Defining the Force between Separated Sources on a Light Front

TL;DR

This paper proposes a new way to define the force between separated sources on a light front by minimizing gauge field energy with sources modeled as lines, leading to Coulomb law at weak coupling and insights into asymptotic freedom.

Contribution

It introduces a novel definition of force on a light front using line sources and demonstrates its consistency with Coulomb law and asymptotic freedom in gauge theories.

Findings

Derives Coulomb force law at weak coupling.

Shows emergence of asymptotic freedom via one-loop S-matrix evaluation.

Suggests potential applications to confinement problems.

Abstract

The Newtonian character of gauge theories on a light front requires that the longitudinal momentum P^+, which plays the role of Newtonian mass, be conserved. This requirement conflicts with the standard definition of the force between two sources in terms of the minimal energy of quantum gauge fields in the presence of a quark and anti-quark pinned to points separated by a distance R. We propose that, on a light front, the force be defined by minimizing the energy of gauge fields in the presence of a quark and an anti-quark pinned to lines (1-branes) oriented in the longitudinal direction singled out by the light front and separated by a transverse distance R. Such sources will have a limited 1+1 dimensional dynamics. We study this proposal for weak coupling gauge theories by showing how it leads to the Coulomb force law. For QCD we also show how asymptotic freedom emerges by evaluating the S-matrix through one loop for the scattering of a particle in the N_c representation of color SU(N_c) on a 1-brane by a particle in the \bar N_c representation of color on a parallel 1-brane separated from the first by a distance R<<1/Lambda_{QCD}. Potential applications to the problem of confinement on a light front are discussed.