SCET sum rules for $B\to D_1(2420)$ and $B\to D_1'(2430)$ form factors at next-to-leading order

Abstract

We present the first calculation of the $B \to D_1(2420)$ and $B \to D'_1(2430)$ transition form factors at ${\cal O}(\alpha_s)$ using light-cone sum rules within the framework of soft-collinear effective theory (SCET). We first match the QCD transition currents onto ${\rm SCET_I}$ and then factorize the corresponding vacuum-to-$B$-meson correlation functions in ${\rm SCET_{II}}$. The resulting factorization formulae are used to construct the leading-power sum rules for the effective SCET form factors $\xi^R_{\parallel/\perp}$ and $\Xi^R_{\parallel/\perp}$ ($R=D_1,D'_1$). In particular, we calculate the additional longitudinal form factor $\xi^R_{\parallel,m_c}$ 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 $D'_1$ 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 $R(D_1) = 0.070^{+0.028}_{-0.018}$ and $R(D'_1) = 0.159^{+0.032}_{-0.025}$, which can be confronted with the future measurements at Belle~II and LHCb.