Cluster expansion approach to the effective potential in Φ 2+1 4 -theory

Abstract

We apply a truncated set of dynamical equations of motion for connected equal-time Green functions up to the 4-point level to the investigation of spontaneous ground state symmetry breaking in 4 2+1 quantum eld theory. Within our momentum space discretization we obtain a second order phase transition as soon as the connected 3-point function is included. However, an additional inclusion of the connected 4-point function still shows a signi cant in uence on the shape of the e ective potential and the critical coupling.

Convergence properties of the cluster expansion for equal- time Green functions in scalar theories

“…The derivation of the correlation dynamical equations for Φ 4 -theory has been presented in detail in [15,16]; the resulting equations in (1 + 1) dimensions are given by Eqs. (20)-(31) in appendix A of [15] and in (2 + 1) dimensions by Eqs.…”

“…The calculations in [15,16] aimed at the evaluation of the effective potential V ef f (Φ 0 ), which is given by the minimum of the energy expectation value in the subspace of states with a fixed expectation value of the scalar field Φ = Φ 0 . Since the Φ 4 d+1 CD-equations describe the (equal-time) evolution of the system for a given initial condition, a direct (variational) evaluation of the effectiv potential within this approach would amount to minimizing the energy in the manifold of stationary solutions with the constraint Φ = Φ 0 .…”

“…by adiabatically changing the parameters of the theory in time from a configuration with a well known lowest energy state solution to the actual configuration of interest as described in [15,16]. Whereas this many-body scheme was found to converge quite rapidly in case of nonrelativistic nuclear physics problems [20], it is not clear if this also holds for the quantum field theoretical case.…”

“…including the connected npoint functions with n ≤ 6. In line with the notations in [15,16] this set of equations is denoted by Φ [7,15,21] (cf. appendix C) and switch on the residual interaction terms adiabatically.…”

“…δm 2 contains the mass counterterms, which in (2 + 1) dimensions are given by the tadpole diagram and the setting sun diagram, and in (1+1) dimensions by the tadpole diagram only (cf. [15,16]). …”