[en] Spontaneously broken gauge theories are considered in which the gauge coupling constant g vanishes asymptotically and the quartic self-coupling lambda approaches a finite UV stable fixed point. A third order calculation of the Callan-Symanzik β function for the most general quartic coupling indicates the existence of a UV stable fixed point close to the origin. The gauge coupling constant vanishes asymptotically for all realistic representations of the Higgs scalars, including those for which all gauge fields acquire mass. Under these conditions, the usual logarithmic deviations from exact scaling are recovered for the deep inelastic non-singlet structure functions and to leading order in q² for the singlet ones. (Auth.)