[Paper Review] A model realisation of the Jaffe-Wilczek correlation for pentaquarks

Jo Dudek|arXiv (Cornell University)|Mar 22, 2004

Quantum Chromodynamics and Particle Interactions8 references3 citations

TL;DR

This paper presents a quark potential model with color-spin contact interactions and colored harmonic confinement to realize the Jaffe-Wilczek pentaquark structure, demonstrating that spatially compact diquarks can form and lower the state's mass. However, when the delta-function contact interaction is smeared to realistic scales, the large mass suppression seen in the schematic model is significantly diluted, weakening the predicted exotic state's mass reduction.

ABSTRACT

We discuss a realisation of the pentaquark structure proposed by Jaffe and Wilczek within a simple quark model with colour-spin contact interactions and coloured harmonic confinement, which accurately describes the $Δ-N$ splitting. In this model spatially compact diquarks are formed in the pentaquark but no such compact object exists in the nucleon. The colour-spin attraction brings the Jaffe-Wilczek-like state down to a low mass, compatible with the experimental observation and below that of the naive ground state with all $S$-waves. We find, however, that although these trends are maintained, the extreme effects observed do not survive the required ``smearing'' of the delta function contact interaction. We also demonstrate the weakness of the ``schematic'' approximation when applied to a system containing a $P$-wave. An estimate of the anti-charmed pentaquark mass is made which is in line with the Jaffe-Wilczek prediction and significantly less than the value reported by the H1 collaboration.

Motivation & Objective

• To provide a dynamical realization of the Jaffe-Wilczek pentaquark model within a quark potential framework.
• To investigate whether color-spin contact interactions and colored harmonic confinement can generate spatially compact diquarks as in the Jaffe-Wilczek picture.
• To assess the robustness of the Jaffe-Wilczek mass suppression effect when the idealized delta-function interaction is replaced with a physically realistic smeared version.
• To estimate the mass of the anti-charmed pentaquark within this model and compare it with experimental data and prior predictions.

Proposed method

• Uses a quark potential model with a colored harmonic oscillator potential $ V(q_i q_j) = -a \frac{\vec{\lambda}_i}{2} \cdot \frac{\vec{\lambda}_j}{2} (\vec{r}_i - \vec{r}_j)^2 $ to model quark confinement.
• Introduces a color-spin contact interaction $ V^\sigma(q_i q_j) = -h \frac{\vec{\lambda}_i}{2} \cdot \frac{\vec{\lambda}_j}{2} \vec{S}_i \cdot \vec{S}_j \delta(\vec{r}_{ij}) $ to bind diquarks.
• Applies a spatial smearing to the delta function: $ \delta(\vec{r}) \to \frac{1}{(r_0\sqrt{\pi})^3} \exp(-r^2/r_0^2) $, with $ r_0 \sim 1/(4m_u) $ to $ 1/(6m_u) $, to simulate finite quark size.
• Computes the hyperfine splitting $ E_{\mathrm{hyp}} $ for the nucleon, Jaffe-Wilczek-like state, and a $ \mathbf{210}_{FS} $ state under varying smearing parameters.
• Estimates the anti-charmed pentaquark mass using the $ \Lambda_c - \Lambda $ mass difference to set the charm quark mass.
• Compares the model's predictions with the H1 collaboration's observed anti-charmed pentaquark mass of 3.1 GeV.

Experimental results

Research questions

• RQ1 Can a simple quark potential model with color-spin contact interactions and colored harmonic confinement reproduce the Jaffe-Wilczek pentaquark structure?
• RQ2 Does the schematic approximation, which ignores spatial dependence, accurately describe pentaquarks with P-wave diquark correlations?
• RQ3 How does the mass suppression of the Jaffe-Wilczek-like state change when the delta-function interaction is physically smeared?
• RQ4 What is the predicted mass of the anti-charmed pentaquark in this model, and how does it compare to the H1 observation and Jaffe-Wilczek prediction?

Key findings

• The model successfully forms spatially compact diquarks in the pentaquark, consistent with the Jaffe-Wilczek picture, due to color-spin attraction.
• The schematic approximation fails for P-wave states, as it cannot capture the spatial dependence crucial for accurate hyperfine splitting in such configurations.
• When the delta function is smeared with $ r_0 \sim 1/(4m_u) $ to $ 1/(6m_u) $, the large mass suppression of the Jaffe-Wilczek-like state is significantly reduced.
• After smearing, the Jaffe-Wilczek-like state becomes heavier than the $ \mathbf{210}_{FS} $ state, indicating the extreme mass shift is not robust under physical regularization.
• The model predicts an anti-charmed pentaquark mass of approximately 2.8 GeV, which is within 100 MeV of the Jaffe-Wilczek prediction and close to the DN threshold.
• The state with two scalar diquarks is about 400 MeV heavier than the Jaffe-Wilczek-like state, suggesting the H1 candidate may be an excited state or isovector state.