On the Feynman-Hellmann Theorem in Quantum Field Theory and the Calculation of Matrix Elements

TL;DR

This paper presents an improved method for calculating matrix elements in quantum field theory using the Feynman-Hellmann theorem, enabling better control over systematic uncertainties in lattice QCD computations.

Contribution

The authors develop a novel approach leveraging the Feynman-Hellmann theorem to compute matrix elements from two-point functions, applicable to various currents and momentum transfers.

Findings

Successfully calculated the nucleon axial charge with controlled excited-state systematics.

Demonstrated the method's ability to handle nonzero momentum transfer and flavor-changing currents.

Achieved a value of g_A = 1.213(26) with controlled uncertainties.

Abstract

The Feynman-Hellmann theorem can be derived from the long Euclidean-time limit of correlation functions determined with functional derivatives of the partition function. Using this insight, we fully develop an improved method for computing matrix elements of external currents utilizing only two-point correlation functions. Our method applies to matrix elements of any external bilinear current, including nonzero momentum transfer, flavor-changing, and two or more current insertion matrix elements. The ability to identify and control all the systematic uncertainties in the analysis of the correlation functions stems from the unique time dependence of the ground-state matrix elements and the fact that all excited states and contact terms are Euclidean-time dependent. We demonstrate the utility of our method with a calculation of the nucleon axial charge using gradient-flowed domain-wall valence quarks on the $N_f=2+1+1$ MILC highly improved staggered quark ensemble with lattice spacing and pion mass of approximately 0.15 fm and 310 MeV respectively. We show full control over excited-state systematics with the new method and obtain a value of $g_A = 1.213(26)$ with a quark-mass-dependent renormalization coefficient.