Abstract
We present a model-independent anatomy of the ∆F = 2 transitions K 0 − \( {\overline{K}}^0 \), Bs,d − \( {\overline{B}}_{s,d} \) and D0 − \( {\overline{D}}^0 \) in the context of the Standard Model Effective Field Theory (SMEFT). We present two master formulae for the mixing amplitude [M12]BSM. One in terms of the Wilson coefficients (WCs) of the Low-Energy Effective Theory (LEFT) operators evaluated at the electroweak scale μew and one in terms of the WCs of the SMEFT operators evaluated at the BSM scale Λ. The coefficients \( {P}_a^{ij} \) entering these formulae contain all the information below the scales μew and Λ, respectively. Renormalization group effects from the top-quark Yukawa coupling play the most important role. The collection of the individual contributions of the SMEFT operators to [M12]BSM can be considered as the SMEFT atlas of ∆F = 2 transitions and constitutes a travel guide to such transitions far beyond the scales explored by the LHC. We emphasize that this atlas depends on whether the down-basis or the up-basis for SMEFT operators is considered. We illustrate this technology with tree-level exchanges of heavy gauge bosons (Z′, G′) and corresponding heavy scalars.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.J. Buras and J. Girrbach, Towards the identification of new physics through quark flavour violating processes, Rept. Prog. Phys. 77 (2014) 086201 [arXiv:1306.3775] [INSPIRE].
A.J. Buras, Gauge theory of weak decays, Cambridge University Press, Cambridge, U.K. (2020).
M.K. Gaillard and B.W. Lee, Rare decay modes of the K-mesons in gauge theories, Phys. Rev. D 10 (1974) 897 [INSPIRE].
Z. Bai, N.H. Christ, T. Izubuchi, C.T. Sachrajda, A. Soni and J. Yu, KL -KS mass difference from lattice QCD, Phys. Rev. Lett. 113 (2014) 112003 [arXiv:1406.0916] [INSPIRE].
N.H. Christ, X. Feng, G. Martinelli and C.T. Sachrajda, Effects of finite volume on the KL -KS mass difference, Phys. Rev. D 91 (2015) 114510 [arXiv:1504.01170] [INSPIRE].
Z. Bai, N.H. Christ and C.T. Sachrajda, The KL -KS mass difference, EPJ Web Conf. 175 (2018) 13017 [INSPIRE].
J.M. Gérard, W. Grimus, A. Raychaudhuri and G. Zoupanos, Super Kobayashi-Maskawa CP-violation, Phys. Lett. B 140 (1984) 349 [INSPIRE].
F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, A complete analysis of FCNC and CP constraints in general SUSY extensions of the Standard Model, Nucl. Phys. B 477 (1996) 321 [hep-ph/9604387] [INSPIRE].
UTfit collaboration, Model-independent constraints on ∆F = 2 operators and the scale of new physics, JHEP 03 (2008) 049 [arXiv:0707.0636] [INSPIRE].
G. Isidori, Y. Nir and G. Perez, Flavor physics constraints for physics beyond the Standard Model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900] [INSPIRE].
L. Silvestrini and M. Valli, Model-independent bounds on the Standard Model effective theory from flavour physics, Phys. Lett. B 799 (2019) 135062 [arXiv:1812.10913] [INSPIRE].
L. Calibbi, A. Crivellin, F. Kirk, C.A. Manzari and L. Vernazza, Z′ models with less-minimal flavour violation, Phys. Rev. D 101 (2020) 095003 [arXiv:1910.00014] [INSPIRE].
E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-energy effective field theory below the electroweak scale: operators and matching, JHEP 03 (2018) 016 [arXiv:1709.04486] [INSPIRE].
W. Dekens and P. Stoffer, Low-energy effective field theory below the electroweak scale: matching at one loop, JHEP 10 (2019) 197 [arXiv:1908.05295] [INSPIRE].
M. Endo, T. Kitahara, S. Mishima and K. Yamamoto, Revisiting kaon physics in general Z scenario, Phys. Lett. B 771 (2017) 37 [arXiv:1612.08839] [INSPIRE].
C. Bobeth, A.J. Buras, A. Celis and M. Jung, Yukawa enhancement of Z -mediated new physics in ∆S = 2 and ∆B = 2 processes, JHEP 07 (2017) 124 [arXiv:1703.04753] [INSPIRE].
F. Feruglio, P. Paradisi and A. Pattori, On the importance of electroweak corrections for B anomalies, JHEP 09 (2017) 061 [arXiv:1705.00929] [INSPIRE].
M. González-Alonso, J. Martin Camalich and K. Mimouni, Renormalization-group evolution of new physics contributions to (semi)leptonic meson decays, Phys. Lett. B 772 (2017) 777 [arXiv:1706.00410] [INSPIRE].
D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, B-physics anomalies: a guide to combined explanations, JHEP 11 (2017) 044 [arXiv:1706.07808] [INSPIRE].
J. Kumar, D. London and R. Watanabe, Combined explanations of the b → sμ+ μ− and b → \( c{\tau}^{-}\overline{v} \) anomalies: a general model analysis, Phys. Rev. D 99 (2019) 015007 [arXiv:1806.07403] [INSPIRE].
J. Aebischer, J. Kumar, P. Stangl and D.M. Straub, A global likelihood for precision constraints and flavour anomalies, Eur. Phys. J. C 79 (2019) 509 [arXiv:1810.07698] [INSPIRE].
M. Endo, T. Kitahara and D. Ueda, SMEFT top-quark effects on ∆F = 2 observables, JHEP 07 (2019) 182 [arXiv:1811.04961] [INSPIRE].
J. Aebischer, W. Altmannshofer, D. Guadagnoli, M. Reboud, P. Stangl and D.M. Straub, B-decay discrepancies after Moriond 2019, Eur. Phys. J. C 80 (2020) 252 [arXiv:1903.10434] [INSPIRE].
J. Aebischer, A.J. Buras and J. Kumar, Another SMEFT story: Z′ facing new results on ε′/ε, ∆MK and K → \( \pi v\overline{v} \), arXiv:2006.01138 [INSPIRE].
J. Aebischer and J. Kumar, Flavour violating effects of Yukawa running in SMEFT, JHEP 09 (2020) 187 [arXiv:2005.12283] [INSPIRE].
J. Aebischer, C. Bobeth, A.J. Buras and D.M. Straub, Anatomy of ε′/ε beyond the Standard Model, Eur. Phys. J. C 79 (2019) 219 [arXiv:1808.00466] [INSPIRE].
J. Aebischer, C. Bobeth, A.J. Buras, J.-M. Gérard and D.M. Straub, Master formula for ε′/ε beyond the Standard Model, Phys. Lett. B 792 (2019) 465 [arXiv:1807.02520] [INSPIRE].
A.J. Buras, S. Jäger and J. Urban, Master formulae for ∆F = 2 NLO QCD factors in the Standard Model and beyond, Nucl. Phys. B 605 (2001) 600 [hep-ph/0102316] [INSPIRE].
A.J. Buras and M. Jung, Analytic inclusion of the scale dependence of the anomalous dimension matrix in Standard Model effective theory, JHEP 06 (2018) 067 [arXiv:1804.05852] [INSPIRE].
B. Grzadkowski, M. Iskrzynski, M. Misiak and J. Rosiek, Dimension-six terms in the Standard Model Lagrangian, JHEP 10 (2010) 085 [arXiv:1008.4884] [INSPIRE].
M. Ciuchini, E. Franco, V. Lubicz, G. Martinelli, I. Scimemi and L. Silvestrini, Next-to-leading order QCD corrections to ∆F = 2 effective Hamiltonians, Nucl. Phys. B 523 (1998) 501 [hep-ph/9711402] [INSPIRE].
A.J. Buras, M. Misiak and J. Urban, Two loop QCD anomalous dimensions of flavor changing four quark operators within and beyond the Standard Model, Nucl. Phys. B 586 (2000) 397 [hep-ph/0005183] [INSPIRE].
A.J. Buras, M. Jamin and P.H. Weisz, Leading and next-to-leading QCD corrections to ε parameter and \( {B}^0\hbox{-} {\overline{B}}^0 \) mixing in the presence of a heavy top quark, Nucl. Phys. B 347 (1990) 491 [INSPIRE].
P. Gambino, A. Kwiatkowski and N. Pott, Electroweak effects in the \( {B}^0\hbox{-} {\overline{B}}^0 \) mixing, Nucl. Phys. B 544 (1999) 532 [hep-ph/9810400] [INSPIRE].
A.J. Buras and J.-M. Gérard, Dual QCD insight into BSM hadronic matrix elements for \( {K}^0\hbox{-} {\overline{K}}^0 \) mixing from lattice QCD, Acta Phys. Polon. B 50 (2019) 121 [arXiv:1804.02401] [INSPIRE].
H. Boos, T. Mannel and J. Reuter, The gold plated mode revisited: sin(2β) and B0 → J/ΨKS in the Standard Model, Phys. Rev. D 70 (2004) 036006 [hep-ph/0403085] [INSPIRE].
C. Bobeth, U. Haisch, A. Lenz, B. Pecjak and G. Tetlalmatzi-Xolocotzi, On new physics in ∆Γd, JHEP 06 (2014) 040 [arXiv:1404.2531] [INSPIRE].
A. Lenz and G. Tetlalmatzi-Xolocotzi, Model-independent bounds on new physics effects in non-leptonic tree-level decays of B-mesons, JHEP 07 (2020) 177 [arXiv:1912.07621] [INSPIRE].
A.J. Buras and P.H. Weisz, QCD nonleading corrections to weak decays in dimensional regularization and ’t Hooft-Veltman schemes, Nucl. Phys. B 333 (1990) 66 [INSPIRE].
M.J. Dugan and B. Grinstein, On the vanishing of evanescent operators, Phys. Lett. B 256 (1991) 239 [INSPIRE].
S. Herrlich and U. Nierste, Evanescent operators, scheme dependences and double insertions, Nucl. Phys. B 455 (1995) 39 [hep-ph/9412375] [INSPIRE].
G. Buchalla, A.J. Buras and M.E. Lautenbacher, Weak decays beyond leading logarithms, Rev. Mod. Phys. 68 (1996) 1125 [hep-ph/9512380] [INSPIRE].
J. Aebischer, A.J. Buras and J.-M. Gérard, BSM hadronic matrix elements for ε′/ε and K → ππ decays in the dual QCD approach, JHEP 02 (2019) 021 [arXiv:1807.01709] [INSPIRE].
Flavour Lattice Averaging Group collaboration, FLAG review 2019: Flavour Lattice Averaging Group (FLAG), Eur. Phys. J. C 80 (2020) 113 [arXiv:1902.08191] [INSPIRE].
M. Beneke, G. Buchalla, C. Greub, A. Lenz and U. Nierste, Next-to-leading order QCD corrections to the lifetime difference of Bs mesons, Phys. Lett. B 459 (1999) 631 [hep-ph/9808385] [INSPIRE].
M. Gorbahn, S. Jager, U. Nierste and S. Trine, The supersymmetric Higgs sector and \( B\hbox{-} \overline{B} \) mixing for large tan β, Phys. Rev. D 84 (2011) 034030 [arXiv:0901.2065] [INSPIRE].
R. Babich, N. Garron, C. Hölbling, J. Howard, L. Lellouch and C. Rebbi, \( {K}^0\hbox{-} {\overline{K}}^0 \) mixing beyond the standard model and CP-violating electroweak penguins in quenched QCD with exact chiral symmetry, Phys. Rev. D 74 (2006) 073009 [hep-lat/0605016] [INSPIRE].
SWME collaboration, Kaon BSM B-parameters using improved staggered fermions from Nf = 2 + 1 unquenched QCD, Phys. Rev. D 93 (2016) 014511 [arXiv:1509.00592] [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].
ETM collaboration, ∆S = 2 and ∆C = 2 bag parameters in the Standard Model and beyond from Nf = 2 + 1 + 1 twisted-mass lattice QCD, Phys. Rev. D 92 (2015) 034516 [arXiv:1505.06639] [INSPIRE].
Fermilab Lattice and MILC collaborations, \( {B}_{(s)}^0 \) -mixing matrix elements from lattice QCD for the Standard Model and beyond, Phys. Rev. D 93 (2016) 113016 [arXiv:1602.03560] [INSPIRE].
A. Bazavov et al., Short-distance matrix elements for D0 -meson mixing for Nf = 2 + 1 lattice QCD, Phys. Rev. D 97 (2018) 034513 [arXiv:1706.04622] [INSPIRE].
J. Aebischer et al., WCxf: an exchange format for Wilson coefficients beyond the Standard Model, Comput. Phys. Commun. 232 (2018) 71 [arXiv:1712.05298] [INSPIRE].
ETM collaboration, Kaon mixing beyond the SM from Nf = 2 tmQCD and model independent constraints from the UTA, JHEP 03 (2013) 089 [Erratum ibid. 07 (2013) 143] [arXiv:1207.1287] [INSPIRE].
RBC/UKQCD collaboration, Neutral kaon mixing beyond the Standard Model with nf = 2 + 1 chiral fermions part 1: bare matrix elements and physical results, JHEP 11 (2016) 001 [arXiv:1609.03334] [INSPIRE].
RBC and UKQCD collaborations, Neutral kaon mixing beyond the Standard Model with nf = 2 + 1 chiral fermions. Part 2: non perturbative renormalisation of the ∆F = 2 four-quark operators, JHEP 10 (2017) 054 [arXiv:1708.03552] [INSPIRE].
P. Boyle, N. Garron, J. Kettle, A. Khamseh and J.T. Tsang, BSM kaon mixing at the physical point, EPJ Web Conf. 175 (2018) 13010 [arXiv:1710.09176] [INSPIRE].
ETM collaboration, B-physics from Nf = 2 tmQCD: the Standard Model and beyond, JHEP 03 (2014) 016 [arXiv:1308.1851] [INSPIRE].
A.G. Grozin, R. Klein, T. Mannel and A.A. Pivovarov, \( {B}^0\hbox{-} {\overline{B}}^0 \) mixing at next-to-leading order, Phys. Rev. D 94 (2016) 034024 [arXiv:1606.06054] [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].
A.G. Grozin, T. Mannel and A.A. Pivovarov, \( {B}^0\hbox{-} {\overline{B}}^0 \) mixing: matching to HQET at NNLO, Phys. Rev. D 98 (2018) 054020 [arXiv:1806.00253] [INSPIRE].
D. King, A. Lenz and T. Rauh, Bs mixing observables and |Vtd/Vts| from sum rules, JHEP 05 (2019) 034 [arXiv:1904.00940] [INSPIRE].
L. Di Luzio, M. Kirk, A. Lenz and T. Rauh, ∆Ms theory precision confronts flavour anomalies, JHEP 12 (2019) 009 [arXiv:1909.11087] [INSPIRE].
Particle Data Group collaboration, Review of Particle Physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].
RBC and UKQCD collaborations, Domain wall QCD with physical quark masses, Phys. Rev. D 93 (2016) 074505 [arXiv:1411.7017] [INSPIRE].
S. Dürr et al., Lattice QCD at the physical point: light quark masses, Phys. Lett. B 701 (2011) 265 [arXiv:1011.2403] [INSPIRE].
S. Dürr et al., Lattice QCD at the physical point: simulation and analysis details, JHEP 08 (2011) 148 [arXiv:1011.2711] [INSPIRE].
C. McNeile, C.T.H. Davies, E. Follana, K. Hornbostel and G.P. Lepage, High-precision c and b masses, and QCD coupling from current-current correlators in lattice and continuum QCD, Phys. Rev. D 82 (2010) 034512 [arXiv:1004.4285] [INSPIRE].
MILC collaboration, MILC results for light pseudoscalars, PoS(CD09)007 (2009) [arXiv:0910.2966] [INSPIRE].
Z. Fodor et al., Up and down quark masses and corrections to Dashen’s theorem from lattice QCD and quenched QED, Phys. Rev. Lett. 117 (2016) 082001 [arXiv:1604.07112] [INSPIRE].
A. Bazavov et al., B- and D-meson leptonic decay constants from four-flavor lattice QCD, Phys. Rev. D 98 (2018) 074512 [arXiv:1712.09262] [INSPIRE].
ETM collaboration, Mass of the b quark and B-meson decay constants from Nf = 2 + 1 + 1 twisted-mass lattice QCD, Phys. Rev. D 93 (2016) 114505 [arXiv:1603.04306] [INSPIRE].
HPQCD collaboration, B-meson decay constants from improved lattice nonrelativistic QCD with physical u, d, s, and c quarks, Phys. Rev. Lett. 110 (2013) 222003 [arXiv:1302.2644] [INSPIRE].
C. Hughes, C.T.H. Davies and C.J. Monahan, New methods for B meson decay constants and form factors from lattice NRQCD, Phys. Rev. D 97 (2018) 054509 [arXiv:1711.09981] [INSPIRE].
G. Beall, M. Bander and A. Soni, Constraint on the mass scale of a left-right symmetric electroweak theory from the KL -KS mass difference, Phys. Rev. Lett. 48 (1982) 848 [INSPIRE].
J.A. Bagger, K.T. Matchev and R.-J. Zhang, QCD corrections to flavor changing neutral currents in the supersymmetric Standard Model, Phys. Lett. B 412 (1997) 77 [hep-ph/9707225] [INSPIRE].
M. Blanke, A.J. Buras, B. Duling, S. Gori and A. Weiler, ∆F = 2 observables and fine-tuning in a warped extra dimension with custodial protection, JHEP 03 (2009) 001 [arXiv:0809.1073] [INSPIRE].
A.J. Buras, D. Buttazzo, J. Girrbach-Noe and R. Knegjens, Can we reach the Zeptouniverse with rare K and Bs,d decays?, JHEP 11 (2014) 121 [arXiv:1408.0728] [INSPIRE].
A.J. Buras, New physics patterns in ε′/ε and εK with implications for rare kaon decays and ∆MK, JHEP 04 (2016) 071 [arXiv:1601.00005] [INSPIRE].
M. Artuso, G. Borissov and A. Lenz, CP violation in the \( {B}_s^0 \) system, Rev. Mod. Phys. 88 (2016) 045002 [Addendum ibid. 91 (2019) 049901] [arXiv:1511.09466] [INSPIRE].
S. Esen and A. Lenz, CKM 2018 summary of working group 4: mixing and mixing-related CP-violation in the B system ∆M, ∆Γ, ϕs, ϕ1/β, ϕ2/α, ϕ3/γ, in 10th International Workshop on the CKM Unitarity Triangle, (2019) [arXiv:1901.05000] [INSPIRE].
J. Brod, M. Gorbahn and E. Stamou, Standard-Model prediction of ϵK with manifest quark-mixing unitarity, Phys. Rev. Lett. 125 (2020) 171803 [arXiv:1911.06822] [INSPIRE].
R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization group evolution of the Standard Model dimension six operators III: gauge coupling dependence and phenomenology, JHEP 04 (2014) 159 [arXiv:1312.2014] [INSPIRE].
J. Aebischer, A. Crivellin, M. Fael and C. Greub, Matching of gauge invariant dimension-six operators for b → s and b → c transitions, JHEP 05 (2016) 037 [arXiv:1512.02830] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization group evolution of the Standard Model dimension six operators I: formalism and λ dependence, JHEP 10 (2013) 087 [arXiv:1308.2627] [INSPIRE].
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization group evolution of the Standard Model dimension six operators II: Yukawa dependence, JHEP 01 (2014) 035 [arXiv:1310.4838] [INSPIRE].
Z. Bern, J. Parra-Martinez and E. Sawyer, Structure of two-loop SMEFT anomalous dimensions via on-shell methods, JHEP 10 (2020) 211 [arXiv:2005.12917] [INSPIRE].
J. de Vries, G. Falcioni, F. Herzog and B. Ruijl, Two- and three-loop anomalous dimensions of Weinberg’s dimension-six CP-odd gluonic operator, Phys. Rev. D 102 (2020) 016010 [arXiv:1907.04923] [INSPIRE].
J. Aebischer, J. Kumar and D.M. Straub, Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale, Eur. Phys. J. C 78 (2018) 1026 [arXiv:1804.05033] [INSPIRE].
J. Aebischer, M. Fael, C. Greub and J. Virto, B physics beyond the Standard Model at one loop: complete renormalization group evolution below the electroweak scale, JHEP 09 (2017) 158 [arXiv:1704.06639] [INSPIRE].
E.E. Jenkins, A.V. Manohar and P. Stoffer, Low-energy effective field theory below the electroweak scale: anomalous dimensions, JHEP 01 (2018) 084 [arXiv:1711.05270] [INSPIRE].
C. Bobeth, A.J. Buras, A. Celis and M. Jung, Patterns of flavour violation in models with vector-like quarks, JHEP 04 (2017) 079 [arXiv:1609.04783] [INSPIRE].
C. Bobeth and A.J. Buras, Leptoquarks meet ε′/ε and rare kaon processes, JHEP 02 (2018) 101 [arXiv:1712.01295] [INSPIRE].
P. Arnan, L. Hofer, F. Mescia and A. Crivellin, Loop effects of heavy new scalars and fermions in b → sμ+ μ− , JHEP 04 (2017) 043 [arXiv:1608.07832] [INSPIRE].
P. Arnan, A. Crivellin, M. Fedele and F. Mescia, Generic loop effects of new scalars and fermions in b → sℓ+ℓ− and a vector-like 4th generation, JHEP 06 (2019) 118 [arXiv:1904.05890] [INSPIRE].
J. de Blas, J.C. Criado, M. Pérez-Victoria and J. Santiago, Effective description of general extensions of the Standard Model: the complete tree-level dictionary, JHEP 03 (2018) 109 [arXiv:1711.10391] [INSPIRE].
A.J. Buras and J. Girrbach, Complete NLO QCD corrections for tree level ∆F = 2 FCNC processes, JHEP 03 (2012) 052 [arXiv:1201.1302] [INSPIRE].
M. Bordone, C. Cornella, J. Fuentes-Martin and G. Isidori, A three-site gauge model for flavor hierarchies and flavor anomalies, Phys. Lett. B 779 (2018) 317 [arXiv:1712.01368] [INSPIRE].
V. Gherardi, D. Marzocca, M. Nardecchia and A. Romanino, Rank-one flavor violation and B-meson anomalies, JHEP 10 (2019) 112 [arXiv:1903.10954] [INSPIRE].
A.J. Buras, Weak Hamiltonian, CP-violation and rare decays, in Les Houches summer school in theoretical physics, session 68: probing the Standard Model of particle interactions, (1998), pg. 281 [hep-ph/9806471] [INSPIRE].
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: 2009.07276
Supplementary Information
ESM 1
(PDF 207 kb)
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
Aebischer, J., Bobeth, C., Buras, A.J. et al. SMEFT atlas of ∆F = 2 transitions. J. High Energ. Phys. 2020, 187 (2020). https://doi.org/10.1007/JHEP12(2020)187
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)187