Inclusive Hadronic Decay Rate of the τ Lepton from Lattice QCD: The u¯s Flavor Channel and the Cabibbo Angle

We present a lattice determination of the inclusive decay rate of the process <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>τ</a:mi><a:mo stretchy="false">↦</a:mo><a:msub><a:mrow><a:mi>X</a:mi></a:mrow><a:mrow><a:mi>u</a:mi><a:mi>s</a:mi></a:mrow></a:msub><a:msub><a:mrow><a:mi>ν</a:mi></a:mrow><a:mrow><a:mi>τ</a:mi></a:mrow></a:msub></a:mrow></a:math> in which the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>τ</d:mi></d:math> lepton decays into a generic hadronic state <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>X</f:mi><f:mrow><f:mi>u</f:mi><f:mi>s</f:mi></f:mrow></f:msub></f:math> with <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mover accent="true"><h:mi>u</h:mi><h:mo stretchy="false">¯</h:mo></h:mover><h:mi>s</h:mi></h:math> flavor quantum numbers. Our results have been obtained in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:msub><l:mrow><l:mi>n</l:mi></l:mrow><l:mrow><l:mi>f</l:mi></l:mrow></l:msub><l:mo>=</l:mo><l:mn>2</l:mn><l:mo>+</l:mo><l:mn>1</l:mn><l:mo>+</l:mo><l:mn>1</l:mn></l:mrow></l:math> isosymmetric QCD with full nonperturbative accuracy, without any operator product expansion approximation and, except for the presently missing long-distance isospin-breaking corrections, include a solid estimate of all sources of theoretical uncertainties. This has been possible by using the Hansen-Lupo-Tantalo method [M. Hansen , ] that we have already successfully applied [A. Evangelista , ] to compute the inclusive decay rate of the process <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>τ</n:mi><n:mo stretchy="false">↦</n:mo><n:msub><n:mi>X</n:mi><n:mrow><n:mi>u</n:mi><n:mi>d</n:mi></n:mrow></n:msub><n:msub><n:mi>ν</n:mi><n:mi>τ</n:mi></n:msub></n:math> in the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mover accent="true"><q:mi>u</q:mi><q:mo stretchy="false">¯</q:mo></q:mover><q:mi>d</q:mi></q:math> flavor channel. By combining our first-principles theoretical results with the presently available experimental data, we extract the Cabibbo-Kobayashi-Maskawa matrix element <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo stretchy="false">|</u:mo><u:msub><u:mi>V</u:mi><u:mrow><u:mi>u</u:mi><u:mi>s</u:mi></u:mrow></u:msub><u:mo stretchy="false">|</u:mo></u:math>, the Cabibbo angle, with a 0.9% accuracy, dominated by the experimental error.