The $\phi(2170)$ production in the process $\gamma p\to \eta \phi p$

We have studied the $\gamma p\to \eta \phi p$ reaction within the effective Lagrangian approach, by considering the contribution of the intermediate state $\phi(2170)$ production, and the background contributions of $t$-channel $\pi^0$ and $\eta$ mesons exchanges with the intermediate states $N$ and $N(1535)$. Our calculations show that there may be a peak, at least a bump structure around 2180 MeV associated to the resonance $\phi(2170)$ in the $\eta\phi$ mass distribution. We suggest to search for the resonance $\phi(2170)$ in this reaction, which would be helpful to shed light on its nature.


I. INTRODUCTION
The exotic hadrons beyond the conventional quark model are permitted to exist in the framework of the Quantum Chromodynamics.Some possible candidates for the exotic states have been accumulated experimentally [1][2][3][4][5][6], and there are many different explanations for their natures.Further study on the possible candidates for the exotic states is obviously needed both theoretically and experimentally.
As we know, the associate production of hadrons by photon has been extensively studied since it provides an excellent tool to learn details of the hadron spectrum [27][28][29][30].The intense photon beams can be used to study the strangeonium-like states because of the strong affinity of the photon for ss.For instance, the strangeness φ(1019) and φ(1680) were observed in the photo-production reactions respectively in Refs.[31][32][33][34] and Refs.[35][36][37].The observation of the φ(2170) in φf 0 (980) indicates that the φ(2170) has a substantial ss component.Thus, the photo-production reaction could be also suitable to study the resonance φ(2170).
It should be pointed out that, in Fig. 25 of Ref. [36], which corresponds to the dσ/dM K K distribution for the reaction γN → K + K − N measured by the Omega Photon Collaboration, besides the peak of the φ(1680), another enhancement around 2150 MeV exists, which could be associated to a resonance with the same quantum numbers as φ(1680), i.e.J P C = 1 −− .In addition, BE-SIII also observed a resonant structure around 2200 MeV at the cross section of e + e − → K + K − reaction [38].The K K channel is expected to be one of the main decay channels of the φ(2170) [12][13][14].Thus, it is natural to associate this enhancement structure around 2150 MeV in Fig. 25 of Ref. [36] to the φ(2170), which implies that the φ(2170) photo-production should be accessible experimentally.
Since the φ(2170) was observed in the φf 0 (980) channel [7], it would be straightforward to search for the resonance φ(2170) in the reaction of γp → φf 0 (980)p.However, one would face the possible mixing between the φ(2170) and the threshold effect in γp → φf 0 (980)p reaction because the mass of φ(2170) is close to the threshold of φf 0 (980).The measurement of Γ(φ(2170) → ηφ)/Γ(φ(2170) → φf 0 (980)) = (1.7 ± 0.7 ± 1.3)/(2.5 ± 0.8 ± 0.4) [39] indicates that the coupling of the φ(2170) to the φf 0 (980) channel is of the same order of magnitude as its coupling to ηφ channel, which suggests that the φ(2170) has a sizeable coupling to the ηφ channel.Also, the threshold of ηφ is about 1570 MeV, thus the signal of the φ(2170) can not be misidentified with the threshold effect.These factors encourage us to study the φ(2170) production in the reaction of γp → ηφp within the effective Lagrangian approach.Our main purpose is to propose a possible process of searching for the φ(2170) resonance.
This paper is organized as follows.In Sec.II, we present the mechanisms of the reaction γp → ηφp, our results and discussions are given in Sec.III.Finally, a short summary is given in Sec.IV.

II. FORMALISMS
In this section, we will present the mechanisms for the reaction, by considering the tree level diagram as depicted in Fig. 1. p i (i = 1, 2, 3, 4, 5) are the four-momenta for photon, initial proton, φ, η, and outgoing proton, respectively, and s i (i = 1, 2, 3, 5) correspond to the polarizations of photon, initial proton, φ, and outgoing proton, respectively.We consider the background contribution of the t-channel π 0 and η exchanges with the final state ηp producing through the intermediate states N and N * , as shown in Fig. 1(a).The φ(2170) can be directly produced by t-channel π 0 and η exchanges, and then decays to ηφ, which is shown in Fig. 1(b).
We use the commonly employed Lagrangian densities for πN N and ηN N as follows [40,41]: with the coupling constants g πN N = 13.45 and g ηN N = 2.24, taken from Refs.[40][41][42].For the γφπ and γφη couplings, we take the interaction Lagrangian densities as used in Refs.[43], where e is the electron charge, φ ν , A β , π, and η are the fields of φ, photon, π, and η, respectively.For the intermediate nucleon excited resonances, we only take into account the contribution of the resonances N (1535), since the ηp production from the intermediate resonance N (1535) plays the dominant role near the threshold, as discussed in Refs.[41,44] where more nucleon excited resonances were considered.We also need the interaction Lagrangian densities involving N (1535) (marked as N * ) [45], For the intermediate state φ(2170) production of Fig. 1(b), we only consider the t-channel η exchange, because there is no any information about the radiative decays of the φ(2170), and the process φ(2170) → πγ should be suppressed due to the isospin breaking effect.More comprehensive mechanism can be considered if more experimental information about this reaction is available.The interaction Lagrangian densities involving φ(2170)(≡ φ * ) meson [46] are, The coupling constants in the above Lagrangian densities can be determined from the partial decay widths, for the φ meson, for for the φ(2170), with where m i denotes the mass of the initial state, m f1 and m f2 are the masses of two final states, and λ is the Källen function with λ(x, y, z) = (x − y − z) 2 − 4yz.At present, there is no information about Γ(φ(2170) → ηφ) and Γ(φ(2170) → γη) experimentally.Model predictions on Γ(φ(2170) → ηφ) lie a region from about 1 to 20 MeV [13,14,20,47].As far as we know, the radiative decays of φ(2170) have so far never been investigated in any models.We can roughly estimate the value of Γ(φ(2170) → γη) in the quark model based on the M 1 radiative partial width between the v = n 2S+1 L J and where , m 1 and m 2 are the quark 1 and 2 masses, respectively, Q 1 and Q 2 stand for the quark 1 and 2 charges in units of |e|, respectively.α = 1/137 is the fine-structure constant, E γ is the photon energy, E f is the energy of final meson, M i is the mass of initial state.In the picture of the φ, φ(1680), and φ(2170) as the 1 3 S 1 , 2 3 S 1 , and 3 3 S 1 ss, respectively, employing the wavefunctions obtained from the Godfrey-Isgur relativized quark model [50], we have Γ(φ(2170 which lead to Γ(φ(1680) → γη) ≃ 0.06 MeV and Γ(φ(2170) → γη) ≃ 0.17 MeV with the central value of Br(φ → γη) = 1.303% [39].The Γ(φ(1680) → γη) has been predicted to be about 0.09 MeV in a quark model with Gaussian wavefunctions [51] or 0.14 ± 0.09 MeV in an effective relativistic quantum field theoretical model based on flavor symmetry [46].Our predicted Γ(φ(1680) → γη) ≃ 0.06 MeV is smaller than the result of Ref. [51] and close to the lower limit of the result of Ref. [46], thus, one can conjecture that Γ(φ(2170) → γη) ≃ 0.17 MeV maybe also correspond to the lower limit of other model predictions.In the present work, we shall take Γ(φ(2170) → ηφ) = 1 MeV and Γ(φ(2170) → γη) ≃ 0.17 MeV.We want to test whether the signal of φ(2170) is observable in γp → ηφp with the lower limit of g φ * φη and g φ * γη .The obtained results for the coupling constants are listed in Table I.Since the hadrons are not point-like particles, the form factors are also needed.We adopt the dipole form factor, for exchanged mesons [30,42,52,53], and for the exchanged baryons [27,28,30,54], where the q and M are the four-momentum and the mass of the exchanged hadron, respectively.Indeed, the values of the cutoff parameters can be directly related to the hadron size.Since the question of hadron size is still open, the cutoff parameters are usually adjusted to the related experimental measurement [55,56].The typical value of the cut-off Λ in the born potential is in the region of 1.0 ∼ 2.0 GeV.In our present calculation, we use the cutoff parameters Λ π = Λ η = 1.5 GeV, and Λ N = Λ N * = 2.0 GeV for N and N (1535) as used in Refs.[41,42].For the one of φ(2170), we take Λ φ * = 1.0 GeV.The propagators used in our calculation are for exchanged π, η mesons, for the propagator of spin-1/2 baryon, and for the propagator of φ(2170), where q, M , and Γ stand for the four-momentum, mass, and total width of the resonance, respectively.Indeed, as we discussed in the introduction, there are many interpretations on the nature of φ(2170), and all of them give rise to the similar structures as the Breit-Wigner form of Eq. ( 24).With the above effective Lagrangian densities, the scattering amplitudes for the reaction γp → ηφp can be obtained straightforwardly as follows, for the contribution of the t-channel π 0 exchange with the intermediate nucleon, and for the contribution of the t-channel π 0 exchange with the intermediate N (1535).The amplitudes M η N and M η N * due to the η exchange can be obtained from M η N and M η N * by replacing π by η.
The amplitude of the intermediate φ(2170) production can be written as, q π = p 1 − p 3 is the four-momentum for the exchanged π meson, and q R = p 4 + p 5 is the four-momentum for intermediate N and N (1535).It is easy to show that the above amplitudes respect the gauge invariance [41].Then the differential cross section for the reaction γp → ηφp can be expressed as, with where E 3 , E 4 , and E 5 are the energies of the φ, η, and outgoing proton, respectively, and E γ is the photon energy in the laboratory frame.

III. RESULT AND DISCUSSION
With the above formalisms, we calculate the total and differential cross sections for the γp → ηφp reaction by using a Monte Carlo multi-particle phase space integration program used in Refs.[41,57].
The ηφ mass distribution of the γp → ηφp reaction with E γ = 8 MeV is shown in Fig. 2. It should be pointed out that for the photo-production, there is a contribution from Pomeron exchange, whose effect is dominant at large center-of-mass energy and forward angle.However, in this paper only the ηφ mass distribution is relevant to the signal of φ(2170), and the comprehensive mechanism involved the Pomeron exchange dose not change too much the shape of the ηφ mass distribution.As we can see, there is a bump structure around 2180 MeV, which is associated to the resonance φ(2170).Our model prediction is based on the lower limit of g φ * φη and g φ * γη .One can expect that if a larger value for the product of g φ * φη g φ * γη is used, the signal of the φ(2170) would be more clear.For example, if we take Γ(φ(2170) → ηφ) ≃ 6.6 MeV expected by the recent quark model calculations [14], one can find a significant peak around 2180 MeV as shown in Fig. 3 (see the curve labeled as 'T-1.0').
In addition to the couplings related to the φ(2170) state, the φ(2170) form factor of Eq. ( 20) depended on the Λ φ * would affect the numerical results.We also show the results with cutoff Λ φ * = 1.3 GeV in Fig. 3, where the curves labeled as 'T=1.3'and 'φ * -1.3' correspond to the contributions of the full model and the intermediate φ(2170), respectively.It is found that the results for Λ φ * = 1.3 GeV have little change compared with those for Λ φ * = 1.0 GeV.The exact value of Λ φ * can be ex-tracted from the experimental measurement in future.In Fig. 4, we show the total cross section of the γp → ηφp reaction with the parameters listed in Table I .Very recently, the reaction of γp → ηφp is also suggested to study the nucleon resonances production in Ref. [44] where the total cross section at E γ = 3.8 GeV is around 8 ∼ 10 nb, which is consistent with our prediction.
Finally, it should be noted that the GlueX Collaboration has proposed to search for the φ(2170) in the photoproduction [58], and the γp → ηφp reaction has been selected as a particularly suitable process to search for strangeonium states by the CLAS12 Collaboration [59].Our predictions should be useful for the future experimental study.

IV. CONCLUSIONS
Motivated by the small enhancement around 2150 MeV in the K + K − mass distribution of the γp → K + K − p reaction measured by Omega Photon Collaboration, and the clues that the branching ratio Br(φ(2170) → ηφ) is of the same order as Br(φ(2170) → φf 0 (980)), we propose to search for the resonance φ(2170) in the γp → ηφp reaction.
Our calculations show that there will be a peak, at least a bump structure around 2180 MeV in the ηφ mass distribution of γp → φηp reaction, the The magnitude of our total cross section is consistent with the prediction of Ref. [44] where the same reaction is used to study the nucleon resonances.We suggest our experimental colleagues to search for the resonance φ(2170) in the γp → ηφp reaction, which would be helpful to shed light on its nature.

FIG. 1 :
FIG. 1: Feynman diagrams for the γp → ηφp reaction.(a) the contribution of the t-channel π 0 and η exchanges with the intermediate states N and N * .(b) the contribution of the intermediate states φ(2170) production.

FIG. 2 :FIG. 3 :
FIG.2:The ηφ mass distribution of the γp → ηφp reaction with Eγ = 8 GeV.The curves labeled as 'B' and 'φ * 'stand for the contributions of the background and the intermediate φ(2170) production, respectively.The curve labeled as 'T' corresponds to the total contributions.

TABLE I :
[39]l parameters used in the present work, the masses, widths, and branching ratios are taken from Particle Data Group[39].