Gauge Dependence of Effective Quark Mass and Matrix Elements in Gaugefixed Large $N$ Strong Coupling Lattice QCD

In conjunction with recent numerical \hbox{$\lambda~\partial_0 A_0 + \nabla\cdot\vec{A} =0} ``\lambda$-gauge’’ results reported in a companion paper, we construct an $N\to \mathrm{\infty }$ Wilson loop picture of $\lambda$-gaugefixing in which (I)the $\lambda$-gauge expectation value of a link chain $C$ is the weighted sum over Wilson loops made by joining to $C$ all selfavoiding chains $\tilde{C}$ closing $C$. (II)Weights $A_{\tilde{C}}$, containing all the $\lambda$-dependence, are given by the $\beta =0$ $\lambda$-gauge expectation value of $\tilde{C}$. (III)$A_{\tilde{C}}$ equals path-products of coefficients from the trace expansion of the gaugefixing Boltzmann weight. From (II) and (III) we deduce formulas for $\beta =0$ quark matrix elements. We find that $M_q^{(\lambda )}$ decreases with increasing $\lambda$; the quark propagator dispersion relation is not covariant when $\lambda \mathrm{\ne }1$; and $\mathrm{\Delta }I=1\mathrm{/}2$ matching coefficients are $\lambda$-independent. These strong coupling features are qualitatively consistent with numerical $\beta =5.7$ and $6.0$ results briefly described here for comparison purposes but mainly presented in a companion paper.