Analysis of the strong D 2 (2460)0 D + π and D s2 (2573)+ D + K 0 transitions via QCD sum rules

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1 Eur. Phys. J. C :306 DOI 0.40/epjc/s x Regular Article - Theoretical Physics Analysis of the stron D and D s D K 0 transitions via QCD sum rules K. Azizi,a, Y. Sarac 2,b, H. Sundu 3,c Physics Department, Doğuş University, Acıbadem-Kadıköy, Istanbul, Turkey 2 Electrical and Electronics Engineering Department, Atilim University, Ankara, Turkey 3 Department of Physics, Kocaeli University, 4380 Izmit, Turkey Received: 8 July 204 / Accepted: 26 September 204 The Authors 204. This article is published with open access at Springerlink.com Abstract The stron D and D s D K 0 transitions are analyzed via three-point QCD sum rules. First we calculate the corresponding strong coupling constants 2 D and s2 DK. Then we use them to calculate the corresponding decay widths and branching ratios. Making use of the existing experimental data on the ratio of the decay width in the pseudoscalar D channel to that of the vector D channel, finally, we estimate the decay width and branching ratio of the stron D 200 transition. Introduction Following the first observation, reported in 986 ], the past few decades have been a period for the observations of orbitally excited charmed mesons 2 7]. During this period there have also been several theoretical studies on the masses, strong and electromagnetic transitions of these mesons via various methods for instance see 8 2] and references therein. Among these orbitally excited mesons are the D and D s mesons. The D state has the quantum numbers I J P = 2 2. Being not known exactly, I J P = 02 quantum numbers are favored by the width and decay modes of the Ds state. In this work, it is considered as a charmed strange tensor meson. One may refer to 22 32] and references therein for some experimental and theoretical studies on the properties of the charmed strange mesons. In the literature, compared to the other types of mesons, there are little theoretical works on the properties of the tensor mesons. Especially, their strong transitions are not studa kazizi@dogus.edu.tr b ysoymak@atilim.edu.tr c hayriye.sundu@kocaeli.edu.tr ied much. Studying the parameters of these tensor mesons and the comparison of the attained results with the existing experimental results may provide fruitful information about the internal structures and the natures of these mesons. Considering the appearance of these charmed tensor mesons as intermediate states in studying the B meson decays, the results of this work can also be helpful in this respect. Beside all of these, the possibility for searches on the decay properties of D2 and D s2 mesons at LHC is another motivation for theoretical studies on these states. The present work puts forward the analysis of the strong transitions D D and Ds D K 0. For this aim, first we calculate the strong coupling form factors 2 D and s2 DK via QCD sum rules as one of the most powerful and applicable non-perturbative methods to hadron physics 33,34]. These strong coupling form factors are then used to calculate the corresponding decay widths and branching ratios of the transitions under consideration. Making use of the existing experimental data on the ratio of the decay width in the pseudoscalar D channel to that of the vector D channel, finally, we evaluate the decay width of the strong D D 200 transition. 2 QCD sum rules for the strong coupling form factors 2 D and s2 DK The aim of this section is to present the details of the calculations of the coupling form factors 2 D and s2 DK for which we use the following three-point correlation function: μν p, p, q = i 2 d 4 x d 4 y e ip x e ip y 0 T J D y J K ] 0 J D 2 D s2 ] x 0, μν 23

2 306 Page 2 of 7 Eur. Phys. J. C :306 where T is the time ordering operator and q = p p is transferred momentum. The interpolating currents appearing in this three-point correlation function can be written in terms of the quark field operators as J D y = i dyγ 5 cy, J K ] 0 = iū s]0γ 5 d0, J D 2 D s2 ] μν x = i ū s]xγ μ Dν xcx 2 ] ū s]xγ ν Dμ xcx, 2 with Dμx being the two-side covariant derivative that acts on left and right, simultaneously. The covariant derivative Dμx is defined as Dμx = Dμ x Dμx], 3 2 where D μ x = μ x i g 2 λa A a μ x, D μ x = μ x i g 2 λa A a μ x. 4 Here λ a a =, 2,..., 8 are the Gell-Mann matrices and A a μ x stand for the external gluon fields. These fields are expressed in terms of the gluon field strength tensor using the Fock Schwinger gauge x μ A a μ x = 0, i.e. A a μ x = dααx β G a βμ αx 0 = 2 x βg a βμ 0 3 x ηx β D η G a βμ 0, 5 where we keep only the leading term in our calculations and ignore contributions of the derivatives of the gluon field strength tensor. One follows two different ways to calculate the above mentioned correlation function according to the QCD sum rule approach. It is calculated in terms of hadronic parameters, called the hadronic side. On the other hand, it is calculated in terms of quark and gluon degrees of freedom with the help of the operator product expansion in the deep Euclidean region, called the OPE side. The match of the coefficients of the same structures from both sides provides the QCD sum rules for the intended physical quantities. With the help of a double Borel transformation with respect to the variables p 2 and p 2 one suppresses the contribution of the higher states and the continuum. In the hadronic side, the correlation function in Eq. is saturated with complete sets of appropriate D2 D s2 ], K ], and D hadronic states with the same quantum numbers as the ones of the used interpolating currents. Performing the four-integrals over x and y leads to had μν p, p, q = 0 J K ] K ]q 0 J D Dp D 2 D s2 ]p,ɛ J D 2 D s2 ] μν 0 p 2 m 2 D 2 D s2 ]p 2 m 2 D q2 m 2 K ] K ]qdp D2 D s2 ]p,ɛ, 6 where represents the contributions of the higher states and continuum. The matrix elements appearing in this equation are parameterized as follows: 0 J K ] K ]q =i m2 K ] f K ] m d m us], 7 0 J D Dp m 2 D =i f D, 8 m d m c D 2 D s2 ]p,ɛ J D 2 μν and 0 =m 3 D 2 D s2 ] f D 2 D s2 ] ɛ λ μν, 9 K ]qdp D2 D s2 ]p,ɛ = 2 DDs2 DK] ɛ λ ηθ p η p θ, 0 where f K ], f D and f D 2 D s2 ] are leptonic decay constants of K ], D and D 2 D s2 ] mesons, respectively, and 2 D and s2 DK are the strong coupling form factors among the mesons under consideration. In writing Eq. 0 we have used the following relationships of the polarization tensor ɛ λ ηθ 35]: ɛ λ ηθ = ɛλ θη, ɛλη η = 0, p η ɛ ηθ λ = p θ ɛ ηθ λ = 0, ɛ λ ηθ ɛ λ ηθ = δ λλ. Using of the matrix elements given in Eqs. 7, 8, 9, and 0 in Eq.6, the correlation function takes its final form in the hadronic side, μν had p, p, q = 2 DDs2 DK] m 2 D m2 K ] f D f K ] f D 2 Ds2 ] m c m d m us] m d p 2 m 2 D2 m 2 D s2 ]p 2 D q2 m 2 K ] m D 2 Ds2 ] p p p μ p ν 2 p p 2 m 2 D2 D s2 ] p 2 p μ p ν m 3 3 m D 2 Ds2 ] D2 D s2 ] p μ p ν m D 2 D s2 ] p p p μ p ν m D 2 Ds2 ] m 2 D2 D s2 ] p 2 p p 2 ] g μν,

3 Eur. Phys. J. C :306 Page 3 of where the summation over the polarization tensor has been Sc il applied, i.e. x = i 2 4 d 4 ke ik x { ε μν λ ε λ αβ = 2 T μαt νβ 2 T μβ T να 3 T δ il μνt αβ, 3 g sg αβ il σ αβ k m c k m c σ αβ k m c 4 k 2 m 2 c λ 2 and T μν = g μν p 2 μ p ν 3 α sgg δ k 2 m c k ilm c k m m 2, 7 c 4 D2 D s2 ] and S us] x and S d x are the light quark propagators and are given by Following the application of the double Borel transformation with respect to the initial and final momenta squared, we obtain the hadronic side of the correlation function: B had μν q = 2 DD s2 DK] f D f D 2 D s2 ] f K ] m 2 D m2 K ] m c m d m us] m d m 2 K ] q2 m2 e D 2 D s2 ] M 2 e m2 D M 2 { 2 m D2 D s2 ] m 4 D m2 D2 D s2 ] q2 2 2m 2 D m2 D2 D s2 ] q2 g μν m 4 6m D 2 Ds2 ] D m2 D 4m2 D2 D s2 ] 2q2 m 2 D 2 D s2 ] q2 2] p μ p ν 2 m D 2 D s2 ] m 2 D m2 D 2 D s2 ] q2 p ν p μ m 3 D 2 D s2 ] p μ p ν 2 m D 2 D s2 ] m 2 D m2 D 2 D s2 ] q2 p μ p ν, 5 where M 2 and M 2 are Borel mass parameters. In the OPE side, we calculate the aforesaid correlation function in deep Euclidean region, where p 2 and p 2. Substituting the explicit forms of the interpolating currents into the correlation function Eq. and after contracting out all quark pairs via Wick s theorem, we get μν p, p, q = i 5 d 4 x d 4 ye ip x e ip y 2 { Tr γ 5 S ji d yγ 5Sc il y xγ ] μ DνxS lj us] x μ ν], 6 OPE x q x = i 2 2 x 4 δ ij S ij m q 4 2 x 2 δ ij qq i m q 2 4 x δ ij x2 92 m2 0 qq i m q 6 x δ ij ig sg ij θη xσ θη 32 2 x 2 σ θη x ]. 8 After the insertion of the explicit forms of the heavy and light quark propagators into Eq. 6, we use the following transformations in D = 4 dimensions: y x 2 ] n = y 2 ] m = d D t 2 D e ity x i n 2 D 2n D/2 ƔD/2 n Ɣn d D t 2 D e it y i m 2 D 2m D/2 ƔD/2 m Ɣm t 2 D/2 n, D/2 m t 2 9 and perform the four-x and four-y integrals after the replacements x μ i p μ and y μ i p μ. The four-integrals over k and t are performed with the help of the Dirac delta functions which are obtained from the four-integrals over x and y. The remaining four-integral over t is performed via the Feynman parametrization and d 4 t 2 β t t 2 L α = i2 β α Ɣβ 2Ɣα β 2 Ɣ2Ɣα L] α β In spite of its smallness we also include the contributions coming from the two-gluon condensate in our calculations. The correlation function in the OPE side is written in terms of different structures as OPE μν p, p, q = q 2 p μ p ν 2 q 2 p ν p μ 3 q 2 p μ p ν 4q 2 p μ p ν 5q 2 g μν, 2 where Sc il x represents the heavy quark propagator which is given by 36] where each i q 2 function receives contributions from both the perturbative and non-perturbative parts and can be written 23

4 306 Page 4 of 7 Eur. Phys. J. C :306 as i q 2 = ds ds ρ pert i s, s, q 2 s p 2 s p 2 non-pert i q 2, 22 where the spectral densities ρ i s, s, q 2 are given by the imaginary parts of the i functions, i.e., ρ i s, s, q 2 = Im i]. In the present study, we consider the Dirac structure p μ p ν to obtain the QCD sum rules for the considered strong coupling form factors. The ρ s, s, q 2 and non-pert q 2 corresponding to this Dirac structure are obtained as ρ pert s, s, q 2 = 0 x dx 0 dy 3 8x2 7y 8y 2 7x 6xy 8 2 θls, s, q 2 ], with θ...] being the unit-step function and non-pert q 2 x = dx 0 0 { αs G 2 dyy 8L 4 m cx 3 2x 2y m c m d m q x y m c p 2 x q 2 y x y x y m c p 2 xx y xy y 2 m q x y m d x y p 2 x x y yp 2 x ] q 2 x y 24L 3 x 2 x 2 2x p 2 q 2 p 2 3x 2 xyx q 2 x 4 3x 6x 2 p 2 x 2 7x 24x 2 p 2 3 x 5x 2 6x 3 q 2 y x 8x 2 75x 3 24x 4 xy 2 p 2 57x 90x 2 42x 3 0 p 2 40x 50x 2 8x 3 q 2 y 3 x 5 62x 42x 2 xy 3 p 2 x 42x 9 48xp 2 24x 2 p 2 9p 2 xy 4 p 2 7 8x p x 23 q 2 y x 42x 2 6xy 5 p 2 p 2 3y 5 q 2 8x 7 6y 6 q 2 m 2 c x3 8x 2 7y 8y 2 7x 6xy m c m q xx y 8x 3 3x 2 2x 5y 0xy 8x 2 y 8y 2 m c m q x8x 4 x 3 8x 2 3x 3y 4xy 9x 2 y 6x 3 y 7y 2 2xy 2 ] 8x 2 y 2 4y 3 48L 2 24x 4 x 3 72y 55 3x y 32y 2 y 2 y8 3y 24y 2 8x 75xy 44xy 2 72xy 3]] m 2 0 dd m q 24q 2 m 2 c 9m 4 p 2 4 c 8m3 c m d 2m 2 c p 2 2m c m d p 2 3p 4 m2 0 qq m d 24q 2 m 2 c p2 4 9m 4 c 8m3 c m q 2m 2 c p 2 2m c m q p 2 3p 4, 24 where qq = uu, m q = m u and qq = ss, m q = m s for the initial D2 and D s2 states, respectively, and Ls, s, q 2 = m 2 c x sx sx2 q 2 y q 2 xy sxy s xy q 2 y The final form of the OPE side of the correlation function is obtained after a double Borel transformation as { B OPE μν q2 = ds where B nonpert q 2 ds e s M 2 e s M 2 ρ pert s, s, q 2 p μ p ν, 26 B non-pert q 2 0 = dx m 2 exp c M 4 x m 2 c M4 x M 2 M 2 q 2 x 2 2m 2 c x ] α s G 2 M 2 M 2 M 2 M 2 xx { M 2 x 6 M 2 M 2 x 48 x 2 x 3 u 6 M 2 M 2 0 xm 2 c M 4 M 4 M 2 M 2 ] q 2 x 2 2m 2 c x 23

5 Eur. Phys. J. C :306 Page 5 of M 2 x 6 M 2 M 2 x M 2 q 2 x x 3 u 5 M 2 M 2 9 4M 4 x M 2 q 2 2M 2 x q 2 x M 8 x 4 m x 2 u 4 M 2 M 2 M 2 7 c m d M 6 M 6 x M 2 x m c m d x M 4 M 2 4M 2 x m c m d 2x M 2 M 4 ] m c m d x2 x M 2 7x 5x 2 2 with M 8 M 2 M 2 x x 2 u 3 M 2 M 2 M 2 5 x 4 m c m u M 2] θ M 2 M 2 x ] M 2 M 2 27 u = x M2 M 2 x. 28 M 2 M 2 Equating the coefficients of the same Dirac structure from both sides of the correlation function, we get the following sum rules for the coupling form factors 2 D and s2 DK: 2 DD s2 DK] m 2 D 2 D s2 ] m 2 D 6m = e M 2 e M 2 c m d m d m us] m 2 K ] q2 m D 2 Ds2 ] f D 2 Ds2 ] f D f K ] m 2 D m2 K ] m 4 D m2 D 4m2 D 2 D s2 ] 2q2 m 2 D 2 D s2 ] q2 2] { s0 s ds 0 ds e s M 2 e s M 2 ρ pert m c m us] 2 m c m d 2 s, s, q 2 B non-pert q 2, 29 where s 0 and s 0 are continuum thresholds in D 2 D s2 ] and D channels, respectively, and we have used the quark hadron duality assumption. 3 Numerical results In this section, we numerically analyze the obtained sum rules for the strong coupling form factors in the previous section and search for the behavior of those couplings with respect to Q 2 = q 2. The values of the strong coupling form factors at Q 2 = m 2 K ] give the strong coupling constants whose values are then used to find the decay rate and branching ratio of the strong transitions under consideration. To proceed, we use some input parameters, presented in Table. The next task is to find the working regions for the auxiliary parameters M 2, M 2, s 0, and s 0. As they are not physical parameters, the strong coupling form factors should roughly be independent of these parameters. In the case Table Input parameters used in calculations Parameters Values m c.275 ± GeV 37] m d MeV 37] m u MeV 37] m s 95 ± 5MeV37] m D ,462.6 ± 0.6 MeV 37] m D s ,57.9 ± 0.8 MeV 37] m D, ± 0.5 MeV 37] m ± MeV 37] m K ± 0.06 MeV 37] f D ± ] f D s ± ] f D ± 8.9 MeV37] f 30.4 ± 0.03 ± 0.20 MeV 37] f K 56. ± 0.2 ± 0.8 ± 0.2 MeV37] αs G ± GeV 4 38,39] of the continuum thresholds, they are not completely arbitrary but are related to the energy of the first excited states with the same quantum numbers as the considered interpolating fields. From a numerical analysis, the working intervals are obtained as ] GeV 2 s ] GeV 2 and 4.7GeV 2 s 0 5.6GeV2 for the strong vertex D2 DD s2 DK]. In the case of the Borel mass parameters M 2 and M 2, we choose their working windows such that they guarantee not only the pole dominance but also the convergence of the OPE. If these parameters are chosen too large, the convergence of the OPE is good but the continuum and higher state contributions exceed the pole contribution. On the other hand if one chooses too small values, although the pole dominates the higher state and continuum contributions, the OPE have a poor convergence. By considering these conditions we choose the windows 3 GeV 2 M 2 8GeV 2 and 2GeV 2 M 2 5GeV 2 for the Borel mass parameters. Our analysis shows that, in these intervals, the dependence of the results on the Borel parameters are weak. Now we proceed to find the variations of the strong coupling form factors with respect to Q 2. Using the working regions for the auxiliary parameters we observe that the following fit function well describes the strong coupling form factors in terms of Q 2 : 2 DDs2 DK] Q 2 = c exp Q2 ] c 3, 30 c 2 where the values of the parameters c, c 2, and c 3 for different structures are presented in Tables 2 and 3 for D2 D and Ds2 DK, respectively. From this fit parametrization we obtain the values of the strong coupling constants for each 23

6 306 Page 6 of 7 Eur. Phys. J. C :306 Table 2 Parameters appearing in the fit function of the coupling form factor for D2 D vertex Structure c GeV c 2 GeV 2 c 3 GeV p μ p ν 5.7 ± ± ± 0.6 p μ p ν 8.2 ± ± ± 3.77 p μ p ν.57 ± ± ± 0.34 p μ p ν.57 ± ± ± 0.34 g μν 5.24 ± ± ± 0.00 Table 5 Numerical results for decay width and branching ratio of D D transition obtained via different structures Structure Ɣ GeV BR p μ p ν 6.26 ± ± p μ p ν.25 ± ± p μ p ν 4.73 ± ± p μ p ν 4.73 ± ± g μν 5.0 ± ± Table 3 Parameters appearing in the fit function of the coupling form factor for Ds2 DK vertex Structure c GeV c 2 GeV 2 c 3 GeV p μ p ν 6.43 ± ± ± 0.24 p μ p ν 9.79 ± ± ± 3.7 p μ p ν 2.03 ± ± ± 0.24 p μ p ν 2.03 ± ± ± 0.24 g μν 7.75 ± ± ± Table 4 Value of the 2 DD s2 DK] coupling constant in GeV unit for different structures Structure 2 D Q 2 = m 2 s2 DK Q 2 = m 2 K p μ p ν 4.63 ± ±.84 p μ p ν ± ± 5.5 p μ p ν 2.72 ± ± 3.85 p μ p ν 2.72 ± ± 3.85 g μν 5.30 ± ± 5.48 structure at Q 2 = m 2 K ] as presented in Table 4. The errors appearing in our results belong to the uncertainties in the input parameters as well as errors coming from the determination of the working regions for the auxiliary parameters. From Table 4 we see that the results strongly depend on the selected structure such that the maximum values for the strong couplings in D2 and D s2 channels that belong to the structure p μ p ν are roughly four times greater that those of the minimum values which correspond to the structure p μ p ν. The values obtained using other structures lie between these maximum and minimum values. Note that the coupling constant in the channel has been estimated in a pioneering study via chiral perturbation theory 40]. By converting the parametrization of the coupling constant used in 40] to our parametrization, Falk 40] finds a value of 2 D 6 GeV in the vertex which is close to our prediction obtained via the structure g μν. Our results obtained via the structures p μ p ν and p μ p ν are comparable with that of 40] within the errors. However, our results obtained via the structure p μ p ν are considerably high and our prediction obtained using the structure p μ p ν is very low compared to Table 6 Numerical results for decay width and branching ratio of D s D K 0 transition obtained via different structures Structure Ɣ GeV BR p μ p ν 3.70 ± ± p μ p ν 4.73 ± ± p μ p ν.84 ± ± p μ p ν.84 ± ± g μν 3.72 ± ± the result of 40] for the strong coupling constant associated to the D2 D vertex. The final task in the present work is to calculate the decay rates and branching ratios for the strong D D and Ds D K 0 transitions. Using the amplitudes of these transitions we find Ɣ = Mp 2 40m 2 p, 3 D2 D s2 ] where Mp 2 = g D2 DD s2 DK] 3m 4 m D 2 D p D2 D s2 ] s2 ] 2 m 2 D 4m2 2 D 3m 2 m D 2 D p D2 D s2 ] s2 ] 2 m 2 2m 4 ] D D, 32 3 and p = 2m D 2 Ds2 ] m 4 D 2 D s2 ]m4 D m4 2m2 D 2 D s2 ]m2 K ] 2m2 D m2 K ] 2m2 D 2 D s2 ]m2 D. 33 The numerical values of the decay rates for the transitions under consideration are presented in Tables 5 and 6.Usingthe total widths of the initial particles as Ɣ D = 49.0 ±.3 MeV, Ɣ D s = 7 ± 4 MeV 37] wealsofind the corresponding branching ratios that are also presented in Tables 5 and 6. Using the following experimental ratio in the channel 37,4]: 23

7 Eur. Phys. J. C :306 Page 7 of Table 7 Numerical results for decay width and branching ratio of D D 200 transition obtained via different structures Structure ƔGeV BR p μ p ν 3.84 ± ± p μ p ν 7.67 ± ± p μ p ν 2.90 ± ± p μ p ν 2.90 ± ± g μν 3.2 ± ± ƔD D ] ƔD D ]ƔD D 200 ] = 0.62 ± 0.03 ± 0.02, 34 we also get the values of the decay rate and branching ratio for D D 200 channel for different structures as presented in Table 7. Considering the fact that the dominant decay modes of D are D D and D D, from the values presented in Tables 5 and 7, we see that all structures give the results for the total decay width of the D tensor meson compatible with the experimental data 37] except for the structure p μ p ν, which gives a result roughly one order of magnitude smaller than the experimental values. To sum up, we calculated the strong coupling form factors 2 D q 2 and s2 DKq 2 in the framework of QCD sum rules. Using the obtained working regions for the auxiliary parameters entering the sum rules of the strong form factors, we found the behavior of those form factors in terms of Q 2.UsingQ 2 = m 2 K ], we also found the values of the strong coupling constants 2 D and s2 DK, which have then been used to calculate the decay widths and branching ratios of the strong D D, D D 200, and Ds D K 0 transitions. Our results can be used in analyses of the future experimental data, especially at the K channel. Acknowledgments This work has been supported in part by the Scientific and Technological Research Council of Turkey TUBITAK under the research project 4F08. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original authors and the source are credited. Funded by SCOAP 3 /LicenseVersionCCBY4.0. References 2. H. Albrecht et al. ARGUS Collaboration, Phys. Lett. B 232, H. Albrecht et al. ARGUS Collaboration, Phys. Lett. B 23, H. Albrecht et al. ARGUS Collaboration, Phys. Lett. B 22, H. Albrecht et al. ARGUS Collaboration, Phys. Lett. B 230, H. Albrecht et al. ARGUS Collaboration, Phys. Lett. B 297, J.C. Anjos et al. E69 Collaboration, Phys. Rev. Lett. 62, P.L. Frabetti et al. E687 Collaboration, Phys. Rev. Lett. 72, P. Avery et al. CLEO Collaboration, Phys. Rev. D 4, P. Avery et al. CLEO Collaboration, Phys. Lett. B 33, J.P. Alexander et al. CLEO Collaboration, Phys. Lett. B 303, Y. Kubota et al. CLEO Collaboration, Phys. Rev. Lett. 72, T. Bergfeld et al. CLEO Collaboration, Phys. Lett. B 340, J. Link et al. FOCUS Collaboration, Phys. Lett. B 586, K. Abe et al. BELLE Collaboration, Phys. Rev. D 69, V.M. Abazov et al. D0 Collaboration, Phys. Rev. Lett. 95, R. Aaij et al. LHCb Collaboration, Phys. Lett. B 698, S. Godfrey, Phys. Rev. D 72, H. Sundu, K. Azizi, Eur. Phys. J. A 48, K. Azizi, H. Sundu, J.Y. Süngü, N. Yinelek, Phys. Rev. D 88, Erratum-ibid. D 88, K. Azizi, H. Sundu, A.Y. Türkan, E. Veli Veliev., J. Phys. G Nucl. Part. Phys. 4, F. De Fazio, arxiv: hep-ph] 23. R. Molina et al., AIP Conf. Proc. 322, A. Faessler et al., Phys. Rev. D 76, B. Aubert et al. BaBar Collaboration, Phys. Rev. Lett. 03, B. Aubert et al. BaBar Collaboration, Phys. Rev. D 80, D. Liventsev et al. Belle Collaboration, Phys. Rev. D 77, P. Colangelo et al., Phys. Rev. D 86, B. Aubert et al., Phys. Rev. Lett. 90, D. Besson et al., Phys. Rev. D 68, Erratum-ibid. D ] 3. B. Aubert et al., Phys. Rev. Lett. 97, J. Brodzicka et al., Phys. Rev. Lett. 00, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 47, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 47, H.-Y. Cheng, K.-C. Yang, Phys. Rev. D 83, L.J. Reinders, H. Rubinstein, S. Yazaki, Phys. Rept. 27, Beringer et al. Particle Data Group, Phys. Rev. D 86, V.M. Belyaev, B.L. Ioffe, Sov. Phys. JETP 57, V.M. Belyaev, B.L. Ioffe, Phys. Lett. B 287, A.F. Falk, M. Luke, Phys. Lett. B 292, The BABAR Collaboration, B. Aubert, et al., Phys. Rev. Lett. 03, H. Albrecht et al. ARGUS Collaboration, Phys. Rev. Lett. 56,