Rare radiative charm decays within the standard model and beyond

TL;DR

This paper provides Standard Model estimates for rare charm radiative decays, compares them with experimental data, and explores implications for new physics models like leptoquarks and SUSY, highlighting potential CP asymmetries and angular observables.

Contribution

It offers the first comprehensive SM predictions for $c o u \, ext{gamma}$ processes and analyzes constraints on new physics from current measurements.

Findings

Measured branching ratios exceed SM upper estimates.

Constraints on dipole operators from $D^0$ decay data.

Potential for large CP asymmetries in new physics scenarios.

Abstract

We present standard model (SM) estimates for exclusive $c \to u \gamma$ processes in heavy quark and hybrid frameworks. Measured branching ratios ${\cal{B}}(D^0 \to (\phi, \bar K^{*0}) \gamma)$ are at or somewhat exceeding the upper range of the SM and suggest slow convergence of the $1/m_D, \alpha_s$-expansion. Model-independent constraints on $|\Delta C|=|\Delta U|=1$ dipole operators from ${\cal{B}}(D^0 \to \rho^0 \gamma)$ data are obtained. Predictions and implications for leptoquark models are worked out. While branching ratios are SM-like CP asymmetries $\lesssim 10 \%$ can be induced. In SUSY deviations from the SM can be even larger with CP asymmetries of $O(0.1)$. If $\Lambda_c$-baryons are produced polarized, such as at the $Z$-pole, an angular asymmetry in $\Lambda_c \to p \gamma$ decays can be studied that is sensitive to chirality-flipped contributions.

Tables

Table 1. Table 1: Branching ratios of D → V ​ γ → 𝐷 𝑉 𝛾 D\to V\gamma 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 μ c ∈ [ m c / 2 , 2 ​ m c ] subscript 𝜇 𝑐 subscript 𝑚 𝑐 2 2 subscript 𝑚 𝑐 \mu_{c}\in[m_{c}/\sqrt{2},\sqrt{2}m_{c}] . The branching ratios from the hard spectator interaction plus weak annihilation scale as ( 0.1 GeV ) / λ D ) 2 (0.1\,\text{GeV})/\lambda_{D})^{2} . Also given are data by the Belle Abdesselam:2016yvr and the CLEO (at 90% CL) Asner:1998mv collaborations as well as SM predictions from Fajfer:1997bh ; Fajfer:1998dv , via pole diagrams and VMD Burdman:1995te and QCD sum rules Khodjamirian:1995uc . † Statistical and systematic uncertainties are added in quadrature.

branching ratio	$D^0\to {\rho }^0\gamma$	$D^0\to \omega \gamma$	$D^{+}\to {\rho }^{+}\gamma$	$D_s\to K_{}^{\ast }{}_{}{}^{+}\gamma$
two-loop QCD	$\left(0.14-2.0\right)\cdot {10}^{-8}$	$\left(0.14-2.0\right)\cdot {10}^{-8}$	$\left(0.75-1.0\right)\cdot {10}^{-8}$	$\left(0.32-5.5\right)\cdot {10}^{-8}$
HSI+WA	$\left(0.11-3.8\right)\cdot {10}^{-6}$	$\left(0.078-5.2\right)\cdot {10}^{-6}$	$\left(1.6-1.9\right)\cdot {10}^{-4}$	$\left(1.0-1.4\right)\cdot {10}^{-4}$
hybrid	$\left(0.041-1.17\right)\cdot {10}^{-5}$	$\left(0.042-1.12\right)\cdot {10}^{-5}$	$\left(0.017-2.33\right)\cdot {10}^{-4}$	$\left(0.053-1.54\right)\cdot {10}^{-4}$
Fajfer:1997bh ; Fajfer:1998dv	$\left(0.1-1\right)\cdot {10}^{-5}$	$\left(0.1-0.9\right)\cdot {10}^{-5}$	$\left(0.4-6.3\right)\cdot {10}^{-5}$	$\left(1.2-5.1\right)\cdot {10}^{-5}$
Burdman:1995te	$\left(0.1-0.5\right)\cdot {10}^{-5}$	$0.2\cdot {10}^{-5}$	$\left(2-6\right)\cdot {10}^{-5}$	$\left(0.8-3\right)\cdot {10}^{-5}$
Khodjamirian:1995uc 222Uncertainties not available. We take $a_1=1.3$ and $a_2=-0.55$ Bauer:1986bm .	$3.8\cdot {10}^{-6}$	–	$4.6\cdot {10}^{-6}$	–
data†	$\left(1.77\pm 0.31\right)\cdot {10}^{-5}$		–	–

Table 2. Table 2: Branching ratios of D 0 → ( ϕ , K ¯ ∗ 0 , K ∗ 0 ) ​ γ → superscript 𝐷 0 italic-ϕ superscript ¯ 𝐾 absent 0 superscript superscript 𝐾 0 𝛾 D^{0}\to(\phi,\bar{K}^{*0},{K^{*}}^{0})\gamma , D + → K ∗ + ​ γ → superscript 𝐷 superscript superscript 𝐾 𝛾 D^{+}\to{K^{*}}^{+}\gamma and D s → ρ + ​ γ → subscript 𝐷 𝑠 superscript 𝜌 𝛾 D_{s}\to\rho^{+}\gamma within the SM from weak annihilation and within the hybrid framework Fajfer:1997bh ; Fajfer:1998dv (appendix C ). We vary the decay constants, lifetimes and μ c ∈ [ m c / 2 , 2 ​ m c ] subscript 𝜇 𝑐 subscript 𝑚 𝑐 2 2 subscript 𝑚 𝑐 \mu_{c}\in[m_{c}/\sqrt{2},\sqrt{2}m_{c}] . The branching ratios induced by weak annihilation scale as ( 0.1 GeV ) / λ D ) 2 (0.1\,\text{GeV})/\lambda_{D})^{2} . Also given are available data by the Belle Abdesselam:2016yvr and BaBar Aubert:2008ai collaborations, as well as SM predictions obtained in Fajfer:1997bh ; Fajfer:1998dv , via pole diagrams and VMD Burdman:1995te and QCD sum rules Khodjamirian:1995uc . † Statistical and systematic uncertainties are added in quadrature.

branching ratio	$D^0\to \varphi \gamma$	$D^0\to {\overline{K}}^{\ast 0}\gamma$	$D^0\to K_{}^{\ast }{}_{}{}^0\gamma$	$D^{+}\to K_{}^{\ast }{}_{}{}^{+}\gamma$	$D_s\to {\rho }^{+}\gamma$
WA	$\left(0.0074-1.2\right)\cdot {10}^{-5}$	$\left(0.011-1.6\right)\cdot {10}^{-4}$	$\left(0.032-4.4\right)\cdot {10}^{-7}$	$\left(0.73-1.1\right)\cdot {10}^{-5}$	$\left(1.8-2.9\right)\cdot {10}^{-3}$
hybrid	$\left(0.24-2.8\right)\cdot {10}^{-5}$	$\left(0.26-4.6\right)\cdot {10}^{-4}$	$\left(0.076-1.3\right)\cdot {10}^{-6}$	$\left(0.48-7.6\right)\cdot {10}^{-6}$	$\left(0.11-1.3\right)\cdot {10}^{-3}$
Fajfer:1997bh ; Fajfer:1998dv	$\left(0.4-1.9\right)\cdot {10}^{-5}$	$\left(6-36\right)\cdot {10}^{-5}$	$\left(0.03-0.2\right)\cdot {10}^{-5}$	$\left(0.03-0.44\right)\cdot {10}^{-5}$	$\left(20-80\right)\cdot {10}^{-5}$
Burdman:1995te	$\left(0.1-3.4\right)\cdot {10}^{-5}$	$\left(7-12\right)\cdot {10}^{-5}$	$0.1\cdot {10}^{-6}$	$\left(0.1-0.3\right)\cdot {10}^{-5}$	$\left(6-38\right)\cdot {10}^{-5}$
Khodjamirian:1995uc 333Uncertainties not available. We use $a_1=1.3$ and $a_2=-0.55$ Bauer:1986bm .	–	$1.8\cdot {10}^{-4}$	–	–	$4.7\cdot {10}^{-5}$
Belle Abdesselam:2016yvr †	$\left(2.76\pm 0.21\right)\cdot {10}^{-5}$	$\left(4.66\pm 0.30\right)\cdot {10}^{-4}$	–	–	–
BaBar Aubert:2008ai †444We update the normalization Olive:2016xmw .	$\left(2.81\pm 0.41\right)\cdot {10}^{-5}$	$\left(3.31\pm 0.34\right)\cdot {10}^{-4}$	–	–	–

Table 3. Table 3: Couplings to eq. ( IV.2 ) with flavor indices made explicit. The lepton index l 𝑙 l also represents neutrinos in the models S 2 subscript 𝑆 2 S_{2} (for κ ′ superscript 𝜅 ′ \kappa^{\prime} ), S 3 subscript 𝑆 3 S_{3} (with an additional factor 2), V ~ 2 subscript ~ 𝑉 2 \tilde{V}_{2} and V 3 subscript 𝑉 3 V_{3} .

LQ	$\kappa$	${\kappa }^{\prime }$	$\nu$	${\nu }^{\prime }$
$S_1$	${\lambda }_R^{\left(cl\right)}{\left({\lambda }_R^{\left(ul\right)}\right)}^{\ast }$	${\lambda }_L^{\left(cl\right)}{\left({\lambda }_L^{\left(ul\right)}\right)}^{\ast }$	${\lambda }_L^{\left(cl\right)}{\left({\lambda }_R^{\left(ul\right)}\right)}^{\ast }$	${\lambda }_R^{\left(cl\right)}{\left({\lambda }_L^{\left(ul\right)}\right)}^{\ast }$
$S_2$	${\left({\lambda }_R^{\left(cl\right)}\right)}^{\ast }{\lambda }_R^{\left(ul\right)}$	${\left({\lambda }_L^{\left(cl\right)}\right)}^{\ast }{\lambda }_L^{\left(ul\right)}$	${\left({\lambda }_L^{\left(cl\right)}\right)}^{\ast }{\lambda }_R^{\left(ul\right)}$	${\left({\lambda }_R^{\left(cl\right)}\right)}^{\ast }{\lambda }_L^{\left(ul\right)}$
$S_3$	–	${\lambda }^{\left(cl\right)}{\left({\lambda }^{\left(ul\right)}\right)}^{\ast }$	–	–
$V_1$	–	$-{\left({\lambda }^{\left(c\nu \right)}\right)}^{\ast }{\lambda }^{\left(u\nu \right)}$	–	–
${\tilde{V}}_1$	$-{\left({\lambda }^{\left(cl\right)}\right)}^{\ast }{\lambda }^{\left(ul\right)}$	–	–	–
$V_2$	${\lambda }^{\left(cl\right)}{\left({\lambda }^{\left(ul\right)}\right)}^{\ast }$	–	–	–
${\tilde{V}}_2$	–	${\lambda }^{\left(cl\right)}{\left({\lambda }^{\left(ul\right)}\right)}^{\ast }$	–	–
$V_3$	–	$-2{\left({\lambda }^{\left(cl\right)}\right)}^{\ast }{\lambda }^{\left(ul\right)}$	–	–

Table 4. Table 4: Leptoquark induced coefficients for M = 1 ​ TeV 𝑀 1 TeV M=1\,\text{TeV} from eq. ( IV.2 ). For M → 10 ​ TeV → 𝑀 10 TeV M\to 10\,\text{TeV} the effective coupling scales as λ ( c ​ l ) ​ λ ( u ​ l ) / ( M = 1 ​ TeV ) 2 → 0.9 ​ λ ( c ​ l ) ​ λ ( u ​ l ) / ( M = 10 ​ TeV ) 2 → superscript 𝜆 𝑐 𝑙 superscript 𝜆 𝑢 𝑙 superscript 𝑀 1 TeV 2 0.9 superscript 𝜆 𝑐 𝑙 superscript 𝜆 𝑢 𝑙 superscript 𝑀 10 TeV 2 \lambda^{(cl)}\lambda^{(ul)}/(M=1\,\text{TeV})^{2}\to 0.9\,\lambda^{(cl)}\lambda^{(ul)}/(M=10\,\text{TeV})^{2} . The lepton index l 𝑙 l also represents neutrinos in the models S 2 , 3 subscript 𝑆 2 3 S_{2,3} , V ~ 2 subscript ~ 𝑉 2 \tilde{V}_{2} and V 3 subscript 𝑉 3 V_{3} .

LQ	${\delta }_{LQ}{A}_7$	${\delta }_{LQ}{A}_7^{\prime }$
$S_1$	$-0.001{\lambda }_R^{\left(cl\right)}{\left({\lambda }_R^{\left(ul\right)}\right)}^{\ast }+0.04\frac{m_l}{m_{\tau }}{\lambda }_L^{\left(cl\right)}{\left({\lambda }_R^{\left(ul\right)}\right)}^{\ast }$	$-0.001{\lambda }_L^{\left(cl\right)}{\left({\lambda }_L^{\left(ul\right)}\right)}^{\ast }+0.04\frac{m_l}{m_{\tau }}{\lambda }_R^{\left(cl\right)}{\left({\lambda }_L^{\left(ul\right)}\right)}^{\ast }$
$S_2$	$0.002{\left({\lambda }_R^{\left(cl\right)}\right)}^{\ast }{\lambda }_R^{\left(ul\right)}-0.03\frac{m_l}{m_{\tau }}{\left({\lambda }_L^{\left(cl\right)}\right)}^{\ast }{\lambda }_R^{\left(ul\right)}$	$0.004{\left({\lambda }_L^{\left(cl\right)}\right)}^{\ast }{\lambda }_L^{\left(ul\right)}-0.03\frac{m_l}{m_{\tau }}{\left({\lambda }_R^{\left(cl\right)}\right)}^{\ast }{\lambda }_L^{\left(ul\right)}$
$S_3$	0	$-0.003{\lambda }^{\left(cl\right)}{\left({\lambda }^{\left(ul\right)}\right)}^{\ast }$
$V_1$	0	$0.003{\left({\lambda }^{\left(c\nu \right)}\right)}^{\ast }{\lambda }^{\left(u\nu \right)}$
${\tilde{V}}_1$	$0.01{\left({\lambda }^{\left(cl\right)}\right)}^{\ast }{\lambda }^{\left(ul\right)}$	0
$V_2$	$0.005{\left({\lambda }^{\left(cl\right)}\right)}^{\ast }{\lambda }^{\left(ul\right)}$	0
${\tilde{V}}_2$	0	$0.01{\lambda }^{\left(cl\right)}{\left({\lambda }^{\left(ul\right)}\right)}^{\ast }$
$V_3$	0	$0.04{\left({\lambda }^{\left(cl\right)}\right)}^{\ast }{\lambda }^{\left(ul\right)}$

Table 5. Table 5: Scalar and vector leptoquark constraints on couplings to τ 𝜏 \tau ’s scaling as TeV / M TeV 𝑀 \text{TeV}/M and TeV / M TeV 𝑀 \sqrt{\text{TeV}/M} for Δ ​ m D 0 Δ subscript 𝑚 superscript 𝐷 0 \Delta m_{D^{0}} . The kaon constraints are found via Carpentier:2010ue . The vector ( V ~ 1 , 2 subscript ~ 𝑉 1 2 \tilde{V}_{1,2} ) leptoquark couplings are unconstrained by the above observables.

couplings/mass	constraint	observable
$\left|{\lambda }_{S_3}^{\left(u\tau \right)}\right|$	$\sim [0.0,0.2]$	${\tau }^{-}\to {\pi }^{-}{\nu }_{\tau }$
$\mathrm{Re}\left[{\lambda }_{SR}^{\left(u\tau \right)}{\left({\lambda }_{SL}^{\left(u\tau \right)}\right)}^{\ast }\right]$	$\sim [0.00,0.09]$
$\left|{\lambda }_{V_{1,3}}^{\left(u\tau \right)}\right|$	$\sim [0.0,0.4]$
$\mathrm{Re}\left[{\lambda }_{S_{1}L,S_3}^{\left(u\tau \right)}{\left({\lambda }_{S_{1}L,S_3}^{\left(c\tau \right)}\right)}^{\ast }\right]$	$\sim [-0.2,0.2]$	${\tau }^{-}\to K^{-}{\nu }_{\tau }$
$\mathrm{Re}\left[{\lambda }_{SR}^{\left(u\tau \right)}{\left({\lambda }_{SL}^{\left(c\tau \right)}\right)}^{\ast }\right]$	$\sim [-0.07,0.04]$
$\left|\mathrm{Im}\left[{\lambda }_{SR}^{\left(u\tau \right)}{\left({\lambda }_{SL}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0.0,0.7]$
$\mathrm{Re}\left[{\lambda }_{V_{1,3}}^{\left(u\tau \right)}{\left({\lambda }_{V_{1,3}}^{\left(c\tau \right)}\right)}^{\ast }\right]$	$\sim [-0.1,0.1]$
$\left|\mathrm{Im}\left[{\lambda }_{V_{1,3}}^{\left(u\tau \right)}{\left({\lambda }_{V_{1,3}}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0.0,0.9]$
$\left|\mathrm{Re}\left[{\lambda }_{SL,SR}^{\left(u\tau \right)}{\left({\lambda }_{SL,SR}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0,0.02]$	$\Delta m_{D^0}$
$\left|\mathrm{Re}\left[{\lambda }_{S_3}^{\left(u\tau \right)}{\left({\lambda }_{S_3}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0,0.007]$
$\left|\mathrm{Re}\left[{\lambda }_{V_2}^{\left(u\tau \right)}{\left({\lambda }_{V_2}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0,0.8]$
$\left|\mathrm{Re}\left[{\lambda }_{V_3}^{\left(u\tau \right)}{\left({\lambda }_{V_3}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\sim [0,0.4]$
$\mathrm{Re}\left[{\lambda }_{SR}^{\left(u\tau \right)}{\left({\lambda }_{SL}^{\left(c\tau \right)}\right)}^{\ast }\right]$	$\lesssim 0.3$	$D^{+}\to {\tau }^{+}{\nu }_{\tau }$
$\left|\mathrm{Re}\left[{\lambda }_{V_{1,3}}^{\left(u\tau \right)}{\left({\lambda }_{V_{1,3}}^{\left(c\tau \right)}\right)}^{\ast }\right]\right|$	$\lesssim 0.5$
$\mathrm{Re}\left[{\lambda }_{SR}^{\left(c\tau \right)}{\left({\lambda }_{SL}^{\left(c\tau \right)}\right)}^{\ast }\right]$	$\sim [-1,0.09]$	$D_s\to {\tau }^{+}{\nu }_{\tau }$
$\left|{\lambda }_{S_3}^{\left(c\tau \right)}\right|$	$\sim [0.0,0.4]$
$\left|{\lambda }_{V_{1,3}}^{\left(c\tau \right)}\right|$	$\sim [0.0,0.2]$
$\left|{\lambda }_{S_{1}L,S_3}^{\left(u\tau \right)}{\lambda }_{S_{1}L,S_3}^{\left(c\tau \right)}\right|$	$\lesssim 4\cdot {10}^{-4}$	$\left(K^{+}\to {\pi }^{+}\overline{\nu }\nu \right)/\left(K^{+}\to {\pi }^0\overline{e}\nu \right)$
$\left|{\lambda }_{V_3}^{\left(u\tau \right)}{\left({\lambda }_{V_3}^{\left(c\tau \right)}\right)}^{\ast }\right|$	$\lesssim 8\times {10}^{-5}$

Equations

$$\mathcal{B}(D^0\to {\rho }^0\gamma )$$

$$A_{CP}(D^0\to {\rho }^0\gamma )$$

$$A_{CP}(D\to V\gamma )$$

$${\mathcal{L}}_{\text{eff}}^{\text{weak}}=\frac{4G_F}{\sqrt{2}}\left(\sum _{q\in \{d,s\}}{V}_{cq}^{\ast }{V}_{uq}\sum _{i=1}^2{C}_i{Q}_i^{(q)}+\sum _{i=3}^6{C}_i{Q}_i+\sum _{i=7}^8\left(C_i{Q}_i+C_i^{\prime }{Q}_i^{\prime }\right)\right),$$

$$Q_1^{(q)}=({\bar{u}}_L{\gamma }_{{\mu }_1}{T}^a{q}_L)({\overline{q}}_L{\gamma }^{{\mu }_1}{T}^a{c}_L),$$

$$Q_3=({\bar{u}}_L{\gamma }_{{\mu }_1}{c}_L)\sum _{\{q:m_q<{\mu }_c\}}(\overline{q}{\gamma }^{{\mu }_1}q),$$

$$Q_5=({\bar{u}}_L{\gamma }_{{\mu }_1}{\gamma }_{{\mu }_2}{\gamma }_{{\mu }_3}{c}_L)\sum _{\{q:m_q<{\mu }_c\}}(\overline{q}{\gamma }^{{\mu }_1}{\gamma }^{{\mu }_2}{\gamma }^{{\mu }_3}q),$$

$$Q_7=\frac{e{m}_c}{16{\pi }^2}({\bar{u}}_L{\sigma }^{{\mu }_1{\mu }_2}{c}_R)F_{{\mu }_1{\mu }_2},$$

$$Q_8=\frac{g_s{m}_c}{16{\pi }^2}({\bar{u}}_L{\sigma }^{{\mu }_1{\mu }_2}{T}^a{c}_R)G_{{\mu }_1{\mu }_2}^a,$$

$$C_1^{(0)}\in [-1.28,-0.83],C_2^{(0)}\in [1.14,1.06],$$

$$C_8^{(0)\text{eff}}\in [0.47\cdot 10^{-5}-1.33\cdot 10^{-5}i,0.21\cdot 10^{-5}-0.61\cdot 10^{-5}],$$

$$\frac{4}{9}{C}_1^{(0)}+\frac{1}{3}{C}_2^{(0)}\in [-0.189,-0.018],$$

$$C_7^{\text{eff}}\in [-0.00151-(0.00556i{)}_{\text{s}}+(0.00005i{)}_{\text{CKM}},-0.00088-(0.00327i{)}_{\text{s}}+(0.00002i{)}_{\text{CKM}}]$$

$$\Gamma =\frac{m_D^3}{32\pi }{\left(1-\frac{m_V^2}{m_D^2}\right)}^3\left(|A_{\text{PC}}{|}^2+|A_{\text{PV}}{|}^2\right),$$

$$A_{\text{PC/PV}}=\frac{\sqrt{{\alpha }_{e}4\pi }{G}_F{m}_c}{2\sqrt{2}{\pi }^2}(A_7\pm A_7^{\prime })T$$

$$A_7^{(\prime )}=C_7^{(\prime )\text{eff}}+...,$$

$$C_7^{\text{HSI},V}=\frac{{\alpha }_s({\mu }_h)}{4\pi }\left(\sum _{q\in \{d,s\}}{V}_{cq}^{\ast }{V}_{uq}\left(-\frac{1}{6}{C}_1^{(0)}({\mu }_h)+C_2^{(0)}({\mu }_h)\right)H_1^{(q)}+C_8^{(0)\text{eff}}({\mu }_h)H_8\right),$$

$$H_1^{(q)}=\frac{4{\pi }^2{f}_D{f}_V^{\perp }}{27T{m}_D{\lambda }_D}{\int }_0^1\mathrm{d}v{h}_V^{(q)}(\bar{v}){\Phi }_{V\perp }(v),$$

$$h_V^{(q)}=\frac{4m_q^2}{m_c^2{\bar{v}}^2}\left(\mathrm{L}{\mathrm{i}}_2\left[\frac{2}{1-\sqrt{(\bar{v}-4m_q^2/m_c^2+iϵ])/\bar{v}}}\right]+\mathrm{L}{\mathrm{i}}_2\left[\frac{2}{1+\sqrt{(\bar{v}-4m_q^2/m_c^2+iϵ])/\bar{v}}}\right]\right)-\frac{2}{\bar{v}},$$

$$H_8=-\frac{32{\pi }^2{f}_D{f}_V^{\perp }}{27T{m}_D{\lambda }_D}{\int }_0^1\mathrm{d}v\frac{{\Phi }_{V\perp }(v)}{v},$$

$${\Phi }_{V\perp }=6v\bar{v}\left(1+a_1^{V\perp }3(v-\bar{v})+a_2^{V\perp }\frac{3}{2}\left(5(v-\bar{v}{)}^2-1\right)\right).$$

$$\frac{m_D}{{\lambda }_D}={\int }_0^1\mathrm{d}\xi \frac{{\Phi }_D(\xi )}{\xi },$$

$$C_7^{\text{HSI},\rho }\in [0.00051+0.0014i,0.00091+0.0020i]\cdot \frac{\text{GeV}}{{\lambda }_D},$$

$$C_7^{\text{HSI},\omega }\in [0.00030+0.0010i,0.00098+0.0020i]\cdot \frac{\text{GeV}}{{\lambda }_D},$$

$$C_7^{\text{HSI},{K^{\ast }}^{+}}\in [0.00032+0.0013i,0.00096+0.0022i]\cdot \frac{\text{GeV}}{{\lambda }_D}.$$

$$C_7^{\text{WA},{\rho }^0}=-\frac{2{\pi }^2{Q}_u{f}_D{f}_{{\rho }^0}^{(d)}{m}_{\rho }}{T{m}_{D^0}{m}_c{\lambda }_D}{V}_{cd}^{\ast }{V}_{ud}\left(\frac{4}{9}{C}_1^{(0)}+\frac{1}{3}{C}_2^{(0)}\right),$$

$$C_7^{\text{WA},\omega }=\frac{2{\pi }^2{Q}_u{f}_D{f}_{\omega }^{(d)}{m}_{\omega }}{T{m}_{D^0}{m}_c{\lambda }_D}{V}_{cd}^{\ast }{V}_{ud}\left(\frac{4}{9}{C}_1^{(0)}+\frac{1}{3}{C}_2^{(0)}\right),$$

$$C_7^{\text{WA},{\rho }^{+}}=\frac{2{\pi }^2{Q}_d{f}_D{f}_{\rho }{m}_{\rho }}{T{m}_{D^{+}}{m}_c{\lambda }_D}{V}_{cd}^{\ast }{V}_{ud}{C}_2^{(0)},$$

$$C_7^{\text{WA},{K^{\ast }}^{+}}=\frac{2{\pi }^2{Q}_d{f}_{D_s}{f}_{K^{\ast }}{m}_{K^{\ast }}}{T{m}_{D_s}{m}_c{\lambda }_D}{V}_{cs}^{\ast }{V}_{us}{C}_2^{(0)},$$

$$C_7^{\text{WA},{\rho }^0}\in [-0.010,-0.0011]\cdot \frac{\text{GeV}}{{\lambda }_D},$$

$$C_7^{\text{WA},\omega }\in [0.0097,0.0011]\cdot \frac{\text{GeV}}{{\lambda }_D},$$

$$C_7^{\text{WA},{\rho }^{+}}\in [0.029,0.038]\cdot \frac{\text{GeV}}{{\lambda }_D},$$

$$C_7^{\text{WA},{K^{\ast }}^{+}}\in [-0.034,-0.047]\cdot \frac{\text{GeV}}{{\lambda }_D}.$$

$$|C_7^{\text{WA},V^{+}}|>|C_7^{\text{WA},V^0}|\gtrsim |C_7^{\text{HSI}}|>|C_7^{\text{eff}}|.$$

$$C_7^{\prime (c\to u\gamma g)}\sim \frac{1}{3}{C}_2^{(0)}\left(V_{cd}^{\ast }{V}_{ud}{f}^{(c\to u\gamma g)}(m_d^2/m_c^2)+V_{cs}^{\ast }{V}_{us}{f}^{(c\to u\gamma g)}(m_s^2/m_c^2)\right)\frac{{\Lambda }_{\text{QCD}}}{m_c},$$

$$|A_{7,\mathrm{SM}}^{\prime }/A_{7,\mathrm{SM}}|\lesssim 0.2,$$

$$\displaystyle C_{7}^{(\prime)\text{HSI},\rho}\big{|}_{\langle Q_{8}^{(\prime)}\rangle}\in-\frac{\text{GeV}}{\lambda_{D}}\cdot[0.031,0.042]\cdot C_{8}^{(\prime)}\,,$$

$$\displaystyle C_{7}^{(\prime)\text{HSI},\omega}\big{|}_{\langle Q_{8}^{(\prime)}\rangle}\in-\frac{\text{GeV}}{\lambda_{D}}\cdot[0.024,0.040]\cdot C_{8}^{(\prime)}\,,$$

$$\displaystyle C_{7}^{(\prime)\text{HSI},{K^{*}}^{+}}\big{|}_{\langle Q_{8}^{(\prime)}\rangle}\in-\frac{\text{GeV}}{\lambda_{D}}\cdot[0.031,0.039]\cdot C_{8}^{(\prime)}\,.$$

Full text

††preprint: DO-TH 16/20, QFET-2016-18

I Introduction

A multitude of radiative charm decays is accessible at current and future high luminosity flavor facilities Hewett:2004tv ; Aushev:2010bq . In anticipation of the new data we revisit opportunities to test the standard model (SM) with $c\to u\gamma$ transitions Burdman:1995te ; Khodjamirian:1995uc ; Greub:1996wn ; Fajfer:1997bh ; Fajfer:1998dv ; Fajfer:1999dq ; Fajfer:2000zx ; Isidori:2012yx , complementing studies with dileptons, e.g., Paul:2011ar ; Fajfer:2001sa ; deBoer:2015boa . To estimate the beyond the standard model (BSM) reach we detail and evaluate exclusive $D_{(s)}\to V\gamma$ 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 Bosch:2001gv , and a hybrid phenomenological one, updating Fajfer:1997bh ; Fajfer:1998dv . 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 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 Fajfer:2000zx . 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 Abdesselam:2016yvr

[TABLE]

where the CP asymmetry $A_{CP}$ is defined as111The CP asymmetry of $D^{0}\to\rho^{0}\gamma$ is mostly direct, analogous to the time-integrated CP asymmetry in $D^{0}\to K^{+}K^{-}$ Aaij:2016cfh . We thank Alan Schwartz for providing us with this information. In this work, we refer to $A_{CP}$ as the direct CP asymmetry, neglecting the small indirect contribution.

[TABLE]

We compare data (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 $|\Delta C|=|\Delta U|=1$ processes at this order Greub:1996wn ; deBoer:2015boa ; deBoer:2016dcg .

While one expects the heavy quark and $\alpha_{s}$-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 II we calculate weak annihilation and hard scattering contributions to $D\to V\gamma$ decay amplitudes. In section III 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 IV. Section V is on $\Lambda_{c}\to p\gamma$ decays and the testability of a polarized $\Lambda_{c}$-induced angular asymmetry at future colliders Zhao:2002zk ; dEnterria:2016fpc . In section VI we summarize. In appendix A and B we give the numerical input and $D\to V$ 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.

II $D\to V\gamma$ in effective Theory framework

The effective weak Lagrangian and SM Wilson coefficients are discussed in section II.1. We work out and provide a detailed breakdown of the individual contributions to $D\to V\gamma$ amplitudes in the heavy-quark approach. We work out weak annihilation and hard gluon exchange corrections in section II.2, with contributions from the gluon dipole operator given in section II.3. In section II.4 we consider weak annihilation induced modes.

II.1 Generalities

The effective $c\to u\gamma$ weak Lagrangian can be written as deBoer:2015boa

[TABLE]

where $G_{F}$ is the Fermi constant, $V_{ij}$ are CKM matrix elements and the operators read

[TABLE]

where $F_{\mu\nu},G^{a}_{\mu\nu},a=1,..,8$ 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 $\mu_{c}$ of the order of the charm mass $m_{c}$, and $\mu_{c}=1.27\,\text{GeV}$ 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 $\alpha_{s}$ one obtains deBoer:2015boa ; deBoer:2016dcg , respectively,

[TABLE]

where $\mu_{c}$ is varied within $[m_{c}/\sqrt{2},\sqrt{2}m_{c}]$. $C_{8}^{(0)\text{eff}}$ is strongly GIM suppressed in the SM and negligible therein. $C_{1}^{(0)}$ and the color suppressed coefficient of the weak annihilation contribution introduced in section II.2 ,

[TABLE]

are subject to a large scale-uncertainty. Note, at next-to leading order, $4/9\,C_{1}+1/3\,C_{2}\in[-0.042,0.092]$. In this work first (second) entries in intervals correspond to the lower (upper) value of $\mu_{c}$ within $[m_{c}/\sqrt{2},\sqrt{2}m_{c}]$.

The effective coefficient of $Q_{7}$ including the matrix elements of $Q_{1-6}$ at two-loop QCD, see Greub:1996wn ; deBoer:2015boa ; deBoer:2016dcg and appendix D for details, is in the range

[TABLE]

and $C_{7}^{\prime\,\text{eff}}\sim m_{u}/m_{c}\simeq 0$. 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_{7}^{\text{eff}}\big{|}_{\langle Q_{3-6}\rangle}\lesssim 10^{-6}, that is, negligible due to small SM Wilson coefficients $C_{3-6}$ and the GIM suppression.

The $D\to V\gamma$ decay rate can be written as Isidori:2012yx

[TABLE]

where the parity conserving (PC) and parity violating (PV) amplitudes read

[TABLE]

times $1/\sqrt{2}$ for $V^{0}\in\{\rho^{0},\omega\}$. 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 $D\to V$ tensor form factor, see appendix B for details. We stress that the dominant SM contribution to $D_{(s)}\to V\gamma$ branching ratios is independent of $T$. Furthermore,

[TABLE]

where the ellipses indicate additional contributions from within and outside the SM. Corrections from within the SM are obtained in sections II.2, II.3 and II.4. BSM coefficients and amplitudes are denoted by $\delta C$ and $\delta A_{7}^{(\prime)}$, respectively. Note, $A_{7}$ and $A_{7}^{\prime}$ do not mix in eq. (8).

II.2 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.

The leading ($\sim\alpha_{s}^{1}\,(\Lambda_{\text{QCD}}/m_{c})^{0}$) hard spectator interaction within QCD factorization adopted from $b$-physics Bosch:2001gv (also Ali:2001ez ; Beneke:2001at ; Beneke:2004dp ) can be written as

[TABLE]

where we consistently use $C_{1,2,8}^{(0)}$ at leading order in $\alpha_{s}$ due to additional non-factorizable diagrams at higher order and $\mu_{h}\sim\sqrt{\Lambda_{\text{QCD}}m_{c}}$. Furthermore,

[TABLE]

$\bar{v}=1-v$ and $\mathrm{Li_{2}}[x]=-\int_{0}^{x}\mathrm{d}t\,\mathrm{ln}[1-t]/t$ and the decay constants $f_{D},f_{V}^{\perp}$ are given in appendix A. We use $m_{d}=0$. As $Q_{8}^{(\prime)}$-induced HSI contributions are negligible in the SM it follows that $C_{7}^{\rm HSI}$ is driven by $V_{cs}^{*}V_{us}$. The transverse distribution at leading twist is to first order in Gegenbauer polynomials

[TABLE]

Numerical input on the Gegenbauer moments $a_{1,2}^{V\perp}$ is given in appendix A.

The parameter $\lambda_{D}$ is defined as

[TABLE]

that is the first negative moment of the leading twist distribution amplitude $\Phi_{D}$ of the light-cone momentum fraction $\xi$ of the spectator quark within the $D$-meson. In $b$-physics, the first negative moment of the $B$-meson light-cone distribution amplitude, $\lambda_{B}^{\text{HQET}}>0.172\,\text{GeV}$ at 90% C.L. Heller:2015vvm , a positive light-cone wave function yields $\lambda_{B}^{\text{HQET}}\leq 4/3\,\bar{\Lambda}$ Korchemsky:1999qb and by means of light-cone sum rules (LCSR) $\lambda_{B}^{\text{QCD}}\lesssim\bar{\Lambda}$ Ball:2003fq ; Khodjamirian:2005ea , where $\bar{\Lambda}=(m_{B}-m_{b})+\mathcal{O}(\Lambda_{\text{QCD}}^{2}/m_{b})$ and $\lambda_{B}^{\text{HQET}}>\lambda_{B}^{\text{QCD}}$ at one-loop QCD Pilipp:2007sb . We use $\lambda_{D}\sim\Lambda_{\text{QCD}}\sim\mathcal{O}(0.1\,\text{GeV})$.

Taking $\mu_{h}=1\,\text{GeV}$, varying the Gegenbauer moments and decay constants (but not the form factor $T$ as it cancels in the amplitude) we find

[TABLE]

We neglect isospin breaking in the Gegenbauer moments of the $\rho$. Contributions induced by $Q_{8}^{(\prime)}$ are discussed in section II.3.

The leading ($\sim\alpha_{s}^{0}\,(\Lambda_{\text{QCD}}/m_{c})^{1}$) weak annihilation contribution to $D^{0}\to(\rho^{0},\omega)\gamma$, $D^{+}\to\rho^{+}\gamma$ and $D_{s}\to{K^{*}}^{+}\gamma$ can be inferred from $b$-physics Bosch:2001gv ; Bosch:2004nd . We obtain

[TABLE]

where $Q_{u}=2/3$, $Q_{d}=-1/3$ and we consistently use $C_{1,2}^{(0)}$ at leading order in $\alpha_{s}$. 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 $\rho^{0}$ is due to isospin.

Varying the decay constants and $\mu_{c}$ within $[m_{c}/\sqrt{2},\sqrt{2}m_{c}]$ we find

[TABLE]

Note that non-factorizable power corrections (inducing $A_{7}^{\prime}$) could in principle be calculated with LCSR, see, e.g., Ball:2006eu and that non-local corrections to weak annihilation by means of QCD sum rules are additionally power suppressed Kagan:2001zk .

To summarize, we observe the following hierarchies among the SM contributions to $A_{7}$

[TABLE]

The leading SM uncertainties are therefore those stemming from the WA-amplitudes, that is, the $\mu_{c}$-scale and $\lambda_{D}$ uncertainties, followed by the parameters entering HSI-amplitudes, i.e., Gegenbauer moments, decay constants and the $\mu_{h}$-scale. The latter we fixed for simplicity.

Contributions to $A_{7}^{\prime}$ arise in the SM from a $Q_{2}$-induced quark loop with a soft gluon as a power correction Grinstein:2004uu

[TABLE]

which is $\mathcal{O}(10^{-4})$ if the expansion coefficients of $f^{(c\to u\gamma g)}$ in $m_{q}^{2}/m_{c}^{2}$ are order one. Note that the $c\to u\gamma g$ process induces as well a contribution to $A_{7}$, and that the $Q_{1}$-induced quark loop is additionally color suppressed. Note also that $f^{(c\to u\gamma g)}$ could in principle be calculated with LCSR, see e.g., Ball:2006eu , yet $\alpha_{s}$-corrections vanish at leading twist in the limit of massless quarks in the $V$-meson Grinstein:2000pc . To be specific, and in absence of further calculations, we limit the size of the chirality-flipped SM amplitudes in our numerical analysis as

[TABLE]

and take the structure of the weak phases as in (19) into account.

II.3 Contributions from $Q_{8}^{(\prime)}$

We detail here the contributions from $Q_{8}^{(\prime)}$ to $c\to u\gamma$ modes. While in the SM they are negligibly small they can be relevant in BSM scenarios.

Numerically, we find for the $\langle Q_{8}^{(\prime)}\rangle$-induced hard spectator interaction of eq. (11)

[TABLE]

Note that QCD factorization breaks down at subleading power for $\langle Q_{8}\rangle$ hard spectator interaction due to a logarithmic singularity for a soft spectator quark Kagan:2001zk .

Alternatively, LCSR yield the $\langle Q_{8}^{(\prime)}\rangle$ gluon spectator interaction (GSI) Dimou:2012un

[TABLE]

The contributions in eqs. (22) and (21) are similar in size for $\lambda_{D}\sim\mathcal{O}(0.1\,\text{GeV})$. One may compare these to the $\langle Q_{8}^{(\prime)}\rangle$ induced contribution to eq. (7)

[TABLE]

BSM values $\delta C_{8}^{(\prime)}\lesssim 0.1$ can therefore lift the HSI/GSI contributions and the one to $C_{7}^{(\prime)\text{eff}}$ such that

[TABLE]

II.4 Weak annihilation induced modes

The contributions to $A_{7}$ of the weak annihilation induced decays $D^{0}\to(\phi,\bar{K}^{*0},{K^{*}}^{0})\gamma$, $D^{+}\to{K^{*}}^{+}\gamma$ and $D_{s}\to\rho^{+}\gamma$ are obtained as follows

[TABLE]

Here we made the flavor structure of the Wilson coefficients explicit, however, since QCD is flavor symmetric, use $C_{1/2}^{(\bar{q}_{1}q_{2})(\bar{q}_{3}c)}=C_{1/2}^{(0)}$. While the form factor $T$ is process-dependent, it cancels together with $m_{c}$ in the decay amplitude. Numerically, $\text{GeV}/(m_{c}T)\sim 1$ . Varying the decay constants and $\mu_{c}\in[m_{c}/\sqrt{2},\sqrt{2}m_{c}]$ we find

[TABLE]

For $D^{0}\to\phi\gamma$ additional contributions to the decay amplitude can arise, induced by $d\bar{d}+u\bar{u}$-admixture in the $\phi$ or rescattering Isidori:2012yx . Such effects can be parametrized by $y$ as follows

[TABLE]

To estimate $A_{CP}$ we made CKM factors explicit except for the $C_{7}^{\rm eff}$-induced term which can receive large BSM CP violating phases. The amplitudes correspond, in order of appearance, to $C_{7}^{D^{0}\to\phi\gamma}$, and the three contributions eqs. (16), (II.2) and (7) to $D^{0}\to\rho^{0}\gamma$. Note the minus signs due do the $SU(3)$-composition. One obtains, model-independently,

[TABLE]

III SM Phenomenology

We provide SM predictions for various $D_{(s)}\to V\gamma$ modes and compare to existing data. In addition to the QCD-based approach of the previous section we present branching ratios in the phenomenological approach of Fajfer:1997bh ; Fajfer:1998dv . 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 Fajfer:1997bh ; Fajfer:1998dv we rewrite the amplitudes in terms of newly measured parameters and vary (updated) parameters within uncertainties. Analytical expressions for the $D\to V\gamma$ amplitudes are provided in appendix C. The hierarchies of the various $D\to V\gamma$ 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) $c\to u\gamma$ modes. The branching ratio from two-loop QCD eq. (7) is subleading in each case. The branching ratios in the hybrid approach cover the ranges previously obtained in Burdman:1995te ; Khodjamirian:1995uc ; Fajfer:1997bh ; Fajfer:1998dv . The measured $D^{0}\to\rho^{0}\gamma$ branching ratio is somewhat above the SM prediction in the hybrid model.

The branching ratios of $D\to\rho\gamma$ as a function of $\lambda_{D}$ are shown in figure 2.

The $D^{+}\to\rho^{+}\gamma$ SM branching ratio is $\lesssim 2\cdot 10^{-4}$, a measurement would constrain $\lambda_{D}$ efficiently. Specifically, we find $\mathcal{B}(D^{+}\to\rho^{+}\gamma)\simeq[44,2900]\cdot\mathcal{B}(D^{0}\to\rho^{0}\gamma)$ by means of hard spectator interaction plus weak annihilation and in the hybrid model $\mathcal{B}(D^{+}\to\rho^{+}\gamma)\simeq[0.3,280]\cdot\mathcal{B}(D^{0}\to\rho^{0}\gamma)$. The $D^{0}\to\rho^{0}\gamma$ branching ratio can be subject to stronger cancellations between the contributions in eq. (18) than in the hybrid model. Assuming that the phase of each amplitude $A_{\text{PV/PC}}^{\text{I/II/III}}$ is equal for $D^{+}\to\rho^{+}\gamma$ and $D^{0}\to\rho^{0}\gamma$ reduces the possible isospin breaking to $\mathcal{B}(D^{+}\to\rho^{+}\gamma)\simeq[0.6,140]\cdot\mathcal{B}(D^{0}\to\rho^{0}\gamma)$. Note, isospin is already significantly broken by the lifetimes $\tau(D^{0})/\tau(D^{+})\simeq 0.4$ Olive:2016xmw .

The uncertainties in the hybrid model are dominated by the relative strong phases, followed by the phenomenological fit coefficients $a_{1}=1.3\pm 0.1$, $a_{2}=-0.55\pm 0.1$ Bauer:1986bm (also Buras:1994ij ; Cheng:2002wu ).

The branching ratios of $D^{0}\to(\phi,\bar{K}^{*0},{K^{*}}^{0})\gamma$, $D^{+}\to{K^{*}}^{+}\gamma$ and $D_{s}\to\rho^{+}\gamma$ are given in table 2.

The measurements by Belle Abdesselam:2016yvr and BaBar Aubert:2008ai of ${\cal{B}}(D^{0}\to\bar{K}^{*0}\gamma)$ differ by $2.2\sigma$, yet both are in the range of the hybrid model predictions. Interpreted in the QCD framework to the order we are working, ${\cal{B}}(D^{0}\to(\bar{K}^{*0},\phi)\gamma)$ data require a low value of $\lambda_{D}$ below 0.1 GeV or a low charm mass scale $\mu_{c}\sim m_{c}/2$, similar to ${\cal{B}}(D^{0}\to\rho^{0}\gamma)$ data assuming the SM. Quite generally the deficiency in explaining the largely SM-dominated branching ratios in table 2 suggest slow convergence of the $1/m_{D}$, $\alpha_{s}$-expansion. Measurements of branching ratios of color allowed $D^{+},D_{s}$ mesons could shed light on this, as here the scale uncertainty is smaller.

The CP asymmetry for $D^{0}\to\rho^{0}\gamma$ within the SM in the heavy quark-based approach is shown in figure 3 as a function of the SM $D^{0}\to\rho^{0}\gamma$ branching ratio.

Within the SM $|A_{CP}|\lesssim 2\cdot 10^{-2}$, if the branching ratio is $\gtrsim 10^{-9}$ and $|A_{CP}|\lesssim 2\cdot 10^{-3}$, if $\mathcal{B}(D^{0}\to\rho^{0}\gamma)\gtrsim 10^{-6}$, e.g., for $\lambda_{D}\lesssim 0.3$ GeV, and as measured (1) assuming the SM. $A_{CP}^{\rm SM}$ calculated in the hybrid model is $\lesssim 10^{-3}$, and vanishes in the $SU(3)$-limit. The SM CP asymmetry for $D^{0}\to\omega\gamma$ is very similar to $A_{CP}(D^{0}\to\rho^{0}\gamma)$; the CP asymmetries in the SM for the charged decays $D^{+}\to\rho^{+}\gamma$ and $D_{s}\to{K^{*}}^{+}\gamma$ are $\lesssim 2\cdot 10^{-3}$ and $\lesssim 3\cdot 10^{-4}$ 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 $D^{0}\to(\bar{K}^{*0},{K^{*}}^{0})\gamma$, $D^{+}\to{K^{*}}^{+}\gamma$ and $D_{s}\to\rho^{+}\gamma$ vanish due to a single weak phase c.f. eq. (25) and appendix C. The decay $D^{0}\to\phi\gamma$ is special as it can receive contributions at a fraction $y$ similar to $D^{0}\to\rho^{0}\gamma$ decays, and has therefore a finite CP asymmetry, estimated in equation (28). Taking into account a percent level $u\bar{u}+d\bar{d}$ content in the $\phi$ Olive:2016xmw values of $A_{CP}$ up to ${\cal{O}}(10^{-4})$ in the SM and up to ${\cal{O}}(10^{-3})$ in BSM models can arise in $D^{0}\to\phi\gamma$ decays. Effects from rescattering at the $\phi$-mass are roughly $y\lesssim 0.1$, hence corresponding CP asymmetries can reach ${\cal{O}}(10^{-3})$ in the SM and ${\cal{O}}(10^{-2})$ in BSM scenarios. The following asymmetries have been measured Abdesselam:2016yvr ,

[TABLE]

$A_{CP}(D^{0}\to\phi\gamma)$ 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+HS 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 Isidori:2012yx . 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. (18), allowing for possible additional suppressions of CP asymmetries beyond CKM factors.

IV $D\to V\gamma$ beyond the Standard Model

In section IV.1 we work out model-independent constraints on $A_{7}^{(\prime)}$, $C_{7}^{(\prime)}$ and $C_{8}^{(\prime)}$. We calculate BSM Wilson coefficients within leptoquark models in section IV.2 and in SUSY in section IV.3, respectively, and discuss BSM implications.

IV.1 Model-independently

Model-independently, from $\mathcal{B}(D^{0}\to\rho^{0}\gamma)$ data, eq. (1), we obtain

[TABLE]

Constraints from $\mathcal{B}(D^{+}\to\pi^{+}\mu^{+}\mu^{-})$ data are similar deBoer:2015boa . These constraints prohibit that decays $D^{+}\to\rho^{+}\gamma$ and $D_{s}\to{K^{*}}^{+}\gamma$ are dominated by a BSM dipole contribution. Still, a sizable $\delta C_{7}^{(\prime)}$ can give the leading contribution to the neutral modes, causing their branching ratios to be very close to each other, $\mathcal{B}(D^{0}\to\omega\gamma)\sim\mathcal{B}(D^{0}\to\rho^{0}\gamma)$, similar to the SM, see table 1. Non-observation of this correlation indicates the presence of intermediate values of $\delta C_{7}^{(\prime)}$ Fajfer:2000zx .

BSM-induced CP asymmetries can reach $\mathcal{O}(0.1)$. Hence, current data on $A_{CP}(D^{0}\to\rho^{0}\gamma)$ 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 $D^{0}\to\rho^{0}\gamma$ branching ratio and CP asymmetry, we assume hypothetical data with a factor four reduced statistical uncertainty of the current measurements with central values kept Abdesselam:2016yvr , that is,

[TABLE]

In figure 4 corresponding 1 sigma-constraints are shown. The ones for $A_{7}$ (purple) and $A_{7}^{\prime}$ (cyan) are similar. The bounds are stronger for the QCD-based framework, roughly $0.2\lesssim|\delta A_{7}^{(\prime)}|\lesssim 0.3$, $|\mathrm{Im}[\delta A_{7}^{(\prime)}]|\gtrsim\mathcal{O}(0.001)$ (plot to the left) than for the hybrid one, $0.1\lesssim|\delta A_{7}^{(\prime)}|\lesssim 0.4$, $|\mathrm{Im}[\delta A_{7}]|\gtrsim\mathcal{O}(0.001)$, $|\mathrm{Im}[\delta A_{7}^{\prime}]|\gtrsim\mathcal{O}(0.0001)$ (plot to the right), improving on the present situation, eq. (30). In the hybrid model the unknown strong phases prohibit a meaningful calculation of $A_{7}/A_{7}^{\prime}$ in the SM Fajfer:1998dv . We therefore do not employ constraints from eq. (20) in this model.

We consider now the impact of the chromomagnetic dipole operator. The CP asymmetry induced by the matrix element of $Q_{8}^{(\prime)}$ estimated within LCSR reads Lyon:2012fk

[TABLE]

which, together with data eq. (1), yields the constraint $|\mathrm{Im}[\delta C_{8}^{(\prime)}]|\lesssim\mathcal{O}(0.1)$. CP violation in charm is also constrained by data on $\Delta A_{CP}=A_{CP}(K^{+}K^{-})-A_{CP}(\pi^{+}\pi^{-})\sim-2\mathrm{Im}[C_{8}-C_{8}^{\prime}]\sin\delta_{KK-\pi\pi}$ Giudice:2012qq ; Lyon:2012fk , where $\Delta A_{CP}=-0.00134\pm 0.00070$ Amhis:2016xyh . To escape a potentially strong bound on ${\rm Im}[\delta C_{8}-\delta C_{8}^{\prime}]$ at permille level requires suppression by the unknown strong phase difference between $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$, $\delta_{KK-\pi\pi}$, or some cancellations between different sources of BSM CP violation.

The Wilson coefficients $\delta C_{7}^{(\prime)}(M)$ and $\delta C_{8}^{(\prime)}(M)$ are generically related within BSM models, where $M$ denotes the matching scale, of the order of the electroweak scale or higher. In addition, $\delta C_{7,8}^{(\prime)}$ are related by the renormalization group evolution. At one-loop QCD

[TABLE]

where

[TABLE]

Including effects of $\langle Q_{8}^{(\prime)}\rangle$, eq. (23), and neglecting SM contributions, we find

[TABLE]

Additionally, we find the mixing via $Q_{3-6}^{(\prime)}$, deBoer:2016dcg and appendix D, neglecting the SM,

[TABLE]

BSM effects from 4-quark operators are, however, strongly constrained by $\epsilon^{\prime}/\epsilon$ and $D-\bar{D}$ mixing, and we do not consider this possibility any further.

To compete with the SM $\delta C^{(\prime)}(M)\sim{\cal{O}}(0.1-1)$ is required, which is difficult to achieve given the loop factor and possible further flavor suppressions. However, BSM CP asymmetries around a percent require $\delta C(M)$ of a few permille only but need sizable phases. The impact of $\delta C^{\prime}(M)$ 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.

IV.2 Leptoquark models

We consider contributions from scalar $S_{1,2,3}$ and vector $V_{1,2,3},\tilde{V}_{1,2}$ leptoquark representations to $c\to u\gamma$ processes, see Buchmuller:1986zs ; Davidson:1993qk ; deBoer:2015boa ; Dorsner:2016wpm ; Hiller:2016kry for Lagrangians and details 555In deBoer:2015boa the notation differs from the one used here by means of charge conjugated fields. Here we write $q\to\bar{q}^{C}$ for the leptoquarks $S_{1}$, $S_{3}$, $V_{2}$ and $\tilde{V}_{2}$ in deBoer:2015boa and adjust their couplings correspondingly. Moreover, here an additional sign for all vector leptoquarks is accounted for. Conclusions in deBoer:2015boa are unaffected.. In this section we denote by $M$ the mass of the leptoquark and by $\lambda_{L/R}$ leptoquark couplings to left-/right-handed leptons. To simplify the notation this includes also neutrinos with appropriate replacements. For vector-like couplings we omit the chirality index.

A matching at $\mu\sim M$ yields, employing expressions from Lavoura:2003xp , $\delta_{S}C_{7}^{(\prime)}(\mu=M)=1/(2\sqrt{2}G_{F})([-Q_{l}/12-Q_{S}/24]\kappa^{(\prime)}/M^{2}+m_{l}/m_{c}[-Q_{S}/4+Q_{l}(3/4+1/2\ln[\mu_{M}^{2}/M^{2}])]\nu^{(\prime)}/M^{2})$ and $\delta_{V}C_{7}^{(\prime)}(\mu=M)=1/(2\sqrt{2}G_{F})[Q_{l}/3+Q_{V}5/12]\kappa^{(\prime)}/M^{2}$, where $Q_{l}$ denotes the electric charge of the lepton. The electric charge of the leptoquark $Q_{S,V}$ is fixed by charge conservation and the following vector $(V)$ and scalar $(S)$ operators are induced at tree-level

[TABLE]

plus chirality-flipped contributions. Here, $C_{V}^{(l)(\prime)}(\mu=M)=\kappa^{(\prime)}/M^{2}$ and $C_{S}^{(l)(\prime)}(\mu=M)=\nu^{(\prime)}/M^{2}$. The couplings $\kappa^{(\prime)}$ and $\nu^{(\prime)}$ within leptoquark models are given in table 3. At one-loop QCD $C_{V}^{(l)(\prime)}(\mu_{c})=C_{V}^{(l)(\prime)}(M)$, $C_{S}^{(l)(\prime)}(\mu)=(\alpha_{s}(M)/\alpha_{s}(\mu))^{-4\beta_{0}}\,C_{S}^{(l)(\prime)}(M)$ and $\delta C_{7}^{(\prime)}(\mu)=(\alpha_{s}(M)/\alpha_{s}(\mu))^{16/(3\beta_{0})}\delta C_{7}^{(\prime)}(M)-3/56[(\alpha_{s}(M)/\alpha_{s}(\mu))^{16/(3\beta_{0})}-(\alpha_{s}(M)/\alpha_{s}(\mu))^{-4/\beta_{0}}]C_{S}^{(l)(\prime)}(M)/(2\sqrt{2}G_{F})$, where $\beta_{0}=11-2/3\,n_{f}$ and $n_{f}$ is the number of active flavors, hence thresholds need to be taken into account.

At the scale $\mu=m_{\tau}$ the $\tau$ lepton is to be integrated out. Since numerically $m_{\tau}\sim\sqrt{2}m_{c}$ we include the tau-loop contributions in the matrix element of $O_{V,S}^{(l)}$, see figure 5.

The contribution of $O_{V}^{(l)(\prime)}$ vanishes to all orders in $\alpha_{s}$. Including the matrix element of $O_{S}^{(l)(\prime)}$ we obtain

[TABLE]

Here,

[TABLE]

and $a_{7}$ is given in eq. (34). The resulting coefficients are worked out numerically in table 4. Finite contributions are expected at two-loop QED. The corresponding leading order calculation in $\alpha_{e}$ is similar to the $\langle Q_{2}\rangle$ two-loop QCD calculation in Greub:1996wn ; Greub:1996tg , and beyond the scope of our work. Note that $\delta_{LQ}A_{8}^{(\prime)}(\mu_{c})$ is additionally $\alpha_{e}/(4\pi)$ suppressed and will be neglected throughout.

Constraints on $\tau$ couplings are worked out and given in table 5, where we followed deBoer:2015boa and used Olive:2016xmw . Note that $\mathcal{B}(D^{0}\to\rho^{0}\gamma)$ yields no constraint for $\lambda\lesssim 1$. Lepton flavor violating $\tau$ decays constrain couplings with a $\tau$ 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, the largest contributions are found for $l=\tau$ and the constraints given in table 5, and read

[TABLE]

for the chirality-flipping contributions $\propto m_{\tau}/m_{c}$ 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 $\lambda_{R}\lambda_{L}^{*}$ relative to the imaginary ones, the latter of which are weaker constrained experimentally.

For $l=e,\mu$ we find with the constraints given in deBoer:2015boa that $|\delta_{\tilde{V}_{1}}A_{7}|_{e,\mu}\lesssim 0.0006$, $|\delta_{V_{2}}A_{7}|_{e,\mu}\lesssim 0.0003$, $|\delta_{\tilde{V}_{2}}A_{7}^{\prime}|_{e,\mu}\lesssim 0.0006$ and $|\delta_{V_{3}}A_{7}^{\prime}|_{e,\mu}\lesssim 0.003$. The contributions from scalar leptoquarks are negligible, $|\delta_{S_{1,2,3}}A_{7}^{(\prime)}|_{e,\mu}\lesssim\mathcal{O}(10^{-5})$.

The $\tau$ couplings for $S_{1L}$, $S_{3}$ and $V_{3}$ receive their strongest constraint from $K$ decays, $|\delta_{S_{1L,3}}A_{7}^{\prime}|_{\tau}\lesssim\mathcal{O}(10^{-7})$ and $|\delta_{V_{3}}A_{7}^{\prime}|_{\tau}\lesssim\mathcal{O}(10^{-6})$. All other bounds in table 5 can be escaped with phase-tuning ${\rm Im}[\lambda^{(u\tau)}(\lambda^{(c\tau)})^{*}]\gg{\rm Re}[\lambda^{(u\tau)}(\lambda^{(c\tau)})^{*}]$. Corresponding BSM coefficients $\delta A_{7}$ and $\delta A_{7}^{\prime}$ can be read off from table 4 for $\lambda_{L,R}^{(q\tau)}\lesssim 1$. Assuming instead ${\rm Im}[\lambda^{(u\tau)}(\lambda^{(c\tau)})^{*}]\lesssim{\rm Re}[\lambda^{(u\tau)}(\lambda^{(c\tau)})^{*}]$, one obtains from $D$-mixing $|\delta_{S_{1L,2L,2R}}A_{7}^{(\prime)}|_{\tau}\lesssim\mathcal{O}(10^{-5})$. For the $SU(2)$-singlet and -doublet vector leptoquarks we find $|\delta_{V_{1}}A_{7}|_{v_{\tau}}\lesssim 0.0003$, $|\delta_{\tilde{V}_{1}}A_{7}|_{\tau}\lesssim 0.01$, $|\delta_{V_{2}}A_{7}|_{\tau}\lesssim 0.004$, and $|\delta_{\tilde{V}_{2}}A_{7}^{\prime}|_{\tau}\lesssim 0.01$. Note, $\delta_{S_{1},S_{2}}A_{7}^{(\prime)}|_{\tau}$ from chirality-flipping contributions without resummation would be about one order of magnitude larger than eq. (41).

To summarize, within leptoquark models the $c\to u\gamma$ branching ratios are SM-like with CP asymmetries at $\mathcal{O}(0.01)$ for $V_{2}$ and SM-like for $S_{3}$ and $V_{1,3}$. On the other hand, in models $S_{1,2}$ and $\tilde{V}_{1,2}$ $A_{CP}\lesssim\mathcal{O}(10\%)$. 666For vector leptoquarks, which are gauge bosons in a renormalizable model, the coupling matrix is unitary, see e.g. Davidson:1993qk . In this case, $\delta_{V}C_{7,8}^{(\prime)}(M)=0$ as a result of a GIM-like mechanism. The argument could in principle be invalidated if there would be contributions from both lepton chiralities $\propto\lambda_{R}\lambda_{L}^{*}$, however, this does not happen in charm FCNCs for vector leptoquarks (we do not consider right-handed neutrinos). As a result, the corresponding CP asymmetries are SM-like. The largest effects arise from $\tau$-loops.

IV.3 SUSY

Here we consider effects within SUSY, taking into account the leading, gluino induced contributions within the mass insertion approximation Gabbiani:1996hi ; Prelovsek:2000xy

[TABLE]

where $m_{\tilde{q}}$ and $m_{\tilde{g}}$ denote the masses of the squarks and the gluino, respectively, $M\sim m_{\tilde{g},\tilde{q}}$,

[TABLE]

and $x=m_{\tilde{g}}^{2}/m_{\tilde{q}}^{2}$. We neglected terms not subject to $m_{\tilde{g}}/m_{c}$-enhancement.

The mass insertions $(\delta_{12})$ are constrained by data on the $D^{0}-\bar{D}^{0}$ mass difference Gabbiani:1996hi ; Olive:2016xmw . In case of i): $m_{\tilde{g}}^{(i)}=1\,\text{TeV},\,m_{\tilde{q}}^{(i)}=2\,\text{TeV}$ and ii): $m_{\tilde{g}}^{(ii)}=2\,\text{TeV},\,m_{\tilde{q}}^{(ii)}=1\,\text{TeV}$, we find

[TABLE]

and $|\delta_{\text{SUSY}}^{(i)}A_{7}^{(\prime)}|\lesssim 0.2$ and $|\delta_{\text{SUSY}}^{(ii)}A_{7}^{(\prime)}|\lesssim 0.3$, respectively. The upper limits are similar to the model-independent constraints obtained in section IV.1. Note that, barring cancellations, constraints on the imaginary parts of $\delta_{12}$ can be about an order of magnitude stronger Carrasco:2014uya , still permitting to signal BSM CP-violation. Constraints from $\epsilon^{\prime}/\epsilon$ and chargino loops are model-dependent. For realistic, not too low SUSY mass parameters corresponding bounds, e.g. Frank:2003ra , are not stronger than those from $D-\bar{D}$ mixing. We further checked that $|\delta_{\text{SUSY}}^{(i),(ii)}C_{3-6}^{(\prime)}(M)|\lesssim 10^{-5}$ is negligible. Note that $|\delta_{\text{SUSY}}A_{7}^{(\prime)}/\delta_{\text{SUSY}}A_{8}^{(\prime)}|\in[0.75,5.46]$, where the lower limit is for $x\gtrsim 10$ and the upper limit for $x\lesssim 0.01$. Thus, supersymmetric models may induce $c\to u\gamma$ branching ratios and CP asymmetries above the SM predictions, and SUSY parameters are constrained by $\mathcal{B}(D^{0}\to\rho^{0}\gamma)$. Note that additional constraints may apply once the SUSY breaking has been specified Prelovsek:2000xy . A detailed evaluation is beyond the scope of this work.

V On $\Lambda_{c}\to p\gamma$

We investigate possibilities to probe the handedness of the $c\to u\gamma$ current in the decay $\Lambda_{c}\to p\gamma$ with polarization asymmetries, that arise once $\Lambda_{c}$’s are produced polarized. We follow closely related works on $\Lambda_{b}\to\Lambda\gamma$ decays Hiller:2001zj ; Falk:1993rf .

The $\Lambda_{c}\to p\gamma$ branching ratio is not measured to date. Quite generally we assume

[TABLE]

in agreement with naive expectations from $\mathcal{B}(D^{0}\to\rho^{0}\gamma)$ 777Via weak annihilation $r(\Lambda_{c}\to p\gamma/D^{0}\to\rho^{0}\gamma)\sim\sqrt{2}$ due to color counting, via resonances $r(\Lambda_{c}\to p\gamma/D^{0}\to\rho^{0}\gamma)\sim 1$ due to the amplitude $A^{\text{III},\Lambda_{c}\to p\gamma}$ Singer:1996ba and $r(\Lambda_{c}\to p\gamma/D^{0}\to\rho^{0}\gamma)\simeq\sqrt{2}f_{\perp}/T\sim 1$ via SM effective and BSM Wilson coefficients. Here, the form factor $f_{\perp}=f_{\perp}^{T}(0)=f_{\perp}^{T5}(0)$ is defined as in Boer:2014kda and calculated within QCD LCSR Khodjamirian:2011jp ; Li:2016qai , a covariant confined quark model Gutsche:2014zna and a relativistic quark model Faustov:2016yza . Within a constituent quark model $\mathcal{B}_{\Lambda_{c}\to p\gamma}=2.2\cdot 10^{-5}$ Uppal:1992cc in agreement with eq. (45).. Note, we employ equation (45) only to estimate uncertainties. ${\cal{B}}(\Lambda_{c}\to p\gamma)$ should be determined experimentally.

The number of $\Lambda_{c}\to p\gamma$ events $N$, modulo reconstruction efficiencies, can be obtained from

[TABLE]

where $f(c\to\Lambda_{c})\simeq 0.06$ Lisovyi:2015uqa is the fragmentation fraction of charm to $\Lambda_{c}$-baryons and $N(c\bar{c})$ the number of $c\bar{c}$ produced. At the forthcoming Belle II experiment, where $\sigma(e^{+}e^{-}\to\bar{c}c)\simeq 1.3\,\text{nb}$, $L\simeq 5\,\text{ab}^{-1}$ within a year Aushev:2010bq , $N\sim[10^{3},10^{4}]$. At a future $e^{+}e^{-}$-collider running at the $Z$ (FCC-ee), where $N(Z)\sim 10^{12}$ within one year dEnterria:2016fpc and $\mathcal{B}(Z\to c\bar{c})\simeq 0.12$ Olive:2016xmw , $N\sim 10^{5}$. This environment suggests a measurement of the $\Lambda_{c}\to p\gamma$ branching ratio, the $\Lambda_{c}$ polarization and the angular asymmetry $A^{\gamma}$ of $\Lambda_{c}\to p\gamma$ decays. The latter is defined in the $\Lambda_{c}$ rest frame by the angle between the $\Lambda_{c}$ spin and the proton momentum, that is, the forward-backward asymmetry of the photon momentum relative to the $\Lambda_{c}$ boost, and normalized to the width. It reads Hiller:2001zj

[TABLE]

where $P_{\Lambda_{c}}$ denotes the (longitudinal) $\Lambda_{c}$ polarization and $r=A_{7}^{\prime}/A_{7}$. $A^{\gamma}\to P_{\Lambda_{c}}/2$ for $r\to\infty$ and $A^{\gamma}\to-P_{\Lambda_{c}}/2$ for $r\to 0$. Calculating $A_{7},A_{7}^{\prime}$ 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 $A_{7}^{(\prime)}\sim\delta C_{7}^{(\prime)}$ for large BSM effects and $A_{7}^{\prime}\ll A_{7}$ in SM-like situations. $A^{\gamma}$ is measurable in the laboratory frame for a boost $\vec{\beta}=\vec{p}_{\Lambda_{c}}/E_{\Lambda_{c}}$, where $\vec{p}_{\Lambda_{c}}$ denotes the $\Lambda_{c}$ three-momentum in the laboratory frame, as

[TABLE]

Here, $\langle q_{\parallel}\rangle_{|\vec{\beta}|}$ is the average longitudinal momentum of the photon in the laboratory frame relative to the boost axis, $\gamma=1/\sqrt{1-|\vec{\beta}|^{2}}$ and $E_{\gamma}^{*}=(m_{\Lambda_{c}}^{2}-m_{p}^{2})/(2m_{\Lambda_{c}})\simeq 0.95\,\text{GeV}$ is the photon energy in the $\Lambda_{c}$ rest frame.

The $\Lambda_{c}$ polarization can be expressed in terms of the charm quarks’ polarization $P_{c}$ as Falk:1993rf ; Galanti:2015pqa ; Kats:2015zth

[TABLE]

where $A\simeq 1.1$ is extrapolated Galanti:2015pqa from a measurement by the E791 collaboration Aitala:1996cy and $\omega_{1}=0.71\pm 0.13$ is measured by the CLEO collaboration Brandenburg:1996jc . At the $Z$ the polarization of the charm-quark $P_{c}^{(Z)}\simeq-0.65$ Korner:1993dy and one obtains a sizable polarization

[TABLE]

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 $\Lambda_{c}\to\Lambda(\to p\pi)l\nu$ decays with $\mathcal{B}(\Lambda_{c}\to\Lambda(\to p\pi)e\nu_{e})\simeq 0.023$ Olive:2016xmw . Note that the depolarization parameters $A$ and $\omega_{1}$ are measurable at Atlas, BaBar, Belle, CMS and LHCb Galanti:2015pqa . The $\Lambda_{c}$ polarization itself is measurable at Atlas, CMS and LHCb via $pp\to t(\to bW^{+}(\to c\bar{s}))\bar{t}(\to\bar{b}W^{-}(\to l^{-}\bar{\nu}))$ Galanti:2015pqa and $pp\to W^{-}c$ Kats:2015zth , where $P_{c}^{(W^{-})}\simeq-0.97$ Korner:1993dy .

The angular asymmetry is shown in figure 6 for $P_{\Lambda_{c}}$ as expected at the $Z$, eq. (50). In the SM and leptoquark models $r\lesssim 0.2$, and $A^{\gamma}\sim-P_{\Lambda_{c}}/2$ and positive. The statistical uncertainty $\delta A^{\gamma}=\sqrt{1-(A^{\gamma})^{2}}/\sqrt{N}$ 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 $|A^{\gamma}|\lesssim|P_{\Lambda_{c}}|/2$, allowing to signal BSM physics. Prerequisite for an interpretation is an experimental determination of $P_{\Lambda_{c}}$ or the depolarization fraction, if $P_{c}$ is known.

Given the uncertainties present in meson decays our analysis of $\Lambda_{c}$’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.

VI Summary

We worked out SM predictions for various $D_{(s)}\to V\gamma$ branching ratios, which are compiled in tables 1 and 2. The hybrid model predicts values up to a factor $\sim 2-3$ larger than the QCD factorization based approach, the latter being dominated by contributions from weak annihilation. All three branching ratios measured so far, the ones of $D^{0}\to\rho^{0}\gamma,D^{0}\to\phi\gamma$ and $D^{0}\to\bar{K}^{*0}\gamma$ are above the QCD factorization range given, suggesting, to the order we are working, a low value of the parameter $\lambda_{D}\lesssim 0.1$ GeV or low charm mass scale. One has to keep in mind, however, that poor convergence of the $1/m_{c}$ and $\alpha_{s}$-expansion prohibits a sharp conclusion without further study. Decays of charged mesons with color allowed weak annihilation contribution are better suited for extracting $\lambda_{D}$ 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. $D^{0}\to\phi\gamma$ and $D^{0}\to\bar{K}^{*0}\gamma$ 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 $D^{0}\to\phi\gamma$, cf. equations (28), (29).

The measured $D^{0}\to\rho^{0}\gamma$ branching ratio provides a model-independent upper limit on the decay amplitudes given in eq. (30), which is similar to the one from $D\to\pi\mu\mu$ decays deBoer:2015boa . If ${\cal{B}}(D^{0}\to\rho^{0}\gamma)$ is saturated with BSM physics or in the SM, ${\cal{B}}(D^{0}\to(\rho^{0}/\omega)\gamma)$ are very close to each other. For intermediate scenarios the two branching ratios can differ by orders of magnitude Fajfer:2000zx , and indicate BSM physics.

CP asymmetries in $c\to u\gamma$ transitions constitute SM null tests. We find $A_{CP}^{\rm SM}\lesssim{\rm few}\cdot 10^{-3}$ for $D^{0}\to\rho^{0}\gamma$, see figure 3, and similar for other radiative rare charm decays. Among the modes in tables 1 $A_{CP}$ is measured only in $D^{0}\to\rho^{0}\gamma$ decays, eq. (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 CP asymmetries at the level of $\sim{\cal{O}}(0.1)$, 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 $D^{0}\to\rho^{0}\gamma$ branching ratio and CP asymmetry while leptoquark models can’t. In the latter CP asymmetries $\lesssim 10\%$ are possible. The largest effects stem from models with vector leptoquarks $\tilde{V}_{1}$, $\tilde{V}_{2}$ and scalars $S_{1},S_{2}$ with couplings to taus.

If $\Lambda_{c}$-baryons are produced polarized, such as at the $Z$, angular asymmetries in $\Lambda_{c}\to p\gamma$ can probe chirality-flipped contributions, see figure 6. Within SUSY $A^{\gamma}$ can be very different from its SM-value, including having its sign flipped. Prerequisite for an interpretation of $A^{\gamma}$ is a measurement of the $\Lambda_{c}$ polarization, however, irrespective of its precise value, in the SM $A^{\gamma}$ is expected to be positive at the $Z$. The branching ratio of $\Lambda_{c}\to p\gamma$ may be investigated at Belle II Aushev:2010bq , the depolarization fraction at the LHC Kats:2015zth , and at the FCC-ee dEnterria:2016fpc all of this including $A^{\gamma}$.

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 III. 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 $|\Delta C|=|\Delta U|=1$ processes, further efforts are worthwhile and necessary. Largest sources of parametric uncertainty within the SM are the $\mu_{c}$-scale dependence and $\lambda_{D}$ 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 $D\to V$ tensor form factors at $q^{2}=0$ is desirable.

Acknowledgements

We thank Jernej Kamenik, Kamila Kowalska, Stephane Monteil, Stefan Schacht and Alan Schwartz for useful discussions. This work has been supported in part by the DFG Research Unit FOR 1873 “Quark Flavour Physics and Effective Field Theories”.

Appendix A Parameters

The couplings, the ($\overline{\text{MS}}$) masses and the widths are taken from the PDG Olive:2016xmw , with the renormalization scale running as in Chetyrkin:2000yt . The values of the CKM matrix elements are taken from the UTfit collaboration UTfit . The decay constants given by the FLAG read Aoki:2016frl

[TABLE]

and Dimou:2012un ; Straub:2015ica (and references therein)

[TABLE]

at $\mu=1\,\text{GeV}$ for the transverse decay constants.

The Gegenbauer moments read Dimou:2012un (and references therein)

[TABLE]

at $\mu=1\,\text{GeV}$, where the sign of $a_{1}^{K^{*}\perp}$ is due to ${K^{*}}^{+}=(u\bar{s})$ Ball:2006eu .

Appendix B $D\to V$ form factors

We define the $D\to V$ form factors as usual

[TABLE]

where $\epsilon_{0123}=1$ and $q_{\mu}=(p_{D}-p_{V})_{\mu}$. In our analysis we need $V(0),T_{1}(0)$ and $A_{1}(q^{2})$.

The $D\to\rho$ form factors have been measured by the CLEO collaboration CLEO:2011ab as

[TABLE]

where statistical and systematic uncertainties are added in quadrature, consistent with lattice computations Flynn:1997ca (and references therein)

[TABLE]

Here, given in parenthesis are the preliminary computations of Gill:2001jp with doubled uncertainties to account for systematic uncertainties. The $D\to\omega$ form factors have been measured by the BESIII collaboration Ablikim:2015gyp as (statistical and systematic uncertainties are added in quadrature)

[TABLE]

consistent with the values for $D\to\rho$, equation (55).

The (axial-)vector form factors have been obtained from QCD LCSR Wu:2006rd

[TABLE]

and

[TABLE]

where we rounded for easier comparison. The constituent quark model (CQM) Melikhov:2000yu and the covariant light-front quark model (CLFQM) Verma:2011yw provide $q^{2}$-shapes for the form factors. Note that we do not employ the form factors of Fajfer:2005ug within chiral theory as the $D\to\rho,\omega$ form factors at $q^{2}=0$ and the $D\to\pi$ form factors differ from measurements/computations.

In the numerical analysis we employ the following form factor values

[TABLE]

where uncertainties are given in parenthesis. These ranges are consistent with eqs. (55-59), the CQM and the CLFQM as well as the large energy relations Charles:1998dr . $A_{1}(q^{2})$ is shown in figure 7.

Appendix C $D\to V\gamma$ amplitudes in the hybrid approach

We write the resonance-induced contributions to $D\to V\gamma$ amplitudes using Fajfer:1997bh ; Fajfer:1998dv and Casalbuoni:1992dx as

[TABLE]

For each $D\to V\gamma$ transition, the CKM factor $V_{cq}^{*}V_{uq^{\prime}}$ can be inferred from the corresponding weak annihilation contribution, eqs. (16) and (25). The amplitudes $A_{\text{PC}/\text{PV}}^{\text{III}}$ 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 $\rho,\omega$ and $\phi$ to recover $A_{\text{PC}/\text{PV}}^{\text{III}}=0$ in the $SU(3)$-limit. For the weak annihilation modes $A_{\text{PC}/\text{PV}}^{\text{III}}=0$.

For $V=V^{0}\in\{\rho^{0},\omega\}$ the $D^{0}\to V\gamma$ amplitudes read

[TABLE]

and for $D^{+}\to\rho^{+}\gamma$

[TABLE]

where $a_{1,2}$ are given in section III and $f_{+}(0)=(0.1426\pm 0.0019)/|V_{cd}|$ Amhis:2016xyh , where statistical and systematic uncertainties are added in quadrature. For $D_{s}\to{K^{*}}^{+}\gamma$ decays

[TABLE]

As the approximation of Fajfer:1997bh ; Fajfer:1998dv is not applicable for $\mathcal{B}(D_{s}^{*}\to D_{s}\pi^{0})$ as a normalization mode due to isospin breaking Cho:1994zu and $\Gamma_{D_{s}^{*}}$ is not measured, $\lambda^{\prime}$ and $\lambda\tilde{g}_{V}$ in $A_{\text{PC}}^{\text{I},{K^{*}}^{+}}$ are related to

[TABLE]

and the ambiguity is fixed by means of the form factor $V$ Casalbuoni:1992dx

[TABLE]

Furthermore,

[TABLE]

where

[TABLE]

yields $|C_{VV\Pi}|\simeq 0.3$,

[TABLE]

$|A_{\text{PC,PV}}^{\text{I,II},D^{0}\to{K^{*}}^{0}\gamma}|=|V_{cd}^{*}V_{us}/(V_{cs}^{*}V_{ud})|\,|A_{\text{PC,PV}}^{\text{I,II},D^{0}\to\bar{K}^{*0}\gamma}|$,

[TABLE]

and

[TABLE]

Appendix D $C_{7}^{\rm eff}$ from $\langle Q_{3-6}\rangle$ at two-loop QCD

We obtain contributions to $C_{7}^{\text{eff}}$ from the two-loop QCD matrix elements of $Q_{3-6}$ from Buras:2002tp

[TABLE]

where

[TABLE]

[TABLE]

where we use $m_{u,d}=0$,

[TABLE]

and

[TABLE]