Abstract
We present analytical expressions for the 31 five-particle phase-space master integrals in massless QCD as an ϵ-series with coefficients being multiple zeta values of weight up to 12. In addition, we provide computer code for the Monte-Carlo integration in higher dimensions, based on the RAMBO algorithm, that has been used to numerically cross-check the obtained results in 4, 6, and 8 dimensions.
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A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].
O. Gituliar, Master integrals for splitting functions from differential equations in QCD, JHEP 02 (2016) 017 [arXiv:1512.02045] [INSPIRE].
O. Gituliar and S. Moch, Towards three-loop QCD corrections to the time-like splitting functions, Acta Phys. Polon. B 46 (2015) 1279 [arXiv:1505.02901] [INSPIRE].
A.A. Almasy, S. Moch and A. Vogt, On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions, Nucl. Phys. B 854 (2012) 133 [arXiv:1107.2263] [INSPIRE].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
O.V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension, Phys. Rev. D 54 (1996) 6479 [hep-th/9606018] [INSPIRE].
R.N. Lee and K.T. Mingulov, DREAM, a program for arbitrary-precision computation of dimensional recurrence relations solutions and its applications, arXiv:1712.05173 [INSPIRE].
R.N. Lee and K.T. Mingulov, Meromorphic solutions of recurrence relations and DRA method for multicomponent master integrals, JHEP 04 (2018) 061 [arXiv:1712.05166] [INSPIRE].
A.S. Schwarz, Gauge theories on noncommutative spaces, hep-th/0011261 [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
R.N. Lee and K.T. Mingulov, Introducing SummerTime: a package for high-precision computation of sums appearing in DRA method, Comput. Phys. Commun. 203 (2016) 255 [arXiv:1507.04256] [INSPIRE].
H. Ferguson, D. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comp. 68 (1999) 351.
R. Kleiss, W.J. Stirling and S.D. Ellis, A New Monte Carlo Treatment of Multiparticle Phase Space at High-energies, Comput. Phys. Commun. 40 (1986) 359 [INSPIRE].
P.A. Baikov and K.G. Chetyrkin, Four Loop Massless Propagators: An Algebraic Evaluation of All Master Integrals, Nucl. Phys. B 837 (2010) 186 [arXiv:1004.1153] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
S. Laporta, High-precision calculation of the 4-loop contribution to the electron g-2 in QED, Phys. Lett. B 772 (2017) 232 [arXiv:1704.06996] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.
B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].
R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [INSPIRE].
G.P. Lepage, A New Algorithm for Adaptive Multidimensional Integration, J. Comput. Phys. 27 (1978) 192 [INSPIRE].
T. Hahn, CUBA: A Library for multidimensional numerical integration, Comput. Phys. Commun. 168 (2005) 78 [hep-ph/0404043] [INSPIRE].
B. Gough. GNU Scientific Library Reference Manual, 3rd edition, Network Theory Ltd. (2009).
J.C. Collins and J.A.M. Vermaseren, Axodraw Version 2, arXiv:1606.01177 [INSPIRE].
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ArXiv ePrint: 1803.09084
On leave of absence from Joint Institute for Nuclear Research, 141980 Dubna, Russia (A. Pikelner).
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Gituliar, O., Magerya, V. & Pikelner, A. Five-particle phase-space integrals in QCD. J. High Energ. Phys. 2018, 99 (2018). https://doi.org/10.1007/JHEP06(2018)099
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DOI: https://doi.org/10.1007/JHEP06(2018)099