Productions of $f_{1}(1420)$ in pion and kaon induced reactions

The $f_{1}(1420)$ productions via pion and kaon induced reactions on a proton target are investigated in an effective Lagrangian approach. Two treatments of the $t$-channel Born term, the Feynman model and the Regge model, are introduced to calculate the total and differential cross sections of the $\pi^{-}p\rightarrow f_{1}(1420)n$ and $K^{-}p\rightarrow f_{1}(1420)\Lambda $ reactions. The numerical results indicate that the experimental data of the total cross section of the $\pi ^{-}p\rightarrow f_{1}(1420)n$ reaction can be reproduced in both the Feynman and the Regge models. Based on the parameters determined in the pion induced reaction, the cross sections of the $K^{-}p\rightarrow f_{1}(1420)\Lambda $ reaction, about which there is little data found in the literature, are predicted in the same beam momentum range. It is found that the line shapes of the total cross section obtained in a kaon induced reaction with two treatments are quite different. The cross sections for both reactions are at an order of magnitude of $\mu$b, or larger, at a beam momenta up to 10 GeV/c. The differential cross sections for both pion and kaon induced reactions are also present. It is found that in the Regge model, the $t$ channel provides a sharp increase at extreme forward angles. The results suggest that the experimental study of the $f_1(1420)$ in the kaon induced reaction on a proton target is as promising as in the pion induced reaction. Such an experimental measurement is also very helpful to clarify the production mechanism of the $f_1(1420)$.


I. INTRODUCTION
The light meson is an important way to understand the nonperturbative QCD. Many light mesons have been observed and listed in the Review of Particle Physics (PDG) [1]. However, the internal structure of light meson is still a confusing problem due to large nonperturbative effect in light flavor sector. Currently, the electron-positron collision is the most important way to study light meson. It will be very helpful to study the production and properties of light meson in different reactions. A new detector, glueX, was equipped at CEBAF after 12 GeV upgraded, which will focus on light meson spectrum [2]. The pion-induced light meson production is very important in the history of discovery of many light mesons. The secondary pion beam is accessible at J-PARC [3] and COM-PASS [4] with high precision. The kaon beam can be also used to study light meson, which is available at OKA@U-70 [5] and SPS@CERN [6], and J-PARC [7]. The data from future experiments at those facillities will provide a good opportunity to deep our understanding of internal structure of light meson.
In the PDG, the f 1 (1420) is listed as an axial-vector state with quantum number I G (J PC ) = 0 + (1 ++ ) with a suggested mass of 1426.4± 0.9 MeV and a suggested width of 54.9±2.6 MeV [1]. The f 1 (1420) meson was first observed in pion-nucleon interaction in the Lawrence Radiation Laboratory in 1967 [8], and confirmed in other experiments with pion beams about the year 1980 [9][10][11]. The f 1 (1420) was also observed in recent experiments in e + e − and J/ψ decays [12][13][14]. Though the f 1 (1420) is well established in experiment * xywang01@outlook.com † Corresponding author : junhe@njnu.edu.cn as a resonance structure, the internal structure of the f 1 (1420) is still unclear upto now. In the conventional qq picture, the f 1 (1420) can be classified as a partner to the f 1 (1285) in the 3 P 1 nonet of axial mesons, and the mixture of nonstrange f 1q = (uū + dd)/ √ 2 and strange f 1s = ss was also discussed in the literature [15][16][17]. However, a recent study in Ref. [18] suggested that the f 1 (1420) is not a genuine resonance but from the decay modes of the f 1 (1285) in K * K and πa 0 (980).
To determine origin of the f 1 (1420), more precise experimental data are required. In this work, based on the existent old data we will analyze the f 1 (1420) production in pion induced reactions in an effective Lagrangian approach. The kaon induced production will be discussed based on the results of pion induced interaction, which will be helpful to future high-precision experimental studies. Since in the current work we focus on the production mechanism of f 1 (1420), the coupling constants are still determined assuming the f 1 (1420) as a genuine resonance [1]. This paper is organized as follows. After introduction, we present formalism including Lagrangians and amplitudes of the f 1 (1420) production in Section II. The numerical results of cross section follow in Sec. III. Finally, the paper ends with a brief summary.

II. FORMALISM
The reaction mechanism is illustrated in Figs. 1. Usually, the contribution from s channel with nucleon pole is expected to be very small, and will be neglected in the current calculation. The u-channel contribution is usually small and negligible at low momentum [19]. Considering the experiment data points at high beam momentum are obtained by continuation of the t-channel contribution at very forward angles to all an-gles, it is reasonable to calculate the cross section only with the t-channel contribution. Hence, in the present work, we do not include the contributions from the nucleon resonances in the s or u channel.
Since dominant decay of the f (1420) is KK * , it is reasonable to take the K * exchange as the dominant contribution in the t channel of kaon induced production. For pion induced production, we need vertex for decay of the f 1 (1420) with a pion. In Ref. [20], a branch ratio about 5% was reported in the a 0 π channel. Hence, in the current work, we adopt a 0 exchange in the t-channel of pion induced production as in the case of pion-induced f 1 (1285) production [19].
The value of g f 1 K * K can be determined from the decay width with where λ is the Källen function with a definition of λ(x, Since the branching fraction of f 1 (1420) decay to K * K was suggested to be 96% in Ref. [20], one gets For the t channel exchange [23], the form factor F(q 2 ) = (Λ 2 − m 2 )/(Λ 2 − q 2 ) is taken into account. Here, q and m are four-momentum and mass of the exchanged meson, respectively. The value of cutoff Λ will be determined by fitting experimental data.

B. Amplitudes
According to the above Lagrangians, the scattering amplitude of the π − p → f 1 (1420)n or K − p → f 1 (1420)Λ process can be written as where ǫ µ * f 1 is the polarization vector of f 1 meson, andū or u is the Dirac spinor of nucleon or Λ baryon.
For the π − p → f 1 (1420)n reaction, the reduced amplitude where t = (k 1 −k 2 ) 2 is the Mandelstam variables. The coupling constants are fixed with the experimental data and the fitting the pion deuced f 1 (1285) production as addressed above. For the process of K − p → f 1 (1420)Λ, the reduced amplitude A µ is written as with Here, the coupling constants are also fixed as in the pion induced production. Hence, the only free parameter are the cutoff in form factor.

C. Reggeized t-channel
To analyze hadron production at high energies, a more economical approach may be furnished by a Reggeized treatment [25][26][27][28][29][30]. In our previous works [19,28,31], an interpolating Regge treatment was introduced to interpolate the Regge trajectories smoothly to the Feynman propagator at low energy as proposed in Ref. [33] . Because there are only 4 data points, we do not adopt the interpolating Reggiezed treatment, but discuss both the Feynman model and the Regge model. For the Feynman model, the t-channel amplitude in Eqs. (9 and 10) is applied directly. The Regge model can be introduced by replacing the t-channel Feynman propagator with the Regge propagator follows: The scale factor s scale is fixed at 1 GeV. In addition, the Regge trajectories α a 0 (t) and α K * (t) read as [29,30], After the Reggeized treatment introduced, no additional parameter is introduced.

III. NUMERICAL RESULTS
With the preparation in the previous section, the cross section of the π − p → f 1 (1420)n and K − p → f 1 (1420)Λ reactions will be calculated and compared with experimental data [8][9][10][11]. The differential cross section in the center of mass (c.m.) frame is written as where s = (k 1 + p 1 ) 2 , and θ denotes the angle of the outgoing f 1 (1420) meson relative to π/K beam direction in the c.m. frame. k c.m. 1 and k c.m.
2 are the three-momenta of initial π/K beam and final f 1 (1420), respectively.
A. Cross section of the π − p → f 1 (1420)n reaction Here, we minimize χ 2 per degree of freedom (d.o. f.) for the experimental data of the total cross section by fitting the cutoff parameter Λ using a total of 4 data points at the beam momentum P Lab from 3.1 to 13.5 GeV. The fitted cutoff and the χ 2 /do f are listed in Table. I. From Fig. 2, it is found that the experimental data of total cross section of π − p → f 1 (1420)n reaction can be reproduced using the Feynman model or with the traditional Reggeized treatment. The shapes of total cross section in both models  [19,28,31,33], is introduced, the data can be described better, but more precise data are required. It is obvious that the χ 2 is mainly from the two data point around 4GeV. In both models, the upper data point is suggested, which can be checked in future high-precision experiment. In Fig. 3, we present the prediction of differential cross section of π − p → f 1 (1420)n reaction in two schemes at different beam momentum. It can be seen that the differences between the Regge and the Feynman model at low energies are small but become large at higher energies. With increases of the beam energy, the slope of curve in the Regge model is steeper than that in the Feynman model at forward angles, which can be tested by further experiment to clarify the role of the Reggeized treatment. Since there does not exist the experimental data for the K − p → f 1 (1420)Λ reaction, here we give the prediction of the cross section of the K − p → f 1 (1420)Λ reaction. In Ref. [11], the experiment shown that σ(K − p → f 1 (1420)Λ)/σ(π − p → f 1 (1420)n) > 10 at beam momentum P Lab = 32.5 GeV. In our above calculation, the σ(π − p → f 1 (1420)n) ≃ 0.017 (µb) at P Lab = 32.5 GeV by taking Λ = 1.6 GeV in the Regge model. For the K − p → f 1 (1420)Λ reaction, by taking Λ = 1.6, one can get a value of total cross section about 0.14 µb in the Regge model, which means that the Λ = 1.6 GeV is a relatively reasonable value to calculate the cross section of K − p → f 1 (1420)Λ reaction in the Regge model. In Fig. 4 we present the total cross section of K − p → f 1 (1420)Λ reaction within the Regge model.
For the Feynman case, if we choose the cutoff as in pion induced case, the cross section of K − p → f 1 (1420)Λ reaction will increase continuously with the increase of the momentum p Lab in the energy region considered. Though it does not conflict with σ(K − p → f 1 (1420)Λ)/σ(π − p → f 1 (1420)n) > 10 as suggested in Ref. [11], it seems unnatural that total cross section will reach 100 µb. Hence, here we chose a smaller value of cutoff Λ=1.0 GeV. Different from the pion induced case, from Fig. 4 the shape of curve in the Regge model is different from that in the Feynman model. For the Regge case, we notice that the line shape of the total cross section goes up very rapidly and has a peak around P Lab = 3.53 GeV. For the Feynman case, it is seen that the value of total cross section is becoming higher and higher with the increase of the beam momentum upto 20 GeV. The  monotonically increasing behavior may be caused by the K * exchange amplitude [30]. The differences between the Regge and the Feynman model will be useful in clarifying the role of the Reggeized treatment. The differential cross section in two models are illustrated in Fig. 5, which show that the discrepancy of the differential cross sections of two models is small at low beam momenta but become large at higher beam momenta. From Fig. 5, one notice that, relative to the results within Feynman model, the differential cross section in Regge model is very sensitive to the θ angle and gives a considerable contribution at forward angles with the increases of beam momentum.

IV. SUMMARY AND DISCUSSION
We have studied the π − p → f 1 (1420)n and K − p → f 1 (1420)Λ reaction within the Feynman model and Regge model. For the π − p → f 1 (1420)n reaction, both results calculated with the Regge and the Feynman model can reproduce the experimental data, but the differential cross sections at high energies are different. It is found that differential cross section for π − p → f 1 (1420)n reaction within Regge model is very sensitive to the θ angle and gives a considerable contribution at forward angles, which can be checked by further experiment and may be an effective way to examine the validity of the Reggeized treatment.
For the K − p → f 1 (1420)Λ reaction, the shapes of total cross sections obtained in both models are very different. The total cross section will increase continuously in Feynman model in the energy considered in the current work. If the cutoff in the pion induced production is adopted, the cross section will increase to 100 µb. In the Regge model it will decrease at momenta larger than P Lab = 4 GeV. It is consistent with that the Regge model is more suitable to describe the behavior of cross section at high energies. The shape of differential cross section of K − p → f 1 (1420)Λ reaction is similar to the result of π − p → f 1 (1420)n reaction. With the Reggeized treatment, the t channel provides a sharp increase at extreme forward angles.
The pion and kaon beams can be provided at J-PARC and COMPASS. The precision of possible data in future experiments at these facilities will be much higher than old experiments. The above theoretical results may provide valuable information for possible experiment of searching for the f 1 (1420) at these facilities.