[Paper Review] The Phase Diagram of Strong Coupling QCD including Gauge Corrections

Philippe de Forcrand, Jens Langelage+2 more|arXiv (Cornell University)|Dec 2, 2013

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper develops a reweighting method to compute leading-order gauge corrections to the phase diagram of strong coupling lattice QCD with staggered fermions, using Monte Carlo simulations with the worm algorithm. It finds that the chiral transition temperature decreases with increasing gauge coupling β, with a slope of −0.446(7) at zero density, and shows that the shift in the phase boundary weakens with increasing chemical potential, vanishing at the tricritical point.

ABSTRACT

The strong coupling limit of lattice QCD with staggered fermions has been studied for decades, both via Monte Carlo and via mean field theory. In this model, the finite density sign problem can be made mild and the full phase diagram can be obtained, even in the chiral limit. It is however desirable to understand the effect of a finite lattice gauge coupling $β$ on the phase diagram in the $μ-T$ plane in order to understand how it evolves into the phase diagram of continuum QCD. Here we discuss how to construct a partition function for non-zero lattice coupling, exact to $\mathcal{O}(β)$, and present corresponding Monte Carlo results, in particular for corrections to the chiral susceptibility and to the phase diagram.

Motivation & Objective

• To understand how the phase diagram of strong coupling QCD evolves into the continuum QCD phase diagram as the gauge coupling β increases from zero.
• To address the challenge of finite-β corrections in the presence of a mild sign problem, particularly for chiral and nuclear transitions.
• To compute the leading-order gauge corrections to the chiral susceptibility and transition temperature using reweighting techniques.
• To determine the β-dependence of the chiral phase boundary, especially near the tricritical point.
• To validate the method via finite-size scaling and comparison with mean field results.

Proposed method

• Reweighting is used to compute the leading-order (O(β)) correction to the chiral susceptibility, with the Taylor coefficient cχ derived from the connected correlation between the susceptibility and the plaquette operator.
• The partition function is reformulated in terms of hadronic degrees of freedom—monomers, dimers, and baryonic loops—using a dual representation after integrating out Grassmann variables.
• The worm algorithm enables efficient sampling of the 2-point monomer correlation function, which is used to compute the chiral susceptibility χ.
• Finite-size scaling with 3d O(2) critical exponents is applied to extract the chiral transition temperature Tc(β) from the susceptibility data.
• The derivative dTc/dβ is computed via critical scaling, with the slope s = −0.446(7) at μ = 0, using the scaling function and non-universal coefficients.
• The method is extended to finite chemical potential, with phase boundary corrections extrapolated linearly in β, showing weakening β-dependence as μ increases.

Experimental results

Research questions

• RQ1 How does the chiral transition temperature Tc(β) evolve with increasing gauge coupling β in the strong coupling limit?
• RQ2 What is the leading-order effect of finite β on the chiral susceptibility and its critical scaling?
• RQ3 How does the phase boundary shift with β at finite chemical potential, and does this shift vanish at the tricritical point?
• RQ4 To what extent do the results at finite β agree with mean field theory and continuum expectations?
• RQ5 Can the mild sign problem in the dimer representation be effectively managed to compute β corrections at finite density?

Key findings

• The chiral transition temperature decreases with increasing β, with a slope of dTc/dβ = −0.446(7) at zero chemical potential.
• The finite-size scaling of the chiral susceptibility collapses onto a universal function using 3d O(2) critical exponents, with A ≈ 1.001(5) and B ≈ −0.892(5).
• The leading-order correction to the chiral susceptibility is given by cχ = 3N_s^3 N_t (⟨(ψ̄ψ)^2 P⟩ − ⟨(ψ̄ψ)^2⟩⟨P⟩), with P being the plaquette operator.
• The phase boundary shift weakens with increasing μ and vanishes at the tricritical point, suggesting no first-order boundary shift at that location.
• The method is consistent with mean field results and HMC data, with small Nτ dependence for β < 1.
• The ratio Tc(μ=0)/3μc(T=0) in the strong coupling limit (≈0.82) is much larger than the continuum value (≈0.165), indicating a faster drop in Tc with β than in μc.