[Paper Review] Calculating Quark Number Susceptibilities with Domain-Wall Fermions

Prasad Hegde, Frithjof Karsch+1 more|arXiv (Cornell University)|Oct 1, 2008

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper presents lattice QCD calculations of quark number susceptibilities using domain-wall fermions on 2+1-flavor dynamical ensembles, demonstrating that iso-spin and electric charge susceptibilities provide robust, low-error indicators of the deconfinement transition. The results yield a transition coupling βₚ = 2.028(6) from iso-spin and βₚ = 2.030(7) from electric charge susceptibilities, in excellent agreement with the chiral condensate-based value βc = 2.031(5), confirming that deconfinement and chiral symmetry restoration occur in the same narrow temperature range.

ABSTRACT

We present results from calculations of different quark number and hadronic susceptibilities on 2+1-flavor dynamical domain wall ensembles. We find that the iso-spin and electric charge susceptibilities are especially well suited to determine the transition temperature, as these quantities show only small statistical errors. Moreover, the transition values of the coupling obtained from iso-spin and electrical charge susceptibilities are in good agreement with the one obtained from the chiral condensate.

Motivation & Objective

• To calculate quark number and hadronic susceptibilities in 2+1-flavor QCD using domain-wall fermions to study the nature of the deconfinement transition.
• To assess the sensitivity of thermodynamic observables to the improved chiral symmetry of domain-wall fermions compared to staggered fermions.
• To determine the pseudo-critical coupling βₚ from second-order Taylor expansion coefficients of the pressure at zero chemical potential.
• To compare transition temperatures extracted from susceptibilities with those from the chiral condensate, validating consistency across different observables.

Proposed method

• Uses 2+1-flavor dynamical domain-wall fermion ensembles with Nτ = 8 to compute the second-order Taylor expansion coefficients c₂ᵘ, c₂ˢ, c₁₁ᵘᵈ, and c₁₁ᵘˢ of the pressure in the grand canonical potential.
• Constructs iso-spin and electric charge susceptibilities c₂ᴵ and c₂Q from the expansion coefficients using the relations c₂ᴵ = ½(2c₂ᵘ − c₁₁ᵘᵈ) and c₂Q = ¹⁄₉(5c₂ᵘ + c₂ˢ − 2c₁₁ᵘᵈ − c₁₁ᵘˢ).
• Applies a hyperbolic tangent fit, c₂⁽ᴵ,Q⁾ = A tanh{B(β − βₚ)} + C, to the susceptibility data to extract the pseudo-critical coupling βₚ.
• Estimates statistical errors using the jackknife method, carefully accounting for both ensemble size and trace estimation errors from random vectors, especially for disconnected contractions.
• Compares results with previous staggered fermion calculations and the chiral condensate-based determination of βc = 2.031(5) from the RBC Collaboration.
• Evaluates cut-off effects via comparison to the ideal gas limit and acknowledges residual mass effects in domain-wall fermions as a source of systematic uncertainty.

Experimental results

Research questions

• RQ1 Can iso-spin and electric charge susceptibilities serve as reliable, low-error indicators of the QCD deconfinement transition in lattice simulations with domain-wall fermions?
• RQ2 To what extent do the transition temperatures extracted from different susceptibilities (c₂ᴵ, c₂Q) agree with one another and with the chiral condensate?
• RQ3 How do the thermodynamic properties of the QGP, as encoded in the second-order expansion coefficients, compare between domain-wall and staggered fermion formulations?
• RQ4 What is the impact of residual chiral symmetry breaking in domain-wall fermions on the determination of the pseudo-critical coupling βₚ?

Key findings

• The iso-spin and electric charge susceptibilities c₂ᴵ and c₂Q exhibit significantly smaller statistical errors than the individual quark flavor coefficients, making them ideal for transition temperature determination.
• The extracted pseudo-critical coupling βₚ = 2.028(6) from c₂ᴵ and βₚ = 2.030(7) from c₂Q are in excellent agreement with the chiral condensate-based value βc = 2.031(5), confirming consistency across observables.
• The transition region is narrow, with c₂ᴵ and c₂Q showing a smooth, sigmoid-like rise across β ≈ 2.03, consistent with a crossover transition.
• The results suggest that the deconfinement and chiral symmetry restoration transitions occur in the same narrow temperature interval, as indicated by the agreement in βₚ values.
• The susceptibility coefficients show a slight overshoot of the Stefan-Boltzmann ideal gas limit at high temperatures, consistent with known cut-off effects in the domain-wall fermion formulation.
• Residual mass effects and cut-off errors (estimated at ~10% for Nτ = 8) remain a source of systematic uncertainty, requiring larger Ls or Nτ for future refinement.