Testing the standard model with $D_{(s)} \to K_1 (\to K\pi\pi) \gamma$ decays

The photon polarization in $D_{(s)} \to K_1 (\to K\pi\pi) \gamma$ decays can be extracted from an up-down asymmetry in the $K \pi \pi$ system, along the lines of the method known to $B \to K_1 (\to K\pi\pi) \gamma$ decays. Charm physics is advantageous as partner decays exist: $D^+ \to K_1^+ (\to K\pi\pi) \gamma$, which is standard model-like, and $D_s \to K_1^+ (\to K\pi\pi) \gamma$, which is sensitive to physics beyond the standard model in $|\Delta c| =|\Delta u|=1$ transitions. The standard model predicts their photon polarizations to be equal up to U-spin breaking corrections, while new physics in the dipole operators can split them apart at order one level. We estimate the proportionality factor in the asymmetry multiplying the polarization parameter from axial vectors $K_1(1270)$ and $K_1(1400)$ to be sizable, up to the few ${\cal{O}}(10)\%$ range. The actual value of the hadronic factor matters for the experimental sensitivity, but is not needed as an input to perform the null test.

distributions of D + → K + 1 (→ Kππ)γ and D s → K + 1 (→ Kππ)γ decays, as a means to test the SM. Originally proposed for B-decays [1,2], the method is advantageous in charm as one does not have to rely on prior knowledge of the Kππ spectrum and theory predictions of the photon polarization.
Instead, one can use the fact that the spectrum is universal and the photon polarizations of D + and D s decays in the SM are identical in the U-spin limit [3].
Both D (s) → K + 1 γ decays are color-allowed, and are induced by W -exchange "weak annihilation" (WA), which is doubly Cabibbo-suppressed and singly Cabbibo-suppressed in D + and D s decays, respectively. Thus, the ratio of their branching fractions B(D + → K + 1 γ)/B(D s → K + 1 γ) ≈ |V cd /V cs | 2 (τ D /τ Ds ) is about 0.1, taking into account the different CKM elements V ij and life times τ D (s) [4]. While the D + decay is SM-like, the D s decay is a flavor changing neutral current (FCNC) process and is sensitive to physics beyond the SM (BSM) in photonic dipole operators, which can alter the polarization. The photon dipole contributions in the SM are negligible due to the Glashow-Iliopoulos-Maiani (GIM) mechanism. The photon polarization in the SM in c → uγ is predominantly left-handed, however, in the D-meson decays sizable hadronic corrections are expected [3,[5][6][7]. In the proposal discussed in this work the polarization is extracted from the SM-like decay D + → K + 1 γ. We test the SM by comparison to the photon polarization in D s → K + 1 γ decays. Methods to look for new physics (NP) with the photon polarization in c → uγ transitions have been studied recently in [3,8].
The plan of the paper is as follows: General features of the decays D + → K + 1 γ and D s → K + 1 γ are discussed in Sec. II, including angular distributions for an axial-vector K + 1 decaying to Kππ. Predictions in the framework of QCD factorization [9,10] are given, which we use to estimate the NP reach. In Sec. III we analyze K + 1 → K + π + π − and K + 1 → K 0 π + π 0 decay chains. Phenomenological profiles of the up-down asymmetry are worked out in Sec. IV. In Sec. V we conclude. Auxiliary information is given in three appendices.

II. THE DECAYS D
In Sec. II A we give the D (s) → K 1 (→ Kππ)γ angular distribution that allows to probe the photon polarizations and perform the null test. In Sec. II B we discuss dominant SM amplitudes and estimate the D (s) → K 1 (1270)γ and D (s) → K 1 (1400)γ branching ratios. The BSM reach is investigated in Sec. II C. A.
The D (s) → K 1 γ decay rate, where K 1 is an axial-vector meson, can be written as [11] where L, R refers to the left-handed, right-handed polarization state, respectively, of the photon. Here, G F denotes Fermi's constant and α e is the fine structure constant. A The polarization parameter λ and can be extracted from the angular distribution in D (s) → K 1 (→ Kππ)γ decays with the phase space factor Here, s denotes the Kππ invariant mass squared, needed for finite width effects, θ is the angle between the normal n = ( p 1 × p 2 )/| p 1 × p 2 | and the direction opposite to the photon momentum in the rest frame of the K 1 , and s ij = (p i + p j ) 2 with four-momenta p i of the final pseudo-scalars with assignments specified in (18). Note, p 3 refers to the K's momentum. Furthermore, J is a helicity amplitude defined by the decay amplitude A(K 1 → Kππ) ∝ ε µ J µ with a polarization vector ε of the K 1 , see Sec. III for details. J are the spacial components of the four vector J . J is a feature of the resonance decay and as such it is universal for D + and D s decays.
From (3) one can define an integrated up-down asymmetry which is proportional to the polarization parameter, where κ = sgn[s 13 − s 23 ] for K + 1 → K 0 π + π 0 and κ = 1 for K + 1 → K + π + π − . The .. -brackets denote integration over s 13 and s 23 . The reason for introducing κ is explained in Sec III. The up-down asymmetry is maximal for maximally polarized photons, purely left-handed, λ or purely right-handed ones, λ It is clear from Eqs. (3) and (5) that the sensitivity to the photon polarization parameter λ D (s) γ depends on Im[ n · ( J × J * )]. If this factor is zero, or too small, we have no access to λ D (s) γ . As the J -amplitudes are the same for D + and D s , the factor drops out from the ratio In the SM, this ratio equals one in the U-spin limit. Corrections are discussed in Sec. II B.
In general, there is more than one K 1 resonance contributing to Kππ, such as K 1 (1270) and K 1 (1400). Note, the phase space suppression for the K J (1400)-family and higher with respect to the K 1 (1270) is stronger in charm than in B-decays. Therefore, a single-or double-resonance ansatz with the K 1 (1270) or K 1 (1400) is in better shape than in the corresponding B → K 1 (→ Kππ)γ decays.
In the presence of more than one overlapping K 1 resonance, beyond the zero-width approximation, the relation between the polarization and the up-down asymmetry gets more complicated than (5).
The reason is that, ultimately, r D (s) and the polarization are different for K 1 (1270) and K 1 (1400), that is, they vary with s, an effect that can be controlled by cuts. The general formula can be seen in Appendix C. What stays intact, however, is the SM prediction, A D + UD /A Ds UD SM = 1 up to U-spin breaking.

B. SM
Rare c → uγ processes can be described by the effective Hamiltonian [12], where the operators relevant to this work are defined as follows with chiral left (right) projectors L(R), the field strength tensor of the photon, F µν , and the generators of SU (3) c , T a , a = 1, 2, 3. Contributions to D (s) → K 1 γ decays are illustrated in Fig. 1. level [11], and C 7 ∼ m u /m c 0. The D + → K + 1 γ and D s → K + 1 γ decays are therefore expected to be dominated by the four quark operators.
We employ QCD factorization methods [10] to estimate the branching ratios and the BSM sensitivity. The leading SM contribution is shown in the diagram to the left in Fig. 1, with the radiation of the photon from the light quark of the D (s) meson. The other three WA diagrams are suppressed by Λ QCD /m c and are neglected. The corresponding WA amplitudes for D → V γ have been computed in Ref. [11]. We obtain 1 where Q d = −1/3. We also kept explicitly, i.e., did not expand in 1/m D , the factors that correct for the kinematic factors in Γ D (s) , see (1), corresponding to the matrix elements of dipole operators.
Using the range C 2 ∈ [1.06, 1.14] [11] we find where the first (second) value corresponds to the lower (upper) end of the range for the Wilson coefficient C 2 . In each case, parametric uncertainties from the K 1 decay constants (A4), D (s) decay constants from lattice-QCD f D = (212.15 ± 1.45) MeV and f Ds = (248.83 ± 1.27) MeV [13], masses, life times [4] and CKM elements [14] are taken into account and added in quadrature. The parameter λ D (s) ∼ Λ QCD is poorly known, and constitutes a major uncertainty to the SM predictions (10).
Data on D → V γ branching ratios suggest a rather low value for λ D [11]. We use 0.1 GeV as benchmark value for both D and D s mesons.
Despite its V-A structure in the SM contributions to right-handed photons are expected, which we denote by A D (s) R SM . One possible mechanism responsible for λ D (s) γ = −1 is a quark loop with an O 1,2 insertion and the photon and a soft gluon attached [15], at least perturbatively also subject to GIM-suppression [11]. Here we do not need to attempt an estimate of such effects as we take the SM fraction of right-to left-handed photons from a measurement of A D + UD in D + → K + 1 γ decays, which has no FCNC-contribution. (We neglect BSM effects in four quark operators.) U-spin breaking between D and D s meson decays can split the photon polarizations in the SM. While obvious sources such as phase space and CKM factors can be taken into account in a straight-forward manner, there are further effects induced by hadronic physics. Examples for parametric input are the decay constants, and λ D (s) , as in (9). The former has known U -spin splitting of ∼ 0.15 [13], and for the latter, as not much is known, we assume that the spectator quark flavor does not matter beyond that. A measurement of D s → ρ + γ, which is a Cabibbo and color-allowed SM-like mode with branching ratios of order 10 −3 [11] can put this to a test.
Nominal U-spin breaking in charm is O(0.2 − 0.3), e.g. [16][17][18], however, the situation for the photon polarization is favorable, as only the residual breaking on the ratio of left-handed to right-handed amplitude is relevant for the null test. In the BSM study we work with U-spin breaking between r D+ and r Ds within ±20% .

C. BSM
Beyond the SM, the GIM suppression does not have to be at work in general and the dipole coefficients can be significantly enhanced. Model-independently, the following constraints hold obtained from D → ρ 0 γ decays [11,19], and consistent with limits from D → π + µµ decays [12].
The corresponding NP contributions to D s → K + 1 γ decays are given as where From radiative B-decay data [20] one infers that T The SM plus NP decay amplitudes read and In Fig. 2 we illustrate BSM effects that show up in λ Ds γ being different from λ D + γ for NP in C 7 with C 7 = 0 (green curves) and in C 7 with C 7 = 0 (red curves), within the constraints in (11) for the K 1 (1270), central values of input, and for λ D (s) = 0.1 GeV. We learn that NP in the left-or right-handed dipole operator can significantly change the polarization in D + decays from the one in D s decays. Larger values of λ D (s) and T K 1 , and smaller values of f K 1 enhance the BSM effects.
Here we provide input for the K 1 → Kππ helicity amplitude J , which drives the sensitivity to the photon polarization in the up-down asymmetry (5). After giving a general Lorentz-decomposition we resort to a phenomenological model for the form factors, which allows us to estimate J and sensitivities. This section is based on corresponding studies in B decays [2,21]. While being relevant for the sensitivity, we recall that knowledge of J in charm is not needed as a theory input to perform the SM null test.
We consider two K 1 states, K 1 (1270) and K 1 (1400), with spin parity J P = 1 + . For the charged resonance K + 1 two types of charge combinations exist for the final state, K + 1 → K 0 π + π 0 (channel I) and K + 1 → K + π + π − (channel II), both of which we consider in the following.
The K 1 → Kππ decay amplitude can be written in terms of the helicity amplitude J as From here on assumptions are needed to make progress on the numerical predictions of the phenomenological profiles. First, the C 1,2 -functions are modelled by the quasi-two-body decays K 1 → Kρ(→ ππ) and K 1 → K * (→ Kπ)π. Taking into account the isospin factors for each charge mode, K + 1 → K 0 π + π 0 and K + 1 → K + π + π − , C I,II 1,2 can be rewritten in the following form [21] C I 1 = where, using factorization, The definitions of the form factors of the K 1 → V P (V = K * , ρ and P = π, K) decay, f V , h V , and decay constants of the V → P i P j decay, g V P i P j are given in Appendix B. The form factors are obtained in the Quark-Pair-Creation Model (QPCM) [25].
In the presence of two K 1 states, K 1 (1270) and K 1 (1400), this framework can be extended by and the parameter ξ Kres , which allows to switch the states on and off individually. Importantly, in a generic situation with all K 1 -resonances contributing ξ Kres takes into account the differences in their production in the weak decay. Such effects are induced by the K 1 -dependence of hadronic matrix elements, such as f K 1 m K 1 in (9), or T K 1 in (12). For f K 1 (1400) m K 1 (1400) /(f K 1 (1270) m K 1 (1270) ) ∼ 1.1 and T K 1 (1400) /T K 1 (1270) ∼ 0.5 this effect is rather mild. The ansatz (23), which is an approximation of the general formula (C3), allows to compute A UD /λ γ as in (5) in Sec. IV independent of the weak decays. Eq. (23) becomes exact, i.e., coincides with (C3) for universal ξ Kres .
Due to isospin Im[ n · ( J × J * )] in the K + 1 → K 0 π + π 0 channel is antisymmetric in the (s 13 , s 23 )-Dalitz plane. This can be seen explicitly by interchanging s 13 ↔ s 23 in Eq. (21), which implies C 1 ↔ C 2 and therefore Im[ n · ( J × J * )] ∝ Im[C 1 C * 2 ] changes sign when crossing the s 13 = s 23 line, see the plot to the right in Fig. 3. Therefore, in order to have a non-zero up-down asymmetry after s 13 , s 23 -integration, one has to define the asymmetry with sgn(s 13 − s 23 )Im[ n · ( J × J * )] in Eq. (5). In the K + 1 → K + π − π + channel and with only one K 1 , the border, at which A UD changes sign, is a straight line in the (s 13 , s 23 )-plane, see the plot to the left in Fig. 3 Figure 3: Dalitz contour plots of Im[ n · ( J × J * )] for K + π + π − (plot to the left) and K 0 π + π 0 (plot to the right) at m 2 Kππ = m 2 K1(1270) . Red (blue) areas correspond to positive (negative) values of Im[ n · ( J × J * )].

IV. UP-DOWN ASYMMETRY PROFILES
In the following we work out estimates for the up-down asymmetry in units of the photon polarization parameter A UD /λ γ , as in (5). The crucial ingredient for probing the photon polarization is the hadronic factor Im[ n · ( J × J * )]. Using (23), and for two interfering resonances a, b, e.g., a = K 1 (1270) and b = K 1 (1400), dropping channel I, II superscripts and kinematic variables to ease notation, it reads which shows the necessity of having relative strong phases for a non-zero up-down asymmetry. Such phases can come from the interference between K * π and Kρ channels inside of C 1,2 , as well as from the interference between the K 1 resonances. Due to the larger number of interfering amplitudes (18), we quite generally expect larger phases in the K + 1 → K 0 π + π 0 channel. While the K 1 (1270) decays both to Kρ and K * π, the K 1 (1400) decays predominantly to K * π. We therefore expect the pure K 1 (1400) contribution to A UD /λ γ in the K + π + π − channel to be very small.
In Fig. 4 we show the m Kππ dependence of | J | 2 (plots to the left) and A UD /λ γ (plots to the right). The different colors refer to different ratios of the K 1 (1270) and K 1 (1400) contributions.
Since the up-down asymmetry is sensitive to complex phases in the K 1 decay amplitudes, we test several possible sources apart from the ones coming from the Breit-Wigner functions of the K 1 , K * and the ρ. As expected, it turns out that such phases have only a negligible effect on the | J | 2 distributions, and we do not show corresponding plots. The Belle collaboration in the analysis of B + → J/ψK + π + π − and B + → ψ K + π + π − decays signals a non-zero phase,   6. Note that δ ρ and δ D vanish in the QPCM and are therefore termed "off-set" phases.
We learn from Figs. 4 -6 that A UD /λ γ profiles with ξ K 1 (1400) = 0.5, 1 (red, green curves, respectively) can be of the order ∼ 0.05 − 0.1 (channel II) and ∼ 0.2 − 0.3 (channel I), which are, as expected, larger for K 0 π + π 0 than for K + π + π − final states. Adding phenomenological strong phases such as δ ρ and δ D has a significant effect for channel II. As zero-crossings can occur it may be disadvantageous to not use m Kππ bins, in particular, for channel II. The position of the zeros, however, cannot be firmly predicted, although the one at m K + π + π − 1GeV, whose origin is discussed at the end of Sec. III, is quite stable, as well as the one at m K + π + π − 1.3GeV. The latter stems from K 1 (1270) and K 1 (1400) interference.
Strong phases and, related to this, K 1 -mixing, constitute the main sources of uncertainty. Figs [−13, +24]% (channel II) of Ref. [28], which are based on K 1 (1270) dominance. Note that Ref. [28] uses κ = sgn(s 13 − s 23 ) for both channels. Our prediction for channel II in this convention reads We stress that the estimates are subject to sizable uncertainties and serve as a zeroth order study to explore the BSM potential in D s → K 1 γ decays. Kππ profiles from the B-sector can be linked to charm physics, and vice versa.

V. CONCLUSIONS
New physics may be linked to flavor, and K, D, and B systems together are required to decipher its family structure. Irrespective of this global picture, SM tests in semileptonic and radiative c → u transitions are interesting per se, and quite unexplored territory today: present bounds on short-distance couplings are about two orders of magnitude away from the SM [11,12].
We study a null test of the SM in radiative rare charm decays based on the comparison of the up-down asymmetry in D + → K + 1 (→ Kππ)γ, which is SM-like, to the one in D s → K + 1 (→ Kππ)γ, which is an FCNC. The up-down asymmetry depends on the photon polarization, subject to BSM effects in the |∆c| = |∆u| = 1 transition.