Order 1/m_b^3 corrections to B\to X_c\ell\barνdecay and their implication for the measurement of \barΛand λ_1

TL;DR

The paper addresses the $O(1/m_b^3)$ nonperturbative corrections to inclusive $B\to X_c\ell\bar\nu$ decays within the HQET/OPE framework. It derives both local (two) and nonlocal (four) dimension-six operator contributions, detailing their impact on the lepton and hadronic spectra and on the total rate, and connects these to the physical $B$-meson state via nonlocal matrix elements ${\cal T}_1$–${\cal T}_4$. The study shows that while the total rate is only mildly affected, the extraction of $\bar{\Lambda}$ and $\lambda_1$ from the lepton spectrum can be strongly biased by $O(1/m_b^3)$ terms, with uncertainties dominated by the unknown higher-dimension matrix elements. It uses hadronic invariant-mass moments to bound $Br(B\to D^{**}\ell\bar\nu)$ and discusses implications for consistency checks between leptonic and hadronic observables, highlighting the need for additional theoretical or experimental input on the dimension-six matrix elements.

Abstract

We compute the order 1/m_b^3 nonperturbative contributions to the inclusive differential B\to X_c\ell\barνdecay rate. They are parametrized by the expectation values of two local and four nonlocal dimension-six operators. We use our results to estimate part of the theoretical uncertainties in the extraction of matrix elements \barΛand λ_1 from the lepton spectrum in the inclusive semileptonic B decay and find them to be very large. We also compute the 1/m_b^3 corrections to the moments of the hadronic invariant mass spectrum in this decay, and combine them with the extracted values of \barΛand λ_1 to put an upper bound on the branching fraction Br(B\to D^{**}\ell\barν).

Figures (2)

• Figure 1: (a) The relevant term in the operator product expansion. Wavy lines denote the insertions of left-handed currents. (b) does not contribute to $b\rightarrow c$ decay.
• Figure 2: Impact of $1/m_b^3$ corrections on the extraction of $\bar{\Lambda},\lambda_1$. Shaded region: Higher order matrix elements estimated by dimensional analysis. Cross-hatched region: $\rho_1=0.13{\rm GeV}^3,\,\rho_2=0$. Cross and ellipse show the values of $\bar{\Lambda},\lambda_1$ extracted without $1/m_b^3$ corrections but including the experimental statistical error.