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 $|\mathrm{\Delta }c|=|\mathrm{\Delta }u|=1$ 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 $K_1(1400)$, and find it to be sizable, up to the few $\mathcal{O}(10)\%$ 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.

Figure 1

Weak annihilation (left) and photon dipole (right) contributions to $D_{(s)}\to K_1\gamma$ decays. In the weak annihilation diagram, the crosses indicate where the photon can be attached.

Figure 2

BSM reach of ${\lambda }_{\gamma }^{D_s}$ for given ${\lambda }_{\gamma }^{D^{+}}$ for NP in $C_7^{\prime }$ (with $C_7=0$, green curves) and NP in $C_7$ (with $C_7^{\prime }=0$ red curves), within (11) for the $K_1(1270)$, central values of input, $f_{K_1}=170\mathrm{MeV}$, $T^{K_1}=0.8$, and for ${\lambda }_{D_{(s)}}=0.1\mathrm{GeV}$. The black dashed line denotes the SM in the flavor limit, and the gray shaded area illustrates $\pm 20\%$ U-spin breaking between $r_{D^{+}}$ and $r_{D_s}$.

Figure 3

Dalitz contour plots of $\mathcal{I}m[\overrightarrow{n}·(\overrightarrow{\mathcal{J}}\times {\overrightarrow{\mathcal{J}}}^{*})]$ for $K^{+}{\pi }^{+}{\pi }^{-}$ (plot to the left) and $K^0{\pi }^{+}{\pi }^0$ (plot to the right) at $m_{K\pi \pi }^2=m_{K_1(1270)}^2$. Red (blue) areas correspond to positive (negative) values of $\mathcal{I}m[\overrightarrow{n}·(\overrightarrow{\mathcal{J}}\times {\overrightarrow{\mathcal{J}}}^{*})]$. Grey bands represent the $K^{*}(\rho )$ resonance $[(m_{K^{*}(\rho )}-{\mathrm{\Gamma }}_{K^{*}(\rho )}{)}^2,(m_{K^{*}(\rho )}+{\mathrm{\Gamma }}_{K^{*}(\rho )}{)}^2]$ intervals.

Figure 4

Invariant $K^{+}{\pi }^{+}{\pi }^{-}$ (upper plots) and $K^0{\pi }^{+}{\pi }^0$ (lower plots) mass dependence of $|\overrightarrow{\mathcal{J}}{|}^2$ (plots to the left), multiplied by the four-body phase space factor (4), and ${\mathcal{A}}_{\mathrm{UD}}/{\lambda }_{\gamma }$ (plots to the right) for $K_1(1270,1400)$ resonances separately and with relative fraction of the $K_1(1400)$ contribution, ${\xi }_{K_1(1400)}$; see the text for details.

Figure 5

Invariant $K^{+}{\pi }^{+}{\pi }^{-}$ (plot to the left) and $K^0{\pi }^{+}{\pi }^0$ (plot to the right) mass dependence of ${\mathcal{A}}_{\mathrm{UD}}/{\lambda }_{\gamma }$ for $K_1(1270,1400)$ resonances separately and with the relative fraction of the $K_1(1400)$ contribution, ${\xi }_{K_1(1400)}$. Solid lines correspond to all “offset” phases equal to zero, i.e., the pure quark model prediction. Dashed lines represent the offset phase ${\delta }_{\rho }=\arg [\mathcal{M}(K_1(1270)\to K\rho {)}_S/\mathcal{M}(K_1(1270)\to (K^{*}\pi {)}_S)]=-40°$.

Figure 6

The same as Fig. 5 for ${\delta }_{\rho }=0$ and with dotted lines representing the offset phase ${\delta }_D=\arg [\mathcal{M}(K_1(1270)\to (K^{*}\pi {)}_D)/\mathcal{M}(K_1(1270)\to (K^{*}\pi {)}_S)]=90°$.