[Paper Review] Meson masses in large Nf QCD from the Bethe-Salpeter equation

Masayasu Harada, Masafumi Kurachi+1 more|arXiv (Cornell University)|May 2, 2003

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper solves the homogeneous Bethe-Salpeter equation for meson bound states in large-$N_f$ QCD using an improved ladder approximation in the Landau gauge, with consistent quark mass functions derived from the Schwinger-Dyson equation. Remarkably, due to the analytic two-loop running coupling in large-$N_f$ QCD, the equations can be solved directly in the time-like region. The key result is that scalar, vector, axial-vector, and pseudoscalar meson masses all vanish with identical scaling near the chiral phase transition point, suggesting a novel, universal manifestation of chiral symmetry restoration.

ABSTRACT

We solve the homogeneous Bethe-Salpeter (HBS) equation for the scalar, pseudoscalar, vector, and axial-vector bound states of quark and anti-quark in large Nf QCD with the improved ladder approximation in the Landau gauge. The quark mass function in the HBS equation is obtained from the Schwinger-Dyson (SD) equation in the same approximation for consistency with the chiral symmetry. Amazingly, due to the fact that the two-loop running coupling of large Nf QCD is explicitly written in terms of an analytic function, large Nf QCD turns out to be the first example in which the SD equation can be solved in the complex plane and hence the HBS equation directly in the time-like region. We find that approaching the chiral phase transition point from the broken phase, the scalar, vector, and axial-vector meson masses vanish to zero with the same scaling behavior, all degenerate with the massless pseudoscalar meson. This may suggest a new type of manifestation of the chiral symmetry restoration in large Nf QCD.

Motivation & Objective

• To investigate the nature of chiral symmetry restoration in large-$N_f$ QCD by studying the masses of quark-antiquark bound states.
• To resolve the technical challenge of solving the Bethe-Salpeter and Schwinger-Dyson equations simultaneously in the complex plane for realistic meson spectra.
• To explore whether all vector, axial-vector, scalar, and pseudoscalar mesons become degenerate and massless at the chiral phase transition point.
• To establish a first-principles framework for studying critical behavior of meson masses near the conformal phase transition in large-$N_f$ QCD.

Proposed method

• Solves the homogeneous Bethe-Salpeter (HBS) equation for scalar, pseudoscalar, vector, and axial-vector mesons in large-$N_f$ QCD using the improved ladder approximation in the Landau gauge.
• Uses a consistent quark mass function obtained from the Schwinger-Dyson (SD) equation in the same approximation, ensuring chiral symmetry consistency.
• Leverages the fact that the two-loop running coupling in large-$N_f$ QCD is analytically expressible, enabling solution of the SD equation directly in the complex plane and thus the HBS equation in the time-like region.
• Employs numerical discretization of the integral equations with UV and IR cutoffs, adjusting discretization size to minimize numerical uncertainty.
• Applies normalization conditions for Bethe-Salpeter amplitudes and computes decay constants via matrix elements of quark currents.
• Performs systematic checks on numerical convergence by varying discretization parameters and confirming stability of mass and decay constant results.

Experimental results

Research questions

• RQ1 Do scalar, vector, axial-vector, and pseudoscalar mesons all become massless at the chiral phase transition point in large-$N_f$ QCD?
• RQ2 What is the scaling behavior of meson masses as the system approaches the critical $N_f$ from the broken phase?
• RQ3 Can the Bethe-Salpeter and Schwinger-Dyson equations be consistently solved in the complex plane for meson bound states in large-$N_f$ QCD?
• RQ4 Do all meson states exhibit identical critical scaling near the phase transition, indicating a universal chiral restoration mechanism?

Key findings

• Scalar, vector, axial-vector, and pseudoscalar meson masses all vanish at the same rate as the system approaches the chiral phase transition point from the broken phase.
• All four meson states become degenerate with the massless pseudoscalar meson at the critical point, indicating a universal scaling behavior.
• The two-loop running coupling in large-$N_f$ QCD allows analytic continuation into the complex plane, enabling direct solution of the SD and HBS equations in the time-like region without model-dependent approximations.
• Numerical results show that the vector meson mass $M_V$ and decay constant $F_V$ are stable under variation of discretization size when $(N_{BS,U}, N_{BS,X}) = (20,55)$, indicating convergence.
• The ratios $F_V/F_P$, $F_A/F_P$, and $G_S/F_P$ all exhibit the same scaling as $F_P$, confirming universal critical behavior across all meson channels.
• The study identifies a new type of chiral symmetry restoration in large-$N_f$ QCD, where multiple meson states become massless simultaneously, distinct from linear sigma model or vector manifestation scenarios.