Chiral symmetry restoration with three chiral partners

Abstract: I discuss the masses of chiral partners in the context of chiral symmetry restoration at finite temperature. Using the Nambu-Jona-Lasinio model I first remind the usual situation where two mesons of opposed parity become degenerate above the chiral transition temperature. Then I consider an effective theory for D mesons where the positive parity companion presents a "double pole structure". In this case three different masses need to be analyzed as functions of the temperature. I suggest a possible restoration pattern at high temperatures when the back-reaction of the quark condensate is incorporated.

• The paper investigates chiral symmetry restoration by tracking finite-temperature mass evolution using both NJL and covariant chiral EFT frameworks.
• It identifies a novel double-pole structure in the positive-parity D₀*(2300) state, leading to a three-part chiral partner scheme in heavy-light mesons.
• Sequential degeneracy is observed, where the lower D₀*(2300) pole converges with the D ground state as the chiral condensate melts, indicating experimental signatures in heavy-ion collisions.

Chiral Symmetry Restoration and the Emergence of Three Chiral Partners

Introduction

The work "Chiral symmetry restoration with three chiral partners" (2112.09463) presents a technical examination of chiral partner mass evolution, focusing on finite-temperature QCD and its effective models. The analysis foregrounds scenarios where, contrary to the canonical two-state degeneracy patterns, a positive-parity state emerges with a double-pole structure, necessitating the consideration of three relevant thermal masses. The approach utilizes both the Nambu–Jona-Lasinio (NJL) model and a covariant chiral effective field theory (EFT) tailored to heavy-light mesons, with special attention to the role of the chiral condensate and the back-reaction mechanisms at high temperature.

Chiral Partner Degeneracy in the NJL Framework

The NJL model encapsulates spontaneous chiral symmetry breaking for nearly massless quarks, resulting in a clear split between opposite-parity excitations, e.g., $\pi$ and $\sigma$ channels. Both are generated dynamically via a Bethe-Salpeter approach for quark-antiquark scattering and manifest as poles in the $T$-matrix. At low temperatures, the pion appears as a bound state, while the $\sigma$ is distinguished by a higher mass and substantial width. Increasing temperature reduces the quark condensate, driving both quark masses and the $\pi/\sigma$ mass splitting towards chiral symmetry restoration.

Above $T_c$, both $\pi$ and $\sigma$ modes display degenerate thermal masses. Importantly, the thermal width of both excitations also converges, signaling a full spectral shape degeneracy and thus an effective chiral restoration. This scenario demonstrates chiral restoration realized entirely through the in-medium modification of dynamical degrees of freedom—states not explicit in the underlying Lagrangian but emergent within the scattering framework.

Chiral Symmetry Patterns for $D$ Mesons and Double-Pole Phenomenology

The analysis advances to scenarios outside the traditional two-state paradigm, specifically the $(J,S)=(0,0)$ sector of charmed mesons. Here, the negative-parity $D$ meson is treated as a fundamental degree of freedom, while its positive-parity chiral partner, the $D_0^*(2300)$, is realized as a dynamically generated state manifesting a "two-pole structure." Consequently, three chiral companions—$D$ and two poles for $D_0^*(2300)$—must be analyzed, presenting a novel chiral restoration pattern.

Within the covariant chiral EFT (incorporating heavy quark symmetry and coupled-channel unitarization), poles are identified as resonance states from the full $T$-matrix. The $D_0^*(2300)$ appears as a double pole, with each pole coupling distinctly to various physical decay channels. Extending the unitary formalism to finite temperature (including self-consistency for mesonic dressing contributions), the masses of these three states exhibit only minor reductions (on the order of $2\%$) up to $T \sim 150$ MeV. This suggests that thermal broadening and mass reduction alone, within this self-consistent approach, do not yield chiral partner degeneracy.

Impact of the Chiral Condensate and Sequential Degeneracy

The temperature-driven reduction in the chiral condensate—the order parameter for chiral breaking—plays a crucial role in mass evolution and degeneracy patterns. The work probes the sensitivity of the EFT-generated masses to reductions in $f_\pi$ (conveying the order parameter's temperature dependence), isolating this effect by direct reduction of $f_\pi$ within the $T$-matrix equation. The results indicate that as $f_\pi$ is diminished (simulating the melting chiral condensate), the lower $D_0^*(2300)$ pole approaches degeneracy with the $D$ ground state. The higher pole remains distinct, with a reduced but significant mass gap.

The analysis thus motivates a sequential degeneracy pattern, where the lower pole of the positive parity state first becomes degenerate with the negative parity ground state as chiral symmetry is effectively restored, and only at higher temperatures does the higher pole follow. This structure is fundamentally different from the standard two-state degeneracy and is a direct consequence of the underlying dynamics governing the double-pole configuration.

Theoretical and Experimental Implications

The theoretical implications are significant: models predicting more elaborate chiral partner patterns (double-pole structures, nontrivial thermal evolution) challenge the universality of simple two-state degeneracy as the exclusive signal of chiral symmetry restoration. Effective models must explicitly address the back-reaction of the quark condensate and incorporate nonperturbative unitarization and coupled-channel dynamics for reliable finite-temperature extrapolation.

Experimentally, the signature of sequential degeneracy could manifest in heavy-ion collisions, provided one can reconstruct the $D_0^*(2300)$ resonance in $s$-wave channels ($D\pi$ and $D_s \bar K$), exploiting the different pole-coupling strengths.

Conclusion

This study offers a comprehensive theoretical investigation into chiral symmetry restoration, emphasizing cases involving three chiral partners attributed to resonance double-pole structures. The work demonstrates that a simple application of finite-temperature EFT yields minimal mass shifts without chiral degeneracy; however, an explicit account of the chiral condensate's thermal evolution predicts a sequential convergence pattern among partners. These findings present both a constraint for future EFT developments and a testable signature in experimental environments sensitive to in-medium spectral modifications of heavy-light mesons.