Multiple-Scale Analysis and Renormalization of Quenched Second Order Phase Transitions

Details Multiple-Scale Analysis and Renormalization of Quenched Second Order Phase Transitions Multiple-Scale Analysis and Renormalization of Quenched Second Order Phase Transitions

2001-05-04

arxiv.org/abs/hep-ph/0105045v2

Phys.Rev. D64 (2001) 105026

Physics

High Energy Physics

High Energy Physics - Phenomenology

RevTex 19 pages. A paragraph on the limitations of the Hartree approx. has been added. Version to appear in Phys.Rev.D

Scientific paper

10.1103/PhysRevD.64.105026

A quenched second order phase transition is modeled by an effective $\Phi^4$-theory with a time-dependent Hamiltonian $\hat{H} (t)$, whose symmetry is broken spontaneously in time. The quantum field evolves out of equilibrium (nonequilibrium) during the phase transition as the density operator significantly deviates from $\hat{\rho} (t) = e^{- \beta \hat{H} (t)}/Z_H$. The recently developed Liouville-von Neumann (LvN) method provides various quantum states for the phase transition in terms of a complex solution to the mean-field equation, which is equivalent to the Gaussian effective potential in the static case and to the time-dependent Hartree-Fock equation in the nonequilibrium case. Using the multiple scale perturbation theory (MSPT) we solve analytically the mean-field equation to the first order of coupling constant and find the quantum states during the quenched second order phase transition. We propose a renormalization scheme during the process of phase transition to regularize the divergences, which originate from the mode coupling between hard and hard modes or between the soft and hard modes. The effect of mode coupling is discussed.

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