Strong Couplings of Three Mesons with Charm(ing) Involvement

We determine the strong couplings of three mesons that involve, at least, one ηc or J/ψmeson, within the framework of a constituent-quark model by means of relativistic dispersion formulations. For strong couplings of J/ψmesons to two charmed mesons, our approach leads to predictions roughly twice as large as those arising from QCD sum rules. 1 Three-meson strong coupling from meson–meson transition amplitudes We determine the strong couplings of three mesons at least one of which is one of the charmonia ηc and J/ψ, generically called gPP′V and gPV ′V for pseudoscalar mesons P of mass MP and vector mesons V of mass MV and polarization vector εμ and defined, for momentum transfer q ≡ p1−p2, by the amplitudes 〈P(p2) V(q)|P(p1)〉 = − gPP′V 2 (p1 + p2) ε∗μ(q) , 〈V (p2) V(q)|P(p1)〉 = −gPV ′V ǫμνρσ ε (q) ε(p2) p ρ 1 p σ 2 , from the residues of poles situated at the masses MPR and MVR of (appropriate) pseudoscalar and vector resonances PR and VR and contributing to transition form factors FP≻P ′ + (q 2), VP≻V (q2) and AP≻V 0 (q 2), in terms of vector quark currents jμ ≡ q̄1 γμ q2 and axial-vector quark currents jμ ≡ q̄1 γμ γ5 q2 defined by 〈P(p2)| jμ|P(p1)〉 = F ′ + (q ) (p1 + p2)μ + · · · , F ′ + (q ) ∣


Three-meson strong coupling from meson-meson transition amplitudes
We determine the strong couplings of three mesons at least one of which is one of the charmonia η c and J/ψ, generically called g PP ′ V and g PV ′ V for pseudoscalar mesons P of mass M P and vector mesons V of mass M V and polarization vector ε µ and defined, for momentum transfer q ≡ p 1 −p 2 , by the amplitudes P ′ (p 2 ) V(q)|P(p 1 ) = − g PP ′ V 2 (p 1 + p 2 ) µ ε * µ (q) , V ′ (p 2 ) V(q)|P(p 1 ) = −g PV ′ V ǫ µνρσ ε * µ (q) ε * ν (p 2 ) p ρ 1 p σ 2 , from the residues of poles situated at the masses M P R and M V R of (appropriate) pseudoscalar and vector resonances P R and V R and contributing to transition form factors F P≻P ′ + (q 2 ), V P≻V (q 2 ) and A P≻V 0 (q 2 ), in terms of vector quark currents j µ ≡q 1 γ µ q 2 and axial-vector quark currents j 5 µ ≡q 1 γ µ γ 5 q 2 defined by where the P or V decay constants f P,V parametrize the matrix elements of the interpolating currents j (5) Such strong-coupling results may prove to be useful for studies of long-distance QCD effects in hadron decays involving charmed mesons or charmonia in the final state of a kind similar to the one in Ref. [1].

Quark-model-underpinned dispersion analysis of transition form factors
We describe the relevant properties of the involved strongly coupling mesons by means of a relativistic constituent-quark model [2][3][4]. Of course, this requires us to match the QCD currents j (5) µ to associated constituent-quark currents, which is, for heavy quarks, easily effected by introducing form factors g V,A , [5] but, for light quarks, rendered rather involved [6,7], for instance, if embedding partial axial-current conservation. For the radial meson wave functions, Gaussian shapes with slopes β P,V given, together with all relevant mesonic features, in Table 1 [8][9][10][11][12][13], turn out to suffice for our purposes. Table 2 lists the numerical values adopted for the masses of the constituent quarks Q.
Within the framework of a relativistic dispersion formalism (reviewed, e.g., in Ref. [14]), we represent each transition form factor F (q 2 ) = F P≻P ′ + (q 2 ), V P≻V (q 2 ), A P≻V 0 (q 2 ) by a double dispersion integral of a double spectral density ∆ F (s 1 , s 2 , q 2 ) the one-loop contributions to which derive from Feynman graphs like the ones in Fig. 1 and each decay constant f P,V by a dispersion integral of a spectral density ρ P,V (s),   and of the wave functions of all mesons entering the corresponding one-or two-meson matrix elements

Three-meson strong coupling: determination from transition amplitudes
We fix the slopes β P,V such that the decay constants f P,V are reproduced by their spectral representation. Equipped with these β P,V values, we deduce all strong couplings from the spectral representation of the relevant form factors F (q 2 ) derived sufficiently off the resonances at M R (R = P R , V R ), by interpolating pointwise given momentum dependences of F (q 2 ) by three-parameter (σ 1,2 , F (0)) ansätze of the form and extrapolating F (q 2 ) to the poles at q 2 = M 2 R , where the strong couplings emerge from the residues. Using σ 1,2 , F (0), and M R as fit parameters, all arising masses M R come close to the known resonances. Quite generally, a given strong coupling may show up in and therefore can be extracted from more than one meson-meson transition form factor, for example, g η c η c ψ from F η c ≻η c + or A η c ≻ψ 0 (see Fig. 2) [15][16][17], g DDψ from F D≻D + or A D≻ψ 0 (see Fig. 3(a)) [15][16][17] and g DD * η c from F η c ≻D + , A η c ≻D * 0 or A D≻D * 0 (see Fig. 3(b)) [15][16][17]; for further examples of such multiple involvements, consult Tables III, V, and VI of Ref. [15].

Strong coupling predictions from relativistic constituent-quark approach
We collect our emerging strong-coupling findings -extracted, in the case of multipresence of one and the same three-meson coupling in more than one meson-meson transition amplitude, by a combined fit -in Table 3: Strange quark content instead of a down quark implies a reduction of the involved strong couplings, by roughly 10%. Confronting, in Table 4, our D (s) -D ( * ) (s) -J/ψ predictions with QCD sum-rule outcomes [18][19][20], the QCD sum-rule estimates prove to be lower than ours [15][16][17] by a factor of two. Table 3. Charm(ing) three-meson strong couplings: quark-model-based dispersion-approach outcomes [15][16][17].