Perturbative QCD- and Power-Corrected Hadron Spectra and Spectral Moments in the Decay $B \to X_s \ell^+ \ell^-$

Abstract

We compute the leading order (in $\alpha_s$) perturbative QCD and power ($1/m_b^2)$ corrections to the hadronic invariant mass and hadron energy spectra in the decay $B \to X_s \ell^+ \ell^-$ in standard model. This is done both by using the heavy quark expansion technique (HQET) and a perturbative-QCD improved Fermi motion (FM) model which takes into account $B$-meson wave-function effects. The corrections in the hadron energy ($E_H$) spectrum are found to be small over a good part of this spectrum in both the methods. However, the expansion in $1/m_b$ in HQET fails near the lower kinematic end-point and at the $c\bar{c}$ threshold. The hadronic invariant mass ($S_H$) spectrum is calculable only over a limited range $S_H > \bar{\Lambda}m_B$ in the heavy quark expansion, where $\bar{\Lambda} \simeq m_B-m_b$. We also present results for the first two hadronic moments $< S_H^n>$ and $< E_H^n>$, $n=1,2$, working out their sensitivity on the HQET and FM model parameters. For equivalent values of these parameters, the moments in these methods are remarkably close to each other. Using the FM model, we study the effect of the experimental cuts, used recently by the CLEO collaboration in searching for the decay $B \to X_s \ell^+ \ell^-$, on the hadron spectra and hadronic invariant mass moments. The constraints following from assumed values of $< S_H^n>$ on the HQET parameters $\lambda_1$ and $\bar{\Lambda}$ are worked out. Data from the forthcoming B facilities could be used to measure the short-distance contribution in $B \to X_s \ell^+ \ell^-$ and determine the HQET parameters $\lambda_1$ and $\bar{\Lambda}$. This could be combined with complementary constrains in $B \to X \ell \nu_\ell$ to determine them precisely.