Up-down asymmetries and angular distributions in $D\to K_1(\to K\pi \pi ){\ell }^{+}{\nu }_{\ell }$

Using the helicity amplitude technique, we derive differential decay widths and angular distributions for the decay cascade $D\to K_1(1270,1400){\ell }^{+}{\nu }_{\ell }\to (K\pi \pi ){\ell }^{+}{\nu }_{\ell }(\ell =e,\mu )$, in which the electron and muon mass is explicitly included. Using a set of phenomenological results for $D\to K_1$ form factors, we calculate partial decay widths and branching fractions for $D^0\to K_1^{-}{\ell }^{+}{\nu }_{\ell }$ and $D^{+}\to K_1^0{\ell }^{+}{\nu }_{\ell }$, but find that results for $\mathcal{B}(D\to K_1(1270)e^{+}{\nu }_e)$ are about a factor 1.5 larger than recent BESIII measurements. We further demonstrate that the measurement of up-down asymmetry in $D\to K_1{e}^{+}{\nu }_e\to (K\pi \pi )e^{+}{\nu }_e$ and angular distributions in $D\to K_1{\ell }^{+}{\nu }_{\ell }\to (K\pi \pi ){\ell }^{+}{\nu }_{\ell }$ can help to determine the hadronic amplitude requested in $B\to K_1(\to K\pi \pi )\gamma$.

Figure 1

The kinematics for $D\to K_1{\ell }^{+}{\nu }_{\ell }\to (K\pi \pi ){\ell }^{+}{\nu }_{\ell }$. In the lepton pair ${\ell }^{+}{\nu }_{\ell }$ rest frame, the angle ${\theta }_{\ell }$ is defined by the ${\ell }^{+}$ flight direction and the opposite of the $D$ meson flight direction. In the $K_1$ rest frame the $\overrightarrow{n}$ is defined as the normal direction of the $K_1$ decay plane, and ${\theta }_K$ is the relative angle between $\overrightarrow{n}$ and the opposite of the $D$ flight direction. Since $\overrightarrow{n}$ is made of the cross product of two momenta, its direction will not be altered under parity transformation. But since the $D$ flight direction will be reversed, the ${\theta }_K$ will be changed to $\pi -{\theta }_K$ under parity transformation, implying that the $\cos {\theta }_K$ is parity odd.

Figure 2

Differential decay widths for $D\to K_1(1270){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$) and $D\to K_1(1400){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$) based on the form factors from Ref. [33]. The dotted and dashed lines correspond to the longitudinal and transverse polarizations, while the solid line gives the total differential decay widths. The shadowed region arises from the uncertainties in the mixing angle ${\theta }_K$.

Figure 3

Differential decay widths for $D\to K_1(1270){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$) and $D\to K_1(1400){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$) based on the form factors from Ref. [30]. The dotted and dashed lines correspond to the longitudinal and transverse polarizations, while the solid line gives the total differential decay widths. The shadowed region arises from the uncertainties in the mixing angle ${\theta }_K$.

Figure 4

Differential decay widths for $D\to K_1(1270){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$) and $D\to K_1(1400){\ell }^{+}{\nu }_{\ell }$ (in units of ${10}^{-15}{\mathrm{GeV}}^{-1}$), based on the form factors from QCD sum rules [34]. The dotted and dashed lines correspond to the longitudinal and transverse polarizations, while the solid line gives the total differential decay widths. The shadowed region arises from the uncertainties in the mixing angle ${\theta }_K$.

Figure 5

Dependence of branching ratios $\mathcal{B}(D^0\to K_1^{-}{\ell }^{+}{\nu }_{\ell })$ (in units of ${10}^{-3}$) on the mixing angle ${\mathrm{\Theta }}_K$ in the range $30°<{\mathrm{\Theta }}_K<60°$ based on the form factors from LFQM in Ref. [33] (the two upper lines), and Ref. [30] (the two lower curves). Dotted and dashed curves correspond to the electron and muon mode, respectively.

Figure 6

Dependence of branching ratios $\mathcal{B}(D^0\to K_1^{-}{\ell }^{+}{\nu }_{\ell })$ (in units of ${10}^{-3}$) on the mixing angle ${\mathrm{\Theta }}_K$ in the range $-47°<{\mathrm{\Theta }}_K<-21°$ based on the form factors from QCD sum rules [34]. Dotted and dashed curves correspond to the electron and muon mode, respectively.

Figure 7

Results for up-down asymmetry $A_{UD}^{\prime }$ (in unit of $H_{K_1}$) in $D\to K_1(\to K\pi \pi ){\ell }^{+}{\nu }_{\ell }$. The solid curve corresponds to the electron final state, in which the lepton mass is negligible and the result is very close to unity. The results shadowed in red (upper) and blue (lower) correspond to the muon mode with form factors from light-front quark model in Ref. [30, 33], respectively, while the green band (middle) is obtained using the QCD sum rules in Ref. [34], respectively. The nonzero mass of $\mu$ provides sizable corrections as shown in Eq. (29).

Figure 8

Fit to $1.0\times {10}^5$ MC events with 3 components using two models: $m_l=0$ (Red) and $m_l\ne 0$ (Blue)