# Exotic Baryons in Chiral Soliton Models

## Abstract

**:**

## 1. Introduction

## 2. Chiral Soliton Models

## 3. Collective Coordinate Quantization with Three Flavors

## 4. $\mathrm{\Theta}$ Mass and Width

- (1)
- The field equations are solved in two steps, first for the soliton and second for the (constraint) fluctuations. These are the two leading orders in the $1/{N}_{C}$ counting. Consequently, the above-mentioned inconsistency with PCAC is not an issue for the RVA.
- (2)
- The interaction between the collective models and the constrained fluctuations can be written as a separable potential that is similar to a resonance exchange. Reaction matrix theory can be applied to this potential to compute the exchange phase shift ${\delta}_{E}={\delta}_{E}\left({N}_{C}\right)$. The ${N}_{C}$ dependence originates from the matrix elements of the collective coordinate operators in the separable potential. As indicated after Equation (5) they need to be taken between elements of representations that vary with ${N}_{C}$.
- (3)
- Up to a small (a few MeV) pole shift, ${\delta}_{E}$ passes through $\frac{\pi}{2}$ exactly at the momentum given by the mass difference between the ${\mathrm{\Theta}}^{+}$ and the nucleon predicted by the collective coordinate Hamiltonian, Equation (5). Modulo flavor symmetry breaking effects, this difference decreases by a factor of two when going from ${N}_{C}=3$ to ${N}_{C}\to \infty $.
- (4)
- The reaction matrix formalism also derives a width function from the separable potential. Up to flavor symmetry breaking this width function contains the matrix element of a single collective coordinate operator between the nucleon and the ${\mathrm{\Theta}}^{+}$. It is, therefore, impossible that substantial cancellations between matrix elements of several operators occur as was argued in Ref. [5].
- (5)
- The BSA gives exact results in the ${N}_{C}\to \infty $ limit. It is possible to modify the BSA such that the scattering fluctuations are orthogonal to the rotations described by the flavor rotations in Equation (2). For both BSA versions scattering phase shifts can be computed and their difference is the resonance phase shift ${\delta}_{R}$.

## 5. Heavy Baryons

## 6. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

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1 | Listing the references on the mean field description for heavy baryons in soliton models can hardly be exhaustive. The interested reader my trace them from the review [15]. |

2 | |

3 | To the author’s knowledge the occurrence of these exotic baryons in the collective quantization scheme was first noticed by Biedenharn and Dothan [28]. |

4 | |

5 | For an infinitely heavy quark, they are elements of a single multiplet [45]. |

**Figure 2.**(Color online) Young tableaux and particle content the anti-decuplet and the 27-plet $SU\left(3\right)$ representations. Red circles indicate exotic baryons that cannot be built from three quarks.

**Figure 3.**(Color online) Skyrme model results for phase shifts and decay widths of kaon nucleon scattering in the ${\mathrm{\Theta}}^{+}$ channel.

**Figure 5.**(Color online) Young tableaux and particle content of the anti-decapentaplet ($\overline{\mathbf{15}}$). Black circles denote ordinary diquarks whose quantum numbers relate to two quark states. Double circles indicate two states with different R quantum numbers. Red circles denote exotic states that cannot be built from two quarks.

**Figure 6.**(Color online) Flavor symmetry breaking eigenvalues for states that at $\lambda =0$ are pure $\overline{\mathbf{15}}$ states, cf. Figure 5. The parameter $\lambda $ measures the strength of symmetry breaking.

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Weigel, H.
Exotic Baryons in Chiral Soliton Models. *Universe* **2018**, *4*, 142.
https://doi.org/10.3390/universe4120142

**AMA Style**

Weigel H.
Exotic Baryons in Chiral Soliton Models. *Universe*. 2018; 4(12):142.
https://doi.org/10.3390/universe4120142

**Chicago/Turabian Style**

Weigel, Herbert.
2018. "Exotic Baryons in Chiral Soliton Models" *Universe* 4, no. 12: 142.
https://doi.org/10.3390/universe4120142