Semileptonic $W$ Decay to the $B$ Meson with Lepton Pairs in Heavy Quark Effective Theory Factorization upto $\mathcal{O}$$(\alpha_s)$

Motivated by the study of heavy-light meson production within the framework of heavy quark effective theory (HQET) factorization, we extend the factorization formalism for a rather complicated process $W^+\to B^+\ell^+\ell^-$ in the limit of a non-zero invariant squared-mass of dilepton, $q^2$, at the lowest order in $1/m_b$ up to $\mathcal{O}(\alpha_s)$. The purpose of the current study is to extend the HQET factorization formula for the $W^+\to B^+\ell^+\ell^-$ process and subsequently compute the form factors for this channel up to next-to-leading-order corrections in $\alpha_s$. We explicitly show the amplitude of the $W^+\to B^+\ell^+\ell^-$ process can also be factorized into a convolution between the perturbatively calculable hard-scattering kernel and the non-perturbative yet universal light-cone distribution amplitude (LCDA) defined in HQET. The validity of HQET factorization depends on the assumed scale hierarchy $m_W \sim m_b \gg \Lambda_{\mathrm{QCD}}$. Within the HQET framework, we evaluate the form factors associated with the $W^+ \rightarrow B^+\ell^+\ell^-$ process, providing insights into its phenomenology. In addition, we also perform an exploratory phenomenological study on $W^+ \rightarrow B^+\ell^+\ell^-$ by employing an exponential model for the LCDAs for $B^+$ meson. Our findings reveal that the branching ratio for $W^+ \rightarrow B^+\ell^+\ell^-$ is below $10^{-10}$. Although the branching ratios are small, this channel in high luminosity LHC experiments may serve to further constraints the value of $\lambda_B$.