Loop space

Abstract

Several topics in the loop-space formulation of non-Abelian gauge theories are considered. The basic objects dealt with are the unrenormalized dimensionally regularized gauge-invariant loop functions $W({C}^{i};g, \ensuremath{\epsilon})$, where ${C}^{i}$ is a set of loops, $g$ is the unrenormalized coupling constant, and $\ensuremath{\epsilon}$ is the deviation from four space-time dimensions. The renormalization-group equations satisfied by the corresponding renormalized loop functions are derived and, using asymptotic freedom, used to determine the exact behavior of the functions when the length $L$ of the loops approaches zero. The result is ${(\ensuremath{-}\mathrm{ln}L\ensuremath{\mu})}^{a(\ensuremath{\gamma})}$, where $\ensuremath{\mu}$ is the subtraction mass and $\ensuremath{\gamma}$ represents the cusp and cross-point angles of the loops. The function $a(\ensuremath{\gamma})$ is exactly computable and several examples are given. The equivalent result may be stated as the exact behavior of the renormalization-constant matrix ${Z}^{\mathrm{ij}}(\ensuremath{\gamma}, {g}_{R}, \ensuremath{\epsilon})$ for $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ with fixed renormalized coupling constant ${g}_{R}$, or as the exact behavior of the unrenormalized loop function for $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ and ${g}_{R}$ fixed. It is shown next that the $W({C}^{i};g, \ensuremath{\epsilon})$ satisfy dimensionally regularized Makeenko-Migdal equations in all orders of perturbation theory. The proof makes detailed use of dimensional regularization, Becchi-Rouet-Stora symmetry, gauge-field combinatorics, and properties of the area functional derivative of path-ordered multiple line integrals. Doubt is cast on the existence of such useful equations when other regularizations are used or when renormalization is performed. The Mandelstam constraints are considered next. Among other things, it is shown that the loop-function renormalization may be performed such that the renormalized functions satisfy a constraint which has the same form as the unrenormalized constraint $\ensuremath{\Sigma}{i=1}^{(N+1)\ensuremath{\uparrow}}{a}_{i}W({C}^{i})=0$, for the $\mathrm{U}(N)$ gauge group. The paper concludes with illustrations of how observable matrix elements of physical (color singlet, quark bilinear) flavor currents may be expressed in terms of loop functions. Among other topics discussed in the paper are the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, two-dimensional QCD, and normalization conditions on the renormalized loop functions.