# A Faddeev Calculation for Pentaquark $\Theta^+$ in Diquark Picture with Nambu-Jona-Lasinio Type Interaction

## TL;DR

This study uses a Bethe-Salpeter-Faddeev approach with NJL interactions to investigate the possibility of a bound pentaquark state, finding it unlikely under realistic coupling constants.

## Contribution

It applies a three-body Faddeev calculation with NJL model to the pentaquark $ heta^+$ in a diquark framework, providing new insights into its binding conditions.

## Key findings

- ${ar s}D$ interaction is attractive
- $DD$ interaction is repulsive
- No bound $ heta^+$ state with realistic couplings

## Abstract

A Bethe-Salpeter-Faddeev (BSF) calculation is performed for the pentaquark $\Theta^+$ in the diquark picture of Jaffe and Wilczek in which $\Theta^+$ is a diquark-diquark-${\bar s}$ three-body system. Nambu-Jona-Lasinio (NJL) model is used to calculate the lowest order diagrams in the two-body scatterings of ${\bar s}D$ and $D D$. With the use of coupling constants determined from the meson sector, we find that ${\bar s}D$ interaction is attractive while $DD$ interaction is repulsive, and there is no bound $\frac 12^+$ pentaquark state. A bound pentaquark $\Theta^+$ can only be obtained with unphysically strong vector mesonic coupling constants.