Resolving the structure of bound states using lattice quantum field theories

TL;DR

This paper introduces a lattice quantum field theory approach to analyze bound states and their form factors, demonstrating the importance of finite-volume effects especially for shallow-bound states, and providing a detailed methodology for such calculations.

Contribution

It presents the first lattice calculation of a two-to-two particle matrix element of a local current using a pionless effective field theory, and develops a formalism to connect finite-volume spectra to infinite-volume bound-state form factors.

Findings

Finite-volume effects are negligible for deep-bound states.

Finite-volume formalism is essential for shallow-bound states.

Good agreement with anomalous threshold predictions for shallow states.

Abstract

This work presents the first lattice calculation of a two-to-two particle matrix element of a local current. This exploratory calculation is performed using a leading-order pionless effective field theory of two nucleons in a finite 3D spatial volume, where the Hamiltonian can be diagonalized exactly for moderate volumes. By considering a range of couplings where the theory supports a deuteron-like bound state, we determine the finite-volume spectra and matrix elements of the conserved local vector current. Using the L\"uscher formalism, we constrain the infinite-volume, purely hadronic amplitude for this theory. Using previously derived formalism, we then map the finite-volume matrix elements to scattering amplitudes describing a reaction coupling two-particle states via a current insertion, $\2+\Jc \to \2$. We then use a recently derived relation between this class of amplitudes and the bound-state elastic form factor to directly constrain the infinite-volume form factor. By varying over a range of values of the coupling of the theory, we explore the effects of this analysis for deep-bound states and shallow-bound states. We reproduce the expected result that for deep bound states, the finite-volume formalism is largely unnecessary, while for shallow bound states, it is absolutely critical to obtain a sensible result. We present a detailed outline of the analysis of this class of matrix elements, including the determination of the charge radius of the bound state. In the shallow bound state limit, we find good agreement with the prediction stemming from the anomalous threshold.