Study of decay dynamics and $CP$ asymmetry in $D^+ \to K^0_L e^+ ν_e$ 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 $\mathcal{B}(D^+ \to K^0_L e^+ ν_e) = (4.481 \pm 0.027(\mathrm{stat.}) \pm 0.103(\mathrm{sys.}))\%$ and the $CP$ asymmetry $A_{CP}^{D^+ \to K^0_L e^+ ν_e} = (-0.59 \pm 0.60(\mathrm{stat.}) \pm 1.48(\mathrm{sys.}))\%$. From the $D^+ \to K^0_L e^+ ν_e$ 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 $f_{+}^{K}(0)$, $|V_{cs}| = 0.975 \pm 0.008(\mathrm{stat.}) \pm 0.015(\mathrm{sys.}) \pm 0.025(\mathrm{LQCD})$.