${\overline{B}}^{(*)}{\overline{B}}^{(*)}$ interactions in chiral effective field theory

In this work, the intermeson interactions of double-beauty $\overline{B}\overline{B}$, $\overline{B}{\overline{B}}^{*}$, and ${\overline{B}}^{*}{\overline{B}}^{*}$ systems have been studied with heavy meson chiral effective field theory. The effective potentials are calculated with Weinberg’s scheme up to one-loop level. At the leading order, four-body contact interactions and one-pion exchange contributions are considered. In addition to two-pion exchange diagrams, we include the one-loop chiral corrections to contact terms and one-pion exchange diagrams at the next-to-leading order. The behaviors of effective potentials, both in momentum space and coordinate space, are investigated and discussed extensively. We notice the contact terms play important roles in determining the characteristics of the total potentials. Only the potentials in $I(J^P)=0(1^{+})$ $\overline{B}{\overline{B}}^{*}$ and ${\overline{B}}^{*}{\overline{B}}^{*}$ systems are attractive, and the corresponding binding energies in these two channels are solved to be $\mathrm{\Delta }{E}_{\overline{B}{\overline{B}}^{*}}\simeq -{12.6}_{-12.9}^{+9.2}\mathrm{MeV}$ and $\mathrm{\Delta }{E}_{{\overline{B}}^{*}{\overline{B}}^{*}}\simeq -{23.8}_{-21.5}^{+16.3}\mathrm{MeV}$, respectively. The masses of $0(1^{+})$ $\overline{B}{\overline{B}}^{*}$ and ${\overline{B}}^{*}{\overline{B}}^{*}$ states lie above the threshold of their electromagnetic decay modes $\overline{B}\overline{B}\gamma$ and $\overline{B}\overline{B}\gamma \gamma$, and thus they can be reconstructed via electromagnetic interactions. Our calculation not only provides some useful information to explore exotic doubly bottomed molecular states for future experiments, but also is helpful for the extrapolations of Lattice QCD simulations.

Figure 1

Two types of 2PR Feynman diagrams of $N$-$N$ scattering, which will also appear in this work.

Figure 2

One-loop corrections to the contact vertex at $\mathcal{O}({\epsilon }^2)$. The thin solid, thick solid, and dashed lines represent $\overline{B}$ meson, ${\overline{B}}^{*}$ meson, and pion, respectively.

Figure 3

Two pion exchange diagrams at $\mathcal{O}({\epsilon }^2)$. There is one football diagram ($F_{1.1}$), two triangle diagrams ($T_{1.1}$ and $T_{1.2}$), one box diagram ($B_{1.1}$), and one crossed diagram ($R_{1.1}$). Notations same as in Fig. 2.

Figure 4

(a) Represents the TPE potential of $1(0^{+})$ $\overline{B}\overline{B}$ system in momentum space. The potentials in coordinate space are shown in (b), where the dot-dashed line describes the TPE potential, the dotted line denotes the sum of FBCI potentials at $\mathcal{O}({\epsilon }^0)$ and $\mathcal{O}({\epsilon }^2)$, and the solid line stands for the total potential.

Figure 5

The diagrams for $\overline{B}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^0)$, where $X_{2.1}$ depicts the contact interaction, and $H_{2.1}$ is OPE diagram. Notations same as in Fig. 2.

Figure 6

One-loop corrections to the FBCI of $\overline{B}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$, which include the corrections to the $\overline{B}{\overline{B}}^{*}\overline{B}{\overline{B}}^{*}$ contact vertex ($g_{2.1}\sim h_{2.4}$), and wave function renormalizations of external legs ($z_{2.1}\sim z_{2.3}$). Notations same as in Fig. 2.

Figure 7

One-loop corrections to the OPE diagram of $\overline{B}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$, which include the renormalizations of pion line ($p_{2.1}$), $\overline{B}{\overline{B}}^{*}\pi$ vertex ($c_{2.1}\sim c_{2.6}$, $c_{2.10}\sim c_{2.11}$), and external legs ($c_{2.7}\sim c_{2.9}$). Notations same as in Fig. 2.

Figure 8

Two-pion exchange diagrams of $\overline{B}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$. Notations same as in Fig. 2.

Figure 9

(a) Represents the OPE and TPE potentials of $1(1^{+})$ $\overline{B}{\overline{B}}^{*}$ system in momentum space, where the dashed line denotes the OPE potential at $\mathcal{O}({\epsilon }^0)$, the dotted line is the OPE potential at $\mathcal{O}({\epsilon }^2)$, the dot-dashed line describes the TPE potential, and the sum of these three parts are depicted by the solid line. (b) shows the potentials in coordinate space, where the dashed line stands for the sum of OPE potentials at $\mathcal{O}({\epsilon }^0)$ and $\mathcal{O}({\epsilon }^2)$, and other basic notations are the same as in Fig. 4.

Figure 10

The OPE and TPE potentials of $0(1^{+})$ $\overline{B}{\overline{B}}^{*}$ system in momentum space (a) and the effective potentials in coordinate space (b). Notations same as in Fig. 9.

Figure 11

The dependence of the total potentials for the $0(1^{+})$ $\overline{B}{\overline{B}}^{*}$ system on cutoff parameter $\mathrm{\Lambda }$, where three different results with $\mathrm{\Lambda }=0.6$, 0.7, and 0.77 GeV are illustrated.

Figure 12

The diagrams for ${\overline{B}}^{*}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^0)$. Notations same as in Fig. 2.

Figure 13

One-loop corrections to the FBCI of ${\overline{B}}^{*}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$. Notations same as in Fig. 2.

Figure 14

One-loop corrections to the OPE diagram of ${\overline{B}}^{*}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$. Notations same as in Fig. 2.

Figure 15

Two-pion exchange diagrams of ${\overline{B}}^{*}{\overline{B}}^{*}$ system at $\mathcal{O}({\epsilon }^2)$. Notations same as in Fig. 2.

Figure 16

The OPE and TPE potentials of $1(0^{+})$ ${\overline{B}}^{*}{\overline{B}}^{*}$ system in momentum space (a) and the effective potentials in coordinate space (b). Notations same as in Fig. 9.

Figure 17

The OPE and TPE potentials of $1(2^{+})$ ${\overline{B}}^{*}{\overline{B}}^{*}$ system in momentum space (a) and the effective potentials in coordinate space (b). Notations same as in Fig. 9.

Figure 18

The OPE and TPE potentials of $0(1^{+})$ ${\overline{B}}^{*}{\overline{B}}^{*}$ system in momentum space (a) and the effective potentials in coordinate space (b). Notations same as in Fig. 9.

Figure 19

The effective potentials of $0(1^{+})$ $\overline{B}{\overline{B}}^{*}$ (a) and ${\overline{B}}^{*}{\overline{B}}^{*}$ (b) systems in strict heavy quark limit. Notations same as in Fig. 9.