The radiative decay $D^0 \to \bar{K}^{*0} \gamma$ with vector meson dominance

Motivated by the experimental measurements of $D^0$ radiative decay modes we have proposed a model to study the $D^0\to \bar{K}^{*0}\gamma$ decay, by establishing a link with $D^0\to \bar{K}^{*0}V$ $(V=\rho^0,\, \omega)$ decays through the vector meson dominance hypothesis. In order to do this properly, we have used the Lagrangians from the local hidden gauge symmetry approach to account for $V\gamma$ conversion. As a result, we have found the branching ratio $\mathcal{B}[D^0\to \bar{K}^{*0}\gamma] =(1.55 - 3.44)\times 10^{-4}$, which is in fair agreement with the experimental values reported by Belle and Babar collaborations.


I. INTRODUCTION
The heavy hadron weak decays have become an important source of information not only in the quest for new physics beyond standard model but also to understand in a deeper way the hadron dynamics behind those processes. For instance, the B meson decays have been experimentally measured by LHCb [1]. A large signal was found for the f 0 (980) resonance in the B s → J/ψπ − π + channel, while no peak associated with f 0 (500) was reported. The same analysis has been done by CDF [2], Belle [3] and D0 [4] collaborations. In contrast, in Ref. [5] the f 0 (500) was seen in the B → J/ψπ − π + decay mode, while only a small fraction for f 0 (980) was observed. Theoretically, these results were understood [6,7] through the chiral unitary approach, which implements chiral theory with some unitary and symmetry techniques [8,9]. Furthermore, the application of chiral unitary techniques to study the B decays into J/ψ and light vector mesons (V ) provided results which gave support to the interpretation of the vector mesons within the qq quark picture [10,11]. On the other hand, concerning to B radiative decays, there is little experimental information available on their branching ratios [12,13], while the theoretical predictions associated with those ratios differ at least by 2 orders of magnitude, requiring more investigation in order to shed light on this issue [14][15][16]. Regarding this point, the B radiative decays were studied in Ref. [17], where accurate results for these ratios were obtained. More concretely, since the long-distance effects might be dominant in those decays, the authors presented a mechanism where a link between B → J/ψV and the B → J/ψγ decay was established by means of the vector meson dominance hypothesis (VMD) [18]. The implementation of VMD was done using the Lagrangians from local hidden gauge symmetry [19][20][21]. The results found in [17] were in a good agreement with the upper limits set for the branching ratios aforementioned.
The charm radiative decays are even more dramatic and have been less discussed in the literature. The amount of theoretical work follows the same line of the experimental counterpart, i. e. the lack of experimental results associated with radiative decays of charmed mesons does not motivate many theoretical studies since most of them are dedicated to the search for new physics beyond the standard model. It turns out that the charm radiative decays are completely dominated by long-distance effects and this feature makes them not so attractive to new physics practitioners. On the other hand, concerning the hadronic systems, this same feature makes these charmed radiative decays an interesting issue to investigate the hadron dynamics as well as to make predictions to be tested by the experimental facilities. This might be a good scenario to test the successful chiral unitary theory and other nonperturbative models related to the description of hadron dynamics.
As mentioned previously, the amount of experimental information for the charm radiative decays is scarce. For instance, the first branching ratio measurement for D 0 →K * 0 γ radiative decay was performed in 2008 by the Babar collaboration [22], with B(D 0 →K * 0 γ) = (3.8 ± 0.20 ± 0.27) × 10 −4 , where the first error is statistical and the second one systematic.
Recently, the Belle collaboration has also measured that same branching ratio [3], obtaining to what happens to the short-range contributions from one model to the other. In view of this, in this work we adopt a different perspective and look at what happens to the hadron dynamics in these decays, and we shall propose a model based on the mechanism of Ref. [17] to estimate the D 0 →K * 0 γ branching ratio. Although the short-range contributions play an important role in B meson decays, in some cases, as that shown by the authors of Ref. [17], the long-range physics is the main ingredient and may help to provide more accurate results, as it was discussed in that work. Since in the charm sector the radiative decays are largely dominated by the long-range physics [3,23,24], we expect to get reasonably accurate results.
The starting point in our approach is to establish a link between the D 0 →K * 0 V decays, with the vector meson V related to the ρ and ω mesons, and the radiative D 0 →K * 0 γ decay via VMD hypothesis. In our case, the VMD is implemented using the hidden gauge Lagrangians [19][20][21], describing the V γ conversion. In the next section, we show the details on how to do this properly and also how to get the branching ratios we are concerned with.
We also present arguments that support the suppression of the short-range effects in the amplitudes contributing to the branching ratio we are interested in.

II. THEORETICAL FRAMEWORK
In order to calculate the radiative decay D 0 →K * 0 γ rate, we follow the approach used in Ref. [17], where the authors combine vector meson dominance, through hidden gauge Lagrangians, with a novel mechanism, proposed in Ref. [7] for B 0 (B 0 s ) → J/ψV , to describe the B 0 (B 0 s ) → J/ψγ decays. In the following, we shall describe briefly this mechanism extended to our problem.
The D 0 meson decays weakly intoK * 0 meson in addition to a ρ 0 or ω meson, denoted by V . At the quark level, this process is illustrated in Fig. 1. According to this figure, a c quark converts into a strange quark by emission of a W boson, that subsequently coalesces into adu pair. As a result, we have aK * 0 meson, related to the sd pair, while the remaining uū can be related to the vector mesons, ρ 0 or ω. It is worth to emphasize at this point that we adhere the qq picture for vector mesons. In fact, studies have shown that wave functions for the low-lying vector mesons are essentially dominated by qq components [10,[25][26][27][28][29][30][31].
Therefore, in terms of quarks the wave functions for vector mesons are given by 1 Since there is no ss pair in the process of Fig. 1, we do not have φ meson contribution.
In order to write the D 0 →K * 0 V amplitudes we restrain ourselves to factorize the weak vertices in terms of a factor V ′ p , which contains weak vertices, Cabibbo angles, etc. The factor V ′ p gets canceled since we are interested in ratios of decay rates. A similar assumption was done in Ref. [7], where the decay rates related toB 0 → J/ψK * 0 andB 0 s → J/ψK * 0 channels were evaluated, with results in good agreement with the experimental ones [32].
Hence, the amplitudes for theK * 0 V production are where polarization vectors in each expression above are omitted (we shall come back later on about the spin structure).
Once we have determined the amplitudes associated with the production ofK * 0 V , we have to go a step further and let the V meson convert into a photon γ, according to VMDhypothesis [18]. In order to implement VMD properly, we use the Lagrangians from the Local hidden gauge approach [19][20][21], which for the V γ vertex are given by where e is the electron charge, e 2 /4π ≈ 1/137, and g is the universal coupling in the hidden gauge Lagrangian, defined by g = M V /(2f π ), with f π the pion decay constant (f π = 93 1 In general, the physical isoscalars φ and ω are mixtures of the SU(3) wave functions ψ 8 and ψ 1 : where θ is the nonet mixing angle and: A µ is associated with the photon field and V µ is the matrix below Eq. (4) can be simplified if we defineṼ µ as denoting the ρ 0 , ω and φ fields and C V γ standing for their respective constants 1/ √ 2, 1/3 √ 2, −1/3. Therefore, we have with C V γ given by Now that we have determined theK * 0 V production amplitude as well as the Lagrangian that describes the V γ vertex, we can write down the amplitude for the photon production, which is depicted in Fig. 2.
and knowing that p V · ǫ(V ) = 0 (Lorentz condition), with p V µ the momentum of the vector meson that is equal to the photon one, after a bit of algebra Eq. (7) can be written as where we have used the approximation M ρ ≈ M ω ≈ M V , as it is usual in the hidden gauge approach.
In order to estimate the ratios we need the decay formulas associated with the D 0 → K * 0 ρ 0 (ω) and D 0 →K * 0 γ channels. They are given by where p ρ(ω) and p γ are the ρ 0 (ω) meson and the photon momenta in the D 0 rest frame. Using Eqs. (1) and (9) into Eq. (10), we get the following expression for the ratio Γ D→K * 0 γ /Γ D 0 →K * 0 ρ 0 As we mentioned before, the parametrization of the weak vertex defined as V ′ p does not play a role in our approach since it gets canceled, as can be seen by looking at the ratio in Eq. (11).
In a general context the mechanism that we have adopted here is considered as a long range process in Refs. [33][34][35][36][37]. In these works, the B radiative decays involving a K * and ρ mesons were addressed. They were separated into long and short range processes and their contribution was estimated. As a result, the short range contribution, considered in those works as the dominant one, for the B → K * γ process was bigger (by a factor 30) than the upper bounds for the B → J/ψγ case, indicating that the equivalent short range contribution could not be dominant in the J/ψγ case, as discussed in Ref. [17]. Furthermore, in the charm sector, it was pointed out in Ref. [37] that the short range diagrams provided results smaller than the one related to the its long range counterpart. In our case the short range diagram gives no contribution since there is noK * 0 (sd) production in the final state, as can be seen in Fig. 3(a).
In Fig. 3 we show all the diagrams associated with short and long range processes. As we have mentioned previously, the diagram of Fig. 3(a) related to the short range contribution, does not contribute in our case, which is represented by the diagram (b), since it produces ρ 0 γ or ωγ but notK * 0 γ. The remaining ones, Fig. 3(c)-(d), are suppressed with respect to that in Fig. 3(b). This happens, because they have a weak process involving two quarks of the original D 0 meson and, according to the discussion in Ref. [38], this kind of processes are penalized with respect to those involving just one quark.
According to Ref. [17] we have to take into account the polarization structure of the D 0 →K * 0 γ vertex. In weak decay processes we can have parity violation as well as parity conservation. In order to take this feature into account in our model, we are going to follow the procedure of Ref. [17] and define both parity conserving (P C) and parity violating (P V ) structures, which are often used in weak decay studies [14][15][16]35]. They are where ǫ ′ (V ) and q ′ are the polarization vector as well as the momentum of the vector meson (ρ 0 or ω) to be converted into γ through VMD. In the case of photon production in both Eqs. (12) and (13), ǫ ′ as well as q ′ stand for the vector polarization and momentum of the photon, respectively.
Note that both structures are gauge invariant. In fact, when we use the Lagrangians from local hidden gauge approach to deal with vector-vector interactions V V and also V γ conversion, gauge invariant amplitudes are obtained, as discussed in Refs. [19-21, 39, 40].
In order to take into account the polarization structure of the weak vertices, as discussed previously, we have to sum the Eqs. (12) and (13) over the polarizations of the vector meson or the photon. Summing up over the polarization provides where q ′ 2 = M 2 V for vector production or 0 in the case of photon production, while with M 2 V = 0 for the case of photon production. With this, we can obtain the following factors and Therefore, the polarization structures discussed above are taken into account in our calculation simply by plugging them in Eq. (11), which now reads where in the left-hand side we have divided the numerator as well as the denominator by Γ total in order to convert the widths into branching fractions.

III. RESULTS
In order to estimate our results, we use the following values for the meson masses: MeV, MK * = 891.6 MeV and M D 0 = 1864.8 MeV. Furthermore, we also use as an input for Γ D 0 →K * ρ (ω) an average value from the following experimental results, extracted from PDG [32], which in our approach should be equal. We have These results are compatible, within errors, providing an average value of (1.4 ± 0.4) × 10 −2 .
Therefore, from Eq. (18), using the values defined above, we get the following result for the branching fraction associated with D 0 →K * 0 γ channel where the uncertainties are obtained from the experimental errors. The average experimental value in the PDG [32] is We can see that the theoretical result with P V is compatible with the experimental number.
The one with P C is somewhat smaller. An equal mixture of both the P C and P V modes would give which is compatible with the experimental value within errors.
In Ref. [23] the authors have used a model related to the extensions of the standard model in order to look for new physics in the charm rare decays. They have done calculations for the long range distance D → V γ amplitudes (see Table IV of that reference), where for the D 0 →K * 0 γ a ratio of about (4.6 − 18) × 10 −5 was obtained. Using a different approach called light-front quark model, a similar result was found in Ref. [24]. The value obtained in this latter work for the same ratio was (4.5 − 19) × 10 −5 . Note that both results are smaller than our result, given by Eqs. (20), (21) and (23), as well as the experimental one. Note also that the range of allowed values is much bigger than in our case, and the lower bound is about one order of magnitude smaller than our results.

IV. CONCLUSIONS
Using a mechanism adopted in Ref. [17] we have established a link between the D 0 → K * 0 V , with V = ρ 0 , ω mesons, and D 0 →K * 0 γ radiative decays. Concretely, after calculating the amplitude for V meson production, we use the vector meson dominance hypothesis in order to convert the vector mesons produced in our mechanism into a photon. This was done using the Lagrangians from the local hidden gauge approach, which provides a gauge invariant amplitude when the vector polarization structure is taken into account. Thus, we have obtained an expression in which both parity violation and conservation contributions are considered. As a result, we have obtained a value for the branching ratio B[D 0 →K * 0 γ] in a fairly agreement with the experimental value quoted in the PDG [32], while other estimations using different approaches provide results with large uncertainties, with some values one order of magnitude smaller than our findings.
We should mention that our evaluation is done using as input the experimental rates for D 0 →K * 0 ρ 0 (ω). Alternative calculations that use other experimental information to fix unknown parameters of the theory [23,24] lead to larger uncertainties. Note that other terms, like loop corrections, that in other approaches must be calculated explicitly, are incorporated empirically in our approach when using the empirical values of the D 0 → K * 0 ρ 0 (ω) rates [17]. In this sense, once one shows that short range terms in the process studied do not contribute, or are small, the method used here proves to be rather accurate for evaluating this kind of radiative decays.