Abstract
We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six. We also construct the basis of irrational constants being harmonic polylgarithms of arguments e kiπ/3.
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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].
V.A. Smirnov, Asymptotic expansions in limits of large momenta and masses, Commun. Math. Phys. 134 (1990) 109 [INSPIRE].
V.A. Smirnov, Asymptotic expansions in momenta and masses and calculation of Feynman diagrams, Mod. Phys. Lett. A 10 (1995) 1485 [hep-th/9412063] [INSPIRE].
J.A.M. Vermaseren, The Symbolic manipulation program FORM, KEK-TH-326, KEK-PREPRINT-92-1, (1992).
M. Steinhauser, MATAD: A program package for the computation of MAssive TADpoles, Comput. Phys. Commun. 134 (2001) 335 [hep-ph/0009029] [INSPIRE].
L.V. Avdeev, Recurrence relations for three loop prototypes of bubble diagrams with a mass, Comput. Phys. Commun. 98 (1996) 15 [hep-ph/9512442] [INSPIRE].
L. Avdeev, J. Fleischer, S. Mikhailov and O. Tarasov, 0(αα 2 s ) correction to the electroweak ρ parameter, Phys. Lett. B 336 (1994) 560 [Erratum ibid. B 349 (1995) 597] [hep-ph/9406363] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, Corrections of order \( \mathcal{O}\left({G}_F{M}_t^2{\alpha}_s^2\right) \) to the ρ parameter, Phys. Lett. B 351 (1995) 331 [hep-ph/9502291] [INSPIRE].
M. Faisst, J.H. Kuhn, T. Seidensticker and O. Veretin, Three loop top quark contributions to the rho parameter, Nucl. Phys. B 665 (2003) 649 [hep-ph/0302275] [INSPIRE].
K.G. Chetyrkin, J.H. Kuhn and M. Steinhauser, Three loop polarization function and O(αS 2 ) corrections to the production of heavy quarks, Nucl. Phys. B 482 (1996) 213 [hep-ph/9606230] [INSPIRE].
K.G. Chetyrkin, M. Misiak and M. Münz, β-functions and anomalous dimensions up to three loops, Nucl. Phys. B 518 (1998) 473 [hep-ph/9711266] [INSPIRE].
A.V. Bednyakov, A.F. Pikelner and V.N. Velizhanin, Higgs self-coupling β-function in the Standard Model at three loops, Nucl. Phys. B 875 (2013) 552 [arXiv:1303.4364] [INSPIRE].
K.G. Chetyrkin and M.F. Zoller, β-function for the Higgs self-interaction in the Standard Model at three-loop level, JHEP 04 (2013) 091 [Erratum ibid. 1309 (2013) 155] [arXiv:1303.2890] [INSPIRE].
M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
T. Luthe, A. Maier, P. Marquard and Y. Schröder, Complete renormalization of QCD at five loops, JHEP 03 (2017) 020 [arXiv:1701.07068] [INSPIRE].
D.J. Broadhurst, Massive three - loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, Eur. Phys. J. C 8 (1999) 311 [hep-th/9803091] [INSPIRE].
E. Kummer, Ueber die transcendenten, welche aus wiederholten integrationen rationaler formeln entstehen, Journal für die reine und angewandte Mathematik 21 (1840) 74.
J. Lappo-Danilevsky, Mémoire sur la théorie des systémes des Ãl’quation différetielles linéaries, Chelsea reprint (1953).
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
M. Yu. Kalmykov and O. Veretin, Single scale diagrams and multiple binomial sums, Phys. Lett. B 483 (2000) 315 [hep-th/0004010] [INSPIRE].
A.I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B 699 (2004) 3 [hep-th/0303162] [INSPIRE].
J. Fleischer, A.V. Kotikov and O.L. Veretin, Analytic two loop results for selfenergy type and vertex type diagrams with one nonzero mass, Nucl. Phys. B 547 (1999) 343 [hep-ph/9808242] [INSPIRE].
S. Weinzierl, Expansion around half integer values, binomial sums and inverse binomial sums, J. Math. Phys. 45 (2004) 2656 [hep-ph/0402131] [INSPIRE].
M. Yu. Kalmykov and B.A. Kniehl, ‘Sixth root of unity’ and Feynman diagrams: Hypergeometric function approach point of view, Nucl. Phys. Proc. Suppl. 205-206 (2010) 129 [arXiv:1007.2373] [INSPIRE].
J. Ablinger, J. Blümlein, C.G. Raab and C. Schneider, Iterated Binomial Sums and their Associated Iterated Integrals, J. Math. Phys. 55 (2014) 112301 [arXiv:1407.1822] [INSPIRE].
J. Ablinger, J. Blümlein and C. Schneider, Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials, J. Math. Phys. 52 (2011) 102301 [arXiv:1105.6063] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six, Nucl. Phys. B 919 (2017) 315 [arXiv:1512.08389] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek, Combinatorial aspects of multiple zeta values, math/9812020.
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68 (1999) 351.
R.N. Lee, DRA method: Powerful tool for the calculation of the loop integrals, J. Phys. Conf. Ser. 368 (2012) 012050 [arXiv:1203.4868] [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].
A.I. Davydychev and J.B. Tausk, Two loop selfenergy diagrams with different masses and the momentum expansion, Nucl. Phys. B 397 (1993) 123 [INSPIRE].
A.I. Davydychev and M. Yu. Kalmykov, Some remarks on the ϵ-expansion of dimensionally regulated Feynman diagrams, Nucl. Phys. Proc. Suppl. 89 (2000) 283 [hep-th/0005287] [INSPIRE].
R.N. Lee and I.S. Terekhov, Application of the DRA method to the calculation of the four-loop QED-type tadpoles, JHEP 01 (2011) 068 [arXiv:1010.6117] [INSPIRE].
J. Fleischer and M. Yu. Kalmykov, Single mass scale diagrams: Construction of a basis for the ϵ-expansion, Phys. Lett. B 470 (1999) 168 [hep-ph/9910223] [INSPIRE].
A.I. Davydychev and M. Yu. Kalmykov, New results for the ϵ-expansion of certain one, two and three loop Feynman diagrams, Nucl. Phys. B 605 (2001) 266 [hep-th/0012189] [INSPIRE].
M. Yu. Kalmykov, About higher order ϵ-expansion of some massive two- and three-loop master-integrals, Nucl. Phys. B 718 (2005) 276 [hep-ph/0503070] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop integrals for massive form factors, JHEP 12 (2016) 144 [arXiv:1611.06523] [INSPIRE].
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ArXiv ePrint: 1705.05136
On leave of absence from Joint Institute for Nuclear Research, 141980 Dubna, Russia. (A. F. Pikelner)
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Kniehl, B.A., Pikelner, A.F. & Veretin, O.L. Three-loop massive tadpoles and polylogarithms through weight six. J. High Energ. Phys. 2017, 24 (2017). https://doi.org/10.1007/JHEP08(2017)024
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DOI: https://doi.org/10.1007/JHEP08(2017)024