Abstract
We compute the two-loop master integrals for non-leptonic heavy-to-heavy decays analytically in a recently-proposed canonical basis. For this genuine two-loop, two-scale problem we first derive a basis for the master integrals that disentangles the kinematics from the space-time dimension in the differential equations, and subsequently solve the latter in terms of iterated integrals up to weight four. The solution constitutes another valuable example of the finding of a canonical basis for two-loop master integrals that have two different internal masses, and assumes a form that is ideally suited for a sub-sequent convolution with the light-cone distribution amplitude in the framework of QCD factorisation.
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References
LHCb collaboration, Letter of Intent for the LHCb Upgrade, CERN-LHCC-2011-001, LHCC-I-018 (2011).
D. Zeppenfeld, SU(3) Relations for B Meson Decays, Z. Phys. C 8 (1981) 77 [INSPIRE].
Y.-Y. Keum, H.-N. Li and A.I. Sanda, Fat penguins and imaginary penguins in perturbative QCD, Phys. Lett. B 504 (2001) 6 [hep-ph/0004004] [INSPIRE].
M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for B → ππ decays: Strong phases and CP-violation in the heavy quark limit, Phys. Rev. Lett. 83 (1999) 1914 [hep-ph/9905312] [INSPIRE].
M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization for exclusive, nonleptonic B meson decays: general arguments and the case of heavy light final states, Nucl. Phys. B 591 (2000) 313 [hep-ph/0006124] [INSPIRE].
M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, QCD factorization in B → πK, ππ decays and extraction of Wolfenstein parameters, Nucl. Phys. B 606 (2001) 245 [hep-ph/0104110] [INSPIRE].
M. Beneke and M. Neubert, QCD factorization for B → PP and B → PV decays, Nucl. Phys. B 675 (2003) 333 [hep-ph/0308039] [INSPIRE].
G. Bell, NNLO vertex corrections in charmless hadronic B decays: Imaginary part, Nucl. Phys. B 795 (2008) 1 [arXiv:0705.3127] [INSPIRE].
G. Bell, NNLO vertex corrections in charmless hadronic B decays: Real part, Nucl. Phys. B 822 (2009) 172 [arXiv:0902.1915] [INSPIRE].
M. Beneke, T. Huber and X.-Q. Li, NNLO vertex corrections to non-leptonic B decays: tree amplitudes, Nucl. Phys. B 832 (2010) 109 [arXiv:0911.3655] [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].
G. Bell, M. Beneke, T. Huber and X.-Q. Li, Two-loop current-current operator contribution to the non-leptonic QCD penguin amplitude, in preparation.
T. Huber and S. Kränkl, Towards NNLO corrections in B → Dπ, arXiv:1405.5911 [INSPIRE].
A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [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, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [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, 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].
M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [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 \( q\overline{q} \) → 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].
A. von Manteuffel, R.M. Schabinger and H.X. Zhu, The two-loop soft function for heavy quark pair production at future linear colliders, arXiv:1408.5134 [INSPIRE].
M. Höschele, J. Hoff and T. Ueda, Adequate bases of phase space master integrals for gg → h at NNLO and beyond, JHEP 09 (2014) 116 [arXiv:1407.4049] [INSPIRE].
H.X. Zhu, On the calculation of soft phase space integral, JHEP 02 (2015) 155 [arXiv:1501.00236] [INSPIRE].
R.N. Lee, Reducing differential equations for multiloop master integrals, arXiv:1411.0911 [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
Analytic results to order \( \mathcal{O} \)(ϵ 4) of all integrals, including their mass-flipped counterparts, are attached in electronic form to the arXiv submission of the present article.
F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [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].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].
A.V. Smirnov, Algorithm FIRE - Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys. A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
V.A. Smirnov, Analytical result for dimensionally regularized massless on shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [INSPIRE].
J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].
T. Huber, On a two-loop crossed six-line master integral with two massive lines, JHEP 03 (2009) 024 [arXiv:0901.2133] [INSPIRE].
M. Czakon, http://mbtools.hepforge.org/.
T. Huber and D. Maˆıtre, HypExp: A Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [INSPIRE].
T. Huber and D. Maˆıtre, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [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.
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [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].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
J. Carter and G. Heinrich, SecDec: a general program for sector decomposition, Comput. Phys. Commun. 182 (2011) 1566 [arXiv:1011.5493] [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].
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Huber, T., Kränkl, S. Two-loop master integrals for non-leptonic heavy-to-heavy decays. J. High Energ. Phys. 2015, 140 (2015). https://doi.org/10.1007/JHEP04(2015)140
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DOI: https://doi.org/10.1007/JHEP04(2015)140