[Paper Review] Derivation of Brown-Rho scaling from scale-chiral perturbation theory

Yanling Li, Yong-Liang Ma|arXiv (Cornell University)|Oct 8, 2017

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper derives Brown-Rho scaling from scale-chiral perturbation theory ($\chi$PT$_\sigma$) by incorporating the $f_0(500)$ scalar meson as a Nambu-Goldstone boson of spontaneously and explicitly broken scale symmetry. It shows that leading-order scale symmetry (LOSS) in the scale-chiral effective theory reproduces Brown-Rho scaling, with medium-modified hadron properties scaling via the dilaton condensate $\Phi$, and establishes a framework for higher-order scale-chiral corrections beyond leading order.

ABSTRACT

The medium modified hadron properties are studied by using the scale-chiral perturbation theory $χ$PT$_σ$ in which the lightest scalar meson $f_0(500)$ is included as an explicit degree of freedom by regarding it as the Nambu-Goldstone boson of scale symmetry both spontaneously broken and explicitly broken by the QCD trace anomaly. We derive Brown-Rho scaling at the leading order of scale symmetry from $χ$PT$_σ$ and formulate how to make higher-order scale-chiral corrections going beyond the leading order of scale symmetry. By taking into account the intrinsic density dependence given by the dilaton condensate in the "bare" parameters of the Lagrangian, the medium modified hadron properties are investigated. Relying on available experimental information and certain reasonable assumptions, we arrive at the leading order scale symmetric effective theory that can be confronted with nature .

Motivation & Objective

• To derive Brown-Rho scaling from scale-chiral perturbation theory ($\chi$PT$_\sigma$) by treating the $f_0(500)$ as a Nambu-Goldstone boson of scale symmetry breaking.
• To formulate higher-order scale-chiral corrections beyond the leading order of scale symmetry, enabling systematic improvements.
• To incorporate intrinsic density dependence (IDD) in bare parameters of the effective Lagrangian via the dilaton condensate, reflecting QCD's vacuum modification in medium.
• To establish a consistent framework for connecting experimental observables to effective field theory parameters through Wilsonian matching of correlation functions.
• To validate the leading-order scale symmetric effective theory against experimental data and theoretical constraints, particularly near nuclear matter density.

Proposed method

• Formulate a scale-chiral effective theory ($\chi$PT$_\sigma$) where the $f_0(500)$ is treated as the Nambu-Goldstone boson of scale symmetry, both spontaneously and explicitly broken by the QCD trace anomaly.
• Introduce intrinsic density dependence (IDD) in the bare parameters of the Lagrangian via the dilaton condensate $\Phi$, which encodes medium effects from QCD vacuum modification.
• Perform mean-field calculations within the 'bare' effective Lagrangian, equivalent to Landau Fermi-liquid theory via single-decimation Wilsonian renormalization group.
• Derive scaling laws for hadron masses, decay constants, and couplings in medium, all expressed as powers of $\Phi$, with exponents determined by scaling dimensions and $\beta'$-expansion.
• Use flavor symmetry ($SU(3)$ for pseudoscalars, $U(3)$ for vectors) to generalize scaling relations across hadron species.
• Ensure consistency with experimental data by matching correlation functions between QCD and the effective theory at a matching scale, accounting for all quantum corrections.

Experimental results

Research questions

• RQ1 Can Brown-Rho scaling for medium-modified hadron properties be derived from a fundamental scale-chiral effective field theory?
• RQ2 How does intrinsic density dependence (IDD) in bare parameters—originating from the dilaton condensate—give rise to scaling behavior in hadron properties?
• RQ3 What is the role of the $f_0(500)$ as a Nambu-Goldstone boson of scale symmetry in reproducing Brown-Rho scaling at leading order?
• RQ4 How can higher-order scale-chiral corrections be systematically constructed beyond the leading-order scale symmetric (LOSS) limit?
• RQ5 To what extent is the leading-order scale symmetric effective theory valid, especially near the skyrmion crystal phase transition at $n_{1/2} \simeq 2n_0$?

Key findings

• Brown-Rho scaling for hadron masses and couplings is derived from scale-chiral perturbation theory at leading order of scale symmetry, with all scaling behaviors expressed as powers of the dilaton condensate $\Phi$.
• The scaling of the $\sigma$ meson mass is $m_{\sigma}^*/m_{\sigma} = \Phi^{\beta' / 2 + 1}$, which matches previous results in the $\beta' \ll 1$ limit.
• The scaling of the $\rho$ and $B$ (baryon) masses and decay constants is $m_{\rho}^*/m_{\rho} = m_B^*/m_B = f_\pi^*/f_\pi = \Phi$, consistent with Brown-Rho scaling.
• The scaling of the $\sigma\pi\pi$ coupling is $g_{\sigma\pi\pi}^*/g_{\sigma\pi\pi} = \Phi^{-1}$, while $g_{\sigma\rho\rho}^*/g_{\sigma\rho\rho} = \Phi$, showing distinct scaling behaviors for different vertices.
• The scaling of $h_5^*/h_5 = \Phi^{4 + 2\beta'}$ and $h_6^*/h_6 = \Phi^{4 + \beta'}$ indicates that higher-dimension couplings scale more strongly with density.
• The framework is valid up to densities $\sim n_{1/2} \simeq 2n_0$, beyond which topological changes in skyrmion crystal structure invalidate the current scaling ansatz.