Radiative (phi ->f_0(980) gamma) decay in light cone QCD sum rules

The light cone QCD sum rules method is used to calculate the transition form factor for the radiative (phi ->f_0 gamma) decay, assuming that the quark content of the (f_0) meson is pure (bar{s}s) state. The branching ratio is estimated to be Br(phi ->f_0 gamma) = 3.5 x (1 pm 0.3) x 10^(-4). A comparison of our prediction on branching ratio with the theoretical results and experimental data existing in literature is presented.


Introduction
According to the quark model, mesons are interpreted as pureqq states. Scalar mesons constitute a remarkable exception to this systematization and their nature is not well established yet [1]- [4].
In the naiveqq picture, one can treat the isoscalar f 0 (980) either as the meson that exists mostly as nonstrange and almost degenerate with the isovector a 0 (980) or as mainlȳ ss, in analogy to the puress vector meson φ(1020).
In order to understand the content of the f 0 meson several alternatives have been suggested, such as, the analysis of the f 0 → 2γ decay [5,6]; study of the ratio Γ(a 0 → f 0 γ)/Γ(φ → f 0 γ) [7]. However, among these, the (φ → f 0 γ) decay occupies a special place, since the branching ratio expected of this decay, is essentially dependent on the content of f 0 . For example B(φ → f 0 γ) is as high as ∼ 10 −4 if it were composed ofqqqq, and ∼ 10 −5 if f 0 were a puress state.
It has been known for a long time that f 0 (980) couples significantly through itsss content, from its detection as a peak in the J/ψ → φf 0 [8] and D s → πf 0 [9] decays, as discussed in [10] and [11] (see also [12]). For this reason, in this work we assume that quark content of both φ and f 0 mesons are puress. In the present paper we analyze the radiative φ → f 0 γ decay in framework of the light cone QCD sum rules (about light cone QCD sum rules and its applications, see for example [13]). Note also that the φ → f 0 γ decay is analyzed in framework of the 3-point sum rules in [14]. In order to calculate the transition form factor describing the φ → f 0 γ decay in light cone QCD sum rules, we consider the following correlator where J s =ss and J φ µ =sγ µ s are interpolating currents for f 0 and φ mesons, respectively, and γ is the background electromagnetic field (for more about external field technique in QCD see [15,16]).
The physical part of the correlator can be obtained by inserting a complete set of one meson states into the correlator, where φ and f 0 are the quantum numbers and p 1 = p+q with q being the photon momentum.
where ε φ µ is the φ meson polarization vector. The coupling of the f 0 (980) to the scalar current J s =ss is defined in terms of a constant λ f The relevant matrix element describing the transition φ → f 0 induced by an external electromagnetic current can be parametrized in the following form: where ε is the photon polarization and we have used (εq) = 0. From gauge invariance we have and since the photon is real in the decay under consideration, we need the values of the form factors only at the point q 2 = 0. Using Eq. (6) the matrix element f 0 |φ γ takes the following gauge invariant form, Using Eqs. (1)-(4) and (7), for the phenomenological part of the correlator we have In order to construct the sum rule, calculation of the correlator from QCD side (theoretical part) is needed. From Eq. (1) we get where S s is the full propagator of the strange quark (see below). Theoretical part of the correlator contains two pieces, perturbative and nonperturbative. Perturbative part corresponds to the case when photon is radiated from the freely propagating quarks. Its expression can be obtained by making the following replacement in each one of the quark propagators in Eq. (9) where the Fock-Schwinger gauge x µ A µ (x) = 0 is used and S f ree s is the free s-quark propagator S f ree s (x) = i x/(2π 2 x 4 ) and the remaining one is the full quark propagator. The nonperturbative piece of the theoretical part can be obtained from Eq. (9) by replacing each one of the propagators with where A i is the full set of Dirac matrices and sum over A i is implied and the other quark propagator is the full propagator, involving pertubative and nonperturbative contributions. In order to calculate perturbative and nonperturbative parts to the correlator function (1), expression of the s-quark propagator in external field is needed. The complete light cone expansion of the light quark propagator in external field is presented in [16]. The propagator receives contributions from the nonlocal operatorsqGq, qGGq,qqqq, where G is the gluon field strength tensor. In the present work we consider operators with only one gluon field and neglect terms with two gluonsqGGq, and four quarksqqqq and formal neglect of these these terms can be justified on the basis of an expansion in conformal spin [17]. In this approximation full propagator of the s-quark is given as It follows from Eqs. (11) and (9) that in calculating the QCD part of the correlator, as is generally the case, we are left with the matrix elements of the gauge invariant nonlocal operators, sandwiched in between the photon and the vacuum states γ(q) |sA i s| 0 . These matrix elements define the light cone photon wave functions. The photon wave functions up to twist-4 are [17,18] The path-ordered gauge factor Pexp ig s 1 0 du x µ A µ (ux) is emitted since the Schwinger-Fock gauge x µ A µ (x) = 0 is used. The functions φ(u), ψ(u) are the leading twist-2 photon wave functions, while g 1 (u) and g 2 (u) are the twist-4 photon wave functions. Note that twist-3 photon wave functions are neglected in the calculations, since their contributions are small and change the result by 5%. In Eq. (13) χ is the magnetic susceptibility of the quark condensate and e q is the quark charge. The theoretical part is obtained by substituting photon wave functions and expression for the s-quark propagators into Eq. (9). The sum rules is obtained by equating the phenomenological and theoretical parts of the correlator. In order to suppress higher states and continuum contribution (for more details see [19,20]) double Borel transformations of the variables p 2 1 = p 2 and p 2 2 = (p + q) 2 are performed on both sides of the correlator, after which the following sum rule is obtained where s 0 is the continuum threshold dy lny e −y , which have been used to subtract continuum, and where M 2 1 and M 2 2 are the Borel parameters in φ and f 0 channels, respectively, Λ is the QCD scale parameter and γ E is the Euler constant. Since the masses of φ and f 0 are very close to each other we will set M 2 1 = M 2 2 ≡ 2M 2 , obviously from which it follows that u 0 = 1/2.
It is clear from Eq. (14) that the values of λ f 0 and f φ are needed in order to determine F (0). The coupling of the f 0 (980) to the scalarss current is determined by the constant λ f 0 and in the two-point QCD sum rules its value is found to be λ f 0 = (0.18 ± 0.0015) GeV [14]. In further numerical analysis we will use f φ = 0.234 GeV which is obtained from the experimental analysis of the φ → e + e − decay [21].
Having the values of λ f 0 and f φ , our next and final attempt is the calculation of transition form factor F 1 (0). As we can easily see from Eq. (15) the main input parameters of the light cone QCD sum rules is the photon wave function. It is known that the leading photon wave function receive only small corrections from the higher conformal spin [17,19,22], so that they do not deviate much from the asymptotic form. The photon wave functions we use in our numerical analysis are given as GeV, m f 0 = 0.98 GeV . The transition form factor is a physical quantity and therefore it must be independent of the auxiliary continuum threshold s 0 and and the Borel mass M 2 parameters. So our main concern is to find a region where the transition form factor F 1 (0) is practically independent of the parameters s 0 and M 2 . For this aim in Fig. (1) we present the dependence of the transition form factor Using the matrix element (7) for the decay width of the considered process, we obtain Using the experimental value Γ tot (φ) = 4.458 MeV [21], and Eqs. (16) and (17), we get for the branching ratio Our result on the branching ratio is obtained under the assumption that f 0 meson is represented as a puress component. How does the result change if we assume that φ and f 0 mesons can be represented as a mixing ofss andnn = (ūu +dd)/ √ 2 state, i.e., φ = cos αss + sin αnn , f 0 = sin βss + cos βnn ?
As the final words we would like to point out that our prediction given in Eq. (18), is in a very good agreement with the existing experimental result B(φ → f 0 γ) ≃ (3.4 ± 1.1) × 10 −4 [21].