Production of high-orbital kaon excited states in the $K^{-}p$ reaction

Technical Summary: Production of High-Orbital Kaon Excited States in the $K^-p$ Reaction

Problem Statement
While meson-beam experiments have been instrumental in establishing the spectrum of light hadrons, the spectroscopic picture of the kaon family remains incomplete, particularly for high-orbital excitation states. Although low-lying kaons were discovered decades ago, experimental data for high-lying states (high orbital quantum numbers $L$) are scarce. Recent observations by the COMPASS Collaboration of new resonances, $K'_3(2120)$ and $K_4(2210)$, have enriched the spectrum, yet the production mechanisms for these high-orbital kaons in meson-nucleon reactions remain a largely underexplored theoretical area. This work addresses the lack of theoretical understanding regarding the production dynamics of high-orbital kaons in $K^-p$ reactions, aiming to provide a framework for their observation in future experiments.

Methodology
The authors employ an effective Lagrangian approach to investigate the production of high-orbital kaon states ($1D$, $1F$, and $1G$ waves) in the $K^-p \to K^* + N$ reaction.

• Reaction Mechanism: The study focuses on $t$-channel exchange processes. Contributions from $s$-channel and $u$-channel diagrams are neglected due to kinematic suppression and the unimportance of baryon-antibaryon decay modes for these states.
• Lagrangian Construction: Effective interaction Lagrangians are constructed for the upper vertices (coupling the produced strange meson to the incident kaon and exchanged light meson) and lower vertices (coupling exchanged mesons to nucleons). The interaction structures satisfy Lorentz covariance and the specific spin-parity quantum numbers of the participating mesons.
• Exchange Mechanisms: The dominant exchanged mesons ($\pi, \rho, \omega$) are selected based on the known or predicted dominant decay channels of the produced kaon states. For instance, pion exchange dominates for states with significant $K\pi$ decay modes, while vector meson exchanges ($\rho, \omega$) are considered for states decaying into vector-pseudoscalar channels.
• Form Factors and Parameters: A phenomenological form factor, $F_t(q)$, is introduced at the interaction vertices to account for finite-size effects. The cutoff parameter $\Lambda_t$ is the sole adjustable parameter, determined by fitting existing experimental total cross-section data for the $K^-p \to K^*_3(1780)p$ reaction, yielding $\Lambda_t = 1.5 \pm 0.2$ GeV.
• Mixing Angles: For mixed states (e.g., $K_2(1770)$ and $K_2(1820)$ from $1D$ waves, and $K'_3(2120)$ and $K_3(1F)$ from $1F$ waves), the mixing angle is constrained by strong decay analyses and production cross-section data, with a value of $\theta_{1D} = -30^\circ$ found to be consistent with experimental measurements.
• Reggeization: To account for high-energy behavior, the Feynman propagators are replaced with Regge propagators, incorporating Regge trajectories for the exchanged mesons.

Key Contributions and Results
The paper systematically calculates total and differential production cross-sections for a wide range of high-orbital kaon states:

1. $1D$-Wave States:
   • The model successfully reproduces the measured total cross-sections for $K^*_3(1780)$, $K_2(1820)$, and $K_2(1770)$ using the single fitted parameter $\Lambda_t$.
   • The mixing angle $\theta_{1D} = -30^\circ$ is validated as it simultaneously describes the production of both $K_2(1770)$ and $K_2(1820)$.
   • Predictions are provided for the $K^*(1680)$ state, which is dominated by $\pi$-exchange, showing a sizable cross-section and forward-peaked angular distribution.
2. $1F$-Wave States:
   • The production of $K^*_4(2045)$ is calculated via $\pi$-exchange, with results consistent with available experimental data at $\sqrt{s} = 4.08$ GeV.
   • Predictions are made for the newly observed $K'_3(2120)$ and the unobserved partner $K_3(1F)$. Both are found to have measurable cross-sections (peaking around $1.1-1.6$ $\mu$b) with strong forward peaking, primarily driven by $\omega$-exchange.
   • The $K^*_2(1980)$ production is analyzed, revealing a dominance of $\pi$-exchange despite smaller $K\pi$ decay widths, with a peak cross-section of $\sim 0.5$ $\mu$b.
3. $1G$-Wave States:
   • Calculations are performed for $K^*_5(2380)$, $K'_4(1G)$, $K_4(2210)$, and $K^*_3(1G)$.
   • The $K^*_5(2380)$ shows the largest cross-section among the $1G$ states ($\sim 6.5$ $\mu$b), dominated by $\pi$-exchange.
   • The newly observed $K_4(2210)$ and its partner $K'_4(1G)$ are predicted to have cross-sections around $0.3$ $\mu$b, driven by $\omega$-exchange.
   • The unobserved $K^*_3(1G)$ is predicted to have a peak cross-section of $\sim 0.46$ $\mu$b.

Significance and Claims
The paper claims that the effective Lagrangian approach provides a unified and reliable framework for describing high-orbital kaon production. The primary significance of the work lies in:

• Validation: The model's ability to reproduce existing experimental data for $1D$-wave states without introducing additional free parameters validates the theoretical framework.
• Predictive Power: The study provides the first systematic theoretical predictions for the production cross-sections of several high-orbital states, including the recently observed $K'_3(2120)$, $K_4(2210)$, and various unobserved states ($K_3(1F)$, $K^*_3(1G)$, etc.).
• Experimental Guidance: A consistent finding across all calculated states is the characteristically forward-peaked angular distribution, a hallmark of $t$-channel exchange. The authors assert that these states possess "sizable cross sections" and that forward-angle measurements are the most favorable kinematic window for their observation in future kaon-beam experiments at facilities such as J-PARC and HIAF.
• Spectroscopic Insight: The work highlights the general trend that production cross-sections decrease as the orbital angular momentum increases ($\sigma_{1G} < \sigma_{1F} < \sigma_{1D}$), yet remain large enough to be experimentally accessible, thereby offering a pathway to refine the incomplete spectroscopic picture of the kaon family.