### Abstract

We study the phase transition of a real scalar φ4 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 [A. Arrizabalaga and U. Reinosa, Nucl. Phys. A785, 234 (2007)] 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 "renormalization group-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 value φ̄ to an external field h, changes from 3 to 5, getting then closer to its expected value 4.789, obtained from accurate numerical estimates [A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002)].

Original language | English |
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Article number | 085031 |

Journal | Physical Review D - Particles, Fields, Gravitation and Cosmology |

Volume | 86 |

Issue number | 8 |

DOIs | |

Publication status | Published - okt. 19 2012 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

**Broken phase effective potential in the two-loop Φ-derivable approximation and nature of the phase transition in a scalar theory.** / Markó, Gergely; Reinosa, Urko; Szép, Z.

Research output: Article

*Physical Review D - Particles, Fields, Gravitation and Cosmology*, vol. 86, no. 8, 085031. https://doi.org/10.1103/PhysRevD.86.085031

}

TY - JOUR

T1 - Broken phase effective potential in the two-loop Φ-derivable approximation and nature of the phase transition in a scalar theory

AU - Markó, Gergely

AU - Reinosa, Urko

AU - Szép, Z.

PY - 2012/10/19

Y1 - 2012/10/19

N2 - We study the phase transition of a real scalar φ4 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 [A. Arrizabalaga and U. Reinosa, Nucl. Phys. A785, 234 (2007)] 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 "renormalization group-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 value φ̄ to an external field h, changes from 3 to 5, getting then closer to its expected value 4.789, obtained from accurate numerical estimates [A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002)].

AB - We study the phase transition of a real scalar φ4 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 [A. Arrizabalaga and U. Reinosa, Nucl. Phys. A785, 234 (2007)] 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 "renormalization group-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 value φ̄ to an external field h, changes from 3 to 5, getting then closer to its expected value 4.789, obtained from accurate numerical estimates [A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002)].

UR - http://www.scopus.com/inward/record.url?scp=84867798177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867798177&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.86.085031

DO - 10.1103/PhysRevD.86.085031

M3 - Article

AN - SCOPUS:84867798177

VL - 86

JO - Physical review D: Particles and fields

JF - Physical review D: Particles and fields

SN - 1550-7998

IS - 8

M1 - 085031

ER -