An exploratory calculation of $K_{\rm L}\toμ^+μ^-$ decay from Lattice QCD at physical pion mass

TL;DR

The paper tackles the long-distance LD2γ contribution to $K_{\rm L}\rightarrow\mu^+\mu^-$ by performing a first-principles lattice QCD calculation augmented with QED$_{\infty}$, enabling a complex amplitude determination from Euclidean correlators. It develops and implements a framework to extract the physical amplitude via a subtraction scheme that removes unphysical exponential growth, and uses isospin decomposition to separate $\pi^0$- and $\eta$-related channels, with a detailed treatment of diagram topologies and intermediate states. The authors achieve about 25% statistical precision on the dispersive part and identify the eta intermediate state's impact as a major systematic, studying three subtraction schemes and providing matrix-element inputs for renormalization and operator matching, while noting that missing GIM counterterms will be addressed in a follow-up work. The results indicate a destructive interference between the LD2γ dispersive part and the short-distance SD$G_F^2$ amplitude, emphasizing the importance of first-principles determinations for robust Standard Model tests, and set the stage for improved calculations on larger, finer lattices and with complete operator bases in future work.

Abstract

We compute the complex, long-distance two-photon-exchange amplitude which contributes to the rare $K_{\rm L}\rightarrowμ^+μ^-$ decay from lattice QCD. We use a $24^3\times 64$ physical-pion-mass gauge field ensemble at an inverse lattice spacing of $1.023$ GeV and a QED${}_\infty$-based formalism. Our implementation strategies for all five non-SU$(3)$-flavor-suppressed diagram topologies are given in detail. We achieve a 25% statistical precision on the dispersive part of this long-distance amplitude. This calculation is carried out with 2+1 quark flavors and therefore requires the addition of counter terms to compensate for the absence of the Glashow-Iliopoulos-Maiani mechanism. These counter terms are not included in the current calculation and will be the subject of a second paper. The precision of our results is limited by the reconstruction of the physical contribution of the $η$ intermediate state, for which various strategies are tested and compared.

Figures (16)

• Figure 1: Feynman graph representing the LD2$\gamma$ amplitude computed.
• Figure 2: Relevant separations involved in our formalism. Here we denote the position of the weak Hamiltonian with '$x$' instead of using the position '0' as in Eqs. \ref{['eq:Result_4pt']} and \ref{['eq:Result_sub']}. This more general location for $\mathcal{H}_{\textrm{W}}(x)$ is also used for clarity in some other places in the paper, e.g. Appendix \ref{['sect:comp']}.
• Figure 3: The five types of Wick-contraction topology evaluated in this paper. From left to right, top to bottom: Type-1, Type-2, Type-3, Type-4 and Type-5. The vertical dashed line represents the Coulomb-gauge-fixed kaon interpolating operator, the pair of circled crosses the four-quark weak operator and the vertices labeled $u$ and $v$ the two EM currents.
• Figure 4: The real (left) and imaginary (right) parts of the $I=1$ amplitude as a function of $\delta_{\max}$, the upper limit on the integral over the time separation between the closest EM current and the weak operator.
• Figure 5: The effective masses for $\eta$ and $\eta^\prime$ measured by solving the GEVP with different bin sizes.