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Isospin symmetry breaking and gluon anomaly

A. A. Osipov

TL;DR

This work analyzes electromagnetic isospin breaking in the pseudoscalar meson nonet within the framework of large-$N_c$ chiral perturbation theory, explicitly including singlet–octet mixing and the $U(1)_A$ gluon anomaly. By organizing the theory in an $oldsymbol{ ext{O}}(oldsymbol{oldsymbol{ au}})$ expansion with $ ext{O}(m_q)= ext{O}(p^2)= ext{O}(1/N_c)= ext{O}(oldsymbol{ au})$ and $ ext{O}(e^2)= ext{O}(oldsymbol{ au})$, the authors compute electromagnetic self-energies, masses, decay constants, and mixing angles for neutral pseudoscalars at next-to-leading order. A key finding is that the $U(1)_A$ anomaly enhances EM contributions to $oldsymbol{π^0}$–$oldsymbol{η}$ and $oldsymbol{π^0}$–$oldsymbol{η’}$ mixing, driving isospin restoration in the neutral-meson spectrum at NLO, while the EM effect on the $oldsymbol{η}$–$oldsymbol{η’}$ mixing angle remains small (about 3%). The analysis yields a two-angle description for the $oldsymbol{η}$–$oldsymbol{η’}$ system with $oldsymbol{ϑ_8} oughly -21^ ext{o}$ and $oldsymbol{ϑ_0} oughly -2^ ext{o}$, and a near-Gross–Treiman–Wilczek value for $oldsymbol{π^0}$–$oldsymbol{η}$ mixing, demonstrating a subtle interplay between electromagnetic, quark-mass, and anomaly effects in isospin violation.

Abstract

The contribution of the electromagnetic interaction to the self-energy of pseudo-Goldstone bosons, as well as to the mixing angles and weak decay constants, is calculated in the first nonleading order in the simultaneous expansion in powers of $1/N_c$, momenta and quark masses. Particular attention is paid to the isospin symmetry breaking which is generated by means of $η$ and $η'$ admixtures to the pion. It is shown that the gluon anomaly enhances the electromagnetic contribution to the $π^0$-$η$ and $π^0$-$η'$ mixing angles, which facilitates the restoration of isospin symmetry in the spectrum of pseudo-Goldstone states at next-to-leading order. The contribution of the electromagnetic interaction to the $η$-$η'$ mixing angle is shown to be only about 3\%.

Isospin symmetry breaking and gluon anomaly

TL;DR

This work analyzes electromagnetic isospin breaking in the pseudoscalar meson nonet within the framework of large- chiral perturbation theory, explicitly including singlet–octet mixing and the gluon anomaly. By organizing the theory in an expansion with and , the authors compute electromagnetic self-energies, masses, decay constants, and mixing angles for neutral pseudoscalars at next-to-leading order. A key finding is that the anomaly enhances EM contributions to – and – mixing, driving isospin restoration in the neutral-meson spectrum at NLO, while the EM effect on the – mixing angle remains small (about 3%). The analysis yields a two-angle description for the – system with and , and a near-Gross–Treiman–Wilczek value for – mixing, demonstrating a subtle interplay between electromagnetic, quark-mass, and anomaly effects in isospin violation.

Abstract

The contribution of the electromagnetic interaction to the self-energy of pseudo-Goldstone bosons, as well as to the mixing angles and weak decay constants, is calculated in the first nonleading order in the simultaneous expansion in powers of , momenta and quark masses. Particular attention is paid to the isospin symmetry breaking which is generated by means of and admixtures to the pion. It is shown that the gluon anomaly enhances the electromagnetic contribution to the - and - mixing angles, which facilitates the restoration of isospin symmetry in the spectrum of pseudo-Goldstone states at next-to-leading order. The contribution of the electromagnetic interaction to the - mixing angle is shown to be only about 3\%.

Paper Structure

This paper contains 9 sections, 83 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Effective Lagrangian to order $\mathcal{O}(\delta)$
  3. Decay constants and masses of charged mesons
  4. Fixing parameters
  5. Neutral mesons: kinetic term and decay constants
  6. Weak-decay coupling constants in the singlet-octet bases
  7. Neutral mesons: masses and mixing angles
  8. Conclusions
  9. Acknowledgments

Figures (1)

  • Figure 1: Photon loop contribution to the self-energy of $\pi^\pm$ and $K^\pm$ mesons. Diagram (a) corresponds to formula (\ref{['loops']}). Diagram (b) is zero in dimensional regularization.