Measurement of lepton momentum moments in the decay $\overline{B}\to X\mathcal{l}\overline{\nu }$ and determination of the heavy quark expansion parameters and $|V_{\mathrm{cb}}|$

We measure the primary lepton momentum spectrum in $\overline{B}\to X\mathcal{l}\overline{\nu }$ decays, for $p_{\mathcal{l}}>~1.5\mathrm{GeV}/c$ in the B rest frame. From this, we calculate various moments of the spectrum. In particular, we find $R_0\equiv {\int }_{1.7\mathrm{GeV}}(d\Gamma {/dE}_{\mathrm{sl}}{)dE}_{\mathcal{l}}/{\int }_{1.5\mathrm{GeV}}(d\Gamma {/dE}_{\mathrm{sl}}{)dE}_{\mathcal{l}}=0.6187\mathrm{}\pm {0.0014}_{\mathrm{stat}}\pm {0.0016}_{\mathrm{sys}}$ and $R_1\equiv {\int }_{1.5\mathrm{GeV}}{E}_{\mathcal{l}}(d\Gamma {/dE}_{\mathrm{sl}}{)dE}_{\mathcal{l}}/{\int }_{1.5\mathrm{GeV}}(d\Gamma {/dE}_{\mathrm{sl}}{)dE}_{\mathcal{l}}=(1.7810\mathrm{}\pm {0.0007}_{\mathrm{stat}}\pm {0.0009}_{\mathrm{sys}})\mathrm{GeV}.$ We use these moments to determine non-perturbative parameters governing the semileptonic width. In particular, we extract the heavy quark expansion parameters $\overline{\Lambda }=(0.39\mathrm{}\pm {0.03}_{\mathrm{stat}}\pm {0.06}_{\mathrm{sys}}\pm {0.12}_{\mathrm{th}})\mathrm{GeV}$ and ${\lambda }_1=(-0.25\pm {0.02}_{\mathrm{stat}}\pm {0.05}_{\mathrm{sys}}\pm {0.14}_{\mathrm{th}}){\mathrm{GeV}}^2.$ The theoretical constraints used are evaluated through order ${1/M}_B^3$ in the non-perturbative expansion and ${\beta }_0{\alpha }_s^2$ in the perturbative expansion. We use these parameters to extract $|V_{\mathrm{cb}}|$ from the world average of the semileptonic width and find $|V_{\mathrm{cb}}|=(40.8\mathrm{}\pm {0.5}_{{\Gamma }_{\mathrm{sl}}}\pm {0.4}_{({\lambda }_1,\overline{\Lambda }{)}_{\exp }}\pm {0.9}_{\mathrm{th}})\times {10}^{-3}.$ In addition, we extract the short range b-quark mass $m_b^{1\mathrm{S}}=(4.82\mathrm{}\pm {0.07}_{\exp }\pm {0.11}_{\mathrm{th}})\mathrm{GeV}{/c}^2.$ Finally, we discuss the implications of our measurements for the theoretical understanding of inclusive semileptonic processes.