Journey to the Bound States

TL;DR

This paper develops a perturbative bound state method for QED and QCD, tested on positronium and mesons, revealing insights into confinement, frame independence, and chiral symmetry breaking.

Contribution

It introduces a novel perturbative approach to bound states in gauge theories, applicable to both QED and QCD, with explicit calculations and implications for confinement and symmetry breaking.

Findings

Validated on positronium with hyperfine splitting and form factors

Demonstrated frame independence of deep inelastic scattering distributions

Revealed confining potentials and Regge trajectories in QCD

Abstract

Guided by the observed properties of hadrons I formulate a perturbative bound state method for QED and QCD. The expansion starts with valence Fock states ($e^+e^-,\ q\bar q,\ qqq,\ gg$) bound by the instantaneous interaction of temporal gauge ($A^0=0$). The method is tested on Positronium atoms at rest and in motion, including hyperfine splitting at $O(\alpha^4)$, electromagnetic form factors and deep inelastic scattering. Relativistic binding is studied for QED in $D=1+1$ dimensions, demonstrating the frame independence of the DIS electron distribution and its sea for $x_{bj} \to 0$. In QCD a homogeneous solution of Gauss' constraint in $D=3+1$ implies $O(\alpha_s^0)$ confining potentials for $q\bar q,\ q\bar qg,\ qqq$ and $gg$ states, whereas $q\bar q\,q\bar q$ is unconfined. Meson states lie on linear Regge trajectories and have the required frame dependence. A scalar bound state with vanishing four-momentum causes spontaneous chiral symmetry breaking when mixed with the vacuum. These lecture notes assume knowledge of field theory methods, but not of bound states. Brief reviews of existing bound state methods and Dirac electron states are included. Solutions to the exercises are given in the Appendix.