Precision corrections to dispersive bounds on form factors

We present precision corrections to dispersion relation bounds on form factors in bottom hadron semileptonic decays and analyze their effects on parametrizations derived from these bounds. We incorporate QCD two-loop and nonperturbative corrections to the two-point correlator, consider form factors whose contribution to decay rates is suppressed by lepton mass, and implement more realistic estimates of truncation errors associated with the parametrizations. We include higher resonances in the hadronic sum that, together with heavy quark symmetry relations near zero recoil, further tighten the sum rule bounds. Utilizing all these improvements, we show that each of the six form factors in $\overline{B}\to D\mathcal{l}\overline{\nu }$ and $\overline{B}\to D^{*}\mathcal{l}\overline{\nu }$ can be described with $3\%$ or smaller precision using only the overall normalization and one unknown parameter. A similar one-coefficient parametrization of one of the ${\Lambda }_b\to {\Lambda }_c\mathcal{l}\overline{\nu }$ form factors, together with heavy quark symmetry relations valid to order ${1/m}^2$, describes the differential baryon decay rate in terms of one unknown parameter and the phenomenologically interesting quantity ${\overline{\Lambda }}_{\Lambda }\approx M_{{\Lambda }_b}-m_b.$ We discuss the validity of slope-curvature relations derived by Caprini and Neubert, and present weaker, corrected relations. Finally, we present sample fits of current experimental $\overline{B}\to D^{*}\mathcal{l}\overline{\nu }$ and $\overline{B}\to D\mathcal{l}\overline{\nu }$ data to the improved one-parameter expansion.