Kinetic Mixing and Axial Charges in the Parity-Doublet Model

TL;DR

This paper addresses the failure of the standard parity-doublet model (PDM) to reproduce the nucleon axial charge $g_A$ by introducing kinetic (derivative) mixing between parity partners. The extended PDM adds two derivative couplings, parameterized by $h$ and $\Delta h$, and two new mixing angles $\theta_1$, $\theta_2$, leading to $\sin(\theta_1-\theta_2)=2h f_\pi$ and a generalized diagonalization via a Nambu–Gorkov formalism that yields independent axial charges for the nucleon, its parity partner, and their transition. The framework recovers the standard GT relations and, crucially, can realize $|g_A| \approx 1.28$ with a smaller $g_A^*$, as well as accommodate ABJ anomaly matching through $g_A - g_A^* = 1$ by tuning $\Delta h$ (with $h \neq 0$). The paper also discusses how to determine the remaining parameters from nucleon and $N^*(1535)$ masses and decays, and outlines future extensions to three flavors and in-medium contexts. Overall, the kinetic-mixing extension provides a viable, symmetry-consistent route to a vacuum phenomenology that better matches observed axial properties and maintains a consistent description of chiral dynamics.

Abstract

The standard parity doublet model with its mass-mixing mechanism fails to describe the axial charge $g_A$ of the nucleon. While $g_A = 1$ in the original Gell-Mann--Levy model, which reproduces the Adler-Bell-Jackiw anomaly of QCD, in the presence of a chirally invariant baryon mass the mass mixing leads to $g_A < 1 $ whereas phenomenologically it is about 1.28. We propose to remedy this problem by introducing kinetic-mixing terms corresponding to meson-baryon derivative couplings, similar in spirit to the two-mixing-angle scenario of the $η$-$η'$ mixing. This extended parity doublet model contains five parameters in the effective baryonic Lagrangian. Three of them can be determined by using the empirical results for the axial charge of the nucleon together with the masses of the nucleon and its parity partner, the $N^*(1535)$ resonance. We discuss various options how to determine the remaining parameters, touching upon the mass of both parity partners if the chiral condensate is put to zero; the mass of the nucleon in the chiral limit; and the values of meson-baryon coupling constants related to the decays of the resonance to pion-nucleon and sigma-nucleon.

Figures (4)

• Figure 1: Masses of nucleon (blue) and $N^*(1535)$ (orange) in the standard PDM (with $h=0$) over the $\sigma$-field expectation value. Left: for chirally invariant baryon masses $m_0 = \{ 0,\, 321, \, 558, \, 722, \, 857\}$ MeV, corresponding to nucleon masses in the chiral limit, where $\langle\sigma\rangle = f_\pi^0$, of $m_N^0 = \{ 885,\, 890, \, 900, \, 910, \, 920\}$ MeV (from bottom up). Right: for $m_0 = \{ 618,\, 920 \}$ MeV, such that $|g_{\pi NN^*}| = 0.64$ is adjusted to the $N^* \to \pi N$ decay as discussed in Sec. \ref{['sec:matchpd-coupl']}, leading to the special parameter sets in gray with $m_N^0 = \{ 903, \, 925\}$ MeV in Tab. \ref{['tab:PDM-Masses-NoKM']}.
• Figure 2: Masses of nucleon (blue) and $N^*(1535)$ (orange) over the $\sigma$-field expectation value, including kinetic mixing with couplings $h$ and $\Delta h$ chosen to adjust the axial charge of the nucleon to $g_A =1.28$. Left: with nucleon mass in the chiral-limit fixed to $m^0_N=880\,$MeV, comparing solutions where $g_{\pi NN^*} = \pm 0.64$ and one where the axial charge of the $N^*$ is fixed from the anomaly matching condition $g_A - g_A^* = 1$. Right: four solutions with $|g_{\pi NN^*}| = 0.64$ and $|g_{\sigma NN^*}| = 5.24$ discussed in Sec. \ref{['sec:matchpd-coupl']}.
• Figure 3: Calculation of the $\sigma$ self-energy beyond one-loop by allowing the pions to rescatter. Figure taken from Leupold:2009nv.
• Figure 4: Left: Spectral distribution ${\cal A}$ for the $\sigma$ meson as a function of the energy $\sqrt{s}$. Right: Comparison of our result for the pion phase shift (blue) to the dispersive analysis of GarciaMartin:2011cn. The underlying figure is taken from GarciaMartin:2011cn. The width of the blue lines (our results) in the two panels is obtained by varying the pion mass between the isospin average (our original choice for the calculations) and the charged pion mass (the choice of GarciaMartin:2011cn).