Theta dependence of the vacuum energy density in chiral effective Lagrangian models at finite temperature, above Tc

Enrico Meggiolaro

doi:10.1103/physrevd.99.114009

Abstract

In this work, extending a previous study at zero temperature ($T=0$), we perform a systematic study of the modifications to the QCD vacuum energy density $\epsilon_{vac}$ in the finite-temperature case, above the chiral transition at $T_c$, caused by a nonzero value of the parameter $\theta$, using two different effective Lagrangian models which implement the $U(1)$ axial anomaly of the fundamental theory and which are both well defined also above $T_c$. In particular, we derive (and critically compare) the expressions for the topological susceptibility $\chi$ and for the second cumulant $c_4$ starting from the $\theta$ dependence of $\epsilon_{vac}(\theta)$ in the two models.

Highlights

It is well known that, at temperatures above a certain critical temperature Tc ≈ 150 MeV, thermal fluctuations break up the chiral condensate hqqi, causing the complete restoration of the SUðLÞL ⊗ SUðLÞR chiral symmetry of QCD with L light quarks (L 1⁄4 2 and L 1⁄4 3 being the physically relevant cases): this leads to a phase transition called chiral transition
In this work, extending a previous study at zero temperature (T 1⁄4 0) [3], we perform a systematic study of the modifications to the QCD vacuum energy density εvac in the finite-temperature case, above the chiral transition at Tc, caused by a nonzero value of the parameter θ, using two different effective Lagrangian models which implement the Uð1Þ axial anomaly of the fundamental theory and which are both well defined above Tc
P2ffiffiBm 2λ2π B2π p2ffiffiBm 2λ2π B2π det M: ð3:7Þ. Comparing these last results with those that we have found in the ELσ model, we see that they coincide with each other provided that the parameter κ in Eqs. (2.7) and (2.8) is identified with κ1FX=4 [and is proportional to the Uð1Þ axial condensate]

Summary

INTRODUCTION

It is well known (mainly by lattice simulations [1]) that, at temperatures above a certain critical temperature Tc ≈ 150 MeV, thermal fluctuations break up the chiral condensate hqqi, causing the complete restoration of the SUðLÞL ⊗ SUðLÞR chiral symmetry of QCD with L light quarks (L 1⁄4 2 and L 1⁄4 3 being the physically relevant cases): this leads to a phase transition called chiral transition. In this work, extending a previous study at zero temperature (T 1⁄4 0) [3], we perform a systematic study of the modifications to the QCD vacuum energy density εvac in the finite-temperature case, above the chiral transition at Tc, caused by a nonzero value of the parameter θ, using two different effective Lagrangian models which implement the Uð1Þ axial anomaly of the fundamental theory and which are both well defined above Tc. In particular, we derive (and critically compare) the expressions for the topological susceptibility χ and for the second cumulant c4 starting from the θ dependence of εvacðθÞ in the two models. [This way, the term including log U in the potential vanishes, eliminating the problem of the divergence, at least as far as the vev of the field X is different from zero or, in other words, as far as the Uð1Þ axial symmetry remains broken above Tc] As it was already observed in Refs. In the last section we shall draw our conclusions, summarizing (and critically commenting) the results obtained in this work and discussing some possible future developments

RESULTS

A κ1FX 2

SUMMARY AND ANALYSIS OF THE RESULTS