Conjecture on the infrared structure of the vacuum Schrödinger wave functional of QCD

Abstract

The Schr\"odinger wave functional $\ensuremath{\psi}=\mathrm{exp}(\ensuremath{-}S{{\mathcal{A}}_{i}^{a}(\stackrel{\ensuremath{\rightarrow}}{x})})$ for the $d=3+1$ QCD vacuum is a partition function constructed in $d=4$; the exponent $2S$ [in $|\ensuremath{\psi}{|}^{2}=\mathrm{exp}(\ensuremath{-}2S)$] plays the role of a $d=3$ Euclidean action. We start from a simple conjecture for $S$ based on dynamical generation of a gluon mass $M$ in $d=4$, then use earlier techniques of the author to extend (in principle) the conjectured form to full non-Abelian gauge invariance. We argue that the exact leading term, of $\mathcal{O}(M)$, in an expansion of $S$ in inverse powers of $M$ is a $d=3$ gauge-invariant mass term (gauged nonlinear sigma model); the next-leading term, of $\mathcal{O}(1/M)$, is a conventional Yang-Mills action. The $d=3$ action that is (twice) the sum of these two terms has center vortices as classical solutions. The $d=3$ gluon mass ${m}_{3}$, which we constrain to be the same as $M$, and $d=3$ coupling ${g}_{3}^{2}$ are related through the conjecture to the $d=4$ coupling strength, but at the same time the dimensionless ratio ${m}_{3}/{g}_{3}^{2}$ can be estimated from $d=3$ dynamics. This allows us to estimate the $d=4$ coupling ${\ensuremath{\alpha}}_{s}({M}^{2})$ in terms of the strictly $d=3$ ratio ${m}_{3}/{g}_{3}^{2}$; we find a value of about 0.4, in good agreement with an earlier theoretical value but somewhat low compared to the QCD phenomenological value of $0.7\ifmmode\pm\else\textpm\fi{}0.3$. The wave functional for $d=2+1$ QCD has an exponent that is a $d=2$ infrared-effective action having both the gauge-invariant mass term and the field-strength squared term, and so differs from the conventional QCD action in two dimensions, which has no mass term. This conventional $d=2$ QCD would lead in $d=3$ to confinement of all color-group representations. But with the mass term (again leading to center vortices), only $N\mathrm{\text{\ensuremath{-}}}\mathrm{ality}\ensuremath{\not\equiv}0$ mod $N$ representations can be confined [for gauge group $SU(N)$], as expected.