Brown, Michael J., and Whittingham, Ian B. (2015) Symmetry improvement of 3PI effective actions for O(N) scalar field theory. Physical Review D (Particles, Fields, Gravitation and Cosmology), 91 (8). 085020. pp. 1-27.

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Abstract

N-particle irreducible effective actions (nPIEA) are a powerful tool for extracting nonperturbative and nonequilibrium physics from quantum field theories. Unfortunately, practical truncations of nPIEA can unphysically violate symmetries. Pilaftsis and Teresi (PT) addressed this by introducing a "symmetry improvement" scheme in the context of the 2PIEA for an O (2) scalar theory, ensuring that the Goldstone boson is massless in the broken symmetry phase [A. Pilaftsis and D. Teresi, Nucl. Phys. B874, 594 (2013)]. We extend this idea by introducing a symmetry improved 3PIEA for O(N) theories, for which the basic variables are the one-, two- and three-point correlation functions. This requires the imposition of a Ward identity involving the three-point function. We find that the method leads to an infinity of physically distinct schemes, though a field theoretic analogue of d'Alembert's principle is used to single out a unique scheme. The standard equivalence hierarchy of nPIEA no longer holds with symmetry improvement, and we investigate the difference between the symmetry improved 3PIEA and 2PIEA. We present renormalized equations of motion and counterterms for two- and three-loop truncations of the effective action, though we leave their numerical solution to future work. We solve the Hartree-Fock approximation and find that our method achieves a middle ground between the unimproved 2PIEA and PT methods. The phase transition predicted by our method is weakly first order and the Goldstone theorem is satisfied, while the PT method correctly predicts a second-order phase transition. In contrast, the unimproved 2PIEA predicts a strong first-order transition with large violations of the Goldstone theorem. We also show that, in contrast to PT, the two-loop truncation of the symmetry improved 3PIEA does not predict the correct Higgs decay rate, although the three-loop truncation does, at least to leading order. These results suggest that symmetry improvement should not be applied to nPIEA truncated to <n loops. We also show that symmetry improvement schemes are compatible with the Coleman-Mermin-Wagner theorem, giving a check on the consistency of the formalism.

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