[Paper Review] Dirac-mode expansion of quark number density and its implications of the confinement-deconfinement transition

Takahiro Doi, Kouji Kashiwa|arXiv (Cornell University)|Jun 2, 2017

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper investigates the quark number density at finite imaginary chemical potential using Dirac-mode expansion to probe the confinement-deconfinement transition. It finds that in the heavy quark mass limit, the quark number density is fully expressible via the Polyakov loop and its conjugate, with no dominant Dirac modes; in the light quark regime, the absolute value depends strongly on low-lying Dirac modes, but the sign—critical for the Roberge-Weiss transition—does not, supporting the quark number holonomy as a robust, gauge-invariant order parameter for topological confinement-deconfinement transitions.

ABSTRACT

We investigate the quark number density at finite imaginary chemical potential by using the Dirac-mode expansion. In the large quark mass region, it is found that the quark number density can be expressed by the Polyakov loop and its conjugate in all order of the large quark mass expansion. Then, there are no specific Dirac-modes which dominantly contribute to the quark number density. In comparison, the small quark mass region is explored by using the quenched lattice QCD simulation. We found that the absolute value of the quark number density strongly depends on the low-lying Dirac-modes, but its sign does not. This means that the existence of the Roberge-Weiss transition which is characterized by the singular behavior of the quark number density is not sensitive to low-lying Dirac-modes. This property enables us to discuss the confinement-deconfinement transition from the behavior of the quark number density via the quark number holonomy.

Motivation & Objective

• To clarify the role of Dirac eigenmodes in quark number density at finite imaginary chemical potential.
• To investigate whether low-lying Dirac modes dominate the quark number density in the light and heavy quark mass regimes.
• To assess the viability of the quark number holonomy as a topological order parameter for the confinement-deconfinement transition.
• To compare the quark number holonomy with other proposed order parameters such as the dual quark condensate and QCD monopole in terms of gauge invariance and calculational robustness.

Proposed method

• Employing the heavy quark mass expansion to analytically express the quark number density in terms of Dirac eigenmodes.
• Using Dirac-mode expansion to decompose the quark number density and identify dominant contributions from specific eigenmodes.
• Performing quenched lattice QCD simulations to compute the quark number density in the small quark mass region.
• Analyzing the sign and magnitude of the quark number density across different Dirac modes to assess sensitivity to low-lying states.
• Defining the quark number holonomy as the contour integral of the quark number susceptibility over θ = 0 to 2π to detect topological phase transitions.
• Comparing the quark number holonomy with other order parameters (e.g., dual quark condensate, QCD monopole) in terms of gauge invariance and boundary condition dependence.

Experimental results

Research questions

• RQ1 Which Dirac modes dominate the quark number density in the heavy and light quark mass limits?
• RQ2 How does the sign of the quark number density at μ_I/T = π/3 relate to the Roberge-Weiss transition and low-lying Dirac modes?
• RQ3 Can the quark number holonomy serve as a reliable, gauge-invariant order parameter for the confinement-deconfinement transition?
• RQ4 How does the quark number holonomy compare to other proposed order parameters in terms of robustness and calculational consistency?
• RQ5 To what extent is the quark number density in the light quark regime sensitive to the structure of low-lying Dirac eigenmodes?

Key findings

• In the heavy quark mass limit, the quark number density is fully expressible in terms of the Polyakov loop and its conjugate, with no dominant contribution from any specific Dirac mode.
• In the small quark mass region, the absolute value of the quark number density strongly depends on low-lying Dirac modes, but its sign does not.
• The sign of the quark number density, which determines the Roberge-Weiss transition, behaves similarly to the Polyakov loop in its dependence on Dirac eigenmodes.
• The quark number holonomy is found to be sensitive to confinement properties because it captures the topological structure of the quark number density's θ-dependence.
• The quark number holonomy is superior to the dual quark condensate and QCD monopole as an order parameter due to its gauge invariance and independence from ambiguous boundary condition choices.
• The quark number holonomy provides a robust, non-perturbative, and gauge-invariant probe of the confinement-deconfinement transition, particularly in the presence of dynamical quarks.