Photoproduction of the scalar mesons $f_0(500)$ and $f_0(980)$ off the nucleon

We investigate photoproduction of the scalar mesona off the nucleon, using the effective Lagrangians and the Regge approach. We first study $f_0(980)$ photoproduction, replacing the Feynman propagator with the Regge ones for the $\rho$-meson exchange in the $t$-channel. The model parameters are fixed by reproducing the experimental data on the differential cross section. We then apply the same method to $f_0(500)$ or $\sigma$ photoproduction with the same parameters for the Regge propagator. Since the threshold energy of the $f_0(500)$ production is rather low, $N^*$ resonances, which can decay into the nucleon and two pions in the isoscalar and scalar state, can come into play in the $s$-channel. To examine the effects of the $N^*$ resonances, we set up two models to look into the respective contribution of the $N^*$ resonances. We find that in particular $N^*(1440)$ and $N^*(1680)$ play important roles in $f_0(500)$ photoproduction. We discuss the physical implications of the present results.


I. INTRODUCTION
Understanding the structure of low-lying scalar mesons has been one of the most challenging issues in hadronic physics. Their internal structure is still under debate. That the f 0 (500) scalar meson, which is also known as σ, is not an ordinary meson consisting of a quark and an anti-quark is more or less in consensus. Recent studies suggest that these scalar mesons may belong to the flavor SU(3) non-qq nonet (see reviews [1,2], a "note on scalar mesons below 2 GeV " in Ref. [3], and references therein. A recent review provides also various information on the structure of the scalar meson [4], including a historical background of the σ meson). The f 0 (500) is also interepreted as one of the glueballs or gluonia, mixed with theqq state [5][6][7], though this idea is criticized because the same analysis is rather difficult to be applied to explaining the strange scalar meson K * 0 (800) or κ, which is also considered as a member of the nonet. The f 0 (500) is often regarded as a tetraquark state in a broad sense [8]. The f 0 (500) as a tetraquark state has a multiple meaning: It can be described as a diquark-antidiquark correlated state [9,10],qqqq state [11], or correlated 2π state [12,13] arising from ππ scattering. This nonqq feature was employed in various theoretical approaches such as QCD sum rules [14], effective Lagrangians [15], and lattice QCD [16][17][18].
While there was a great deal of theoretical works on the structure of the f 0 (500), its reacion mechanism was less investigated. Recently, the CLAS Collaboration has reported the first analysis of the S-wave photoproduction of π + π − pairs in the region of the f 0 (980) at photon energies between 3.0 and 3.8 GeV and momentum transfer squared −t between 0.4 GeV 2 and 1 GeV 2 [31,32]. While the differential cross section for the γp → π + π − p process in the S-wave shows an evident signal for the f 0 (980) production, the f 0 (500) was not seen clearly. However, there is still a hint for the existence of the f 0 (500) in π + π − p photoproduction measured at different kinematic conditions [33]. Thus, it is of great interest to study the γp → π + π − p reaction in the scalar and isoscalar channel. Since these two pions are strongly correlated, one has to consider the rescattering effects of these two pions to describe γp → π + π − p in the scalar and isoscalar channel, which are essential in order to explain the production mechanism of the scalar and isoscalar mesons f 0 quantitatively in π + π − p photoproduction. Moreover, it is crucial to take into account the KK channel in addition [34], since its threshold is open in the vicinity of the f 0 (980) mass. In order to take into account the effects of the KK channel, one has to introduce the coupled-channel formalism, which requires the fully coupled ππ and KK amplitudes.
However, before we carry out the investigation on the γp → π + π − p reaction, we need to examine the related twobody process γp → f 0 p as a first step toward more complicated correlated ππ photoproduction. Moreover, since the CLAS Collaboration already presented the differential cross section for f 0 (980) photoproduction in the photon energy range E γ = (3.0 − 3.8) GeV, it is important to study f 0 photoproduction theoretically as well before we examine the γp → π + π − p process with pion pairs in the S-wave. In addition to f 0 (980) photoproduction, we study in the present work the f 0 (500) production by photon beams, based on effective Lagrangians and a Regge approach. The Regge exchange in f 0 (980) photoproduction was already applied in Ref. [35] with the same Regge trajectory but a different set of parameters. We will first compute the differential cross section for f 0 (980) photoproduction and compare the results with the CLAS experimental data such that we can fix parameters for the t-channel Reggeon exchange. Since there is no excited nucleon that decays into π + π − pairs in the S-wave beyond the f 0 (980)N threshold, we consider only the N exchange in the s channel. Then we will proceed to study f 0 (500) photoproduction with the same parameters for the Reggeon, which is fixed in f 0 (980) production. As far as f 0 (500) photoproduction is concerned, we need to consider several excited nucleons above the threshold energy, which can decay into π + π − N , where the pion pairs are in the isoscalar and scalar wave.
Upon computing the transition amplitude for the f 0 (500) photoproduction, there are ambiguities to which we have to pay attention carefully. Firstly, the width of the f 0 (500) is very large, so that the value of the f 0 (500) mass is quite uncertain. Thus, we have to look into the dependence of the results on the f 0 (500) mass. Secondly, the photocoupling constant g γσρ is experimentally not much known. Though there are several theoretical suggestions on its value, the agreement has not been reached yet. Experimentally, two relevant decay channels are known: ρ 0 → π + π − γ [36,37] and ρ 0 → π 0 π 0 γ [38,39]. To see the contribution of the f 0 (500) in these decay processes, one has to reply on models. So, it is required to examine uncertainties arising from the coupling constant g γσρ . In principle, ω-meson exchange could be considered. However, the branching ratio of ω → π + π − γ is not much known: its experimental upper bound is given as < 3.6 × 10 −3 with CL=95 %. On the other hand, ρ → π + π − γ is experimentally known to be (9.9 ± 1.6) × 10 −3 [3]. Thus, the value of the γf 0 ρ coupling constant is expected to be much larger than that of ω → f 0 γ, based on the experimental data given above. We have confirmed numerically that the effect of ω-exchange is indeed much smaller than that of ρ-exchange. So, we will ignore in this work the contribution from ω-meson exchange. Thirdly, the final state in the N * → (ππ) I=0 S−wave N decay should contain both the background ππ and the f 0 (500) resonance. It indicates that it is rather difficult to determine the coupling constants for the f 0 (500)N N * unambiguously. Considering these points that will bring about the uncertainties of the present work, we have to introduce certain assumptions before we proceed to investigate f 0 (500) photoproduction. Though we will take 500 MeV as a main value for the f 0 (500) mass in this work, we will carefully examine the dependence of the results for the total cross section on the mass of the lowest-lying scalar meson. Lastly, we will regard π + π − pairs in the S-wave as the f 0 (500) meson, which are produced in the course of the N * → (π + π − ) I=0 S−wave N decays, so that we are able to determine the strong coupling constants for the N * → f 0 (500)N transitions. Since the f 0 (500) resonance is the most dominant one in ππ scattering in the scalar-isoscalar channel, this approximation is rather plausible.
In addition to the Roper resonance, we want to consider other N * resonances that can decay into (ππ) I=0 S−wave N . Referring to Ref. [3], we find that 10 excited nucleon resonances have the decay channel of N * → (ππ) I=0 S−wave N . However, there are not enough data for N * → (ππ) I=0 S−wave except for N (1440), N (1680), and N (1880). Since N (1880) has an overall status 2 star, we will not consider it (see the review "N and ∆ resonances" in Ref. [3]). Thus, we expect that the main contribution will come from N (1440) and N (1680). As will see later, N (1680) provides predominantly a large contribution to the total cross section. Thus, we will set up two different models to delve into each contribution from the N * resonances. In Model I, we will include those with spin 1/2 and an overall status 3 or 4 stars. Thus, we consider N (1535)1/2 − , N (1650)1/2 − , and N (1710)1/2 + in addition to N (1440)1/2 + , though their data are not much known. In Model II, we further take into account N (1520)3/2 − , N (1675)5/2 − and N (1680)5/2 + together with those included in Model I.
The present work is sketched as follows: In Section II, we explain the general formalism for the f 0 (980) and f 0 (500) photoproductions. In Section III, we present the numerical results separately for Model I and Model II, and discuss their physical implications. The last Section is devoted to the summary and the conclusion of the present work. We also discuss perspectives of future works in the last Section.

II. GENERAL FORMALISM
Feynman diagrams for f0 photoproduction at the tree level. Each diagram corresponds to the s channel, the t channel, and the u channel in order.
We start with the tree-level Feynman diagrams relevant to the γp → f 0 (980)p and γp → f 0 (500)p reactions based on the effective Lagrangian approach, as depicted in Fig. 1. Note that we will not consider any contribution from N * resonances to f 0 (980) photoproduction, since the threshold energy of its production is rather high. There is no N * above 1900 MeV, which can decay into f 0 (980)N . Thus, we take only into account nucleon exchange both in the s-and u-channels. As far as the t-channel diagram is concerned, ρ-meson exchange comes into play. However, the γN → f 0 (980)N reaction was experimentally measured at higher photon energies (E γ = 3.0 − 3.8 GeV) [31,32], which is quite far beyond threshold. Since the method of effective Lagrangians is devised for describing the mechanism of hadronic reactions near threshold, we need to revise it to explain f 0 (980) photoproduction at higher E γ . We will employ a hybridized Regge approach in which the Feynman propagator in the t channel is replaced with the Regge propagator for ρ-meson exchange, while the coupling constants and spin structures are taken from the effective Lagrangians. This approach was successfully used for describing the productions of strange and charmed hadrons [40][41][42]. A virtue of using this hybridized Regge model is that we can use the reggezied ρ meson in the t channel also for f 0 (500) photoproduction with the same parameters fixed in f 0 (980) photoproduction.
As for the γN → f 0 (500)N reaction, we include the ρ Reggeon in the t channel with the same parameters used in f 0 (980) photoproduction. In the s-and u-channels, we consider nucleon exchange. Since the threshold energy of f 0 (500) photoproduction is about 1.4 − 1.5 GeV, we need to introduce in the s channel the N * resonances that decay only into (ππ) I=0 S−wave N . To study the contributions of the N * resonances, we develop two different models: In Model I, we include N (1440)1/2 + , N (1535)1/2 − , N (1650)1/2 − , and N (1710)1/2 + , while Model II further contains N (1520)3/2 − , N (1675)5/2 − and N (1680)5/2 + with higher spins in addition to those in Model I. Kinematics of the γp → f 0 p reactions are given as shown in Fig. 1: k 1 and k 2 stand for the four momenta respectively for the photon and the nucleon in the initial state, whereas p 1 and p 2 designate those respectively for the f 0 and the nucleon in the final state.

A. Model I
As we have already briefly mentioned in Introduction, there are ten excited nucleon resonances that can decay into f 0 (500)N . However, if we consider N * with overall status 3 or 4 star, then there are only seven N * resonances. In order to scrutinize the effects of the N * resonances carefully, we first introduce the pertinent N * resonances only with spin 1/2 to describe f 0 (500) photoproduction. We call this Model I.
To compute the Feynman invariant amplitudes for f 0 photoproduction, we use the following effective Lagrangians [43][44][45][46]: where A, N , ρ, and f 0 denote the photon, the nucleon, the ρ(770, 1 − ), and the f 0 fields, respectively. The N * represents a field for the excited nucleon. The values of the coupling constants given in the Lagrangians will be discussed later. The matrix Γ (±) depends on the parity of the N * resonance and is defined as Based on the effective Lagrangians in Eq. (1), we can compute the Feynman invariant amplitudes for each channel as follows: where the momentum transfers are given as q t = p 1 − k 1 , q s = k 1 + k 2 , and q u = p 2 − k 1 . e N is the electric charge of the nucleon and µ indicates the polarization vector of the photon. The u(p 2 ) and u(k 2 ) denote the Dirac spinors of the outgoing nucleon and the incoming nucleon, respectively. The m N and m ρ stand for the masses of the nucleon and ρ meson, respectively. t, s, and u represent the Mandelstam variables and are defined as The effective Lagrangian approach has been successfully used to describe hadronic reaction in the low-energy region. However, when the photon energy increases, the results from ths effective Lagrangians start to deviate from the data and do not even satisfy the unitarity [41]. Since the CLAS data on f 0 (980) photoproduction was conducted at E γ = 3.0 GeV and E γ = 3.8 GeV, which is far from threshold, the effective Lagrangian method is not suitable to explain the data. Thus, we employ a hybridized Regge model as already mentioned previously. In this approach, the Reggeon in the t-channel is governed by the Regge trajectory of the ρ meson, which is well known already [47][48][49][50][51][52]. We replace the Feynman propagator with the Regge one P R ρ in the t channel [47,51,52 where the Regge propagator is defined as .
Here, α ρ (t) denotes the Regge trajectory for the ρ meson. s ρ indicates the energy scale parameter for the corresponding Reggeon and is set to be equal to 1.0 GeV 2 . The Regge trajectory for the t-channel is taken from Ref. [52] α ρ (t) = 0.55 + 0.8t.
In addition, we introduce the scaling factor for the ρ meson Reggeon exchange which are often included to explain experimental values of the cross sections. However, we find that the results are not sensitive to the parameters a ρ and Λ ρ of the scaling factor.  While f 0 (980) photoproduction does not require any contribution from the N * resonances because of its high threshold energy, we need to consider them for the explanation of the γp → f 0 (500)p reaction. As mentioned earlier, we will take into account the spin 1/2 resonances in Model I: N (1440)1/2 + , N (1535)1/2 − , N (1650)1/2 − , and N (1710)1/2 + as listed in Table I. Since all of them have rather large widths, we will include the finite width in each propagator for the N * . Then, the Feynman amplitudes are derived as where the first term is the amplitude for the N * with positive parity whereas the second one represents that for the negative-parity N * . The coupling constants f 1 and g f0N N * denote respectively the generic photocouplings and strong coupling constants for the corresponding N * resonances. m N * and Γ N * represent the corresponding masses and the decay widths of the N * , respectively. Note that, for the propagators in Eq.(9), we consider the pole positions of the N * resonances in the complex plane.

B. Model II
In Model II, we additionally consider the N * resonances with spins 3/2 and 5/2 in the s-channel. We include the N (1520)3/2 − , N (1675)5/2 − , and N (1680)5/2 + in the s channel as listed in Table II [3]. The relevant effective Lagrangians for the N * resonances are given as follows where f 1 , f 2 , and g f0N N * denote the photocouplings and the strong coupling constants, respectively. They can be determined by using the experimental data on the photon decay amplitudes and the decay widths Γ N * →f0(500)N . Then, the Feynman invariant amplitudes for the s-channel are derived as where ∆ µα and ∆ ρσ;µα indicate the Rarita-Schwinger propagators for the N * resonances with spin 3/2 and 5/2, respectively, defined as [55][56][57][58] Here, S µα ,ḡ µα , andγ µ are expressed as

C. Parameters and form factors
The coupling constant for the ρN N vertex is the most important parameter to describe f 0 (980) photoproduction, since the t-channel governs the production mechanism of the γN → f 0 (980)N reaction. The coupling constant g ρN N and κ ρ are well known from N N potentials. For example, g ρN N = 3.25 was used in the full Bonn potential [59], while the Nijmegen group employed g ρN N = 2.76 [60]. On the other hand, smaller coupling constants g ρN N = 2.6 and κ ρ = 3.7 have been exploited in Regge models for photoproduction of the pion and of the charged ρ ± meson [63,64]. We use g ρN N = 3.25 and the ratio of the vector and tensor couplings κ ρ = 6.1 [59].
The f 0 (980)N N coupling constants is taken from Ref. [65]: g f0(980)N N = 5.8. The f 0 (500)N N coupling constant g f0(500)N N is a crucial one which explains the mid-range strong attraction in the N N interaction. Its value is given in a wide numerical range. For example, the full Bonn potential suggests g σ N N = 8.46 [59] with m σ = 550 MeV. The notation σ in Ref. [59] was introduced to emphasize the fact that σ represents only an effective description of correlated 2π exchange in S wave. In the one-boson-exchange (OBE) Bonn NN potential, two different σs were introduced, i.e. σ 1 in the isovector (T = 1) channel and σ 2 in the isoscalar channel (T = 0). The charge-dependent OBE Bonn N N potential has even several different values for different partial waves in N N scattering [66]. One has to keep in mind that the coupling constant for the σN N vertex describes effectively whole 2π and even πρ exchanges. On the other hand, the Nijmegen soft-core (NSC) potential includes two different scalar mesons, flavor SU(3) symmetry and ideal mixing being used [67]. Depending on whether the scalar mesons constitute quark-antiquark pairs or tetraquarks, Ref. [67] suggested two different values for the σN N and f 0 (980)N N coupling constants with different flavor content. Since the f 0 (500) in the present work corresponds to the S-wave correlated 2π, we will take the value of g σ N N from the full Bonn potential, i.e. g f0(500)N N = 8.46. This choice is reasonable, since f 0 (500) in the present work indeed corresponds to the resonance in the S-wave correlated 2π channel. Note that σ -exchange in the full Bonn potential was later replaced by the explicit S-wave correlated 2π-exchange [61,62]. Of course, there is one caveat: Since the f 0 (500) meson has a very broad width, a single coupling constant is a rather crude approximation. A more complete work considering f 0 (500) as S-wave correlated 2π resonance will be considered in a future work.
The mass of the f 0 (500) meson brings out another ambiguity in dealing with f 0 (500) photoproduction. Because of the broad width of f 0 (500) (Γ = (400 − 700) MeV), it is rather difficult to determine its mass exactly. The Particle Data Group estimated the pole mass of f 0 (500) to be (400 − 500) − i(200 − 350) MeV [3]. It implies that we need to examine carefully the f 0 (500) mass dependence of observables. If it is taken to be larger than m f0(500) = 500 MeV, the threshold energy can be larger than the mass of the N (1440). Thus, the total cross section of the γN → f 0 (500)N could reveal different behavior from that with the lower value of the f 0 (500) mass. We will discuss in detail the dependence of the total cross section on m f0(500) in the next Section.
In order to determine the strong coupling constants for the excited baryons, we assume that the f 0 (500) meson consists mainly of the correlated 2π state in S-wave. Then, we can use the decay modes of N * → (π + π − ) I=0 S−wave to fix the corresponding coupling constants by using the partial decay widths defined as where the partial decay width, Γ(N * → f 0 N ) is given as Γ BW N * × Br(N * → f 0 N ). Γ BW N * denotes the Breit-Wigner total decay width and Br(N * → f 0 N ) the branching ratio. |p| represents the magnitude of the final-state momentum defined as The results of the strong coupling constants are shown in Table. III Concerning the photocoupling constants for ρ-meson exchange, we use g γf0(500)ρ = 0.25 [38] from the measurement of the SND Collaboration, and g γf0(980)ρ ≈ 0.21 [68,69] from the molecular KK model for Γ f0→ργ [70]. On the other hand, we can utilize the helicity amplitudes given in the Particle Data Group [3] to find the photocoupling constants f 1 and f 2 for excited nucleons. Since the helicity amplitude are expressed as [71] where k γ stands for the photon decay momentum in the center of mass (CM) frame and is expressed as we obtain the photocouplings for the electromagnetic transitions N * → γN as listed in Table IV. 2γN N (1675) f 1γN N (1680) f 2γN N (1680 The photon coupling constants for the N * resonances whose spin is 1/2 are listed in the first row. In the second and the third row the photon coupling constants for the N * resonances whose spin is 3/2 and 5/2, respectively, are listed.
A hadron has a spatial size, which can be characterized by the phenomenological form factors. Hence, one has to introduce them at each baryon-baryon-meson vertex. Note that each amplitude of N exchanges does not satisfy the gauge invariance in the s-and u-channels, but the sum does. To restore the gauge invariance, we modify the Born scattering amplitudes as where a common form factor is introduced as [72] with Here, q indicates the off-shell four-momentum for an exchanged hadron, m ex stands for its mass, and Λ denotes a cutoff parameter. As for the form factors in N * exchange, we also use Eq. (20). However, it is not sufficient to preserve the unitarity for the N * resonances with spin 3/2 and 5/2, since the corresponding amplitudes show much stronger q 2 dependence so that they are divergent as E γ increases. Thus, in order to tame the divergence, we will employ a Gaussian type of the form factor for both N (1520)3/2 − , N (1675)5/2 − and N (1680)5/2 + -exchange in the s-channel, defined as We are now in a position to present the numerical results of our work. Since there exists the experimental data on the differential cross section of the γN → f 0 (980)N reaction [31], we start with f 0 (980) photoproduction.
A. γN → f0(980)N In order to describe the experimental data on dσ/dt of the γN → f 0 (980)N process [31], we fix the scaling factor in Eq. (8) to be a ρ = 9.0. As is well known, while the Regge approach describes the energy dependence very well, it cannot determine the absolute magnitude of the cross sections. Thus, it is inevitable to introduce the scale parameter a ρ to fit the cross sections. The value of the cutoff parameter Λ ρ is selected to be 1 GeV that is a typical order of the cut-off value. We do not fit it to avoid additional ambiguity. The cutoff parameters in N -and N * -exchanges are chosen as Λ N = 0.8 GeV and Λ N * = 1.0 GeV. The values of Λ ρ,N * are also typical ones. Note that the cutoff value for the backward scattering amplitude is usually smaller than that for the forward scattering one in the literature [44,45]. Furthermore, the experimental data for the backward region are currently not available. So, we simply choose these values without any fitting procedure.   [31], where dσ/dt were measured within the range of the photon energy E γ = (3.0 − 3.8) GeV. In order to compare the present results with the data, we present the results as the shaded band of which the width represents the corresponding region of E γ . Considering the large experimental uncertainty, the results describe the data very well. Note that the t-dependence is governed by ρ-Reggeon exchange in the t channel.
In Fig. 3, we depict the contribution of each channel to the total cross section of the γN → f 0 (980)N reaction. The ρ-Reggeon exchange dominates over the s-and u-channel diagrams, whereas the N exchanges have small effects in the whole energy region. The parameters for f 0 (500) photoproduction in the t, s, and u channels are kept to be the same as those in the case of the γN → f 0 (980)N reaction. However, the N * resonances play essential roles in describing the γN → f 0 (500)N reaction in particular in the vicinity of threshold.  We need to delve into the physical reasons for Model I. The N (1440)1/2 + and N (1535)1/2 − resonances increase the total cross section of the γN → f 0 (500)N reaction near threshold. As shown in the left panel of Fig. 4, the most dominant contribution arises from the N (1440)1/2 + resonance. Since the N (1650)1/2 − and N (1710)1/2 + resonances have relatively smaller strong coupling constants as well as photocouplings as listed in Tables III and IV, the effect of these resonances is tiny. Interestingly, the contribution of the ρ Reggeon in the t channel is rather suppressed. The effect of N exchanges in the s and u channels is much smaller than that of ρ-Reggeon exchange through the whole energy region. The magnitude of the total cross section for f 0 (500) photoproduction is about 40 times larger than that for f 0 (980) production in the case of Model I.
The right panel of Fig. 4 depicts the results of the total cross section for the γN → f 0 (500)N reaction from Model II, where the N * resonances with higher spins are added in the s channel in addition. The N (1680)5/2 + resonance in the s channel yields a remarkably large contribution to the total cross section of f 0 (500) photoproduction, so that its magnitude reaches even about 80 µb around E γ ≈ 1.1 GeV. There is at least one clear reason for this large contribution of the N (1680)5/2 + . Firstly, the photocouplings of the N (1680)5/2 + are very large, as shown in Table IV, which come from the large values of the experimental data on the photon decay amplitudes A 1/2 and A 3/2 [3]. The value of f 1γN N (1680) is even about 32 times larger than that of f 1γN N (1440) . Moreover, the size of the strong coupling constant for the f 0 (500)N N (1680) vertex is comparable to that of g f0(500)N N (1440) . Note that even though the value of g f0(500)N N (1675) is rather large, the N (1675)5/2 − resonance has almost no effect on the total cross section because of its negative parity. The results are not so sensitive to variations of the cut-off masses. In this regard, the contribution of the N (1680)5/2 + resonance discussed here seems to be robust. Thus, it would be indeed of great interest if one could justify experimentally whether the N (1680)5/2 + resonance plays such a dominant role in describing the γN → f 0 (500)N reaction. As mentioned briefly in Introduction, the uncertainty in the mass of f 0 (500) is so large on account of its broad width, it is quite unclear to settle the threshold energy. The PDG data has it that the pole mass of f 0 (500) is [3]. In fact, the estimated mass of f 0 (500) is given in a wide range of its values, as listed in Ref. [3]. Thus, we have to examine the dependence of the total cross section on the mass of f 0 (500). In the left panel of Fig. 5, we draw the results of the total cross section of the γN → f 0 (500)N reaction from Model I with various values of m f0 given between 0.4 GeV and 0.6 GeV. As expected, the smaller values of m f0 produce the larger magnitudes of the total cross section. Note that if one uses the value of m f0 larger than 500 MeV, the N (1440)1/2 + will be excluded because of the larger threshold energy. Thus, the total cross section starts to get reduced when the value of m f0 is larger than 500 MeV. In the present work, we will take m f0(500) = 500 MeV from now on.
In Figs. 4 and 5, we mainly have examined the total cross section of f 0 (500) photoproduction in the vicinity of the threshold energy. We now delve into the dependence of the total cross section on E γ from the threshold energy through 10 GeV in the log scale. In the left panel of Fig. 6, we show the behavior of each contribution from Model I as E γ increases. As expected, all the resonance effects are diminished quickly with the photon energy increased, while the ρ-Reggeon in the t channel takes over the contributions of all the N * resonances around E γ ≈ 2 GeV and then dictates the dependence of the total cross section on E γ . The t-channel Reggeon ensures the unitarity of the total cross section, as shown in Fig. 6. In the limit of s → ∞, the unpolarized sum of the Regge amplitude complies with Thus, if one further increases E γ , the contribution of the ρ meson startes to decrease, satisfying the asymptotic behavior of Eq. (22). The right panel of Fig. 6 shows the results from Model II. Similarly, the contributions of most N * resonances fall off very fast as E γ increases. The effect of the N (1680)5/2 + lessens continuously after E γ ≈ 1.2 GeV, and then becomes smaller than those of the ρ Reggeons around E γ ≈ 2 GeV. Thus, the N * resonances come into play only near the threshold region as anticipated.
In the upper panel of Fig. 7, the results of the differential cross section dσ/d cos θ from Model I are plotted as functions of cos θ, as the photon energy E γ is varied from 0.8 GeV through 3.5 GeV. Usually, the s-channel contributions including all N * resonances with spin 1/2 do not show any cos θ dependence. Note that Model I does not contain any N * resonances with higher spins. Nevertheless, the results of dσ/d cos θ exhibit different peculiarities. The differential cross section dσ/d cos θ in the forward region grows as E γ increases. However, the value of dσ/d cos θ almost vanishes at the very forward angle at higher values of E γ . This arises from the structure of the amplitude of ρ exchange in Eq. (3). Model II yields rather different results from those based on Model I. Since the N (1680)5/2 + resonance is the most dominant one from the threshold energy through 1.5 GeV as shown in Fig. 4, and it has spin 5/2 with positive parity, we expect that it will have a certain effect on the cos θ dependence of the differential cross section. Indeed, it steers dσ/d cos θ up to E γ ≈ 2.1 GeV, as shown in the lower panel of Fig. 7. As E γ increases more than 1.7 GeV, the ρ Reggeon gains control of the cos θ dependence of the differential cross section, so that we have more or less the same results as in the upper panel of Fig. 7. Figure 8 draws the results of the differential cross section dσ/dt as a function of −t in the range of E γ = (3.0 − 3.8) GeV, so that we can directly compare them with those of f 0 (980) photoproduction. The t dependence is almost the same as that of the f 0 (980) case, because we have exactly the same ρ-Reggeon and N exchanges. The only differences come from the coupling constants. The contributions of the N * resonances are all suppressed in this region of the photon energies. The magnitude of dσ/dt is approximately 2 times larger than that of f 0 (980) photoproduction on account of different coupling constants. Note that the differential cross section dσ/dt obeys the following asymptotic behavior and the result shown in Fig. 8

IV. SUMMARY AND CONCLUSION
We aimed in this work at investigating f 0 (500) and f 0 (980) photoproduction, based on a hybridized Regge model. We first described the differential cross section dσ/dt for the γN → f 0 (980)N reaction, compared with the recent experimental data on it. We fixed the relevant Regge parameters by reproducing the data. We introduced the N * contribution in the s channel to study the production mechanism of the γN → f 0 (500)N reaction. Since its threshold energy is much smaller than f 0 (980) photoproduction, there exist several N * resonances that can decay into (ππ) I=0 S−wave N . Assuming that f 0 (500) is much stronger than the background of the (ππ) I=0 S−wave channel, we were able to find the strong coupling constants for the f 0 (500)N N * vertices. The photocouplings of the N * resonances were determined by using the experimental data on the corresponding photon decay amplitudes. The cut-off masses for the form factors were fixed to be 1.0 GeV to avoid additional ambiguity. In dealing with these N * resonances, we constructed Model I and Model II. Model I included those with spin 1/2 only, while Model II was built in such a way that more N * resonances with higher spins were added to Model I. Near threshold, we found that the N (1440)1/2 + and N (1535)1/2 − were dominant ones in Model I, whereas the effects of other N * resonances were almost negligible. In Model II, the contribution of the N (1680)5/2 + dictates the total cross section of the γN → f 0 (500)N reaction. Remarkably, the N (1680)5/2 + resonance enhances the magnitude of the total cross section up to about 80 µb. The main reason comes from the large value of its photoncouplings. The strong coupling constant of the N (1680)5/2 + is also relatively large. Since the mass of the f 0 (500) meson is not precisely fixed because of its large width, we examined the dependence of the total cross section on its mass in the range of (400 − 600) MeV. As expected, small the f 0 (500) mass was, the larger the total cross section was in Model I and Model II. If E γ increases, then the N * contribution fade away very fast, so that ρ-Reggeon exchange takes over the control as in the case of f 0 (980) photoproduction. We also computed the differential cross section dσ/d cos θ. While the contributions of the N * resonances in the s channel are rather flat in the case of Model I, N (1680)5/2 + governs cos θ dependence again because of its high spin. Finally, we computed the differential cross section dσ/dt for the γN → f 0 (500)N reactoin. The results showed that the t dependence and the magnitude looked very similar to those for f 0 (980) photoproduction.
Though it is very difficult to study f 0 (500) photoproduction experimentally, it is still of great importance to study the production mechanism of the γN → f 0 (500)N reaction, since it cast light on the structure of the N * resonances as investigated in the present work. It also provides a certain clue in studying more complicated processes with three-particle final states such as γN → (ππ) I=0 S−wave N in the future.