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The Rare Decay D^0 ->gamma gamma

We present a calculation of the rare decay mode D^0 ->gamma gamma, in which the long distance contributions are expected to be dominant. Using the Heavy Quark Chiral Perturbation Theory Lagrangian with a strong g coupling as recently determined by CLEO from the D^* ->D pi width, we consider both the anomaly contribution which relates to the annihilation part of the weak Lagrangian and the one-loop pi, K diagrams. The loop contributions which are proportional to g and contain the a_1 Wilson coefficient are found to dominate the decay amplitude, which turns out to be mainly parity violating. The branching ratio is then calculated to be (1.0+-0.5)x10^(-8). Observation of an order of magnitude larger branching ratio could be indicative of new physics.


Introduction
With the new data coming and expected from the B-factories, there is very strong emphasis and activity in the field of B-physics in all its aspects. This includes the rare decays of B-mesons, which are considered as an attractive source for possible signals of new physics. In contradistinction to the growing efforts to understand the decay mechanisms of rare B decays, studies of rare D decays have received less attention in recent years. Partially this is because theoretical investigations of D weak decays are rather difficult, also due to the presence of many resonances close to this energy region. The penguin effects on the other hand, which are very important in B and also in K decays, are usually suppressed in the case of charm mesons due to the presence of d, s, b quarks in the loop with the respective values of CKM elements.
Nevertheless, D meson physics has produced some interesting results in the past year. Experimental results on time dependent decay rates of D 0 → K + π − by CLEO [1] and D 0 → K + K − and D 0 → K − π + by FOCUS [2] have stimulated several studies on the D 0 −D 0 oscillations [3]. The recently measured D* decay width by CLEO [4] has provided the long expected information on the value of D * Dπ coupling. Among the rare D decays, the decays D → V γ and D → V (P )l + l − are subjects of CLEO and FERMILAB searches [5]. On the theoretical side, these rare decays of charm mesons into light vector meson and photon or lepton pair have been considered lately by several authors (see, e.g., [6]- [11] and for radiative leptonic D meson decay see [12]). The investigations of D → V γ showed that certain branching ratios can be as large as 10 −5 , like for D 0 →K * 0 γ, D + s → ρ + γ [6,11]. However, the decays which are of some relevance to the D 0 → 2γ mode studied here, like D 0 → ρ 0 γ, D 0 → ωγ, are expected with branching ratios in the 10 −6 range [13]. Thus, it is hard to believe that the branching ratio of the D 0 → 2γ decay mode can be as high as 10 −5 in the Standard Model (SM), as found by [14]. Apart from this estimation, there is no other detailed work on D 0 → 2γ in the literature, to the best of our knowledge.
On the other hand, in the B and K meson systems there are numerous studies of their decays to two photons. For example, the B s → γγ decay has been studied with various approaches within SM and beyond. In SM, the short distance (SD) contribution [15] leads to a branching ratio B(B s → γγ) ≃ 3.8 × 10 −7 . The QCD corrections enhance this rate to 5 × 10 −7 [16]. On the other hand, in some of the SM extensions the branching ratio can be considerably larger. The two Higgs doublet scenario, for example, could enhance this branching ratio by an order of magnitude [17]. Such "new physics" effects could at least in principle be dwarfed by long distance (LD) effects. However, existing calculations show that these are not larger than the SD contribution [18], which is typical of the situation in radiative B decays [19]. In the K 0 system the situation is rather different. Here, the SD contribution is too small to account for the observed rates of K S → 2γ, K L → 2γ by factors of ∼ 3 − 5 [20], although it could be of relevance in the mechanism of CPviolation. Many detailed calculations of these processes have been performed over the years (see recent refs. [20]- [23] and refs. therein), especially using the chiral approach to account for the pole diagrams and the loops. These LD contributions lead to rates which are compatible with existing measurements.
Motivated by the experimental efforts to observe rare D meson decays, as well as by the lack of detailed theoretical treatments, we undertook an investigation of the D 0 → γγ decay. The short distance contribution is expected to be rather small, as already encountered in the one photon decays [6,7], hence the main contribution would come from long distance interactions. In order to treat the long distance contributions, we use the heavy quark effective theory combined with chiral perturbation theory (HQχPT) [24]. This approach was used before for treating D * strong and electromagnetic decays [25]- [27]. The leptonic and semileptonic decays of D meson were also treated within the same framework(see [25] and references therein).
The approach of HQχPT introduces several coupling constants that have to be determined from experiment. The recent measurement of the D * decay width [4] has determined the D * Dπ coupling, which is related to g, the basic strong coupling of the Lagrangian. There is more ambiguity, however, concerning the value of the anomalous electromagnetic coupling, which is responsible for the D * Dγ decays [26,27], as we shall discuss later.
Let us address now some issues concerning the theoretical framework used in our treatment. For the weak vertex we shall use the factorization of weak currents with nonfactorizable contributions coming from chiral loops. The typical energy of intermediate pseudoscalar mesons is of order m D /2, so that the chiral expansion p/Λ χ (for Λ χ 1 GeV) is rather close to unity. Thus, for the decay under study here we extend the possible range of applicability of the chiral expansion of HQχPT, compared to previous treatments like D * → Dπ, D * → Dγ [26] or D * → Dγγ [27], in which a heavy meson appears in the final state, making the use of chiral perturbation theory rather natural. The suitability of our undertaking here must be confronted with experiment, and possibly other theoretical approaches.
At this point we also remark that the contribution of the order O(p) does not exist in the D 0 → γγ decay, and the amplitude starts with contribution of the order O(p 3 ). At this order the amplitude receives an annihilation type contribution proportional to the a 2 Wilson coefficient, with the Wess-Zumino anomalous term coupling light pseudoscalars to two photons. As we will show, the total amplitude is dominated by terms proportional to a 1 that contribute only through loops with Goldstone bosons. Loop contributions proportional to a 2 vanish at this order. We point out that any other model which does not involve intermediate charged states cannot give this kind of contribution. Therefore, the chiral loops naturally include effects of intermediate meson exchange.
The chiral loops of order O(p 3 ) are finite, as they are in the similar case of K → γγ decays [20]- [23]. The next to leading terms might be almost of the same order of magnitude compared to the leading O(p 3 ) term, the expected suppression being approximately p 2 /Λ 2 χ . The inclusion of next order terms in the chiral expansion is not straightforward in the present approach. We include, however, terms which contain the anomalous electromagnetic coupling, and appear as next to leading order terms in the chiral expansion, in view of their potentially large contribution (as in B * (D * ) → B(D)γγ decays considered in [27]). As it turns out, these terms are suppressed compared to the leading loop effects, which at least partially justifies the use of HQχPT for the decay under consideration. Contributions of the same order could arise from light resonances like ρ, K * , a 0 (980), f 0 (975). Such resonances are sometimes treated with hidden gauge symmetry (see, e.g., [25]), which is not compatible with chiral perturbation symmetry. Therefore, a consistent calculation of these terms is beyond our scheme and we disregard their possible effect.
Our paper is organized as follows: in Section 2 we present the basic features of the model. We give the results and their discussion in Section 3 and conclude with a summary in Section 4.

The theoretical framework
The invariant amplitude for D 0 → γγ decay can be written using gauge and Lorentz invariance in the following form: where M (−) is a parity violating and M (+) a parity conserving part of the amplitude, while k 1(2) , ǫ 1 (2) are respectively the four momenta and the polarization vectors of the outgoing photons.
In the discussion of weak radiative decays q ′ → qγγ or q ′ → qγ decays, usually the short (SD) and long distance (LD) contribution are separated. The SD contribution in these transitions is a result of the penguin-like transition, while the long distance contribution arises in particular pseudoscalar meson decay as a result of the nonleptonic four quark weak Lagrangian, when the photon is emitted from the quark legs. Here we follow this classification. In the case of b → sγγ decay [28] it was noticed that without QCD corrections the rate Γ(b → sγγ)/Γ(b → sγ) is about 10 −3 . One expect that a similar effect will show up in the case of c → uγγ decays. Namely, according to the result of [28] the largest contribution to c → uγγ amplitude would arise from the photon emitted either from c or u quark legs in the case of the penguin-like transition c → uγ. Without QCD corrections the branching ratio for c → uγ is rather suppressed, being of the order 10 −17 [7,8]. The QCD corrections [29] enhance it up to order of 10 −8 .
In our approach we include the c → uγ short distance contribution by using the Lagrangian where m c is a charm quark mass. In our analysis we follow [29,30] and we take The main LD contribution will arise from the effective four quark nonleptonic ∆C = 1 weak Lagrangian given by where Γ µ = γ µ (1 − γ 5 ), a i are effective Wilson coefficients [31], and V q i q j are CKM matrix elements. At this point it is worth pointing out that long distance interactions will contribute only if the SU (3) flavor symmetry is broken, i.e. if m s = m d . Namely, due to V ud V * cd ≃ −V us V * cs , if m d = m s the contributions arising from the weak Lagrangian (3) cancel. Now, we turn to describe some of the basic features of the HQχPT. This model would serve us as a hadronized counterpart of the quark effective weak Lagrangian. One has the usual O(p 2 ) chiral Lagrangian for the light pseudoscalar mesons where Σ = exp (2iΠ/f ) with Π = j 1 √ 2 λ j π j containing the Goldstone bosons π, K, η, while the trace tr runs over flavor indices and M q = diag(m u , m d , m s ) is the current quark mass matrix. From this Lagrangian, we can deduce the light weak current of the order O(p) corresponding to the quark current j X µ =q L γ µ λ X q L . (λ X is an SU(3) flavor matrix.) In the heavy meson sector interacting with light mesons we have the following lowest order O(p) chiral Lagrangian while the trace Tr runs over Dirac indices. Note that in (6) and the rest of this section a and b are flavor indices.
The vector and axial vector fields V µ and A µ in (6) are given by: where ξ = exp (iΠ/f ). The heavy meson field H a contains a spin zero and spin one boson with P ± = (1 ± γ µ v µ )/2 being the projection operators. The field P 5 (P † 5 ) annihilates (creates) a pseudoscalar meson with a heavy quark having velocity v, and similar for spin one mesons.
For a decaying heavy quark, the weak current is given by where L = (1 − γ 5 )/2 and Q is the heavy quark field in the full theory, in our case a c-quark, and q is the light quark field. From symmetry grounds, the heavy-light weak current is bosonized in the following way [24] J where α is related to the physical decay constant f D through the well known matrix element Note that the current (11) is O(p 0 ) in the chiral counting.
In the calculation of short distance contribution (2) there appears the operator (ūσ µν 1 2 (1+γ 5 )c). Using heavy quark symmetry this operator can be translated into an operator containing meson fields only [36] ūσ µν The photon couplings are obtained by gauging the Lagrangians (4), (6) and the light current (5) with the U (1) photon field B µ . The covariant derivatives are then D µ (for our case of c quark). The vector and axial vector fields (7) change after gauging and now they read . The light weak current (5) contains after gauging the covariant derivative D µ instead of ∂ µ . However, the gauging procedure alone does not introduce a coupling between heavy vector and pseudoscalar meson fields and the photon without emission or absorption of additional Goldstone boson, which is needed to account, for example, for D * → Dγ. To describe this electromagnetic interaction we follow [26] introducing an additional gauge invariant contact term with an unknown coupling β of dimension -1. where The first term concerns the contribution of the light quarks in the heavy meson and the second term describes emission of a photon from the heavy quark. Its coefficient is fixed by heavy quark symmetry. From this "anomalous" interaction, both H * Hγ and H * H * γ interaction terms arise. Even though the Lagrangian (14) is formally 1/m Q ∼ m q suppressed, we do not neglect it, as it has been found that it gives a sizable contribution to D * (B * ) → D(B)γγ decays [27]. In the case of D 0 → γγ it gives largest contribution to the parity conserving part of the amplitude, however it does not contribute to the decay rate by more than 10%, as will be shown later. The Lagrangian (14) in principle receives a number of other contributions at the order of 1/m Q , however, these can be absorbed in the definition of β for the processes considered [26].

Results
The decay width for the D 0 → γγ decay can be obtained using the amplitude decomposition in (1): The short distance contribution to D 0 → γγ decay width is estimated using c → uγ effective Lagrangian (2), (13) with one photon emitted from D 0 leg via L β term (14). The parity violating part of the short distance amplitude is while the parity conserving part of the amplitude is where ∆ * = m D 0 * − m D 0 . Turning now to the long distance contributions, we depict in Figs. 1 and 2 the loop diagrams arising to leading order O(p 3 ) by using Eqs.(4)- (6) and (11). The circled cross indicates the weak interaction. In Figure 1 we grouped all diagrams which vanish by symmetry considerations. All nonvanishing contributions are assembled in Fig. 2. We denote the gauge invariant sums corresponding to nonvanishing diagrams of Fig. 2 by M  part of the amplitude, as in (15). The parity violating sums, which arise from the first term in (3) are with The parity conserving parts of the amplitude M  (20) are gauge invariant and finite. This is expected, since one cannot generate counterterms at this order. There is no µ dependence apart from the one hidden in a 1 even though µ appears in the above functions, but it cancels out completely. Note also, that the one loop chiral corrections vanish in the exact SU (3) limit, i.e. when m K → m π , as it is expected. One should note that taking the chiral limit (i.e. m s , m d → 0) is not unambiguous. Namely, in the combined heavy quark effective theory and the chiral perturbation theory, beside chiral logarithms there are also functions of the form F (m q /∆) whose value depend on the way one takes the limit (see e.g. Ref. [37]).
We remark that there exist additional diagrams of the same order in the chiral expansion as ones given on Fig. 2, but proportional to a 2 part of the effective weak Lagrangian (3). In these additional diagrams the chiral loop is attached to the light current in the factorized vertex, while the photons are emitted from the pseudoscalars in the loop, or they come from the weak vertex. However, the amplitudes of these diagrams vanish due to Lorentz symmetry.
The contribution coming from the anomalous coupling π 0 γγ, ηγγ, η ′ γγ (Fig. 3) is where θ = −20 o ± 5 o is the η − η ′ mixing angle and we have set f π = f η 8 = f η 0 . This choice of parameters reproduces the experimental results for the π 0 → γγ, η → γγ, and η ′ → γγ decay width [34]. In the numerical evaluation we use the values of α and g obtained within the same framework as in [25,26,35,36,37]. The coupling g is extracted from existing experimental data on D * → Dπ. Recently CLEO Collaboration has obtained the first measurement of D * + decay width Γ(D * + ) = 96 ± 4 ± 22 keV [4] by studying the D * → D 0 π + . Using the value of decay width together with branching ratio Br(D * + → D 0 π + ) = (67.7 ± 0.5)% one immediately finds at tree level that g = 0.59 ± 0.08. The chiral corrections to this coupling were found to contribute about 10% [25,26]. In order to obtain the α coupling, we use present experimental data on D s leptonic decays. Namely, at the tree level there is a relation f D = f Ds = α/ √ m D . From the experimental branching ratio D s → µν µ and the D s decay width [34] one gets f Ds = 0.23 ± 0.05 GeV and α = 0.31 ± 0.04 GeV 3/2 . The SU (3) breaking effects in the form of chiral loops and the counterterms can change the extracted value of α. One chiral loop corrections can amount to about 40% when g is taken to be 0.59. This value might be changed by the finite part of the counterterms. However, the contributions coming from counterterms are not known and due to the lack of experimental data they cannot be fixed yet. In our calculation we take α = 0.31 GeV 3/2 , keeping in mind that the chiral corrections might be important (for instance setting counterterms to zero in one loop calculation one gets α = 0.21 ± 0.04 GeV 3/2 using g = 0.59, for details see Appendix B of [32]). For the Wilson coefficients a 1 we take 1.26 and a 2 = −0.47 [31]. We present the numerical results for the one loop amplitudes in Table 1. Table 1: Table of  i.j ). The numbers 1, 2, 3 denote the row of diagrams on the Fig. 2. In the last line the sum of all amplitudes is given.
In the determination of D * → Dγγ and B * → Bγγ a sizable contribution from β-like electromagnetic terms (14) has been found [27]. Therefore we have to investigate their effect in the D 0 → γγ decay amplitude. The considerations of Eq. (14) gives us additional diagrams which are given in Fig. 4 and 5, where the β vertex is indicated by •. The nonzero parity violating parts of the one loop diagrams containing β coupling are (Fig. 5) while the parity conserving parts of the amplitudes arising from the one loop diagrams with β coupling are The amplitudes with β coupling are not finite and have to be regularized. We use the strict M S prescription∆ = 1 as in [26] (note that in [25]∆ = 0 has been used). We take µ = 1 GeV ≃ Λ χ as in [26].
In order to obtain the value of β we use the available experimental data from D * + → D + γ, D * 0 → D 0 γ decays. For instance, one can use the recently determined D * + decay width Γ(D * + ) = 96 ± 4 ± 22 keV [33] together with the branching ratio Br(D * + → D + γ) = (1.6 ± 0.4)% [34]. At tree level one has ) the momentum of outgoing photon. Using the experimental data and m c = 1.4 GeV one arrives at β = 2.9 ± 0.4 GeV −1 * , where the errors reflect the experimental errors.
On the other hand one can also use the ratio of partial decay width in D * 0 system Γ(D * 0 → D 0 γ) : Γ(D * 0 → D 0 π 0 ) = (38.1 ± 2.9) : (61.9 ± 2.9), where the experimental errors are considerably smaller than in the previous case. At tree level one has with k γ ,k π the momenta of outgoing photon and pion respectively. Using m c = 1.4 GeV, g = 0.59, f = f π = 132 MeV one arrives at β = 2.3 ± 0.2 GeV −1 , † where errors quoted again reflect experimental errors only. The β coupling coming from from D * + (31) and D * 0 (32) are in fair agreement, but not equal. This signals that other contributions coming from chiral loops and higher order terms that would alter our determination of β might be important. Since the contribution of chiral loops to Γ(D * + → D + γ) are approximately 50%, while for D * 0 → D 0 γ are about 20% [26], we use in our numerical calculations the value of β = 2.3 GeV −1 obtained from Γ(D * 0 → D 0 γ). Results are shown in Table 2  Due to the suppression of a 2 in comparison to a 1 , we do not include diagrams proportional to a 2 in the calculation of terms with β.
Using short distance contributions, the finite one loop diagrams and the anomaly parts of the amplitudes (shown in Figs. 2, 3 and with numerical values of the amplitudes as listed in Table 1), one obtains This result is slightly changed when one takes into account the terms dependent on β (14). The branching ratio obtained when we sum all contributions is By varying β within 1 GeV −1 ≤ β ≤ 5 GeV −1 and keeping g = 0.59 ± 0.08, the branching ratio is changed by at most 10%. On the other hand, one has to keep in mind that the loop contributions involving beta are not finite and have to be regulated. We have used MS scheme, with the divergent parts being absorbed by counterterms. The size of these is not known, so they might influence the error in our prediction of the branching ratio. Note also that changing α would affect the predicted branching ratio. For instance, if the chiral corrections do decrease the value of α by 30% this would decrease the predicted branching ratio down to 0.5 × 10 −8 .

Summary
We have presented here a detailed calculation of the decay amplitude D 0 → γγ, which includes short distance and long distance contributions, by making use of the theoretical tool of Heavy Quark Chiral Perturbation Theory Lagrangian. Within this framework, the leading contributions are found to arise from the charged π and K mesons running in the chiral loops, and are of order O(p 3 ). These terms are finite and contribute only to the parity violating part of the amplitude. The inclusion of terms of higher order in the chiral expansion is unfortunately plagued with the uncertainty caused by the lack of knowledge of the counterterms. As to the parity conserving part of the decay, it is given by terms coming from the short distance contribution, the anomaly and from loop terms containing the beta coupling, the latter giving most of the amplitude. The size of this part of the amplitude is approximately one order of magnitude smaller than the parity violating amplitude, thus contributing less than 20% to the decay rate. Therefore, our calculation predicts that the D → 2γ decay is mostly a parity violating transition.
In addition to the uncertainties we have mentioned, there is the question of the suitability of the chiral expansion for the energy involved in this process; the size of the uncertainty related to this is difficult to estimate. Altogether, our estimate is that the total uncertainty is not larger than 50%. Accordingly, we conclude that the predicted branching ratio is Br(D 0 → γγ) = (1.0 ± 0.5) × 10 −8 .
The reasonability of this result can be deduced also from a comparison with the calculated decay rates for the D 0 → ρ(ω)γ, which are found to be expected with a branching ratio of approximately 10 −6 [6,7,13].
We look forward to experimental attempts of detecting this decay. Our result suggests that the observation of D → 2γ at a rate which is an order of magnitude larger than (35), could be a signal for the type of "new physics", which leads to sizable enhancement [30] of the short-distance c → uγ transition.
where I 2 (m, ∆) is defined in (37) and N 0 (m, k 2 ) in (49). In calculation we also need several other integrals that have been calculated for the case k µ wherē with δ > 0 an infinitesimal positive parameter. The chiral loops needed are