Measurement of $\phi$(1020) meson leptonic width with CMD-2 detector at VEPP-2M Collider

The $\phi$(1020) meson leptonic width has been determined from the combined analysis of 4 major decay modes of the resonance ($\phi\to K^+ K^-,K^0_LK^0_S,\pi^+\pi^-\pi^0,\eta\gamma$) studied with the CMD-2 detector at the VEPP-2M $e^+e^-$ collider. The following value has been obtained: $\Gamma(\phi\to e^+e^-) = 1.235\pm 0.006\pm 0.022$ keV. The $\phi(1020)$ meson parameters in four main decay channels have been also recalculated: $B(\phi\to K^+K^-) = 0.493\pm 0.003\pm 0.007$, $B(\phi\to K_LK_S) = 0.336\pm 0.002\pm 0.006$, $B(\phi\to\pi^+\pi^-\pi^0) = 0.155\pm 0.002\pm 0.005$, $B(\phi\to\eta\gamma) = 0.0138\pm 0.0002\pm 0.0002$.


Introduction
The simplest decay of any quarkonium vector state occurs through its annihilation into a virtual photon, which produces a lepton or quark-antiquark pair.A leptonic width of a vector state offers a measure of wave function overlap at the origin thus providing information about interactions of quarks composing the vector meson.For these reasons, decays to lepton pairs are heavily studied and used to characterize the most basic features of each vector state.
The present analysis is devoted to a measurement of the φ(1020) leptonic width Γ(φ → e + e − ).Previously it has been measured by various methods in a number of experiments.In Refs.[1,2] the leptonic width was determined in a direct study of φ → e + e − and φ → µ + µ − decays, while in Ref. [3] Γ(φ → e + e − ) was evaluated from the simultaneous fit of four major decay modes of the φ(1020) meson.
The cross sections of the processes e + e − → K + K − , K L K S , π + π − π 0 , ηγ, previously measured in the experiments [4,5,6,7] are listed in Tables 1 -4.The errors of the cross sections in the Tables are statistical only.

Analysis
To determine the leptonic width of the φ(1020) meson, we perform a simultaneous fit of the four φ(1020) major decay modes with a leptonic width as a fit parameter.To fit the experimental cross sections in different channels, we use the same functions and fit parameters as in the corresponding dedicated studies: where s is the center-of-mass (c.m.) energy squared, q = s/4 − m 2 K -momentum of charged (neutral) kaon, m V , Γ V are the mass and total width of the vector meson V, respectively, is the propagator of the vector meson V and energy dependence of the meson V total width is chosen according to [10], Γ(φ → e + e − ) is the φ(1020) meson leptonic width, B(V → e + e − ) is the branching ratio of the decay V → e + e − , B(V → X) -branching ratio of the vector meson decay into a final state X.Here W (s) is the function [10] describing the phase space of the π + π − π 0 final state, 3 -phase space factor for the vector meson V decay into a pseudoscalar meson P and photon, ψ φ -the phase of the φ-ω interference in the φ → π + π − π 0 decay channel.
The function Z(s) given by the relation s describes the Coulombic interaction of charged kaons in the final state [11].It should be mentioned that ρ and ω mesons are below the K K production threshold and their contributions to e + e − → K K have been calculated according to the SU(2) and SU(3) model predictions [10].The branching fractions of different channels, φ(1020)-meson leptonic width, the resonance mass and total width as well as the phase of the φ-ω interference in the φ → π + π − π 0 decay channel are parameters of the fit.To determine branching fractions of the four major φ(1020)-meson decays, we use a constraint: To estimate systematic errors of the parameters, one should take into account correlations between systematic errors of the K + K − and K 0 L K 0 S as well as between π + π − π 0 and ηγ decay channels because of common contributions (like, e.g., from luminosity measurement and radiative corrections).In Tables 5, 6 contributions to a systematic error of each channel are presented.The correlated systematic errors for the K + K − and K L K S as well as for π + π − π 0 and ηγ cross sections due to luminosity measurement and radiative corrections are equal to 1.1% and 2.2%, respectively.A difference in systematic errors due to luminosity measurement is caused by different detector conditions during data taking periods.
To determine the leptonic width and branching fractions taking into account systematic errors, we use the Maximum Likelihood method with the following likelihood function: , where f data ı is the experimental value of the cross section for the process ı -the value of the theoretical cross section for the process ı, subscript j counts an "individual" part of a systematic error in the cross section of the process ı and ∆ 1 denotes a common part of the systematic error in measurements of the kaon cross sections, while ∆ 2 means a common part of systematic errors for the π + π − π 0 and ηγ studies.The following values have been obtained from the maximization procedure: To determine the statistical errors of the parameters separately, the same fit has been performed with ∆ k and δ j fixed at zero.The following values have been obtained: As one can see, the central values of the parameters from the last fit are slightly shifted with respect to the results of the previous fit.Using Monte-Carlo simulation it was checked that a variation of the shape of the likelihood function shifts the "true" value of the leptonic width by −(0.0051 ± 0.0001) keV, while taking into account correlations between the systematic errors leads to changing the Γ ee input value by −(0.0287 ± 0.0002) keV.So, the obtained value of the Γ ee = 1.206 ± 0.022 keV should be corrected by this shift.Thus our final result for the φ(1020) leptonic width is: Γ ee = 1.235 ± 0.022 keV.
The assumption of SU(2) and SU(3) symmetry [10] used to calculate the ρ, ω → K K contributions is valid within ∼20% accuracy.To estimate a systematic error due to the choice of the fitting model we performed a fit with the φ(1020) contribution only in the e + e − → K K channels.The obtained differences in the values of the fitting parameters were less than 0.5% and used as a model systematic error.
Contributions to the systematic error due to uncertainties in parameters used as fit constants (m ρ , m ω , Γ ρ , Γ ω , ...) are at the level of 10 −5 .
In plots of Fig. 1(a-d) one can see the c.m. energy dependence of the cross sections for the processes under study along with the corresponding fitting curves.In Fig. 2(a-d) the differences between the cross sections and the values of the fitting curves for all the processes are presented.So, our final results are: where the first error is statistical and the second is systematic.
In Fig. 3 the results of different measurements of Γ ee are shown along with the result of the present analysis.The shaded region corresponds to the leptonic width value from [12] with its accuracy.As can be seen, the result of our analysis is in good agreement with results of other measurements and has better precision.
The obtained value of the φ(1020) leptonic width is smaller than the value in the previous CMD-2 measurement [3] by about one experimental error reflecting a decrease of the total width of the φ(1020) meson.
The value of the φ(1020) meson leptonic width obtained here is correlated to the values of the four major φ(1020) branching fractions and therefore should not be included in the constrained fit performed by PDG.
All parameters (Γ ee and B(φ → X)) are in good agreement with results of other experiments.

Table 2
Cross section of the process e + e − → φ → K 0

Table 3
[6]ss section of the process e + e − → φ → π + π − π 0 obtained in the analysis[6].The errors of the cross section are statistical only.

Table 5
Contributions to the systematic errors of φ → K + K − and φ → K L K S cross sections.Common contributions of both errors are denoted with ⋆.Source K + K − K L K S