Decay widths of radially excited vector mesons of the $U(3)$ nonet with production of three pseudoscalar mesons in the extended NJL model

Technical Summary: Decay Widths of Radially Excited Vector Mesons in the Extended NJL Model

Problem Statement
The paper addresses the theoretical description of the strong interactions of the first radially excited vector mesons ($\rho', \omega', \phi', K^{*\prime}$) within the U(3) nonet. Specifically, it focuses on calculating the decay widths of these mesons into three pseudoscalar meson final states ($3P$, where $P = \pi, K, \eta$). While experimental data exists for the masses and total widths of these excited states (derived from $e^+e^-$ annihilation and $\tau$ lepton decays), individual partial decay widths into specific three-body channels remain largely unmeasured. Theoretical descriptions of these processes are complicated by the non-perturbative nature of Quantum Chromodynamics (QCD) at low energies, where standard perturbation theory is inapplicable due to large coupling constants.

Methodology
The authors employ the extended Nambu–Jona-Lasinio (NJL) model, a phenomenological framework based on effective chirally symmetric four-quark interactions. Key methodological features include:

• Model Extension: The model incorporates the first radially excited states of pseudoscalar, vector, and axial-vector mesons using polynomial form factors dependent on the transverse momentum of quarks.
• Lagrangian Construction: The interaction Lagrangian includes vertices for quark-meson couplings, mixing between ground and excited states (defined by mixing angles $\theta_M$ and $\theta'_M$), and transitions involving axial-vector mesons.
• Calculation of Amplitudes: The decay amplitudes are calculated by summing contributions from two distinct topological classes of diagrams:
  1. Box Diagrams: Representing the direct production of the three final pseudoscalar mesons.
  2. Vector Channels: Representing processes with intermediate vector mesons in both the ground ($\rho, \omega, \phi, K^*$) and first radially excited ($\rho', \omega', \phi', K^{*\prime}$) states. These intermediate states are treated using Breit-Wigner propagators.
• Renormalization: The model utilizes one-loop quark contributions to fix coupling constants and includes renormalization constants ($Z$) to account for transitions between axial-vector and pseudoscalar mesons.
• Parameters: The calculations rely on constituent quark masses ($m_u=m_d=270$ MeV, $m_s=420$ MeV), a cutoff parameter ($\Lambda_3 = 1030$ MeV), and mixing angles fixed by meson masses.

Key Contributions
The primary contribution of this work is the systematic calculation of decay widths for the processes $\rho', \omega', \phi', K^{*\prime} \to 3P$. The authors provide explicit analytical expressions for the amplitudes of various decay channels, including:

• $\rho' \to K^+K^-\pi^-$, $\pi^+\pi^-\eta$, and $\pi^-\pi^0\eta$.
• $\omega' \to \pi^0K^+K^-$ and $\pi^+\pi^-\pi^0$.
• $\phi' \to K^+K^-\pi^0$ and $K^+K^-\eta$.
• $K^{*\prime} \to K^-\pi^0\eta$, $K^-\pi^+\pi^-$, and related charge conjugate/neutral variants.

The study distinguishes between the contributions of direct box diagrams and those mediated by intermediate vector mesons, quantifying the relative importance of each mechanism for different decay modes.

Results
The numerical results for the decay widths (in MeV) are summarized in Table 2 of the paper. Key findings include:

• Dominant Channel: The decay $\omega' \to 3\pi$ ($\pi^+\pi^-\pi^0$) exhibits the largest width, calculated at $\Gamma = 358.31$ MeV. This theoretical value is noted to be in agreement with the experimental total width of the $\omega'$ ($590 \pm 90$ MeV), considering the model's estimated 15% theoretical error.
• Vector Channel Dominance: For most decays, the contributions from intermediate vector channels significantly exceed those from direct box diagrams. For instance, in $\rho' \to \pi^+\pi^-\eta$, the vector channel contributes 15.48 MeV compared to 1.16 MeV from the box diagram.
• Specific Predictions: The paper presents predictions for channels with currently insufficient experimental data, such as $\phi' \to K^+K^-\pi^0$ ($\Gamma \approx 82.18$ MeV) and $K^{*\prime} \to K^-\pi^+\pi^-$ ($\Gamma \approx 31.09$ MeV).
• Consistency Check: The model's prediction for the $\phi'$ mass ($1682$ MeV) and its partial widths into $KK\pi$ and $KK\eta$ are consistent with values fitted from $e^+e^-$ annihilation data.

Significance and Claims
The authors position this work as a predictive tool for future experimental investigations. They claim that:

1. Predictive Power: The extended NJL model provides a unified method for describing meson interactions without introducing arbitrary parameters beyond those fixed by the ground state spectrum.
2. Experimental Utility: The calculated partial widths serve as predictions for future experiments at lepton colliders (Belle II, BES III, Super c-$\tau$) and hadron factories, where individual channel widths are not yet measured.
3. Theoretical Consistency: The results complement previous NJL calculations of two-particle decays and align with other theoretical approaches (Chiral Perturbation Theory, nonrelativistic quark models) where comparisons are possible.
4. Limitations: The authors modestly note that the theoretical error is estimated at 15%, primarily due to chiral symmetry breaking effects and uncertainties in meson masses. They emphasize that direct experimental verification of the specific three-body partial widths is currently lacking, making the presented results theoretical benchmarks rather than confirmed experimental facts.