Partial widths from analytical extension of the wave function: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>P</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math> states
Zi-Yang Lin, Jianbo Cheng, Bo-Lin Huang, Shi-Lin Zhu
Abstract
A new approach to partial widths is proposed through the analytical extension of the wave function in momentum space. By including the residue of the wave function, the Schr\"odinger equation is extended to the second Riemann sheet. As a result, the partial width is associated with the pole of the wave function. The resonance wave function is convergent in momentum space and can be used to evaluate other observables. This approach is applied to a coupled-channel analysis for ${P}_{c}$ states, involving the contact interactions and one-pion-exchange potential with the three-body effects. Under the reasonable assumption that the off-diagonal contact interactions are small, the ${J}^{P}$ quantum numbers of the ${P}_{c}(4440)$ and the ${P}_{c}(4457)$ are ${\frac{1}{2}}^{\ensuremath{-}}$ and ${\frac{3}{2}}^{\ensuremath{-}}$ respectively. The low energy constants are fitted using the experimental masses and widths as input. The ${P}_{c}(4312)$ is found to decay mainly to ${\mathrm{\ensuremath{\Lambda}}}_{c}{\overline{D}}^{*}$, while the branching ratios of the ${P}_{c}(4440)$ and ${P}_{c}(4457)$ in different channels are comparable. Three additional ${P}_{c}$ states at 4380 MeV, 4504 MeV and 4516 MeV, together with their branching ratios, are predicted. Additionally, a deduction for the revised one-pion-exchange potential involving the on-shell three-body intermediate states is provided.