SCET sum rules for $B\to D_1(2420)$ and $B\to D_1^{\mathrm{\prime }}(2430)$ form factors at next-to-leading order

We present the first calculation of the $B\to D_1(2420)$ and $B\to D_1^{\mathrm{\prime }}(2430)$ transition form factors at $\mathcal{O}({\alpha }_s)$ using light-cone sum rules within the framework of soft-collinear effective theory (SCET). We first match the QCD transition currents onto $\mathrm{S}\mathrm{C}\mathrm{E}{\mathrm{T}}_{\mathrm{I}}$ and then factorize the corresponding vacuum-to-$B$-meson correlation functions in $\mathrm{S}\mathrm{C}\mathrm{E}{\mathrm{T}}_{\mathrm{I}\mathrm{I}}$. The resulting factorization formulae are used to construct the leading-power sum rules for the effective SCET form factors ${\xi }_{\parallel \mathrm{/}\perp }^R$ and ${\mathrm{\Xi }}_{\parallel \mathrm{/}\perp }^R$ ($R=D_1,D_1^{\mathrm{\prime }}$). In particular, we calculate the additional longitudinal form factor ${\xi }_{\parallel ,m_c}^R$ induced by the finite charm-quark mass, whose contribution depends only on the $B$-meson light-cone distribution amplitude ${\phi }_B^{+}(\omega ,\mu )$. To disentangle the mixed $D_1$ and $D_1^{\mathrm{\prime }}$ states, we introduce dedicated combinations of interpolating currents, with their decay constants determined via the equations of motion. To isolate the orbitally excited states from ground-state contamination, we subtract the ground-state contribution from the total sum rules and examine the stability of the resulting sum rules. Furthermore, the $q^2$-dependence of the physical form factors is extrapolated over the full kinematic region using the Bourrely–Caprini–Lellouch parameterization. Finally, we provide phenomenological predictions for the branching fractions, differential decay widths, and lepton flavor universality ratios. Numerically, we obtain $R(D_1)=0.070_{-0.018}^{+0.028}$ and $R(D_1^{\mathrm{\prime }})=0.159_{-0.025}^{+0.032}$, which can be confronted with the future measurements at Belle~II and LHCb.