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Pole determination of X(3960) and X0(4140) in the decay B+D+sDsK+

  • Two near-threshold peaking structures with spin-parities of JPC=0++ were recently discovered by the LHCb Collaboration in the D+sDs invariant mass distribution of the decay process B+D+sDsK+. In our study, we employed a coupled-channel model to fit the experimental results published by the LHCb Collaboration, simultaneously fitting the model to the invariant mass distributions of MD+sDs, MD+sK+, and MDsK+. We utilized a coupled-channel model to search for the poles of X(3960) and X0(4140). The determination of the poles is meaningful in itself, and it also lays a foundation for future research on X(3960) and X0(4140). Upon turning off the coupled-channel and performing another fit, we observed a change in the fitting quality, and the effect was almost entirely due to the peak of X(3960). Therefore, we suggest that X(3960) may not be a kinematic effect.
  • 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 and qqˉqˉq. In addition, more exotic structures have been observed in experiments; see Refs. [18]. 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) and Zcs(4000) [9], X(4140) [10, 11] in B+J/ψϕK+, and X0(2900) and X1(2900) in B+D+DK+ 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. [1417].

    Very recently, the LHCb Collaboration reported a new near-threshold structure named X(3960) in the D+sDs invariant mass distribution of the decay B+D+sDsK+. The peak structure is very close to the D+sDs threshold with a statistical significance larger than 12σ. The mass, width, and quantum numbers of this structure were measured to be M=3956±5±10 MeV, Γ=43±13±8 MeV, and JPC=0++. The LHCb analysis indicates that this structure is an exotic candidate consisting of csˉcˉs constituents. In addition, when checking the data of the D+sDs invariant mass distribution, a dip is observed at approximately 4.14 GeV; the LHCb interpreted it as another structure named X0(4140) with a mass of M=4133±6±6 MeV, width of Γ=67±17±7 MeV, and quantum numbers of JPC=0++ [18]. As analyzed by the LHCb Collaboration, X0(4140) might be caused by either a new resonance with the 0++ assignment or a D+sDsJ/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 [1926]. To determine the origin and structure of X(3960) in decay B+D+sDsK+, scholars have proposed many explanations for the possibility of this structure. Because its mass is close to the D+sDs threshold, this structure can be interpreted as a possible hadronic molecule. Refs. [27, 28] proposed to treat X(3960) as the molecular state of D+sDs with JPC=0++ in the QCD sum rules approach. Another calculation with QCD two-point sum rules [29] results in the assignment that X(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 of X(3960) using the coupled-channel method. The authors of Ref. [31] performed a coupled-channel calcuation of the interaction DˉDD+sDs in the chiral unitary approach and interpreted X(3960) as a hadronic molecule in the coupled DˉDD+sDs system [3133]. The author of Ref. [34] interpreted X(3960) as a csˉcˉs state, whereas in Ref. [35], X(3960) was interpreted as 0++ csˉcˉs tetraquark states using an improved chromomagnetic interaction model. In addition, another study suggested that X(3960) probably has the mixed characteristics of a cˉc confining state and DsˉDs continuum [36]. Some theoretical and experimental research has been conducted on X0(4140), but its origins are still debated. For instance, in Ref. [35], X0(4140) was also interpreted as csˉcˉs 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 of X0(4140).

    In this study, we analyzed the decay process B+D+sDsK+ as published by the LHCb Collaboration. We simulated a coupled-channel model to analyze the data [37] using the default model and fitting the MD+sDs, MD+sK+, and MDsK+ 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 of X(3960) and X0(4140) and (ii) whether the production of X(3960) is solely due to a kinematic effect.

    The LHCb data reveal visible X(3960) and X0(4140) structures around the DsˉDs and DsˉDs 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

    Figure 1.  Contributions of three mechanisms in the decay B+D+sDsK+. (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 weak B+DsˉDsK+ vertex is

    v1=cα,B+K+f0DsˉDsF0K+B+.

    (1)

    For the vertex of process B+DsˉDsK+, there are two cases of parity conservation and parity violation. For the former, the vertex of B+(0)DsˉDs(0+)K+(0) is

    vpc1=cDsˉDs,B+K+ϵDsϵˉDsf0DsˉDsF0K+B+,

    (2)

    in the latter case, the vertex of B+(0)DsˉDs(1+)K+(0) is

    vpv1=cDsˉDs,B+K+pK+(ϵDs×ϵˉDs)f0DsˉDsF0K+B+.

    (3)

    The energy, momentum, and polarization vector of a particle x are denoted by Ex, px, and ϵx , respectively, and particle masses are obtained from Ref. [39]. cα,B+K+ is a complex coupling constant, which reprensnt cDsˉDs,B+K+ and cDsˉDs,B+K+. We introduce the form factors fLij and FLkl, defined by

    fLij=1EiEj(Λ2Λ2+q2ij)2+L2,

    (4)

    FLkl=1EkEl(Λ2Λ2+˜p2k)2+L2,

    (5)

    where qij is the momentum of i in the ij center-of-mass frame, and ˜pk is the momentum of k in the total center-of-mass frame. Λ is a cutoff, and Λ=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 for DsˉDs(DsˉDs)DsˉDs are given by s-wave separable interactions. For DsˉDs(0+)D+sDs(0+),

    v2=hD+sDs,DsˉDsf0D+sDsf0DsˉDs,

    (6)

    and for DsˉDs(0+)D+sDs(0+),

    v2=hD+sDs,DsˉDsϵDsϵˉDsf0D+sDsf0DsˉDs.

    (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; for Ds(0)ˉDs(0)Ds(0)ˉDs(0),

    v3=GDsˉDs,DsˉDs(MD+sDs),

    (8)

    for Ds(0)ˉDs(0)Ds(1)ˉDs(1),

    v3=ϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs),

    (9)

    for Ds(1)ˉDs(1)Ds(0)ˉDs(0),

    v3=ϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs),

    (10)

    and for Ds(1)ˉDs(1)Ds(1)ˉDs(1),

    v3=ϵDsϵˉDsϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs)=3GDsˉDs,DsˉDs(MD+sDs).

    (11)

    We introduce [G1]βα(E)=[δβαhβ,ασα(E)], where hβ,α is a coupling constant, and α and β are label interaction channels, with

    σDsˉDs(E)=dqq2[f0DsˉDs(q)]2EEDs(q)EˉDs(q)+iε,

    (12)

    σDsˉDs(E)=dqq2[f0DsˉDs(q)]2EEDs(q)EˉDs(q)+iε.

    (13)

    With the above ingredients, the amplitudes for the Fig. 1(a) are respectively given by

    A=4πf0D+sDs(pD+s)F0K+B+DsˉDs,DsˉDsαDsˉDs,DsˉDsβ×cα,B+K+Gβα(MD+sDs)hD+sDs,βσβ.

    (14)

    Regarding the direct decay mechanism of Fig. 1(b),

    Adir=cDsˉDs,B+K+f0D+sDsF0K+B+.

    (15)

    Finally, we consider the Breit-Weigner mechanism of Fig. 1(c):

    A1ψ(4260)=cψ(4260)pK+pD+sf1D+sDs,ψf1ψK+,B+EEK+Eψ+i2Γψ(4260),

    (16)

    A1ψ(4660)=cψ(4660)pK+pD+sf1D+sDs,ψf1ψK+,B+EEK+Eψ+i2Γψ(4660),

    (17)

    where pK+ is the B+ CM, and pD+s is the D+sDs CM; the form factor defined by

    f1D+sDs,ψ=1ED+sEDsmψ(Λ2Λ2+q2D+sDs)52,

    (18)

    f1ψK+,B+=1EψEK+E(Λ2Λ2+q2ψK+)52,

    (19)

    with constants cψ(4260) and cψ(4660).

    We simultaneously fit the invariant mass distributions of MD+sDs, MD+sK+, and MDsK+ 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 includes cDsˉDs,B+K+, cDsˉDs,B+K+, cψ(4260), cψ(4660), hDsˉDs,DsˉDs, hDsˉDs,DsˉDs, hDsˉDs,DsˉDs, and hDsˉDs,DsˉDs. To reduce the number of fitting parameters, we set hDsˉDs,DsˉDs=hDsˉDs,DsˉDs, 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.

    Table 1

    Table 1.  Parameter values for B+D+sDsK+ models. The second and third columns are for the default and no couple-channel effect models.
    cDsˉDs,B+K+−0.07+0.23i−0.39+0.45i
    cDsˉDs,B+K+−0.13+0.09i−0.01+0.58i
    cψ(4260)2.26−2.46i8.21−0.77i
    cψ(4660)−3.27+5.30i−8.98−13.08i
    hDsˉDs,DsˉDs13.18+5.94i3.83+16.02i
    hDsˉDs,DsˉDs−17.10+18.21i0
    hDsˉDs,DsˉDs−15.14−11.10i−1.47+10.20i
    Λ/MeV1000 (fixed)1000 (fixed)
    DownLoad: CSV
    Show Table

    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 χ2/ndf=(55.67+44.66+56.08)/(127-8)1.31, where three χ2 values result from three different distributions; "ndf" is the number of bins (43 for D+sDs, 42 for D+sK+, and 42 for DsK+) subtracted by the number of fitting parameters.

    Figure 2

    Figure 2.  (color online) (a)D+sDs, (b)D+sK+, (c)DsK+ invariant mass distributions for B+D+sDsK+.

    We also show the different contributions of the chart in Fig. 2. The solid orange curves represent the contribution of D+sDs single channel, and the dotted green curves represent the contribution of DsˉDs single channel. Generally, the solid orange curves plays a dominant role throughout the entire process, particularly in relation to the peak of X(3960). This behavior can be attributed to the fact that the X(3960) peak primarily results from the threshold of D+sDs. 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], ψ(4260) and ψ(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 DsˉDsDsˉDs coupled-channel scattering amplitude using analytic continuation. We observed the poles of X(3960) and X0(4140), which are summarized in Table 2. Additionally, in the table, we also list the Riemann sheets of the poles by (DsˉDsDsˉDs), where sα=p indicates that the pole is located on the physical p sheet of the channel, whereas sα=u indicates that it is on the unphysical u sheet of the channel. As shown in the table, we can obtain the positions of X(3960) and X0(4140). Based on this, we can suggest that X(3960) is a resonance state and X0(4140) is a virtual state [42, 43]. This observation is consistent with the results shown in Fig. 2, which clearly indicate that the formation of X(3960) is primarily due to the interaction of VD+sDs,D+sDs. Even without considering the contribution of VD+sDs,DsˉDs and VDsˉDs,DsˉDs, the state of X(3960) can be understood as a bound state of D+sDs. The behavior of the green dotted curves in Fig. 2(a) further supports the notion that if X0(4140) is a virtual state, the contribution of VDsˉDs,DsˉDs is weak.

    Table 2

    Table 2.  X(3960) and X0(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.
    X(3960)3952.48+12.46i(pu)
    X0(4140)4142.48(pp)
    DownLoad: CSV
    Show Table

    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 the DsˉDs 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 χ2 is small, it is evident that the height of the first peak undergoes a significant change, and the change in χ2 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.

    Figure 3

    Figure 3.  (color online) Comparison of different models. The blue line in the figure is for the default model, whereas the orange line is for the model that turns off the couple-channel effect.

    We analyze the observations of the LHCb Collaboration on the decay process B+D+sDsK+. 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 ψ(4260), but the peak value of ψ(4260) is slightly earlier than the jump position in the data of the invariant mass spectrum of D+sDs, and this position is very close to the threshold of DˉD. Therefore, we can reasonably expect that building a new model based on our current model and adding DˉD this coupled-channel will be useful in explaining the jump in the invariant mass spectrum of D+sDs. Our default model fits the MD+sDs, MD+sK+, and MDsK+ of these three different invariant mass distributions simultaneously, and the final fitting quality is χ2/ndf1.31.

    Without adding resonance states directly, we search the poles of X(3960) and X0(4140) using the coupled-channel model and finally determine the positions of X(3960) and X0(4140). From this, we suggest that X(3960) may be a resonance state and X0(4140) may be a virtual state. The determination of the pole positions is meaningful, which provides information for the research on X(3960) and X0(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 of X(3960), and almost all the changes of χ2 result from the X(3960) peak. Therefore, we suggest that the pure kinematic effect is insufficient to form the X(3960) peak structure. This conclusion provides certain reference value for future research.

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  • [1] H.-X. Chen, W. Chen, X. Liu et al., Phys. Rept. 639, 1 (2016), arXiv: 1601.02092[hep-ph] doi: 10.1016/j.physrep.2016.05.004
    [2] A. Hosaka, T. Iijima, K. Miyabayashi et al., PTEP 062, 062C01 (2016), arXiv: 1603.09229[hep-ph] doi: 10.1093/ptep/ptw045
    [3] R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), arXiv: 1610.04528[hepph] doi: 10.1016/j.ppnp.2016.11.003
    [4] A. Esposito, A. Pilloni, and A. D. Polosa, Phys. Rept. 668, 1 (2017), arXiv: 1611.07920[hep-ph] doi: 10.1016/j.physrep.2016.11.002
    [5] A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017), arXiv: 1706.00610[hep-ph] doi: 10.1016/j.ppnp.2017.08.003
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Jialiang Lu, Mao Song, Gang Li and Xuan Luo. Pole determination of X(3960) and X0(4140) in decay B+D+sDsK+[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad9898
Jialiang Lu, Mao Song, Gang Li and Xuan Luo. Pole determination of X(3960) and X0(4140) in decay B+D+sDsK+[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad9898 shu
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Pole determination of X(3960) and X0(4140) in the decay B+D+sDsK+

  • Anhui University, Hefei 230601, China

Abstract: Two near-threshold peaking structures with spin-parities of JPC=0++ were recently discovered by the LHCb Collaboration in the D+sDs invariant mass distribution of the decay process B+D+sDsK+. In our study, we employed a coupled-channel model to fit the experimental results published by the LHCb Collaboration, simultaneously fitting the model to the invariant mass distributions of MD+sDs, MD+sK+, and MDsK+. We utilized a coupled-channel model to search for the poles of X(3960) and X0(4140). The determination of the poles is meaningful in itself, and it also lays a foundation for future research on X(3960) and X0(4140). Upon turning off the coupled-channel and performing another fit, we observed a change in the fitting quality, and the effect was almost entirely due to the peak of X(3960). Therefore, we suggest that X(3960) may not be a kinematic effect.

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    I.   INTRODUCTION
    • 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 and qqˉqˉq. In addition, more exotic structures have been observed in experiments; see Refs. [18]. 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) and Zcs(4000) [9], X(4140) [10, 11] in B+J/ψϕK+, and X0(2900) and X1(2900) in B+D+DK+ 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. [1417].

      Very recently, the LHCb Collaboration reported a new near-threshold structure named X(3960) in the D+sDs invariant mass distribution of the decay B+D+sDsK+. The peak structure is very close to the D+sDs threshold with a statistical significance larger than 12σ. The mass, width, and quantum numbers of this structure were measured to be M=3956±5±10 MeV, Γ=43±13±8 MeV, and JPC=0++. The LHCb analysis indicates that this structure is an exotic candidate consisting of csˉcˉs constituents. In addition, when checking the data of the D+sDs invariant mass distribution, a dip is observed at approximately 4.14 GeV; the LHCb interpreted it as another structure named X0(4140) with a mass of M=4133±6±6 MeV, width of Γ=67±17±7 MeV, and quantum numbers of JPC=0++ [18]. As analyzed by the LHCb Collaboration, X0(4140) might be caused by either a new resonance with the 0++ assignment or a D+sDsJ/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 [1926]. To determine the origin and structure of X(3960) in decay B+D+sDsK+, scholars have proposed many explanations for the possibility of this structure. Because its mass is close to the D+sDs threshold, this structure can be interpreted as a possible hadronic molecule. Refs. [27, 28] proposed to treat X(3960) as the molecular state of D+sDs with JPC=0++ in the QCD sum rules approach. Another calculation with QCD two-point sum rules [29] results in the assignment that X(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 of X(3960) using the coupled-channel method. The authors of Ref. [31] performed a coupled-channel calcuation of the interaction DˉDD+sDs in the chiral unitary approach and interpreted X(3960) as a hadronic molecule in the coupled DˉDD+sDs system [3133]. The author of Ref. [34] interpreted X(3960) as a csˉcˉs state, whereas in Ref. [35], X(3960) was interpreted as 0++ csˉcˉs tetraquark states using an improved chromomagnetic interaction model. In addition, another study suggested that X(3960) probably has the mixed characteristics of a cˉc confining state and DsˉDs continuum [36]. Some theoretical and experimental research has been conducted on X0(4140), but its origins are still debated. For instance, in Ref. [35], X0(4140) was also interpreted as csˉcˉs 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 of X0(4140).

      In this study, we analyzed the decay process B+D+sDsK+ as published by the LHCb Collaboration. We simulated a coupled-channel model to analyze the data [37] using the default model and fitting the MD+sDs, MD+sK+, and MDsK+ 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 of X(3960) and X0(4140) and (ii) whether the production of X(3960) is solely due to a kinematic effect.

    II.   FRAMEWORK
    • The LHCb data reveal visible X(3960) and X0(4140) structures around the DsˉDs and DsˉDs 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+sDsK+. (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 weak B+DsˉDsK+ vertex is

      v1=cα,B+K+f0DsˉDsF0K+B+.

      (1)

      For the vertex of process B+DsˉDsK+, there are two cases of parity conservation and parity violation. For the former, the vertex of B+(0)DsˉDs(0+)K+(0) is

      vpc1=cDsˉDs,B+K+ϵDsϵˉDsf0DsˉDsF0K+B+,

      (2)

      in the latter case, the vertex of B+(0)DsˉDs(1+)K+(0) is

      vpv1=cDsˉDs,B+K+pK+(ϵDs×ϵˉDs)f0DsˉDsF0K+B+.

      (3)

      The energy, momentum, and polarization vector of a particle x are denoted by Ex, px, and ϵx , respectively, and particle masses are obtained from Ref. [39]. cα,B+K+ is a complex coupling constant, which reprensnt cDsˉDs,B+K+ and cDsˉDs,B+K+. We introduce the form factors fLij and FLkl, defined by

      fLij=1EiEj(Λ2Λ2+q2ij)2+L2,

      (4)

      FLkl=1EkEl(Λ2Λ2+˜p2k)2+L2,

      (5)

      where qij is the momentum of i in the ij center-of-mass frame, and ˜pk is the momentum of k in the total center-of-mass frame. Λ is a cutoff, and Λ=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 for DsˉDs(DsˉDs)DsˉDs are given by s-wave separable interactions. For DsˉDs(0+)D+sDs(0+),

      v2=hD+sDs,DsˉDsf0D+sDsf0DsˉDs,

      (6)

      and for DsˉDs(0+)D+sDs(0+),

      v2=hD+sDs,DsˉDsϵDsϵˉDsf0D+sDsf0DsˉDs.

      (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; for Ds(0)ˉDs(0)Ds(0)ˉDs(0),

      v3=GDsˉDs,DsˉDs(MD+sDs),

      (8)

      for Ds(0)ˉDs(0)Ds(1)ˉDs(1),

      v3=ϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs),

      (9)

      for Ds(1)ˉDs(1)Ds(0)ˉDs(0),

      v3=ϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs),

      (10)

      and for Ds(1)ˉDs(1)Ds(1)ˉDs(1),

      v3=ϵDsϵˉDsϵDsϵˉDsGDsˉDs,DsˉDs(MD+sDs)=3GDsˉDs,DsˉDs(MD+sDs).

      (11)

      We introduce [G1]βα(E)=[δβαhβ,ασα(E)], where hβ,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 the B^+ CM, and \vec{p}_{D_s^+} is the D_s^+D_s^- CM; the form factor defined by

      f_{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)} and c_{\psi(4660)} .

    III.   RESULTS
    • We simultaneously fit the invariant mass distributions of M_{D_s^+D_s^-} , M_{D_s^+K^+} , and M_{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 includes c_{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} , and h_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} . To reduce the number of fitting parameters, we set 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} , 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)} 2.26−2.46i8.21−0.77i
      c_{\psi(4660)} −3.27+5.30i−8.98−13.08i
      h_{D_s\bar{D}_s,D_s\bar{D}_s} 13.18+5.94i3.83+16.02i
      h_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} −17.10+18.21i0
      h_{D_s^{\ast}\bar{D}_s^{\ast},D_s\bar{D}_s} −15.14−11.10i−1.47+10.20i
      Λ/MeV1000 (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 for D_s^+D_s^- , 42 for D_s^+K^+ , and 42 for D_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 for B^+\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 of D_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 of X(3960) . This behavior can be attributed to the fact that the X(3960) peak primarily results from the threshold of D_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 of X(3960) and X_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}) , where s_\alpha=p indicates that the pole is located on the physical p sheet of the channel, whereas s_{\alpha}=u indicates that it is on the unphysical u sheet of the channel. As shown in the table, we can obtain the positions of X(3960) and X_0(4140) . Based on this, we can suggest that X(3960) is a resonance state and X_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 of X(3960) is primarily due to the interaction of V_{D_s^+D_s^-,D_s^+D_s^-} . Even without considering the contribution of V_{D_s^+D_s^-,D_s^{\ast}\bar{D}_s^{\ast}} and V_{D_s^{\ast}\bar{D}_s^{\ast},D_s^{\ast}\bar{D}_s^{\ast}} , the state of X(3960) can be understood as a bound state of D_s^+D_s^- . The behavior of the green dotted curves in Fig. 2(a) further supports the notion that if X_0(4140) is a virtual state, the contribution of V_{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) and X_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 the D_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.

      Figure 3.  (color online) Comparison of different models. The blue line in the figure is for the default model, whereas the orange line is for the model that turns off the couple-channel effect.

    IV.   CONCLUSION
    • 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 of D_s^+D_s^- , and this position is very close to the threshold of D\bar{D} . Therefore, we can reasonably expect that building a new model based on our current model and adding D\bar{D} this coupled-channel will be useful in explaining the jump in the invariant mass spectrum of D_s^+D_s^- . Our default model fits the M_{D_s^+D_s^-} , M_{D_s^+K^+} , and M_{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) and X_0(4140) using the coupled-channel model and finally determine the positions of X(3960) and X_0(4140) . From this, we suggest that X(3960) may be a resonance state and X_0(4140) may be a virtual state. The determination of the pole positions is meaningful, which provides information for the research on X(3960) and X_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 of X(3960) , and almost all the changes of \chi^2 result from the X(3960) peak. Therefore, we suggest that the pure kinematic effect is insufficient to form the X(3960) peak structure. This conclusion provides certain reference value for future research.

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