On the paper "Dynamical Symmetry Restoration and Constraints on Masses and Coupling Constants in Gauge Theories" by A. D. Linde JETP Lett. 23 (1976) 64-67

I. V. Polyubin

Landau Institute for Theoretical Physics,

Institute of Theoretical and Experimental Physics

In 1973 S. Coleman and E. Weinberg investigated the effect of radiative corrections on the possibility of spontaneous symmetry breaking in seminal paper [1]. A. D. Linde studied some

physical consequences of spontaneous breaking of gauge symmetry at one-loop level. The author considered a complex scalar field minimally coupled to U(1) gauge field with potential

$V=\lambda(\varphi^*\varphi)^2-{\mu}^2{\varphi}^*{\varphi}$. Renormalization conditions for effective potential fixed vacuum expectation value and mass of excitataions to its classical values.

The effective potential has the second minimum at $<\varphi>=0$. The condition $V_{eff}(\sigma)< V_{eff}(0)$ imposed a constraint on self-interaction coupling :

$$\lambda>\frac {3}{32{\pi}^2}g^4$$

Even if classical value of $\lambda$ is very small, due to one-loop correction it becomes equal to $\lambda_{eff}=\lambda+\frac{1}{2{\pi}^2}g^4$.

These inequalities for coupling constant were rewritten in terms of masses of scalar and vector particles. The lower bound on the Higgs mass was estimated as $m_H>5 GeV$. This approach was generalized soon to non-abelian case by S. Weinberg [2]. In this case the lower bound on Higgs mass is $m_H>7.4 GeV$ taking into account contribution of $W,Z $-bosons to effective potential. This lower bound is known as Weinberg-Linde value [3]. It is worth to mention that fermions propagating in the loop diminish WL value ( due to the opposite statistics). t-quark contribution makes the effective

potential unbounded from below. So one loop approximation is not valid. At two loops [4,5] nonsymmetric vacuum approaches the metastability boundary $V_{eff}(\sigma)\simeq V_{eff}(0)$ at

$m_H\simeq126 GeV$, $m_t\simeq174 GeV$ [6].

[1] S. R. Coleman, E. J. Weinberg, Phys. Rev. **D7** (1973), 1888-1910

[2] S. Weinberg, Phys. Rev. Lett. **36** (1976), 294-296

[3] Hung P. Q. Phys. Rev. Lett.** 42** (1979), 873-876

[4] Casas J. A., Espinosa J. R., Quiros M. and Riotto A., Nucl. Phys. **B436** (1995)3Ö29

[5] Hambye T. and Riesselmann K., Phys. Rev. **D55** (1997) 7255-7262

[6] Isidori G., Ridolfi G. and Strumia A., Nucl. Phys. **B609** (2001) 387-409