Subleading shape function contributions to the hadronic invariant mass spectrum in B -> X_u l νdecay

TL;DR

The paper systematically derives the subleading O(Λ_QCD/m_b) corrections to the hadronic invariant-mass spectrum in B -> X_u l ν and its doubly differential form, using a nonlocal OPE framework and a bilocal operator basis. It shows how the leading and subleading shape functions enter the spectra and provides explicit expressions connecting the hadronic spectrum to the radiative B -> X_s γ photon spectrum, enabling a more robust extraction of |Vub|. Through simple shape-function models, the authors quantify that subleading effects are at the few-percent level—dominated by kinematic corrections—while the leading-subleading interplay largely cancels in the hadronic-photon relation. They conclude that the theoretical uncertainty on |Vub| from higher-order shape functions is smaller than that from lepton-energy cuts and remains manageable with current experimental capabilities.

Abstract

We study the O(Lambda/mb) corrections to the singly and doubly differential hadronic invariant mass spectra dΓ/dsH and dΓ/dsH dq^2 in b -> u decays, and discuss the implications for the extraction of the CKM matrix element |Vub|. Using simple models for the subleading shape functions, the effects of subleading operators are estimated to be at thefew percent level for experimentally relevant cuts. The subleading corrections proportional to the leading shape function are larger, but largely cancel in the relation between the hadronic invariant mass spectrum and the photon spectrum in B -> X_s γ. We also discuss the applicability of the usual prescription of convoluting the partonic level rate with the leading light-cone wavefunction of the b quark to subleading order.

Figures (7)

• Figure 1: Feynman rules for the operators $Q_i(\omega, \Gamma)$ in $n\cdot A=0$ gauge. We have defined $\delta_\pm(x) = \delta(\omega+n \cdot \hat{k}+x) \pm \delta(\omega+n \cdot \hat{k})$.
• Figure 2: Full-theory forward scattering diagrams.
• Figure 3: Plot of the kinematic corrections to the hadronic invariant mass spectrum, Eq. (\ref{['subleadingkin1']}). The dashed line is the leading order result (\ref{['leadingordersH']}), while the solid line includes the full set of kinetic corrections. The dotted line corresponds to the expansion of the results of dFN to subleading order, while the dot-dashed line also includes the contribution from the $m_b^5$ term. The difference between the dot-dashed and solid curves is due to the expansion of the heavy quark spinors.
• Figure 4: Three models of $h_1(\omega)$: model 1 (solid curve), model 2 (dashed curve) and model 3 (dot-dashed curve).
• Figure 5: Model calculations of the hadronic invariant mass spectrum $d\Gamma/ds_H$, for $m_b=4.8\,\hbox{GeV}$ (a) and $m_b=4.5\,\hbox{GeV}$ (b). The dotted curve is the leading order result; the other curves are the results in the models discussed in the text. The curves are denoted as in Fig. \ref{['threemodels']}. The right vertical line denotes the kinematic limit $s_H=m_D^2$; the left line denotes the B A B A R cut $s_H=(1.55\,\hbox{GeV})^2$.