$D^* D_s K$ and $D_s ^* D K$ vertices in a QCD Sum Rule approach

We calculate the strong form factors and coupling constants of $ D^* D_s K$ and $D_s^* D K$ vertices using the QCD sum rules technique. In each case we have considered two different cases for the off-shell particle in the vertex: the ligthest meson and one of the heavy mesons. The method gives the same coupling constant for each vertex. When the results for different vertices are compared, they show that the SU(4) symmetry is broken by around 40%.

We calculate the strong form factors and coupling constants of D * DsK and D * s DK vertices using the QCD sum rules technique. In each case we have considered two different cases for the off-shell particle in the vertex: the ligthest meson and one of the heavy mesons. The method gives the same coupling constant for each vertex. When the results for different vertices are compared, they show that the SU(4) symmetry is broken by around 40%. The knowledge of the form factors in hadronic vertices is of crucial importance to estimate hadronic amplitudes when hadronic degrees of freedon are used. When all the particles in a hadronic vertex are on-mass-shell, the effective fields of the hadrons describe pointlike physics. However, when at least one of the particles in the vertex is off-shell, the finite size effects of the hadrons become important.
In this work we study the D * D s K and D * s DK vertices, which are fundamental to the evaluation of the dissociation cross section of J/Ψ by kaons when using effective Lagrangians. The supression of charmonium production is one of the most tradicional signatures of the quark-gluon plasma (QGP) formation in relativistic heavy ion collisions [1]. The dissociation of charmoniums in the QGP due to color screening would lead to a reduction of its production in such collisions. However, using the charmonium suppression as a signature of QGP formation requires the understanding of J/Ψ production and absortion mechanisms in hadronic matter, because this suppression may be due to the interactions with the comovers during the collision [2,3].
One of the approaches used to study the interaction of charmonium with the hadronic medium, mainly in the low energy region ( √ s < 10 GeV), is based on effective SU(4) Lagrangians [4,5,6,7,8,9]. This technique, however, requires the detailed knowledge of the form factors in the hadronic vertices. The calculated cross section may change by a factor of two if a soft, instead of a hard, form factor is used in the vertices containing charmed mesons.
This situation gave us the motivation to start a program to calculate charmed form factors and coupling constants, using the QCD sum rules approach [10]. We have been continuously working on this problem and computing different vertices [11,12,13,14,15,16,17]. An interesting subproduct of such calculations, [12,13,14,17], was the understanding of the behavior of the off-shell particle probing of the vertex: heavier particles resolves better the structure of the vertex, while lighter particles are more suitable for measuring its size. This conclusion is also supported in the present work.
As a part of this project we evaluate, in the present calculation, the form factors in the vertices D * D s K and D * s DK, and compare the results with the predictions from the exact SU(4) symmetry [9].
Following the QCDSR formalism described in our previous works [11,12,13,14,15,16,17], we write the three-point correlation function associated with the D * D s K vertex, which is given by for K meson off-shell, where the interpolating currents are j D * µ =cγ µ d, j K = isγ 5 d and j Ds = icγ 5 s, and for D s meson off-shell, with the interpolating currents j K µ =ūγ µ γ 5 s, j Ds = icγ 5 s, j D * µ =ūγ µ c, with u, d, s and c being the up, down, strange and charm quark fields respectively. In both cases, each one of these currents has the same quantum numbers as the corresponding mesons.
Using the above currents to evaluate the correlation functions (1) and (2), the theoretical or QCD side is obtained. The framework to calculate the correlators in the QCD side is the Wilson operator product expansion (OPE). The Cutkosky's rule allows us to obtain the double discontinuity of the correlation function for each one of the Dirac structures appearing in the correlation function. Calling ρ i the spectral density for the Dirac structure i, we can write the correlation function as a double dispersion relation over the virtualities p 2 and p ′ 2 , holding Q 2 = −q 2 fixed. Therefore, the amplitudes Γ i are given by: where the spectral density ρ i (s, u, Q 2 ) equals the double discontinuity of the amplitude Γ i (p 2 , p ′ 2 , Q 2 ). The amplitudes receive contributions from all terms in the OPE. The leading contribution comes from the perturbative term, shown in Fig. 1. The phenomenological side of the sum rule, which is written in terms of the mesonic degrees of freedom, is parametrized in terms of the form factors, meson decay constants and meson masses. The QCD sum rule is obtained by matching both representations, using the universality principle. The matching is improved by performing a double Borel transform on both sides. The perturbative contribution for both Eqs. (1) and (2), written in terms of Eq.(3), is given by for K off-shell, and where , mq ms mc mK mD s mD * fK [19] fD s [20] fD * [21] 0.0 0. 13   for K off-shell, and The phenomenological side of the vertex functions is obtained considering the contributions of the D s and D * mesons to the matrix element in Eq. (1) and the D * and K mesons to the matrix element in Eq. (2). We introduce the meson decay constants f K , f Ds and f D * , which are defined by the following matrix elements: where ǫ ν is the polarization vector of the D * meson.
In principle, we can work with any Dirac structure appearing in the amplitude in Eqs. (1) and (2). However, there are some points that one must follow: (i) the chosen structure must also appear in the phenomenological side and (ii) the chosen structure must have a stability that guarantees a good match between the two sides of the sum rule. The structures that obey these two points are p ′ µ , in the case K off-shell, and p ′ µ p ′ ν in the case D s off-shell. The corresponding phenomenological amplitudes in these structures are for the K off-shell, and for D s off-shell. In the case of K off-shell the contribution of the quark condensate vanishes after the double Borel transform. In the case of the D s off-shell, the quark condensate does not contribute to the chosen structure.
To write the sum rules we equate each phenomenological amplitude in Eqs. (9)-(10), with the expression obtained by substituting the corresponding spectral density in Eqs. (4)-(5) into Eq. (3). The matching between both sides is improved by performing a double Borel transformation [18] in the variables We get then the final form of the sum rule, which allow us to obtain the form factors g  Fig. 1).
Using ∆ s = ∆ u = 0.5 GeV for the continuum thresholds and fixing Q 2 = 1 GeV 2 , we found a good stability of the form factor g (K) D * DsK , as a function of the Borel mass M 2 , in the interval 3 < M 2 < 5 GeV 2 , as can be seen in Fig. 2. In the case of the form factor g (Ds) D * DsK the interval for stability of the sum rule is 2 < M 2 < 5 GeV 2 , as can be seen in Fig. 3. Fixing ∆ s = ∆ u = 0.5 GeV and M 2 = 3 GeV 2 , we evaluate the momentum dependence of both form factors. The results are shown in Fig. 4, where the squares corresponds to the g (K) D * DsK (Q 2 ) form factor in the interval where the sum rule is valid. The triangles are the result of the sum rule for the g (Ds) D * DsK (Q 2 ) form factor. In the case of the K meson off-shell, our numerical results can be parametrized by an exponencial function (dotted line in Fig. 4): As in Ref. [13], we define the coupling constant as the value of the form factor at Q 2 = −m 2 M , where m M is the mass of the off-shell meson. For the K off-shell case the resulting coupling constant is: In the case when the D s meson is off-shell, our sum rule results can be parametrized by a monopole formula (solid line in Fig. 4): giving the following coupling constant, obtained at the D s pole: Comparing the results in Eqs. (12) and (14) we see that the method used to extrapolate the QCDSR results in both cases, K and D s off-shell, allows us to extract values for the coupling constant which are in very good agreement with each other.
In order to study the dependence of these results with the continuum threshold, we vary ∆ s = ∆ u in the interval 0.4 ≤ ∆ s = ∆ u ≤ 0.6 GeV, as can be seen in Fig 5. This procedure give us uncertainties in such a way that the final results for the couplings in each case are: g   for K meson off-shell, where the interpolating currents are j D * s µ =cγ µ s, j K = iūγ 5 s and j D = icγ 5 u, and for D meson off-shell, with the interpolating currents j K µ =ūγ µ γ 5 s, j D * s ν =cγ ν s, and j D = iūγ 5 c. See Fig. 6 for understanding the perturbative contribution with these currents.
For each correlation function, Eqs. (15) and (16), the corresponding perturbative spectral density which enters in Eq. (3) is: for D off-shell, where k · p = for K off-shell, and The phenomenological side of the vertex functions are obtained by considering the contributions of D and D * s mesons to the matrix element in Eq. (15) and D * s and K mesons to the matrix element in Eq. (16). We introduce the decay constants f D and f D * s , which are defined by the following matrix elements:  where ǫ ν is the polarization vector of the D * s meson. The f K decay constant was already defined in Eq. (6). Again we have to choose one Dirac structure for each case in Eqs. (17)- (18). Following the points discussed before, the chosen Dirac structures are p ′ µ for the off-shell K, and p ′ µ p ′ ν for the off-shell D. The corresponding phenomenological amplitudes in these structures are for K off-shell, and for D off-shell. As in the case of the D * D s K vertex, the quark condensate does not contribute to the sum rule for these structures. The procedure to obtain the QCD sum rule is the same used in the case of the D * D s K vertex studied before. In this case we use the following relations between the Borel masses:  Table II, where we have used the relation f D * s = f D * f Ds /f D and the value of f Ds /f D from Ref. [22] in order to obtain the D * s decay constant. Using ∆ s = ∆ u = 0.5 GeV for the continuum thresholds and fixing Q 2 = 1 GeV 2 , we found a good stability of the sum rule for g (K) D * s DK , as a function of the Borel mass M 2 , in the interval 2 < M 2 < 5 GeV 2 , as can be seen in Fig. 7. In the case of g (D) D * s DK , the interval for stability is also 2 < M 2 < 5 GeV 2 , as can be seen in Fig. 8. Fixing ∆ s = ∆ u = 0.5 GeV and M 2 = 3 GeV 2 in both cases, we calculate the momentum dependence of the form factors which are shown in Fig. 9. The squares corresponds to the g The coupling constant was obtained as the value of the form factor at Q 2 = −m 2 K . In this case the resulting coupling constant is In the case when the D meson is off-shell, the sum rule result is represented by the triangles in Fig. 9, and they can be parametrized by a monopole formula (solid line in the figure):  giving the following coupling constant, obtained at the D pole: Studing the dependence of our results with the continuun threshold, for ∆ s,u varying in the interval 0.4 ≤ ∆ s,u ≤ 0.6 GeV, as can be seen in Fig. 10, we obtain the following values, with errors, for the couplings in each case: g   Concluding, we have estudied the form factors and coupling constants of D * D s K and D * s DK vertices in a QCD sum rule calculation. For each case we have considered two particles off-shell, the lightest and one of the heavy ones: the K and D s mesons for the D * D s K vertex, and the K and D mesons for the D * s DK vertex. In the two situations, the off-shell particles give compatible results for the coupling constant in each vertex. The results are: g D * DsK = 3.02 ± 0.14 (27) g D * s DK = 2.84 ± 0.31 We can compare our result with the prediction of the exact SU(4) symmetry [5,7,9], which would give the following relation among these numbers [9]: g D * DsK = g D * s DK = 5. Eqs. (27) and (28) shows that the coupling constants in the vertices D * D s K and D * s DK are consistent one with the other, but that they are relatively far from the value given by the SU(4) symmetry in the cited reference. Therefore, we conclude that the SU(4) symmetry is broken by approximately 40% in the calculation performed here. We can also extract the cutoff parameter, Λ, from the paramentrizations in Eqs. (11) and (23) for K off-shell, Eq. (13) for D s off-shell and Eq. (25) for D off-shell. We get Λ ≈ 2.07 GeV for the K meson off-shell , Λ ≈ 2.61 GeV for the D s meson off-shell, and Λ ≈ 2.51 GeV for the D meson off-shell. Comparing the values of the cutoffs, we see that the form factor is harder if the off-shell meson is heavier, implying that the size of the vertex depends on the mass of the exchanged meson: the heavier is the meson, the more as a point like particle is its behavior when probing the target, as observed in Refs. [12,13,14,17].