Three-loop vertex integrals at symmetric point. (English) Zbl 1466.81047

Summary: This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the \(\epsilon \)-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.


81S40 Path integrals in quantum mechanics
81T17 Renormalization group methods applied to problems in quantum field theory
81T25 Quantum field theory on lattices
81V05 Strong interaction, including quantum chromodynamics
Full Text: DOI arXiv


[1] Baikov, PA; Chetyrkin, KG; Kühn, JH, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett., 118, 082002 (2017)
[2] Luthe, T.; Maier, A.; Marquard, P.; Schröder, Y., The five-loop β-function for a general gauge group and anomalous dimensions beyond Feynman gauge, JHEP, 10, 166 (2017) · Zbl 1383.81343
[3] Herzog, F.; Ruijl, B.; Ueda, T.; Vermaseren, JAM; Vogt, A., The five-loop β-function of Yang-Mills theory with fermions, JHEP, 02, 090 (2017) · Zbl 1377.81103
[4] K. G. Chetyrkin, G. Falcioni, F. Herzog and J. A. M. Vermaseren, Five-loop renormalisation of QCD in covariant gauges, JHEP10 (2017) 179 [Addendum ibid.12 (2017) 006] [arXiv:1709.08541] [INSPIRE]. · Zbl 1383.81336
[5] G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B61 (1973) 455 [INSPIRE].
[6] Celmaster, W.; Gonsalves, RJ, The Renormalization Prescription Dependence of the QCD Coupling Constant, Phys. Rev. D, 20, 1420 (1979)
[7] Gracey, JA, Three loop QCD MOM β-functions, Phys. Lett. B, 700, 79 (2011)
[8] Almeida, LG; Sturm, C., Two-loop matching factors for light quark masses and three-loop mass anomalous dimensions in the RI/SMOM schemes, Phys. Rev. D, 82, 054017 (2010)
[9] Gracey, JA, RI’/SMOM scheme amplitudes for quark currents at two loops, Eur. Phys. J. C, 71, 1567 (2011)
[10] Gracey, JA, Two loop renormalization of the N = 2 Wilson operator in the RI’/SMOM scheme, JHEP, 03, 109 (2011) · Zbl 1301.81298
[11] Gracey, JA, Amplitudes for the N = 3 moment of the Wilson operator at two loops in the RI/’SMOM scheme, Phys. Rev. D, 84, 016002 (2011)
[12] Bednyakov, A.; Pikelner, A., Quark masses: N3LO bridge from RI/SMOM to \(\overline{\text{MS}}\) scheme, Phys. Rev. D, 101, 091501 (2020)
[13] Bednyakov, A.; Pikelner, A., Four-loop QCD MOM β-functions from the three-loop vertices at the symmetric point, Phys. Rev. D, 101, 071502 (2020)
[14] Kniehl, BA; Veretin, OL, Moments N = 2 and N = 3 of the Wilson twist-two operators at three loops in the RI′/SMOM scheme, Nucl. Phys. B, 961, 115229 (2020)
[15] Kniehl, BA; Veretin, OL, Bilinear quark operators in the RI/SMOM scheme at three loops, Phys. Lett. B, 804, 135398 (2020) · Zbl 1435.81241
[16] Chetyrkin, KG; Seidensticker, T., Two loop QCD vertices and three loop MOM β-functions, Phys. Lett. B, 495, 74 (2000)
[17] Tkachov, FV, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B, 100, 65 (1981)
[18] Chetyrkin, KG; Tkachov, FV, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B, 192, 159 (1981)
[19] Davydychev, AI, Recursive algorithm of evaluating vertex type Feynman integrals, J. Phys. A, 25, 5587 (1992) · Zbl 0767.65005
[20] Usyukina, NI; Davydychev, AI, New results for two loop off-shell three point diagrams, Phys. Lett. B, 332, 159 (1994)
[21] Birthwright, TG; Glover, EWN; Marquard, P., Master integrals for massless two-loop vertex diagrams with three offshell legs, JHEP, 09, 042 (2004)
[22] Usyukina, NI; Davydychev, AI, Exact results for three and four point ladder diagrams with an arbitrary number of rungs, Phys. Lett. B, 305, 136 (1993)
[23] Chavez, F.; Duhr, C., Three-mass triangle integrals and single-valued polylogarithms, JHEP, 11, 114 (2012) · Zbl 1397.81071
[24] E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. Thesis, Humboldt University, Berlin Germany (2015) [arXiv:1506.07243] [INSPIRE]. · Zbl 1344.81024
[25] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601 (2013)
[26] von Manteuffel, A.; Panzer, E.; Schabinger, RM, A quasi-finite basis for multi-loop Feynman integrals, JHEP, 02, 120 (2015) · Zbl 1388.81378
[27] Laporta, S., High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A, 15, 5087 (2000) · Zbl 0973.81082
[28] A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
[29] Gorishnii, SG; Larin, SA; Surguladze, LR; Tkachov, FV, Mincer: Program for Multiloop Calculations in Quantum Field Theory for the Schoonschip System, Comput. Phys. Commun., 55, 381 (1989)
[30] S. A. Larin, F. V. Tkachov and J. A. M. Vermaseren, The FORM version of MINCER, NIKHEF-H-91-18 (1991).
[31] Gehrmann, T.; Glover, EWN; Huber, T.; Ikizlerli, N.; Studerus, C., Calculation of the quark and gluon form factors to three loops in QCD, JHEP, 06, 094 (2010) · Zbl 1288.81146
[32] Lee, RN; Smirnov, AV; Smirnov, VA, Analytic Results for Massless Three-Loop Form Factors, JHEP, 04, 020 (2010) · Zbl 1272.81196
[33] von Manteuffel, A.; Panzer, E.; Schabinger, RM, On the Computation of Form Factors in Massless QCD with Finite Master Integrals, Phys. Rev. D, 93, 125014 (2016)
[34] Lee, RN, Reducing differential equations for multiloop master integrals, JHEP, 04, 108 (2015) · Zbl 1388.81109
[35] Prausa, M., epsilon: A tool to find a canonical basis of master integrals, Comput. Phys. Commun., 219, 361 (2017) · Zbl 1411.81019
[36] A. B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE]. · Zbl 0919.11080
[37] Henn, JM; Smirnov, AV; Smirnov, VA, Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six, Nucl. Phys. B, 919, 315 (2017) · Zbl 1361.81105
[38] F. Dulat and B. Mistlberger, Real-Virtual-Virtual contributions to the inclusive Higgs cross section at N3LO, arXiv:1411.3586 [INSPIRE].
[39] Lee, RN; Smirnov, AV; Smirnov, VA, Solving differential equations for Feynman integrals by expansions near singular points, JHEP, 03, 008 (2018) · Zbl 1388.81927
[40] Harlander, R.; Seidensticker, T.; Steinhauser, M., Complete corrections of O(αα_s) to the decay of the Z boson into bottom quarks, Phys. Lett. B, 426, 125 (1998)
[41] Seidensticker, T., Automatic application of successive asymptotic expansions of Feynman diagrams, in 6th International Workshop on New Computing Techniques in Physics Research: Software Engineering (1999), Artificial Intelligence Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation: Heraklion Greece, Artificial Intelligence Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation
[42] Harlander, R., Asymptotic expansions: Methods and applications, Acta Phys. Polon. B, 30, 3443 (1999)
[43] Dlapa, C.; Henn, J.; Yan, K., Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, JHEP, 05, 025 (2020)
[44] Henn, J.; Mistlberger, B.; Smirnov, VA; Wasser, P., Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP, 04, 167 (2020)
[45] Lee, RN; Smirnov, AV; Smirnov, VA, Master Integrals for Four-Loop Massless Propagators up to Transcendentality Weight Twelve, Nucl. Phys. B, 856, 95 (2012) · Zbl 1246.81057
[46] Panzer, E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun., 188, 148 (2015) · Zbl 1344.81024
[47] Borowka, S., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun., 222, 313 (2018)
[48] Kniehl, BA; Pikelner, AF; Veretin, OL, Three-loop massive tadpoles and polylogarithms through weight six, JHEP, 08, 024 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.