[Paper Review] Meson mass spectrum at finite temperature and density in the strong coupling limit of lattice QCD for color SU(3)

Noboru Kawamoto, Kohtaroh Miura+1 more|ArXiv.org|Oct 9, 2007

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper analytically derives the meson mass spectrum in the strong coupling limit of SU(3) lattice QCD at finite temperature and chemical potential using a mean field approximation. By relating meson masses to the chiral condensate via the effective action and solving for the propagator pole, it shows that meson masses—especially non-pion states—rapidly approach zero as temperature or chemical potential nears the critical values, signaling chiral restoration.

ABSTRACT

We investigate the meson mass spectrum in the strong coupling limit of lattice QCD with one species of staggered fermion for the SU(N_c) color gauge group, including N_c=3. We analytically derive meson masses as functions of temperature and chemical potential via chiral condensates. We show that meson masses quickly decrease to zero when the chemical potential or the temperature approaches to the critical value.

Motivation & Objective

• To derive analytical expressions for meson masses as functions of temperature (T) and chemical potential (μ) in the strong coupling limit of SU(3) lattice QCD.
• To overcome the sign problem in lattice QCD at high density by using a strong coupling expansion with mean field approximation.
• To study the behavior of meson masses near the chiral phase transition point, particularly their dependence on T and μ.
• To confirm the validity of the PCAC relation in the chiral limit and verify the critical scaling of meson masses near phase transitions.
• To provide a benchmark for future lattice Monte Carlo simulations in the strong coupling regime.

Proposed method

• Formulates the lattice QCD action with one staggered fermion flavor and introduces finite T and μ via temporal link variables and chemical potential terms.
• Performs path integral over spatial link variables and retains leading-order terms in 1/d expansion, reducing the partition function to a fermionic and bosonic effective action.
• Applies bosonization to the chiral condensate σ, introducing a Hubbard-Stratanovich transformation to decouple the fermion interaction.
• Derives the effective action for σ, including the determinant of the fermion matrix Rσ, which depends on μ and T through boundary conditions.
• Expands the effective action to second order in fluctuations δσ around the mean-field condensate σ̄, yielding a quadratic form for the meson propagator.
• Uses a Fourier transform and a prescription (ω, k) = (iM, 0) + (δν)π to extract meson masses as poles of the propagator, leading to an analytical expression in terms of σ̄(T, μ).

Experimental results

Research questions

• RQ1 How do meson masses depend on temperature and chemical potential in the strong coupling limit of SU(3) lattice QCD?
• RQ2 What is the behavior of meson masses near the chiral phase transition, particularly as T or μ approaches its critical value?
• RQ3 Does the pion mass satisfy the PCAC relation in the chiral limit under this analytical framework?
• RQ4 How do different meson species (π, ρ, a₁, δ) characterized by taste quantum numbers κ differ in their T and μ dependence?
• RQ5 Can the analytical meson mass spectrum reproduce expected critical scaling behavior near second- and first-order phase transitions?

Key findings

• Meson masses, particularly non-pion states, rapidly decrease to zero as the chemical potential μ approaches its critical value μc, especially in the second-order phase transition region.
• At high temperatures (T > Ttcp), meson masses smoothly and quickly vanish as μ → μc, consistent with chiral symmetry restoration.
• For fixed μ < μc, meson masses also rapidly approach zero as temperature T approaches its critical value Tc, indicating a smooth restoration of chiral symmetry.
• The pion mass satisfies the PCAC relation Mπ ≈ 2√(σ̄m₀) in the chiral limit (m₀ → 0), confirming the analytical consistency of the approach.
• The dependence of meson masses on T and μ is weak in the chiral broken phase away from criticality, but becomes strongly sensitive near Tc or μc.
• The analytical expression for meson masses is derived as Mκ(σ̄; T, μ) = arccosh[2(σ̄ + m₀)((d+κ)/d σ̄ + m₀) + 1], with κ representing different meson species via taste quantum numbers.