Broken phase effective potential in the two-loopΦ-derivable approximation and nature of the phase transition in a scalar theory

Abstract

We study the phase transition of a real scalar ϕ 4 theory in the two-loop Φ-derivable approximation using the imaginary time formalism, extending our previous (analytical) discussion of the Hartree approximation. We combine Fast Fourier Transform algorithms and accelerated Matsubara sums in order to achieve a high accuracy. Our results confirm and complete earlier ones obtained in the real time formalism [1] but which were less accurate due to the integration in Minkowski space and the discretization of the spectral density function. We also provide a complete and explicit discussion of the renormalization of the two-loop Φ-derivable approximation at finite temperature, both in the symmetric and in the broken phase, which was already used in the real-time approach, but never published. Our main result is that the two-loop Φ-derivable approximation suffices to cure the problem of the Hartree approximation regarding the order of the transition: the transition is of the second order type, as expected on general grounds. The corresponding critical exponents are, however, of the mean-field type. Using a "RG-improved" version of the approximation, motivated by our renormalization procedure, we find that the exponents are modified. In particular, the exponent δ, which relates the field expectation valueφ to an external field h, changes from 3 to 5, getting then closer to its expected value 4.789, obtained from accurate numerical estimates [2].

Thermodynamics and phase transition of theO(N)model from the two-loopΦ-derivable approximation

“…(14), one recovers the N ¼ 1 bare parameters of Ref. [9]. It is also simple to obtain the expressions for the bare parameters in the Hartree-Fock approximation.…”

“…[8] and illustrated in Ref. [9], the fact that the gap masses at zero momentum are different from the curvature masses requires the presence of two distinct bare masses m 0 and m 2 . Those are fixed by means of the usual renormalization condition…”

“…[9], it is possible to prove implicitly that the bare parameters given above renormalize the gap and field equations, as well as the effective potential (up to a temperature and field independent divergence for this latter quantity). By ''implicit proof,'' we mean that certain steps require some assumptions on the properties of a function, the spectral function, which is defined implicitly.…”

“…1 Finally, following Ref. [9], we have replaced the bare couplings in the highest loop diagrams of Eq. (1) by a coupling ?…”

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

“…In the end of this section, it should be noted that from the equations (26) and (37), the sigma mass is only dependent on the critical temperature or the sigma mass at vacuum, when submitting these equations into their gap equations, e.g. Eqs.…”