Rare radiative charm decays within the standard model and beyond

We present standard model (SM) estimates for exclusive c → uγ processes in heavy quark and hybrid frameworks. Measured branching ratios ℬD0→ϕK¯∗0γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{B}}\left({D}^0\to\ \left(\phi, {\overline{K}}^{\ast 0}\right)\gamma \right) $$\end{document} are at or somewhat exceeding the upper range of the SM and suggest slow convergence of the 1/mD, αs-expansion. Model-independent constraints on |ΔC| = | ΔU | = 1 dipole operators from ℬ(D0 → ρ0γ) data are obtained. Predictions and implications for leptoquark models are worked out. While branching ratios are SM-like CP asymmetries ≲ 10% can be induced. In SUSY deviations from the SM can be even larger with CP asymmetries of O(0.1). If Λc-baryons are produced polarized, such as at the Z-pole, an angular asymmetry in Λc → pγ decays can be studied that is sensitive to chirality-flipped contributions.


Introduction
A multitude of radiative charm decays is accessible at current and future high luminosity flavor facilities [1,2]. In anticipation of the new data we revisit opportunities to test the standard model (SM) with c → uγ transitions [3][4][5][6][7][8][9][10], complementing studies with dileptons, e.g., [11][12][13]. To estimate the beyond the standard model (BSM) reach we detail and evaluate exclusive D (s) → V γ decay amplitudes, where V is a light vector meson. We employ two frameworks, one based on the heavy quark expansion and QCD, adopting expressions from b-physics [14], and a hybrid phenomenological one, updating [5,6]. The latter combines chiral perturbation and heavy quark effective theory and vector meson dominance (VMD). Both frameworks have considerable systematic uncertainties, leaving individual charm branching ratios without clear-cut interpretation unless the deviation JHEP08(2017)091 from the SM becomes somewhat obvious. On the other hand, considering several observables, correlations can shed light on hadronic parameters or on the electroweak model [7]. The interpretation of asymmetries is much easier, as (approximate) symmetries of the SM make them negligible compared to the experimental precision for a while. In particular, we discuss implications of the recent measurements by Belle [15] B(D 0 → ρ 0 γ) = (1.77 ± 0.30 ± 0.07) · 10 −5 , A CP (D 0 → ρ 0 γ) = 0.056 ± 0.152 ± 0.006 , (1.1) where the CP asymmetry A CP is defined as 1 We compare data (1.1) to the SM predictions and derive model-independent constraints on BSM couplings. We further discuss two specific BSM scenarios, leptoquark models and the minimal supersymmetric standard model with flavor mixing (SUSY). For the former we point out that large logarithms from the leading 1-loop diagrams with leptons and leptoquarks require resummation. The outcome is numerically of relevance for the interpretation of radiative charm decays.
We further obtain analytical expressions for the contributions from the QCD-penguin operators to the effective dipole coefficient at 2-loop QCD. This extends the description of radiative and semileptonic |∆C| = |∆U | = 1 processes at this order [3,11,17].
While one expects the heavy quark and α s -expansion to perform worse than in b-physics an actual quantitative evaluation of the individual contributions in radiative charm decays has not been done to date. Our motivation is to fill this gap and detail the expansion's performance when compared to the hybrid model, and to data. In view of the importance of charm for probing flavor in and beyond the SM seeking after opportunities for any, possibly data-driven improvement of the theory-description is worthwhile.
The organization of this paper is as follows: in section 2 we calculate weak annihilation and hard scattering contributions to D → V γ decay amplitudes. In section 3 we present SM predictions for branching ratios and CP asymmetries in this approach and in the hybrid model. We present model-independent constraints on BSM physics and look into leptoquark models and SUSY within the mass insertion approximation in section 4. Section 5 is on Λ c → pγ decays and the testability of a polarized Λ c -induced angular asymmetry at future colliders [18,19]. In section 6 we summarize. In appendix A and B we give the numerical input and D → V form factors used in our analysis. Amplitudes in the hybrid model are provided in appendix C. Details on the 2-loop contribution from QCD-penguin operators are given in appendix D. 1 The CP asymmetry of D 0 → ρ 0 γ is mostly direct, analogous to the time-integrated CP asymmetry in D 0 → K + K − [16]. We thank Alan Schwartz for providing us with this information. In this work, we refer to ACP as the direct CP asymmetry, neglecting the small indirect contribution.

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2 D → V γ in effective theory framework The effective weak Lagrangian and SM Wilson coefficients are discussed in section 2.1. We work out and provide a detailed breakdown of the individual contributions to D → V γ amplitudes in the heavy-quark approach. We work out weak annihilation and hard gluon exchange corrections in section 2.2, with contributions from the gluon dipole operator given in section 2.3. In section 2.4 we consider weak annihilation induced modes.

Generalities
The effective c → uγ weak Lagrangian can be written as [11] L weak where G F is the Fermi constant, V ij are CKM matrix elements and the operators read where F µν , G a µν , a = 1, . . . , 8 denote the electromagnetic, gluonic field strength tensor, respectively, and T a are the generators of QCD. In the following all Wilson coefficients are understood as evaluated at the charm scale µ c of the order of the charm mass m c , and µ c = 1.27 GeV unless otherwise explicitly stated.
For the SM Wilson coefficients of Q 1,2 and the effective coefficient of the chromomagnetic dipole operator at leading order in α s one obtains [11,17], respectively, 2 ∈ [1.14, 1.06] , C is strongly GIM suppressed in the SM and negligible therein. C (0) 1 and the color suppressed coefficient of the weak annihilation contribution introduced in section 2.2 ,  Figure 1. Diagrams driven by C eff 7 , weak annihilation and hard spectator interaction. The crosses indicate photon emission. Diagrams not shown are additionally power suppressed.
The effective coefficient of Q 7 including the matrix elements of Q 1−6 at two-loop QCD, see [3,11,17] and appendix D for details, is in the range Here, we give contributions to the imaginary parts separately: the ones with subscript "s" correspond to strong phases, whereas the ones with label "CKM" stem from the weak phases in the CKM matrix. As a new ingredient we provide in this work the 2-loop QCD matrix element of Q 3−6 , see appendix D for details. Numerically, C eff 7 Q 3−6 10 −6 , that is, negligible due to small SM Wilson coefficients C 3−6 and the GIM suppression.
The D → V γ decay rate can be written as [4] where the parity conserving (PC) and parity violating (PV) amplitudes read Here, m D and m V denote the mass of the D and the vector meson, respectively, and T = T 1 (0) = T 2 (0) is a D → V tensor form factor, see appendix B for details. We stress that the dominant SM contribution to D (s) → V γ branching ratios is independent of T . Furthermore,

Corrections
In this section we calculate the hard spectator interaction (HSI) and weak annihilation (WA) contributions shown in figure 1 as corrections to the leftmost diagram in the figure.

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The leading (∼ α 1 s (Λ QCD /m c ) 0 ) hard spectator interaction within QCD factorization adopted from b-physics [14] (also [20][21][22]) can be written as (2.9) where we consistently use C (0) 1,2,8 at leading order in α s due to additional non-factorizable diagrams at higher order and µ h ∼ Λ QCD m c . Furthermore, is driven by V * cs V us . The transverse distribution at leading twist is to first order in Gegenbauer polynomials Numerical input on the Gegenbauer moments a V ⊥ 1,2 is given in appendix A. The parameter λ D is defined as that is the first negative moment of the leading twist distribution amplitude Φ D of the light-cone momentum fraction ξ of the spectator quark within the D-meson. In b-physics, the first negative moment of the B-meson light-cone distribution amplitude, λ HQET B > 0.172 GeV at 90% C.L. [23], a positive light-cone wave function yields λ HQET B ≤ 4/3Λ [24] and by means of light-cone sum rules (LCSR) λ QCD B Λ [25,26] at one-loop QCD [27]. We use λ D ∼ Λ QCD ∼ O(0.1 GeV).
Taking µ h = 1 GeV, varying the Gegenbauer moments and decay constants (but not the form factor T as it cancels in the amplitude) we find

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We neglect isospin breaking in the Gegenbauer moments of the ρ. Contributions induced by Q ( ) 8 are discussed in section 2.3. The leading (∼ α 0 s (Λ QCD /m c ) 1 ) weak annihilation contribution to D 0 → (ρ 0 , ω)γ, D + → ρ + γ and D s → K * + γ can be inferred from b-physics [14,28]. We obtain where Q u = 2/3, Q d = −1/3 and we consistently use C 1,2 at leading order in α s . We neglect weak annihilation contributions from Q 3−6 as the corresponding Wilson coefficients in the SM are strongly GIM suppressed. The minus sign for ρ 0 is due to isospin.
Varying the decay constants and Note that non-factorizable power corrections (inducing A 7 ) could in principle be calculated with LCSR, see, e.g., [29] and that non-local corrections to weak annihilation by means of QCD sum rules are additionally power suppressed [30].
To summarize, we observe the following hierarchies among the SM contributions to A 7 The leading SM uncertainties are therefore those stemming from the WA-amplitudes, that is, the µ c -scale and λ D uncertainties, followed by the parameters entering HSI-amplitudes, i.e., Gegenbauer moments, decay constants and the µ h -scale. The latter we fixed for simplicity.
Contributions to A 7 arise in the SM from a Q 2 -induced quark loop with a soft gluon as a power correction [31] C (c→uγg) 7 which is O(10 −4 ) if the expansion coefficients of f (c→uγg) in m 2 q /m 2 c are order one. Note that the c → uγg process induces as well a contribution to A 7 , and that the Q 1 -induced JHEP08(2017)091 quark loop is additionally color suppressed. Note also that f (c→uγg) could in principle be calculated with LCSR, see e.g., [29], yet α s -corrections vanish at leading twist in the limit of massless quarks in the V -meson [32]. To be specific, and in absence of further calculations, we limit the size of the chirality-flipped SM amplitudes in our numerical analysis as and take the structure of the weak phases as in (2.17) into account.

Contributions from Q ( ) 8
We detail here the contributions from Q ( ) 8 to c → uγ modes. While in the SM they are negligibly small they can be relevant in BSM scenarios.
Numerically, we find for the Q ( ) 8 -induced hard spectator interaction of eq. (2.9) Note that QCD factorization breaks down at subleading power for Q 8 hard spectator interaction due to a logarithmic singularity for a soft spectator quark [30].

Weak annihilation induced modes
The contributions to A 7 of the weak annihilation induced decays D 0 → (φ,K * 0 , K * 0 )γ, D + → K * + γ and D s → ρ + γ are obtained as follows (2.23) Here we made the flavor structure of the Wilson coefficients explicit, however, since QCD is flavor symmetric, use C 1/2 . While the form factor T is process-dependent, it cancels together with m c in the decay amplitude. Numerically, GeV/(m c T ) ∼ 1 . Varying the decay constants and For D 0 → φγ additional contributions to the decay amplitude can arise, induced by dd + uū-admixture in the φ or rescattering [4]. Such effects can be parametrized by y as follows (2.25) To estimate A CP we made CKM factors explicit except for the C eff 7 -induced term which can receive large BSM CP violating phases. The amplitudes correspond, in order of appearance, to C D 0 →φγ 7 , and the three contributions eqs. (2.14), (2.13) and (2.5) to D 0 → ρ 0 γ. Note the minus signs due do the SU (3)-composition. One obtains, model-independently, (2.26)

SM phenomenology
We provide SM predictions for various D (s) → V γ modes and compare to existing data. In addition to the QCD-based approach of the previous section we present branching ratios JHEP08(2017)091 Uncertainties not available. We take a1 = 1.3 and a2 = −0.55 [34]. Table 1. Branching ratios of D → V γ within the SM at two-loop QCD, from the hard spectator interaction plus weak annihilation and the hybrid approach. We vary the form factors, decay constants, lifetimes, Gegenbauer moments, relative strong phases and The branching ratios from the hard spectator interaction plus weak annihilation scale as (0.1 GeV)/λ D ) 2 . Also given are data by the Belle [15] and the CLEO (at 90% CL) [35] collaborations as well as SM predictions from [5,6], via pole diagrams and VMD [8] and QCD sum rules [9]. † Statistical and systematic uncertainties are added in quadrature.
in the phenomenological approach of [5,6]. This model is a hybrid of factorization, heavy quark effective and chiral perturbation theory, where the SU (3) flavor symmetry is broken by measured parameters. Compared to [5,6] we rewrite the amplitudes in terms of newly measured parameters and vary (updated) parameters within uncertainties. Analytical expressions for the D → V γ amplitudes are provided in appendix C. The hierarchies of the various D → V γ amplitudes are predominantly set by CKM factors and large-N C counting, taken care of in both the heavy quark and the hybrid frameworks.
The SM branching ratios and presently available data are given in table 1. We learn the following: the branching ratios induced by hard spectator interaction plus weak annihilation are typically smaller than (similar to) the ones obtained in the hybrid approach for neutral (charged) c → uγ modes. The branching ratio from two-loop QCD eq. (2.5) is subleading in each case. The branching ratios in the hybrid approach cover the ranges previously obtained in [5,6,8,9]. The measured D 0 → ρ 0 γ branching ratio is somewhat above the SM prediction in the hybrid model.
The uncertainties in the hybrid model are dominated by the relative strong phases, followed by the phenomenological fit coefficients a 1 = 1.3 ± 0.1, a 2 = −0.55 ± 0.1 [34] (also [37,38]). The upper orange curves are for D + → ρ + γ and the lower blue curves are for D 0 → ρ 0 γ. The solid curves represent the bands of two-loop QCD and hard spectator interaction plus weak annihilation, the dashed lines are the maximal predictions in the hybrid approach and the cyan band depicts the measured branching ratio [15]. We vary the form factors, decay constants, lifetimes, Gegenbauer moments, relative strong phases and
The branching ratios of D 0 → (φ,K * 0 , K * 0 )γ, D + → K * + γ and D s → ρ + γ are given in table 2. The measurements by Belle [15] and BaBar [39] of B(D 0 →K * 0 γ) differ by 2.2σ, yet both are in the range of the hybrid model predictions. Interpreted in the QCD framework to the order we are working, B(D 0 → (K * 0 , φ)γ) data require a low value of λ D JHEP08(2017)091 The CP asymmetry for D 0 → ρ 0 γ within the SM in the heavy quark-based approach is shown in figure 3 as a function of the SM D 0 → ρ 0 γ branching ratio. Within the SM |A CP | 2 · 10 −2 , if the branching ratio is 10 −9 and |A CP | 2 · 10 −3 , if B(D 0 → ρ 0 γ) 10 −6 , e.g., for λ D 0.3 GeV, and as measured (1.1) assuming the SM. A SM CP calculated in the hybrid model is 10 −3 , and vanishes in the SU(3)-limit. The SM CP asymmetry for D 0 → ωγ is very similar to A CP (D 0 → ρ 0 γ); the CP asymmetries in the SM for the charged decays D + → ρ + γ and D s → K * + γ are 2 · 10 −3 and 3 · 10 −4 in the heavy quark-based approach and the hybrid model, respectively. The enhancement of A CP possible together with very low values of the branching ratio for D 0 decays originates from cancellations between different amplitudes.
The SM CP asymmetries for the pure WA modes D 0 → (K * 0 , K * 0 )γ, D + → K * + γ and D s → ρ + γ vanish due to a single weak phase cf. eq. (2.23) and appendix C. The decay D 0 → φγ is special as it can receive contributions at a fraction y similar to D 0 → ρ 0 γ decays, and has therefore a finite CP asymmetry, estimated in equation (2.26). Taking into account a percent level uū + dd content in the φ [36] values of A CP up to O(10 −4 )

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in the SM and up to O(10 −3 ) in BSM models can arise in D 0 → φγ decays. Effects from rescattering at the φ-mass are roughly y 0.1, hence corresponding CP asymmetries can reach O(10 −3 ) in the SM and O(10 −2 ) in BSM scenarios. The following asymmetries have been measured [15], (3.1) A CP (D 0 → φγ) exhibits presently a mild tension with zero.
We stress that in our numerical evaluations we vary all relative strong (unknown) phases, including those between the WA+HSI contributions and the perturbative ones. In view of the appreciable uncertainties we refrain from putting an exact upper limit on the SM-induced CP asymmetries, but consider, to be specific, CP asymmetries at percent-level and higher as an indicator of BSM physics, consistent with [4]. This is supported by the large measured branching fractions, which indicate unsuppressed WA topologies. For the FCNC decays this suggests no large cancellations between the contributions in eq. (2.16), allowing for possible additional suppressions of CP asymmetries beyond CKM factors.

D → V γ beyond the Standard Model
In section 4.1 we work out model-independent constraints on A   Constraints from B(D + → π + µ + µ − ) data are similar [11]. These constraints prohibit that decays D + → ρ + γ and D s → K * + γ are dominated by a BSM dipole contribution. Still, a sizable δC  [7]. BSM-induced CP asymmetries can reach O(0.1). Hence, current data on A CP (D 0 → ρ 0 γ) are too uncertain to provide further constraints. There is essentially no BSM pattern for CP asymmetries apart from their sign. Turning this around, it is possible to have a sizable value of A CP in one mode but a small one in another.
To illustrate the impact of improved measurements of the D 0 → ρ 0 γ branching ratio and CP asymmetry, we assume hypothetical data with a factor four reduced statistical uncertainty of the current measurements with central values kept [15], that is,   1). In the hybrid model the unknown strong phases prohibit a meaningful calculation of A 7 /A 7 in the SM [6]. We therefore do not employ constraints from eq. (2.18) in this model.
We consider now the impact of the chromomagnetic dipole operator. The CP asymmetry induced by the matrix element of Q ( ) 8 estimated within LCSR reads [40] which, together with data eq. (1.1), yields the constraint |Im[δC [40,41], where ∆A CP = −0.00134 ± 0.00070 [42]. To escape a potentially strong bound on Im[δC 8 − δC 8 ] at permille level requires suppression by the unknown strong phase difference between K + K − and π + π − , δ KK−ππ , or some cancellations between different sources of BSM CP violation. where   BSM effects from 4-quark operators are, however, strongly constrained by / and D −D mixing, and we do not consider this possibility any further.

The Wilson coefficients δC
To compete with the SM δC ( ) (M ) ∼ O(0.1 − 1) is required, which is difficult to achieve given the loop factor and possible further flavor suppressions. However, BSM CP asymmetries around a percent require δC(M ) of a few permille only but need sizable phases. The impact of δC (M ) on CP asymmetries is suppressed due to the hierarchy between the left and right-chiral SM contribution and since there is no interference between them in the branching ratio.
Due to the light leptons in the loop δ LQ C plus chirality-flipped contributions. Here, schematically, C S (µ) = 2 In [11] the notation differs from the one used here by means of charge conjugated fields. Here we write q →q C for the leptoquarks S1, S3, V2 andṼ2 in [11] and adjust their couplings correspondingly. Moreover, here an additional sign for all vector leptoquarks is accounted for. Conclusions in [11] are unaffected. LQ κ κ ν ν  (4.9) Here, Q l denotes the electric charge of the leptons. The couplings ν ( ) within leptoquark models are given in table 3. Note that δ LQ A ( ) 8 (µ c ) is additionally α e /(4π) suppressed and will be neglected throughout.
Constraints on τ couplings are worked out and given in table 4, where we followed [11] and used [36]. The representations V 2,3 turn about to be not relevant for c → uγ decays and no constraints are given. Note that B(D 0 → ρ 0 γ) yields no constraint for λ 1. Lepton flavor violating τ decays constrain couplings with a τ and a light lepton; we do not take these constraints into account as they can be evaded with a flavor suppression of the light leptons.
Numerically, we find for l = τ and the constraints given in table 4 for the chirality-flipping contributions ∝ m τ /m c of the leptoquark representations S 1 , S 2 . As we are interested in CP asymmetries we allow here for a mild suppression of the real parts of λ R λ * L relative to the imaginary ones, the latter of which are weaker constrained JHEP08(2017)091 couplings/mass constraint observable |λ (uτ ) experimentally. At this order, all other leptoquark contributions vanish. Finite contributions are, however, expected for other couplings and representations at two-loop QED. The corresponding leading order calculation in α e is similar to the Q 2 two-loop QCD calculation in [3,48], and beyond the scope of our work. Instead we employ expressions from [49] for fixed order results in the full SM plus scalar (S) and gauge vector (V) leptoquark theory and obtain, respectively, where the electric charge of the leptoquark Q S,V is fixed by charge conservation, and the couplings ν ( ) , κ ( ) are given in table 3. Note the large logarithm ln[m 2 l /M 2 ] proportional to m l in the scalar contribution. It resembles the (resummed) logarithm in equation (4.9) and originates from the inclusion of a light mass, which we drop in the following. The resulting coefficients are worked out numerically in table 5. There are no contributions to A ( ) 7 in leptoquark models V 2,3 . For l = e, µ we find with the constraints given in [11]    The τ couplings for S 3 and S 1L receive their strongest constraint from K decays,

Assuming instead Im
. For vector leptoquarks we find | τ from chirality-flipping contributions without resummation would be about one order of magnitude larger than eq. (4.10).
To summarize, within leptoquark models the c → uγ branching ratios are SM-like with CP asymmetries at O(0.01) for S 1,2 andṼ 2 and SM-like for S 3 . On the other hand, in modelṼ 1 A CP O(10%). The largest effects arise from τ -loops.

SUSY
Here we consider effects within SUSY, taking into account the leading, gluino induced contributions within the mass insertion approximation [50,51] δ SUSY C ( ) where mq and mg denote the masses of the squarks and the gluino, respectively, M ∼ mg ,q , and x = m 2 g /m 2 q . We neglected terms not subject to mg/m c -enhancement. The mass insertions (δ 12 ) are constrained by data on the D 0 −D 0 mass difference [36,50]. In case of i): m The upper limits are similar to the model-independent constraints obtained in section 4.1. Note that, barring cancellations, constraints on the imaginary parts of δ 12 can be about an order of magnitude stronger [52], still permitting to signal BSM CP-violation. Constraints from / and chargino loops are model-dependent. For realistic, not too low SUSY mass parameters corresponding bounds, e.g. [53], are not stronger than those from D −D mixing. We further checked that |δ 75, 5.46], where the lower limit is for x 10 and the upper limit for x 0.01. Thus, supersymmetric models may induce c → uγ branching ratios and CP asymmetries above the SM predictions, and SUSY parameters are constrained by B(D 0 → ρ 0 γ). Note that additional constraints may apply once the SUSY breaking has been specified [51]. A detailed evaluation is beyond the scope of this work.

On Λ c → pγ
We investigate possibilities to probe the handedness of the c → uγ current in the decay Λ c → pγ with polarization asymmetries, that arise once Λ c 's are produced polarized. We follow closely related works on Λ b → Λγ decays [54,55].

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where P Λc denotes the (longitudinal) Λ c polarization and r = A 7 /A 7 . A γ → P Λc /2 for r → ∞ and A γ → −P Λc /2 for r → 0. Calculating A 7 , A 7 in the SM is a difficult task and beyond the scope of our work. In the subsequent estimates of BSM sensitivity we assume that approximately A ( ) 7 ∼ δC ( ) 7 for large BSM effects and A 7 A 7 in SM-like situations. A γ is measurable in the laboratory frame for a boost β = p Λc /E Λc , where p Λc denotes the Λ c three-momentum in the laboratory frame, as Here, q | β| is the average longitudinal momentum of the photon in the laboratory frame relative to the boost axis, γ = 1/ 1 − | β| 2 and E * γ = (m 2 Λc − m 2 p )/(2m Λc ) 0.95 GeV is the photon energy in the Λ c rest frame.
The Λ c polarization can be expressed in terms of the charm quarks' polarization P c as [55,64,65] P Λc P c where A 1.1 is extrapolated [64] from a measurement by the E791 collaboration [66] and ω 1 = 0.71 ± 0.13 is measured by the CLEO collaboration [67]. At the Z the polarization of the charm-quark P (Z) c −0.65 [68] and one obtains a sizable polarization The polarization is negative since c-quarks from the Z are predominantly left-handed. Ultimately, its value needs to be determined experimentally, e.g., from Λ c → Λ(→ pπ)lν decays with B(Λ c → Λ(→ pπ)eν e ) 0.023 [36]. Note that the depolarization parameters A and ω 1 are measurable at Atlas, BaBar, Belle, CMS and LHCb [64]. The Λ c polarization itself is measurable at Atlas, CMS and LHCb via pp → t(→ bW + (→ cs))t(→bW − (→ l −ν )) [64] and pp → W − c [65], where P (W − ) c −0.97 [68]. The angular asymmetry is shown in figure 6 for P Λc as expected at the Z, eq. (5.6). In the SM and leptoquark models r 0.2, and A γ ∼ −P Λc /2 and positive. The statistical uncertainty δA γ = 1 − (A γ ) 2 / √ N is represented by the bands corresponding to N = 10 3 (orange) and N = 10 5 (purple). Within SUSY already for N = 10 3 the angular asymmetry could be observed essentially everywhere within |A γ | |P Λc |/2, allowing to signal BSM physics. Prerequisite for an interpretation is an experimental determination of P Λc or the depolarization fraction, if P c is known.
Given the uncertainties present in meson decays our analysis of Λ c 's is clearly explorative, pointing out an opportunity with future polarization measurements with baryons. More work is needed to detail wrong-chirality contributions in the SM.

Summary
We worked out SM predictions for various D (s) → V γ branching ratios, which are compiled in tables 1 and 2. The hybrid model predicts values up to a factor ∼ 2 − 3 larger than the JHEP08(2017)091 QCD factorization based approach, the latter being dominated by contributions from weak annihilation. All three branching ratios measured so far, the ones of D 0 → ρ 0 γ, D 0 → φγ and D 0 →K * 0 γ are above the QCD factorization range given, suggesting, to the order we are working, a low value of the parameter λ D 0.1 GeV or low charm mass scale. One has to keep in mind, however, that poor convergence of the 1/m c and α s -expansion prohibits a sharp conclusion without further study. Decays of charged mesons with color allowed weak annihilation contribution are better suited for extracting λ D as there is lesser chance for large cancellations, see also figure 2. The measured branching ratios are close to the top end of the ones obtained in the hybrid model. D 0 → φγ and D 0 →K * 0 γ belong to the class of those decays with no direct contribution from electromagnetic dipole operators. Corresponding decays are listed in table 2, their branching ratios have essentially no sensitivity to BSM physics unlike the CP asymmetry in D 0 → φγ, cf. equations (2.26), (3.1).
The measured D 0 → ρ 0 γ branching ratio provides a model-independent upper limit on the decay amplitudes given in eq. (4.1), which is similar to the one from D → πµµ decays [11]. If B(D 0 → ρ 0 γ) is saturated with BSM physics or in the SM, B(D 0 → (ρ 0 /ω)γ) are very close to each other. For intermediate scenarios the two branching ratios can differ by orders of magnitude [7], and indicate BSM physics.
CP asymmetries in c → uγ transitions constitute SM null tests. We find A SM CP few · 10 −3 for D 0 → ρ 0 γ, see figure 3, and similar for other radiative rare charm decays. Among the modes in tables 1 A CP is measured only in D 0 → ρ 0 γ decays, eq. (1.1), consistent with zero and the SM. Uncertainties on A CP are presently too large to provide phenomenologically useful constraints. However, CP-violating BSM can induce significant JHEP08(2017)091 CP asymmetries at the level of ∼ O(0.1), and as demonstrated in figure 4, already a factor four reduction of statistical uncertainty (with central values kept) shows that constraints can be improved significantly.
We worked out implications for two BSM models, SUSY and leptoquark ones. We find that SUSY can saturate the measured D 0 → ρ 0 γ branching ratio and CP asymmetry while leptoquark models can't. In the latter CP asymmetries 10% are possible. The largest effects stem from models with vector leptoquarksṼ 1 followed byṼ 2 and scalars S 1 , S 2 with couplings to taus.
If Λ c -baryons are produced polarized, such as at the Z, angular asymmetries in Λ c → pγ can probe chirality-flipped contributions, see figure 6. Within SUSY A γ can be very different from its SM-value, including having its sign flipped. Prerequisite for an interpretation of A γ is a measurement of the Λ c polarization, however, irrespective of its precise value, in the SM A γ is expected to be positive at the Z. The branching ratio of Λ c → pγ may be investigated at Belle II [2], the depolarization fraction at the LHC [65], and at the FCC-ee [19] all of this including A γ .
We analyzed radiative rare charm decays in the SM and beyond with presently available technologies. Both the heavy quark and the hybrid framework share qualitatively similar phenomenology. The reason is the dominance of weak annihilation and corresponding color and CKM factors. A closer look exhibits numerical differences between the two frameworks for branching ratios and CP asymmetries, detailed in section 3. Despite the considerable uncertainties we demonstrated that existing charm data are already informative on BSM physics. Future measurements can improve theoretical frameworks and allow to check for patterns. Given the unique window into flavor in the up-sector that is provided by |∆C| = |∆U | = 1 processes, further efforts are worthwhile and necessary. Largest sources of parametric uncertainty within the SM are the µ c -scale dependence and λ D in the heavy quark approach and a 1,2 and the form factor A 1 (q 2 ) in the hybrid model. For a BSM interpretation in scenarios with enhanced dipole operators improved knowledge of D → V tensor form factors at q 2 = 0 is desirable.

B D → V form factors
We define the D → V form factors as usual where 0123 = 1 and q µ = (p D − p V ) µ . In our analysis we need V (0), T 1 (0) and A 1 (q 2 ). The D → ρ form factors have been measured by the CLEO collaboration [73] as C D → V γ amplitudes in the hybrid approach We write the resonance-induced contributions to D → V γ amplitudes using [5,6] and [82] as For each D → V γ transition, the CKM factor V * cq V uq can be inferred from the corresponding weak annihilation contribution, eqs. (2.14) and (2.23). The amplitudes A III PC/PV originate from the long-distance penguin estimated with VMD. They contain terms with different CKM factors, allowing for CP violation. We adjusted the relative sign between the VMD contributions from ρ, ω and φ to recover A III PC/PV = 0 in the SU (3)  × a 1 f ρ Γ(D * + → D + γ) Γ(D * + → D + π 0 ) f + (0) , where a 1,2 are given in section 3 and f + (0) = (0.1426 ± 0.0019)/|V cd | [42], where statistical and systematic uncertainties are added in quadrature. For D s → K * + γ decays √ α e (m 2 Ds − m 2 K + )(m 2 K * + − m 2 K + ) 3/2 a 1 f Ds f K Γ(K * + → K + γ) , |A II,K * + PV | = 2(m Ds + m φ )m K * + 3m φ (m 2 Ds − m 2 K * + ) As the approximation of [5,6] is not applicable for B(D * s → D s π 0 ) as a normalization mode due to isospin breaking [83] and Γ D * s is not measured, λ and λg V in A I,K * + PC are related to