Glueballs and fractional anomalous determinants at nonzero $\theta$, and the decays of the X(2370)

Technical Summary: Glueballs and Fractional Anomalous Determinants at Nonzero $\theta$

Problem Statement
Quantum Chromodynamics (QCD) features two fundamental quantum anomalies: the trace anomaly, which breaks classical scale invariance and generates the mass scale for glueballs, and the axial $U(1)_A$ anomaly, which is tied to the topological structure of the $\theta$-vacuum. While the breaking of $U(1)_A$ is traditionally described by the Kobayashi-Maskawa-'t Hooft determinant (an integer power of the determinant of the chiral field), the behavior of the theory at nonzero $\theta$ angles—particularly in the pure gauge sector and its coupling to mesons—requires a more generalized framework. Specifically, there is a need to reconcile the $\theta$-dependence of the vacuum energy and scalar glueball mass with lattice data, and to explain the decay patterns of the pseudoscalar glueball candidate $X(2370)$ observed by the BESIII collaboration. Standard models often struggle to simultaneously reproduce the lattice results for the pure gauge theory and the specific ratios of three-pseudoscalar decay channels of $X(2370)$.

Methodology
The authors construct an effective field theory model that couples scalar and pseudoscalar glueballs to mesons using fractional anomalous determinants. The methodology proceeds in two main stages:

1. Pure Gauge Theory ($\theta$-dependence):
   The authors start with the pure Yang-Mills Lagrangian including the $\theta$ term. They introduce a dilaton field $G$ to model the trace anomaly and couple it to an even function $f_G(\theta/N_c)$ to account for the $\theta$-dependence. This function is expanded as a Fourier series involving terms like $(1 - \cos(n\theta/N_c))$. This approach allows for the reproduction of the vacuum energy $E_{vac}(\theta)$ and the scalar glueball mass $M_G(\theta)$ as functions of $\theta$. The model incorporates the large-$N_c$ scaling properties and the periodicity of the $\theta$-vacuum, leading to a multi-branch structure where the physical vacuum is the minimum of these branches.

2. Coupling to Mesons and Fractional Determinants:
   To include dynamical quarks, the $\theta$ angle is shifted to $\Theta = \theta - i \ln(\det \Phi / |\det \Phi|)$, where $\Phi$ is the chiral field containing scalar and pseudoscalar mesons. The key innovation is the introduction of fractional powers of the anomalous determinant, $(\det \Phi / |\det \Phi|)^{n/N_c}$, where $n$ is an integer and $N_c=3$. This generalizes the standard integer-power determinant interactions.
   The interaction Lagrangian for the pseudoscalar glueball $P$ (identified with $X(2370)$) is derived by expanding these fractional determinants. Crucially, the authors retain the field dependence of the modulus $|\det \Phi|$ in the denominator, rather than freezing it to its vacuum value. This leads to specific cubic interaction terms involving traces of the meson fields, specifically $\text{Tr}(\Phi_0^{-1} P)^3$ and $(\text{Tr}(\Phi_0^{-1} P))^3$.

Key Contributions and Results

• Pure Gauge Sector:

  • The model successfully reproduces the $\theta$-dependence of the vacuum energy and the scalar glueball mass derived from lattice QCD.
  • It predicts the coefficients $b_2$ and $m_2$ in the expansion of vacuum energy and mass in powers of $\theta^2$, yielding values ($b_2 \approx -0.0245$, $m_2 \approx -0.0076$) that are in excellent agreement with lattice simulations.
  • The model predicts a first-order phase transition at $\theta = \pi$ for $N_c=3$, characterized by a jump in the topological charge density. The calculated jump $|\Delta Q| \approx 6.1 \times 10^{-3} \text{ GeV}^4$ is consistent with other theoretical estimates and lattice extrapolations.
  • The gluon condensate's $\theta$-dependence is also calculated, showing agreement with semiclassical analyses.

• Phenomenology of $X(2370)$:

  • The model is applied to the decays of the $0^{-+}$ glueball candidate $X(2370)$ into three pseudoscalar mesons.
  • By using the fractional determinant interactions with field-dependent moduli, the authors calculate the decay amplitudes for four measured channels: $\pi K \bar{K}$, $\pi \pi \eta$, $\pi \pi \eta'$, and $K \bar{K} \eta'$.
  • Fit Quality: A single normalization parameter is sufficient to describe all four BESIII data points with a $\chi^2/\text{d.o.f.} = 1.82/3$. This is a significant improvement over models using "frozen-modulus" integer powers of the determinant (e.g., $n=1, 2, 3$), which yield poor fits ($\chi^2/\text{d.o.f.} \approx 10\text{--}18$).
  • The success of the fit supports the interpretation of $X(2370)$ as a pseudoscalar glueball.

• Predictions:

  • The model provides specific predictions for the unmeasured (or currently limited) decay channels $\eta \eta \eta$ and $\eta \eta \eta'$.
  • For the $\eta \eta \eta'$ channel, the model predicts values for $n=1$ and $n=2$ that lie below the current BESIII 90% confidence level upper limit ($9.2 \times 10^{-6}$), while the $n=3$ prediction is slightly above.
  • The authors emphasize that measuring the $\eta \eta \eta$ and $\eta \eta \eta'$ decay rates is a stringent test of the model, as these channels depend on the specific harmonic content (the value of $n$) of the fractional determinant.

Significance
The paper claims that the $\theta$-dependence of the QCD vacuum naturally leads to a "tower of fractional determinant phases." This framework provides a unified description of anomalous meson and glueball interactions. The primary significance lies in demonstrating that fractional anomalous determinants, rather than standard integer powers, are necessary to simultaneously reproduce lattice results for the pure gauge sector and the phenomenological decay patterns of the $X(2370)$. The model suggests that the field dependence of the determinant modulus is crucial for correctly calculating decay vertices. The authors conclude that their approach opens new directions for studying the QCD phase diagram, CP violation, and axion physics, while offering a concrete, testable hypothesis for the nature of the $X(2370)$ resonance.