$K \rightarrow ππ$ $ΔI=3/2$ decay amplitude in the continuum limit

Abstract: We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2 = 1.50(4)\mathrm{stat}(14)\mathrm{syst}\times 10^{-8}$ GeV; Im$A_2 = -6.99(20)\mathrm{stat}(84)\mathrm{syst}\times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $K\to\pi\pi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}\simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $\Delta I=1/2$ rule, Re$A_0$/Re$A_2\simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $\epsilon^\prime/\epsilon$ is Re($\epsilon^\prime/\epsilon)_\textrm{EWP}=-(6.6\pm 1.0)\times 10^{-4}$.

Overview of the Paper: $K\rightarrow \pi\pi$ $\Delta I = 3/2$ Decay Amplitude in the Continuum Limit

This paper presents a detailed lattice QCD analysis of the $K \rightarrow \pi\pi$ decay amplitude with a focus on the isospin $I=2$ component, denoted as $A_2$. The efforts are part of the ongoing research by the RBC and UKQCD Collaborations aimed at explaining key features of kaon decays such as the $\Delta I = 1/2$ rule and $CP$ violation within the Standard Model.

Calculation Methodology

The calculation advances previous works by incorporating results from two lattice ensembles generated at physical quark masses with much finer lattice spacings. Crucially, it allows for a continuum extrapolation, significantly reducing the systematic uncertainties primarily attributed to discretization errors in past studies. This refinement is supported by the use of M\"obius domain-wall fermions, which offer enhanced chiral symmetry properties compared to predecessors.

The operators contributing to the process are renormalized using the RI-SMOM scheme, matched perturbatively to the $\overline{\mathrm{MS}}$ scheme used for Wilson coefficients. The authors carefully separate the contributions of different operators which govern the real and imaginary parts of $A_2$, demonstrating a noteworthy cancellation effect in the real part that aligns with interpretations of the $\Delta I = 1/2$ rule's characteristics.

Numerical Results

The paper reports the real part of $A_2$ as Re($A_2$) = 1.50(4)$\mathrm{stat}$(14)$\mathrm{syst}$ $\times 10^{-8}$ GeV, a result consistent with experimental observations. The imaginary part, previously undisclosed in phenomenological analyses, is determined as Im($A_2$) = -6.99(20)$\mathrm{stat}$(84)$\mathrm{syst}$ $\times 10^{-13}$ GeV.

Implications

The calculated imaginary part of $A_2$ facilitates an estimate of the electroweak penguin component of $\epsilon'/\epsilon$, yielding $(\epsilon'/\epsilon)_{\mathrm{EWP}} \approx -6.6(10) \times 10^{-4}$. This value contributes importantly towards understanding observed $CP$-violation in kaon decays.

The findings underscore the potential inaccuracies in phenomenological models based on na\"ive factorization, while lending support to theoretical frameworks that allow for operator cancellations.

Future Directions

A significant impediment remains in finalizing the calculation for the $I=0$ component, $A_0$, particularly the implementation of G-parity boundary conditions to manage isospin breaking effects. Future work will focus on overcoming these technical challenges. Successful computation of $A_0$ is expected to provide further clarity on the $\Delta I = 1/2$ rule and deliver a comprehensive lattice-based assessment of direct $CP$ violation, enhancing theoretical predictions for $\epsilon'/\epsilon$.

The paper opens avenues for improving lattice QCD accuracy, notably by enhancing nonperturbative renormalization via step scaling or through higher-order perturbative matching, which may reduce reliance on current Wilson coefficients' perturbative estimates.

In summary, this study marks a critical step in understanding kaon decay mechanisms, illuminating pathways for theoretical and computational innovations in lattice QCD and offering insights consistent with experimental phenomena.