QCD corrections to the Darwin coefficient in inclusive semileptonic B→Xuℓν¯ℓ decays

In this paper, we compute <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>α</a:mi><a:mi>s</a:mi></a:msub></a:math> corrections to the matching coefficients of the dimension-six operators in the heavy quark expansion of the inclusive semileptonic heavy hadron decay rate and leptonic invariant mass spectrum with a massless quark and both a massive or massless lepton in the final state, analytically. The obtained results can be applied to the inclusive semileptonic <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>B</c:mi><c:mo stretchy="false">→</c:mo><c:msub><c:mi>X</c:mi><c:mi>u</c:mi></c:msub><c:mo>ℓ</c:mo><c:msub><c:mover accent="true"><c:mi>ν</c:mi><c:mo stretchy="false">¯</c:mo></c:mover><c:mo>ℓ</c:mo></c:msub></c:math> (<h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mo>ℓ</h:mo><h:mo>=</h:mo><h:mi>e</h:mi></h:math>, <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mrow><j:mi>μ</j:mi></j:mrow></j:math>, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mi>τ</l:mi></l:mrow></l:math>) and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>D</n:mi><n:mo stretchy="false">→</n:mo><n:mi>X</n:mi><n:mo>ℓ</n:mo><n:msub><n:mover accent="true"><n:mi>ν</n:mi><n:mo stretchy="false">¯</n:mo></n:mover><n:mo>ℓ</n:mo></n:msub></n:math> (<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>ℓ</s:mo><s:mo>=</s:mo><s:mi>e</s:mi></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>μ</u:mi></u:mrow></u:math>) decays. The main application of our results is the background subtraction of the <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>B</w:mi><w:mo stretchy="false">→</w:mo><w:msub><w:mi>X</w:mi><w:mi>u</w:mi></w:msub><w:mo>ℓ</w:mo><w:msub><w:mover accent="true"><w:mi>ν</w:mi><w:mo stretchy="false">¯</w:mo></w:mover><w:mo>ℓ</w:mo></w:msub></w:math> decay in the measurement of the <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:mi>B</bb:mi><bb:mo stretchy="false">→</bb:mo><bb:msub><bb:mi>X</bb:mi><bb:mi>c</bb:mi></bb:msub><bb:mo>ℓ</bb:mo><bb:msub><bb:mover accent="true"><bb:mi>ν</bb:mi><bb:mo stretchy="false">¯</bb:mo></bb:mover><bb:mo>ℓ</bb:mo></bb:msub></bb:math> decay, which is important for the precise extraction of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>V</gb:mi><gb:mrow><gb:mi>c</gb:mi><gb:mi>b</gb:mi></gb:mrow></gb:msub></gb:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>R</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:msup><ib:mi>D</ib:mi><ib:mrow><ib:mo stretchy="false">(</ib:mo><ib:mo>*</ib:mo><ib:mo stretchy="false">)</ib:mo></ib:mrow></ib:msup><ib:mo stretchy="false">)</ib:mo></ib:math>. They also play a role in the computation of lifetimes of heavy hadrons.