Study of $X(3872)$ from Effective Field Theory with Pion-Exchange Interaction

We study $D{\overline{D}}^{*}$ ($D^{*}\overline{D}$) scattering in the framework of unitarized heavy meson chiral perturbation theory with pion exchange and a contact interaction. ${}^{3}S_1-{}^{3}D_1$ mixing effects are taken into account. A loosely bound state $X(3872)$, with the pole position being $M_{\mathrm{pole}}=(3871.70-i0.39)\mathrm{MeV}$, is found. The result is not sensitive to the strength of the contact interaction. Our calculation provides a theoretical confirmation of the existence of the $1^{++}$ state $X(3872)$. The light quark mass dependence of the pole position indicates it has a predominately $D{\overline{D}}^{*}$ ($D^{*}\overline{D}$) molecular nature. When the $\pi$ mass is larger than 142 MeV, the pole disappears, which makes impossible the lattice simulation of this state at large quark mass.

Figure 1

Tree level Feynman diagrams. (a) is for the contact interaction and (b) is for the one-pion exchange interaction. Double lines indicate the vector $D^{*0}$ or ${\overline{D}}^{*0}$ mesons, solid lines pseudoscalar $D^0$ or ${\overline{D}}^0$ mesons. The dashed line represents the pion.

Figure 2

One loop Feynman diagrams labelled as (a)–(i). The notations are the same as Fig. 1.

Figure 3

Pole trajectory as increasing the strength of contact interaction.

Figure 4

Upper plot: Pion mass dependence of the mass of $X(3872)$ (solid line) and $D^{*}\overline{D}$ threshold (dashed line); Lower plot: Pion mass dependence of ${\Gamma }_{X(3872)}/2$.