Approximate Solutions of Exact Renormalization Group Equations

We study exact renormalization group equations in the framework of the effective average action. We present analytical approximate solutions for the scale dependence of the potential in a variety of models. These solutions display a rich spectrum of physical behaviour such as fixed points governing the universal behaviour near second order phase transitions, critical exponents, first order transitions (some of which are radiatively induced) and tricritical behaviour. e-mail: [email protected] e-mail: [email protected] Introduction: The solution of an exact renormalization group equation [1]-[7] is a particularly difficult task. The reason is that such an equation describes the scale dependence of an effective action, which is characterized by infinitely many couplings multiplying the invariants consistent with the symmetries of the model under consideration. As a result an exact renormalization group equation corresponds to infinitely many evolution equations for the couplings of the theory. The crucial step is developing efficient approximation schemes which can reduce the complexity of the problem while capturing the essential aspects of the physical system. Perturbative expansions have been used for proofs of perturbative renormalizability [3, 8], while the powerful ǫ-expansion [1, 9] has been employed for the study of fixed points governing second order phase transitions in three dimensions. More recently, evolution equations for truncated forms of the effective action have been solved through a combination of analytical and numerical methods. A full, detailed and transparent picture of second and first order phase transitions for a variety of models has emerged [10]-[13]. Also, numerical solutions for the fixed point potential of three-dimensional scalar theories have been computed in ref. [14]. Fully analytical solutions have not been obtained, with the exception of ref. [11], where an exact solution for the three-dimensional O(N)-symmetric scalar theory in the large N limit is given. In this letter we present analytical approximate solutions of evolution equations for truncated forms of the effective action. We work in the framework of the effective average action Γk [5, 15], which results from the integration of quantum fluctuations with characteristic momenta q ≥ k. The effective average action Γk interpolates between the classical action S for k equal to the ultraviolet cutoff Λ of the theory (no integration of modes) and the effective action Γ for k = 0 (all the modes are integrated). Its dependence on k is given by an exact renormalization group equation with the typical form (t = ln(k/Λ)) ∂ ∂t Γk = 1 2 Tr { (Γ (2) k +Rk) −1 ∂ ∂t Rk }