Photoproduction of hidden-bottom pentaquark and related topics

Due to the discovery of the hidden-charm pentaquark Pc states by the LHCb collaboration, the interests on the candidates of hidden-bottom pentaquark Pb states are increasing. They are anticipated to exist as the analogues of the Pc states in the bottom sector and predicted by many models. We give an exploration of searching for a typical Pb in the γp→ Υp reaction, which shows a promising potential to observe it at an electron-ion collider. The possibility of searching for Pb in open-bottom channels are also briefly discussed. Meanwhile, the t-channel non-resonant contribution, which in fact covers several interesting topics at low energies, is systematically investigated. ∗ caoxu@impcas.ac.cn † fkguo@itp.ac.cn ‡ liangyt@impcas.ac.cn § wujiajun@ucas.ac.cn ¶ xiejujun@impcas.ac.cn ∗∗ xieyaping@impcas.ac.cn †† zhiyang@impcas.ac.cn ‡‡ zoubs@itp.ac.cn 1 ar X iv :1 91 2. 12 05 4v 1 [ he pph ] 2 7 D ec 2 01 9


I. INTRODUCTION
Since the discovery of the X(3872) by the Belle collaboration [1], a rich spectrum of exotic states has been emerging, see comprehensive reviews for references [2][3][4][5][6][7][8][9]. They not only shed new insights into the study of hadron spectrum and structure, but also deepen our understanding of nonperturbative properties of Quantum Chromodynamics (QCD). Among these states, the charged Z c (3900) and Z c (4020) found respectively in the J/ψπ ± [10,11] and h c π ± [12] systems seem to be surely exotic since they must contain at least one additional light quark and anti-quark pair besides the hidden pair of cc to match the electric charge. Their partners in the bottomonium sector, namely the Z b (10610) and Z b (10650), were firmly established by Belle in several different decay modes [13]. The spin and parity of these states are determined unambiguously to be 1 + by the amplitude analysis of BE-SIII [14] and Belle [15], except for the Z c (4020), which is believed to be of the same quantum numbers by most of the models. Their masses are very close to the S-wave thresholds of the corresponding open-flavor channels DD ( * ) and BB ( * ) , respectively. As for their strange partner Z s , so far the BESIII Collaboration did not find a signal in the φπ spectrum of e + e − → φππ [18].
In the baryon sector, the hidden-strangeness pentaquark P s states containing only light quarks are expected in constituent quark models [19,20], and in models considering the QCD van der Waals force [21,22]. But they are not explicitly found at present after a long searching for them in πN and γN reactions [23]. Other reactions and decays were suggested to study them from the theoretical side [24][25][26][27][28]. Interestingly, no narrow peaks were found in total cross section of near threshold γp → φp, but a non-monotonic structure, found in the differential cross section by LEPS Collaboration [29], would imply a very wide ∼ 500 MeV states [30,31]. There is also no any evident signal in the φp energy spectrum of the process Λ + c → φpπ 0 [32], which was shown in Ref. [33] to be not a good choice for the search of P s due to the presence of triangle singularities (for a recent review, see Ref. [9]) and the tiny phase space. However, in the charm sector, the astonishing observation of P c states by the LHCb Collaboration [34,35] has provided us an insightful place to study the exotic baryons in the charm sector, the existence of which were anticipated by several models [36][37][38][39]. The photoproduction reactions of these states with two-body final states, first proposed in Ref. [40] and followed by other works [41,42], are an exceptional platform to exclude their non-resonant possibility, because the on-shell conditions required by the triangle singularities discussed in Refs. [43][44][45][46][47] cannot be satisfied. The upper limit of the P c photoproduction cross section in γp → J/ψp was determined recently by the GlueX Collaboration [48], constraining the branching ratios of the P c decays into the J/ψp mode together with the results at LHCb [49]. Due to the null results in the GlueX data, double polarization observables were proposed to be a benchmark in the search of pentaquark photoproduction [50]. Although the nature of these exotic states is under discussion [51][52][53][54][55][56][57], they motivated the speculation from heavy-quark spin symmetry that there should be seven molecular pentaquarks in two spin multiplets [58][59][60]. Motivated by the heavy quark flavor symmetry for the potential between heavy mesons and baryons, the correspondence of these states in the bottom sector, label as P b here, are expected to be surely existing [42,[61][62][63].
Unlike the P c , they cannot be produced through the decay of heavier baryons. Therefore, they can only be directly produced in high-energy processes, such as the ep, γp scattering and the pp collisions.
In this paper we will discuss the possibilities of searching for one of typical P b states, the bottom analogs of P c , in the photoproduction of the the bottomonium channel γp → Υp at electron-ion colliders (EICs). To this end, we first explore the non-resonant contribution to the γ * p → Υp in Sec. II. This is very meaningful on its own right because several subjects are relevant to it. The detailed investigation of the P b contribution is presented in Sec. III.
At last we finish with a short summary in Sec. IV. LHCb [64] and CMS [68] were used in the fit in order to determine the overall normalization but are not shown. The models include the DVMP empirical formula (FGHK) [69], two-gluon exchange model (BCHL) [70], the parameterization in Ref. [71] (GV), and the soft dipole Pomeron [72,73].

II. NON-RESONANT CONTRIBUTION
The main purpose of studying the photo-and electro-production of vector heavy quarkonia off the nucleon is to study the gluon component within the nucleon probed by heavy quarks. The low energies are also important for several other topics which are critically relevant. First, the near-threshold region would provide clue for the quarkonium-nucleon interaction. The measured cross sections have been used to extract the J/ψp scattering length [71,74], whereas the Υp scattering length is rarely known due to the lack of data.
Second, it was proposed to be a promising platform to probe the trace anomaly term in the QCD energy-momentum tensor and the proton mass decomposition, resulting into a deep exploration of the origin of the nucleon mass [75,76].
Since the discovery of the J/ψ, its photoproduction has attracted plenty of interests both from experimental and theoretical aspects. Because the bottom quark is heavier than the charm one, the Υ photoproduction has its own merits. The multipole expansion [75,77] converges more quickly. Relative uncertainties of the current quark mass and the running coupling constant are much smaller. This is an essential advantage for the theoretical calculation because the amplitudes are expected to be proportional to powers of these quantities. At last but not at least, it is ideal to search for hidden-bottom pentaquark candidates. The GlueX Collaboration has searched for the P c in the near-threshold region of the γp → J/ψp [48] as mentioned above. The P b states, whose lowest mass in many theoretical models is expected to be lying above Υp production threshold, making γp → Υp as a perfect place to hunt for them. However, the data of the Υ production below 100 GeV has never been measured up to now, and it becomes one of the main concerns of the proposed Electron-Ion Collider in China (EicC), as proposed in the white paper [78].
In order to explore the possibility of studying the Υ production at relative low energies, we need to estimate its cross section as a premise, with the help of a reliable model to extrapolate from high to low energies. The non-resonant contribution would come from the t-channel two-gluon or Pomeron exchange, as shown in Fig. 1. A rough evaluation of total cross section reads as which is suggested by the empirical formula of the deeply virtual meson production (DVMP) γ * p → V p [69]. Here M V is the Υ mass, W the γp center of mass (c.m.) energy and Q 2 the photon virtuality. The advantage of this simple parameterization is that it is applicable to all DVMP processes with proper Q 2 dependence. The parameters α and β have been determined by the DVMP data to be α = 0.31 ± 0.02 and β = 0.13 ± 0.01 by Favart, Guidal, Horn and Kroll (FGHK) [69]. Correspondingly, δ(Q 2 = 0) = 0.89±0.05, confronted with the perturbative QCD prediction δ ∼ 1.7 [79] and ZEUS results δ = 0.69 ± 0.02 ± 0.03 [80]. The normalization N is determined by the data of γp → Υp at high energies to be 2.62 ± 0.38, where the experimental uncertainty of W is not taken into account. The result is shown as the grey band in Fig. 2, together with those from a few other models. As can be seen, the general trend of high energy data with large errors follows the exponential behavior in Eq.
(1). Note that the data above 300 GeV up to 2 TeV from LHCb [64] and CMS [68] were used in the fit in order to determine the overall normalization; they also follow the exponential behavior though not shown in the figure.
Gryniuk and Vanderhaeghen (GV) adopted the following parametrization for the cross section [71]: where f V is the vector meson decay constant, This simple parametrization generally preserves the exponential trend at high energies, and surprisingly agrees very well with the data above 100 GeV.
The two-gluon exchange model proposed by Brodsky, Chudakov, Hoyer and Laget (BCHL) suggests the following t-dependent cross sections [70]: , and the transfer-momentum squared t. We use the slope parameter b = 1.13 GeV −2 in the original scheme, which is compatible with the measured one b = 1.25 ± 0.20 GeV −2 at W = 11 GeV for the J/ψ production [81].
The corresponding result is shown as the green band in Fig. 2. The normalization N 2g is adjusted to the data around 100 GeV because obviously this model cannot describe the data at high energies. The same authors also proposed the form of three-gluon exchange with an unknown normalization. It is premature to discuss such a contribution at present because of lack of data below 100 GeV.
Several Pomeron models have been constructed [82], but few of them have been used to study the case of the Υ. The soft dipole Pomeron model, put forward by Martynov, Predazzi and Prokudin [72,73], preserves unitarity bounds with a double Regge pole with an intercept equal to 1. By fitting to all the available data of γ * p → V p at that moment, the model predicts the behavior of γ * p → Υp, which is consistent with the measured data afterwards, see the black curves in Fig. 2. Besides the usual exponential tendency at high energies, additional small fluctuations are observed. The shoulder shape around 20 GeV is caused by a Regge pole mainly contributing to low energies. The trough around 30 GeV is from the interference between two Regge poles. We also show the Q 2 dependence of the cross sections, which tend to be more moderate with larger Q 2 as expected from the (Q 2 + M 2 V ) −1 behavior. Sibirtsev et al. also concluded that two Regge trajectories were required to describe the data of γ * p → J/ψp over a wide energy range after comparing various models [83]. This is different from most of the Pomeron models with only one Regge trajectory [40,84]. Figure 2 shows that various models can describe the data at high energies comparably well, except the two-gluon exchange model, which is designed to focus on the near-threshold region. However, the inserted subfigure in Fig. 2 shows that the deviations between different models are large at low energies, which are covered by the proposed EicC. The empirical formula of DVMP, as a guideline and a rough upper limit, does not take into account the influence from phase space, which is significant at low energies as one can easily anticipate.
The soft dipole Pomeron model overlaps with the two-gluon exchange one within uncertainties, but is larger in the very close-to-threshold range. The GV parametrization is smaller than the other models below 20 GeV.
In a short conclusion, the soft dipole Pomeron model and the GV parametrization are both compatible with high energy data and give the expected behavior of the phase space at low energies. So they serve as a good input for the study of the non-resonant contribution to the ep → epΥ process. Because the parameters in the soft dipole Pomeron model is from a global fit to all the data, in the next section we will use it as the non-resonant contribution to γp → Υp. We also use the empirical formula from DVMP as a crude estimation of the upper limit of the non-resonant contribution. Besides, other models that are available to calculate the cross section of the γ * p → J/ψp can also be extended to the case of the γ * p → Υp.
However, most of them have more undetermined parameters owing to lack of data of the Υ production, so we do not consider them at present.

III. P b AS A RESONANCE IN PHOTOPRODUCTION
We list the properties of a typical P b predicted by various phenomenological models which used the P c as inputs in Tab. I. We do not attempt to collect all the models here because of the still increasing literature. We would like to point out that nearly all models predict a resonant state with a mass around 11.12 GeV which couples to the Υp channel, while the total width differs due to detailed constructions of the models, ranging from 30 MeV to 300 MeV. In this paper we will adopt the mass of 11.12 GeV with two possible width values 30 MeV and 300 MeV. Later on they are debbed as the narrow P b and the wide P b , respectively. The spin J = 1/2 is used here as a representative choice. Other P b states with different quantum numbers can be similarly calculated since the production cross section is proportional to 2J + 1 in our prescription.
The production cross section of the exotic P b in the reaction γp → Υp, as shown by the Feynman diagram in Fig. 1, can be written as Here k in is the magnitude of the initial-state three momentum in the c.m. frame, and s 1 and s 2 are the spins of initial photon and proton, respectively. Because the mass M of P b is very large, this formula is a very good approximation even for the wide P b . If assuming that the P b → γp is dominated by only the heavy vector meson in the vector meson dominance model, e.g., V = Υ in Fig. 1, the branching ratio B(P b → γp) is proportional to If all closed channels included, it is 10.304 (1/2 − ) and 10.382 (3/2 − ). † Roughly estimation from phase space ratio Γ(P b )/Γ(P c ) = k out (P b )/k out (P c ). ‡ Assume the same width with P c (4450) at LHCb. [41,42,49]: which has assumed the lowest orbital excitation L = 0 between the Υ and the proton. Here α is the fine structure constant, k out is the magnitude of final-state three momentum in the c.m. frame, and the dilepton width Γ(Υ → e + e − ) = 1.34 keV [91]. As a result, we have It shall be noted that the intermediate vector meson V = Υ in Fig. 1 is highly off-shell, so a form factor would be present with a possible strong suppression, as pointed out in Ref. [92]. At present, the branching fraction B(P b → γp) is not directly measured so the magnitude of this form factor is unknown. As a result, B(P b → γp) above needs to be understood as an effective branching ratio with this factor absorbed. Recently, the measurement of GlueX at JLab Hall-D has given the upper limit of B(P + c → J/ψp) to be several per cent without considering this off-shell factor. The LHCb results indicate a stringent lower limit of B(P + c → J/ψp) to be 0.05% ∼ 0.5% [49]. We use these values of P c as a reference and adopt 0.5% < B(P b → Υp) < 5% for P b . The calculated values in most of the models in Tab. I are within this chosen range, except one of them approaching to about 0.01% [87].
The non-resonant contribution studied in Sec. II is considered as the smooth background of P b . The interference effect between them in the total and differential cross sections is not significant because the t-channel Pomeron exchange contributes only to the forward angles while the s-channel resonances are present in full angles. The calculation of γp → J/ψp confirm this expectation [92]. The hereafter error bands are from the uncertainty of nonresonant contribution but does not include the errors of the mass M and width Γ of P b , just because it is too premature to consider them at this stage.
The calculated results are presented in Fig. 3 (a) with B(P b → Υp) = 5% and the nonresonant contribution of the DVMP empirical formula in Eq. (1). The background is smooth within the EicC energies in the range of 0.01 ∼ 0.02 nb. The peak cross section of the narrow P b is around 0.1 nb at most. The effects of both the narrow and wide P b are prominent, as can be seen. This is contrary to the decay of Λ 0 b → K − J/ψp at LHCb, where a wide resonance is much harder to be identified due to the more complicated background. Notice that the DVMP parameterization does not consider the phase space, so that the results need to be considered as an upper limit in the low energy region as already mentioned.
We show the results with the non-resonant contribution of the soft dipole Pomeron model in Fig. 3 (b) with 0.5% < B(P b → Υp) < 5%. The background varies rapidly in the range of the EicC energies because of the phase space. The P b signal is still clearly visible if B(P b → Υp) > 1.0%. It would be difficult to find the P b with B(P b → Υp) as small as 0.5% in an unpolarized measurement, and therefore polarization observables are needed. The formalism for a detailed calculation of polarized measurements is well established [23,40,92].
But we will not pursue that aspect in this paper, because the interference is definitely essential but out of control due to lack of low energy data. The t-dependence of the nonresonant contribution in the soft Pomeron model is very close to e b t with the same value for the slope b in Eq. (3), because this slope is mainly driven by the data of the J/ψ production in the soft dipole Pomeron model. It would be very interesting to look into the slope for the Υ once data are available in the future.
As shown in Fig. 3 (a), the non-resonant Υ photon-production at EicC energies is around 0.02 nb at most, and a reduction factor of about five is introduced by the two-body phase space, see Fig. 3 (b). The resonant P b photon-production in the peak energy is around 0.1 nb. For reactions at electron ion colliders, a roughly two orders of magnitude smaller cross section is anticipated for the electro-production comparing to above photon-production.
Take the EicC as an example, about 5 × 10 4 signal events of ep → eP b → eΥp are expected with an integrated luminosity of 50 fb −1 . Even after considering the small leptonic decay branching fraction of the Υ and the detection efficiency, the observation of this channel is still optimistic at the EicC. The produced P b is not far away from the central rapidity region at the EicC energies, which is good for detection. A detailed simulation is under investigation and will be soon available for publication [78].

IV. SUMMARY AND CONCLUSION
In this paper, we made a detailed exploration of the non-resonant contribution to the γp → Υp, with the aim to find a reasonable estimation of the production rate at relatively low energies, where no data are available up to now. An extrapolation from the energies of the LHC and HERA data to low energies by several models gives us a reasonable estimate of the cross section below 100 GeV. We emphasize that this non-resonant contribution to the γp → Υp is related to several appealing topics. It may give access to the Υp scattering length, which is a key parameter for understanding of whether a bottomonium can be bound with the nucleon and light nuclei. Our results in Fig. 2 in fact can be used to roughly estimate the scattering length, as done for that of J/ψp [74]. It could be also decisive for extracting the information of the trace anomaly contribution to the nucleon mass, so finally solve the problem of the proton mass decomposition [76]. We would like to further remark that the larger mass of the Υ than the J/ψ could make it a better place for studying these issues, because the relative uncertainties of current quark mass and running coupling constants are smaller at high energies [91] These issues may be notably clarified by measurements at the electron proton colliders.
After the study of the non-resonant contribution, we conducted a careful estimation of the production of P b in γp → Υp under the assumption that the P b naturally inherits features from the P c . If it is found in the photo-and electro-production in the future, the P b will be firmly established as a genuine resonant state because resonant-like structures from triangle singularities are inapplicable to this reaction. We estimated the production yield at EIC machines based on the calculated cross sections and found that if the B(P b → Υp) is larger than 1.0%, the P b states should be observable at the EicC through γp → Υp process. On the other hand, if B(P b → Υp) is smaller than 1.0% as predicted by Ref. [87], then the P b states need to be searched for in the dominant decay channelsB ( * ) Λ b final states. Since the cross section of the semi-inclusive γp → bbX at high energies is found to be two orders of magnitude larger than that of the γp → Υp by experiments [93,94] and the next-toleading order QCD calculation [95,96], the open bottom channelsB ( * ) Λ b are expected to have larger cross section than that of Υp channels. So these P b states may be observed at