A discussion on the anomalous threshold enhancement of J/ψψ(3770) couplings and X(6900) peak

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Ye Lu, Chang Chen, Guang-You Qin and Han-Qing Zheng. A discussion on the anomalous threshold enhancement of J/ψψ(3770) couplings and X(6900) peak[J]. Chinese Physics C. doi: 10.1088/1674-1137/ad2361
Ye Lu, Chang Chen, Guang-You Qin and Han-Qing Zheng. A discussion on the anomalous threshold enhancement of J/ψψ(3770) couplings and X(6900) peak[J]. Chinese Physics C.  doi: 10.1088/1674-1137/ad2361 shu
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Received: 2023-12-19
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A discussion on the anomalous threshold enhancement of J/ψψ(3770) couplings and X(6900) peak

  • 1. Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University,Wuhan 430079, China
  • 2. Department of Physics, Peking University, Beijing 100871, China
  • 3. College of Physics, Sichuan University, Chengdu 610065, China

Abstract: The attractive interaction between J/ψ and ψ(3770) has to be strong enough if X(6900) is of the molecule type. We argue that since ψ(3770) decays predominantly into a DˉD pair, the interactions between J/ψ and ψ(3770) may be significantly enhanced owing to the three point DˉD loop diagram. The enhancement originates from the anomalous threshold located at t=1.288 GeV2, whose effect propagates into the s-channel partial wave amplitude in the vicinity of s6.94 GeV. This effect may be helpful in the formation of the X(6900) peak.

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  • TheX(6900) peak observed by the LHCb Collaboration in the di-J/ψ invariant mass spectrum [1, 2], and later in the Jψψ(3686) invariant mass spectrum [3], has stimulated many discussions of theoretical aspects (see for example Ref. [4] for an incomplete list of references). Moreover, X(6900) is close to the threshold of J/ψψ(3770), J/ψψ2(3823), J/ψψ3(3842), and χc0χc1, whereas X(7200) is close to the threshold of J/ψψ(4160) and χc0χc1(3872)). Inspired by this, Ref. [5] studied the properties of X(6900) and X(7200) by assuming the X(6900) coupling to J/ψJ/ψ,J/ψψ(3770), J/ψψ2(3823), J/ψψ3(3842), and χc0χc1 channels and the X(7200) coupling to J/ψJ/ψ, J/ψψ(4160), and χc0χc1(3872) channels. For the S-wave J/ψJ/ψ coupling, the pole counting rule (PCR) [6], which has been applied to the studies of "XYZ" physics in Refs. [710], was employed to analyze the nature of the two structures. It was found that the di-J/ψ data alone are not sufficient to judge the intrinsic properties of the two states. It was also pointed out that X(6900) is unlikely a molecule of J/ψψ(3686) [5], a conclusion drawn before the discovery of Ref. [3]. More recently, Refs. [4, 11, 12] investigated X(6900) using a combined analysis of di-J/ψ and J/ψψ(3686) data and concluded that X(6900) cannot be a J/ψψ(3686) molecule.

    Nevertheless, as already stressed in Ref. [4], even though X(6900) is very unlikely a molecule of J/ψψ(3686), this does not mean that it has to be an "elementary state" (i.e., a compact ˉcˉccc tetraquark state). It was pointed out that it is possible that X(6900) be a molecular state composed of other particles, such as J/ψψ(3770), which form thresholds closer to X(6900) if the channel coupling is sufficiently large 1.

    This note will discuss a possible mechanism for the enhancement of the J/ψψ(3770) channel coupling. The DˉD (orDˉD) component inside ψ(3770) may play an important role, so far ignored in the literature, in explaining the X(6900) resonant peak through the anomalous threshold emerged from the triangle diagram generated by the D (D) loop, as depicted in Fig. 1.

    Figure 1.  DˉD triangle diagram in the J/ψψ scattering process.

    Noticing that ψ(3770) or ψ couples dominantly to DˉD, we start from the Feynman diagram as depicted in Fig. 1 by assuming that it contributes to J/ψψ elastic scatterings near the J/ψψ threshold 2. Assuming an interaction Lagrangian 3

    L=ig(D0μˉD0ˉD0μD0)ψμig(D+μDDμD+)ψμ+gD0ˉD0J/ψμJ/ψμ+gD+DJ/ψμJ/ψμ ,

    (1)

    after performing the momentum integration, the amplitude as depicted by Fig. 1 is

    iM=(16g2g)(ϵ2ϵ4)ϵμ1ϵν3dDk×kμkν((kp1)2m2D)((kp3)2m2D)(k2m2D)(16g2g)(ϵ2ϵ4)ϵμ1ϵν3Aμν

    (2)

    where

    Aμν=i16π210dx1x0dy{pμ3pν1xyΔgμν4Γ(ϵ)1Δϵ}i16π2{pμ3pν1B+10dx1x0dy(gμν4Γ(ϵ)1Δϵ)} ,

    (3)

    and

    B(t)=10dxx2M2(2(M2(12x)+tx)×ArcTan(M2txΛ(t,x))ArcTan(M2(2x1)txΛ(t,x))Λ(t,x)+lnm2+t(x1)xm2+M2(x1)x) .

    (4)

    Here, Λ(t,x)=4m2M2M4+4M2tx22M2txt2x2, Δ=M2(x2+y2)+(2M2t)xyM2(x+y)+m2, and Γ(ϵ)1Δϵ=1ϵlnΔγ+ln4π+O(ϵ). On the right hand side of Eq. (3), only the B term will be considered since the rest will be absorbed by the contact interactions to be introduced latter. M is the mass of ψ(3770), and m is the mass of D(ˉD). Parameter g is the coupling strength of the ψDˉD three point vertex, and g is the coupling strength of the J/ψJ/ψDˉD four point vertex; {t=(p2p4)2}. Parameter g can be determined by the ψD0ˉD0 decay process ,

    iMψDD=igϵ(ψ)[p(D0)p(ˉD0)] ,Γ=18π|iMψDD|2q(DD)M2ψ=16πg2q(DD)3M2ψ ,

    (5)

    where q(DD) is the norm of the three-dimensional momentum of D0 orˉD0 in the final state. The PDG value ΓψD0ˉD027.2×52%×103 [16] determines g12 4. Parameter g is unknown and is left as a free parameter. The amplitude, Eq. (4), contains a rather complicated singularity structure, especially the well-known anomalous threshold, which was discovered by Mandelstam who used it to explain the looseness of the deuteron wave function [18]. The anomalous threshold is located at

    sA=4m2(M22m2)2m2 .

    (6)

    Considering the mass of the ψ and D0 mesons, one obtains sA=1.28 GeV2 (for the D+ loop, –0.98 GeV2). Numerically, function B is plotted in Fig. 2(a), where one clearly sees the anomalous threshold beside the normal one at t=4m2. Note that if M2 is smaller than 2m2, the anomalous branch point is located below the physical threshold, but on the second sheet. It touches the physical threshold 4m2 and turns up to the physical sheet if the value of M2 increases to 2m2. With a further increase in M2, the anomalous threshold moves towards the left on the real axis, passes the origin when M2=4m2, and finally reaches the physical value, i.e., –1.28 GeV2. The situation is depicted in Fig. 2(b). Note that sA here is negative, contrary to what occurs with the deuteron, because the latter is a bound state with a normalizable wave function, whereas ψ is an unstable resonance.

    Figure 2.  (color online) Left: triangle diagram contribution (the y axis label is arbitrary). Right: the trajectory of the anomalous threshold with respect to the variation of M2.

    To proceed, one needs to make the partial wave projection of M and obtain

    TJμ1μ2μ3μ4(s)=132π12q2(s)04q2(s)dtdJμμ×(1+t2q2(s))Mμ1μ2μ3μ4(t) ,

    (7)

    where the channel momentum square reads q2(s)=[s(MMJ)2][s(M+MJ)2]/4s, MJ is the mass of J/ψ,μi denotes the corresponding helicity configuration, and μ=μ1μ2, μ=μ3μ4. The key observation is that the integral interval in Eq. (7) will coversA ifs>6.96 GeV (for the D+ loop,s>6.94 GeV). In other words, the partial wave amplitude will be enhanced in the vicinity of the X(6900) peak by the anomalous threshold enhancement in the t channel, as can be observed in Fig. 3.

    Figure 3.  (color online) Enhancement of the J/ψψ scattering amplitude from the triangle diagram.

    Based on the above observation, it is suggested that the X(6900) peak may at least partly be explained by the anomalous threshold generated by the triangle diagram depicted in Fig. 1. Furthermore, to obtain the L=0 (s wave) amplitudes, we need the relation between the s wave and helicity amplitudes (see Refs. [11, 12] for further discussions):

    T0L=0(s)=13[2T0++++(s)+2T0++(s)2T0++00(s)2T000++(s)+T00000(s)],

    (8)

    T2L=0(s)=115[T2++++(s)+T2++(s)]+615[T2+++(s)+T2+++(s)+T2+++(s)+T2+++(s)]+315[T2+++0(s)+T2+0++(s)+T2++0+(s)+T20+++(s)+T2++0(s)+T20++(s)+T2++0(s)+T20++(s)]+15[T2+00+(s)+T20++0(s)+T2+00(s)+T20+0(s)+T2+0+0(s)+T20+0+(s)+T20+0(s)+T2+00(s)]+25[T2++0(s)+T2+0+(s)+T2+0+(s)+T20++(s)+T2++0(s)+T2+0+(s)+T2+0+(s)+T20++(s)]+215[T2++00(s)+T200++(s)+T20000(s)]+2615[T2+00(s)+T200+(s)]+25[T2++(s)+T2++(s)]+2315[T2+000(s)+T200+0(s)+T20+00(s)+T2000+(s)] ,

    (9)

    In practice, it is found that the anomalous enhancement gives a more prominent effect to the J=2 amplitude than the J=0 amplitude. Furthermore, to estimate the triangle diagram contribution, a combined fit with J/ψJ/ψ and J/ψψ(3686) data is made. A coupled–channel K-matrix unitarization scheme is employed including J/ψJ/ψ, J/ψψ(3686), and J/ψψ(3770). The tree level amplitudes are also taken into account from the following contact interaction Lagrangian [19]:

    Lc=c1VμVαVμVα+c2VμVαVμVα+c3VμVαVμVα+c4VμVμVαVα+c5VμVαVμVα+c6VμVαVμVα+c7VμVμVαVα+c8VμVαVμVα+c9VμVμVαVα,

    (10)

    and these tree level amplitudes are as follows:

    iMJ/ψJ/ψJ/ψJ/ψ=i8c1(ϵ1μϵ2αϵμ3ϵα4+ϵ1μϵ2αϵα3ϵμ4+ϵ1μϵμ2ϵ3αϵα4),

    (11)

    iMJ/ψJ/ψJ/ψψ(2S)=i2c2(ϵ1μϵ2αϵμ3ϵα4+ϵ1μϵμ2ϵ3αϵα4+ϵ1αϵ2μϵμ3ϵα4),

    (12)

    iMJ/ψψ(2S)J/ψψ(2S)=i4c3(ϵ1μϵ2αϵμ3ϵα4)+i2c4(ϵ1μϵμ2ϵ3αϵα4+ϵ1μϵα2ϵ3αϵμ4),

    (13)

    iMJ/ψJ/ψJ/ψψ(3770)=i2c5(ϵ1μϵ2αϵμ3ϵα4+ϵ1μϵμ2ϵ3αϵ4α+ϵ1αϵ3μϵμ3ϵ4),

    (14)

    iMJ/ψψ(3770)J/ψψ(3770)=i4c6(ϵ1μϵ2αϵμ3ϵ+α4)+i2c7(ϵ1μϵμ2ϵ3αϵα4+ϵ1μϵα2ϵ3αϵμ4),

    (15)

    iMJ/ψψ(2S)J/ψψ(3770)=ic9(ϵ1μϵμ2ϵ3αϵα4+ϵ1αϵμ2ϵ3μϵα4)+i2c8(ϵ1μϵ2αϵμ3ϵα4).

    (16)

    After the same partial wave projection process of Eqs. (7) – (9), the coupled–channel partial wave amplitudes at the tree level, MJ,ijL=0(J=0,2 and i,j=1,2,3), are determined. By taking into account K-matrix unitarization and final state interaction, the unitarized partial wave amplitude is

    FJi(s)=3k=1αk(s)TJ,kiL,U(s) ,

    (17)

    where

    TJL,U(s)=[1iρKJL]1 .

    (18)

    αk(s) is the real polynomial function in general and is set to be constant here, and we set α1(s)2=1. Particularly, as for the case i=j=3 and J=2, the triangle diagram needs to be taken into consideration. That is to say KJ=2,i=3j=3L=0=MJ=2,i=3j=3L=0,tree+TJ=2triangle, in which the triangle diagram contribution comes from Fig. 1 5. For other cases, KJ,ijL=0=MJ,ijL=0,tree. Further, to fit the experimental data, one has

    dEventsids=Nipi(s)|Fi|2,

    (19)

    where pi(s) refers to the abs of three momenta for the corresponding channel. According to partial wave convention, for the J=0 and J=2 case, they have a total scale factor [11, 12]:

    |Fi|2=|FJ=0i|2+5|FJ=2i|2 .

    (20)

    The fit is overdone since there are many parameters. One solution is shown in Fig. 4, and the fit parameters are listed in Table 1 for illustration. In this fit, we set c1,c2,,c4 to be negligible simply because they are not directly related to the J/ψ(3770) channel and the fit can be performed reasonably well without them. The error band in Fig. 4 is rather large; this is due to the two normalization factors, N1 and N2, which contain rather large error bars.

    Figure 4.  (color online) Fit results from Table 1. Left: data from Ref. [1]. Right: data from Ref. [3].

    Parameterχ2/d.o.fgN1N2α2α3
    Fit0.9318.3±1.623±100.34±0.234.94±0.023.97±0.05
    c5 c6c7c8c9
    11.34±0.0550.79±0.0535.01±0.1964.71±0.071.529±0.002

    Table 1.  Fit parameters of Fig. 4. The ci parameters are defined in Eq. (10). The errors are statistical only.

    During the fit, many solutions exist. Nevertheless, it was found that the triangle diagram contributions are all small. This behavior is unclear; however, one possible reason could be that the peak position through the anomalous threshold contribution, as shown in Fig. 3, is approximately 60 – 80 MeV above the X(6900) peak; hence, the fit becomes difficult. One possible way to resolve the problem is to adopt another parameterization in which the background contributions are more flexible to be tuned. Thus, the interference between the background and the anomalous enhancement can lead to the shift of the peak position by a few tens of MeV. Another possible mechanism for the suppression of the triangle diagram is that the ψ(3770)DˉD vertex is in the p-wave form; hence, it may provide another suppression factor due to (non-relativistic) power counting. [20]. We defer this investigation for future studies.

ACKNOWLEDGEMENTS
  • We would like to thank De-Liang Yao and Ling-Yun Dai for the very helpful discussions.

Reference (20)

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