$\text{QED}_\text{r}$: a finite-volume QED action with redistributed spatial zero-momentum modes

TL;DR

The paper tackles finite-volume QED corrections in precision lattice QCD+QED by introducing QED$_r$, a redistribution of the spatial zero-mode onto neighboring momentum shells to eliminate the $O(1/L^3)$, kinematics-independent contamination. In the minimal implementation (R=1), this scheme sets the zero-mode coefficient $ar{c}_0$ to zero, reducing leading finite-volume effects on pseudoscalar masses to $O(1/L)$ and $O(1/L^2)$, while ensuring that velocity-dependent $O(1/L^3)$ terms in leptonic decays can be addressed either through clever kinematics or stochastic averaging over directions. The authors derive the corresponding finite-volume corrections for pseudoscalar masses and leptonic decays, highlighting how QED$_r$ removes the most problematic contributions and outlining practical strategies to suppress remaining terms, with clear implications for CKM precision tests. They also discuss extending the framework to multiple shells and emphasize the need for multi-volume simulations to validate the infinite-volume limit in lattice calculations.

Abstract

We present a finite-volume QED action designed to improve the infinite-volume extrapolation of hadronic observables in precision lattice QCD+QED calculations. The action proposed in this work, which we call $\text{QED}_\text{r}$, can be seen as a particular case of the infrared-improved QED actions introduced by Davoudi et al. in 2019, and is specifically designed to remove kinematics-independent finite-volume corrections that appear at $\mathrm{O}(1/L^3)$ in the commonly used $\text{QED}_\text{L}$ formulation, where $L$ is the spatial extent of the physical volume. For a number of key observables, these effects depend on the internal structure of the hadrons and are difficult to evaluate non-perturbatively, making an analytical subtraction of the finite-volume effects impractical. We explicitly study the $\text{QED}_\text{r}$ electromagnetic finite-size effects on hadron masses and leptonic decay rates, relevant for Standard Model precision tests using the Cabibbo-Kobayashi-Maskawa matrix elements. In addition, we propose methods to remove the kinematics-dependent $\mathrm{O}(1/L^3)$ effects in leptonic decays. The removal of such contributions, shifting the leading contamination to $\mathrm{O}(1/L^4)$, will help to reduce the systematic uncertainties associated with finite-volume effects in future lattice QCD+QED calculations.

Figures (5)

• Figure 1: Diagrammatic representation of the mass shift in \ref{['eq:massshiftcompton']}, where the Compton tensor enters in the grey blob. The wiggly line corresponds to the photon.
• Figure 2: Analytical structure of the integrand function in \ref{['eq:massshiftcompton']} in the upper half of the $k_0$-plane. The photon pole is here labelled as $\textrm{pp}$ and the pseudoscalar pole as $\textrm{psp}$.
• Figure 3: Diagrammatic representation of the leptonic decay, with the photon represented by the wiggly line. The grey blob contains the weak current mediating the decay.
• Figure 4: The analytic structure of the $k_0$ integrand for (a) the factorisable contribution in \ref{['eq:corr_fact']}, and (b) the non-factorisable contribution in \ref{['eq:corr_nonfact']}. The poles have been labeled pp (photon pole), psp (pseudoscalar pole) and lp (lepton pole). Single poles are denoted by regular circles, double poles by squares and branch cuts by zigzag lines. The positions of the singularities in general depend on $\mathbf{k}$.
• Figure 5: Angular dependence of $\bar{c}_0(\mathbf{v}_\ell)$ for different values of $|\mathbf{v}_\ell|$. Positive and negative values are shown in blue and red, respectively. White regions correspond to directions for which $\bar{c}_0(\mathbf{v}_\ell)=0$.