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For a long time, we have been asking the question, "What type of matter can be formed by quark models?" The traditional quark model successfully explains that baryons are complexes of three quarks and mesons are combinations of quarks and antiquarks. With the advancements in experimental methods, the recent discovery of candidates for the pentaquark and tetraquark states in experiments has expanded the scope of our study of traditional hadrons, which include qualitatively different
qqqˉqˉq andqqˉqˉq . In addition, more exotic structures have been observed in experiments; see Refs. [1−8]. To answer the appeal question, we must determine whether the pentaquark and tetraquark states exist.In determining the strange state of quantum chromodynamics (QCD), the decay process of B mesons will be an important and effective platform. In this process, many candidates for strange hadron states can be observed. Over the past few years, major laboratories have successively discovered candidates for strange hadron states in the decay of B mesons, such as
Zcs(4000) andZcs(4000) [9],X(4140) [10, 11] inB+→J/ψϕK+ , andX0(2900) andX1(2900) inB+→D+D−K+ decay [12, 13]. Referring to these experiments, we can observe that the three-body decay of B mesons can provide much information on hadron resonance; see Refs. [14−17].Very recently, the LHCb Collaboration reported a new near-threshold structure named
X(3960) in theD+sD−s invariant mass distribution of the decayB+→D+sD−sK+ . The peak structure is very close to theD+sD−s threshold with a statistical significance larger than 12σ. The mass, width, and quantum numbers of this structure were measured to beM=3956±5±10 MeV,Γ=43±13±8 MeV, andJPC=0++ . The LHCb analysis indicates that this structure is an exotic candidate consisting ofcsˉcˉs constituents. In addition, when checking the data of theD+sD−s invariant mass distribution, a dip is observed at approximately4.14 GeV; the LHCb interpreted it as another structure namedX0(4140) with a mass ofM=4133±6±6 MeV, width ofΓ=67±17±7 MeV, and quantum numbers ofJPC=0++ [18]. As analyzed by the LHCb Collaboration,X0(4140) might be caused by either a new resonance with the0++ assignment or aD+sD−sJ/ψϕ coupled-channel effect, but no firm conclusion has been reached [18].Many theoretical studies have shown much interest in X resonances. In recent years, many studies have used different models and technical methods to analyze the characteristics of exotic mesons
csˉcˉs [19−26]. To determine the origin and structure ofX(3960) in decayB+→D+sD−sK+ , scholars have proposed many explanations for the possibility of this structure. Because its mass is close to theD+sD−s threshold, this structure can be interpreted as a possible hadronic molecule. Refs. [27, 28] proposed to treatX(3960) as the molecular state ofD+sD−s withJPC=0++ in the QCD sum rules approach. Another calculation with QCD two-point sum rules [29] results in the assignment thatX(3960) is a scalar diquark-antidiquark state. The calculations in the one-boson-exchange model [30] also favor the molecule interpretation. It can also be analyzed through the characteristics ofX(3960) using the coupled-channel method. The authors of Ref. [31] performed a coupled-channel calcuation of the interactionDˉD−D+sD−s in the chiral unitary approach and interpretedX(3960) as a hadronic molecule in the coupledDˉD−D+sD−s system [31−33]. The author of Ref. [34] interpretedX(3960) as acsˉcˉs state, whereas in Ref. [35],X(3960) was interpreted as0++ csˉcˉs tetraquark states using an improved chromomagnetic interaction model. In addition, another study suggested thatX(3960) probably has the mixed characteristics of acˉc confining state andDsˉDs continuum [36]. Some theoretical and experimental research has been conducted onX0(4140) , but its origins are still debated. For instance, in Ref. [35],X0(4140) was also interpreted ascsˉcˉs tetraquak states. The discussion about mass and width in Ref. [29] enabled us to consider that the model is also acceptable. Because different computational models suggest different explanations, forming with new concepts and insights into this state can aid us in further understanding the origin ofX0(4140) .In this study, we analyzed the decay process
B+→D+sD−sK+ as published by the LHCb Collaboration. We simulated a coupled-channel model to analyze the data [37] using the default model and fitting theMD+sD−s ,MD+sK+ , andMD−sK+ of these three different invariant mass distributions. Using the amplitude provided by the coupled channel model, we address the following problems: (i) the pole position ofX(3960) andX0(4140) and (ii) whether the production ofX(3960) is solely due to a kinematic effect. -
The LHCb data reveal visible
X(3960) andX0(4140) structures around theDsˉDs andD∗sˉD∗s thresholds, respectively. Thus, we can reasonably assume that the structures are caused by the threshold cusps that are further enhanced or suppressed by hadronic rescatterings and the associated poles [37, 38]; see Fig. 1(a). Meanwhile, for the two peaks at 4260 and 4660 MeV, we refer to the suggestions given by the LHCb Collaboration and add two Breit-Weigner effects, as shown in Fig. 1(c). We assume that other possible mechanisms are absorbed by the direct decay mechanism in Fig. 1(b).Figure 1. Contributions of three mechanisms in the decay
B+→D+sD−sK+ . (a) Coupled-channel; (b) Direct production; (c) Breit-Weigner effects.First, we present the amplitude for Fig. 1(a). The first vertex
v1 is a weak interaction, and the initial weakB+→DsˉDsK+ vertex isv1=cα,B+K+f0DsˉDsF0K+B+.
(1) For the vertex of process
B+→D∗sˉD∗sK+ , there are two cases of parity conservation and parity violation. For the former, the vertex ofB+(0−)→D∗sˉD∗s(0+)K+(0−) isvpc1=cD∗sˉD∗s,B+K+→ϵD∗s⋅→ϵˉD∗sf0D∗sˉD∗sF0K+B+,
(2) in the latter case, the vertex of
B+(0−)→D∗sˉD∗s(1+)K+(0−) isvpv1=cD∗sˉD∗s,B+K+→pK+⋅(→ϵD∗s×→ϵˉD∗s)f0D∗sˉD∗sF0K+B+.
(3) The energy, momentum, and polarization vector of a particle x are denoted by
Ex ,px , andϵx , respectively, and particle masses are obtained from Ref. [39].cα,B+K+ is a complex coupling constant, which reprensntcDsˉDs,B+K+ andcD∗sˉD∗s,B+K+ . We introduce the form factorsfLij andFLkl , defined byfLij=1√EiEj(Λ2Λ2+q2ij)2+L2,
(4) FLkl=1√EkEl(Λ2Λ2+˜p2k)2+L2,
(5) where
qij is the momentum of i in theij center-of-mass frame, and˜pk is the momentum of k in the total center-of-mass frame. Λ is a cutoff, andΛ=1 GeV. We use a common value of the cutoff for all the interaction vertices.The second vertex
v2 is hadron scattering; the perturbative interactions forDsˉDs(D∗sˉD∗s)→DsˉDs are given by s-wave separable interactions. ForDsˉDs(0+)→D+sD−s(0+) ,v2=hD+sD−s,DsˉDsf0D+sD−sf0DsˉDs,
(6) and for
D∗sˉD∗s(0+)→D+sD−s(0+) ,v2=hD+sD−s,D∗sˉD∗s→ϵD∗s⋅→ϵˉD∗sf0D+sD−sf0D∗sˉD∗s.
(7) Another vertex exists between the two vertices, which is the coupling of the two loops, which we denote as
v3 . The coupling of different loops is similar in form; forDs(0−)ˉDs(0−)→Ds(0−)ˉDs(0−) ,v3=GDsˉDs,DsˉDs(MD+sD−s),
(8) for
Ds(0−)ˉDs(0−)→D∗s(1−)ˉD∗s(1−) ,v3=→ϵD∗s⋅→ϵˉD∗sGD∗sˉD∗s,DsˉDs(MD+sD−s),
(9) for
D∗s(1−)ˉD∗s(1−)→Ds(0−)ˉDs(0−) ,v3=→ϵD∗s⋅→ϵˉD∗sGDsˉDs,D∗sˉD∗s(MD+sD−s),
(10) and for
D∗s(1−)ˉD∗s(1−)→D∗s(1−)ˉD∗s(1−) ,v3=→ϵD∗s⋅→ϵˉD∗s→ϵD∗s⋅→ϵˉD∗sGD∗sˉD∗s,D∗sˉD∗s(MD+sD−s)=3GD∗sˉD∗s,D∗sˉD∗s(MD+sD−s).
(11) We introduce
[G−1]βα(E)=[δβα−hβ,ασα(E)] , wherehβ,α is a coupling constant, and α and β are label interaction channels, with\sigma_{D_s\bar{D}_s}(E)=\int {\rm d} qq^2\dfrac{[f_{D_s\bar{D}_s}^0(q)]^2}{E-E_{D_s}(q)-E_{\bar{D}_s}(q)+{\rm i} \varepsilon},
(12) \sigma_{D_s^*\bar{D}_s^*}(E)=\int {\rm d} qq^2\dfrac{[f_{D_s^*\bar{D}_s^*}^0(q)]^2}{E-E_{D_s^*}(q)-E_{\bar{D}_s^*}(q)+{\rm i}\varepsilon}.
(13) With the above ingredients, the amplitudes for the Fig. 1(a) are respectively given by
\begin{aligned}[b] A=\; &4\pi f_{D_s^+D_s^-}^0(p_{D_s^+})F_{K^+B^+}^0\sum_{\alpha}^{D_s\bar{D}_s,D_s^*\bar{D}_s^*}\sum_{\beta}^{D_s\bar{D}_s,D_s^*\bar{D}_s^*}\\ & \times c_{\alpha,B^+K^+}G_{\beta\alpha}(M_{D_s^+D_s^-})h_{D_s^+D_s^-,\beta}\sigma_\beta. \end{aligned}
(14) Regarding the direct decay mechanism of Fig. 1(b),
A_{\rm{dir}}=c_{D_s\bar{D}_s,B^+K^+}f_{D_s^+D_s^-}^0F_{K^+B^+}^0.
(15) Finally, we consider the Breit-Weigner mechanism of Fig. 1(c):
A_{\psi(4260)}^{1^-}=c_{\psi(4260)}\dfrac{\vec{p}_{K^+}\cdot\vec{p}_{D_s^+}f_{D_s^+D_s^-,\psi}^1f_{\psi K^+,B^+}^1}{E-E_{K^+}-E_\psi+\dfrac{i}{2}\Gamma_{\psi(4260)}},
(16) A_{\psi(4660)}^{1^-}=c_{\psi(4660)}\dfrac{\vec{p}_{K^+}\cdot\vec{p}_{D_s^+}f_{D_s^+D_s^-,\psi}^1f_{\psi K^+,B^+}^1}{E-E_{K^+}-E_\psi+\dfrac{i}{2}\Gamma_{\psi(4660)}},
(17) where
\vec{p}_{K^+} is theB^+ CM, and\vec{p}_{D_s^+} is theD_s^+D_s^- CM; the form factor defined byf_{D_s^+D_s^-,\psi}^1=\frac{1}{\sqrt{E_{D_s^+}E_{D_s^-}m_\psi}}\left(\frac{\Lambda^2}{\Lambda^2+q_{D_s^+D_s^-}^2}\right)^{\frac{5}{2}},
(18) f_{\psi K^+,B^+}^1=\frac{1}{\sqrt{E_\psi E_{K^+}E}}\left(\frac{\Lambda^2}{\Lambda^2+q_{\psi K^+}^2}\right)^{\frac{5}{2}},
(19) with constants
c_{\psi(4260)} andc_{\psi(4660)} . -
We simultaneously fit the invariant mass distributions of
M_{D_s^+D_s^-} ,M_{D_s^+K^+} , andM_{D_s^-K^+} from the LHCb Collaboration using the amplitudes of Eq. (14). The amplitude includes the vertices of the weak interaction and the adjustable coupling constant resulting from the hadron interaction; this includesc_{D_s\bar{D}_s,B^+K^+} ,c_{D_s^{\ast}\bar{D}_s^{\ast},B^+K^+} ,c_{\psi(4260)} ,c_{\psi(4660)} ,h_{D_s\bar{D}_s,D_s\bar{D}_s} ,h_{D_s\bar{D}_s,D_s^{\ast}\bar{D}_s^{\ast}} ,h_{D_s^{\ast}\bar{D}_s^{\ast},D_s\bar{D}_s} , andh_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} . To reduce the number of fitting parameters, we seth_{D_s\bar{D}_s,D_s^{\ast}\bar{D}_s^{\ast}}=h_{D_s^{\ast}\bar{D}_s^{\ast},D_s\bar{D}_s} , as making them different does not significantly affect the quality of the fit. Because the coupling and interaction constants of hadron scattering are consistent, we can further reduce the fitting parameters. Finally, because the magnitude and phase of the full amplitude are arbitrary, our default model has a total of nine fitting parameters. Our default model has a total of 8 (7+1) fitting parameters, in addition to these seven parameters as constants, the last parameter added is the overall factor. The parameters obtained from the final fit are shown in Table 1.c_{D_s\bar{D}_s,B^+K+} −0.07+0.23i −0.39+0.45i c_{D_s^{\ast}\bar{D}_s^{\ast},B^+K+} −0.13+0.09i −0.01+0.58i c_{\psi(4260)} 2.26−2.46i 8.21−0.77i c_{\psi(4660)} −3.27+5.30i −8.98−13.08i h_{D_s\bar{D}_s,D_s\bar{D}_s} 13.18+5.94i 3.83+16.02i h_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} −17.10+18.21i 0 h_{D_s^{\ast}\bar{D}_s^{\ast},D_s\bar{D}_s} −15.14−11.10i −1.47+10.20i Λ/MeV 1000 (fixed) 1000 (fixed) Table 1. Parameter values for
B^+\to D_s^+D_s^-K^+ models. The second and third columns are for the default and no couple-channel effect models.We show the default model by the solid blue curves in Fig. 2, which closely matches the LHCb data. We can clearly observe the peak at 3960 MeV and a dip at 4140 MeV. The fitting quality is
\chi^2 /ndf=(55.67+44.66+56.08)/(127-8)\simeq 1.31, where three\chi^2 values result from three different distributions; "ndf" is the number of bins (43 forD_s^+D_s^- , 42 forD_s^+K^+ , and 42 forD_s^-K^+ ) subtracted by the number of fitting parameters.Figure 2. (color online) (a)
D_s^+D_s^- , (b)D_s^+K^+ , (c)D_s^-K^+ invariant mass distributions forB^+\to D_s^+D_s^-K^+ .We also show the different contributions of the chart in Fig. 2. The solid orange curves represent the contribution of
D_s^+D_s^- single channel, and the dotted green curves represent the contribution ofD_s^{\ast}\bar D_s^{\ast} single channel. Generally, the solid orange curves plays a dominant role throughout the entire process, particularly in relation to the peak ofX(3960) . This behavior can be attributed to the fact that theX(3960) peak primarily results from the threshold ofD_s^+D_s^- . For the peaks at 4260 and 4660 MeV, we adopted the same method as the LHCb Collaboration and introduced two Breit-Weigner effects [40, 41],\psi(4260) and\psi(4660) , which are represented by purple and brown dotted curves, respectively. The analysis here is generally consistent with the analysis given by LHCb; for two peaks near 4260 and 4660 MeV, the final fitting results have been significantly improved.In our study, we conducted a search for poles in the default
D_s\bar D_sD_s^{\ast}\bar D_s^{\ast} coupled-channel scattering amplitude using analytic continuation. We observed the poles ofX(3960) andX_0(4140) , which are summarized in Table 2. Additionally, in the table, we also list the Riemann sheets of the poles by(D_s\bar D_sD_s^{\ast}\bar D_s^{\ast}) , wheres_\alpha=p indicates that the pole is located on the physical p sheet of the channel, whereass_{\alpha}=u indicates that it is on the unphysical u sheet of the channel. As shown in the table, we can obtain the positions ofX(3960) andX_0(4140) . Based on this, we can suggest thatX(3960) is a resonance state andX_0(4140) is a virtual state [42, 43]. This observation is consistent with the results shown in Fig. 2, which clearly indicate that the formation ofX(3960) is primarily due to the interaction ofV_{D_s^+D_s^-,D_s^+D_s^-} . Even without considering the contribution ofV_{D_s^+D_s^-,D_s^{\ast}\bar{D}_s^{\ast}} andV_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} , the state ofX(3960) can be understood as a bound state ofD_s^+D_s^- . The behavior of the green dotted curves in Fig. 2(a) further supports the notion that ifX_0(4140) is a virtual state, the contribution ofV_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} is weak.X(3960) 3952.48+12.46i (pu) X_0(4140) 4142.48 (pp) Table 2.
X(3960) andX_0(4140) poles in the default model. Pole positions (in MeV) and their Riemann sheets (see the text for notation) are given in the second and third columns, respectively.In order to investigate the threshold effect of the kinematic effect in the vicinity and determine whether the
X(3960) peak structure is solely caused by theD_s\bar D_s threshold, we disabled the coupled-channel effect, equivalent to directly finding the monocyclic graph contribution of the Fig. 1. The data was then re-fitted, as shown in the Table 2. In Fig. 3, although the overall change in\chi^2 is small, it is evident that the height of the first peak undergoes a significant change, and the change in\chi^2 is primarily due to this peak. Therefore, we can conclude that the pure kinematic effect alone is insufficient to form a peak structure. The peak structure should indicate a state that actually exists. -
We analyze the observations of the LHCb Collaboration on the decay process
B^+\to D_s^+D_s^-K^+ . Note that when calculating the total amplitude, we refer to the work of the LHCb Collaboration and introduce the Breit-Weigner effect of the resonance state\psi(4260) , but the peak value of\psi(4260) is slightly earlier than the jump position in the data of the invariant mass spectrum ofD_s^+D_s^- , and this position is very close to the threshold ofD\bar{D} . Therefore, we can reasonably expect that building a new model based on our current model and addingD\bar{D} this coupled-channel will be useful in explaining the jump in the invariant mass spectrum ofD_s^+D_s^- . Our default model fits theM_{D_s^+D_s^-} ,M_{D_s^+K^+} , andM_{D_s^-K^+} of these three different invariant mass distributions simultaneously, and the final fitting quality is\chi^2/ndf\simeq1.31 .Without adding resonance states directly, we search the poles of
X(3960) andX_0(4140) using the coupled-channel model and finally determine the positions ofX(3960) andX_0(4140) . From this, we suggest thatX(3960) may be a resonance state andX_0(4140) may be a virtual state. The determination of the pole positions is meaningful, which provides information for the research onX(3960) andX_0(4140) and lays a certain foundation for the study of their properties in the future. By turning off the coupled-channel effect and fitting the data again, we find that the overall fitting quality does not change sigificantly. However, the final fitting result shows that the influence is relatively large at the position ofX(3960) , and almost all the changes of\chi^2 result from theX(3960) peak. Therefore, we suggest that the pure kinematic effect is insufficient to form theX(3960) peak structure. This conclusion provides certain reference value for future research.
