Calculating the two-photon exchange contribution to $K_L\rightarrow\mu^+\mu^-$ decay

TL;DR

This paper develops a lattice QCD framework to calculate the complex two-photon exchange amplitude in the rare decay $K_L\rightarrow\mu^+\mu^-$, crucial for testing the Standard Model's predictions for second-order weak processes.

Contribution

It introduces a novel lattice QCD approach to compute both real and imaginary parts of the two-photon exchange amplitude in $K_L$ decay, addressing a key theoretical challenge.

Findings

The real part of the two-photon amplitude is comparable to second-order weak process contributions.

Finite volume effects from three-particle states are estimated to be manageable at a few percent.

Neglecting three-particle states is justified at 10% target accuracy in lattice calculations.

Abstract

We present a theoretical framework within which both the real and imaginary parts of the complex, two-photon exchange amplitude contributing to $K_L\rightarrow\mu^+\mu^-$ decay can be calculated using lattice QCD. The real part of this two-photon amplitude is of approximately the same size as that coming from a second-order weak strangeness-changing neutral-current process. Thus a test of the standard model prediction for this second-order weak process depends on an accurate result of this two-photon amplitude. A limiting factor of our proposed method comes from low-energy three-particle $\pi\pi\gamma$ states. The contribution from these states will be significantly distorted by the finite volume of our calculation -- a distortion for which there is no available correction. However, a simple estimate of the contribution of these three-particle states suggests their contribution to be at most a few percent allowing their neglect in a lattice calculation with a 10% target accuracy.