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

Abstract

We present the first calculation of the $B\to D_1(2420)$ and $B\to D_1^{\prime }(2430)$ 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 $\mathrm{SCE}{\mathrm{T}}_{\mathrm{I}}$ and then factorize the corresponding vacuum-to-$B$-meson correlation functions in $\mathrm{SCE}{\mathrm{T}}_{\mathrm{II}}$. The resulting factorization formulae are used to construct the leading-power sum rules for the effective SCET form factors ${\xi }_{\parallel /\perp }^R$ and ${\Xi }_{\parallel /\perp }^R$ ($R=D_1,D_1^{\prime }$). 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 ${\varphi }_B^{+}(\omega ,\mu )$. To disentangle the mixed $D_1$ and $D_1^{\prime }$ 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 $R(D_1^{\prime })=0.159_{-0.025}^{+0.032}$, which can be confronted with the future measurements at Belle~II and LHCb.

Comments: 40 pages, 9 figures, 3 tables