Abstract
We provide details to several technical aspects which are important for the calculation of next-to-next-to-leading order corrections to the mixing of neutral B mesons. This includes the computation of the master integrals for finite charm and bottom quark masses, projectors for products of up to 22 γ matrices and tensor integrals with up to rank 11.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Lenz and U. Nierste, Theoretical update of Bs − \( {\overline{B}}_s \) mixing, JHEP 06 (2007) 072 [hep-ph/0612167] [INSPIRE].
A. Buras, Gauge Theory of Weak Decays, Cambridge University Press (2020) [https://doi.org/10.1017/9781139524100] [INSPIRE].
R.J. Dowdall et al., Neutral B-meson mixing from full lattice QCD at the physical point, Phys. Rev. D 100 (2019) 094508 [arXiv:1907.01025] [INSPIRE].
M. Kirk, A. Lenz and T. Rauh, Dimension-six matrix elements for meson mixing and lifetimes from sum rules, JHEP 12 (2017) 068 [Erratum ibid. 06 (2020) 162] [arXiv:1711.02100] [INSPIRE].
D. King, A. Lenz and T. Rauh, SU(3) breaking effects in B and D meson lifetimes, JHEP 06 (2022) 134 [arXiv:2112.03691] [INSPIRE].
M. Beneke et al., Next-to-leading order QCD corrections to the lifetime difference of B(s) mesons, Phys. Lett. B 459 (1999) 631 [hep-ph/9808385] [INSPIRE].
M. Ciuchini et al., Lifetime differences and CP violation parameters of neutral B mesons at the next-to-leading order in QCD, JHEP 08 (2003) 031 [hep-ph/0308029] [INSPIRE].
M. Beneke, G. Buchalla, A. Lenz and U. Nierste, CP asymmetry in flavor specific B decays beyond leading logarithms, Phys. Lett. B 576 (2003) 173 [hep-ph/0307344] [INSPIRE].
M. Gerlach, U. Nierste, V. Shtabovenko and M. Steinhauser, Two-loop QCD penguin contribution to the width difference in Bs − \( {\overline{B}}_s \) mixing, JHEP 07 (2021) 043 [arXiv:2106.05979] [INSPIRE].
M. Gerlach, U. Nierste, V. Shtabovenko and M. Steinhauser, The width difference in B − \( \overline{B} \) mixing at order αs and beyond, JHEP 04 (2022) 006 [arXiv:2202.12305] [INSPIRE].
H.M. Asatrian, A. Hovhannisyan, U. Nierste and A. Yeghiazaryan, Towards next-to-next-to-leading-log accuracy for the width difference in the Bs − \( {\overline{B}}_s \) system: fermionic contributions to order (mc/mb)0 and (mc/mb)1, JHEP 10 (2017) 191 [arXiv:1709.02160] [INSPIRE].
H.M. Asatrian et al., Penguin contribution to the width difference and CP asymmetry in Bq-\( {\overline{B}}_q \) mixing at order \( {\alpha}_s^2{N}_f \), Phys. Rev. D 102 (2020) 033007 [arXiv:2006.13227] [INSPIRE].
A. Hovhannisyan and U. Nierste, Addendum to: Towards next-to-next-to-leading-log accuracy for the width difference in the Bs − \( {\overline{B}}_s \) system: fermionic contributions to order (mc/mb)0 and (mc/mb)1, JHEP 06 (2022) 090 [arXiv:2204.11907] [INSPIRE].
M. Gerlach, U. Nierste, V. Shtabovenko and M. Steinhauser, Width Difference in the B-B− System at Next-to-Next-to-Leading Order of QCD, Phys. Rev. Lett. 129 (2022) 102001 [arXiv:2205.07907] [INSPIRE].
P. Nogueira, Automatic Feynman Graph Generation, J. Comput. Phys. 105 (1993) 279 [INSPIRE].
P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
P. Maierhöfer and J. Usovitsch, Kira 1.2 Release Notes, arXiv:1812.01491 [INSPIRE].
J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun. 266 (2021) 108024 [arXiv:2008.06494] [INSPIRE].
M. Gerlach, U. Nierste, P. Reeck, V. Shtabovenko and M. Steinhauser, Next-to-next-to-leading order QCD corrections to the B-meson mixing, in preparation.
K.G. Chetyrkin, M. Misiak and M. Munz, |∆F| = 1 nonleptonic effective Hamiltonian in a simpler scheme, Nucl. Phys. B 520 (1998) 279 [hep-ph/9711280] [INSPIRE].
T. Peraro and L. Tancredi, Tensor decomposition for bosonic and fermionic scattering amplitudes, Phys. Rev. D 103 (2021) 054042 [arXiv:2012.00820] [INSPIRE].
L. Tancredi, Tensor decomposition for multiloop, multileg helicity amplitudes, PoS LL2022 (2022) 020 [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
A. Pak, The Toolbox of modern multi-loop calculations: novel analytic and semi-analytic techniques, J. Phys. Conf. Ser. 368 (2012) 012049 [arXiv:1111.0868] [INSPIRE].
R. Mertig, M. Bohm and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman amplitudes, Comput. Phys. Commun. 64 (1991) 345 [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun. 207 (2016) 432 [arXiv:1601.01167] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun. 256 (2020) 107478 [arXiv:2001.04407] [INSPIRE].
V. Shtabovenko, R. Mertig and F. Orellana, FeynCalc 10: Do multiloop integrals dream of computer codes?, arXiv:2312.14089 [INSPIRE].
V. Shtabovenko, FeynHelpers: Connecting FeynCalc to FIRE and Package-X, Comput. Phys. Commun. 218 (2017) 48 [arXiv:1611.06793] [INSPIRE].
R. Lewis, Fermat, https://home.bway.net/lewis.
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Berlin, Heidelberg (2012) [https://doi.org/10.1007/978-3-642-34886-0] [INSPIRE].
J. Fleischer and M.Y. Kalmykov, ON-SHELL2: FORM based package for the calculation of two loop selfenergy single scale Feynman diagrams occurring in the standard model, Comput. Phys. Commun. 128 (2000) 531 [hep-ph/9907431] [INSPIRE].
E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015) 148 [arXiv:1403.3385] [INSPIRE].
H. Cheng and T.T. Wu, Expanding protons: Scattering at high-energies, MIT Press (1987) [INSPIRE].
E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].
E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt-Universität zu Berlin, 12489 Berlin, Germany (2015) [arXiv:1506.07243] [INSPIRE].
A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP 02 (2015) 120 [arXiv:1411.7392] [INSPIRE].
S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313 [arXiv:1703.09692] [INSPIRE].
S. Borowka et al., A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec, Comput. Phys. Commun. 240 (2019) 120 [arXiv:1811.11720] [INSPIRE].
G. Heinrich et al., Expansion by regions with pySecDec, Comput. Phys. Commun. 273 (2022) 108267 [arXiv:2108.10807] [INSPIRE].
G. Heinrich et al., Numerical scattering amplitudes with pySecDec, Comput. Phys. Commun. 295 (2024) 108956 [arXiv:2305.19768] [INSPIRE].
A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
A.V. Smirnov, N.D. Shapurov and L.I. Vysotsky, FIESTA5: Numerical high-performance Feynman integral evaluation, Comput. Phys. Commun. 277 (2022) 108386 [arXiv:2110.11660] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
O. Schnetz, HyperLogProcedures, https://www.math.fau.de/person/oliver-schnetz.
D. Broadhurst, Multiple Deligne values: a data mine with empirically tamed denominators, arXiv:1409.7204 [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [INSPIRE].
M. Beneke and V.A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321 [hep-ph/9711391] [INSPIRE].
A. Pak and A. Smirnov, Geometric approach to asymptotic expansion of Feynman integrals, Eur. Phys. J. C 71 (2011) 1626 [arXiv:1011.4863] [INSPIRE].
B. Jantzen, A.V. Smirnov and V.A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [INSPIRE].
C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with nonAbelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, Comput. Phys. Commun. 247 (2020) 106877 [arXiv:1901.07808] [INSPIRE].
A.V. Smirnov and M. Zeng, FIRE 6.5: Feynman integral reduction with new simplification library, Comput. Phys. Commun. 302 (2024) 109261 [arXiv:2311.02370] [INSPIRE].
M. Fael, F. Lange, K. Schönwald and M. Steinhauser, A semi-analytic method to compute Feynman integrals applied to four-loop corrections to the \( \overline{MS} \)-pole quark mass relation, JHEP 09 (2021) 152 [arXiv:2106.05296] [INSPIRE].
M. Fael, F. Lange, K. Schönwald and M. Steinhauser, Massive Vector Form Factors to Three Loops, Phys. Rev. Lett. 128 (2022) 172003 [arXiv:2202.05276] [INSPIRE].
M. Fael, F. Lange, K. Schönwald and M. Steinhauser, Singlet and nonsinglet three-loop massive form factors, Phys. Rev. D 106 (2022) 034029 [arXiv:2207.00027] [INSPIRE].
M. Fael, F. Lange, K. Schönwald and M. Steinhauser, Massive three-loop form factors: Anomaly contribution, Phys. Rev. D 107 (2023) 094017 [arXiv:2302.00693] [INSPIRE].
A.I. Davydychev and V.A. Smirnov, Threshold expansion of the sunset diagram, Nucl. Phys. B 554 (1999) 391 [hep-ph/9903328] [INSPIRE].
K. Melnikov, private communication.
M. Egner, M. Fael, K. Schönwald and M. Steinhauser, Revisiting semileptonic B meson decays at next-to-next-to-leading order, JHEP 09 (2023) 112 [arXiv:2308.01346] [INSPIRE].
X. Liu and Y.-Q. Ma, AMFlow: A Mathematica package for Feynman integrals computation via auxiliary mass flow, Comput. Phys. Commun. 283 (2023) 108565 [arXiv:2201.11669] [INSPIRE].
K.G. Chetyrkin and M.F. Zoller, Four-loop renormalization of QCD with a reducible fermion representation of the gauge group: anomalous dimensions and renormalization constants, JHEP 06 (2017) 074 [arXiv:1704.04209] [INSPIRE].
Acknowledgments
We thank Ulrich Nierste, Gurdun Heinrich, Erik Panzer, Oliver Schnetz, Kay Schönwald and Alexander Smirnov for useful discussions. This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant 396021762 — TRR 257 “Particle Physics Phenomenology after the Higgs Discovery”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2405.14698
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Reeck, P., Shtabovenko, V. & Steinhauser, M. B meson mixing at NNLO: technical aspects. J. High Energ. Phys. 2024, 2 (2024). https://doi.org/10.1007/JHEP08(2024)002
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2024)002