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Light quarks masses and condensates in QCD

Jan Stern

TL;DR

The paper investigates the possibility that spontaneous chiral symmetry breaking in QCD can occur with a small quark condensate, developing a generalized chiral perturbation theory (GχPT) framework to explore light-quark masses and condensates. It analyzes quark-mass ratios through $r=\frac{m_s}{\hat{m}}$ and $R=\frac{m_s-\hat{m}}{m_d-m_u}$, contrasts standard and generalized χPT, and derives how condensates and order parameters like $F_0$ influence Goldstone-boson masses and the Gell-Mann–Oakes–Renner relation via $X_{GOR}$. The work also connects quark masses to condensates through QCD sum rules, discusses the role and normalization of spectral functions, and proposes experimental tests (notably in $\pi\!-\pi$ scattering and $\tau$ decays) to discriminate between large- and small-condensate scenarios. Overall, it provides a theoretical framework and phenomenological analyses to test the size of the quark condensate and refine light-quark mass determinations, with significant implications for our understanding of chiral symmetry breaking and nonperturbative QCD.

Abstract

We review some theoretical and phenomenological aspects of the scenario in which the spontaneous breaking of chiral symmetry is not triggered by a formation of a large condensate <\bar{q} q>. Emphasis is put on the resulting pattern of light quark masses, on the constraints arising from QCD sum rules and on forthcoming experimental tests.

Light quarks masses and condensates in QCD

TL;DR

The paper investigates the possibility that spontaneous chiral symmetry breaking in QCD can occur with a small quark condensate, developing a generalized chiral perturbation theory (GχPT) framework to explore light-quark masses and condensates. It analyzes quark-mass ratios through and , contrasts standard and generalized χPT, and derives how condensates and order parameters like influence Goldstone-boson masses and the Gell-Mann–Oakes–Renner relation via . The work also connects quark masses to condensates through QCD sum rules, discusses the role and normalization of spectral functions, and proposes experimental tests (notably in scattering and decays) to discriminate between large- and small-condensate scenarios. Overall, it provides a theoretical framework and phenomenological analyses to test the size of the quark condensate and refine light-quark mass determinations, with significant implications for our understanding of chiral symmetry breaking and nonperturbative QCD.

Abstract

We review some theoretical and phenomenological aspects of the scenario in which the spontaneous breaking of chiral symmetry is not triggered by a formation of a large condensate <\bar{q} q>. Emphasis is put on the resulting pattern of light quark masses, on the constraints arising from QCD sum rules and on forthcoming experimental tests.

Paper Structure

This paper contains 16 sections, 2 theorems, 46 equations, 3 figures.

Table of Contents

  1. Light quark masses and condensates in QCD
  2. Introduction
  3. QCD vacuum as a disordered system
  4. Quark mass ratios
  5. The standard picture
  6. Expansion of Goldstone boson masses
  7. The ratio $r=m_{s}/\widehat{m}$ in $G\chi PT$
  8. The ratio $R=\frac{ m_{s}-\widehat{m}}{ m_{d}-m_{u}}$ in $G\chi PT$: dismiss of the ellipse
  9. The value of the strange quark mass
  10. Consequences for parameters $B$ and $A_{0}$
  11. Natural size of $A_{0}$
  12. QCD sum rules
  13. Two-point function of $\partial^\mu A_{\mu}$
  14. QCD - Hadron duality
  15. Relation between quark masses and condensates
  16. ...and 1 more sections

Key Result

Theorem 1

If $N_{c}\geq 3,\ N_{f}\geq 3$, and provided quarks are confined (no coloured physical states), the chiral symmetry $SU_{L}(N_{f})\times SU_{R}(N_{f})\times U_{V}(1)$ is necessarily broken down to its diagonal subgroup $U_{V}(N_{f})$ generated by vector currents.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2