Complete $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections to the leptonic invariant mass spectrum in $$ b\to {X}_cl{\overline{\nu}}_l $$ decay

A bstract In the determination of the Cabibbo-Kobayashi-Maskawa matrix element | V cb | from inclusive semileptonic B -meson decays, moments of the leptonic invariant mass spectrum constitute valuable observables. To evaluate them with sufficient precision, perturbative $$ \mathcal{O}\left({\alpha}_s^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mfenced> </mml:math> corrections to the analogous spectrum in the partonic $$ b\to {X}_cl{\overline{\nu}}_l $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>b</mml:mi> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mi>l</mml:mi> <mml:msub> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>l</mml:mi> </mml:msub> </mml:math> decay are necessary. In the present paper, we compute such perturbative corrections in a complete manner, including contributions from the triple-charm channel, namely from the $$ cc\overline{c}l{\overline{\nu}}_l $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>cc</mml:mi> <mml:mover> <mml:mi>c</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>l</mml:mi> <mml:msub> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>l</mml:mi> </mml:msub> </mml:math> final states. We present our results in terms of numerical fits in both the single- and triple-charm cases. We confirm the recently found results for the single-charm correction, and analyze the triple-charm channel impact on centralized moments of the spectrum.