Operator-product expansion and the asymptotic behavior of spontaneously broken scalar field theories

We reexamine a recent study of the operator-product expansion in spontaneously broken scalar field theories. First, the asymptotic behavior of the propagator in a spontaneously broken $\lambda {\phi }^4$ theory is calculated to lowest nontrivial order. The use of the operator-product expansion in the "naive" vacuum, with operators developing nonvanishing vacuum expectation values, is found to correctly reproduce the usual perturbative analysis of the shifted theory when carried out to the same order. The renormalization-group improvement of this result is studied. We find that ${\gamma }_{{\phi }^2}$, the renormalization-group coefficient of the operator ${\phi }^2$, is nonzero at first order in $\lambda$, This contradicts the result of the study of Gupta and Quinn. The generalization of this analysis to all Green's functions at all orders in perturbation theory is outlined. We argue that the renormalization-group improvement of the perturbation theory should yield the same answer for the two methods of calculating the asymptotic limit. Finally, we discuss the implications of this study for gauge theories.