$X(3872)$ Relevant $D{\bar{D}}^{\ast }$ Scattering in $N_f=2$ Lattice QCD

Abstract

We study the $S$-wave $D{\bar{D}}^{\ast }(I=0)$ scattering at four different pion masses $m_{\pi }$ ranging from 250 MeV to 417 MeV from $N_f=2$ lattice QCD. Three energy levels $E_{2,3,4}$ are extracted at each $m_{\pi }$. The analysis of $E_{2,3}$ using the effective range expansion (ERE) comes out with a shallow bound state below the $D{\bar{D}}^{\ast }$ threshold, and the phase shifts at $E_{3,4}$ indicate the possible existence of a resonance near 4.0 GeV. We also perform a joint analysis to $E_{2,3,4}$ through the $K$-matrix parameterization of the scattering amplitude. In this way, we observe a $D{\bar{D}}^{\ast }$ bound state whose properties are almost the same as that from the ERE analysis. At each $m_{\pi }$, this joint analysis also results in a resonance pole with a mass slightly above 4.0 GeV and a width around 40-60 MeV, which are compatible with the properties of the newly observed ${\chi }_{c1}(4010)$ by LHCb. More scrutinized lattice QCD calculations are desired to check the existence of this resonance.

Comments: 19 pages, 14 figures. Substantially revised. A section is added to address the joint analysis of the $E_{2,3,4}$ based on the $K$-matrix parameterization of scattering amplitude. A resonance pole is observed with a mass slightly $>4.0$ GeV and width around 40-60 MeV, compatible with the newly observed ${\chi }_{c1}(4010)$ by LHCb. The conclusion on the existence of a bound state does not change