Eigenvalue conditions and asymptotic freedom for Higgs-scalar gauge theories

Abstract

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...

Non-Abelian Higgs models: Paving the way for asymptotic freedom

“…This is not necessarily true: already a standard perturbative analysis [3,8,[23][24][25][26][27][28][29] reveals the existence of gauged Yukawa models, where also the seemingly problematic scalar self-interaction ∼ λφ 4 can become asymptotically free as well. This happens along suitable RG trajectories depending on the precise matter content of the model.…”

“…This line of model building has been pursued since the early days of asymptotic freedom [3,8,[23][24][25][26][27][28][29]. However in this work, we stay within the class of nonabelian Higgs models, and intend to construct asymptotically free trajectories without further degrees of freedom.…”

Constraints on Finite Soft Supersymmetry Breaking Terms

“…Thus, the idea of gauge-Yukawa unification (GYU) [4]- [6] relies not only on a symmetry principle, but also on the principle of reduction of couplings [7,8] (see also [9]). This principle is based on the existence of RGI relations among couplings, which do not necessarily result from a symmetry, but nevertheless preserve perturbative renormalizability or even finiteness.…”

Asymptotic Behaviour in Quantum Field Theory

“…It depends on the validity of assumption (I), because the factor exp JG depends on the values of S(x) and yG(x) in the region 0 < x < g. Also, a (ln n)cG/ 2 b singularity cannot be generated by finite-order perturbation theory for G(nq), because in general, the power cG/2b is not an integer, Of course, if there is no wave-function renormalization for G [i,e, yG(x) = yG(O)], the leading singularity is the same as in free-field theory (parton model), In particular, the ratio Nevertheless, asymptotically free theories with scalar mesons do exist 54 ) and in particular, there are models 55 ) in which all perturbative states are massive. An immediate reason for not pursuing this line further is that it involves an unrealistic perturbative constraint.…”