Finite Temperature Field Theory and Phase Transitions *

We review different aspects of field theory at zero and finite temperature, related to the theory of phase transitions. We discuss different renormalization conditions for the effective potential at zero temperature, emphasizing in particular the MS renormalization scheme. Finite temperature field theory is discussed in the real and imaginary time formalisms, showing their equivalence in simple examples. Bubble nucleation by thermal tunneling, and the subsequent development of the phase transition is described in some detail. Some attention is also devoted to the breakdown of the perturbative expansion and the infrared problem in the finite temperature field theory. Finally the application to baryogenesis at the electroweak phase transition is done in the Standard Model and in the Minimal Supersymmetric Standard Model. In all cases we have translated the condition of not washing out any previously generated baryon asymmetry by upper bounds on the Higgs mass. IEM-FT-187/99 January 1999 Based on lectures given at the Summer School in High Energy Physics and Cosmology, ICTP, Trieste (Italy) 29 June–17 July 1998. 1 The effective potential at zero temperature The effective potential for quantum field theories was originally introduced by Euler, Heisenberg and Schwinger , and applied to studies of spontaneous symmetry breaking by Goldstone, Salam, Weinberg and Jona-Lasinio. Calculations of the effective potential were initially performed at one-loop by Coleman and E. Weinberg and at higher-loop by Jackiw and Iliopoulos, Itzykson and Martin . More recently the effective potential has been the subject of a vivid investigation, especially related to its invariance under the renormalization group. I will try to review, in this section, the main ideas and update the latest developments on the effective potential. 1.1 Generating functionals To fix the ideas, let us consider the theory described by a scalar field φ with a lagrangian density L{φ(x)} and an action S[φ] = ∫ dxL{φ(x)} (1) The generating functional (vacuum-to-vacuum amplitude) is given by the pathintegral representation, Z[j] = 〈0out | 0in〉j ≡ ∫ dφ exp{i(S[φ] + φj)} (2) where we are using the notation φj ≡ ∫ dxφ(x)j(x) (3) Using (2) one can obtain the connected generating functional W [j] defined as, Z[j] ≡ exp{iW [j]} (4) and the effective action Γ[φ] as the Legendre transform of (4) Γ[φ] = W [j]− ∫ dx δW [j] δj(x) j(x) (5) where φ(x) = δW [j] δj(x) (6)