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The
X(6900) peak observed by the LHCb Collaboration in the di-J/ψ invariant mass spectrum [1, 2], and later in theJψψ(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 ofJ/ψψ(3770) ,J/ψψ2(3823) ,J/ψ ψ3(3842) , andχc0χc1 , whereasX(7200) is close to the threshold ofJ/ψψ(4160) andχc0 χc1(3872) ). Inspired by this, Ref. [5] studied the properties ofX(6900) andX(7200) by assuming theX(6900) coupling toJ/ψJ/ψ ,J/ψψ(3770) ,J/ψψ2(3823) ,J/ψψ3(3842) , andχc0χc1 channels and theX(7200) coupling toJ/ψJ/ψ ,J/ψψ(4160) , andχc0χc1(3872) channels. For the S-waveJ/ψJ/ψ coupling, the pole counting rule (PCR) [6], which has been applied to the studies of "XYZ " physics in Refs. [7−10], 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 thatX(6900) is unlikely a molecule ofJ/ψψ(3686) [5], a conclusion drawn before the discovery of Ref. [3]. More recently, Refs. [4, 11, 12] investigatedX(6900) using a combined analysis of di-J/ψ andJ/ψψ(3686) data and concluded thatX(6900) cannot be aJ/ψψ(3686) molecule.Nevertheless, as already stressed in Ref. [4], even though
X(6900) is very unlikely a molecule ofJ/ψψ(3686) , this does not mean that it has to be an "elementary state" (i.e., a compactˉcˉccc tetraquark state). It was pointed out that it is possible thatX(6900) be a molecular state composed of other particles, such asJ/ψψ(3770) , which form thresholds closer toX(6900) if the channel coupling is sufficiently large1 .This note will discuss a possible mechanism for the enhancement of the
J/ψψ(3770) channel coupling. TheDˉD (orD∗ˉD∗ ) component insideψ(3770) may play an important role, so far ignored in the literature, in explaining theX(6900) resonant peak through the anomalous threshold emerged from the triangle diagram generated by the D (D∗ ) loop, as depicted in Fig. 1.Noticing that
\psi(3770) or\psi'' couples dominantly toD\bar D , we start from the Feynman diagram as depicted in Fig. 1 by assuming that it contributes toJ/\psi\psi'' elastic scatterings near theJ/\psi \psi'' threshold2 . Assuming an interaction Lagrangian3 \begin{aligned}[b] \mathcal{L} =& -{\rm i}g(D^0\partial_\mu\bar{D}^0 - \bar{D}^0\partial_\mu D^0) \psi''^\mu -{\rm i}g (D^+\partial_\mu D^- - D^-\partial_\mu D^+) \psi''^\mu\\ &+ g' D^0\bar{D}^0 J/\psi^\mu J/\psi_\mu+ g' D^+D^- J/\psi^\mu J/\psi_\mu\ , \end{aligned}
(1) after performing the momentum integration, the amplitude as depicted by Fig. 1 is
\begin{aligned}[b] {\rm i}\mathcal{M} =& (-16 g^2 g') \; (\epsilon_2 \cdot \epsilon_4) \; \epsilon_1^\mu \epsilon_3^\nu \int {\rm d}^Dk\\&\times \frac{k_\mu k_\nu}{((k-p_1)^2-m_D^2)((k-p_3)^2-m_D^2)(k^2-m_D^2)}\\ \equiv& (-16 g^2 g') \; (\epsilon_2 \cdot \epsilon_4) \; \epsilon_1^\mu \epsilon_3^\nu \; \mathcal{A}_{\mu\nu} \end{aligned}
(2) where
\begin{aligned}[b] \mathcal{A}^{\mu\nu}& =\frac{-{\rm i}}{16\pi^2}\int_0^1 {\rm d}x\int_0^{1-x}{\rm d}y \; \left\{\frac{p_3^\mu p_1^\nu \; xy}{\Delta} - \frac{g^{\mu\nu}}{4} \Gamma(\epsilon) \frac{1}{\Delta^\epsilon} \right\}\\ & \equiv \frac{-{\rm i}}{16\pi^2} \left\{ p_3^\mu p_1^\nu\; \mathcal{B} + \int_0^1 {\rm d}x\int_0^{1-x}{\rm d}y \left(- \frac{g^{\mu\nu}}{4} \Gamma(\epsilon) \frac{1}{\Delta^\epsilon} \right)\right\}\ , \end{aligned}
(3) and
\begin{aligned}[b] \mathcal{B}(t) = &\int_0^1 {\rm d}x \frac{x}{2M^2} \Bigg( 2(M^2(1-2x)+tx)\\&\times\frac{\mathrm{ArcTan}\left(\dfrac{M^2-tx}{\Lambda(t,x)}\right)-\mathrm{ArcTan}\left(\dfrac{M^2(2x-1)-tx}{\Lambda(t,x)}\right)}{\Lambda(t,x)} \\ &+\left. \ln \frac{m^2+t(x-1)x}{m^2+M^2(x-1)x}\right)\ . \end{aligned}
(4) Here,
\Lambda(t, x)= \sqrt{ 4 m^2 M^2 - M^4 + 4 M^2 t x^2 - 2 M^2 t x - t^2 x^2} ,\Delta = M^2(x^2+y^2)+(2M^2-t)xy-M^2(x+y)+m^2 , and\Gamma(\epsilon) \dfrac{1}{\Delta^\epsilon} = \dfrac{1}{\epsilon} - \ln \Delta - \gamma + \ln 4\pi + \mathcal{O}(\epsilon) . On the right hand side of Eq. (3), only the\mathcal{B} term will be considered since the rest will be absorbed by the contact interactions to be introduced latter. M is the mass of\psi(3770) , and m is the mass ofD(\bar D) . Parameter g is the coupling strength of the\psi'' \;\; D\bar D three point vertex, andg' is the coupling strength of theJ/\psi J/\psi D\bar D four point vertex; {t=(p_2-p_4)^2 }. Parameter g can be determined by the\psi''\to D^0\bar{D}^0 decay process ,\begin{aligned}[b] {\rm i} \mathcal{M}_{\psi'' DD} =& {\rm i} \,g \,\epsilon(\psi'') \cdot [p(D^0)-p(\bar{D}^0)]\ ,\\ \Gamma =& \frac{1}{8\pi}\,|{\rm i} \mathcal{M}_{\psi'' DD}|^2\,\frac{q(DD)}{M_{\psi''}^2} = \frac{1}{6 \pi} \, g^2 \, \frac{q(DD)^3}{M_{\psi''}^2}\ , \end{aligned}
(5) where
q(DD) is the norm of the three-dimensional momentum ofD^0 or\bar{D}^0 in the final state. The PDG value\Gamma_{\psi''\to D^0\bar{D}^0} \sim 27.2\times 52{\text%} \times10^{-3} [16] determinesg \sim 12 4 . Parameterg' 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 ats_A = 4m^2-\frac{(M^2-2m^2 )^2 }{m^2}\ .
(6) Considering the mass of the
\psi'' andD^0 mesons, one obtainss_A=-1.28 GeV^2 (for theD^+ loop, –0.98 GeV^2 ). Numerically, function\cal{B} is plotted in Fig. 2(a), where one clearly sees the anomalous threshold beside the normal one att=4m^2 . Note that ifM^2 is smaller than2m^2 , the anomalous branch point is located below the physical threshold, but on the second sheet. It touches the physical threshold4m^2 and turns up to the physical sheet if the value ofM^2 increases to2m^2 . With a further increase inM^2 , the anomalous threshold moves towards the left on the real axis, passes the origin whenM^2=4m^2 , and finally reaches the physical value, i.e., –1.28 GeV^2 . The situation is depicted in Fig. 2(b). Note thats_A here is negative, contrary to what occurs with the deuteron, because the latter is a bound state with a normalizable wave function, whereas\psi'' 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
M^2 .To proceed, one needs to make the partial wave projection of
\mathcal{M} and obtain\begin{aligned}[b] T^J_{\mu_1\mu_2\mu_3\mu_4}(s)=&\frac{1}{32\pi}\frac{1}{2q^2(s)}\int^0_{-4q^2(s)}\,{\rm d}t\, d^J_{\mu\mu'}\\&\times\left(1+\frac{t}{2q^2(s)}\right)\mathcal{M}_{\mu_1\mu_2\mu_3\mu_4}(t)\ , \end{aligned}
(7) where the channel momentum square reads
q^2(s)= [s - (M - M_J)^2] [s - (M + M_J)^2]/4s ,M_J is the mass ofJ/\psi ,\mu_i denotes the corresponding helicity configuration, and\mu = \mu_1-\mu_2 ,\mu' = \mu_3-\mu_4 . The key observation is that the integral interval in Eq. (7) will covers_A if\sqrt{s}>6.96 GeV (for theD^+ loop,\sqrt{s}>6.94 GeV). In other words, the partial wave amplitude will be enhanced in the vicinity of theX(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/\psi\psi'' 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 theL = 0 (s wave) amplitudes, we need the relation between the s wave and helicity amplitudes (see Refs. [11, 12] for further discussions):\begin{aligned}[b] T_{L=0}^{0}(s)=&\frac{1}{3}\Big[2 T_{++++}^{0}(s)+2 T_{++-}^{0}(s)-2 T_{++00}^{0}(s)\\&-2 T_{00++}^{0}(s)+T_{0000}^{0}(s)\Big], \end{aligned}
(8) \begin{aligned}[b]T_{L=0}^{2}(s)=& \frac{1}{15}[T_{++++}^{2}(s)+T_{++-}^{2}(s)]\\ &+\frac{\sqrt{6}}{15}[T_{+++-}^{2}(s)+T_{+-++}^{2}(s)+T_{++-+}^{2}(s)+T_{-+++}^{2}(s)] \\ & +\frac{\sqrt{3}}{15}[T_{+++0}^{2}(s)+T_{+0++}^{2}(s)+T_{++0+}^{2}(s)+T_{0+++}^{2}(s)\\ &+T_{++-0}^{2}(s)+T_{-0++}^{2}(s)+T_{++0-}^{2}(s)+T_{0-++}^{2}(s)] \\ & +\frac{1}{5}[T_{+00+}^{2}(s)+T_{0++0}^{2}(s)+T_{+00-}^{2}(s)+T_{0-+0}^{2}(s)\\ &+T_{+0+0}^{2}(s)+T_{0+0+}^{2}(s)+T_{0+0-}^{2}(s)+T_{+0-0}^{2}(s)] \\ & +\frac{\sqrt{2}}{5}[T_{+-+0}^{2}(s)+T_{+0+-}^{2}(s)+T_{+-0+}^{2}(s)+T_{0++-}^{2}(s)\\ &+T_{-++0}^{2}(s)+T_{+0-+}^{2}(s)+T_{-+0+}^{2}(s)+T_{0+-+}^{2}(s)] \\ & +\frac{2}{15}[T_{++00}^{2}(s)+T_{00++}^{2}(s)+T_{0000}^{2}(s)]+\frac{2 \sqrt{6}}{15}[T_{+-00}^{2}(s)\\ &+T_{00+-}^{2}(s)]+\frac{2}{5}[T_{+-+-}^{2}(s)+T_{+-+}^{2}(s)] +\frac{2 \sqrt{3}}{15}[T_{+000}^{2}(s)\\ &+T_{00+0}^{2}(s)+T_{0+00}^{2}(s)+T_{000+}^{2}(s)]\ , \end{aligned}
(9) In practice, it is found that the anomalous enhancement gives a more prominent effect to the
J=2 amplitude than theJ=0 amplitude. Furthermore, to estimate the triangle diagram contribution, a combined fit withJ/\psi J/\psi andJ/\psi\psi(3686) data is made. A coupled–channel K-matrix unitarization scheme is employed includingJ/\psi J/\psi ,J/\psi\psi(3686) , andJ/\psi\psi(3770) . The tree level amplitudes are also taken into account from the following contact interaction Lagrangian [19]:\begin{aligned}[b] \mathcal{L}_c= & c_1 V_\mu V_\alpha V^\mu V^\alpha+c_2 V_\mu V_\alpha V^\mu V^{\prime \alpha}+c_3 V_\mu V_\alpha^{\prime} V^\mu V^{\prime \alpha}\\ &+c_4 V_\mu V^{\prime \mu} V_\alpha V^{\prime \alpha}+c_5 V_\mu V_\alpha V^\mu V^{\prime \prime \alpha} +c_6 V_\mu V_\alpha^{\prime \prime} V^\mu V^{\prime \prime \alpha}\\ &+c_7 V_\mu V^{\prime \prime \mu} V_\alpha V^{\prime \prime \alpha}+c_8 V_\mu V_\alpha^{\prime} V^\mu V^{\prime \prime \alpha}+c_9 V_\mu V^{\prime \mu} V_\alpha V^{\prime \prime \alpha}, \end{aligned}
(10) and these tree level amplitudes are as follows:
\begin{aligned}[b] {\rm i} M_{J / \psi J / \psi \rightarrow J / \psi J / \psi}= &{\rm i}\, 8\, c_1\Big(\epsilon_{1 \mu} \epsilon_{2 \alpha} \epsilon_3^{\dagger \mu} \epsilon_4^{\dagger \alpha}\\ &+\epsilon_{1 \mu} \epsilon_{2 \alpha} \epsilon_3^{\dagger \alpha} \epsilon_4^{\dagger \mu}+\epsilon_{1 \mu} \epsilon_2^\mu \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\dagger \alpha}\Big), \end{aligned}
(11) \begin{aligned}[b] {\rm i} M_{J / \psi J / \psi \rightarrow J / \psi \psi(2 S)}=&{\rm i} \, 2\, c_2\Big(\epsilon_{1 \mu} \epsilon_{2 \alpha} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \dagger \alpha}\\ &+\epsilon_{1 \mu} \epsilon_2^\mu \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \dagger \alpha}+\epsilon_{1 \alpha} \epsilon_{2 \mu} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \dagger \alpha}\Big), \end{aligned}
(12) \begin{aligned}[b] {\rm i} M_{J / \psi \psi(2 S) \rightarrow J / \psi \psi(2 S)}=&{\rm i} \, 4 \, c_3\left(\epsilon_{1 \mu} \epsilon_{2 \alpha}^{\prime} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \dagger \alpha}\right)\\ &+{\rm i} \, 2 \, c_4\left(\epsilon_{1 \mu} \epsilon_2^{\prime \mu} \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \dagger \alpha}+\epsilon_{1 \mu} \epsilon_2^{\prime \alpha} \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \dagger \mu}\right), \end{aligned}
(13) \begin{aligned}[b] {\rm i} M_{J / \psi J / \psi \rightarrow J / \psi \psi(3770)} =&{\rm i} \, 2\, c_5\Big(\epsilon_{1 \mu} \epsilon_{2 \alpha} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \prime \dagger \alpha}\\ &+\epsilon_{1 \mu} \epsilon_2^\mu \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \prime}{ }^{\dagger \alpha}+\epsilon_{1 \alpha} \epsilon_{3 \mu} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \prime}\Big), \end{aligned}
(14) \begin{aligned}[b] {\rm i} M_{J / \psi \psi(3770) \rightarrow J / \psi \psi(3770)} =&{\rm i}\, 4 \, c_6\left(\epsilon_{1 \mu} \epsilon_{2 \alpha}^{\prime \prime} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \prime+\alpha}\right)\\ &+{\rm i}\, 2\, c_7\left(\epsilon_{1 \mu} \epsilon_2^{\prime \prime \mu} \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \prime \dagger \alpha} + \epsilon_{1 \mu} \epsilon_2^{\prime \prime \alpha} \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \prime \dagger \mu}\right), \end{aligned}
(15) \begin{aligned}[b] {\rm i} M_{J / \psi \psi(2 S) \rightarrow J / \psi \psi(3770)}=&{\rm i} \, c_9 \left(\epsilon_{1 \mu} \epsilon_2^{\prime \mu} \epsilon_{3 \alpha}^{\dagger} \epsilon_4^{\prime \prime \dagger \alpha}+\epsilon_{1 \alpha} \epsilon_2^{\prime \mu} \epsilon_{3 \mu}^{\dagger} \epsilon_4^{\prime \prime \dagger\alpha}\right)\\ &+ {\rm i} \, 2 \, c_8 \left(\epsilon_{1 \mu} \epsilon_{2 \alpha}^{\prime} \epsilon_3^{\dagger \mu} \epsilon_4^{\prime \prime \alpha}\right). \end{aligned}
(16) After the same partial wave projection process of Eqs. (7) – (9), the coupled–channel partial wave amplitudes at the tree level,
M^{J,ij}_{L=0} (J=0,2 andi,j=1,2,3 ), are determined. By taking into account K-matrix unitarization and final state interaction, the unitarized partial wave amplitude isF^J_i(s) = \sum\limits^3_{k = 1} \alpha_k(s) T_{L,\,\mathcal{U}}^{J,ki}(s)\ ,
(17) where
T_{L,\,\mathcal{U}}^{J}(s) = [1-{\rm i} \rho K_L^{J}]^{-1}\ .
(18) \alpha_k(s) is the real polynomial function in general and is set to be constant here, and we set\alpha_1(s)^2 = 1 . Particularly, as for the casei = j = 3 andJ = 2 , the triangle diagram needs to be taken into consideration. That is to sayK_{L=0}^{J=2,i=3\,j=3} = M^{J=2,i=3\,j=3}_{L=0,\, {\rm tree}}+ T_{\rm triangle}^{J=2} , in which the triangle diagram contribution comes from Fig. 15 . For other cases,K_{L=0}^{J,\,ij} = M^{J,\,ij}_{L=0,\, {\rm tree}} . Further, to fit the experimental data, one has\frac{{\rm d} Events_i}{{\rm d}\sqrt{s}} = N_i\; p_{i}(s)\; |F_i|^2\; ,
(19) where
p_{i}(s) refers to the abs of three momenta for the corresponding channel. According to partial wave convention, for theJ=0 andJ=2 case, they have a total scale factor [11, 12]:\begin{array}{*{20}{l}} |F_i|^2 = |F_i^{J=0}|^2 + 5\; |F_i^{J=2}|^2\ . \end{array}
(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
c_1,\,c_2, \cdots, c_4 to be negligible simply because they are not directly related to theJ/\psi(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,N_1 andN_2 , which contain rather large error bars.Parameter \chi^2/d.o.f g' N_1 N_2 \alpha_2 \alpha_3 Fit 0.93 -18.3\pm 1.6 23\pm 10 0.34\pm 0.23 -4.94\pm 0.02 3.97\pm 0.05 c_5 c_6 c_7 c_8 c_9 11.34\pm 0.05 50.79\pm0.05 -35.01\pm0.19 -64.71\pm 0.07 1.529\pm 0.002 Table 1. Fit parameters of Fig. 4. The
c_i 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\psi(3770)D\bar D 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.
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We would like to thank De-Liang Yao and Ling-Yun Dai for the very helpful discussions.