Radiative Leptonic Decays of D_s^\pm and D^\pm Mesons

In this work, we investigate the radiative leptonic decays D_(s)^- \to \gamma \ell \bar\nu (\ell = e, \mu) at tree level within the non-relativistic consistuent quark model and the effective Lagrangian for the heavy flavor decays. We find that the contributions for the three Feynman diagrams are all important. With the full calculation, the decay branching ratios are of the order of 10^-5 for D_s^- \to \gamma \ell \bar\nu and 10^-6 for D^- \to \gamma \ell \bar\nu, respectively. These decays can be measured at B factories and future CLEO-C experiments to determine the decay constants f_D_s and f_D.


Introduction
The pure-leptonic decays of heavy mesons are useful to determine the meson decay constants, and they are also sensitive to new physics beyond the Standard Model(SM) [1]. But it is well known that the decays of D − (s) into light lepton pairs (see, Fig.1) are helicity suppressed by m 2 ℓ /m 2 D (s) : Here V CKM is the corresponding Cabibbo-Kobayashi-Maskawa matrix element. In the case of D − s (D − ) decay, it is V cs (V cd ). Fortunately the helicity suppression can be overcome by a photon radiated from the charged particles at the cost of the electromagnetic suppression with coupling constant α. It is possible that the radiative decays may be comparable or even larger than the corresponding pure leptonic decays [2].
Several years ago, D. Atwood et al. calculated B ± (D ± s ) → γℓν in a non-relativistic quark model [3], with a large branching ratio. In fact, they only considered one dominant diagram, neglecting other diagrams. They found the order of 10 −4 for the branching ratio of D ± s → γℓν decay.
Later on, Gregory P. Korchemsky et al. calculated these decays in perturbative QCD approach [4]. They gave larger branching ratios. And C. Q. Geng et al. did the calculation in the light front quark model [5]. Their branching ratios are rather smaller.
In this Letter, We will study the radiative leptonic decays D − (s) → γℓν (ℓ = e, µ) carefully at tree level, using the non-relativistic constituent quark model, similar to Ref. [3], but including all diagrams. In the following section, we will calculate the processes D − (s) → γℓν (ℓ = e, µ) in the framework of the constituent quark model (see, for example [6]). In the third section, we will compare the result with some of the previous calculations [3,4,5]. At last we will conclude the calculation briefly.
We begin with the quark diagram calculation of D − s → γℓν (ℓ = e, µ). There are four charged particle lines in Fig.1, which correspond to four Feynman diagrams contributing to the radiative decays D − s → γℓν (ℓ = e, µ) at tree level, as shown in Fig.2. However when the photon line is attached to the internal charged line of W boson such as Fig.2d, there is a suppression factor of m 2 c /m 2 W . Thus we neglect it for simplicity. To be consistent in the following calculation, we will always neglect the terms suppressed by the factor m 2 c /m 2 W .
The decay amplitudes corresponding to the other three diagrams are As mentioned in the Introduction, we will use the constituent quark model to reduce the amplitudes into the 'hadronic level'. In this simple model, both of the quark and anti-quark inside the meson move with the same velocity. Thus we have We use further the interpolating field technique [7] which relate the hadronic matrix elements to the decay constants of the mesons. The decay constant f P for a charged pseudoscalar meson is defined by [8]: The whole decay amplitude for D − s → γℓν (ℓ = e, µ) is derived from eqs.(2,3,5) by neglecting the terms suppressed by m l /m c In the D − s rest frame, the differential decay width [8] is Neglecting the mass of light leptons, we get the differential decay width: with Theŝ,t are defined asŝ = (p ℓ + p ν ) 2 ,t = (p ℓ + p γ ) 2 . Integrating eqn. (8) in phase space, we obtain the decay width Using α = 1/137, m c = 1.5 GeV, m Ds = 1.97 GeV, |V cs | = 0.974 [8], we get For the lifetime τ (D s ) = 0.5 × 10 −12 s [8], and the decay constant used as f Ds = 230MeV [3], the branching ratio is found to be 1.8 × 10 −5 . From eqs.(9,10), we can easily see that the decay width is sensitive to the decay constant f 2 Ds , and the constituent quark mass m c (or m s ). Any changes of the two input parameters, will result in a big change in the decay amplitude. Therefore the prediction of branching ratios remain the accuracy at the order of magnitude, unless we can precisely determinate the input parameters.
1 By neglecting terms proportional to m l /m c , we drop the infrared divergence terms, which should be canceled by the radiative corrections of the pure leptonic decay D − s → ℓν [9].
It is worthy of considering the differential spectrum for experimental purposes. Deriving from eqn. (8), we obtain with λ γ = E γ /m Ds . This result is the same as Ref. [3]. We show the photon energy spectrum in Fig.3 as the solid line. This is clearly distinct from the bremsstrahlung photon spectrum.
The lepton energy distributions are where λ ν = E ν /m Ds , λ ℓ = E ℓ /m Ds . We show the neutrino energy spectrum 1 Γ dΓ dλν , and the charged lepton (e, µ) energy spectrum 1 Γ dΓ dλ ℓ in Fig.3 as dashed and dash-dotted lines, respectively. Eqs.(13,14) are consistent with Ref. [3], if we consider only the diagram in Fig.2a with the photon connecting the strange quark line like the case in that paper.
The formulas above can be applied to the case of D − decay i.e. D − → γℓν (ℓ = e, µ), directly. We get the decay width easily: with Using m d = 0.37 GeV, m D − = 1.87 GeV, |V cd | = 0.22 [8], we get For τ (D − ) = 1.05×10 −12 s [8], the decay branching ratio is 4.6×10 −6 with the decay constant f D − = 230MeV [5]. Again, without the precise determination of the constituent quark mass m d , the decay branching ratio is only meaningful at the order of magnitude.
In the case of D − decay, the formulas for the differential spectra of photon and lepton energy distribution are the same as the D s decay, except replacing x s with x d in eqs. (12,13,14).
Their numerical results are shown in Fig.4 as solid, dashed and dash-dotted lines for γ, ν, ℓ respectively. From Fig.3 and Fig.4, we can see that the differential spectra of D ± s and D ± radiative leptonic decays are very similar. Only the endpoints of leptonic energy spectra are different.

Comparison with other calculations
In Ref. [3], D. Atwood et. al. made the calculation within the non-relativistic constituent quark model like us. As stated in the introduction part that they just considered the contribution of the emission of photon from the strange quark, i.e. Fig.2a, for they made an analogy with B − decay directly. After our careful calculation, we conclude that the contribution of the Feynman diagram where the photon is emitted from the initial light quark is dominant enough to neglect the other diagrams in the case of B − decay, but not for the case of D − s or D − . It can be seen at Table.1 that the contribution of the other two diagrams corresponding to Fig.2b and 2c must be considered because the interference among the three Feynman diagrams is large and destructive. That is the reason why their branching ratio of Br(D − s → γℓν)(ℓ = e, µ) decay [3] is about four times of ours.
Gregory P. Korchemsky et al. used the perturbative QCD method to calculate B and D meson radiative decays. Their result for D − decay is very large. In fact, the perturbative QCD approach [10] is good for the B meson decays since the energy release is very large there, but may not be good for the lighter D meson decays. In addition, out result is consistent with that of [5] within the light front quark model in the case of D − s → γeν. It is instructive to calculate with various models, and the accuracy of various models will be tested in future experiments. Table 1: Decay width with different diagrams and their relative size. Considering only the diagram where the photon is emitted from the initial light quark (Fig.2a), heavy quark ( Fig.2b) or lepton (Fig.2c), we get Γ a , Γ b , Γ c , respectively. And considering the three diagrams together, we get Γ a+b+c .

Summary
We have calculated D − (s) → γℓν (ℓ = e, µ) decay in non-relativistic constituent quark model. We included all the three Feynman diagrams, and found that none of them is small in D (s) decays. We obtained the decay branching ratios of D − s → γℓν , D − → γℓν(ℓ = e, µ) are of order 10 −5 and 10 −6 respectively. Such a branching ratio for the radiative leptonic decays can be measured in the two B factories and the future CLEO-C Experiments.
Eqs. (10,15) indicate that the decay rate of D − (s) → γℓν (ℓ = e, µ) is proportional to f 2 D (s) , so one can use it to determine the decay constant f D (s) . On the other hand, it is seen that these processes can also be used to test the |V cs | and |V cd | if f Ds and f D are known.