Photoproduction of strange hidden-charm and hidden-bottom states

Recently BESIII collaboration discovered a charged strange hidden-charm states $Z_{cs}$(3985) in the $D_s^-D^{*0} + D_s^{*-}D^{0}$ spectrum. A higher $Z'_{cs}$ state coupling to $\bar{D}_s^{*-}D^{*0}$ is expected to exist by SU(3)-flavor symmetry, and their bottom partners are anticipated by heavy quark flavor symmetry. Here we study the photoproduction of these exotic states and explore the possibility of searching for them in future electron-ion colliders (EIC), where the background from Pomeron exhcange is carefully investigated. Our results also indicate that the maximal photoproduction cross section of strange partner is around 1 $\sim$ 2 orders of magnitude smaller than that of the corresponding non-strange states.


I. INTRODUCTION
Exotic hadron candidates challenge our understanding of the strong interaction and hence shed light on the underlying mechanism constrained by Quantum Chromodynamics (QCD) [1][2][3][4][5][6][7][8][9]. The states in the charm and bottom region are arranged into the well organized patterns in the spectroscopy, constrained by a high degree heavy quark spin and flavor symmetry (HQSS and HQFS). The most successful and evident of these patterns applies to the pentaquark P c in light of the latest LHCb data [10]. The scattering potential involving only short-range operators constrained by heavy quark spin symmetry can manifest the first example of full and complete HQSS multiplet of baryon-like P c [11][12][13]. The one-pion exchange as well as the long range interaction would also play an important role [13,14]. The strange hidden-charm pentaquark was investigated by several theoretical methods [15][16][17][18][19][20][21].
Until recently it was announced in a talk at the Implications Workshop [22].
A particular feature in the strange sector is that the one-pion exchange is forbidden by parity and isospin for D − s D * 0 /D * − s D 0 interaction. Other mesons, e.g. σ, φ and ω, are allowed with the effective interaction range less than 0.5 fm, much shorter than π-exchange [43,44].
Whether the induced strength by these heavier meson is enough to bind the D − s D * 0 /D * − s D 0 system is questioned because of the OZI suppression [45,46]. The bound between two heavy mesons is much tighter than that for non-strange case so the underlying scenario would be dominant by short range contact interaction as discussed in Ref. [47], which also proposes a hypothesis to compute the coupling constant. Thus the mechanism of Z cs (3985) formation needs further investigation. The J/ψK decay channel of Z cs was thought to be important in the earlier studies, e.g. QCD sum rule [48,49], initial K-meson emission mechanism [50] and the hadrocharmonium picture [51]. A rich spectrum of Z cs and Z bs coupled to the hidden channels were predicted in the compact tetraquark [52]. However, a null result in e + e − → K + K − J/ψ channel from Belle [53,54] at center-of-mass (c.m.) energies between the threshold and 6.0 GeV and BESIII [55] with lower c.m. energies indicates a relatively small J/ψK partial decay width, calling for higher statistics to study in future BESIII and BelleII experiments.
The photoproduction of some exotic states, e.g. Z c (4430) and pentaquarks P c,b , have attracted a lot of theoretical attention [56][57][58][59][60][61][62][63][64][65][66][67][68], due to the contribution of kinematic effect to the resonance structure [69,70]. Particularly, though the triangle diagrams could be present in this reaction, it is hardly possible to satisfy the on-shell conditions of the triangle singularity [71]. Thus their contribution is expected to be negligible. On the other hand, the discovery potential in photoproduction is also very essential. For example, the exotic Y (3940) [72], X(3915) [73], Z c (3900) [74], and Z c (4200) [75] through photoproduction were discussed theoretically. Experimental effort was also devoted to the photoproduction of P c [76] by GlueX collaboration, Z c (3900) [77] and X(3872) [78] by COMPASS collaboration. Therefore it is naturally speculate if it is possible to study strange exotic states in photoproduction. In this paper we study the Z
In the vector meson dominant (VMD) model [79,80], the coupling of a vector meson V with mass M V to photon γ is written as in these diagrams.  where f V is the decay constant of the vector meson, determined by the dilepton decay width with α = e 2 /(4π) = 1/137 being the fine-structure constant. We obtain f J/ψ = 11.16, Particle Data Group (PDG) [23]. While the effective Lagrangian for the coupling of N KΛ is taken from Ref. [81] as where the coupling constant g N KΛ = 14.0 [82].
As discussed in Sec. I among several possible exotic state assignments to Z cs , the Swave D − s D * 0 + D * − s D 0 molecular state is the most popular explanation with the spin-parity J P = 1 + . In the following formula, we use Z and V to denote Z ( ) cs/bs and vector quarkonium, respectively. Then the effective Lagrangian of the ZV K coupling is [67] The dimensionless coupling constant g ZV K can be determined by the corresponding decay where p cm and E cm are the three-vector momentum and energy of the K-meson in the Z rest frame, respectively. The masses and widthes, and the coupling strengths of Z ( ) are summarized in Table I We introduce a form factor F s to regularize the off-shell Z resonance, Here, the cutoff is set as the mass of the intermediate vector meson, e.g. Λ Z = M ψ(nS) or M Υ(nS) [83]. An alternative form factor usually are used in literature, but will not change the final results in charm sector much because we concentrate on the energies where Z ( ) cs is nearly on-shell. But for Z ( ) bs → Υ(2S)K vertex, this form factor encounters a non-physical pole [81], which shall be avoided. Thus the form factor in Eq. (6) is a more reasonable choice. A monopole form factor is introduced to regularize the off-shell effect in the t-channel propagator: where Λ t is the corresponding cutoff parameter.
With the above prescription, we obtain the differential cross section of the γp → Z cs /Z bs Λ: where t is the square of the four-momentum transfer from initial proton to final Λ-baryon,  [84]. This has been adopted in the photoproduction of Z c (4430) [68] and Z c (4200) [75]. Since we concentrate on the threshold region, the choice of ordinary meson propagator in above equation is favored.
The amplitude of γp → V KΛ in Fig. 3 reads generally as, with γµ and V ν being the polarization tensor of initial photon γ and final vector quarkonium  Fig. 3(a) is where q, p, p , p V , p K and p Z = p V + p K are the momentum of photon, proton, Λ, J/ψ, bs signal. This diffractive process with a Pomeron trajectory G P (s, t ) = −i(α s) +α t could be proceeded by final or initial emission of K-meson, as depicted in Fig. 3(b) and Fig. 3(c), respectively. The parameters of Pomeron trajectory are well known, e.g. = 0.08 and α = 0.25 GeV −2 [84,85]. The γV P vertex can be described by gauge invariant coupling 2β c V (t )T µαν [86,87] with here µ 0 = 1.2 GeV and β 2 c = 0.8 GeV 2 for charmonium, and β 2 b = 0.1 GeV 2 for bottomonium. The Pomeron-nucleon interaction in Fig. 3(b) can be written in a manner of vector coupling 3β 0 f (t )γ µ where β 0 = 2 GeV is the coupling constant between Pomeron and constituent quark within nucleon. The f (t ) is the parameterized nucleon electromagnetic form factor (EFF) [85]: with the squared energy transfer t = (p V − q) 2 in the unit of GeV 2 . This prescription is widely used in J/ψ photoproduction and describes well the data in wide energies [88,89].
Then the amplitude in Fig. 3(b) is calculated as (15) where p N and M N are the four momentum and mass of intermediate proton, respectively.
The form factor F t (Λ N , M N ) is presented to take into account the off-shell effect of nucleon with Λ N the cut-off parameter. Other nucleon excited states N * with strong coupling to KΛ would be also contribute to this diagram. We neglect them for the moment because of the unknown coupling strength of Pomeron-N * interaction. The Pomeron-Λ interaction is expected to be similar to that of Pomeron-nucleon. However, both its coupling strength β 0 and the Λ EFF [90] are poorly known. Hence here we neglect the contribution in Fig. 3(c).
As a matter of fact, the initial emission of K-meson is kinematically unfavored and is  expected to be small, similar to the initial emission of π-meson in the study of various Z c photoproduction [67,74,75].
We can evaluate the cross sections and distributions of final particles in three-body phase space with the amplitudes in Eq. (10). In order to compare with the results in the bottom sector, we also present the results of Z ( ) b states, which has been estimated by JPAC [91]. The formalism is quite analogous to above one with the exception of isospin factors, which can be easily found in references [67,74]. The relevant parameters of Z ( ) b are adopted from PDG [23] and listed in Table I.

III. NUMERICAL RESULTS AND DISCUSSION
We show the cross sections of the γp → ΛZ ( ) cs in Fig. 4. As explained in Sec. II, we plot the σ/B(Z → J/ψK) becasue of the unknown B(Z → J/ψK). The differential cross sections in Fig. 4(a) are featured by a typical behavior of t-channel meson exchange, decreasing rapidly with larger |t|. So the Z ( ) cs are produced in the forward beam direction. This is also applicable to Z ( ) bs states, since they are driven by the similar production mechanism. The magnitude of Z ( ) cs production cross sections is two orders smaller than that of the Z ( ) c states [73,74], as can be seen from Fig. 4(b). Here we need to point out that for Z   Table. I. since the coupling constants g V ZM are calculated with the measured branching fractions, the cross section distributions in Fig. 4 (b) and Fig. 5 do not divide by B(Z → V K) again.
Because of the close masses, the maximal production of Z cs and Z cs is nearly the same and at around 7.0 GeV.
We exhibit in Fig. 5 the differential and total cross sections of γp → BZ The same is true for Z ( ) bs photoproduction in γp → ΛΥK. This is not considered by previous study of Z c photoproduction in the γp → pJ/ψπ [67,68,72,74,75] mainly because of the unknown relative phases of various amplitudes. To further proceed we have to assume that these relative phases are zero and various contributions are constructively interfered. In bs ) is present. In Fig. 7(a,c), we present the Dalitz plot and J/ψK invariant mass spectrum of the γp → ΛJ/ψK reaction at √ s =8.0 GeV. It is hopeful to separate the Z cs and Z cs in this channel if enough events are accumulated. In Fig. 7(b,d), we show the Dalitz plot and ΥK invariant mass spectrum of the γp → ΛΥK reaction at 16.76 GeV, which is the optimal energy of proposed EicC [92]. It can be seen that it is very challenging to distinguish the very narrow Z bs and Z bs because the invariant mass M 2 ΥK covers a very wide in this kinematic range, so a very fine resolution of mass is required.

IV. SUMMARY AND CONCLUSION
Stimulated by the newly observed charged strange hidden-charm state Z cs (3985) and the charm-strange states X 0 /X 1 (2900) withcsud, a full axial-vector heavy quark Z spectroscopy is believed to be emerging. The HQSS has spoken that exotica come with pairs. In this work we investigate the photoproduction of Z bs are below 1 nb and 0.1 nb, respectively. The electroproduction cross sections of these states would be further reduced by two orders of magnitude due to the electromagnetic coupling. The Dalitz plots and invariant mass spectra indicate that a fine mass resolution of J/ψK in the γp → ΛJ/ψK and ΥK in γp → ΛΥK would be prerequisite to identify these narrow states experimentally. As a result, it is very challenged to search for them at EicC and a higher luminosity design is favored, e.g. US-EIC. Our estimations give a natural order hierarchy for the photoproduction of strange and non-strange states photoproduction in charm and bottom sector, as already found in the light quark sector, e.g. pp collisions [81,94] and γp reactions [95]. Then we can expect that the cross section of X 0 /X 1 (2900) photoproduction is very tiny, e.g. through γp → X 0/1 Λ + cK 0 .
Before taking seriously these results, it is worth pointing out that several sources of model uncertainty shall be keep in mind. First, the mentioned destructive interference of nearby twin states would reduce the cross sections. Second, the VMD model is not careful inspected in the heavy quark sector, as noticed by the previous studies [58,64,66]. Third, the cut-off in form factor is not well scaled because of the unavailable data. Thus the search for heavy exotic states through photoproduction would help to examine these aspects in one hand, and gain more insights on themselves on the other hand.