Couplings in coupled channels versus wave functions: application to the X(3872) resonance

TL;DR

This paper analytically explores the relationship between couplings and wave functions in coupled channel problems, applying it to the X(3872) resonance to clarify its isospin composition and channel probabilities.

Contribution

It establishes an analytical link between couplings from the scattering matrix and wave functions, specifically applied to the X(3872) resonance, enhancing interpretation of coupled channel models.

Findings

Couplings reflect wave functions at the origin in coordinate space.

X(3872) has nearly equal couplings to D0D*0 and D+D*- channels.

The state exhibits good isospin I=0 despite larger probability in the loosely bound channel.

Abstract

We perform an analytical study of the scattering matrix and bound states in problems with many physical coupled channels. We establish the relationship of the couplings of the states to the different channels, obtained from the residues of the scattering matrix at the poles, with the wave functions for the different channels. The couplings basically reflect the value of the wave functions around the origin in coordinate space. In the concrete case of the X(3872) resonance, understood as a bound state of $\ddn$ and $\ddc$ (and $c.c.$), with the $\ddn$ loosely bound, we find that the couplings to the two channels are essentially equal leading to a state of good isospin I=0 character. This is in spite of having a probability for finding the $\ddn$ state much larger than for $\ddc$ since the loosely bound channel extends further in space. The analytical results, obtained with exact solutions of the Schr\"odinger equation for the wave functions, can be useful in general to interpret results found numerically in the study of problems with unitary coupled channels methods.