aa r X i v : . [ h e p - ph ] J un Applications of FIESTA
M. Tentyukov ∗ Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), D-76128Karlsruhe, GermanyE-mail: [email protected]
A.V. Smirnov
Scientific Research Computing Center, Moscow State University, 119992 Moscow, RussiaE-mail: [email protected]
Sector decomposition in its practical aspect is a constructive method used to evaluate Feynmanintegrals numerically. We present a new program performing the sector decomposition and in-tegrating the expression afterwards. The program can be also used in order to expand Feynmanintegrals automatically in limits of momenta and masses with the use of sector decompositionsand Mellin–Barnes representations. The program is parallelizable on modern multicore com-puters and even on multiple computers. Also we demonstrate some new numerical results forfour-loop massless propagator master integrals. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ pplications of FIESTA
M. Tentyukov
1. Introduction
Originally sector decomposition was used as a tool for analyzing the convergence and prov-ing theorems on renormalization and asymptotic expansions of Feynman integrals [1, 2, 3, 4, 5].After [6], the sector decomposition approach has become an efficient tool for numerical evaluatingFeynman integrals (see Ref. [7] for a recent review). At present, there are two public codes per-forming the sector decomposition [8] and [9]. The latter one was developed by the present authors;it is named
FIESTA which stands for “Feynman Integral Evaluation by a Sector decomposiTionApproach”. Recently
FIESTA was greatly improved in various aspects [10].During the last year
FIESTA was widely used, some of application are listed in [11]. In [12]we used
FIESTA in order to confirm numerically the recent analytic results for master integrals (MI’s) for four-loop massless propagators which recently were analytically evaluated in [13]. Herewe provide some more numerical results for extra orders in epsilon expansions for these MI’s.
2. Theoretical background and software structure
FIESTA calculates Feynman integrals with the sector decomposition approach. After per-forming Dirac and Lorentz algebra one is left with a scalar dimensionally regularized Feynmanintegral [14] F ( a , . . . , a n ) R · · · R d d k ... d d k l E a ... E ann , where d = − e is the space-time dimension, a n areindices, l is the number of loops and 1 / E n are propagators. We work in Minkowski space wherethe standard propagators are the form 1 / ( m − p − i ) . Other propagators are permitted, see [9].Substituting E aii = e ai p / G ( a ) R ¥ d aa a i − e − iE i a , after usual tricks [9], performing the decomposition ofthe integration region into the so-called primary sectors [6] and making a variable replacement,one results in a linear combination of integrals R x j = dx i . . . dx n ′ (cid:16) (cid:213) n ′ j = x a j − j (cid:17) U A − ( l + ) d / F A − ld / If the functions U A − ( l + ) d / F A − ld / had no singularities in e , one would be able to perform the expansionin e and perform the numerical integration afterwards. However, in general one has to resolvethe singularities first. Thus, one starts a process the sector decomposition aiming to end with asum of similar expressions, but with new functions U and F which have no singularities (all thesingularities are now due to the part (cid:213) nj = x ′ a ′ j − j ). The way sector decomposition is performed iscalled a sector decomposition strategy ([6, 8, 9]) and is an essential part of the algorithm (let usalso mention a geometrical approach to sector decomposition [15] which is rather complicated inimplementation as a strategy on a computer but promises to be the optimal one).After the sector decomposition one resolves the singularities by evaluating the first terms ofthe Taylor series: in those terms one integration is taken analytically. Afterwards the e -expansioncan be performed and finally one can do the numerical integration. FIESTA is written in
Mathematica [17] and C. The user is not supposed to use the Cpart directly as it is launched from
Mathematica via the Mathlink protocol in order to per-form a numerical integration. To run
FIESTA , the user has to load the
FIESTA package into
Mathematica
SDEvaluate[UF[loop_momenta,propagators,subst], indices,order] , where loop_momenta is a list of all loop momenta, propagators is a list of all propagators, subst is a list of substitutions for external momenta, masses and other values. For example,2 pplications of FIESTA
M. Tentyukov
SDEvaluate[UF[{k},{-k ,-(k+p ) ,-(k+p +p ) ,-(k+p +p +p ) },{p → → →
0, p p → -s/2,p p → -t/2,p p → -(s+t)/2,s → -3,t → -1}], {1,1,1,1},0] evaluats the massless on-shell box diagram with Mandelstam variables equal to − −
3. Numerical results for four-loop massless propagators M , e M , e M , e M , e M , e M , e M , e M , e M , e M , e M , e M , e M , e N , e Figure 1: M – M : the thirteen complicated four-loop master integrals according to [13]. The two MI’s M and M can be identically expressed through the three-loop nonplanar MI N . In [18] a full set of four-loop massless propagator-like MI’s was identified. There are 28 inde-pendent MI’s. Analytical results for these integrals were obtained in [13]. The most complicatedMI’s are demonstrated on Fig. 1. e m after M i j stands for the maximal term in e -expansion of M i j which one needs to know for evaluation of the contribution of the integral to the final result for afour-loop integral after reduction is done, see [13]. Two of the complicated integrals ( M and M )are related by a simple factor with the three-loop MI N [12] so it is enough to evaluate remainingeleven complicated MI’s M – M as well as first three terms of the e -expansion of N .We calculated them (for q = −
1) using
FIESTA with the
Cuba [16]
Vegas integrator and1 500 000 sampling points for integration. Our results alongside with the corresponding analyticalexpressions (transformed to the numerical form) from [13] look like follows : M e − : 0.08333 ± e − : 0.916667 ± e − : 5.64251 ± e − : 27.6413 ± e : 98.638 ± e :342.736 ± e : 857.88 ± e : 2659.84 ± e : 4344.28 ± e : 17483.1 ± M e − : 0.601028 ± e − : 7.4231 ± e : 44.9127 ± e : 217.023 ± e : 780.436 ± e :2678.13 ± e : 7195.9 ± Please, note that the overall normalization used by
FIESTA is different from the one employed by the authors of[13], see [12]. pplications of FIESTA M. Tentyukov M e − : 5.184645 ± e : 38.8948 ± e : 240.069 ± e : 948.623 ± e : 3679.77 ± M e − : 20.73860 ± e : 102.033 ± e : 761.60 ± e : 2326.18 ± e : 12273.6 ± M e − : 20.73860 ± e : 145.381 ± e : 985.91 ± e : 3930.65 ± e : 17486.6 ± M e : 55.58537 ± e : 175.325 ± e : 1496.52 ± M e : 52.0181 ± e : 175.50 ± e : 1475.272 ± e : 2623.5 ± M e − : -5.184651 ± e : -32.0962 ± e : -91.158 ± e : 119.06 ± e : 2768.6 ± N e : 20.73857 ± e : 190.60 ± e : 1049.20 ± e : 4423.84 ± e : 16028.8 ± M e − : -10.36931 ± e : -70.990 ± e : -21.650 ± e : 2832.69 ± M e − : -10.36933 ± e : -58.6187 ± e : 244.681 ± M e − : -5.18467 ± e : 14.3989 ± e : 739.979 ± e -expansion in comparison (inparentheses) with the known from [13] analiycal results (if any). As we can see, our calculationsreproduce the result of [13] with 3-4 correct digits. The extra terms in the e -expansion of each MIwhich are currently unavailable analytically but are necessary for future five-loop calculations.
4. Conclusion
Usually, analytical evaluation of multiloop MI is a kind of art. It requires a lot of efforts (andCPU time). In many situations, independent checkup is hardly any possible in reasonable time.That is why the simple in use tools for numerical evaluation like
FIESTA are important.
Acknowledgments.
This work was supported in part by DFG through SBF/TR 9 and the Rus-sian Foundation for Basic Research through grant 08-02-01451.
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