Photoproduction of $f_0(980)$ and $f_0(1500)$ resonances off a proton target

We study the $\gamma p \to p f_0$ [$f_0(980)$ and $f_0(1500)$] reaction close to threshold within an effective Lagrangian approach. The production process is described by $s$-channel nucleon pole or $t$-channel $\rho$ and $\omega$ exchange. The $K^0 \bar{K}^0$ invariant mass distributions of the $\gamma p\to f_0(980)p \to K^0 \bar{K}^0$ and $\gamma p\to f_0(1500)p \to K^0 \bar{K}^0 p$ reactions are investigated, where the two kaons have been separated in $S$ wave decaying from $f_0(980)$ and $f_0(1500)$. It is shown that the $s$-channel process is favored for the production of $f_0(980)$, while for the $f_0(1500)$ production, the experimental measurements can be described quite well by the $t$-channel process. It is expected that the theoretical results can be tested by further experiments at CLAS.


I. INTRODUCTION
The study of the structure of low-lying scalar mesons is a topic of high interest in hadronic physics and is attracting much attention [1,2]. For the scalar meson f 0 (980), it is now widely accepted that the simplest picture, where it is described as an orbital excitation of quark-antiquark pairs, is not compatible with the experimental observations on its decay modes. Thus, the f 0 (980) is thought to be a molecule state formed from the interaction of pseduoscalar mesons [3][4][5][6][7][8][9][10], and it couples strongly to the KK channel [11], which is its dominant component. The f 0 (1500), on the other hand, with a mass of 1504±6 MeV and a width of 109±7 MeV [12], is a candidate for having much glueball content [13,14]. Photoproduction of these scalar resonances provides a unique place to probe their nature.
On the experimental side, photoproduction of f 0 (980) meson on protons was measured by CLAS collaboration in Refs. [15,16] at the photon energy region of E γ = 3.0 − 3.8 GeV, where f 0 (980) was detected via its decay in the π + π − channel by performing a partial wave analysis of the reaction γp → pπ + π − . However, the production rate of f 0 (980) is much smaller than the one for the ρ meson. Very recently, a partial wave analysis is performed for the γp → pK + K − reaction by the CLAS collaboration [17], where the production amplitudes have been parametrized using a Regge-theory inspired model. There were also pioneering measurements [18,19] for the photoproduction of K + K − pairs. After that, there were several theoretical calculations about the scalar mesons production in the process of γp scattering. A combined analysis of ππ and KK photoproduction in S-wave is conducted in Ref. [20], while * Electronic address: gli@mail.qfnu.edu.cn the f 0 (980) and a 0 (980) photoproduction for photon energies close to the KK production threshold was studied in Ref. [21] using tools of chiral unitary approach. In Ref. [22], within a model based on the Regge approach, a theoretical analysis of the data on photoproduction of the f 0 (980) was done, where it was shown that the radiative decay rate for f 0 (980) → γV is important in the theoretical predictions. In Ref. [23] the γp → a 0 (980)p and γp → f 0 (980)p reactions were investigated with the main aim for studying the possibility of observing a 0 (980)-f 0 (980) mixing in these processes. In Ref. [24], the a 0 (980) and f 0 (980) photoproduction was investigated by considering the Regge-cut effects which were fixed from π 0 photoproduction. With the Regge theory, the f 0 (1500) photoproduction was also studied in Ref. [25] at E γ = 9 GeV.
Recently, the reaction γp → pX → pK 0 S K 0 S was investigated by the CLAS Collaboration [26] with photon energies from 2.7 − 5.1 GeV, where it was found that the angular distributions of the data suggest that most of the K 0 S K 0 S decay is from scalar mesons in S wave. In particular, a clear peak is seen at 1500 MeV in the invariant mass spectra of K 0 S K 0 S , and the mass and width of this peak is consistent with that of the scalar meson f 0 (1500), while the enhancement close to K 0 S K 0 S threshold is due to the f 0 (980) decay. In addition, there is no clear signals for contributions from the baryon resonances.
In the present work, based on the new measurements of CLAS collaboration [26], we reanalyze the γp → f 0 (980)p → K 0K 0 p and γp → f 0 (1500)p → K 0K 0 p reactions 1 within an effective Lagrangian method near threshold. As in Refs. [22,23] we consider the contributions from t-channel ρ 0 and ω exchange. Since the cou- 1 We take |K 0 >= 1 where we ignore the CP violation.
plings of f 0 to V γ channel is scarce [12], we take these results obtained in Refs. [27,28], where meson loops were considered, and the f 0 (980) was taken as a dynamically generated state. On the other hand, possible s-channel proton pole process, which was not included in all these above theoretical calculations, is also investigated in this work. It is shown that the new measurements of Ref. [26] may indicate the dominant s-channel contribution for the f 0 (980) photoproduction. In this respect, we show in this work how the CLAS measurements could be used to determine the reaction mechanisms of the photoproduction of these scalar mesons.
In the next section, we will give the formalism and ingredients in this work, then numerical results and discussions are given in Sec. III. A short summary is given in the last section.

II. FORMALISM AND INGREDIENTS
The effective Lagrangian method is widely used to calculate cross sections for different reactions in the resonance production region. In this section, we introduce theoretical formalism and ingredients to calculate the scalar mesons photoproduction off protons within the effective Lagrangian method.

A. Interaction Lagrangian densities and scattering amplitudes
We first consider the basic t-channel tree level diagram for the γp → pf 0 [f 0 ≡ f 0 (980) or f 0 (1500)] reaction as shown in Fig. 1. This includes the contributions from ρ 0 and ω meson exchange terms. Following Ref. [28], we can write down the amplitude for the f 0 → V γ deacy as from where, we can obtain the partial decay width of the f 0 meson into a vector meson and a photon, (2) With masses (M f0(980) = 990 MeV, M f0(1500) = 1504 MeV, and m ρ = m ω = m V = 780 MeV), and the partial decay widths of the scalar f 0 (980) and f 0 (1500) mesons radiative decay into a vector meson and a photon as obtained in Refs. [27,28], we obtain these coupling constants as list in Table I.
To compute the scattering amplitudes of the diagrams shown in Fig. 1, we need also the effective interactions for the ρN N and ωN N vertices. We take the interaction Lagrangian densities as used in Refs. [29,30]: We use the coupling constants g ρN N = 3.36, κ ρ = 6.1, g ωN N = 15.85 and κ ω = 0 of Refs. [31,32]. Then we can write the ρN N and ωN N vertices as, B. γp → pf0 scattering amplitudes With ingredients given above, we can easily obtain the t-channel γp → f 0 p reaction invariant scattering amplitude: where G µν is the Feynman propagator of ρ or ω meson which has the following form: Since hadrons are not point-like particles, the form factor of hadrons need to be taken into account [32,33]: with t = k 2 and Λ c a free cut-off parameter.

C. Differential cross section
The differential cross section for the γp → pf 0 reaction by the exchanged ρ 0 and ω mesons can be expressed as where s is the invariant mass square of the γp system, and p 1 denotes the photon three momentum in the center of mass (c.m.) frame. The total invariant scattering amplitude M is given by On the other hand, we can generalize the two body process as in Eq. (10) by considering the situation which allows the f 0 to decay into a K 0 and aK 0 as shown in Fig. 2. By working out the three-body phase space of the γp → f 0 p → pK 0K 0 reaction, we find where Γ f0 is the total decay width and we take Γ f0 = 100 MeV 2 and 109 MeV for f 0 (980) and f 0 (1500), respectively. M inv represents the invariant mass of K 0K 0 . For f 0 (1500), Γ f0→K 0K 0 is given by 2 We take a relative large value for the total decay width of the f 0 (980), which is favored by the new CLAS measurements [26].

III. NUMERICAL RESULTS AND DISCUSSION
In this section we will show the numerical results for the γp → pf 0 reaction. We firstly show the theoretical results for the case of f 0 (1500) photoproduction.
A. Invariant mass distributions for the γp → pf0(1500) → pK 0K 0 reaction We compare our theoretical calculations for the invariant K 0K 0 mass distributions as a function of M inv with the recent CLAS data of Ref. [26]. The theoretical dσ/dM inv is calculated by with E max γ = 5.1 GeV and E min γ = 2.7 GeV, which are the photon energy region of Ref. [26].
In Fig. 3, we show the theoretical results, c 1 dσ/dM inv , for the K 0K 0 invariant mass distributions for the γp → pf 0 (1500) → pK 0K 0 reaction, comparing with the experimental measurements of Ref. [26], where c 1 = 2.2 and Λ c = 1.7 GeV has been adjusted to the strength of the experimental data reported by the CLAS Collaboration [26] at its peak around M inv = 1500 MeV. One can see that, we can describe quite well the experimental measurements for the γp → pf 0 (1500) → pK 0K 0 reaction by considering the t-channel ρ 0 and ω exchange, especially for the case of |t| < 1 GeV 2 . This may indicate that the t-channel process is dominant for the photoproduction of the f 0 (1500) resonance. We first present the theoretical results for the γp → pf 0 (980) → pK 0K 0 reaction by including the t-channel ρ 0 and ω exchange. The numerical results of the K 0K 0 invariant mass distributions obtained with c 1 = 0.9 and Λ c = 1.07 GeV, are shown in Fig. 4. The peak of the K 0K 0 invariant mass distributions is around 1020 MeV, very close to the mass threshold (995 MeV) of K 0K 0 . One can see that the model cannot describe simultaneously both the experimental data for |t| < 1 GeV 2 and |t| > 1 GeV 2 . At M inv = 1020 MeV and E γ = 3.9 GeV, the values of t is −5.36 GeV 2 < t < −0.02 GeV 2 , 3 from where we find that the phase space for |t| > 1 GeV 2 is more than four times larger than the case of |t| < 1 GeV 2 . However, the t-channel form factor F 1 = ( c −t ) 2 with Λ c ∼ 1.07 GeV will contribute a suppression with factor about 14 for the case of |t| > 1 GeV 2 than that of |t| < 1 GeV 2 . Hence, it is expected that, with the values of c 1 = 0.9 and Λ c = 1.07 GeV, the results for 3 The values of t for the production of f 0 (1500) is −3.89 GeV 2 < t < −0.14 GeV 2 .
|t| > 1 GeV 2 should be much smaller than the ones for |t| < 1 GeV 2 . But, the experimental data of Ref. [26] tell us that the values for |t| > 1 GeV 2 are even larger than those for |t| < 1 GeV 2 . This may indicate that the tchannel exchange mechanism is not enough to explain the experimental measurements of CLAS Collaboration [26]. We have also performed calculations for the γp → pf 0 (980) → pK 0K 0 reaction with different values of c 1 and Λ c . It turns out that we can also reproduce the experimental measurements with c 1 = 0.012 and a large Λ c = 5 GeV. Thus, the inclusion of other reaction mechanism is needed to achieve a good description of the new CLAS experimental measurements.
Next, we study another kind of reaction mechanism for γp → pf 0 (980) → pK 0K 0 reaction, which is depicted in Fig. 5, where we have considered the contribution from the s-channel nucleon pole term. To compute the contribution of this term, the interaction Lagrangian densities for γpp and f 0 (980)pp vertexes are needed. We take them as used in Refs. [23,36]: where κ p = 1.5. Then one can easily write down the corresponding amplitude for s-channel nucleon pole term as, with and which is obtained from a contact term and for keeping the scattering amplitude M s gauge invariant [36,37]. The theoretical results of K 0K 0 invariant mass distributions of the γp → f 0 (980)p → K 0K 0 p reaction with the contribution from s-channel nucleon pole are shown in Fig. 6, from where we can see that we can explain the experimental measurements for both |t| < 1 GeV 2 and |t| > 1 GeV 2 cases quite well, since there is no so strong t dependence factor F 1 in the s-channel process. The theoretical numerical results shown in Fig. 6 are obtained with c 1 g f0pp = 7.5 and Λ s = 1.1 GeV.
One might think that the inclusion of higher nucleon excitations might improve the situation, since they have large mass and will give large contributions. However, at one certain photon energy E γ , the propagator of the s-channel process is then just a constant. The estimation of the K 0K 0 invariant mass distributions in our model is only sensitive to the production rate of the f 0 (980), and the nucleon pole term is sufficient for this purpose. By neglecting contribution from other N * resonances, we can present a more general picture of the s-channel f 0 (980) production processes, though our results are more general than this would suggest.
On the other hand, we calculate dσ/dt for γp → pf 0 (980) reaction with the above two different reaction mechanisms at the photon energy E γ = 3.4 GeV. The numerical results 4 are shown in Fig. 7, comparing with the experimental data taken from Refs. [15,17]. The solid and dashed lines represent the result from t-and s-channel process, respectively. One can see that both t-channel mechanism and s-channel process can describe fairly well the current experimental data. However, the line shapes of these two different reaction mechanisms are sizably different. The slope of the results for the schannel process is more flat than the case of t-channel ρ 0 and ω exchange. We hope that this feature may be used to determine the reaction mechanism of γp → pf 0 (980) reaction.

IV. SUMMARY
In this work, we have studied the γp → pf 0 [f 0 (980), f 0 (1500)] → pK 0K 0 reactions near threshold within an effective Lagrangian approach. The K 0K 0 invariant mass distributions are evaluated, where the two kaons have been separated in S wave decaying from the scalar mesons f 0 (980) and f 0 (1500). It is shown that the t-channel ρ 0 and ω exchange processes can describe the experimental data on the γp → pf 0 (1500) → pK 0K 0 reaction, while the s-channel process is favored for the γp → pf 0 (980) → pK 0K 0 reaction, since the t-channel mechanism for the f 0 (980) photoproduction fails to reproduce the experimental measurements. Furthermore, it is found that the theoretical numerical results for the γp → pf 0 (980) differential cross section, dσ/dt, of the two different reaction mechanisms are sizeably different. It is expected that the theoretical results can be tested by further experimental measurements at CLAS [26].
Finally, we would like to stress that, thanks to the important role played by the non t-channel process in the γp → pf 0 (980) reaction, accurate data for this reaction can be used to improve our knowledge about the reaction mechanism of this reaction and also the nature of f 0 (980). This work constitutes a first step in this direction.