Study of decay dynamics and $CP$ asymmetry in $D^+ to K^0_L e^+ u_e$ decay

Using 2.92~fb$^{-1}$ of electron-positron annihilation data collected at $sqrt{s} = 3.773$~GeV with the BESIII detector, we obtain the first measurements of the absolute branching fraction $mathcal{B}(D^+ to K^0_L e^+ u_e) = (4.481 pm 0.027(mathrm{stat.}) pm 0.103(mathrm{sys.}))%$ and the $CP$ asymmetry $A_{CP}^{D^+ to K^0_L e^+ u_e} = (-0.59 pm 0.60(mathrm{stat.}) pm 1.48(mathrm{sys.}))%$. From the $D^+ to K^0_L e^+ u_e$ differential decay rate distribution, the product of the hadronic form factor and the magnitude of the CKM matrix element, $f_{+}^{K}(0)|V_{cs}|$, is determined to be $0.728 pm 0.006(mathrm{stat.}) pm 0.011(mathrm{sys.})$. Using $|V_{cs}|$ from the SM constrained fit with the measured $f_{+}^{K}(0)|V_{cs}|$, $f_{+}^{K}(0) = 0.748 pm 0.007(mathrm{stat.}) pm 0.012(mathrm{sys.})$ is obtained, and utilizing the unquenched LQCD calculation for $f_{+}^{K}(0)$, $|V_{cs}| = 0.975 pm 0.008(mathrm{stat.}) pm 0.015(mathrm{sys.}) pm 0.025(mathrm{LQCD})$.