# On the microscopic structure of $\pi NN$, $\pi N\Delta$ and $\pi\Delta\Delta$ vertices

## TL;DR

This paper develops a relativistic constituent-quark model to analyze the microscopic structure of pion-nucleon and pion-delta vertices, revealing coupling ratios that align with hadronic models but differ from traditional SU(6) symmetry predictions.

## Contribution

It introduces a hybrid relativistic quark model with dynamical pions to compute vertex couplings and form factors, providing insights beyond non-relativistic SU(6) symmetry.

## Key findings

- Coupling ratios resemble hadronic coupled-channel models
- Ratios deviate from non-relativistic SU(6) predictions
- Relativistic effects significantly impact vertex structures

## Abstract

We use a hybrid constituent-quark model for the microscopic description of $\pi N N$, $\pi N \Delta$ and $\pi \Delta \Delta$ vertices. In this model quarks are confined by an instantaneous potential and are allowed to emit and absorb a pion, which is also treated as dynamical degree of freedom. The point form of relativistic quantum mechanics is employed to achieve a relativistically invariant description of this system. Starting with an $SU(6)$ spin-flavor symmetric wave function for $N_0$ and $\Delta_0$, i.e. the eigenstates of the pure confinement problem, we calculate the strength of the $\pi N_0 N_0$, $\pi N_0 \Delta_0$ and $\pi \Delta_0 \Delta_0$ couplings and the corresponding vertex form factors. Interestingly the ratios of the resulting couplings resemble strongly those needed in purely hadronic coupled-channel models, but deviate significantly from the ratios following from SU(6) spin-flavor symmetry in the non-relativistic constituent-quark model.