Boosting QED and QCD bound states in the path integral formalism

TL;DR

This paper develops a Lorentz covariant formalism for bound states in QED and QCD using the path integral approach, demonstrating how boosted wave functions and energies behave under Lorentz transformations and deriving implications for parton distributions.

Contribution

It introduces a method to obtain Lorentz covariant bound state wave functions and energies in the path integral formalism, including boosted states and their applications to form factors and parton distributions.

Findings

Boosted energy eigenvalues follow the relativistic form $E= \,\sqrt{\veP^2+M_0^2}$.

Wave functions contract under Lorentz boosts, affecting parton distributions.

Form factors decrease with increasing momentum transfer, consistent with quark counting rules.

Abstract

Wave functions and energy eigenvalues of the path integral Hamiltonian are studied in Lorentz frame moving with velocity $v$. The instantaneous interaction produced by the Wilson loop is shown to be reduced by an overall factor $\sqrt{1-(\frac{v}{c})^2}$. As a result one obtains the boosted energy eigenvalues in the Lorentz covariant form $E= \sqrt{\veP^2+M^2_0}$, where $M_0$ is the c.m. energy, and this form is tested for two free particles and for the Coulomb and linear interaction.Using Lorentz contracted wave functions of the bound states one obtains the scaled parton wave functions and valence quark distributions for large $P$. Matrix elements containing wave functions moving with different velocities strongly decrease with growing relative momentum, e.g. for the time-like formfactors one obtains $F_h(Q_0)\sim (\frac{M_h}{Q_0})^{2 n_h} $ with $n_h = 1$ and 2 for mesons and baryons, as in the "quark counting rule".