Shallow $T_{bc}$ states from an EFT analysis of $B^{(*)} \bar D^{(*)}$ scattering on the lattice

TL;DR

The authors develop two complementary EFT frameworks to analyze lattice QCD results for coupled-channel $B^{(*)}\bar{D}^{(*)}$ scattering and to study shallow $T_{bc}$ states. The first approach (EFT1) uses a low-energy, diagonal-contact EFT near the $B\bar{D}$ ($J=0$) and $B^*\bar{D}$ ($J=1$) thresholds constrained by HQSS, while the second (EFT2) treats all relevant momenta as soft and includes both contact interactions and one-pion exchange (OPE), introducing left-hand cuts from two-pion exchange. Fits to lattice finite-volume spectra show consistent near-threshold bound states across both formalisms, with HQSS predicting additional near-threshold partners near $B\bar{D}^*$ and $B^*\bar{D}^*$; OPE has a modest effect on the spectra and ground-state pole positions, though resonance poles are model-dependent. The analyses yield a large compositeness for near-threshold states (indicative of molecular structure) and motivate future lattice studies to test these HQSS-based predictions and to refine the chiral and coupled-channel dynamics.

Abstract

We present an effective field theory (EFT) framework for coupled-channel $B^{(*)}\bar D^{(*)}$ scattering, applying it to recent lattice QCD results by Alexandrou et al. [Phys. Rev. Lett. 132, 151902 (2024)]. Two complementary EFT approaches are developed: (1) A low-energy theory near the $B \bar D$ ($J=0$) and $B^* \bar D$ ($J=1$) thresholds, where coupled-channel effects are integrated out; (2) A coupled-channel formulation, where all relevant momentum scales are treated as soft, incorporating contact interactions and one-pion exchange (OPE). Importantly, OPE contributes to the lowest channels only through off-diagonal transitions, thus resulting in the appearance of the left-hand cut from two-pion exchange. The two approaches yield mutually consistent results, supporting the existence of shallow bound states in both channels, in agreement with the lattice findings. The finite-volume spectra and extracted pole positions show a near-degeneracy in $J=0$ and $J=1$ channels, consistent with heavy-quark spin symmetry (HQSS). Using HQSS, we predict additional shallow bound states near the $B \bar{D}^*$ and $B^* \bar{D}^*$ thresholds, which are accessible to future lattice simulations. The effect of OPE on the finite volume spectra is found to be small, with only moderate impact on HQSS partners.

Figures (8)

• Figure 1: Results of the single-channel EFT (EFT1) for the finite-volume spectra and the infinite-volume S-wave phase shifts $\delta$. Left panel: FV spectra in the $A_1^+[0]$ ($J=0$) and the $T_1^+[0]$ ($J=1$) irreps. Right panel: predictions for the infinite-volume $k \cot \delta$. The three fits discussed in the text---fit$^{(4+0)}$, fit$^{(0+4)}$, and fit$^{(4+4)}$---are presented in panels ($a$1), ($b$1), and ($c$1), respectively. Lattice data points from $A_1^+[0]$ used in the fits are shown in green, those from $T_1^+[0]$ in orange, while data points not included in the fits are indicated in gray. The fit range in each fit is also indicated by a cyan ellipse. The EFT results are shown in red, with error bands indicating the 68% ($1\sigma$) statistical uncertainty. Each panel also reports the $\chi^2/{\rm dof}$ and the fitted values of parameters defined in Eq. \ref{['def_alpha_1']}. The noninteracting energy levels (left panel) are shown as dashed lines: gray for the lowest channel ($B\bar{D}$ or $B^*\bar{D}$, depending on $J$), black for $B\bar{D}^*$, and blue for $B^*\bar{D}^*$. The predicted energy levels are displayed up to approximately 30 MeV above the $B^* \bar{D}^*$ threshold. The predicted FV levels from different two-meson channels are horizontally shifted so that levels from each channel are easily distinguishable. For $A_1^+[0]$ irrep, the $B^* \bar{D}^*$ levels are placed to the right of the $B \bar{D}$ levels. For $T_1^+[0]$, the ordering from left to right is $B^*\bar{D}$, $B\bar{D}^*$ and $B^*\bar{D}^*$. The gray dashed lines in the right panels correspond to $ik = \pm |k|$ from unitarity.
• Figure 2: Residual cutoff dependence of $k \cot \delta$ from fits with one (CT1+0+0, green band) and two (CT2+0+0, red band) contact terms in EFT1. For $k \cot \delta >0$, the upper edge of each band corresponds to the lowest cutoff $\Lambda= 0.8$ GeV and the lower edge to the highest cutoff $\Lambda = 1.2$ GeV; for $k \cot \delta < 0$, the ordering is reversed. The green dashed line corresponds to $ik = \pm |k|$ from unitarity.
• Figure 3: Comparison of the EFT2 results obtained using different interaction terms. Panel (a) shows the fit with two diagonal contact terms (CT2+0+0); panel (b) displays the fit including the OPE interaction together with the off-diagonal term ${\cal C}_f$ (CT2+1+0&OPE); panel (c) shows the fit including the off-diagonal term ${\cal D}_f$ (CT2+0+1) and panel (d) presents the fit with three diagonal contact terms (CT3+0+0), as discussed in text. All results correspond to a cutoff of 1.2 GeV. The color coding is the same as in Fig. \ref{['Fig_EFT1']}. For fits with diagonal interactions, as in Fig. \ref{['Fig_EFT1']}, FV levels from different two-meson channels are horizontally shifted so that levels from each channel are easily distinguishable. See the caption of Fig. \ref{['Fig_EFT1']} for further details.
• Figure 4: Convergence of the results from fits using two (blue) and three (red) diagonal contact interactions. The FV spectra in the $A_1^+(0)$ and $T_1^+(0)$ irreps are shown in the left panel. Right panel shows corresponding $k \cot \delta$ in the infinite volume. Bands indicate 1$\sigma$ statistical errors. The results are shown for a cutoff of 1.2 GeV. Lattice data points from $A_1^+[0]$ used in the fits are shown in green, those from $T_1^+[0]$ in orange, while data points not included in the fits are indicated in gray.
• Figure 5: QQ plot for the fit$^{(4+4)}$ EFT1 calculation (blue dots), as described in the main text. A straight 45$^\circ$ line is shown to guide the eye. The area between the green (orange) lines corresponds to the 68% (95%) confidence interval.