Symmetries of hadrons after unbreaking the chiral symmetry

Abstract: We study hadron correlators upon artificial restoration of the spontaneously broken chiral symmetry. In a dynamical lattice simulation we remove the lowest lying eigenmodes of the Dirac operator from the valence quark propagators and study evolution of the hadron masses obtained. All mesons and baryons in our study, except for a pion, survive unbreaking the chiral symmetry and their exponential decay signals become essentially better. From the analysis of the observed spectroscopic patterns we conclude that confinement still persists while the chiral symmetry is restored. All hadrons fall into different chiral multiplets. The broken U(1)_A symmetry does not get restored upon unbreaking the chiral symmetry. We also observe signals of some higher symmetry that includes chiral symmetry as a subgroup. Finally, from comparison of the Δ- N splitting before and after unbreaking of the chiral symmetry we conclude that both the color-magnetic and the flavor-spin quark-quark interactions are of equal importance.

• The paper demonstrates that restoring chiral symmetry via Dirac mode truncation maintains confinement, as most hadrons (excluding the pion) persist with improved mass signals.
• The study employs dynamical fermions with extended Gaussian sources and a variational method to extract ground and excited states, revealing parity doubling in the spectrum.
• The findings challenge the traditional view linking the quark condensate to mass generation, suggesting hadronic masses arise from mechanisms beyond spontaneous chiral symmetry breaking.

Artificial Restoration of Chiral Symmetry in Hadrons

Introduction and Motivation

The relationship between chiral symmetry breaking and hadron structure stands at the core of non-perturbative QCD. Parity doubling, apparent in highly excited hadrons, suggests effective chiral symmetry restoration, but its precise role in hadron mass generation and the persistence of confinement remains debated. Utilizing lattice QCD, the study "Symmetries of hadrons after unbreaking the chiral symmetry" (1205.4887) examines hadron correlators under the artificial restoration of spontaneously broken chiral symmetry. This is achieved by systematically removing the lowest-lying eigenmodes of the Dirac operator from the quark propagators, thereby eliminating the quark condensate, the principal order parameter for chiral symmetry breaking via the Banks–Casher relation. Through this method, the paper interrogates whether hadronic bound states and the confining regime persist once chiral symmetry is "unbroken" and analyzes the resulting spectroscopic and symmetry patterns.

Lattice Methodology and Symmetry Restoration Procedure

The simulation employs two dynamical CI fermions on $16^3 \times 32$ lattices with a spatial lattice spacing of $a=0.144$ fm and pion mass $m_\pi \approx 322$ MeV. The lowest 128 eigenmodes of the Hermitian Dirac operator $D_5$ are calculated and, upon truncation, systematically removed from the valence quark sector of the hadron propagators. This truncation progressively restores chiral symmetry in the valence sector, decoupling hadronic structure from the underlying quark condensate.

Extended Gaussian sources and multiple interpolator structures, supplemented by derivative sources, are used in conjunction with a variational method to robustly extract both ground and excited states of baryons ($N$, $\Delta$) and isovector mesons ($\rho$, $a_1$, $b_1$, etc.). The analysis focuses on the behavior of masses and spectroscopic multiplet structures as a function of the number and energy scale of the subtracted Dirac eigenmodes.

Numerical Results and Spectroscopic Analysis

Universal Observations:

• Upon removal of the lowest Dirac modes (chiral symmetry restoration), exponential decay signals in the correlators for all hadrons—except the pion—persist and typically become cleaner, yielding improved effective mass plateaus.
• The pion, a pseudo-Goldstone boson associated with spontaneous chiral symmetry breaking, vanishes from the spectrum with symmetry restoration, in line with theoretical expectations.

Confinement and Mass Generation:

• The survival of hadronic states (excluding the pion) post-truncation conclusively demonstrates that confinement remains intact even when the quark condensate is eliminated from the valence sector. There is no universal scaling of hadron masses (i.e., not all hadron masses simply track 2$m_0$ for mesons and 3$m_0$ for baryons, with $m_0$ a constituent-like mass), and excited states remain discernible. This precludes a scenario in which the lattice signals originate from unbound quarks, providing direct computational evidence of confinement without chiral symmetry breaking (1205.4887).

Symmetry Patterns and Multiplet Structure:

• The $\rho$ and $a_1$ mesons become degenerate under mode removal, consistent with parity-chiral multiplet formation and $SU(2)_L \otimes SU(2)_R$ chiral symmetry restoration.
• The $b_1$ meson remains nondegenerate with $\rho$ and $a_1$, indicating that $U(1)_A$ symmetry is not restored by the removal of the low-lying Dirac modes. This decouples the anomalous breaking of $U(1)_A$ from the spontaneous breaking manifested via the quark condensate.
• Higher symmetry is suggested by the near-degeneracy of states such as $\rho$ and its radial excitation $\rho'$, not explained solely by chiral symmetry restoration and pointing toward an enlarged symmetry pattern in the chirally restored regime.

Baryon Sector:

• Nucleon parity doublets systematically emerge as degeneracies between positive and negative parity states, while similar doublets and quartets in the $\Delta$ sector show non-uniform degeneracy, indicating more nuanced symmetry dynamics.
• Comparisons of $\Delta$ and $N$ hyperfine splitting pre- and post-chiral restoration reveal that both the color-magnetic and flavor-spin quark-quark interactions contribute equally to the physical $\Delta-N$ splitting, with the former persisting as chiral symmetry is restored.

Theoretical and Practical Implications

This study yields several significant implications for non-perturbative QCD:

• Chiral Symmetry Restoration and Confinement: The persistence of confined hadrons in the artificial absence of chiral symmetry breaking demonstrates the independence of confinement and the quark condensate, challenging models that vigorously tie confining dynamics to spontaneous chiral symmetry breaking.
• Origin of Hadron Mass: The substantial mass of hadrons, particularly vector mesons, in the chirally restored regime invalidates the assertion that their mass is predominantly generated by the quark condensate.
• Role of $U(1)_A$ Breaking: The non-restoration of $U(1)_A$ symmetry upon mode removal underscores the essential role of anomaly-related topological fluctuations distinct from the quark condensate in hadron structure.
• Spectral Symmetry Structure: The emergence of higher degeneracies in the spectrum hints at unexplored symmetry groups in QCD at high excitation or in the chirally restored regime, influencing expectations for QCD matter under extreme conditions.

Outlook and Prospects for Future Research

The results motivate further exploration of symmetry structures beyond $SU(2)_L \otimes SU(2)_R$, including the nature of the higher symmetry revealed by the observed multiplet degeneracies. Future developments could analyze isoscalar states, extend studies to larger lattice volumes and lighter pion masses, and examine the interplay between confinement, chiral symmetry, and anomaly-induced symmetry breaking under varying temperature and density, relevant for understanding QCD matter in heavy-ion collisions and astrophysical environments.

Conclusion

Artificial restoration of chiral symmetry via Dirac eigenmode truncation in lattice QCD reveals that confinement persists in the absence of the quark condensate; hadrons, except the pion, survive and manifest restored chiral symmetry multiplet structure, without restoration of the $U(1)_A$ symmetry. Certain parity doublets become degenerate, indicating a possibly larger symmetry. Hyperfine baryon mass splittings point to comparable importance of color-magnetic and flavor-spin interactions. These results refine conceptual understanding of mass generation, the structure of hadronic bound states, and underlying QCD symmetries, establishing a methodological foundation for further theoretical and numerical studies of strong interaction dynamics (1205.4887).