Abstract
We calculate the full set of the two-loop master integrals for heavy-to-light form factors of two different massive fermions for arbitrary momentum transfer in NNLO QCD or QED corrections. These integrals allow to determine the two-loop QCD or QED corrections to the amplitudes for heavy-to-light form factors of two massive fermions in a full analytical way, without any approximations. The analytical results of the master integrals are derived using the method of differential equations, along with a proper choosing of canonical basis for the master integrals. All the results of master integrals are expressed in terms of Goncharov polylogarithms.
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K.G. Chetyrkin, R. Harlander, T. Seidensticker and M. Steinhauser, Second order QCD corrections to Γ(t → W b), Phys. Rev. D 60 (1999) 114015 [hep-ph/9906273] [INSPIRE].
I.R. Blokland, A. Czarnecki, M. Slusarczyk and F. Tkachov, Heavy to light decays with a two loop accuracy, Phys. Rev. Lett. 93 (2004) 062001 [hep-ph/0403221] [INSPIRE].
A. Czarnecki, M. Slusarczyk and F.V. Tkachov, Enhancement of the hadronic b quark decays, Phys. Rev. Lett. 96 (2006) 171803 [hep-ph/0511004] [INSPIRE].
M. Brucherseifer, F. Caola and K. Melnikov, \( \mathcal{O}\left({\alpha}_s^2\right) \) corrections to fully-differential top quark decays, JHEP 04 (2013) 059 [arXiv:1301.7133] [INSPIRE].
J. Gao, C.S. Li and H.X. Zhu, Top quark decay at next-to-next-to leading order in QCD, Phys. Rev. Lett. 110 (2013) 042001 [arXiv:1210.2808] [INSPIRE].
R. Bonciani and A. Ferroglia, Two-loop QCD corrections to the heavy-to-light quark decay, JHEP 11 (2008) 065 [arXiv:0809.4687] [INSPIRE].
H.M. Asatrian, C. Greub and B.D. Pecjak, NNLO corrections to \( \overline{B}\to X(u)l\overline{\nu} \) in the shape-function region, Phys. Rev. D 78 (2008) 114028 [arXiv:0810.0987] [INSPIRE].
M. Beneke, T. Huber and X.Q. Li, Two-loop QCD correction to differential semi-leptonic b → u decays in the shape-function region, Nucl. Phys. B 811 (2009) 77 [arXiv:0810.1230] [INSPIRE].
G. Bell, NNLO corrections to inclusive semileptonic B decays in the shape-function region, Nucl. Phys. B 812 (2009) 264 [arXiv:0810.5695] [INSPIRE].
A. Pak and A. Czarnecki, Mass effects in muon and semileptonic b → c decays, Phys. Rev. Lett. 100 (2008) 241807 [arXiv:0803.0960] [INSPIRE].
A. Pak and A. Czarnecki, Heavy-to-heavy quark decays at NNLO, Phys. Rev. D 78 (2008) 114015 [arXiv:0808.3509] [INSPIRE].
M. Dowling, A. Pak and A. Czarnecki, Semi-leptonic b-decay at intermediate recoil, Phys. Rev. D 78 (2008) 074029 [arXiv:0809.0491] [INSPIRE].
A. Arbuzov and K. Melnikov, O(α 2 ln(m μ /m e )) corrections to electron energy spectrum in muon decay, Phys. Rev. D 66 (2002) 093003 [hep-ph/0205172] [INSPIRE].
C. Anastasiou, K. Melnikov and F. Petriello, The electron energy spectrum in muon decay through O(α 2), JHEP 09 (2007) 014 [hep-ph/0505069] [INSPIRE].
A. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158.
A. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123.
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
M. Argeri and P. Mastrolia, Feynman diagrams and differential equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
M. Argeri et al., Magnus and dyson series for master integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].
X. Liu, Y.-Q. Ma and C.-Y. Wang, A systematic and efficient method to compute multi-loop master integrals, arXiv:1711.09572 [INSPIRE].
J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].
J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].
T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for \( q\overline{q}\to VV \), JHEP 06 (2014) 032 [arXiv:1404.4853] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP 09 (2014) 043 [arXiv:1404.5590] [INSPIRE].
S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP 09 (2014) 148 [arXiv:1408.3107] [INSPIRE].
G. Bell and T. Huber, Master integrals for the two-loop penguin contribution in non-leptonic B-decays, JHEP 12 (2014) 129 [arXiv:1410.2804] [INSPIRE].
T. Huber and S. Kränkl, Two-loop master integrals for non-leptonic heavy-to-heavy decays, JHEP 04 (2015) 140 [arXiv:1503.00735] [INSPIRE].
R. Bonciani et al., Next-to-leading order QCD corrections to the decay width H → Zγ, JHEP 08 (2015) 108 [arXiv:1505.00567] [INSPIRE].
T. Gehrmann, S. Guns and D. Kara, The rare decay H → Zγ in perturbative QCD, JHEP 09 (2015) 038 [arXiv:1505.00561] [INSPIRE].
A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].
R. Bonciani, S. Di Vita, P. Mastrolia and U. Schubert, Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering, JHEP 09 (2016) 091 [arXiv:1604.08581] [INSPIRE].
P. Mastrolia, M. Passera, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs, JHEP 11 (2017) 198 [arXiv:1709.07435] [INSPIRE].
R. Bonciani, P. Mastrolia and E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level, Nucl. Phys. B 661 (2003) 289 [Erratum ibid. B 702 (2004) 359] [hep-ph/0301170] [INSPIRE].
T. Huber, On a two-loop crossed six-line master integral with two massive lines, JHEP 03 (2009) 024 [arXiv:0901.2133] [INSPIRE].
G. Bell, Higher order QCD corrections in exclusive charmless B decays, arXiv:0705.3133 [INSPIRE].
G. Bell, NNLO vertex corrections in charmless hadronic B decays: Imaginary part, Nucl. Phys. B 795 (2008) 1 [arXiv:0705.3127] [INSPIRE].
A.V. Smirnov, Algorithm FIRE —- Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput. 33 (2002) 1 [cs/0004015].
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
D. Maître, Extension of HPL to complex arguments, Comput. Phys. Commun. 183 (2012) 846 [hep-ph/0703052] [INSPIRE].
H. Frellesvig, D. Tommasini and C. Wever, On the reduction of generalized polylogarithms to Li n and Li 2,2 and on the evaluation thereof, JHEP 03 (2016) 189 [arXiv:1601.02649] [INSPIRE].
L.-B. Chen and C.-F. Qiao, Two-loop QCD corrections to B c meson leptonic decays, Phys. Lett. B 748 (2015) 443 [arXiv:1503.05122] [INSPIRE].
M. Argeri, P. Mastolia and E. Remiddi, The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass, [Nucl. Phys. B 631 (2002) 388.
L.-B. Chen, Y. Liang and C.-F. Qiao, Two-loop integrals for CP-even heavy quarkonium production and decays, JHEP 06 (2017) 025 [arXiv:1703.03929] [INSPIRE].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
J. Gluza, K. Kajda and T. Riemann, AMBRE: a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun. 177 (2007) 879 [arXiv:0704.2423] [INSPIRE].
J. Gluza, K. Kajda, T. Riemann and V. Yundin, Numerical evaluation of tensor Feynman integrals in euclidean kinematics, Eur. Phys. J. C 71 (2011) 1516 [arXiv:1010.1667] [INSPIRE].
J. Blümlein et al., Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums, PoS(LL2014)052 [arXiv:1407.7832] [INSPIRE].
A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090 [arXiv:1312.3186] [INSPIRE].
A.V. Smirnov, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
S. Borowka, J. Carter and G. Heinrich, Numerical evaluation of multi-loop integrals for arbitrary kinematics with SecDec 2.0, Comput. Phys. Commun. 184 (2013) 396 [arXiv:1204.4152] [INSPIRE].
S. Borowka, G. Heinrich, S.P. Jones, M. Kerner, J. Schlenk and T. Zirke, SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].
G. Degrassi and A. Vicini, Two loop renormalization of the electric charge in the standard model, Phys. Rev. D 69 (2004) 073007 [hep-ph/0307122] [INSPIRE].
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Chen, LB. Two-loop master integrals for heavy-to-light form factors of two different massive fermions. J. High Energ. Phys. 2018, 66 (2018). https://doi.org/10.1007/JHEP02(2018)066
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DOI: https://doi.org/10.1007/JHEP02(2018)066