[Paper Review] Chiral Lagrangian and spectral sum rules for two-color QCD at high density

Takuya Kanazawa, Tilo Wettig+1 more|arXiv (Cornell University)|Oct 13, 2009

Quantum Chromodynamics and Particle Interactions14 references3 citations

TL;DR

This paper constructs a low-energy effective chiral Lagrangian for two-color QCD at high baryon density, where BCS-type diquark pairing breaks chiral symmetry. It identifies a new epsilon-regime in finite volume and derives exact Leutwyler-Smilga-type spectral sum rules for the complex eigenvalues of the Dirac operator, showing the spectrum is governed by the fermion gap Δ rather than the chiral condensate, providing testable predictions for lattice QCD simulations.

ABSTRACT

We report on our analytical study of two-color QCD with an even number of flavors at high baryon density. Based on the pattern of chiral symmetry breaking induced by BCS-type diquark pairing we construct the low-energy effective Lagrangian for the Nambu-Goldstone bosons. We also identify a new epsilon-regime at high baryon density and derive Leutwyler-Smilga-type spectral sum rules for the complex eigenvalues of the Dirac operator in terms of the fermion gap. Our results can in principle be tested in lattice QCD simulations.

Motivation & Objective

• To construct a low-energy effective chiral Lagrangian for two-color QCD with even Nf at high baryon density.
• To identify a new finite-volume ε-regime specific to the BCS superfluid phase at high density.
• To derive exact spectral sum rules for the complex eigenvalues of the Dirac operator in terms of the fermion gap Δ.
• To establish that the Dirac spectrum at high density is governed by the BCS gap Δ, not the chiral condensate, contrasting with the μ=0 case.
• To provide analytically tractable predictions testable in lattice QCD simulations, where the sign problem is absent.

Proposed method

• Construct the low-energy effective Lagrangian based on the chiral symmetry breaking pattern induced by diquark pairing in the BCS channel.
• Identify a new ε-regime in finite volume where the partition function can be exactly computed from the effective theory.
• Derive spectral sum rules for inverse powers of the complex Dirac eigenvalues using the massive partition function in the ε-regime.
• Introduce double-microscopic variables to define universal spectral densities ρs and ρs^(Nf) in the ε-regime.
• Use the relation between the microscopic spectral density and the partition function to obtain sum rules involving modified Bessel functions I0, I1, I2, I3.
• Establish that the sum rules depend explicitly on the fermion gap Δ and quark masses, not on the chiral condensate.

Experimental results

Research questions

• RQ1 What is the low-energy effective Lagrangian for two-color QCD at high baryon density with even Nf, based on BCS diquark pairing?
• RQ2 How does the finite-volume spectrum of the Dirac operator behave in the new ε-regime at high density?
• RQ3 What spectral sum rules govern the complex eigenvalues of the Dirac operator in this regime?
• RQ4 How does the Dirac spectrum at high density differ from that at μ=0 in terms of governing order parameters?
• RQ5 Can these sum rules be tested in lattice QCD simulations, given the absence of the fermion sign problem?

Key findings

• The low-energy effective Lagrangian is constructed based on the chiral symmetry breaking pattern induced by BCS diquark pairing in two-color QCD at high density.
• A new finite-volume ε-regime is identified for the BCS superfluid phase at high baryon density, where exact results can be derived.
• Spectral sum rules are derived for inverse powers of the complex Dirac eigenvalues, explicitly depending on the fermion gap Δ and quark masses.
• For two flavors, the sum rule ⟨⟨∑' 1/(zn² + m̃₁²)⟩⟩ = (m̃₂²/4) × (I₀(x) - I₂(x))/I₀(x) is obtained with x = m̃₁m̃₂.
• For four degenerate flavors, the sum rule ∑' 1/(zn² + m̃²) = [2I₀(y)I₁(y) - 3I₁(y)I₂(y) + I₂(y)I₃(y)] / [4(3I₀(y)² - 4I₁(y)² + 3I₂(y)²)] is derived with y = m̃².
• The spectral sum rules show that the Dirac spectrum at high density is governed by the BCS gap Δ, not the chiral condensate, distinguishing it from the μ=0 case.