# Proper-time Quantum Mechanics for Multi-Quark System and Composite-Hadron Spectroscopy

## TL;DR

This paper develops a proper-time quantum mechanics framework for multi-quark systems to establish a Lorentz-invariant classification of composite hadrons, introducing new symmetry concepts and explaining Regge trajectories in hadron spectroscopy.

## Contribution

It introduces a proper-time quantum mechanics approach for multi-quark systems, incorporating chiral symmetry and a novel Regge trajectory formulation based on mass-squared versus quantum number.

## Key findings

- Regge trajectories follow a linear relation with quantum number N
- Intrinsic hadron spin originates solely from quark spin
- Phenomenological insights into light and heavy quarkonium systems

## Abstract

One of the most important problem in hadron physics is to establish the Lorentz-invariant classification scheme of composite hadrons, extending the framework of non-relativistic quark model. We present an attempt, by developing proper-time $\tau$ quantum mechanics on a multi-quark system in particle frame (with constant boost velocity $\boldsymbol{v}$). We start from the variational method on a classical mechanics action where a constituent quark has Pauli-type $SU(2)_{\sigma}$ spin. Then the $SU(2)_{\mathfrak{m}}$ symmetry, concerning the sign-reversal on quark mass, has arisen with the basic vectors, the normal Dirac spinor with $J^{P}=(1/2)^{+}$ and the chiral one with $J^{P}=(1/2)^{-}$, appearing as a "shadow" of the former. Herewith, the mass reversal between these basic vectors become equivalent to the chirality, which is a symmetry of the standard gauge theory. We describe the role of chirality in hadron spectroscopy and regard it as attribute {$\chi$} of "elementary" hadrons in addition to {$J, P, C$}. A novel feature of our hadron spectroscopy is, in the example of $q\bar{q}$ meson system, that the "Regge trajectories", are given by mass-squared vs. the number of quantum $N$ ; where $M^2 =M_{0}^2 +2N\Omega$ ($N=2n$, $n$ the radial quantum number, $\Omega$ the oscillator quantum), and the intrinsic spin of hadrons $\boldsymbol{J}$ comes only from quark spin $\boldsymbol{S}$, $\boldsymbol{J}=\boldsymbol{S}$. Some phenomenological facts crucial to its validity are pointed out on the light-through-heavy quarkonium system.