Renormalization group flow equations and the phase transition in \(O(N)\)-models
type: Item
instance of: scholarly article
zbMATH Open document ID: 1012.81037
DOI: 10.1142/S0217751X0100502X
author: O. Bohr
author: B.-J. Schaefer
author: Jochen Wambach
published in: International Journal of Modern Physics A
publication date: 2003-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-ph/0007098
Mathematics Subject Classification ID: 81T17
Mathematics Subject Classification ID: 34K28
Mathematics Subject Classification ID: 65Z05
zbMATH DE Number: 1686671
zbMATH Keywords: transition temperature
zbMATH Keywords: Matsubara formalism
zbMATH Keywords: numerical study
MaRDI profile type: Publication
arXiv ID: hep-ph/0007098
OpenAlex ID: W3104939239
title: Renormalization group flow equations and the phase transition in \(O(N)\)-models
review text: Self-consistent flow equations for a general \(O(N)\)-symmetric effective potential without any polynomial truncation are derived, and their numerical solutions are examined. The effective Lagrangian for the \(O(N)\)-model at the scale \(\Lambda\) of the ultraviolet region is NEWLINE\[NEWLINE{\mathfrak L}_\Lambda= {1\over 2} (\partial_\mu \Phi)^2+ V(\Phi^2); \quad V(\Phi^2)= {\lambda\over 4}(\Phi^2-\Phi_0^2)^2,NEWLINE\]NEWLINE \(\Phi=(\Phi_1, \dots, \Phi_N)\) and the negative sign signals the broken phase. The expansion of the effective action of this model up to order \(O (\partial^4)\) is written NEWLINE\[NEWLINE\Gamma[\Phi=]\int d^dx\Bigl\{ V(\Phi^2)+\frac 12 Z_1 (\Phi^2) (\partial_\mu \Phi)^2+ \frac 12 Z_2(\Phi^2) (\Phi\partial_\mu \Phi)^2 \Bigr\}.NEWLINE\]NEWLINE By using a heat kernel, the effective action can be written as NEWLINE\[NEWLINE \Gamma [\Phi]=-{1\over 2}\int d^dx \int^\infty_0 {d\tau\over \tau}f_k \int {d^dp \over (2\pi)^d} \text{tr} e^{-\tau \bigl(p^2-2i p_\mu\partial_\mu-\partial^2 + V''_{ij} (\Phi)\bigr)},NEWLINE\]NEWLINE \(V_{ij}''= \lambda (\Phi^2- \Phi^2_0) \delta_{ij}+ 2\lambda\Phi_i \Phi_j\). The technical details of the derivation of this expression is given in Appendix A. Introducing differential equations for the blocking functions \(f_k^{(i)}\) the solutions of these equations are given by NEWLINE\[NEWLINEf_k^{(i)}= {2^i (d-2)!! \over\Gamma (d/2)(d-2+2i)!!} \Gamma\left({d \over 2}+ i,\tau Zk^2 \right).NEWLINE\]NEWLINE Then using the abbreviations \(m^2_\sigma= \lambda(3 \Phi^2- \Phi^2_0)\), \(m^2_\pi= \lambda(\Phi^2- \Phi^2_0)\), the renormalization flow equations are derived as NEWLINE\[NEWLINE\begin{aligned} \partial_tV & =S_d {k^d\over d}\left[ {1\over 1+2v'+4 \Phi^2 v''} +{N-1\over 1+2v'} \right],\\ -{1\over Z}\partial_tZ & ={2S_d\over \Phi^2 Zk^2} {k^d\over d}\left. \left[1+{1\over 1+4 \Phi^2v'')^2}+ {1\over 2 \Phi^2 v''}\left( {1\over 1+4\Phi^2v''}-1\right) \right] \right |_{ \Phi^2 =\Phi_0^2},\\ \partial_tV & =k{\partial V\over \…
reviewed by: Akira Asada
description: scientific article; zbMATH DE number 1686671