Evidence for quark-diquark structure of baryons from fluctuations of conserved charges

Technical Summary: Evidence for quark-diquark structure of baryons from fluctuations of conserved charges

Problem Statement
Fluctuations and correlations of conserved charges (net-baryon number, strangeness, and electric charge) serve as sensitive probes of the degrees of freedom in Quantum Chromodynamics (QCD) matter. While high-precision Lattice QCD (LQCD) calculations have mapped these susceptibilities, particularly around the chiral crossover temperature ($T_c \simeq 155$ MeV), a quantitative description of these observables in the confined phase remains challenging. Standard descriptions using the Hadron Resonance Gas (HRG) model, which relies on the experimentally established spectrum from the Particle Data Group (PDG), systematically underestimate higher-order fluctuations and correlations near $T_c$. This discrepancy suggests that the effective hadronic spectrum relevant for thermodynamics may differ from the spectroscopic spectrum, potentially indicating missing states or a different underlying structural picture of hadrons.

Methodology
The authors employ a string-based description of the hadronic mass spectrum to model the thermodynamics of the confined phase. In this framework:

• Mesons are modeled as open relativistic strings with quark-antiquark ($q\bar{q}$) endpoints.
• Baryons are modeled as open relativistic strings with a quark and a diquark ($q$-$[qq]$) attached to the endpoints.

This construction posits that the string dynamics governing baryons are identical to those of mesons, with differences arising solely from endpoint quantum numbers and physical threshold masses. The density of states follows an exponential Hagedorn form, $\rho(m) \propto m^a e^{m/T_H}$, where the Hagedorn temperature $T_H$ is determined by the string tension $\sigma$.

The study proceeds in two stages:

1. Spectroscopic Constraint: The continuous Hagedorn spectrum is fitted to the discrete cumulative spectrum of experimentally established hadrons (PDG data) for both mesons and baryons. This yields a Hagedorn temperature of $T_H \approx 340$ MeV.
2. Thermodynamic Extraction: Recognizing the discrepancy between the PDG-constrained model and LQCD data, the authors extract the effective Hagedorn temperature directly by fitting the string-based baryonic spectrum to the second-order net-baryon number susceptibility ($\hat{\chi}_B^2$) from LQCD in the temperature range $T \leq 155$ MeV.

Using these spectra, the authors calculate generalized susceptibilities (fluctuations and correlations) for net-baryon number, strangeness, and electric charge up to fourth order, comparing the results against LQCD data.

Key Contributions and Results

• Discrepancy Identification: The study confirms that a continuous Hagedorn spectrum constrained solely by PDG spectroscopy ($T_H \approx 340$ MeV) systematically underestimates the fluctuations of conserved charges (specifically $\hat{\chi}_B^2$, $\hat{\chi}_B^4$, $\hat{\chi}_S^2$, and $\hat{\chi}_Q^2$) compared to LQCD results. This supports the hypothesis that the thermodynamic spectrum contains more states than currently established experimentally.
• Parameter-Free Fit: By extracting $T_H$ directly from LQCD data, the authors obtain a consistent Hagedorn temperature of $T_H \simeq 323$ MeV. This lower temperature implies a steeper growth in the density of states compared to the PDG fit.
• Improved Agreement: Using the $T_H \simeq 323$ MeV spectrum, the model provides a significantly improved, parameter-free description of a broad set of LQCD observables, including net-baryon fluctuations and electric charge fluctuations.
• Residual Deviations:
  • The baryon-strangeness correlator ($\hat{\chi}_{BS}^{11}$) remains underestimated. The authors attribute this to the extraction method (based on $\hat{\chi}_B^2$, dominated by non-strange baryons) and suggest the possible existence of unconfirmed strange baryons.
  • The baryon-charge correlator ($\hat{\chi}_{BQ}^{11}$) overshoots LQCD results. This is attributed to the ideal-gas approximation used, which neglects repulsive interactions known to reduce this correlator in more sophisticated S-matrix approaches.

Significance
The paper claims that the successful description of LQCD fluctuations using a string-based spectrum with a quark-diquark baryon structure provides thermodynamic evidence supporting this specific picture of baryons in the confined phase. The results suggest that the effective degrees of freedom in QCD thermodynamics are consistent with a universal Hagedorn temperature governing both mesons and baryons, where baryons behave as open strings with diquark endpoints. The work demonstrates that thermodynamic observables can constrain the hadronic mass spectrum more effectively than spectroscopy alone, offering a consistent framework for understanding the transition from hadronic matter to the quark-gluon plasma.