[Paper Review] Continuum limit and universality of the Columbia plot

Philippe de Forcrand, Massimo D’Elia|arXiv (Cornell University)|Feb 1, 2017

High-Energy Particle Collisions Research14 references4 citations

TL;DR

This paper investigates the continuum limit and universality of the QCD phase transition in the Columbia plot using 4-flavor staggered fermions on lattices with 4 to 10 temporal slices. It finds that cutoff effects are extremely large, even at fine lattices, implying that the critical pion mass for a first-order transition in the continuum limit is likely much smaller than previously estimated—potentially approaching zero—challenging earlier expectations based on 3-flavor studies and suggesting that both Wilson and staggered fermion formulations face severe systematic errors in extrapolating to the continuum.

ABSTRACT

Results on the thermal transition of QCD with 3 degenerate flavors, in the lower-left corner of the Columbia plot, are puzzling. The transition is expected to be first-order for massless quarks, and to remain so for a range of quark masses until it turns second-order at a critical quark mass. But this critical quark mass and resulting "pion" mass disagree violently between Wilson and staggered fermions at finite lattice spacing, and decrease sharply with the lattice spacing, for staggered fermions at least. To clarify this puzzle and eliminate potential systematic effects from rooting, we study the 4-flavor theory with staggered fermions, on lattices with 4 to 10 time-slices. Our results are qualitatively similar to the 3-flavor case, so that rooting is not an issue. However, dramatic cutoff effects are visible, even on our finest lattices. Universality implies that cutoff effects for Wilson fermions are even more dramatic. In order to obtain a first-order thermal transition in the continuum theory, extremely light quarks are needed.

Motivation & Objective

• To resolve the longstanding discrepancy in the critical pion mass for the QCD chiral phase transition between Wilson and staggered fermions at finite lattice spacing.
• To test whether the observed discrepancy in 3-flavor QCD is due to rooting artifacts or a generic feature of lattice cutoff effects.
• To investigate the universality of the continuum limit by studying the 4-flavor QCD theory with staggered fermions, which avoids the complications of rooting.
• To determine whether the dramatic reduction of the critical pion mass with decreasing lattice spacing is a universal feature, implying that extremely light quarks are required for a first-order transition in the continuum.
• To assess the implications for the QCD phase diagram at finite chemical potential, particularly the existence and location of a chiral critical point.

Proposed method

• Simulating 4-flavor QCD with improved staggered fermions on lattices with temporal extent $N_t = 4$ to $10$, using a non-perturbatively improved action to reduce cutoff effects.
• Employing a combination of zero-temperature ($20^3 \times 40$) and finite-temperature ($20^3 \times 10$) lattices to compute the critical pion mass $m_{\pi}^c$ via the Binder cumulant of the chiral condensate.
• Performing finite-size scaling analysis of the Binder cumulant to locate the critical point for each $N_t$.
• Conducting a continuum extrapolation of $m_{\pi}^c / T_c$ using both linear and quadratic fits to the lattice spacing $a^2$, with data from $N_t = 4$ to $10$.
• Comparing results with existing $N_f=3$ data and Wilson fermion results to assess universality and the role of fermion discretization.
• Analyzing the impact of Matsubara frequency cutoffs on thermodynamic observables, using the free boson pressure as a benchmark to assess lattice artifacts.

Experimental results

Research questions

• RQ1 Does the large discrepancy in the critical pion mass between Wilson and staggered fermions in 3-flavor QCD arise from rooting artifacts or from universal lattice cutoff effects?
• RQ2 How do cutoff effects in the chiral phase transition manifest in 4-flavor QCD with staggered fermions, and do they persist at fine lattices ($N_t = 10$)?
• RQ3 Is the observed reduction of $m_{\pi}^c / T_c$ with decreasing lattice spacing a universal feature, implying that the continuum limit may have $m_{\pi}^c / T_c \to 0$?
• RQ4 Can the observed dramatic cutoff effects be explained by the Matsubara frequency cutoff in thermal field theory, rather than by taste symmetry breaking or fermion doublers?
• RQ5 What are the implications of these findings for the existence of a QCD chiral critical point at finite chemical potential, given that the $\mu=0$ critical line may lie at extremely small quark masses?

Key findings

• The $N_f=4$ staggered results show a dramatic reduction of $m_{\pi}^c / T_c$ with decreasing lattice spacing, qualitatively mirroring the behavior seen in $N_f=3$ simulations, indicating that the discrepancy between staggered and Wilson fermions is not due to rooting.
• Even at $N_t = 10$, corresponding to a lattice spacing of $a \sim 0.13$ fm, the value of $m_{\pi}^c / T_c$ remains significantly reduced, suggesting that the continuum limit is approached very slowly.
• The observed cutoff effects are so large that the continuum extrapolation of $m_{\pi}^c / T_c$ is compatible with zero, raising the possibility that the critical pion mass may vanish in the continuum limit.
• The results imply that both Wilson and staggered fermion formulations suffer from severe cutoff effects, with the latter showing similar trends despite different doublers and taste symmetry properties.
• The authors suggest that the Matsubara frequency cutoff in thermal lattice field theory may be a dominant source of error, as it leads to a significant deficit in the pressure compared to the continuum Stefan-Boltzmann law.
• The findings suggest that future simulations must use much finer temporal lattices ($N_t \gtrsim 16$) or anisotropic actions to reliably determine the continuum value of $m_{\pi}^c / T_c$, especially for $N_f=3$ and $N_f=2+1$ QCD.