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

The Schrödinger wave functional $\psi =\exp (-S\{{\mathcal{A}}_i^a(\overrightarrow{x})\})$ for the $d=3+1$ QCD vacuum is a partition function constructed in $d=4$; the exponent $2S$ [in $|\psi {|}^2=\exp (-2S)$] 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 ${\alpha }_s(M^2)$ 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 $0.7\pm 0.3$. 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 $N\mathrm{\text{−}}\mathrm{ality}\not\equiv 0$ mod $N$ representations can be confined [for gauge group $SU(N)$], as expected.