Applications of FIESTA
Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), D-76128Karlsruhe, GermanyE-mail: A.V. Smirnov
Scientific Research Computing Center, Moscow State University, 119992 Moscow, Russia

Sector decomposition in its practical aspect is a constructive method used to evaluate Feynman integrals 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 Feynman integrals automatically in limits of momenta and masses with the use of sector decompositions and Mellin–Barnes representations. The program is parallelizable on modern multicore com- puters and even on multiple computers. Also we demonstrate some new numerical results for four-loop massless propagator master integrals.
13th International Workshop on Advanced Computing and Analysis Techniques in Physics ResearchFebruary 22-27, 2010Jaipur, India Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence.
Applications of FIESTA Originally sector decomposition was used as a tool for analyzing the convergence and prov- ing theorems on renormalization and asymptotic expansions of Feynman integrals [, .
After [], the sector decomposition approach has become an efficient tool for numerical evaluating Feynman integrals (see Ref. for a recent review). At present, there are two public codes per- forming the sector decomposition [and . The latter one was developed by the present authors; it is named FIESTA which stands for "Feynman Integral Evaluation by a Sector decomposiTion Approach". Recently FIESTA was greatly improved in various aspects [].
During the last year FIESTA was widely used, some of application are listed in [. In ] 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 . Here we 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 Feynman integral [] F(a1, . . , an) R · · · R ddk1.ddkl , where d = 4 − 2ε is the space-time dimension, a E 1 .Ean indices, l is the number of loops and 1/En are propagators. We work in Minkowski space wherethe standard propagators are the form 1/(m2 − p2 − i0). Other propagators are permitted, see .
Substituting 1 dααai−1eiEi , after usual tricks , performing the decomposition of the integration region into the so-called primary sectors and making a variable replacement, one results in a linear combination of integrals R 1 i . . dxnj=1 x j If the functions UA−(l+1)d/2 had no singularities in ε, one would be able to perform the expansion in ε 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 a sum of similar expressions, but with new functions U and F which have no singularities (all the singularities are now due to the part ∏nj=1 xaj−1). The way sector decomposition is performed is called a sector decomposition strategy ([) and is an essential part of the algorithm (let us also mention a geometrical approach to sector decomposition [] which is rather complicated in implementation 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 of the Taylor series: in those terms one integration is taken analytically. Afterwards the ε-expansioncan be performed and finally one can do the numerical integration.
FIESTA is written in Mathematica [and C. The user is not supposed to use the C part 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 6 or 7. In order to evaluate a Feynman integral one has to use the command 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, Applications of FIESTA SDEvaluate[UF[{k},{-k2,-(k+p1)2,-(k+p1+p2)2,-(k+p1+p2+p4)2},{p21 →0,p22 →0,p24 →0, p1 p2 →-s/2,p2 p4 →-t/2,p1 p4 →-(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 and −1.
3. Numerical results for four-loop massless propagators
Figure 1: M61–M43: the thirteen complicated four-loop master integrals according to [. The two MI's
M52 and M43 can be identically expressed through the three-loop nonplanar MI N0.
In [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 . The most complicated MI's are demonstrated on Fig. εm after Mi j stands for the maximal term in ε-expansion of Mi jwhich one needs to know for evaluation of the contribution of the integral to the final result for a four-loop integral after reduction is done, see []. Two of the complicated integrals (M43 and M52)are related by a simple factor with the three-loop MI N0 [so it is enough to evaluate remainingeleven complicated MI's M61–M36 as well as first three terms of the ε-expansion of N0.
We calculated them (for q2 = −1) using FIESTA with the Cuba[Vegas integrator and 1 500 000 sampling points for integration. Our results alongside with the corresponding analytical expressions (transformed to the numerical form) from [look like follows1: : 0.08333 ± 0 (0.08333); ε−3: 0.916667 ± 0.000018 (0.91666); ε−2: 5.64251 ± 0.00022 (5.6425109); ε−1: 27.6413 ± 0.00077 (27.6412581); ε0: 98.638 ± 0.0034 (98.637928); ε1:342.736 ± 0.012 (342.7349920); ε2: 857.88 ± 0.048 (857.8735165); ε3: 2659.84 ± 0.19(2659.825402); ε4: 4344.28 ± 0.75 (unknown); ε5: 17483.1 ± 5.7 (unknown).
: 0.601028 ± 0.000012 (0.601028); ε−1: 7.4231 ± 0.00024 (7.423055); ε0: 44.9127 ± 0.00073 (44.91255); ε1: 217.023 ± 0.0037 (217.0209); ε2: 780.436 ± 0.013 (780.432); ε3:2678.13 ± 0.053 (unknown); ε4: 7195.9 ± 0.3 (unknown).
1Please, note that the overall normalization used by FIESTA is different from the one employed by the authors of Applications of FIESTA : 5.184645 ± 0.000042 (5.1846388); ε0: 38.8948 ± 0.00039 (38.8946741); ε1: 240.069 ± 0.0019 (240.0684359); ε2: 948.623 ± 0.0091 (unknown); ε3: 3679.77± 0.06 (unknown).
: 20.73860 ± 0.00023(20.7385551); ε0: 102.033 ± 0.003 (102.0326759); ε1: 761.60 ± 0.011 (761.5969858); ε2: 2326.18 ± 0.062 (unknown); ε3: 12273.6 ± 0.4 (unknown).
: 20.73860 ± 0.00024 (20.7385551); ε0: 145.381 ± 0.0029 (145.3808999); ε1: 985.91 ± 0.014 (985.9082306); ε2: 3930.65 ± 0.076 (unknown); ε3: 17486.6 ± 0.6 (unknown).
ε0: 55.58537 ± 0.00031 (55.5852539); ε1: 175.325 ± 0.004 (unknown); ε2: 1496.52 ± 0.02 : 52.0181 ± 0.0003 (52.0178687); ε1: 175.50 ± 0.0036 (175.496447); ε2: 1475.272 ± 0.0098 (unknown); ε3: 2623.5 ± 0.1 (unknown).
: -5.184651 ± 0.000048 (-5.184638); ε0: -32.0962 ± 0.00057 (-32.09614); ε1: -91.158 ± 0.0052 (-91.1614); ε2: 119.06 ± 0.043 (unknown); ε3: 2768.6 ± 0.45 (unknown).
: 20.73857 ± 0.00026 (20.7385551); ε1: 190.60 ± 0.0023 (190.600238); ε2: 1049.20 ± 0.014 (1049.194196); ε3: 4423.84 ± 0.072 (unknown); ε4: 16028.8 ± 0.5 (unknown).
: -10.36931 ± 0.00006 (-10.3692776); ε0: -70.990 ± 0.0011 (-70.99081719); ε1: - 21.650 ± 0.013 (-21.663005); ε2: 2832.69 ± 0.096 (unknown).
: -10.36933 ± 0.00006 (-10.36927); ε0: -58.6187 ± 0.0013(-58.6210); ε1: 244.681 ± 0.015 (unknown).
: -5.18467 ± 0.000042 (-5.184638); ε0: 14.3989 ± 0.00081 (14.39739); ε1: 739.979 ± 0.0099 (unknown).
Here for each MI we provide our numerical result for coefficients of ε-expansion in comparison (inparentheses) with the known from [analiycal results (if any). As we can see, our calculations reproduce the result of ] with 3-4 correct digits. The extra terms in the ε-expansion of each MIwhich are currently unavailable analytically but are necessary for future five-loop calculations.
Usually, analytical evaluation of multiloop MI is a kind of art. It requires a lot of efforts (and CPU 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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