On the energy momentum dispersion in the lattice regularization

TL;DR

This paper investigates the energy-momentum dispersion relations in lattice regularizations of free scalar and U(1) gauge theories, revealing finite size effects and a novel relation between ground state energy and mass parameter.

Contribution

It provides a detailed analysis of finite volume corrections to dispersion relations and derives an exact relation between ground state energy and mass parameter in lattice theories.

Findings

Squared dispersion energy can be negative for small lattices.

Finite size corrections do not remove the difference between ground state energy and mass parameter.

Derived an exact relation: $E_0=\cosh^{-1}(1+M^2/2)$, independent of lattice size.

Abstract

For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by $E_{\rm dis}^2=E_{\vec{k}}^2-E_0^2-4\sum_{i=1}^{d-1}\sin(k_i/2)^2$ is in both cases negative ($d$ is the Euclidean space-time dimension and $E_{\vec{k}}$ the energy of momentum $\vec{k}$ eigenstates). Observation of $E_{\rm dis}^2=0$ has been an accepted method to demonstrate the existence of a massless photon ($E_0=0$) in 4D lattice gauge theory, which we supplement here by a study of its finite size corrections. A surprise from the lattice regularization of the free field is that infrared corrections do {\it not} eliminate a difference between the groundstate energy $E_0$ and the mass parameter $M$ of the free scalar lattice action. Instead, the relation $E_0=\cosh^{-1} (1+M^2/2)$ is derived independently of the spatial lattice size.