Behavior in strong fields of Euclidean gauge theories. II

Functional determinants resulting from functional integration in quantum gauge theories are studied. We derive an expansion around the constant field strength for the (renormalized) spinor determinant detMF in QED. We show that, if the field strength F is large and its derivatives are bounded, then detMF≡exp(-W)~exp(c∫F²lnF²), where $c>0$. Hence, the effective action W in (four-dimensional) QED is unbounded from below. Moreover, we prove that exp(-W) is not integrable. A similar result is established in the Yukawa model [detMY~exp(∫φ⁴lnφ⁴)]. We estimate the scalar determinant detMA² for a non-Abelian gauge field. We show that (like in the Abelian case studied earlier) detMA²=exp[c∫|F|²ln|F|²+r₂(F,DF,DDF)], where $c>0$ and r₂ is bounded by a quadratic form of the gauge-invariant variables $|F|$, |DF|, and |DDF|. We investigate the effect of gluon self-interaction on the stability of models with broken gauge symmetry G→H (we discuss in detail the Georgi-Glashow model). We sum up (in an approximation) the contribution of massive gluons to the O(2)-invariant effective action. It is shown that this effective action is bounded from below for slowly varying fields, if the couplings are asymptotically free at the one-loop level.