Coupling $g_{f_0 K^+ K^-}$ and the structure of $f_0(980)$

We use light-cone QCD sum rules to evaluate the strong coupling $g_{f_0 K^+ K^-}$ which enters in several analyses concerning the scalar $f_0(980)$ meson. The result: $6.2\le g_{f_0 K^+ K^-}\le 7.8$ GeV is larger than in previous determinations.


Introduction
Light scalar mesons are the subject of an intense and continuous scrutiny aimed at clarifying several aspects of their nature that still need to be unambiguously established [1,2].
From the experimental point of view, these particles are difficult to resolve because of the strong overlap with the continuum background. On the other hand, the identification is made problematic since both quark-antiquark (qq) and non qq scalar states are expected to exist in the energy regime below 2 GeV. For example, lattice QCD and QCD sum rule analyses indicate that the lowest lying glueball is a 0 ++ state with mass in the range 1.5-1.7 GeV [3]. Actually, the observed light scalar states are too numerous to be accomodated in a single qq multiplet, and therefore it has been suggested that some of them escape the quark model interpretation. In addition to glueballs, other interpretations include multiquark states and admixtures of quarks and gluons.
Particularly debated is the nature of the meson f 0 (980). Among the oldest suggestions, there is the proposal that quark confinement could be explained through the existence of a state with vacuum quantum numbers and mass close to the proton mass [4]. On the other hand, following the quark model and considering the strong coupling to kaons, f 0 (980) could be interpreted as an ss state [5,6,7,8]. However, this does not explain the mass degeneracy between f 0 (980) and a 0 (980) interpreted as a (uu − dd)/ √ 2 state. A four quark qqqq state interpretation has also been proposed [9]. In this case, f 0 (980) could either be nucleon-like [10], i.e. a bound state of quarks with symbolic quark structure f 0 = ss(uu + dd)/ √ 2, the a 0 (980) being a 0 = ss(uu − dd)/ √ 2, or deuteron-like, i.e. a bound state of hadrons. If f 0 is a bound state of hadrons, it is usually referred to as a KK molecule [11,12,13,14]. In the former of these two possibilities the mesons are treated as point-like, while in the latter they should be considered as extended objects.
The identification of the f 0 and of the other lightest scalar mesons with the Higgs nonet of a hidden U(3) symmetry has also been suggested [15]. Finally, a different interpretation consists in considering f 0 (980) as the result of a process in which strong interaction enriches a pureqq state with other components, such as KK , a process known as hadronic dressing [6,16]; such an interpretation is supported in [2,5,6,8,17,18,19]. In ref. [20] it has been shown that the experimentally observed lightest scalar particles in the I=1 and I=1/2 sectors can be reproduced in this way, starting from a bare qq and sq structure respectively (q being a light non strange quark). On the other hand, I=0 states are the most elusive ones, since there are two possible bare structures, qq and ss, which could not only undergo hadronic dressing, but also mix through hadronic loops. The resulting picture strongly depends on the couplings of the bare structures to the hadronic channels.
Several experimental analyses aimed at discriminating among the different possibilities. In particular, the radiative φ → f 0 γ decay mode has been identified as an effective tool for such a purpose [10,12,21]. As a matter of fact, if f 0 has a pure strangeness component f 0 = ss, the dominant φ → f 0 γ decay mechanism is the direct transition, while in the four-quark scenario φ → f 0 γ is expected to proceed through kaon loops with a branching fraction depending on the specific bound state structure [12,21].
An important hadronic parameter entering in several analyses involving f 0 (980) is the strong coupling g f 0 K + K − . Indeed, the kaon loop diagrams contributing to φ → f 0 γ are expressed in terms of g f 0 K + K − , as well as in terms of the coupling g φK + K − which can be inferred from experimental data on φ meson decays. The coupling g f 0 K + K − can be obtained from various processes, and we shall present an overview of the determinations in the last part of this paper. It is interesting to carry out a calculation in a framework based on QCD, trying to point out what is a distinctive feature of the scalar particles, i.e. their large couplings to the hadronic states.
The present study is devoted to a determination of g f 0 K + K − by light-cone QCD sum rules, a method applied to the calculation of several hadronic parameters both in the light, both in the heavy quark sector [22,23]. The analysis and the numerical results are presented in Section 2, while a summary of the experimental data and of other theoretical determinations is given in Section 3.
2 Coupling g f 0 K + K − by light-cone QCD sum rules In order to evaluate the strong coupling g f 0 K + K − , defined by the matrix element: we consider the correlation function The quark currents J K µ and J f 0 represent the axial-vector J K µ =ūγ µ γ 5 s and the scalar J f 0 =ss current, respectively, while the external kaon state has four momentum q, with The choice of the J f 0 =ss current does not imply that f 0 (980) has a puress structure, but it simply amounts to assume that J f 0 has a non-vanishing matrix element between the vacuum and f 0 [19,24]. Such a matrix element, as mentioned below, has been derived by the same sum rule method.
Exploiting Lorentz invariance, T µ can be written in terms of two independent invariant functions, T 1 and T 2 : The analysis of the correlation function in eq.(2), following the general strategy of QCD sum rules, allows us to obtain a quantitative estimate of g f 0 K + K − . The method consists in representing T µ in terms of the contributions of hadrons (one-particle states and the continuum) having non-vanishing matrix elements with the vacuum and the currents J K µ and J f 0 , and matching such a representation with a QCD expression computed in a suitable region of the external momenta p and p + q [25].
Let us consider, in particular, the invariant function T 1 in eq.(3) that can be represented by a dispersive formula in the two variables p 2 and (p + q) 2 : The hadronic spectral density ρ had gets contribution from the single-particle states K and f 0 , for which we define current-particle matrix elements: as well as from higher resonances and a continuum of states that we assume to contribute in a domain D of the s, s ′ plane, starting from two thresholds s 0 and s ′ 0 . Therefore, neglecting the f 0 width, the spectral function ρ had can be modeled as: where ρ cont includes the contribution of the higher resonances and of the hadronic continuum. The resulting expression for T 1 is: . (8) We do not consider possible subtraction terms in eq.(4) as they will be removed by a Borel transformation.
For space-like and large external momenta (large −p 2 , −(p + q) 2 ) the function T 1 can be computed in QCD as an expansion near the light-cone x 2 = 0. The expansion involves matrix elements of non-local quark-gluon operators, which are defined in terms of kaon distribution amplitudes of increasing twist. 1 The first few terms in the expansion are retained, since the higher twist contributions are suppressed by powers of 1/(−p 2 ) or 1/(−(p + q) 2 ). As a result, the following expression for T 1 is obtained to twist four accuracy: The functions ϕ p and ϕ σ appearing in eq.(9) are kaon distribution amplitudes defined by the matrix elements m s being the strange quark mass (we put to zero the mass of the light quarks). 2 Moreover, g 2 (u) is defined by the matrix element Kaon matrix elements of quark-gluon operators also contribute to eq.(9); they are parameterized in terms of twist three and twist four distribution amplitudes: 1 The short-distance expansion of the 3-point vacuum correlation function of one scalarss and two pseudoscalarsiγ 5 q densities has been considered in [26]. The present calculation mainly differs for the possibility of incorporating an infinite series of local operators [23]. 2 The path-ordered gauge factor P exp ig s 1 0 dtx µ A µ (tx) is not included in the matrix elements having chosen the gauge x µ A µ = 0. and The operatorG αβ is the dual of G αβ : The function ϕ 3K is twist three, while the distribution amplitudes in (14) and (15) are twist four. The functionsψ andφ appearing in eq.(9) are defined in terms of ϕ ⊥ , ϕ ,φ ⊥ andφ as follows: The sum rule for g f 0 K + K − follows from the approximate equality of eqs. (8) and (9).
Invoking global quark-hadron duality, the contribution of the continuum in (8) can be identified with the QCD contribution above the thresholds s 0 , s ′ 0 . This allows us to isolate the pole contribution in which the coupling appears. The matching between the expressions in (8) and (9) can be improved performing two independent Borel transformations with respect to the variables −p 2 and −(p + q) 2 . Defining M 2 1 and M 2 2 as the Borel parameters associated to the channels p 2 and (p + q) 2 , respectively, and using the identity: with , we get the following expression for the Borel tranformed eq.(9): Analogously, a double Borel transformation can be carried out for the hadronic representation eq.(8): As shown by (18), the Borel transformation exponentially suppresses the contribution of the higher states and of the continuum; furthermore, it removes possible subtraction terms in (4) which depend only on p 2 or (p + q) 2 . The second term in (18) represents the continuum contribution. In order to identify it with part the QCD term (17), a prescription has been proposed in [27]. It consists in considering the symmetric points M 2 1 = M 2 2 = 2M 2 (corresponding to u 0 = 1/2) in the (M 2 1 , M 2 2 ) plane and performing the continuum subtraction through the substitution e −m 2 0 M 2 in the leading-twist term in (17). Such a prescription is not adeguate in our case, where the Borel parameters correspond to channels with different mass scales and should not be constrained to be equal. Here we can exploit the property of the amplitudes ϕ p (u) and ϕ σ (u) of being polynomials in u (or 1 − u): in order to compute their contribution in the duality region D. As for the other terms in (17), they represent a small contribution to the QCD side of the sum rule, and therefore the calculation can leave them unaffected.
The final expression for g f 0 K + K − reads: The main nonperturbative quantities constituting the input information in the sum rule (20) are the kaon light-cone wave functions. A theoretical framework for their determination relies on an expansion in terms of matrix elements of conformal operators [28].
For the function ϕ p , conformal expansion results in the expression with ξ = 2u − 1, a p 0 = 1 and C ℓ k the Gegenbauer polynomials. In (21) we have included the normalization scale dependence of the distribution amplitude ϕ p , which appears in the multiplicatively renormalizable coefficients a p k (µ). The nonperturbative information is encoded in the coefficients, which are peculiar for the various mesons. In the case of kaon, the asymmetry between the strange and nonstrange quark momentum distribution in the meson can be taken into account by non-vanishing odd-order coefficients a p k . Such SU(3) flavour violating effects have not been investigated so far for distribution amplitudes of twist larger than two, and we neglect them in the following, with consequences that we shall mention below. As for the even order coeficients, their updated values are reported in [27,29] , worked out in [29]. Analogously, the ϕ σ distribution amplitude can be expressed as with For ϕ 3K (α i ) and for the other higher twist distribution amplitudes we refer to the expressions reported in [27,29].
The result for g f 0 K + K − versus the Borel parameters M 2 1 and M 2 2 is depicted in fig.1.   fig.2, which can be quoted as 6.2 ≤ g f 0 K + K − ≤ 7.8 GeV.
Let us briefly discuss the uncertainties affecting the numerical result. As for the SU(3) F breaking effects rendering the kaon distribution amplitudes asymmetric with respect to the middle point, the neglect should have a minor role in our approach, due to the possibility of exploring wide ranges of the variable u and smoothing the effects of the actual shapes of the wave functions. Another uncertainty is related to the value of the strange quark mass, m s ; since the dependence of the sum rule on m s mainly involves the ratio M 2 K /m s , one can fix this ratio using chiral perturbation theory, obtaining results in the same range quoted for We can compare now our result with the available experimental determinations of g f 0 K + K − , as well as with the results of other calculations. We shall see how complex the scenario is. 3 Other determinations of g f 0 K + K − As discussed in the Introduction, g f 0 K + K − can be considered in connection with the radiative φ → f 0 γ decay mode. Several analyses go through this decay channel. KLOE Collaboration at the DAΦNE collider in Frascati has examined the decay channel φ → π 0 π 0 γ measuring the branching fraction: B(φ → π 0 π 0 γ) = (1.09 ± 0.03 stat ± 0.05 syst ) × 10 −4 [32]. The decay mode is supposed to proceed through ρπ intermediate state and through kaon loop processes, with the kaons annihilating into scalar resonances that subsequently decay to π 0 π 0 . Different fits of the two pion invariant mass spectrum dΓ dM ππ are performed in order to measure the parameters of the scalar states. In a first fit (A) only the contribution of the intermediate state f 0 (980) is considered, and the three parameters In a second fit (B) the contribution of a possible broad scalar σ state is included, and the coupling g φσγ is considered as a further parameter. It is assumed that the two pion decay modes saturate the f 0 width, and that B(f 0 → π + π − ) = 2 B(f 0 → π 0 π 0 ) invoking isospin symmetry. sizeable values for g f 0 K + K − are obtained; they are reported in Table 1.
Considering the central f 0 production in pp collisions, the WA102 experiment at CERN gets: [35]. On the other hand, analyzing the f 0 production in D s decays to three pions, the Collaboration E791 at Fermilab finds a value compatible with zero [36]. These results are also reported in Table 1. In Ref. [10] the coupling constant is evaluated for different values of the phase shift of the elastic background in the ππ → ππ reaction, of the ratio R = g 2 f 0 K + K − /g 2 f 0 π + π − and according to different scenarios for the f 0 structure, obtaining results in a wide range: The analysis of the decay channel J/ψ → φKK(ππ) has been carried out in Ref. [37]. The f 0 pole is described as a Breit-Wigner resonance coupled to two channels. Two fits of the experimental data are performed depending upon the ππ phase shift data used, obtaining g f 0 K + K − = 2.5 ± 0.15 GeV and g f 0 K + K − = 2.0 ± 0.06 GeV, respectively.
A prediction for g f 0 K + K − based on chiral symmetry and the linear sigma model, when no mixing with the σ is considered, is: g f 0 K + K − = 2.24 GeV [38], to be compared to old determinations g f 0 K + K − = 2.74 GeV [39]. Using the method of the T-matrices, the value g f 0 K + K − = 3.8 GeV is obtained [40].
Considering all the above results one sees that a general consensus on g f 0 K + K − has not been reached, so far. In particular, experimental analyses of different processes produce contradicting results. The outcome from φ → f 0 γ points towards sizeable values of the coupling, consistent with the light-cone sum rule result. One has to say that the error quoted for the experimental determinations, which in general looks small, mainly accounts for the statistical uncertainties; one could infer the size of the systematical uncertainties comparing different determinations.
As for g f 0 K + K − from D s decays, presumably the determination will be improved at the B factories by experiments such as BaBar at SLAC, thanks to large available samples of D s mesons. In these measurements g f 0 K + K − is expected to be determined by coupled channel analyses, with D s decaying to final states containing kaons as well as pions [41]. Table 1: Experimental determinations of g f 0 K + K − using different physical processes. Double items refer to two different fits (see text).
Collaboration process

Conclusions
The purpose of this paper was the evaluation of the strong coupling constant g f 0 K + K − , the value of which is rather controversial, as it emerges comparing different experimental and theoretical determinations. In particular, the KLOE Collaboration measured a larger value than in other determinations, with a greater accuracy as well. However, such a result stems from the investigation of φ → f 0 γ, and therefore it is mandatory to wait for the study of unrelated processes, namely the combined analysis of D s decays to pions and kaons. The outcome of light-cone QCD sum rules is in keeping with a large value for the coupling. The uncertainty affecting the result is intrinsic of the method and does not allow a better comparison with data. However, the analysis confirms a peculiar aspect of the scalar states, i.e. their large hadronic couplings, thus pointing towards a scenario in which the process of hadronic dressing is favoured.