Abstract
As an important step towards a complete next-to-leading (NLO) QCD analysis of the ratio ε′/ε within the Standard Model Effective Field Theory (SMEFT), we present for the first time the NLO master formula for the BSM part of this ratio expressed in terms of the Wilson coefficients of all contributing operators evaluated at the electroweak scale. To this end we use the common Weak Effective Theory (WET) basis (the so-called JMS basis) for which tree-level and one-loop matching to the SMEFT are already known. The relevant hadronic matrix elements of BSM operators at the electroweak scale are taken from Dual QCD approach and the SM ones from lattice QCD. It includes the renormalization group evolution and quark-flavour threshold effects at NLO in QCD from hadronic scales, at which these matrix elements have been calculated, to the electroweak scale.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. J. Buras, The ϵ′/ϵ-story: 1976–2021, Acta Phys. Polon. B 52 (2021) 7 [arXiv:2101.00020] [INSPIRE].
NA48 collaboration, A precision measurement of direct CP-violation in the decay of neutral kaons into two pions, Phys. Lett. B 544 (2002) 97 [hep-ex/0208009] [INSPIRE].
KTeV collaboration, Measurements of direct CP-violation, CPT symmetry, and other parameters in the neutral kaon system, Phys. Rev. D 67 (2003) 012005 [Erratum ibid. 70 (2004) 079904] [hep-ex/0208007] [INSPIRE].
KTeV collaboration, Precise measurements of direct CP-violation, CPT symmetry, and other parameters in the neutral kaon system, Phys. Rev. D 83 (2011) 092001 [arXiv:1011.0127] [INSPIRE].
RBC, UKQCD collaboration, Direct CP-violation and the ∆I = 1/2 rule in K → ππ decay from the standard model, Phys. Rev. D 102 (2020) 054509 [arXiv:2004.09440] [INSPIRE].
A. J. Buras and J.-M. Gérard, Upper bounds on ε′/ε parameters \( {B}_6^{\left(1/2\right)} \) and \( {B}_8^{\left(3/2\right)} \) from large N QCD and other news, JHEP 12 (2015) 008 [arXiv:1507.06326] [INSPIRE].
V. Cirigliano, H. Gisbert, A. Pich and A. Rodríguez-Sánchez, Theoretical status of ε′/ε, J. Phys. Conf. Ser. 1526 (2020) 012011 [arXiv:1912.04736] [INSPIRE].
A. Pich and A. Rodríguez-Sánchez, SU(3) analysis of four-quark operators: K → ππ and vacuum matrix elements, JHEP 06 (2021) 005 [arXiv:2102.09308] [INSPIRE].
V. Cirigliano, H. Gisbert, A. Pich and A. Rodríguez-Sánchez, Isospin-violating contributions to ϵ′/ϵ, JHEP 02 (2020) 032 [arXiv:1911.01359] [INSPIRE].
A. J. Buras and J.-M. Gérard, Isospin-breaking in ε′/ε: impact of η0 at the dawn of the 2020s, Eur. Phys. J. C 80 (2020) 701 [arXiv:2005.08976] [INSPIRE].
J. Aebischer, C. Bobeth and A. J. Buras, ε′/ε in the standard model at the dawn of the 2020s, Eur. Phys. J. C 80 (2020) 705 [arXiv:2005.05978] [INSPIRE].
A. J. Buras, ϵ′/ϵ–2018: a Christmas story, arXiv:1812.06102 [INSPIRE].
J. Aebischer, C. Bobeth and A. J. Buras, On the importance of NNLO QCD and isospin-breaking corrections in ε′/ε, Eur. Phys. J. C 80 (2020) 1 [arXiv:1909.05610] [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].
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].
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].
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].
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].
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].
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].
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].
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].
J. Aebischer, C. Bobeth, A. J. Buras, J. Kumar and M. Misiak, General non-leptonic ∆F = 1 WET at the NLO in QCD, JHEP 11 (2021) 227 [arXiv:2107.10262] [INSPIRE].
J. Aebischer, C. Bobeth, A. J. Buras and J. Kumar, SMEFT ATLAS of ∆F = 2 transitions, JHEP 12 (2020) 187 [arXiv:2009.07276] [INSPIRE].
A. J. Buras, M. Gorbahn, S. Jäger and M. Jamin, Improved anatomy of ε′/ε in the standard model, JHEP 11 (2015) 202 [arXiv:1507.06345] [INSPIRE].
V. Cirigliano, A. Pich, G. Ecker and H. Neufeld, Isospin violation in ε′, Phys. Rev. Lett. 91 (2003) 162001 [hep-ph/0307030] [INSPIRE].
V. Cirigliano, G. Ecker, H. Neufeld and A. Pich, Isospin breaking in K → ππ decays, Eur. Phys. J. C 33 (2004) 369 [hep-ph/0310351] [INSPIRE].
J. Bijnens and F. Borg, Isospin breaking in K → 3π decays III: Bremsstrahlung and fit to experiment, Eur. Phys. J. C 40 (2005) 383 [hep-ph/0501163] [INSPIRE].
M. Misiak and M. Münz, Two loop mixing of dimension five flavor changing operators, Phys. Lett. B 344 (1995) 308 [hep-ph/9409454] [INSPIRE].
T. Blum et al., K → ππ ∆I = 3/2 decay amplitude in the continuum limit, Phys. Rev. D 91 (2015) 074502 [arXiv:1502.00263] [INSPIRE].
A. J. Buras, J.-M. Gérard and W. A. Bardeen, Large N approach to kaon decays and mixing 28 years later: ∆I = 1/2 rule, \( {\hat{B}}_K \) and ∆MK, Eur. Phys. J. C 74 (2014) 2871 [arXiv:1401.1385] [INSPIRE].
ETM collaboration, K → π matrix elements of the chromomagnetic operator on the lattice, Phys. Rev. D 97 (2018) 074501 [arXiv:1712.09824] [INSPIRE].
A. J. Buras and J.-M. Gérard, K → ππ and K − π matrix elements of the chromomagnetic operators from dual QCD, JHEP 07 (2018) 126 [arXiv:1803.08052] [INSPIRE].
J. Aebischer, A. J. Buras and J. Kumar, Another SMEFT story: Z′ facing new results on ϵ′/ϵ, ∆MK and K → \( \pi v\overline{v} \), JHEP 12 (2020) 097 [arXiv:2006.01138] [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].
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: 2107.12391
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. BSM master formula for ε′/ε in the WET basis at NLO in QCD. J. High Energ. Phys. 2021, 43 (2021). https://doi.org/10.1007/JHEP12(2021)043
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)043