Consistent analysis of $f_1 (1285)$ meson form factors

Parameterization of the form factors of $f_1 (1285)$ meson is proposed. This parameterization is consistent with the available experimental data on the cross sections of $f_1 (1285)$ meson production in the processes $e^+e^- \to f_1 (1285)$ and $e^+e^- \to e^+e^-f_1 (1285)$, as well as on the widths of the decays $f_1 (1285)\to e^+e^-$, $f_1 (1285)\to \rho^0\gamma$, $f_1 (1285)\to \rho^0\pi^+\pi^-$, and $f_1 (1285)\to 2\pi^+2\pi^-$. Our parameterization is also consistent with the predictions for the asymptotic behavior of these form factors.


I. INTRODUCTION
The experimental and theoretical investigations of two-photon production of f 1 (1285) meson is very interesting since a particle with the spin S = 1 cannot be produced in a collision of two real photons due to their identity [1]. However, f 1 (1285) meson can be produced in a collision of two virtual photons or one virtual photon and one real photon.
Therefore, the probability of these processes can be sensitive to the f 1 (1285) meson internal structure, i.e., to the dependence of the form factors on photon virtualities. At present, there are a few experimental [2][3][4][5][6][7] and theoretical [8][9][10][11][12][13][14][15][16][17] results on production and decays of f 1 (1285) meson. Unfortunately, QCD cannot predict now the shapes of the corresponding form factors at moderate photon virtualities. Some predictions for the form factors exist only in the region of very large virtualities, though even in this case the particular shape of the form factors depends on the unknown wave functions of f 1 (1285) meson [8]. Therefore, to understand the features of f 1 (1285) meson production processes, it is necessary to use the phenomenological parameterization of the form factors, which should be consistent with all available experimental data. This is the goal of our work.
For the f 1 (1285) meson decay into virtual ρ 0 meson with the momentum k 1 and a virtual photon with the momentum k 2 , we follow the logic of the vector dominance model and write the corresponding form factors as Here µ 2 ρ = m 2 ρ −im ρ Γ ρ , where m ρ and Γ ρ are the mass and the width of ρ 0 meson, respectively, ef ρ is the constant of the ρ 0 meson-photon transition. This quantity can be expressed via the width Γ ρ→ee of the ρ 0 meson decay into e + e − pair, where e is the electron charge, α = e 2 ≈ 1/137 is the fine structure constant, and = c = 1.
At last, the form factors for the amplitude of the f 1 (1285) meson decay into two virtual photons read in rest frame of f 1 (1285) meson. In this frame Then we have for the spiral amplitudes , Here subscripts "+, −, 0" stand for the corresponding helicities of the virtual photons. Using this formula and the parameterization of the form factors (4), we can describe various processes, extract the parameters of the model, and compare our predictions with the experimental data available.

III. PARAMETERS OF THE MODEL
It is seen from Eq. (6) that the amplitudes M ±0 and M 0± are independent of the form factor F (k 2 1 , k 2 2 ). Therefore, the experimental data, which are related solely to these amplitudes, allow us to extract the constant g 2 .
Using this result and Eq. (6) we find The uncertainty in this formula comes from experimental uncertainties of all quantities in it, though the main contribution is given by the uncertainty of Γ γγ .
VES Collaboration has studied the process f 1 (1285) → ρ 0 γ followed by the decay of ρ 0 meson into a pair of pions [4]. Using the angular distribution of pions it was possible to extract the events with the longitudinal polarization of ρ 0 meson and with the transverse polarization. The following result was obtained for the elements of the ρ 0 meson polarization density matrix ρ 00 ρ 11 = 3.9 ± 0.9 (stat.) ± 1.0 (sys.) .

We use this value in the relation
where a = m ρ /M ≈ 0.6. Then it follows from Eq. (3) that We also predict the width Γ ργ of the f 1 (1285) → ρ 0 γ decay, and compare this quantity with the experimental value Γ ργ = (1.2 ± 0.3) MeV [18]. As a result we obtain This value is in good agreement with (8).
The ratio |g 1 /g 2 | can be extracted from Eq. (10). Since g 1 and g 2 are complex numbers, the value |g 1 /g 2 | derived from (10) depends on the relative phase φ of these numbers.
It is seen from Fig. 1 that the value |g 1 /g 2 | is almost constant and small when φ varies in a wide region from π/2 to 3π/2. That means that to fix φ together with |g 1 /g 2 | it is necessary to use another data (see Section V).
Recently, the first observation of f 1 (1285) meson production in e + e − annihilation has been reported [7]. An estimate of the cross section of this process was made in [17] for another parameterization of the form factors. However, the predictions for the cross section of the process e + e − → e + e − f 1 (1285) following from that parameterization differ substantially from the experimental data [5]. Therefore, it is interesting to compare the predictions for this process following from our parameterization of the form factors (see next Section).

VII. CONCLUSION
In this paper we propose parameterization of the f 1 (1285) meson form factors and compare our predictions with the available experimental data. For φ close to π, where φ is the relative phase of the constants g 1 and g 2 in the form factors, we find good agreement between our theoretical predictions and the experimental data.