[Paper Review] $\\epsilon'$/$\\epsilon$: three years later

M. Ciuchini|arXiv (Cornell University)|Jan 13, 1997

Probabilistic and Robust Engineering Design10 citations

TL;DR

This paper reviews the theoretical prediction of the direct CP-violating parameter ǫ′/ǫ in kaon decays, updating estimates after three years of next-to-leading-order QCD calculations. Despite progress, the prediction remains uncertain at ~100% theoretical error due to non-perturbative matrix elements and CKM parameters. The analysis shows that a value of ǫ′/ǫ > 10⁻³ would require either new physics or significant revisions to non-perturbative QCD inputs like the strange quark mass or B-parameters.

ABSTRACT

Three years after the completion of the next-to-leading order calculation, the status of the theoretical estimates of $\\epsilon'/\\epsilon$ is reviewed. In spite of the theoretical progress, the prediction of $\\epsilon'/\\epsilon$ is still affected by a 100% theoretical error. In this paper the different sources of uncertainty are critically analysed and an updated estimate of of $\\epsilon'/\\epsilon$ definitely larger than $10^{-3}$ are also discussed.

Motivation & Objective

• To update the theoretical prediction of ǫ′/ǫ in the Standard Model after three years of next-to-leading-order QCD calculations.
• To critically assess the dominant sources of theoretical uncertainty in ǫ′/ǫ, particularly non-perturbative matrix elements and CKM parameters.
• To evaluate the implications of a future measurement of ǫ′/ǫ > 10⁻³ for the Standard Model and potential new physics.
• To explore how deviations in key non-perturbative parameters—such as the strange quark mass and B-parameters—could enhance ǫ′/ǫ beyond current estimates.

Proposed method

• Uses the next-to-leading-order ∆S = 1 effective Hamiltonian to separate short- and long-distance physics in K⁰ decays.
• Applies the vacuum insertion approximation (VIA) to estimate matrix elements of local four-quark operators, parameterized by B-parameters.
• Incorporates CKM matrix elements via the Wolfenstein parametrization, with Im(V*tsVtd) derived from ǫ and B-meson mixing.
• Evaluates Wilson coefficients at next-to-leading order using QCD renormalization group evolution.
• Performs a global uncertainty analysis by varying input parameters such as ms, B6, and B3/28.
• Considers extreme scenarios—e.g., small ms or large B6 with small B3/28—to test sensitivity of ǫ′/ǫ to non-perturbative inputs.

Experimental results

Research questions

• RQ1 What is the current theoretical uncertainty in the prediction of ǫ′/ǫ, and what are its dominant sources?
• RQ2 How do variations in the strange quark mass or B-parameters affect the predicted value of ǫ′/ǫ?
• RQ3 What would be the theoretical implications if ǫ′/ǫ were measured to be definitively larger than 10⁻³?
• RQ4 Can the observed ǫ′/ǫ values be accommodated within the Standard Model without new physics, given current uncertainties?
• RQ5 How do non-perturbative QCD effects—especially matrix elements of four-quark operators—impact the final prediction?

Key findings

• The theoretical prediction for ǫ′/ǫ remains uncertain at approximately 100% relative error, despite three years of next-to-leading-order progress.
• A value of ǫ′/ǫ = (14 ± 8) × 10⁻⁴ is predicted if the running strange quark mass is reduced to mMSs(2 GeV) = 70 ± 11 MeV.
• An extreme scenario with B6 = 1.50 ± 0.15 and B3/28 = 0.50 ± 0.05 yields ǫ′/ǫ = (12 ± 4) × 10⁻⁴, indicating a significant enhancement.
• Such a large ǫ′/ǫ would marginally exclude ǫ′/ǫ = 0, suggesting that the cancellation between strong and electro-penguin contributions is fragile.
• A measurement of ǫ′/ǫ > 10⁻³ would strongly challenge the current theoretical framework, requiring either new physics or revised non-perturbative inputs.
• The results highlight the critical role of non-perturbative parameters—especially ms and B-parameters—whose uncertainties dominate the theoretical error.