Eigenvalue conditions are obtained from a study of the renormalization-group equations for a non-Abelian gauge theory with Higgs scalars. With these conditions, it is found that the theory is asymptotically free. For the purely leptonic SO(3) model of Georgi and Glashow, the eigenvalue conditions fix completely the parameters of the theory.It has become a common belief very recently that spontaneously broken gauge theories, with Higgs s c a l a r s , a r e not asymptotically free.'" In the work of G r o s s and Wilczek,' an initial investigation of this problem was made, without, however, considering in detail the effect of Yukawa couplings which generate fermion masses. In a subsequent paper by Cheng, Eichten, and Li,3 the full problem was discussed and the possibility of asymptotic freedom was quickly dismissed. In this note we wish to point out a simple eigenvalue condition that was not considered in their investigation, and, with it, we show that asymptotic freedom i s restored f o r the Georgi-Glashow-type gauge theories of weak and electromagnetic intera c t i o n~.~ Theories of the Weinberg-Salam type,5 involving mixing with an Abelian gauge group, a r e not asymptotically free. F o r simplicity, we have considered the GeorgiGlashow model with only leptons present. Let h,, h, be the Yukawa coupling constants such that the lowest-order m a s s e s for e -, E', and X , a r e m, -h,u, m,+h,v, and +h,v/sin~, respectively ( B i s the u,,X mixing angle in the model, while v i s the vacuum expectation of the neutral component of the Higgs s c a l a r field). F o r the muon system, corresponding coupling constants a r e denoted by H,, Hz. Let A be the quartic self-interaction coupling constant f o r the Higgs s c a l a r field [C,= -(h/4!) $:+-. . 1. Then the lowest-order coupled equations for the effective coupling constants read in the usual notation',6 .7 The equation f o r HI2 and Hz2 can, by p-e symmetry, b e obtained from Eqs. ( l b ) and ( l c ) . F o r clarity, let us suppose f i r s t that the hz,Hl, Hz couplings a r e absent in the theory and examine the prototype equation with A, B sfrictly positive constants. Call z2 h2/gZ and u ( t ) 5 Jo d ? g 2 (~) ; then Eq. (2) reduces to As Cheng, Eichten, and Li3 have pointed out, the critical point g2 = 0 i s ultraviolet-stable, for B> So. This solution has h2 vanishing a s t -I i , v = ( B -do),' 3,. Since this implies that h2 vanishes faster than g 2 , in the asymptotic domain, it is the s a m e situation a s that considered by Gross and Wilczek,' and a largely pessimistic conclusion results. However, there exzsts a solution to Eq. (3) which i s identically satisfied, viz., when i2 i s a constant, K = (B -$,)/A. It i s a trivial solution to Eq. (3);however, by virtue of the fact that the proportionality between 1z2 and g 2 holds for all t , it i s an elgenualue condition that must be imposed on the renormalized theory. That i s to say, if the r enormalized Yukawa coupling constant [h2(0)] i s chosen initially to be an a r b i t r a r y value s m a l l e r tha...