[Paper Review] Singularities of QCD in the complex chemical potential plane

Shinji Ejiri, Hiroshi Yoneyama|arXiv (Cornell University)|Nov 11, 2009

High-Energy Particle Collisions Research3 citations

TL;DR

This paper investigates thermodynamic singularities in QCD using lattice simulations with p4-improved staggered fermions, focusing on the complex chemical potential plane to estimate the convergence radius of the Taylor expansion of the thermodynamic potential. By analyzing the complex phase distribution of the quark determinant and employing a reweighting approach with Gaussian approximation, the study identifies singularities (Lee-Yang zeros) in both real and imaginary directions of μq², finding that the convergence radius is smaller in the crossover region than near the critical point.

ABSTRACT

We study the thermodynamic singularities of QCD in the complex chemical potential plane by a numerical simulation of lattice QCD, and discuss a method to understand the nature of the QCD phase transition at finite density from the information of the singularities. The existence of singular points at which the partition function (Z) vanishes is expected in the complex plane. These are called Lee-Yang zeros or Fisher zeros. We investigate the distribution of these singular points using the data obtained by a simulation of two-flavor QCD with p4-improved staggered quarks. The convergence radius of a Taylor expansion of ln Z in terms of the chemical potential is also discussed.

Motivation & Objective

• To understand the nature of the QCD phase transition at finite density using singularities in the complex chemical potential plane.
• To estimate the convergence radius of the Taylor expansion of the thermodynamic potential in μq by identifying the nearest singularity in the complex plane.
• To investigate the distribution of Lee-Yang zeros in QCD using numerical simulations with heavy quarks, despite limitations in universality class.
• To address the sign problem in finite-density QCD by analyzing the complex phase distribution of the quark determinant.
• To explore the connection between the effective potential's double-well structure and the emergence of singularities in the complex μq plane.

Proposed method

• A reweighting method is used to compute the normalized partition function Z_norm = |Z(μq)/Z(0)|, which vanishes at Lee-Yang zeros.
• The complex phase θ of the quark determinant is modeled using a Taylor expansion up to O(μq⁶), and its distribution W(θ) is analyzed.
• The effective potential V_eff is constructed from the phase distribution, and its derivative dV_eff/dP is used to detect non-monotonic behavior indicating singularities.
• The condition d²V_eff/dP² = 0 is used to locate the boundary where the effective potential transitions from single-well to double-well, signaling the onset of singularities.
• The convergence radius is estimated as the distance from μq = 0 to the nearest singularity in the complex (μq/T)² plane.
• The method assumes a Gaussian distribution of the complex phase and uses a parameter shift from β₀ to β to extrapolate results across different β values.

Experimental results

Research questions

• RQ1 Where are the singularities (Lee-Yang zeros) located in the complex chemical potential plane for two-flavor QCD with heavy quarks?
• RQ2 What is the convergence radius of the Taylor expansion of the thermodynamic potential in μq, and how does it relate to the nearest singularity?
• RQ3 How does the complex phase distribution of the quark determinant reflect the presence of singularities and the sign problem?
• RQ4 Does the effective potential develop a double-well structure in the complex μq plane, and what does this imply for phase transitions?
• RQ5 Is the convergence radius larger near the QCD critical point than in the crossover region, even with heavy quarks?

Key findings

• Singularities (Lee-Yang zeros) are found not only in the real μq direction but also in the imaginary direction, indicating that the convergence radius is limited by singularities in both Re(μq²) and Im(μq²) planes.
• The distance from μq = 0 to the nearest singularity in the complex (μq/T)² plane is estimated as the convergence radius of the Taylor expansion of lnZ/(VT³).
• The convergence radius is smaller in the crossover region (T > T_cp) than near the critical point, suggesting better convergence near T_cp.
• The boundary where d²V_eff/dP² = 0 separates single-well and double-well effective potential regions, and this boundary is closer to μq = 0 in the imaginary direction than in the real direction.
• The effective potential becomes S-shaped in the complex μq plane at large |μq|, indicating phase cancellation and the presence of singularities.
• The results suggest that the convergence radius at the real QCD critical point temperature may exceed that in the crossover region, even with heavy quarks.