Spectral-function determination of complex electroweak amplitudes with lattice QCD

TL;DR

This paper introduces a new lattice QCD method to accurately determine both real and imaginary parts of complex electroweak amplitudes, overcoming analytic continuation challenges and demonstrated on a decay process.

Contribution

The novel spectral-function approach extends existing techniques to reliably extract complex amplitudes from lattice QCD, including intermediate state effects.

Findings

Successfully applied to $D_s o ext{leptonic} u_ ext{leptonic} ext{gamma}^*$ decay

Demonstrated the method's effectiveness with a single lattice ensemble

Provided non-perturbative evaluation of a hadronic amplitude

Abstract

We present a novel method to determine on the lattice both the real and imaginary parts of complex electroweak amplitudes involving two external currents and a single hadron or the QCD vacuum in the external states. The method is based on the spectral representation of the relevant time-dependent correlation functions and, by extending the range of applicability of other recent proposals built on the same techniques, overcomes the difficulties related to the analytic continuation from Minkowskian to Euclidean time, arising when intermediate states with energies smaller than the external states contribute to the amplitude. In its simplest form, the method relies on the standard $i \varepsilon$ prescription to regularize the Feynman integrals and at finite $\varepsilon$ it requires to verify the condition $1/L \ll \varepsilon \ll \Delta(E)$, where $L$ is the spatial extent of the lattice and, for any given energy $E$, $\Delta(E)$ represents the typical size of the interval around $E$ in which the hadronic amplitude is significantly varying. In order to illustrate the effectiveness of this approach in a realistic case, we apply the method to evaluate non-perturbatively the hadronic amplitude contributing to the radiative leptonic decay $D_s \to\ell\nu_\ell\,\gamma^*$, working for simplicity with a single lattice ensemble at fixed volume and lattice spacing.