$T_{cc}^+$ and $X(3872)$ with the complex scaling method and $DD(\bar{D})\pi$ three-body effect

TL;DR

This paper investigates the properties of the $T_{cc}^+$ and $X(3872)$ states using the complex scaling method, emphasizing the importance of three-body effects and energy-dependent potentials to accurately determine their widths and isospin breaking effects.

Contribution

The study introduces a complex scaling approach with energy-dependent potentials to analyze double-charm states, providing new insights into their widths and isospin effects.

Findings

The $T_{cc}^+$ has a width of about 80 keV, nearly cutoff independent.

The $X(3872)$ has an even smaller width.

Isospin breaking effects are significant for $X(3872)$ but small for $T_{cc}^+$.

Abstract

We use the leading order (LO) contact interactions and OPE potentials to investigate the newly observed double-charm state $T_{cc}^+$. The $DD\pi$ three-body effect is important in this system since the intermediate states can go on shell. We keep the dependence of the pion propagators on the center-of-mass energy, which results in a unitary cut of the OPE potential at the $DD\pi$ three-body threshold. By solving the complex scaled Schr\"odinger equation, we find a pole corresponding to the $T_{cc}^+$ on the physical Riemann sheet. Its width is around 80 keV and nearly independent of the choice of the cutoff. Assuming the $D\bar{D}\pi$ and $D\bar{D}^*$ channels as the main decay channels, we apply the similar calculations to the $X(3872)$, and find its width is even smaller. Besides, the isospin breaking effect is significant for the $X(3872)$ while its impact on the $T_{cc}^+$ is relatively small.