The photon polarization in radiative D decays, phenomenologically

We work out the phenomenology of untagged time-dependent analysis with radiative D 0 - decays into CP eigenstates, which allows to probe the photon polarization by means of the charm mesons’ ﬁnite width diﬀerence. We show that D 0 → φγ or D 0 → ¯ K 0 ∗ γ decays, which are SM-dominated, or SM-like, respectively, together with U-spin allow to obtain chirality-predictions for radiative decay amplitudes. The order of magnitude of wrong-chirality contributions in the SM can be cross-checked with an up-down asymmetry in D 0 → ¯ K 01 ( → ¯ Kππ ) γ . We explore the sensitivity to new physics in | ∆ c | = | ∆ u | = 1 dipole couplings in the decays D 0 → ρ 0 γ . We point out the possibility to test the SM with D s → K +1 ( → Kππ ) γ decays.

following a proposal for B s -mesons [10] 1 , and briefly discussed in [11] for charm. As first-principle theory predictions have large uncertainties, we propose to use data and U-spin to obtain a datadriven SM prediction for the photon polarization in D 0 → V γ, V =K * 0 , φ, ρ 0 , ω. We work out the phenomenology, and provide predictions in models beyond the SM (BSM). We further suggest to study an up-down asymmetry in D →K 1 (→Kππ)γ along the lines the one known to B-decays [4,[12][13][14], as a consistency check of the SM prediction for the photon polarization. An analogous asymmetry allows to test the SM with D s → K + 1 (→ Kππ)γ decays. The paper is organized as follows: In section II we review time-dependence in decays into CPeigenstates and show how the photon polarization in D → V γ decays can be probed. Features of different charm decay observables and their relations are discussed in section III. In section IV we show how the SM can be tested and give BSM expectations. In section V we summarize. In the appendix we give the angular distribution of D (s) → K 1 γ → Kππγ decays.

II. TIME-DEPENDENCE IN D → V γ
The D → V γ decay amplitudes can be written as where L, R denote the chirality, j labels different amplitudes, A where ξ denotes the CP eigenvalue of the self-conjugate vector meson V , i.e. ξ = +1 for V = ρ 0 , φ,K * 0 (K 0 S π 0 ) and ξ = −1 for V =K * 0 (K 0 L π 0 ). We define the normalized CP asymmetry as usual where Γ(D → V γ) = Γ(D → V γ L ) + Γ(D → V γ R ). The time-dependent decay rate is given as where ζ = +1 for a D meson, ζ = −1 for aD meson and the normalization N can be found in, e.g., [15]. Here, ∆Γ = Γ H − Γ L > 0 and ∆m = m H − m L are the differences between the heavy and light D mass eigenstates and Γ is the mean width. Note that different sign conventions and notations are used in the literature. The direct CP asymmetry A dir CP = C and the observable S [10] can be measured only when the initial flavor is tagged. On the other hand, A ∆ can be observed in untagged time-dependent measurements by means of a finite width difference ∆Γ, as has been shown already for the decays B 0 s → φγ [16]. The observable A ∆ is given in terms of the decay amplitudes as where The 95% C.L. intervals of the D 0 −D 0 mixing parameters read [15] It is instructive to consider A ∆ in the limit q/p 1 and assuming that the decays can be described by only one amplitude per chirality. One obtains in this limit where A a , δ a and φ a denote the modulus, strong and weak phase, respectively of the chirality amplitude A a = A a e iδa e iφa , a = L, R. Eq. (8) holds if there is no CP violation in the decay, or if strong phases are negligible. As CKM-induced CP violation in charm is small due to the GIMmechanism this is a useful approximation within the SM and in models with no BSM sources of CP-violation. Defining the photon polarization fraction r as it follows We discuss expectations for the strong phases δ L,R and relations between D 0 → V γ modes in section III.

III. DECAY ANATOMIES
The decays D → V γ, V =K * 0 , φ, ρ 0 , ω are dominated in the SM by weak annihilation (WA) [6,11,17], see figure 1, plot to the left. While this holds model-independently for V =K * 0 , the final state mesons ρ 0 , ω and, to a lesser degree, the φ allow for additional contributions in and beyond the SM. Here we consider BSM effects in dipole operators, in the effective Lagrangian  [6,11,18], but there is a contribution from mixing onto Q ( ) 7 . It can be absorbed effectively into the coefficient C ( ) 7 , see, e.g., [6] for explicit formulae. Corresponding contributions to D → V γ are illustrated in figure 1, plot to the right. The four-fermion operators Q (q) 1,2 ∼ū L γ µ q LqL γ µ c L are SM-like and induce WA amplitudes. It is possible that chirality-flipped versions of Q (q) 1,2 are present in BSM scenarios. As we neglect CP-violation such contributions are not distinguishable from the V-A ones, and effectively accounted for in our framework.
To test the SM using A ∆ requires sufficient understanding of its SM value -it is the main point of this paper to obtain such a prediction experimentally by relating A ∆ from SM-dominated modes V =K * 0 , φ to A ∆ from V = ρ 0 , ω using U-spin. We show in section IV that this framework describes available data in a consistent way.
In the following we give details on D 0 → V γ decays for V =K * 0 , ρ 0 and φ in section III A, III B and III C, respectively. These decays enter the SM tests described in section IV. In section III D and III E we discuss D 0 →K 0 1 (→Kππ)γ and D s → K + 1 (→ Kππ)γ decays, respectively. The former mode assists the extraction of the SM's photon polarization from A ∆ as argued in section III A as well as serves as a standard candle for BSM searches with the latter, the D s -decays.
A. D 0 →K * 0 γ decays The decay D 0 →K * 0 γ is purely induced by WA and SM-like. Strong phases are small [6,11,17], however, beyond leading order effects could induce non-vanishing phases 2 . Note that predictions for the D 0 →K * 0 γ branching ratio obtained in other, hybrid chiral frameworks [19][20][21] are in line with experimental data given in table I only if interfering amplitudes add coherently [6], i.e., for small relative strong phases. Therefore, eq. (11) simplifies where r 0 denotes the corresponding D 0 →K * 0 γ photon polarization fraction. Theoretical predictions for r 0 are rather uncertain as A R is presently not known, except for being power suppressed with respect to A L [6], Due to the low charm mass corrections can be considerable and r 0 unsuppressed [11]. As r 0 < 1 for a convergent power series a measurement of r 0 allows to probe the performance of the expansion.
In view of the sizeable uncertainties we refrain from using theory input on r 0 and propose to use the value that can be determined experimentally via A ∆ . To do so we assume that the strong phase difference δ L − δ R and r 0 are not both large, because in this case a suppressed A ∆ could not unambiguously point to a suppressed r 0 , see eq. (11). This possibility, although being not the plain-vanilla theory expectation, can only be cross-checked with other measurements: We propose to study an up-down asymmetry in D 0 →K 1 γ withK 1 →Kππ, constructed from the photon with respect to the plane formed by the pions in theK 1 's rest frame, discussed further in section III D, and defined in the appendix. The advantage of measuring the up-down asymmetry in D 0 →K 1 γ decays is that it probes the photon polarization parameter which is independent of the relative phase between A L and A R . On the flip-side, D 0 -tagging is required, because λ γ changes sign between D 0 andD 0 . Note, K − π + π 0 final states are selftagging, unlikeK 0 π + π − ones. This method returns predominantly the polarization fraction of thē K 1 (1270)-resonance, r 0 (K 1 ), rather than of the one of the vectorK * 0 (892), r 0 . As spin, flavor and color counting are identical and masses not too much apart we expect the dynamics to be sufficiently related. There is a doubly-Cabibbo suppressed contamination from D 0 → K 0 1 γ, K 1 → Kππ decays affecting λ γ at order V * cd V us , that is, a few percent. One may also use the doubly-Cabibbo suppressed but color-enhanced modes D + → K + 1 γ to estimate the size of the SM polarization. Another way is to look for large relative (strong) phases with CP-asymmetries in the D 0 → ρ 0 γ time-dependent distribution (4), which are sensitive to phases in a complementary way. The last method requires to establish a finite CP-asymmetry in D 0 → ρ 0 γ. The current measurement [24] is consistent with zero. We note that the phase differences probed in CP-asymmetries are those corresponding to same chirality amplitudes, so the The WA-contributions of D 0 → ρ 0 γ and D 0 →K * 0 γ are related by U-spin. Therefore, in the SM, Here we neglected contributions from the soft gluon operator c → uγg [25], see also [26], to D → ρ 0 γ, where it is GIM-suppressed [6]. The perturbative and hard spectator interaction induced SMamplitudes for c → u transitions are negligible with respect to the WA-amplitude [6].
While the U-spin breaking from differences in masses and CKM elements can be accounted for trivially, the residual one on the left and right-chiral amplitude, denoted by f L , f R , respectively, depends on hadronic physics. Note, f L,R are in general complex-valued. Estimations based on factorization identify the largest WA-contributions as the ones with the photon being radiated off the initial state [6,17,27]. In this case, the breaking in the matrix element is given by the final vector 0.9, an effect within the nominal size of U-spin breaking in charm, O(0.2 − 0.3), e.g., [28][29][30]. We find that in the hybrid model [20,21], also [31], using the expressions compiled in [6], the U-spin breaking is of similar size, f L,R 0.9 ± 0.1, where we varied input parameters.
From (17) follows subject to corrections of the order f R /f L . Eq. (18) provides, once r 0 is known from D 0 →K * 0 γ data, a SM-prediction for D 0 → ρ 0 γ. Hence, up to U-spin breaking, Any sizeable deviation from eq. (19) would signal BSM physics in the c → u transition which contributes to D 0 → ρ 0 γ, but not to D 0 →K * 0 γ. On the other hand, experimental confirmation of eq. (19) would establish c → uγ amplitudes other than WA ones to be subleading.
The decay D 0 → φγ is not a pure WA-induced decay due to the dd+uū admixture, or rescattering [32]. We parameterize such effects by a complex-valued parameter y, and y O(0.1) as follows [6] A L, where r φ denotes the polarization fraction of the photon in D → φγ decays. Therefore, As already discussed in section III B for the ρ 0 , the leading U-spin breaking based on dominance of initial state radiation is given by The up-down asymmetry in D 0 →K 0 1 γ withK 0 1 →Kππ probes the photon polarization by measuring the polarization of the kaon resonanceK 0 1 . The asymmetry is the D-decay version of the one in B → K 1 γ decays [12]. It is proportional to the photon polarization parameter λ γ (16), see the appendix for details. The proportionality factor depends on the details of hadronic decays of theK 1 . As such, it is independent of the resonance's production, hence, is the same for Band D-decays. The rate ofKππ events, of course, differs as well as the relative importance of resonances and their interference effects.
The contribution from the K(1400)-family and higher, which includes resonances which dilute the asymmetry, is phase space suppressed in charm relative to theK 1 (1270) one by about a factor of two. This reduces the impact of interference effects and suggests a single-resonance analysis in terms of theK 1 (1270). As stressed in the more recent B-physics literature [13,14,22,23], insufficient understanding of the hadronic structure of theK 1 -decay prohibits a precision extraction of the photon polarization. While this can be overcome [14,23], here, we merely need to check whether the wrong-chirality amplitudes satisfy A R ∼ A L or not in a SM-like decay.
The decay D s → K + 1 (→ Kππ)γ is color-allowed, hence the sensitivity to BSM physics is suppressed by 1/N C , where N C denotes the number of colors, relative to the one in D 0 → (ρ 0 , ω)γ decays. A similar up-down asymmetry as in D 0 →K 0 1 (→Kππ)γ can be constructed, see appendix. Predictions for the polarization fraction of D s → K + 1 γ decays, r s (K 1 ), are In the latter case, for negligible SM-contribution to A R , λ γs + 1 becomes a null test of the SM, where λ γs denotes the corresponding photon polarization parameter (16) in D s → K + 1 γ decays In the SM it has to be equal to λ γ , eq. (16), up to U-spin corrections. Significant deviations can signal BSM physics.

IV. TESTING THE SM
We provide explicit expressions on how to probe BSM-sensitive contributions in the photon polarization fraction in D 0 → ρ 0 γ decays using D 0 →K * 0 γ in section IV A, and using D 0 → φγ in section IV B. We also show consistency of the framework -WA dominance in SM-dominated modes consistent with leading U-spin breaking -with current data on branching ratios, and give expectations for the photon polarization in D → ρ 0 γ decays in BSM models.
The D 0 →K * 0 γ is a WA-induced mode. Its branching ratio can be written as where, in the notation of the previous sections, a 0 , a 0 correspond to A L , A R , respectively. Then, the ratio of right-to left-handed photons is given as A measurement of B 0 , r 0 returns the magnitude of both amplitudes The BSM-sensitive mode D 0 → ρ 0 γ can be affected by contributions from left-and right-handed Wilson coefficients C 7 and C 7 , respectively. We write the branching ratio as (note, factor 1/2 for isospin) where T denotes the D 0 → ρ 0 dipole form factor at maximum momentum transfer, and f L,R = 1 accounts for U-spin breaking effects beyond phase space and CKM already discussed in section III.
The polarization fraction of D 0 → ρ 0 γ is given as Experimental findings for the reduced branching ratios B 0 , B and B φ , the latter corresponding to D → φγ decays, are given in table I.
Measurement of 4 observables, B, B 0 , r, r 0 determines 4 coefficients, the SM contributions a, a and the BSM ones C 7 , C 7 . By definition, r, r 0 ≥ 0. Presently, only branching ratios are measured, see table I. It would be desirable to have more precise data available, in particular, the discrepancy in D 0 →K * 0 γ between Belle and BaBar should be settled.
In absence and anticipation of future polarization data we discuss the following limiting cases: a) C 7 , C 7 0. This corresponds to the SM, r r 0 , discussed around eq. (18). b) r 0 0. It follows branching ratio   The polarization fraction r is a null test of the SM for negligible r 0 . We can already now make a data-based prediction for r given C 7 irrespective of C 7 . Possible values of r from eq. (32) are illustrated in figure 2, where the blue band displays the one sigma range of B. Within leptoquark models holds |C 7 | 0.02, which, using T = 0.7 [6], implies r 0.09, indicated by the green box. On the other hand, SUSY models can provide significantly higher values |C 7 | 0.3, while model-independently holds |C 7 | 0.5. As r diverges towards C 7 0.15, in both latter cases there is no upper limit on r. Upper limits on the Wilson coefficients are taken from [6].
This allows to predict r if one calculates a. Using the results for the WA-amplitude in the heavy quark limit obtained in [6] we find (for λ D ≥ 0.1 GeV) r ≥ 2. Such large values of r are consistent with the fact that the corresponding SM prediction for the D 0 → ρ 0 γ branching ratio B = 0.005 · (0.1 GeV/λ D ) 2 is significantly below the measured one given in table I. Note that for the SM-dominated modes the agreement is much better, B φ = 0.016 · (0.1 GeV/λ D ) 2 Here, λ D denotes a non-perturbative parameter expected to be of the order Λ QCD .
This will be useful once r 0 in addition to B 0 is measured and, using eq. (30), allows to illustrate viable ranges for r in BSM scenarios.
In the present situation where only branching fraction measurements are available, it is useful to define the ratio where r = |(a + T C 7 )/a|, and we used eq. (30). The ratio R equals one if only WA contributes, that is, r = r 0 and C 7 = C 7 = 0, irrespective of the size of the SM contributions. With BSM physics, the ratio can be larger or smaller than one. Similar ratios have been mentioned in [36] as a test of the SM.
In R only trivial U-spin breaking from phase space and CKM-elements has been accounted for.
We define in addition the ratioR, in which also the leading dynamical one ∝ m V f V is covered.
In figure 3 we show r (plots to the left) for the cases a), c) and d) forR = 1.6 ± 0.3 (upper row) andR = 1.6 ± 0.45 (lower row), illustrating the data's discriminative power. Case b) has already been considered in figure 2. We learn that r can be order one, and that it can be close to r 0 , in which case discrimination from the SM is not possible. In the plots to the right we show 2r/(1 + r 2 ) as a function of 2r 0 /(1 + r 2 0 ), which enter the observables A ∆ (D 0 →K * 0 γ) and A ∆ (D 0 → ρ 0 γ), respectively, for a), c) and d). Within present data onR, scenario d) with no right-handed BSM physics (red band) cannot be sufficiently separated from the SM (black dashed curve), while scenario c) with only C 7 present (green band), exhibits a significant SM-deviation. ForR < 1 the green band corresponding to scenario c) would be below the SM curve while the band corresponding to scenario d) would be above it. The lower plots in figure 3 correspond to a value ofR with 50% larger uncertainty, mimicking larger U-spin corrections. As the upper and lower plots are similar we learn that such effects do not change the picture qualitatively.
We can apply this strategy to probe for new physics in the decay D → ωγ, once its branching ratio and its polarization fraction become available.
Due to the hybrid nature of the φ, one may ask whether the D → φγ branching ratio is consistent with the assumption of a predominantly WA-induced decay amplitude. Corresponding ratios, where B φ is the reduced branching ratio analogous to B for D → ρ 0 γ, eq. (29), and using f = Leading U-spin breaking makes both branching ratios of similar size. This is consistent with which, together with eq. (21), holds in the SM and beyond. While this numerical agreement could be accidental, it does give a consistent picture between the predominantly SM-like modes and the size of U-spin breaking in the range obtained within the lowest order heavy quark expansion [6], sum rules [17] and hybrid models [20,21]. This is beneficial as time-dependent analysis with K * 0 → K S,L π 0 is more difficult than with φ → K + K − . We therefore suggest to use r φ , the polarization fraction of the photon in D → φγ decays, as a SM prediction for r.
We repeat the analysis previously performed with ρ 0 andK * 0 , eqs. (37),(38), for the ρ 0 and the φ. We obtain and for the averageR(ρ 0 /φ) ave = 1.61 ± 0.27. The good agreement seen in the data, between eq. (38) and eq. (44), supports that the φ indeed can be used as a standard candle as long as effects of O(y) can be neglected, and that the working assumption of the leading U-spin breaking is consistent with data.

V. SUMMARY
Untagged, time-dependent analysis into CP eigenstates allows to extract the photon polarization in radiative charm decays. Given a measurement of the photon's polarization fraction, its interpretation requires control of SM contributions to D → V γ decays. We explore the possibility to obtain the size of the SM background to wrong-chirality contributions from data and U-spin.
While there are sizeable uncertainties related to this procedure, there is presently no measurement available and large room for BSM physics.
Specifically, we propose measurement of A ∆ in D 0 → φ(→ K + K − )γ decays to obtain the SM fraction, r φ . While the φ is not purely ss and therefore not purely WA-induced, the final state is advantageous over the one from the pure SM-mode, D 0 →K 0 * (→ K S,L π 0 )γ. If feasible, the latter should be studied experimentally as well.
If r φ , or r 0 , the polarization fraction of D 0 →K 0 * γ decays, is negligible, the photon polarization and therefore A ∆ in D 0 → ρ 0 (→ π + π − )γ becomes a null test of the SM. Possible ranges depending on the BSM model are illustrated in figure 2. A ∆ in D 0 → ρ 0 γ decays can be O(1) in SUSY models, while leptoquark models give SM-like values. The method works as well for D 0 → ωγ decays, however, the branching ratios of the ω into suitable final states such as π + π − are small [35].
We further explored the correlation between the SM and BSM polarization fraction based on ratios of branching fraction measurements, shown in figure 3. Our study shows that U-spin breaking effects of nominal size are not qualitatively changing the picture. In particular, we find that available branching ratios, table I, are consistent with U-spin hierarchies predicted by the heavy quark expansion, and other theory frameworks [17,20,21]. However, uncertainties are large, and more study is needed to achieve a completer picture. This includes the clarification of the discrepancy in the B(D 0 →K * 0 γ) data. A cleaner BSM interpretation would require better knowledge of the dipole form factor T (0).
We point out that another way to probe the photon polarization in radiative charm decays is provided by an up-down asymmetry (A2). As in the time-dependent analysis the SM value can be extracted from a SM-like decay, here D 0 →K 0 1 γ, and then used together with U-spin for a SM test in a BSM-sensitive mode, D s → K + 1 γ decays. Due to limited phase space the K 1 (1270) is more pronounced in charm relative to higher resonances than in B-decays.