Quark Coalescence: Formation of Mesons Including Excited States

TL;DR

This work addresses hadronization by formulating a quantum-mechanical quark coalescence model that includes a complete spectrum of excited meson states within a non-relativistic quark model using a harmonic oscillator potential. It employs a Wigner-phase-space formalism with Gaussian quark wave packets to compute unpolarized meson formation probabilities, yielding analytic expressions that incorporate color, spin, flavor, and spatial overlaps, and extends to excited states up to $N=4$ (and $L=4$). The authors provide extensive mass and decay prescriptions for 330 states (confirmed and predicted), and implement the framework into Hybrid Hadronization within JETSCAPE, validating the approach with an $e^+e^-$ test that reveals excited states significantly enhance recombination yields while leaving total spectra modestly affected. The results enable a universal, self-consistent hadronization description across collision systems, with $N_{\max}$ serving as a tunable knob to study the role of excitations in jet and heavy-ion phenomenology, and offer a foundation for data-driven tuning of shower and fragmentation parameters. The methodology and generated state catalog offer practical utility for Monte Carlo event generators and for interpreting hadron spectra in high-energy collisions.

Abstract

We discuss the quantum mechanics of coalescence of quark-antiquark pairs into mesons using a non-relativistic quark model. We derive the coalescence probabilities assuming a harmonic oscillator potential and generic Gaussian wave packet shapes for the initial quarks and antiquarks. Our particular emphasis is on modeling excited states of the meson spectrum consistently. We provide the formalism to systematically include excited states from the Particle Data Book, and many more predicted by the quark model, up to $L=4$ and masses of about 2.2 GeV for light flavors. We provide estimates of masses and decay branching ratios for unconfirmed states. We use a phase space picture which is appropriate for the quasi-classical nature of the information typically available for the quarks and antiquarks in applications like Monte Carlo simulations. We demonstrate that for typical parton configurations expected in jets, excited meson states are populated abundantly.

Figures (8)

• Figure 1: Spectrum $dN_\text{ch}/dx_E$ of charged hadrons as a function of energy fraction $x_E$ compared to data from the ALEPH collaboration ALEPH:2003obs. All available excited meson states ($N_{\text{max}}=4$) are used in recombination and direct Goldstone boson recombination is off. Hadron decays settings and analysis follow the prescription for the ALEPH data set.
• Figure 2: Primordial multiplicities of mesons from recombination only, with full excitation spectrum in Hybrid Hadronization ($N_{\text{max}}=4$) and before decays. Direct Goldstone boson recombination is off. The distributions of total meson spin $J$, orbital angular momentum $L$, radial excitation number $k$ and spin $S$ are shown.
• Figure 3: Primordial $x_E$-spectrum of mesons from recombination only, with full excitation spectrum in Hybrid Hadronization ($N_{\text{max}}=4$) and before decays. Direct Goldstone boson recombination is off. The spectra of all mesons of a particular allowed value of orbital angular momentum $L$ are shown.
• Figure 4: $x_E$-spectrum of charged pions from recombination only (upper right panel) and from all channels (upper left panel) after strong and electromagnetic decays. Spectra with high excitation allowed in recombination ($N_{\text{max}}=4$) and ground states only ($N_{\text{max}}=0$) are compared. The bottom panels show the ratios of spectra $[N_{\text{max}}=4$]/[$N_{\text{max}}=0]$
• Figure 5: Same as Fig. \ref{['fig:pions']} for charged kaons.