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As mentioned earlier, there are numerous newly observed enhancements located near some
$ S $ -wave open thresholds to which the couplings turn out to be sizeable. We make some general discussions on the emergence of threshold structures and some observed enhancements can be possibly interpreted as hadronic molecules.In the meson sector the lowest-lying "stable" particles are pseudoscalar (P) and vector (V) mesons. Their
$ S $ -wave couplings give access to quantum numbers which can be identified as candidates for hadronic molecules. For some of the low excitations, e.g. the first orbital excitation states, there also exist narrow states with which the strong$ S $ -wave couplings may give access to more quantum numbers. Whether a$ P $ -wave coupling can lead to hadronic molecular structures is still a question under debating. We will comment some issues concerning the$ P $ -wave interactions.In Table 1 the possible quantum numbers accessible via the "stable" states are listed. In the heavy meson sector there are candidates that can fit in the category. The observation of
$ X(3872) $ by Belle in 200[41] is often marked as the first evidence for exotic heavy-quarkonium-like states. Since then, a large number of the so-called "$ XYZ $ states" have been observed in experiment and initiated a lot of theoretical studies. See e.g. recent reviews and different theoretical interpretations therein[16-20]. As mentioned earlier, one interesting features with these$ XYZ $ states is that most of them are located near some$ S $ -wave open channels to which these states seem to have very strong couplings.Two-body channel $ S $-wave ($ L = 0 $) $ P $-wave ($ L = 1 $) Open charm Open bottom PP $ 0^{+(\pm)} $ $ 1^{-(\pm)} $ $ D\bar{D} $ $ B\bar{B} $ PV $ 1^{+(\pm)} $ $ 0^{-(\pm)}, \ 1^{-(\mp)}, \ 2^{-(\pm)} $ $ D^*\bar{D}+c.c. $ $ B^*\bar{B}+c.c. $ VV $ 0^{+(+)} $ $ 1^{-(+)} $ $ D^*\bar{D}^* $ $ B^*\bar{B}^* $ $ 1^{+(-)} $ $ 0^{-(-)} $, $ 1^{-(-)} $, $ 2^{-(-)} $ $ 2^{+(+)} $ $ 1^{-(+)} $, $ 2^{-(+)} $, $ 3^{-(+)} $ PA $ 1^{-(-)} $ $ \dots\dots $ $ D\bar{D}_1+c.c. $ $ B\bar{B}_1+c.c. $ VA $ 0^{-(\pm)} $, $ 1^{-(\mp)} $, $ 2^{-(\pm)} $ $ \dots\dots $ $ D^*\bar{D}_1+c.c. $ $ B^*\bar{B}_1+c.c. $ Table 1. Quantum numbers accessible via the pseudoscalar, vector, and axial vector meson two-body interactions.
As an example to demonstrate the threshold phenomena, we focus on the
$ D^*\bar{D}+c.c. $ threshold where two exotic candidates,$ X(3872) $ [41] and$ Z_c(3900) $ [42-44], have been observed in experiment. With their masses close to the$ D^*\bar{D}+c.c. $ threshold and the quantum numbers measured in experiment, i.e.$ (I^G, \ J^{PC}) = (0^+, \ 1^{++}) $ and$ (1^+, \ 1^{+-}) $ , respectively, they can couple to the$ D^*\bar{D}+c.c. $ via the following configurations with fixed$ C $ -parity:Here,
$ \hat C {\cal M} = \bar{\cal M} $ is adopted for the$ C $ -parity transformation.Another advantage for the heavy meson system with a small binding energy is that it allows the implementation of the non-relativistice effective field theory (NREFT) approaches which are improvable order by order with the inclusion of local operators and pion exchanges, and the short-distance physics is encoded in the coefficients of the local operators. The compositeness theorem by Weinberg then connects the effective coupling constant of the physical state to the open channel continuum with the probability of finding the compact short-distance component in the wavefunction (see Eq. (12)).
The success of the EFT approaches can be recognized via the indication of a universality of the long-distance wavefunction which is insensitive to the binding mechanisms. It should be noted that the physical coupling,
$g_{\rm eff}^2\equiv\lambda^2 g_0^2 = \frac{2\pi\sqrt{2\mu E_{\rm B}}}{\mu^2}(1-\lambda^2)$ , suggests that there must be a short-distance component inside a hadronic molecular state given that$ E_{\rm B}> 0 $ . However, a quantitative determination of the short-distance dynamics is nontrivial. For a bound state the probability of finding the compact component in the physical wavefunction, i.e.$ \lambda^2 $ , is insufficient for disentangling the detailed dynamics since they are hard-scale physics which have been integrated out and absorbed into some unknown coefficients. Thus, observables sensitive to the short-distance physics thus should be identified and investigated.To be slightly specific here, we discuss some interesting threshold phenomena with the exotic candidates
$ X(3872) $ and$ Z_c(3900) $ . We skip details about the experimental status and theoretical models since one can refer to the recent reviews[16-20] and references therein, but only focus on some key points which turn out to be strongly associated with the threshold dynamics and kinematics, and need further attention or investigations. -
The strong attractive
$ S $ -wave interaction between$ D^{*}\bar{D}+c.c. $ was studied by Törnqvist and the$ X(3872) $ with$ (I^G, \ J^{PC}) = (0^+, \ 1^{++}) $ was literally predicted as a$ D^{*}\bar{D}+c.c. $ bound state which is an analogue of the deuteron[45] in a sense of the crucial role played by the long-ranged pion exchange potential. The mass of$ X(3872) $ almost sits at the$ D^{*0}\bar{D}^0+c.c. $ threshold with a small binding energy$ B_X = M_{D^0}+M_{D^{*0}}-M_X = 0.00\pm 0.18 $ MeV[46]. It makes$ X(3872) $ an ideal candidate for the hadronic molecule. In contrast, the potential quark model did not predict any state in such a mass region[6].The small binding energy, or large scattering length for the
$ S $ -wave$ D^{*0}\bar{D^0}+c.c. $ scattering implies a universality of the long-distance wavefunction of$ X(3872) $ to be insensitive to the binding mechanism. Effective field theories were developed to study the dynamics arising from the$ S $ -wave$ D^{*0}\bar{D^0}+c.c. $ scattering in association with the production and decays of$ X(3872) $ [47-50]. However, the case for the$ D^{*}\bar{D}+c.c. $ interactions should be significantly different from the proton-neutron long-range interactions since in the pion exchange potential for the$ D^{*}\bar{D}+c.c. $ interactions the exchanged pion can be on-shell and contribute to the potential with an imaginary part. This gives rise to a three-body cut in the scattering amplitude and makes the potential approximation for the pion exchange invalid[47-50]. A detailed review of the hadronic molecule interpretation of$ X(3872) $ can be found in Ref. [16].It is crucial that the quantum numbers of
$ X(3872) $ have been unambiguously determined as$ (I^G, \ J^{PC}) = (0^+, \ 1^{++}) $ . This makes it natural to include the$ S $ -wave open channel contributions and allows a separation of different energy scales. Thus, the EFT approaches can be implemented.For scatterings between two heavy flavor mesons the heavy quark spin symmetry (HQSS) in the limit of infinitely large mass for the heavy quarks can be implemented. In the limit of infinitely large heavy quark mass the heavy quark spin will decouple from the light ones, thus, conserve in the scattering. For convenience, the heavy-light meson can be labelled by the total angular momentum carried by the light quark, i.e.
$ j^P $ , with$ { j} = { l}+{ s_q} $ and$ P $ for the meson parity. In the HQSS limit,$ D $ and$ D^* $ (similarly$ B $ and$ B^* $ ) form a spin doublet$ j^P = 1/2^- $ . Thus, the light quarks in the$ D^*\bar{D}+c.c. $ system with the relative orbital angular momentum$ { l} = 0 $ can form$ j^P = 0^+ $ and$ 1^+ $ .By assigning
$ X(3872) $ as a hadronic molecule of the$ D^{*0}\bar{D}^0+c.c. $ pair with$ (I^G, \ J^{PC}) = (0^+, \ 1^{++}) $ , it implies the possible existence of accompanying states in the multiplet of$ j = 1 $ with$ J^{PC} = 0^{++}, \ 1^{+-}, \ 2^{++} $ . However, note that such a scenario is only based on the HQSS. In reality, the HQSS is broken and more complicated phenomena are anticipated. Firstly, the$ D $ and$ D^* $ mass splitting will lead to different mass corrections in different channels. Secondly, the role played by the pion exchange potentials is different in different channels. For instance, the pion exchange is forbidden in the$ D\bar{D} $ channel. One need understand how crucial the pion exchange potential is in the formation of hadronic molecule states.Taking the pion exchange potential in the channel of
$ (I^G, \ J^{PC}) = (0^+, \ 1^{++}) $ for$ X(3872) $ as a reference, the light quarks couple to$ j = 1 $ , while the isospin is zero. This gives a sign constraint indicating the attractive pion exchange potential:$ \langle{ \sigma}_1\cdot { \sigma}_2\rangle \langle{ \tau}_1\cdot { \tau}_2\rangle = 1\times (-3) = -3 $ with$ ({ \sigma}_1+ { \sigma}_2)/2 = 1 $ and$ ({ \tau}_1+ { \tau}_2) = 0 $ . Note that$ { \sigma}_1 = 2{ s}_1 $ and$ { \sigma}_2 = { s}_2 $ are the Pauli operators for the interacting light quarks. If the pion exchange also provides an attractive potential for$ Z_c(3900) $ , it should require that$ ({ \sigma}_1+ { \sigma}_2)/ 2 = 0 $ , hence$ \langle{ \sigma}_1\cdot { \sigma}_2\rangle \langle{ \tau}_1\cdot { \tau}_2\rangle = (-3)\times 1 = -3 $ . However, one notices that in such a case$ Z_c(3900) $ belongs to the multiplet with$ j_l = 0 $ . Thus,$ Z_c(3900) $ and$ X(3872) $ cannot be connected with each other by the HQSS. This seems to be able to provide some hints about the production mechanisms for$ X(3872) $ and$ Z_c(3900) $ . Meanwhile, one notices that the$ 1^{+-} $ HQSS partner of$ X(3872) $ is generated by the$ D^*\bar{D}^* $ coupling which can be related to$ Z_c(4020/4025) $ observed in experiment. However, in this case the pion exchange provides a repulsive potential.An interesting consequence is that in the limit of the HQSS the
$ Z_c(3900) $ decays into$ \eta_c\rho $ will be suppressed. However, the recent BESIII measurement shows that the ratio of$ B.R.(Z_c^\pm(3900)\to \rho^\pm\eta_c)/ B.R.(Z_c^\pm(3900)\to J/\psi\pi^\pm) = 2.3\pm 0.8 $ [51], which seems to be contradicting with the expectation of the hadronic molecular picture. Such a conclusion needs to be cautioned because$ \rho $ meson is broad. The decay channel of$ Z_c^\pm(3900)\to \rho^\pm\eta_c $ will be affected significantly by the phase space correction due to the broad width of the$ \rho $ meson. Meanwhile, comparing the rescatterings between$ D^*\bar{D}+c.c.\to \rho\eta_c $ and$ J/\psi\pi $ by exchanging a$ D $ meson, one sees that the exchanged$ D $ meson can become highly off-shell in the$ J/\psi\pi $ channel due to the large mass difference between$ J/\psi $ and$ \pi $ . This will introduce more suppressions into the$ J/\psi\pi $ channel. In addition, the HQSS breaking allows the transition between$ D^*\bar{D}^* $ and$ D^*\bar{D}+c.c. $ and the$ \rho\eta_c $ channel may get further enhanced. Future detailed studies are needed to clarify these issues.The HQSS symmetry breaking also leads to isospin-breaking effects in
$ X(3872) $ decays into$ J/\psi\rho $ . Experimentally, the branching fraction,$ B.R.(X(3872)\!\to\! J/\psi\pi^+\pi^-\pi^0)/B.R.(X(3872)\to J/\psi\pi^+\pi^-)\simeq 1 $ , where the$ \pi^+\pi^-\pi^0 $ and$ \pi^+\pi^- $ are dominantly from the$ \omega $ and$ \rho^0 $ decays, respectively. This result can be understood by the large mass splitting between the$ D^{*0}\bar{D}^0+c.c. $ and$ D^{*+}D^-+c.c. $ thresholds and to both of them the physical state has large couplings. The isospin breaking effects can then be accounted for by the nonvanishing cancellation between the charged and neutral$ D^*\bar{D} $ loop amplitudes. In addition, the$ J/\psi\pi^+\pi^-\pi^0 $ channel is relatively suppressed by the phase space in comparison with the$ J/\psi\pi^+\pi^- $ . This measurement provides a strong evidence for the hadronic molecule interpretation for$ X(3872) $ .In the hadronic molecule scenario observables, which are driven by mechanisms involving the hadronic degrees of freedom, should be useful for probing the corresponding structures in the wavefunction. For instance, the strong decay of
$ X(3872)\to D^{0}\bar{D}^0\pi^0 $ should be dominated by the constituent$ D^{*0} $ decays into$ D\pi $ . The radiative decay of$ X(3872)\to D^{0}\bar{D}^0\gamma $ was also regarded as a probe of the long-distance wavefunction via the radiative decays of$ D^*\to D\gamma $ . However, one should be cautioned by the observation that the long-distance wavefunction contributes to the partial width via the$ M1 $ transition. In contrast, the transition of$ \chi_{c1}(nP)\to \gamma D\bar{D} $ can pick up a$ E1 $ transition operator followed by a strong vector charmonium coupling to$ D\bar{D} $ . So, a small short-distance component of$ c\bar{c} $ may contribute to the radiative decay the same order of magnitude as the long-distance component. This actually raises the question on the role played by the short-distance component of the$ X(3872) $ wavefunction[52]. Proper criteria which are sensitive to the underlying by mechanisms still need to be explored. -
The production mechanisms for both
$ X(3872) $ and$ Z_c(3900) $ have been explored in experiment and theory[16-20]. Instead of going through various model prescriptions, we again focus on some key issues which turn out to be closely related to the threshold phenomena.As candidates for the hadronic molecules the production mechanisms for these two states seem to be very controversial. An intuitive argument would be that their productions should be driven by the productions of their constituent hadrons, i.e.
$ D^*\bar{D}+c.c. $ , in order to feed the long-distance wavefunctions. Namely, their productions should be dominantly via the$ D^*\bar{D}+c.c. $ rescatterings. This argument is based on the wavefunction dominance of the long-distance component and the short-distance part is neglected. However, this is not the case in the production processes. One point to recognize this problem is that the virtual momentum for the charged$ D^*\bar{D}+c.c. $ pair is at the order of 120 MeV, which is lower than the typical$ \Lambda_{QCD} $ scale and should be explicitly considered. Some detailed discussions on the role of the short-distance component can be found in Ref. [16]. Here, we discuss the productions of$ X(3872) $ and$ Z_c(3900) $ in$ B $ meson decays to show that the short-distance component cannot be neglected.For
$ B\to D^*\bar{D}K $ via$ B\to X(3872)K $ the production mechanisms can be illustrated by Fig. 3[52]. The relevant Feynman rules for the$ X(3872) $ in the EFT can be set up following the scheme of Sec. 2.1. The propagator can be expressed asFigure 3. Leading diagrams for
$B\to X(3872)K\to $ $ D^{*0} \bar D^0 K$ [16]. The solid lines in the loop and final state represent the charm and anti-charm mesonswhere
$ \Gamma $ denotes the non-$ D^{*0}\bar D^0 $ partial decay width of$ X(3872) $ . Note that a factor of$ \sqrt{2M_X} $ has been absorbed into the$ X(3872) $ field operator. In this convention a boson field has the dimension of$ 3/2 $ and the Feynman rule for an external boson is$ \sqrt{2M} $ . The vertex coupling for$ XD^{*0}\bar D^0 $ is given aswhere the factor
$ 1/\sqrt{2} $ is due to the definition of the$ C $ -even state$ (D^{*0}\bar D^0+D^0\bar D^{*0})/\sqrt{2} $ .The leading order transition amplitudes in Fig. 3 can be explicitly written as
where
$ { p}_K $ is the momentum of the$ K $ meson in the$ B $ meson rest frame;$ { \epsilon}^* $ is the polarization vector of the outgoing$ D^{*0} $ ;$ {\cal{A}}_{\rm XK} $ and$ {\cal{B}}_{\rm DDK} $ denote the production vertices$ B\rightarrow X(3872)K $ and$ B\rightarrow D^{*0}\bar D^0 K $ , respectively. Near the threshold of$ D^{*0}\bar D^0 $ , the two form factors$ {\cal{A}}_{\rm XK} $ and$ {\cal{B}}_{\rm DDK} $ are treated as constants.The relative strength between
$ {\cal{M}}_a $ and$ {\cal{M}}_b $ can be examined by the NREFT power counting. Note that near threshold the binding momentum$ \gamma = (2\mu B)^{1/2} $ and the three momentum of the charmed meson$ p $ are the same order, i.e.$ \gamma, \ p\sim{\cal{O}}(p) $ . Thus, we count$ E, \ B\sim{\cal{O}}(p^2) $ and$ g\sim {\cal{O}}(p^{1/2}) $ [24-25]. It leads to$ {\cal{M}}_a\sim{\cal{O}}(p^{-3/2}) $ and$ {\cal{M}}_b\sim{\cal{O}}(p^0) $ . From the power counting, one can see that the short-distant production mechanism is more important than the long-distance one. Interestingly, it should be noted that$ {\cal{M}}_a $ is proportional to the factor$ \sqrt{Z} $ . Therefore, it will be suppressed given that$ X(3872) $ is dominated by a molecular component. In the limit of$ Z\to 0 $ the term of$ {\cal{M}}_a $ will vanish. Then, the production of$ X(3872) $ will be via the long-distance production mechanism$ {\cal{M}}_b $ . As shown in Ref. [52], lineshape of the$ D^{*0}\bar{D}^0 $ spectrum in$ B\to D^{*0}\bar{D}^0 K $ is sensitive to the contributions from the short-distance production of$ {\cal{M}}_a $ . This feature naturally accounts for the short-distance production of hadronic molecules in various processes. See e.g. a similar analysis of the production of$ Y(4260) $ as a hadronic molecule of$ D_1(2420)\bar{D}+c.c. $ in$ e^+e^- $ annihilations[37,53-56].The production of
$ Z_c(3900) $ shares some similar features as$ X(3872) $ if it is a hadronic molecule. However, there are certain aspects that its production in$ B $ decays is different. In Fig. 4 we compare the production mechanisms for$ X(3872) $ and$ Z_c(3900) $ via (a) short-distance and (b) long-distance processes. Since$ Z_c(3900) $ is an isovector, its short-distance wavefunction (whatever it could be) cannot be produced directly. Only the long-distance process is allowed for its creation via the$ D^*\bar{D} $ scatterings. In contrast,$ X(3872) $ contains a short-distance component (e.g. a mixture of$ \chi_{c1}(2P) $ state) and a direct production via the$ c\bar{c} $ component is allowed and can be even dominant. This naturally explains why$ X(3872) $ is richly observed in high-energy production processes while$ Z_c(3900) $ so far has not been observed in the exclusive$ B $ decays.Figure 4. (color online) Schematic diagrams for the production of
$X(3872)$ and$Z_c(3900)$ in$B\to D^* {\bar D} K$ . The isoscalar short-distance component of$X(3872)$ allows its production directly via a$c\bar{c}$ configuration in (a) while both can be produced via the long-distance component of$D^*\bar{D}$ scatterings in (b)A novel phenomenon with the productions of
$ X(3872) $ and$ Z_c(3900) $ is associated with the role played by the TS mechanism. Being located in the vicinity of the$ D^*\bar{D}+c.c. $ threshold, these two states can be significantly affected by the TS mechanism in their productions. Following the observations of$ Z_c(3900) $ and$ Z_c(4020) $ in experiment[42-44,57], it was first proposed in Refs. [53,58] that the productions of these two states are closely related to the$ S $ -wave open thresholds,$ D_1(2420)\bar{D}+c.c. $ and$ D_1(2420)\bar{D}^*+ c.c. $ , in$ e^+e^- $ annihilations. These two thresholds have large couplings to the nearby charmonium-like states,$ Y(4260) $ and$ Y(4360) $ , which make it possible to interpret them as hadronic molecule states composed of these constituent hadrons. Furthermore, the productions of$ Z_c(3900) $ and$ Z_c(4020) $ via the decays of$ Y(4260) $ and$ Y(4360) $ will be enhanced by the TS mechanism. This phenomenon has been investigated by a series of works[37,53-56,58-60] which provide a self-consistent prescription of the so-far available experimental observables. It is interesting to note that it seems that all the present signals for$ Z_c(3900) $ are associated with the production of$ Y(4260) $ . Apart from those observed in$ e^+e^- $ annihilations, the D0 Collaboration observed$ Z_c(3900) $ in the decays of$ Y(4260) $ which are produced in semi-inclusive decays of$ b $ -flavored hadrons[61].Similar phenomenon was predicted by Ref. [62] where the TS mechanism plays an important role to enhance the production rate for
$ X(3872) $ in the energy region of$ Y(4260) $ in$ e^+e^- $ annihilations. It was later confirmed by the experimental data[63] which has provided very useful information about the structure of both$ X(3872) $ and$ Y(4260) $ . Recently, it was proposed by Ref. [64] to precisely measure the mass position of$ X(3872) $ in its productions via the TS mechanism. Although$ X(3872) $ has been confirmed in various processes and can be regarded as a well-established state, its mass position still has sizeable uncertainties. A precise measurement of its mass position is crucial for understanding its nature via the pole structure of the scattering amplitude in the energy complex plane. The intriguing idea of Ref. [64] will help experimentalists achieve a high-precision measurement of this quantity.
Threshold Phenomena and Signals for Exotic Hadrons
doi: 10.11804/NuclPhysRev.37.2019CNPC78
- Received Date: 2020-04-27
- Accepted Date: 2020-05-09
- Available Online: 2020-09-30
- Publish Date: 2020-09-20
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Key words:
- Hadron physics /
- hadron spectroscopy /
- QCD exotics /
- hadronic molecule /
- threshold
Abstract: We present a brief review of some topical issues in the study of QCD exotic hadrons. A special emphasis of the threshold phenomena is made by taking into account the implementation of the effective field theory study of hadronic molecules and the impact arising from the triangle singularity. A combined analysis may provide some clues towards a better understanding of the hadron spectroscopy.
More sophisticated approaches including multi-channel effects and keeping unitarity are often called coupled-channel method.Such states are often interpreted as the eigen-states of the potential quark model Hamiltonian which only involves the quark-quark or quark-antiquark interactions, namely, the short-ranged interactions.
Here, we keep the mass term in order to connect the nonrelativistic quark model eigenvalues to the eigenvalues of Hc.
Citation: | Qiang ZHAO. Threshold Phenomena and Signals for Exotic Hadrons[J]. Nuclear Physics Review, 2020, 37(3): 260-271. doi: 10.11804/NuclPhysRev.37.2019CNPC78 |