We start by introducing the strong contribution to the Ds+Ds- amplitude, obtained via the Bethe-Salpeter formalism and used as input to the Ds+Ds- CF. Here we benefit from the analyses done in Refs. [11,37] exploring distinct configurations for exotic states (see also Refs. [45,46]). In particular, Ref. [11] employed a contact potential from a nonrelativistic effective field theory considering heavy quark spin and light-flavor SU(3) symmetries. The low-energy constants (LECs) have been determined by fitting the line shapes of the Ds+Ds- invariant-mass distribution of the B+→Ds+Ds-K+ reaction to the LHCb data reported in Ref. [1]. The conclusion of Ref. [11] was that the X(3960) state might be compatible with either a bound or virtual state in the Ds+Ds- channel. In turn, in the study of the possible femtoscopic signatures of the Zc(3900)/Zcs(3985) states, Ref. [37] used a general S-wave potential up to the next-to-leading order in momentum expansion for the D0D*-/D0Ds*- systems. Accordingly, the LECs have been determined by exploring the scenarios of bound, virtual, and resonant states. In the latter case, the poles reported by the Particle Data Group have been employed, and the former two have been fine-tuned to obtain line shapes similar to those of the resonant state. For the determination of the strong Ds+Ds- amplitude, we adopt the potential and fit the LECs similar to the framework of [37].

We define the S-wave projected strong potential for the process Ds+Ds-→Ds+Ds- as¹

Note that the Coulomb interaction is neglected here and will only be considered in predicting the corresponding femtoscopy correlation function in Sec. III.

The unitarized scattering amplitude matrix of the strong interaction T(S) is then obtained by solving the Bethe-Salpeter equation, T(S)=[1-V(S)G]-1V(S),(4)where G is the loop function for two intermediate mesons in a given channel, which in the cutoff regularization scheme can be written as G(s)=∫0qmaxq2dq2π214ωDs2(q)1s-2ωDs(q)+iϵ,(5)with ωDs(q)=q2+mDs2, and qmax being the cutoff chosen within the range 0.5–1.0 GeV.

We identify virtual, bound, and resonant states through poles of the T(S)-matrix in the Riemann sheets of the s plane [37]. Virtual and bound states correspond to poles on the real axis below threshold in the second and first Riemann sheets, respectively; while resonances appear as complex poles in the second Riemann sheet, with their mass and half-width being given by the real and imaginary parts of the pole position. The second Riemann sheet loop function G(II) is obtained via analytic continuation: G(II)=G+2iρ, where ρ=k/(8πs) is the phase-space factor.

We also calculate two key low-energy scattering observables defined by the effective range expansion (ERE) [37,45–48]: the scattering length a0 and the effective range r0. Using the relation between the T(S)-matrix and phase shift δ0, T(S)∝(kcotδ0-ik)-1, the expansion can be written as -8πsT(S)-1+ik=-1a0+12r0k2+O(k4).(6)From this relation, a0 and r0 are given by a0=T(S)8πs|s=sthr,r0=∂2∂k2(-8πsT(S)-1+ik)|s=sthr.(7)Here, sthr is the squared CM energy at the channel’s threshold. We note that the +ik term in Eqs. (6) and (7) is essential for satisfying the unitarity condition required by the ERE. It ensures that any linear-k imaginary contributions arising from the loop function G cancel in the expansion, leaving a real result. When the derivative is taken at threshold, as in the definition (7) of r0, the loop function G and its derivative are evaluated on the physical sheet. For bound states, this procedure is straightforward. For virtual states and resonances, although the pole resides on the unphysical sheet, the calculation of r0 remains on the physical sheet at threshold, where G is real because the phase-space factor vanishes. Therefore, within our single-channel formalism, the effective range is real.

To determine the LECs, we employ the Ds+Ds- invariant-mass distribution of the B+→Ds+Ds-K+ reaction, which is given by [10,11] dΓdMinv=1(2π)314mB2pK+p˜Ds|T˜|2,(8)where Minv is the invariant mass of the Ds+Ds- system; pK+=λ1/2(mB+2,mK+2,Minv2)/2mB+ and p˜Ds=λ1/2(Minv2,mDs+2,mDs-2)/2Minv are, respectively, the K+ and Ds momenta; and T˜ is the amplitude for the B+→Ds+Ds-K+ transition.

The analytical expression for T˜ can be obtained as follows [10,11]. At the quark level, these B-decays proceed via a Cabibbo-suppressed internal W-emission: b¯(→c¯W+→c¯cs¯)u→c¯c(s¯u). After hadronization of the cc¯ pair, only the Ds+Ds- channel remains. At the hadron level, the production mechanisms considered for B+→Ds+Ds-K+ are the tree-level and one-loop contributions, as depicted in Fig. 1. Consequently, the effects of the state under investigation are produced through the interaction of the intermediate Ds+Ds- pair in the rescattering contribution.

So, the amplitude for the diagrams in Fig. 1 is given by T˜(Minv)=C[1+G(Minv)T(S)(Minv)],(9)where C is an overall constant encoding effectively the information of the weak vertex, G(Minv) is the two-meson loop function given in Eq. (5), and T(S)(Minv) is the unitarized scattering amplitude matrix given in Eq. (4).

Using Eqs. (9) and (8), we fit the LECs a and b of the potential V(S) to reproduce the LHCb data [1] via the least-squares minimization. In the bound- and virtual-state scenarios, the fit considers the first ten data points in the CM energy region from 3.92 to 4.12 GeV (data before the third vertical gray line), as in Ref. [11]. In the resonant-state scenario, only the first four data points (those before the second vertical gray line) are used. These restricted ranges are chosen because our model focuses on the region near the Ds+Ds- threshold and excludes other irrelevant effects or states at higher energies. In the resonant-state scenario, the invariant-mass spectrum is expected to feature a narrow, Breit-Wigner-like peak (as shown in Fig. 2 of Ref. [1]), whose essential characteristics are encapsulated within the first four data points. Fitting exclusively to these points thus results in a small value of χ2 per degree of freedom (χ2/d.o.f.) value, which we consider sufficient for a qualitative discussion of the resonant interpretation.

Figure 2 shows the fitted differential mass distributions for a cutoff parameter qmax=1.0  GeV in the bound, virtual, and resonant scenarios. All three cases produce line shapes compatible with the data, with those for bound and virtual states being practically indistinguishable. Table I provides the fitted values of a and b, along with the resulting pole positions, scattering lengths a0, and effective ranges r0 in the three scenarios. For completeness, we also include results for qmax=0.5  GeV.

The pole positions for the virtual and bound state interpretations are consistent with those obtained in Refs. [3,11] within uncertainties, which are associated with the X(3930) state. The resonant-state scenario yields a pole position consistent with that reported by the LHCb Collaboration [1], where a new state, referred to as X(3960), is claimed.

We emphasize that the parameter C in Eq. (9) only appears in the invariant-mass distribution and does not affect the scattering amplitude or correlation function. The difference between bound- and virtual-state scenarios in the error band widths in Fig. 2 arises not only from the uncertainty of a listed in Table I, but also from that of C. As shown in Table I, the relative uncertainty of C is significantly larger for the bound-state scenario than for the virtual-state scenario, leading to a broader uncertainty band for the bound-state case in Fig. 2.

This also explains why, in Figs. 2(a) and 2(b), the uncertainty bands for the virtual-state scenario are comparable, while the bound-state band is wider in Fig. 2(b) than in Fig. 2(a). For the virtual-state scenario, the relative uncertainty of C is similar at qmax=0.5 and 1 GeV. For the bound-state scenario, it is notably larger at 0.5 GeV, resulting in the broader band in Fig. 2(b).

In other figures (e.g., Figs. 3–5), only the uncertainty of a (and b for the resonant-state scenario) is propagated, as C does not enter the T-matrix or correlation function. Hence, the uncertainty bands in those figures are comparable between bound- and virtual-state scenarios.

For completeness, Fig. 3 shows the squared amplitude and the squared amplitude times the phase-space factor for the Ds+Ds-→Ds+Ds- channel as a function of the CM energy, for a cutoff parameter qmax=1.0  GeV. The inherent kinematic suppression due to the phase-space factor in the near-threshold region makes it difficult to distinguish among the different configurations in the invariant-mass spectrum.

It is most important to emphasize that all three scenarios can fairly reproduce the low-energy experimental data within current uncertainties. This demonstrates that the invariant-mass distribution alone is insufficient to distinguish between these interpretations. Additional observables are therefore necessary to discriminate among the different configurations and clarify this ambiguity.

We now examine how these scenarios manifest in femtoscopic correlation functions.

The CF for two-particle systems is defined as the ratio of the probability of measuring the two-particle state to the product of the probabilities of measuring each particle [49]. According to the Koonin-Pratt approach, after some manipulations, the CF can be given by the following formula [28,50,51]: C(k)=∫d3rS12(r)|Ψ(k;r)|2,(10)where k is the relative momentum in the CM frame of the pair; r is the relative distance between the two particles; Ψ(k;r) is the relative two-particle wave function carrying information of final-state interactions; and S12(r) is the source function encoding the particle-emitting source.

In the present work, we adopt the source function in its usual static Gaussian profile normalized to unity, i.e., [28,29] S12(r)=1(4π)32R3exp(-r24R2),(11)where R is the source size, treated as a free parameter which is chosen in such a way to reproduce the characteristic scale of the particle-emitting source in different collision systems: smaller values (e.g., R∝1  fm) correspond to high-multiplicity pp collisions, while larger values (up to several femtometers) are typical for central AA collisions. We remark that the choice of a Gaussian source is motivated by its simplicity and by the expectation that feed-down contributions from strong decays are minimal for charm mesons, making it a reasonable approximation in this exploratory study. In future experimental studies, the source size can be anchored to proton-proton correlation data by measuring the mT of the Ds+Ds- pair, thereby reducing systematic uncertainties.

A realistic treatment of the |Ds+Ds-⟩ system must include the Coulomb interaction due to its significant role [52]. We therefore employ a relative wave function that incorporates both the S-wave strong interaction and the Coulomb potential [53], Ψ(k;r)=Φ(C)(k;r)-Φ0(C)(k,r)+ψ0(k,r),(12)where Φ(C)(k;r) is the complete Coulomb wave function [54], Φ0(C)(k,r) is its S-wave component, and ψ0(k,r) contains the full strong interaction in the presence of the Coulomb potential.

Using the wave function in Eq. (12) and assuming a spherically symmetric source function from Eq. (11), the Koonin-Pratt formula in Eq. (10) becomes [33,39,45,47,52,55,56] C(k)=∫d3rS12(r)|Φ(C)(k;r)|2+∫0∞4πr2drS12(r)(|ψ0(k,r)|2-|Φ0(C)(k,r)|2).(13)

Following the notation of Refs. [19,57], the complete Coulomb wave function is given by [54] Φ(C)(k;r,z)=e-πγ/2Γ(1+iγ)eikzF11(-iγ;1;ik(r-z)).(14)Here, Γ(z) is the Euler gamma function, F11(a;b;z) is the confluent hypergeometric function (Kummer’s function), and γ is the Sommerfeld parameter, γ=Z1Z2μαk,(15)with Z1Z2 being the product of the charges, α is the fine-structure constant, and μ is the reduced mass of the hadron pair.

The S-wave function ψ0(k,r) is obtained from the Lippmann-Schwinger equation |ψ⟩=|ϕ⟩+GT|ϕ⟩, where |ϕ⟩ is the free wave function [34,53,57], ψ0(k,r)=j0(kr)+∫0qmaxq2dq2π214ωDs2(q)T(k,q;s)j0(qr)s-2ωDs(q)+iϵ.(16)Here, jν(x) is the spherical Bessel function of order ν, and T(k,q;s) is the S-wave scattering amplitude for the charged meson pair including both strong and Coulomb contributions.

Ideally, the amplitude T(k,q;s) should be obtained by simultaneously unitarizing the combined strong and Coulomb amplitudes. Since the Coulomb contribution depends on the meson momenta in both initial and final states, this requires solving the full off-shell Bethe-Salpeter equation. However, following the analysis in Ref. [53], the momentum integral in Eq. (16) is calculated with the Coulomb amplitude up to first order in the Bethe-Salpeter equation combined with the on-shell strong amplitude studied in the previous section. Given the exploratory nature of this study, this approximation has the practical advantage of leaving the fitted strong amplitude unchanged.

Explicitly, the full scattering amplitude is decomposed as T(p,p′;s)=T(S)(s)+T(C)(p,p′;s),(17)where T(S)(s) is the unitarized strong amplitude from Eq. (4) and T(C)(p,p′;s) is the Coulomb amplitude in the Born approximation, T(C)(p,p′;s)=V0(C,rel)(p,p′;s).(18)

To obtain the Coulomb contribution V0(C,rel)(p,p′;s), we begin with the Fourier transform of the potential V(C)=εα/r, where ε=+1(-1) for identical (opposite) charged particles, into momentum space, Vtotal(C)(|p′-p|)=∫0RCd3rei(p′-p)·rεαr=4πεα|p′-p|2(1-cos(|p′-p|RC)),(19)where only the short-range Coulomb interaction (r<RC) is considered to avoid divergences in the integrals. The long-range behavior will be accounted for through the asymptotic wave functions.

The S-wave component of this potential is obtained by projecting it onto the partial wave, V0(C)(p,p′)=12∫-11d(cosθApp′)Vtotal(C)(|p′-p|)(20)=2πεαpp′{Ci[|p′-p|RC]-Ci[|p′+p|RC]+ln(p′+p|p′-p|)},(21)where Ci[x]=∫x∞dt(cost)/t is the cosine integral function. We have tested various values of RC and find that our results stabilize at RC=4  fm, which we therefore adopt in all subsequent calculations.

To consistently incorporate the Coulomb interaction within the relativistic Bethe-Salpeter equation, the S-wave Coulomb potential must be modified with relativistic kinematic factors [53,57], V0(C,rel)(p,p′;s)=2ωDs(p)ξ(p;s)V0(C)(p,p′)⁢ξ(p′;s)2ωDs(p′),(22)where the kinematic factors ξ(p;s) are given by ξ(p;s)=2μs-2ωDs(p)λ(s,m12,m22)4s-p2,(23)with μ=mDs/2 being the reduced mass.

Figure 4 shows the pure strong Ds+Ds- CF as a function of the CM relative momentum k for different values of the parameter R in three scenarios. This contribution is calculated by replacing the Φ(C) and Φ0(C) with eik·r and j0 in Eq. (13) and using the scattering amplitude T(k,q;s)=T(S)(s) in Eq. (16), which contains only the unitarized strong interaction. In the low-momentum region, the pure strong Ds+Ds- CF exhibits distinct and typical behavior for each configuration. Specifically, compared to unity, the virtual- and bound-state scenarios produce, respectively, enhancements and suppressions in the low-momentum correlations, while the resonant configuration yields a moderate enhancement. This distinction is most pronounced for smaller values of the source size parameter R and becomes less significant as R increases. These findings are qualitatively consistent with those obtained in Ref. [37] in investigating femtoscopic signatures for the Zc(3900)/Zcs(3985) states in the D0D*-/D0Ds*- systems. However, a difference can be noted in the resonant interpretation. Because of the larger width given in Table I, the Ds+Ds- CF in Fig. 4 does not exhibit a pronounced dip in the intermediate-momentum region. Specifically, in the momentum region slightly above the resonance pole, i.e., kR≳160  MeV, the CF has an almost plateaulike appearance, unlike the cases of the D0D*-/D0Ds*- systems studied in Ref. [37].

It is worth noting that although the CF of a bound state exhibits a “suppression” in the low-momentum region—similar to the behavior observed in systems with repulsive interactions—the underlying physical mechanisms differ significantly. In the bound-state scenario, pairs that form a bound state are lost to the correlation yield, leading to a suppression in the low-momentum region of the CF. In contrast, for the repulsive interaction case, the repulsive force accelerates the particles apart. This increases their relative momentum and similarly suppresses it at low momentum in the CF.

The complete Ds+Ds- CF, incorporating both the strong and Coulomb interactions, is presented in Fig. 5 as a function of the relative momentum k for different source sizes R and for each of the three scenarios. A significant enhancement is observed at the Ds+Ds- threshold in all cases. This effect has been established in various experimental systems, including p-Ω [58,59] and p-Ξ [59–61] correlations, where the Coulomb interaction plays a significant role at small relative momenta. Our study demonstrates that the Ds+Ds- pair is similarly governed by the Coulomb attraction in the low-k region. Notwithstanding, the low-momentum region retains a distinct and characteristic behavior across interpretations. For small source sizes, the differences are most pronounced: the virtual-state scenario produces a strong enhancement, the bound-state scenario leads to a clear suppression, and the resonant configuration yields a moderate augmentation of the CF, relative to the pure Coulomb CF. This closeness of the resonant CF to the pure Coulomb curve can be understood as follows. While the strong interaction is indeed crucial for forming a resonance, particularly near the resonance energy, its influence near threshold and at low relative momenta is primarily characterized by the scattering length. As shown in Table I, the scattering length for the resonant case has the smallest magnitude (-0.48 in qmax=0.5  GeV and -0.63 in qmax=1  GeV) among the three scenarios, indicating a relatively weak attractive strong interaction. Consequently, the CF for the resonant case remains close to the pure Coulomb result. This interpretation is further supported by Fig. 4, where the resonant CF near threshold is close to unity, reflecting the minimal deviation introduced by the relatively weak strong interaction.

At a relative momentum of k≈100  MeV, the CF exhibits significant deviations from unity: an enhancement of ∼75% for the virtual state, a smaller enhancement of ∼15% for the resonance, and a suppression of ∼40% for the bound state. These pronounced deviations demonstrate the combined effects of Coulomb and strong interactions. Our results indicate that these distinct patterns persist for smaller values of R, corresponding to smaller collision systems such as pp and pA with light nuclei. Furthermore, they exhibit less sensitivity to the choice of qmax, as evidenced in Figs. 4 and 5 for values of 0.5 and 1.0 GeV. This behavior underscores the importance of measuring the Ds+Ds- CF in small collision systems, where the contrast between the interpretations is most pronounced.

Hence, the use of femtoscopy to distinguish between a bound, virtual, and resonant state in the Ds+Ds- system constitutes the central result of this work, providing a valuable tool to clarify the nature of the X(3960) state.