Renormalisability of the 2PI-Hartree approximation of multicomponent scalar models in the broken symmetry phase

Abstract

Non-perturbative renormalisation of a general class of scalar field theories is performed at the Hartree level truncation of the 2PI effective action in the broken symmetry regime. Renormalised equations are explicitly constructed for the one-and two-point functions. The non-perturbative counterterms are deduced from the conditions for the cancellation of the overall and the subdivergences in the complete Hartree-Dyson-Schwinger equations, with a transparent method. The procedure proposed in the present paper is shown to be equivalent to the iterative renormalisation method of Blaizot et al.[1]. MotivationOne of the most popular approximation techniques in many-body quantum theory is the Hartree approximation. In quantum field theory it corresponds to the momentum independent two-loop truncation of the two-particle irreducible effective action. It is used extensively both in equilibrium [2,3,4] and out-of-equilibrium [5,6,7,8] non-perturbative investigations of phase transition phenomena. Its non-perturbative renormalisability was demonstrated as particular case of the general proof of renormalisability of the physical quantities computed in various 2PI approximations [9,1,10]. These proofs are rather involved especially in the broken symmetry phase. For this reason in many practical applications the renormalised equations are not constructed explicitly. For instance, investigations of the finite temperature phase transitions in strongly interacting matter frequently either omit zero temperature quantum corrections in the 2PI approximate equations of the relevant 1-and 2-point functions [3,4,11] or take into account vacuum fluctuations by applying some cut-off [12].The exact generating 2PI-functional Γ[Φ, G] fulfils generalised Ward-Takahashi identities reflecting global internal symmetries of the models. As a consequence the 1PI effective potential Γ[Φ, G(Φ)]

On the quantum stability of Q-balls

“…It is not, however a completely rigorous 2PI renormalisation scheme (as for instance in ref. [14] for renormalisable φ 4 theory). For our purpose, where we do not go to the strict continuum limit, the present renormalisation scheme will be sufficient.…”

“…This may be remedied order by order in a loop expansion, although the fully resummed counterterm procedure of ref. [14] may not generalise to nonrenormalisable potentials.…”

Renormalization of the 2PI-Hartree approximation in a broken phase with nonzero superflow

“…[13,15]. This is based on a scheme in which the divergence structure of a given loop integral is obtained by expanding its integrand around an auxiliary propagator…”

“…III, we present the renormalization of the propagator equations and the field derivative (i.e., basically the field equation). We will put particular emphasis on differences compared to our earlier procedure described in Ref [13]. In Sec.…”

On the symmetry improved CJT formalism in the O(4) linear sigma model

“…Renormalization in many-body approximation schemes is a nontrivial procedure, and this problem has been extensively discussed in Refs. [24,[34][35][36]. In the following discussions, we follow the "counter term" renormalization scheme (CT) adopted in Ref.…”