in the Standard Model at the Dawn of the 2020s

Jason Aebischer; Christoph Bobeth; Andrzej J. Buras

doi:10.1140/epjc/s10052-020-8267-1

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2020) 80: 705
https://doi.org/10.1140/epjc/s10052-020-8267-1

Regular Article - Theoretical Physics

$\varepsilon '/\varepsilon$ in the Standard Model at the Dawn of the 2020s

Jason Aebischer¹^*, Christoph Bobeth² and Andrzej J. Buras³

¹ Department of Physics, University of California at San Diego, La Jolla, CA, 92093, USA
² Physik Department, TU München, James-Franck-Straße, 85748, Garching, Germany
³ TUM Institute for Advanced Study, Lichtenbergstr. 2a, 85748, Garching, Germany

^* e-mail: jaebischer@physics.ucsd.edu

Received: 26 May 2020
Accepted: 21 July 2020
Published online: 6 August 2020

Abstract

We reanalyse the ratio $\varepsilon '/\varepsilon$ in the Standard Model (SM) using most recent hadronic matrix elements from the RBC-UKQCD collaboration in combination with most important NNLO QCD corrections to electroweak penguin contributions and the isospin-breaking corrections. We illustrate the importance of the latter by using their latest estimate from chiral perturbation theory (ChPT) based on the octet approximation for lowest-lying mesons and a very recent estimate in the nonet scheme that takes into account the contribution of $\eta _0$ . We find $(\varepsilon '/\varepsilon )^{(8)}_\text {SM} = (17.4 \pm 6.1) \times 10^{-4}$ and $(\varepsilon '/\varepsilon )^{(9)}_\text {SM} = (13.9 \pm 5.2) \times 10^{-4}$ , respectively. Despite a very good agreement with the measured value $(\varepsilon '/\varepsilon )_\text {exp} = (16.6 \pm 2.3) \times 10^{-4}$ , the large error in $(\varepsilon '/\varepsilon )_\text {SM}$ still leaves room for significant new physics (BSM) contributions to this ratio. We update the 2018 master formula for $(\varepsilon '/\varepsilon )_\text {BSM}$ valid in any extension beyond the SM without additional light degrees of freedom. We provide new values of the penguin parameters $B_6^{(1/2)}(\mu )$ and $B_8^{(3/2)}(\mu )$ at the $\mu$ -scales used by the RBC-UKQCD collaboration and at lower scales $\mathcal {O}(1\, \text {GeV})$ used by ChPT and Dual QCD (DQCD). We present semi-analytic formulae for $(\varepsilon '/\varepsilon )_\text {SM}$ in terms of these parameters and $\hat{\Omega }_\text {eff}$ that summarizes isospin-breaking corrections to this ratio. We stress the importance of lattice calculations of the $\mathcal {O}(\alpha _{\text {em}})$ contributions to the hadronic matrix elements necessary for the removal of renormalization scheme dependence at $\mathcal {O}(\alpha _{\text {em}})$ in the present analyses of $\varepsilon '/\varepsilon$ .