Traces of the $X(3960)$ state in the femtoscopic $D_s^+ D_s^- $ correlations

TL;DR

The paper tackles the ambiguity surrounding the near-threshold X(3960) state observed by LHCb in the $D_s^+ D_s^-$ channel. It develops a Bethe-Salpeter framework with an $S$-wave potential to model strong $D_s^+ D_s^-$ interactions and explores three scenarios—resonant, virtual, and bound—analyzing pole positions, $a_0$, and $r_0$. Although all three interpretations can reasonably fit the invariant mass spectrum near threshold, the study shows that femtoscopic correlations, when Coulomb final-state interactions are included, reveal distinct low-momentum patterns: virtual-state configurations produce large low-$k$ enhancements, bound states lead to suppressions, and resonances give moderate enhancements, with the differences being most pronounced for small source sizes. This suggests that measurements of $D_s^+ D_s^-$ CF in high-multiplicity collisions across $pp$, $pA$, and $AA$ systems can decisively discriminate between the three scenarios and clarify the nature of the X(3960) state.

Abstract

The femtoscopic $ D_s^+D_s^-$ correlations are investigated to predict the signature of the not-yet-established $X(3960)$ state reported by the LHCb Collaboration, in three scenarios: resonant, virtual, or bound. In the last two scenarios, it might also be identified as the state $X(3930)$. The formalism employed to generate this structure dynamically is based on the Bethe-Salpeter equation with a general $S$-wave potential. We investigate how the relevant properties and observables characterizing this state--such as the pole position, scattering length, and effective range--might be affected by variations in the model parameters. The amplitudes encoding the distinct interpretations of the $X(3960)$ state are then used as input to calculate the femtoscopic correlation function of the $D_s^+ D_s^- $ pair, which is analyzed and discussed.

Figures (5)

• Figure 1: Mechanisms contributing to the $B^+ \to D_{s}^+ D_{s}^- K^+$ reaction. (a) Tree-level contribution. (b) Rescattering contribution.
• Figure 2: Differential distributions of the $B^+ \to D_{s}^+ D_{s}^- K^+$ decay taking \ref{['fig_mass_distr-a']}$q_{\text{max}} = 1.0$ GeV and \ref{['fig_mass_distr-b']}$q_{\text{max}} = 0.5$ GeV. The shaded areas represent the uncertainties in the LECs displayed in Table \ref{['table_scatprob']} for the different scenarios. The experimental data are taken from Ref. LHCb:2022aki. The first vertical gray line corresponds to the $D_{s}^+ D_{s}^-$ threshold.
• Figure 3: \ref{['fig_squared_ampl1']}: modulus square of the amplitude of the $D_s^+ D_s^- \rightarrow D_s^+ D_s^-$ channel as a function of the CM energy. \ref{['fig_rho_squared_ampl1']}: modulus square of the amplitude times the phase-space factor of the $D_s^+ D_s^- \rightarrow D_s^+ D_s^-$ channel as a function of the CM energy. The shaded areas represent the uncertainties in the LECs displayed in Table \ref{['table_scatprob']} for the different scenarios.
• Figure 4: \ref{['fig_CF_strong_1fm_0.5GeV']} - \ref{['fig_CF_strong_2fm_1GeV']}: the pure strong contribution of the $D_s^+ D_s^-$ CF as a function of the CM relative momentum $k$, taking different values of the size parameter $R$ in three scenarios. The shaded areas represent the uncertainties in the LECs displayed in Table \ref{['table_scatprob']} for the different scenarios.
• Figure 5: \ref{['fig_CF_total_1fm_0.5GeV']} - \ref{['fig_CF_total_2fm_1GeV']}: the total $D_s^+ D_s^-$ CF as a function of the CM relative momentum $k$, taking different values of the size parameter $R$ in three scenarios. The shaded areas represent the uncertainties in the LECs displayed in Table \ref{['table_scatprob']} for the different scenarios.