On the generalized eigenvalue method for energies and matrix elements in lattice field theory

TL;DR

This paper analyzes the generalized eigenvalue method in lattice gauge theory, demonstrating how to efficiently extract energies and matrix elements, including excited states, with corrections that vanish exponentially as time increases, supported by numerical results for B-mesons.

Contribution

It introduces an optimized application of the generalized eigenvalue method with proven asymptotic correction vanishing, and demonstrates its effectiveness with numerical B-meson results.

Findings

Corrections vanish exponentially with large time separations.

Increasing the number of interpolating fields improves energy gap.

Numerical results for B-mesons validate the method's effectiveness.

Abstract

We discuss the generalized eigenvalue problem for computing energies and matrix elements in lattice gauge theory, including effective theories such as HQET. It is analyzed how the extracted effective energies and matrix elements converge when the time separations are made large. This suggests a particularly efficient application of the method for which we can prove that corrections vanish asymptotically as $\exp(-(E_{N+1}-E_n) t)$. The gap $E_{N+1}-E_n$ can be made large by increasing the number $N$ of interpolating fields in the correlation matrix. We also show how excited state matrix elements can be extracted such that contaminations from all other states disappear exponentially in time. As a demonstration we present numerical results for the extraction of ground state and excited B-meson masses and decay constants in static approximation and to order $1/m_b$ in HQET.