Symmetry-improved CJT effective action

Abstract

The formalism introduced by Cornwall, Jackiw and Tomboulis (CJT) provides a systematic approach to consistently resumming non-perturbative effects in Quantum Thermal Field Theory. One major limitation of the CJT effective action is that its loopwise expansion introduces residual violations of possible global symmetries, thus giving rise to massive Goldstone bosons in the spontaneously broken phase of the theory. In this paper we develop a novel symmetry-improved CJT formalism for consistently encoding global symmetries in a loopwise expansion. In our formalism, the extremal solutions of the fields and propagators to a loopwise truncated CJT effective action are subject to additional constraints given by the Ward identities due to global symmetries. By considering a simple O(2) scalar model, we show that, unlike other methods, our approach satisfies a number of important field-theoretic properties. In particular, we find that the Goldstone boson resulting from spontaneous symmetry breaking of O (2) is massless and the phase transition is a second order one, already in the Hartree-Fock approximation. After taking the sunset diagrams into account, we show how our approach properly describes the threshold properties of the massless Goldstone boson and the Higgs particle in the loops. Finally, assuming minimal modifications to the Hartree-Fock approximated CJT effective action, we calculate the corresponding symmetryimproved CJT effective potential and discuss the conditions for its uniqueness for scalar-field values away from its minimum.

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

“…As a result, in order to recover the Goldstone theorem in the truncated CJT effective action, in analogy to the standard One-Particle-Irreducible effective action, a properly modified formalism for the CJT effective action has been proposed in Ref. [28] through the introduction of a constraint, which has been used to govern the Goldstone theorem in the 1PI effective action. Eventually, in addition to the gap equations (10), the constraint inserted into the stationarity equations has the form…”

“…Once the constrain is included, the stationarity equation (13) derived from the usual CJT formalism certainly needs not to be necessarily obeyed since the CJT effective action or the effective potential has already been modified accordingly as described in Ref. [28].…”

“…However, in comparison, we can find that the constraint proposed in Eq. (14) can be automatically realized in the case of large-N approximation in the O(N ) linear sigma model [6] [7], whereas, it should be artificially introduced in the framework of the symmetry improved CJT formalism [28]. Moreover, the stationarity equation in Eq.…”

“…(33) for the large-N approximation, φ can be explicitly expressed as the function of the temperature T , m and M σ . This means that for both methods there is no self-consistent equations to be determined in the symmetry breaking phase, or the mass-gap equations could be solved analytically [28]. The inconsistency cannot be avoided once the pion is set as a massless Goldstone boson in the symmetry breaking phase.…”

“…Very recently, Pilaftsis and Teresi have developed a novel symmetry improved CJT formalism by consistently encoding global symmetries in loop-wise expansions or truncations of the CJT effective action [28]. This formalism seems to avoid the pathologies of the existing approaches and satisfy many field-theoretic properties.…”