Radiative three-body D-meson decays in and beyond the standard model

We study radiative charm decays $D \to P_1 P_2 \gamma$, $P_{1,2}=\pi,K$ in QCD factorization at leading order and within heavy hadron chiral perturbation theory. Branching ratios including resonance contributions are around $\sim 10^{-3}$ for the Cabibbo-favored modes into $K \pi \gamma$ and $\sim 10^{-5}$ for the singly Cabibbo-suppressed modes into $\pi^+ \pi^- \gamma, K^+ K^- \gamma$, and thus in reach of the flavor factories BES III and Belle II. Dalitz plots and forward-backward asymmetries reveal significant differences between the two QCD frameworks; such observables are therefore ideally suited for a data-driven identification of relevant decay mechanisms in the standard-model dominated $D \to K \pi \gamma$ decays. This increases the potential to probe new physics with the $D \to \pi^+ \pi^- \gamma$ and $D \to K^+ K^- \gamma$ decays, which are sensitive to enhanced dipole operators. CP asymmetries are useful to test the SM and look for new physics in neutral $|\Delta C|=1$ transitions. Cuts in the Dalitz plot enhance the sensitivity to new physics due to the presence of both $s$- and $t,u$-channel intermediate resonances.

of the SM, such as CP and flavor symmetries, or flavor universality [6].
We perform a comprehensive study of theory tools for radiative charm decay amplitudes. A new result is the analysis of D → P 1 P 2 γ at leading order QCD factorization (QCDF), with the P 1 P 2 -form factor as a main ingredient. The framework is formally applicable for light and energetic (P 1 − P 2 ) systems. At the other end of the kinematic spectrum, for large (P 1 − P 2 ) invariant masses, we employ the soft-photon approximation. We also re-derive the heavy-hadron chiral perturbation theory (HHχPT) amplitudes for D → Kπγ decays put forward in Refs. [7,8], and provide results for the FCNC modes D → π + π − γ and D → K + K − γ. We find differences between our results and those in [7] which we detail in Appendix B 2.
We compare the predictions of the QCD methods, with the goal to validate and improve the theoretical description via the study of the SM dominated decays. Then, we work out the NP sensitivities of the FCNC modes D → ππγ and D → KKγ in several distributions and observables.
The paper is organized as follows: In Section II we introduce kinematics and distributions, and use QCD factorization methods (Section II B) and Low's theorem (Section II C) for predictions for small and large P P -invariant masses, respectively. In Section II D we work out the HHχPT amplitudes and Dalitz plots. We provide SM predictions for branching ratios and the forwardbackward asymmetries in all three approaches and compare them in Section III. In Section IV we analyze the maximal impact of BSM contributions on the differential branching ratios and the forward-backward asymmetries. New-physics signals in CP asymmetries are worked out in Section V.
We conclude in Section VI. Auxiliary information on parametric input parameters and form factors is provided in two appendices.

II. RADIATIVE THREE-BODY DECAYS IN QCD FRAMEWORKS
We review the kinematics of the radiative three-body decays D → P 1 P 2 γ in section II A. We then work out the SM predictions using QCD factorization methods in section II B, Low's theorem in section II C, and HHχPT in section II D.

A. Kinematics
The general Lorentz decomposition of the D(P ) → P 1 (p 1 )P 2 (p 2 )γ(k, * ) amplitude reads A(D → P 1 P 2 γ) = A − (s, t) [(p 1 · k)(p 2 · * ) − (p 2 · k)(p 1 · * )] + A + (s, t) µαβγ * µ p 1α p 2β k γ , with parity-even (A + ) and parity-odd (A − ) contributions. The four-momenta of the D, P 1 , P 2 and photon are denoted by P, p 1 , p 2 and k, respectively; the photon's polarization vector is * . Above, s = (p 1 + p 2 ) 2 and t = (p 2 + k) 2 refer to the squared invariant masses of the P 1 -P 2 and P 2 -γ systems, respectively. We denote the negatively charged meson or the K 0 by P 2 . Moreover, µαβγ is the totally antisymmetric Levi-Civita tensor; we use the convention 0123 = +1. The double differential decay rate is then given by where m D is the D-meson mass. We obtain The subscript L(R) refers to the left-(right-)handed polarization state of the photon, and where m 1 (m 2 ) denotes the mass of the P 1 (P 2 ) meson. The single differential distribution in the squared invariant di-meson mass is then given by and (m 1 + m 2 ) 2 ≤ s ≤ m 2 D .

B. QCD Factorization
Rare c → uγ processes can be described by the effective four-flavor Lagrangian [4] L eff = 4G F √ 2   q,q ∈{d,s} Here, G F is Fermi's constant and V ij are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The operators relevant to this work are given by O (q,q ) 1 = u L γ µ T a q L (q L γ µ T a c L ) , O (q,q ) 2 = u L γ µ q L (q L γ µ c L ) , where the subscripts L(R) denote left-(right-)handed quark fields, F µν is the photon field strength tensor, and T a are generators of SU (3) normalized to Tr{T a T b } = δ ab /2, respectively. Because of an efficient cancellation due to the Glashow-Iliopoulos-Maiani mechanism, only the four-quark are induced at the W -scale µ W and receive order-one coefficients at the scale µ c ∼ m c of the order of the charm-quark mass. At leading order in the strong coupling α s , the coefficients are given for The peculiar combination of Wilson coefficientsC arises in the weak annihilation amplitude (see below); note that an accidental numerical cancellation occurs in this combination, leading to a large scale uncertainty (see Table I). This effect is partially mitigated by higher-order QCD corrections which we do not take into account in this work; see, e.g., Ref. [4]. The tiny SM contributions to C 3−8 are a result of renormalization group running and finite threshold corrections at the bottom-mass scale, and can be neglected for the purpose of this work. For instance, the SM contribution of the electromagnetic dipole operator O 7 is strongly suppressed, |C eff 7 | O(0.001) at µ c = m c at next-to-next-to-leading order [6].
In this section we use QCDF methods [9][10][11] to calculate the leading weak annihilation (WA) contribution shown in Fig. 1. We obtain where Q u = 2/3 denotes the electric charge of the up-type quarks, and we decomposed P = v m D .
The nonperturbative parameter λ D ∼ Λ QCD is poorly known and thus source of large theoretical uncertainties. In the following we use λ D = 0.1 GeV [4]. For the final states π + π − γ and K + K − γ, the remaining form factors f P 1 P 2 (q,q ) (s) can be expressed in terms of the electromagnetic pion and kaon form factors [12]. For the final states π + K − γ and π 0 Kγ, we use the form factors extracted from τ − → ν τ K s π 0 decays [13] in combination with isospin relations. We obtain for the non-vanishing

form factors
More details about the form factors are given in appendix B 1. We recall that QCDF holds for light and energetic P 1 -P 2 systems. This limits the validity of the results to s 1.5 GeV 2 . The WA decay amplitudes are independent of t.

C. Soft photon approximation
Complementary to QCDF, we use Low's theorem [14] to estimate the decay amplitudes in the limit of soft photons. This approach holds for photon energies below m 2 P /E P [15], which results in s 2.3 GeV 2 for D → K + K − γ and s 3.4 GeV 2 for decays with a final-state pion. The amplitude is then given by [16] while A Low There is no such contribution to D → π 0K0 γ, since only neutral mesons are involved. The modulus of the D → P 1 P 2 amplitudes can be extracted from branching ratio data using where Γ D is the total width of the D meson. Using the parameters given in appendix A, we obtain A(D → π + π − ) = (4.62 ± 0.04) · 10 −7 GeV 2 , Low's theorem predicts that the differential decay rate behaves as [17] dΓ ds Consequently, there is a singularity at the boundary of the phase space. This corresponds to a vanishing photon energy in the D meson's rest frame. The tail of the singularity dominates the decay rate for small photon energies. We remove these events for integrated rates by cuts in the photon energy, as they are of known SM origin and hamper access to flavor and BSM dynamics.

D. HHχPT
As a third theory description we use the framework of heavy hadron chiral perturbation theory (HHχPT), which contains both the heavy quark and the SU (3) L × SU (3) R chiral symmetry. The effective Lagrangian was introduced in [18][19][20] and extended by light vector resonances by Casalbuoni et al. [21]. We follow the approach of Fajfer et al., who studied radiative two-body decays D → V γ [22,23] and Cabibbo allowed three-body decays D → K − π + γ [7] and D → K 0 π 0 γ [8] in this way.
The light mesons are described by 3 × 3 matrices where f f π is the pion decay constant and g v = 5.9 [24]. To write down the photon interaction with the light mesons in a simple way, we define two currents Here, the covariant derivative acting on u and u † is given by D µ u ( †) = ∂ µ u ( †) + ieB µ Qu ( †) , with the photon field B µ and the diagonal charge matrix Q = diag(2/3, −1/3, −1/3). The even-parity strong Lagrangian for light mesons is then given by [24] L where F µν (ρ) = ∂ µρν − ∂ νρµ + [ρ µ ,ρ ν ] denotes the field strength tensor of the vector resonances.
In general, a is a free parameter, which satisfies a = 2 in case of exact vector meson dominance (VMD). In VMD there is no direct vertex that connects two pseudoscalars and a photon. In this case, the photon couples to pseudoscalars via a virtual vector meson. Analogously, the matrix element P 1 P 2 |qγ µ (1 − γ 5 )q |0 also vanishes. However, we do not use the case of VMD and exact flavor symmetry, but allow for SU (3) breaking effects. Therefore, we choose to set a = 1 and replace the model coupling g v , decay constant f , and vector meson mass m V = a/2g v f in L light with the respective measured masses, decay constants and couplings g v = √ 2m 2 V /g V . They are defined by where j µ K ,K ,K ± ,Φ = qγ µ q and j µ ω,ρ = 1 √ 2 (uγ µ u ± dγ µ d). Here, q and η denote the vector meson's momentum and polarization vector, respectively. For our numerical evaluation we use where f V is the vector meson decay constant with mass dimension one. With these couplings the following V γ interactions arise [23] Instead of the VVP interactions generated by the odd-parity Lagrangian [25], we use effective VPγ and determine the effective coefficients g V P γ from experimental data [7,26] Γ(V → P γ) = α em m 3 The heavy pseudoscalar and vector mesons are represented by 4 × 4 matrices where P a annihilate (create) a heavy spin-one and spin-zero meson h a with quark flavor content cq a and velocity v, respectively. The annihilation operators are normalized as The heavy-meson Lagrangian reads where the covariant derivative is defined as the electric charge of the charm quark Q c = 2/3. The parameter g = 0.59 was determined by experimental data of strong D → Dπ decays [27,28]. The couplingβ seems to be very small and will be neglected [29]. The odd-parity Lagrangian for the heavy mesons is given by with σ µν = i 2 [γ µ , γ ν ]. The couplings λ and λ can be extracted from rations R with data [7]. The partonic weak currents can be expressed in terms of chiral currents as [22,30] (q a Q) µ where the ellipsis denotes higher-order terms in the chiral and heavy-quark expansions. The definition of the heavy-meson decay constants implies α = f h √ m h . The parameters α 1 and α 2 can be extracted Using the D → K form factors [31] we obtain α 1 = 0.188 GeV 1 2 and α 2 = 0.086 GeV 1 2 . The signs in (29) are due to the conventions in [31]. The weak tensor current is given by [32] where, again, the ellipsis denotes higher-order terms in the chiral and heavy-quark expansions.
The parity-even and parity-odd amplitudes are given in terms of four form factors Here, A and B belong to the charged current operator diagrams are shown in Fig. 18 and 19. The non-zero contributions are listed in Appendix B 2, where we also provide a list with differences between our results and those in Ref. [7]. We neglect the masses of the light mesons in the form factors, but consider them in the phase space. To enforce Low's theorem, we remove the bremsstrahlung contributions A 1,2 in (31) and add (12)

III. COMPARISON OF QCD FRAMEWORKS
In this section, we compare the predictions obtained using the different QCD methods in Section II.
We anticipate quantitative and qualitative differences between QCDF to leading order and HHχPT.
First, we study differential and integrated branching ratios in Section III A. In Section III B we propose to utilze a forward-backward asymmetry, defined below in Eq. (32), to help disentangling the resonance contributions to the branching ratios. This subsequently improves the NP sensitivity of the D → P + P − γ decays. We consider the U-spin link, exploited already for polarization-asymmetries in radiative charm decays [33], in Section III C.
(lower left) and D → K + K − γ (lower right) based on HHχPT at µ c = m c .

A. Branching ratios
The branching ratios for the various decay modes, obtained from QCDF (blue bands), HHχPT (green bands) and Low's theorem (red dashed lines), are shown in Fig. 3. The width of the bands represents the theoretical uncertainty due to the µ c dependence of the Wilson coefficients.
The shape of the QCDF results is mainly given by the P 1 − P 2 form factors and their resonance structure. For the D → P + 1 P − 2 γ decays, the high-s regions of the HHχPT predictions are dominated by bremsstrahlung effects. Since we have replaced the model's own bremsstrahlung contributions by those of Low's theorem, the results approach each other asymptotically towards the large-s endpoint.
Without this substitution, the differential branching ratios from HHχPT in this region would be about one order of magnitude larger. For lower s, the impact of the resonances becomes visible.
In the soft photon approximation the photon couples directly to the mesons. Therefore, there is no such contribution for the D → π 0 K 0 γ decay. Its distribution is dominated by the ω resonance which has a significant branching ratio to π 0 γ; this is manifest in the Dalitz plot in Fig. 2.
QCDF is applicable for s 1.5 GeV 2 ; to enable sensible comparison we also provide HHχPT branching ratios with this cut. Also given are HHχPT predictions for E γ ≥ 0.1 GeV, see text for details. The QCDF branching ratios are obtained for Apart from the K * , ρ, and φ peaks, the shapes of the differential branching ratios differ significantly between QCDF and HHχPT, due to the t and u-channel resonance contributions in the latter. This is shown in the Dalitz plot in Fig. 2.
In Table I we give the SM branching ratios for the four decay modes. We employ phase space cuts s ≤ 1.5 GeV 2 , the region of applicability of QCDF, or E γ ≥ 0.1 GeV, corresponding to s ≤ 3.1 GeV 2 , to avoid the soft photon pole. Here, E γ = (m 2 D − s)/(2m D ) is the photon energy in the D meson's rest frame. Applying the same cuts in both cases, the HHχPT branching ratios are generally larger than the QCDF ones, except for the D → K + K − γ mode, where they are of comparable size.
We recall that SM branching ratios within leading order QCDF are proportional to (1/λ D ) 2 .
Since λ D is of the order of Λ QCD and we employ a rather low value λ D = 0.1 GeV [4], the values in Table I should be regarded as maximal branching ratios. The large uncertainty of these values arises from the residual scale dependence of the Wilson coefficientC (9). A measurement of the branching ratios of the SM-like modes D → Kπγ thus provides an experimentally extracted value ofC/λ D .
Color-allowed modes feature Wilson coefficients with significantly smaller scale uncertainty, and allow for a cleaner, direct probe of λ D [4].
Here, θ 2γ is the angle between P 2 and the photon in the P 1 − P 2 center-of-mass frame. In Fig. 4 we  are also shown without or only with individual resonance contributions. The (P 1 P 2 ) res resonances contribute to A FB only via interference terms, since the corresponding form factors depend only on s. For D → π + π − γ and D → K + K − γ the diagrams of the neutral current operator, which contain (P 1 γ) res and (P 2 γ) res resonances, give the same contribution to the amplitude in the forward and backward region of the phase space. For P 1 = P 2 this symmetry does not exist. In case of the charged current operator, these resonances contribute in different ways to the forward and backward region due to the asymmetric factorization of the diagrams B 3 (B20), (B23), (B26). This effect is primarily responsible for the shape of A FB in D → π + π − γ and D → K + K − γ decays.
A FB (D → π 0 K 0 γ) is, like the differential branching ratio shown in Fig. 2, dominated by the ω resonance.
Since the WA form factors are only dependent on s, the SM forward-backward asymmetry vanishes to leading order QCDF. Therefore, we add contributions from t and u-channel resonances using a phenomenological approach. To this end, we combine D → V P amplitudes with the effective V P γ coupling from equation (22). We obtain where the first (second) term in (33) corresponds to the left (right) diagram in Fig. 5. The amplitude for the final state π 0 K 0 γ can be obtained from Eq (33) by substituting C 2 − 1/6C 1 → C 2 /2, m 1 → m 2 , and p 1 ↔ p 2 , and multiplying by the factor −1/ √ 2. The D → P and D → V transition form factors are taken from Ref. [31].

C. The U-spin link
We further investigate the U-spin link between the SM-dominated mode D → K − π + γ and the BSM-probes D → π + π − γ and D → K + K − γ. In practise, a measurement of B(D → K − π + γ) can provide a data-driven SM prediction for the branching ratios of the FCNC decays. The method is phenomenological and serves, in the case of branching ratios, as an order-of-magnitude estimate. The U-spin approximation is expected to yield better results in ratios of observables (which arise already at lowest order in the U-spin limit), such that overall systematics drops out. Useful applications have been made for polarization asymmetries in D → V γ decays [33]. However, three-body radiative decays are considerably more complicated due to the intermediate resonances, and we do not pursue the U-spin link for the forward-backward or CP asymmetries.
A comparison between |V us | 2 /|V ud | 2 dB(D → K − π + γ)/ds with dB(D → K + K − γ)/ds and |V cd | 2 /|V cs | 2 dB(D → K − π + γ)/ds with dB(D → π + π − γ)/ds is shown in Fig. 6. For s 1.  Figure 6: The SM predictions for the differential branching ratios of the decays D → π + π − γ (left) and D → K + K − γ (right) from a direct QCDF computation (blue bands in upper plots), HHχPT computations (green bands in lower plots) and from the D → K − π + γ distribution multiplied by |V cd /V cs | 2 and |V us /V ud | 2 , respectively (red bands). The prediction for the SM-like mode D → K − π + γ in this figure is from the respective models but could be taken from data.

IV. BSM ANALYSIS
BSM physics can significantly increase the Wilson coefficients contributing to c → uγ transitions.
Examples are supersymmetric models with flavor mixing and chirally enhanced gluino loops, or leptoquarks, see Ref. [4] for details. In the following we work out BSM spectra and phenomenology in a model-independent way. Experimental data obtained from D → ρ 0 γ decays provide modelindependent constraints [6,34] These values are in agreement with recent studies of D → πll decays [35]. In Section V A we discuss the implications of CP asymmetries in hadronic charm decays that can lead to constraints on the imaginary parts of the dipole operators.
The D → P 1 P 2 matrix elements of the tensor currents can be parameterized as with the form factors a , b , c , h given in App. B 2. The form factors depend on s and t and satisfy The BSM amplitudes are then obtained as In Figs. 7 and 8 we show differential branching ratios for the FCNC modes based on QCDF and HHχPT, respectively, both in the SM (blue) and in different BSM scenarios. One of the BSM coefficients, C 7 or C 7 , is set to zero while the other one is taken to saturate the limit (37) with CP-phases 0, ±π/2, π. The same conclusions are drawn for both QCD approaches: the D → K + K − γ branching ratio is insensitive to NP in the dipole operators. In particular, the benchmarks for O 7 and the SM prediction are almost identical. For O 7 small deviations occur directly beyond the φ peak.
On the other hand, BSM contributions can increase the differential branching ratio of D → π + π − γ by up to one order of magnitude around the ρ peak. However, due to the intrinsic uncertainties from the Breit-Wigner contributions around the resonance peaks it is difficult to actually claim sensitivity to NP. This is frequently the case in D physics for simple observables such as branching ratios. The NP sensitivity is higher in observables involving ratios, such as CP asymmetries, discussed in the next section.  Figure 7: Comparison of QCDF-based SM predictions of differential branching ratios for D → π + π − γ (upper plots) and D → K + K − γ (lower plots) within different BSM scenarios. One BSM coefficient is set to zero while the other one exhausts the limit (37) with CP-phase 0, ±π/2, π.
The NP impact on A FB is sizable, see Fig. 9 for the HHχPT predictions. However, due to the complicated interplay of s-, tand u-channel resonances further study in SM-like D → Kπγ decays is suggested to understand the decay dynamics before drawing firm conclusions within NP. Since the form factors depend on s and t, the pure BSM contributions (40) induce a forward-backward asymmetry within QCDF, whereas it vanishes in the SM (see Fig. 10).

V. CP VIOLATION
Another observable that offers the possibility to test for BSM physics is the single-or doubledifferential CP asymmetry. It is defined, respectively, by  Here, Γ refers to the decay rate of the CP-conjugated mode. Within the SM, D → K + K − γ is the only decay that contains contributions with different weak phases and thus the only decay mode with a nonvanishing CP asymmetry. A maximum of A SM CP (s) 1.4 · 10 −4 located around the φ peak is predicted by QCDF. Since the φ is a narrow resonance, the CP asymmetry decreases rapidly with increasing s. BSM contributions can contain further strong and weak phases and thus significantly increase the CP asymmetry. In Fig. 11 we show the predictions for the CP asymmetries within the SM and for several different BSM scenarios, based on QCDF. We assign a non-zero value to one of the BSM coefficients and set the weak phase to φ w = ±π/2. The BSM CP asymmetries A CP (s) can, in principle, reach O(1) values. Constraints can arise from data on CP asymmetries in hadronic decays; these are further discussed in Section V A. We emphasize that A CP depends on cuts used in the normalization Γ +Γ. In Fig. 11 we include the contributions up to s = 1.5 GeV 2 .    HHχPT predicts a SM CP asymmetry A SM CP (s) 0.7·10 −4 for the D → K + K − γ decay. In Fig. 12 we show the same BSM benchmarks as before, employing HHχPT. We performed a cut s ≤ 2 GeV to avoid large bremsstrahlung effects in the normalization, which would artificially suppress A CP .
Still, the CP asymmetries obtained using HHχPT are smaller than those using QCDF, since a larger part of the phase space is included in the normalization.
For D → π + π − γ, the contributions of A − and A + to the CP asymmetries are of roughly the same size. Therefore, the relative signs of the dipole Wilson coefficients in (40) results in a constructive increase (for C 7 ) and a cancellation (for C 7 ), respectively, of the CP asymmetry. For the D → K + K − γ mode, the φ resonance contributes only to A + . Therefore, in this case the CP asymmetry is dominated by the parity-even amplitude. In order to get additional strong phases and thus an increase of the CP asymmetry, one could consider further heavy vector resonances such as the φ(1680). Intermediate scalar particles like f 0 (1710) [36] would also add additional strong phases.
We remark that A CP can change its sign in dependence of s; therefore, binning is required to avoid cancellations. A CP is very small beyond the (P 1 P 2 ) res peak due to the cancellation of the (P 1 γ) res and (P 2 γ) res contributions upon integration over t. To avoid this cancellation one could use the sand t-dependent CP asymmetry A CP (s, t) as shown in Fig. 13. Note that part of the resonance contribution to the asymmetry is removed by the bremsstrahlung cut.  Figure 13: Dalitz plot of A CP (s, t) for D → π + π − γ (upper plots) and D → K + K − γ decays (lower plots) based on HHχPT. We have set one BSM coefficient, C 7 or C 7 , to 0 and the other one to 0.1, with weak phase φ w = π/2. We employed a cut s ≤ 2 GeV 2 to avoid large bremsstrahlung contributions in the normalization.
and G µν denotes the chromomagnetic field strength tensor. We do not consider contributions from O ( ) 8 to the matrix element of D → P P γ decays, which is beyond the scope of this work. The corresponding contributions for the D → V γ decays have been worked out in Ref. [4].
The QCD renormalization-group evolution connects the electromagnetic and the chromomagnetic dipole operators at different scales. To leading order we find the following relation [4], which is valid to roughly 20% if Λ, the scale of NP, lies within 1-10 TeV. It follows that CP asymmetries for radiative decays are related to hadronic decays, a connection discussed in [37] in . The latter is measured by LHCb, [38], and implies ∆A NP CP ∼ Im(C 8 − C 8 ) sin δ 2 · 10 −3 for NP from dipole operators, with a strong phase difference δ and Wilson coefficients evaluated at µ = m c . For The BSM CP asymmetries scale linearly with Im C ( ) 7 . We checked explicitly that the CP asymmetries for Im C ( ) 7 2 · 10 −3 agree, up to an overall suppression factor of 50, with those shown in Fig. 13 which are based on Im C ( ) 7 0.1, and are therefore not shown.
Note that the ∆A CP constraint can be eased with a strong phase suppression. In general, it can be escaped in the presence of different sources of BSM CP violation in the hadronic amplitudes.
Yet, our analysis has shown that even with small CP violation in the dipole couplings sizable NP enhancements can occur.

VI. CONCLUSIONS
We worked out predictions for D → P P γ decay rates and asymmetries in QCDF and in HHχPT.
The D → π + π − γ and D → K + K − γ decays are sensitive to BSM physics, while D → Kπγ decays are SM-like and serve as "standard candles". Therefore, a future measurement of the D → Kπγ decay spectra can diagnose the performance of the QCD tools. The forward-backward asymmetry (32) is particularly useful as it vanishes for amplitudes without tor u-channel dependence; this happens, for instance, in leading-order QCDF. On the other hand, tor u-channel resonances are included within HHχPT, and give rise to finite interference patterns, shown in Fig. 4. Within QCDF, the value ofC/λ D can be extracted from the branching ratio.
While branching ratios of D → π + π − γ can be affected by NP, these effects will be difficult to discern due to the large uncertainties. On the other hand, the SM can be cleanly probed with CP asymmetries in the D → π + π − γ and D → K + K − γ decays, which can be sizable, see Figs. 11 and 12. We stress that the sensitivity of the CP asymmetries is maximized by performing a Dalitz analysis or applying suitable cuts in t (see Fig. 13), as otherwise large cancellations occur. Values of the CP asymmetries depend strongly on the cut in s employed to remove the bremsstrahlung contribution. The latter is SM-like and dominates the branching ratios for small photon energies.
The forward-backward asymmetries also offer SM tests, see Fig. 9, but requires prior consolidation of resonance effects.
with the electromagnetic current (B2) In the isospin symmetry limit, only the I = 1 current contributes to F em π , which reads [12] F em where the coefficients c n are given by

(B6)
F em π is shown in Figure 15. The electromagnetic kaon form factor F em K + , defined as is taken from [12] and shown in Figure 16. It can be decomposed into an isospin-one component K + , F s K + , with ω and φ contributions, respectively, The requisite parameters are given by The Kπ − form factors are defined as with s Kπ = (m K + m π ) 2 . The phase δ Kπ 1 (s) is extracted from a two resonance model [13] f Kπ − where The functionH Kπ is a χPT loop integral function [49] H explicit expressions for M r (s) and L(s) can be found in chapter 8 of Ref. [50]: The renormalization scale µ is set to the physical resonance mass µ = m K [13]. The resonance masses and width parameters are unphysical fitting parameters. They are obtained as [13]   (right) versus s in the two resonance models as well as in the dispersive description. The form factor is extracted from τ − → ν τ K s π − decays [13]. ForK 0 π 0 and K + π − , we use isospin relations (11). Figure 18: Feynman diagrams for the D → π + K − γ decay, which contribute to the parity-even form factors A and E. The diagrams for the decays D → π + π − γ and D → K + K − γ are obtained by adjusting the flavors.
We have added the diagrams E 1,2 and E 2,2 (see [7]) to make the amplitude E gauge invariant for any choice of a. Additionally, for each of the diagrams A 1,1 , A 1,2 , A 1,3 , A 2,2 , A 2,3 , A 2,4 , E 1,1 , E 1,2 , E 1,3 , E 2,1 and E 2,3 there is another one where the photon is coupled via a vector meson. Figure 19: Feynman diagrams for the D → π + K − γ decay, which contribute to the parity-odd form factors B and D. The diagrams for the decays D → π + π − γ and D → K + K − γ are obtained by adjusting the flavors.
c. Differences with respect to [7] In the following, we list some differences between our results and those obtained in Ref. [7].
7. We believe that there are diagrams that have not been shown in Ref. [7]: For each of the diagrams A 0 1,1 , A 0 1,2 , A 0 1,3 , A 0 2,2 , A 0 2,3 , A 0 2,4 , C 0 1,1 , C 0 1,2 , C 0 1,3 and C 0 1,4 there is another one in which the photon couples via a vector meson. Moreover, we find two additional diagrams for C 0 . The first one is the same diagram as A 0 2,2 , but with a different factorization. The second is another diagram with a V → P P γ vertex. Only with these two additional diagrams we obtain an expression that is gauge invariant for any value of a. However, we obtain C 0 = 0, as in Ref. [7]. 8. We reproduce A + 1 , but for A 0 1 we get an expression ∼ (q · k) − M (v · k + v · q).

9.
We have an extra factor of 2 in D 0 3 .
10. We obtain a relative minus sign for each vector meson in a diagram; however, we get the same relative signs for R 0/+ γ as given in Eqs. (24) and (25) [22].