MATH] that, for the large majority of strip clusters |
[MATH] , is the signal to noise ratio of the central strip. Increasing the impact position [MATH] the lowest maximum tend to disappears. Similarly at decreasing [MATH] the highest maximum is rapidly reduced. The tails of the maxima differ drastically from gaussian PDF. More complete expressions of eq. contain terms ver... |
The dimensions of the constants [MATH] must be those of the [MATH] for both of them we take directly the ADC counts. The [MATH] -variable (the COG ) is a pure number expressed as a fraction of the strip size, or more precisely, the strip size is the scale of lengths. In the simulated distribution of figure , the partic... |
2.3 The functional dependence from the impact point For our fitting task, the PDFs with constant [MATH] , the noiseless version of the strip signals, and variable [MATH] (the COG noisy values), are irrelevant. We need the PDFs of the impact point [MATH] at constant (noisy) [MATH] . It is clear that the mean values of e... |
, for orthogonal incidence. Identical expressions will be used here. These new pieces of information, contained in the [MATH] extend the PDF of eq. to be function of the impact point. The PDF can be rewritten as: |
[EQUATION] Where [MATH] is the term in square brackets of eq. . Equation is the mathematical infrastructure able to connect the points a), b), c) discussed above. The functions [MATH] |
are the fractions of signal collected by the strips [MATH] in function of [MATH] they are normalized with [MATH] , the total signal of the three strips: the central one with the maximum signal and the two lateral. The extraction of the functions [MATH] from real data described in ref. |
is a delicate operation, but a strict compliance to the described equations produces very reliable results. In any case, slight variations of the [MATH] around the best one give almost identical track parameter distributions, thus their selection is very important but less critical than expected. Our function [MATH] ha... |
The parameters [MATH] are the standard deviations (eq. ) of the three strips considered. Equation can easily handle strips with different [MATH] , but, in the simulations, we will use identical [MATH] s for all the strips of same detector side. To easy the notations, these parameters will not be reported in the future ... |
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