Bound state basics

TL;DR

This paper presents a perturbative approach to bound states in QED and QCD using equal-time Fock expansion in temporal gauge, addressing confinement and Poincaré invariance issues.

Contribution

It introduces a boundary condition on the gauge field that incorporates confinement effects without altering the Lagrangian, ensuring frame-dependent interactions are correctly modeled.

Findings

Bound states can be described perturbatively with a boundary condition on the gauge field.

A universal scale related to the gluon vacuum energy density determines the confining potential.

Color singlet states in QCD exhibit linear confining potentials derived from Poincaré covariance.

Abstract

Perturbative expansions for atoms in QED are developed around interacting states, typically defined by the Schr\"odinger equation. Calculations are nevertheless done using the standard Feynman diagram expansion around free states. The classical $-\alpha/r$ potential is then obtained through an infinite sum of ladder diagrams. The complexity of this approach may have contributed to bound states being omitted from QFT textbooks, restricting the field to select experts. The confinement scale 1 fm of QCD must be introduced without changing the Lagrangian. This can be done via a boundary condition on the gauge field, which affects the bound state potential. The absence of confinement in Feynman diagrams may be due to the free field boundary condition. Poincar\'e invariance is realized dynamically for bound states, i.e., the interactions are frame dependent. Gauge theories have instantaneous interactions, due to gauge fixing at all points of space at the same time. In bound state perturbation theory each order must have exact Poincar\'e invariance. This is non-trivial even for atoms at lowest order. I summarize a perturbative approach to equal time bound states in QED and QCD, using a Fock expansion in temporal ($A^0=0$) gauge. The longitudinal electric field $E_L$ is instantaneous and need not vanish at spatial infinity for the constituents of color singlet states in QCD. Poincar\'e covariance determines the boundary condition for $E_L$ up to a universal scale, characterised by the gluon field energy density of the vacuum. A non-vanishing density contributes a linear term to the $q\bar{q}$ potential, while $qqq,\ q\bar{q}g$ and $gg$ color singlet states get analogous confining potentials.