[Paper Review] The chiral phase transition for two-flavour QCD at imaginary and zero chemical potential

Claudio Bonati, Massimo D’Elia+3 more|arXiv (Cornell University)|Nov 3, 2013

High-Energy Particle Collisions Research21 references3 citations

TL;DR

This study investigates the nature of the chiral phase transition in two-flavor QCD using staggered fermions on $N_t=4$ lattices at imaginary and zero chemical potential. By mapping the chiral critical line via tricritical scaling of the Binder cumulant of the chiral condensate, it finds a finite tricritical point at $(\mu/T)^2_{\text{tric}} = 0.85(5)$, implying a definite first-order transition in the chiral limit on coarse lattices.

ABSTRACT

The chiral symmetry of QCD with two massless quark flavours gets restored in a non-analytic chiral phase transition at finite temperature and zero density. Whether this is a first-order or a second-order transition has not yet been determined unambiguously, due to the difficulties of simulating light quarks. We investigate the nature of the chiral transition as a function of quark mass and imaginary chemical potential, using staggered fermions on N_t=4 lattices. At sufficiently large imaginary chemical potential, a clear signal for a first-order transition is obtained for small masses, which weakens with decreasing imaginary chemical potential. The second-order critical line m_c(mu_i), which marks the boundary between first-order and crossover behaviour, extrapolates to a finite m_c(mu_i=0) with known critical exponents. This implies a definitely first-order transition in the chiral limit on relatively coarse, N_t=4 lattices.

Motivation & Objective

• To determine the order of the chiral phase transition in two-flavor QCD at zero and imaginary chemical potential.
• To resolve the long-standing ambiguity over whether the chiral transition is first-order or second-order in the chiral limit.
• To use the absence of the sign problem at imaginary chemical potential to map the chiral critical line and extrapolate to the chiral limit.
• To test whether tricritical scaling behavior can reliably identify the critical point and distinguish between first- and second-order transitions.
• To provide a controlled extrapolation to the chiral limit using finite-mass data and known critical exponents.

Proposed method

• Employ staggered fermions on $N_t=4$ lattices to simulate QCD at finite imaginary chemical potential and varying quark masses.
• Use the Binder cumulant of the chiral condensate, $B_4(m,\mu) = \langle(\delta X)^4\rangle / \langle(\delta X)^2\rangle^2$, to locate phase transitions.
• Identify critical points via volume collapse of the Binder cumulant, where $B_4 \to 1.604$ in the infinite-volume limit.
• Apply tricritical scaling to the critical line $m_c(\mu_i)$, fitting data to the form $B_4(am, a\mu) = A(am) + B(am)((a\mu)^2 - (a\mu_c)^2)$.
• Extrapolate the critical line to the chiral limit using known critical exponents and the $Z(2)$ universality class.
• Use the $N_t=4$ lattice spacing $a \sim 0.3$ fm to estimate the physical relevance of the result, acknowledging coarse lattice effects.

Experimental results

Research questions

• RQ1 Is the chiral phase transition in two-flavor QCD first-order or second-order in the chiral limit?
• RQ2 Can the chiral critical line at imaginary chemical potential be mapped using tricritical scaling to infer the behavior at zero chemical potential?
• RQ3 Does the observed behavior of the Binder cumulant at finite volume support a first-order transition or a second-order critical point?
• RQ4 What is the value of $(\mu/T)^2$ at the tricritical point in the chiral limit, and does it indicate a first-order transition?
• RQ5 How do the results on coarse $N_t=4$ lattices compare with continuum extrapolations and previous simulations using different fermion actions?

Key findings

• The chiral critical line at imaginary chemical potential exhibits tricritical scaling, with data from six quark masses collapsing onto a single curve.
• The extrapolated tricritical point in the chiral limit is located at $(\mu/T)^2_{\text{tric}} = 0.85(5)$, indicating a first-order transition on $N_t=4$ lattices.
• The critical pion mass at zero chemical potential is estimated to be $m_\pi^c \sim 60$ MeV, consistent with a first-order transition in the chiral limit.
• The Binder cumulant shows clear signatures of a first-order transition at large imaginary chemical potential, weakening with decreasing $\mu_i$.
• The results are consistent with earlier staggered fermion studies and recent overlap fermion simulations, reinforcing the first-order nature of the transition.
• The study provides a controlled method to extrapolate to the chiral limit using tricritical scaling, despite the limitations of coarse lattices.