Natural complex plane for kaon CKM data: Framework, status, and future prospects

Kaon physics can be used to independently determine three out of the four parameters of the CKM matrix, without any B physics input. Treating one parameter, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo stretchy="false">|</a:mo><a:msub><a:mi>V</a:mi><a:mrow><a:mi>u</a:mi><a:mi>s</a:mi></a:mrow></a:msub><a:mo stretchy="false">|</a:mo></a:math>, or alternatively Wolfenstein <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>λ</e:mi></e:math>, as well known, we show that the natural plane for the presentation of kaon CKM information is spanned by the combinations <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">[</g:mo><g:msup><g:mi>A</g:mi><g:mn>2</g:mn></g:msup><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:mo>−</g:mo><g:mover accent="true"><g:mi>ρ</g:mi><g:mo stretchy="false">^</g:mo></g:mover><g:mo stretchy="false">)</g:mo><g:mo>,</g:mo><g:msup><g:mi>A</g:mi><g:mn>2</g:mn></g:msup><g:mover accent="true"><g:mi>η</g:mi><g:mo stretchy="false">^</g:mo></g:mover><g:mo stretchy="false">]</g:mo></g:math>. In this way, the use of B physics inputs is avoided, as well as the artificial inflation of errors due to parametric uncertainties, mainly due to <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo stretchy="false">|</q:mo><q:msub><q:mi>V</q:mi><q:mrow><q:mi>c</q:mi><q:mi>b</q:mi></q:mrow></q:msub><q:mo stretchy="false">|</q:mo></q:math>. We show that the current status of kaon CKM constraints, impacted by recent advances in measurement and theory, is characterized by four allowed regions, and find that incoming data will inevitably disfavor a number of them, either confirming the CKM paradigm as dominant, or discovering a departure from the Standard Model.