[Paper Review] epsilon'/epsilon at the NLO: 10 Years Later

Andrzej J. Buras, Matthias Jamin|arXiv (Cornell University)|Jun 23, 2003

Particle physics theoretical and experimental studies38 references3 citations

TL;DR

This paper re-evaluates the Standard Model prediction for the direct CP-violating parameter ε′/ε at next-to-leading order (NLO), using updated experimental and theoretical inputs. It determines the hadronic matrix element of the dominant QCD penguin operator Q₆ from the measured ε′/ε value, finding R₆ = 1.23 ± 0.16 when R₈ = 1.0 ± 0.2, implying a significant correlation between Q₆ and Q₈ matrix elements that challenges large-Nc expectations.

ABSTRACT

During the last four years several parameters relevant for the analysis of the CP-violating ratio epsilon'/epsilon improved and/or changed significantly. In particular, the experimental value of epsilon'/epsilon and the strange quark mass decreased, the uncertainty in the CKM factor has been reduced, and for a value of the hadronic matrix element of the dominant electroweak penguin operator Q_8, some consensus has been reached among several theory groups. In view of this situation, ten years after the first analyses of epsilon'/epsilon at the next-to-leading order, we reconsider the analysis of epsilon'/epsilon within the SM and investigate what can be said about the hadronic Q_6 matrix element of the dominant QCD penguin operator on the basis of the present experimental value of epsilon'/epsilon and todays values of all other parameters. Employing a conservative range for the reduced electroweak penguin matrix element R_8=1.0+-0.2 from lattice QCD, and present values for all other input parameters, on the basis of the current world average for epsilon'/epsilon, we obtain the reduced hadronic matrix element of the dominant QCD penguin operator R_6=1.23+-0.16 implying _0^NDR(m_c) ~ -0.8 _2^NDR(m_c). We compare these results with those obtained in large-N_c approaches in which generally R_6 ~ R_8 and _0^NDR(m_c) is chirally suppressed relatively to _2^NDR(m_c). We present the correlation between R_6 and R_8 that is implied by the data on epsilon'/epsilon provided new physics contributions to epsilon'/epsilon can be neglected.

Motivation & Objective

• To re-analyze ε′/ε in the Standard Model using improved experimental and theoretical inputs from the prior decade.
• To determine the hadronic matrix element of the dominant QCD penguin operator Q₆ from the measured ε′/ε value, assuming no new physics contributions.
• To assess the consistency of large-Nc approaches with current data, particularly regarding the ratio R₆/R₈.
• To establish the correlation between R₆ and R₈ implied by the experimental ε′/ε value under the SM assumption.

Proposed method

• Uses the NLO Wilson coefficients for penguin and current-current operators from the Munich and Rome groups (1993).
• Applies a phenomenological approach to extract matrix elements, leveraging CP-conserving data and current experimental constraints.
• Imposes the constraint that new physics contributions to ε′/ε are negligible, allowing extraction of Q₆ matrix element from ε′/ε data.
• Employs a conservative range R₈ = 1.0 ± 0.2 from lattice QCD for the electroweak penguin matrix element.
• Performs a global fit to determine R₆ and the ratio R₆/R₈ using the world average ε′/ε = (16.6 ± 1.6) × 10⁻⁴.
• Constructs a correlation plot between R₆ and R₈ to visualize allowed parameter space.

Experimental results

Research questions

• RQ1 What is the value of the hadronic matrix element of the dominant QCD penguin operator Q₆, given the current experimental value of ε′/ε and updated input parameters?
• RQ2 How does the ratio R₆ = ⟨Q₆⟩₀ / ⟨Q₈⟩₂ compare to the large-Nc prediction R₆ = R₈?
• RQ3 What is the allowed correlation between R₆ and R₈ implied by the ε′/ε data under the assumption of no new physics?
• RQ4 Can the large-Nc approach with R₆ = R₈ be reconciled with current data, and if so, under what conditions?
• RQ5 How do different theoretical approaches (lattice QCD, large-Nc, chiral limits) compare in predicting the matrix elements of Q₆ and Q₈?

Key findings

• The analysis yields R₆ = 1.23 ± 0.16 for R₈ = 1.0 ± 0.2, implying ⟨Q₆⟩₀^NDR(m_c) ≈ −0.77⟨Q₈⟩₂^NDR(m_c) with a 19% uncertainty.
• The result R₆ = 1.23 ± 0.16 is inconsistent with the large-Nc prediction R₆ = R₈ = 1.36 ± 0.3, which lies outside the 1σ ellipse in the R₆–R₈ plane.
• The large-Nc approach can only be consistent with data if R₆ = R₈ = 1.36 ± 0.3, requiring a larger ⟨Q₈⟩₂ than most recent estimates.
• The value R₈ = 1.0 ± 0.2 is consistent with recent lattice QCD estimates, supporting the reliability of the input for the electroweak penguin matrix element.
• The correlation between R₆ and R₈ derived from data shows that all three main theoretical approaches—large-Nc, lattice QCD, and chiral limit—can be consistent with data if matrix elements are tuned accordingly.
• If future non-perturbative calculations find R₆ and R₈ outside the allowed region in the R₆–R₈ correlation plot, new physics contributions to ε′/ε would be required.