Cabibbo-Kobayashi-Maskawa unitarity deficit reduction via finite nuclear size

We revisit the extraction of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mrow><a:mo>|</a:mo></a:mrow><a:msub><a:mi>V</a:mi><a:mrow><a:mi>u</a:mi><a:mi>d</a:mi></a:mrow></a:msub><a:mrow><a:mo>|</a:mo></a:mrow></a:mrow></a:math> Cabibbo-Kobayashi-Maskawa (CKM) matrix element from the superallowed transition decay rate of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mmultiscripts><b:mi>Al</b:mi><b:mprescripts/><b:none/><b:mrow><b:mn>26</b:mn><b:mi>m</b:mi></b:mrow></b:mmultiscripts><b:mo>→</b:mo><b:mmultiscripts><b:mi>Mg</b:mi><b:mprescripts/><b:none/><b:mn>26</b:mn></b:mmultiscripts></b:mrow></b:math>, focusing on finite nuclear size effects. The decay rate dependence on the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mmultiscripts><c:mi>Al</c:mi><c:mprescripts/><c:none/><c:mrow><c:mn>26</c:mn><c:mi>m</c:mi></c:mrow></c:mmultiscripts></c:math> charge radius is found to be four times higher than previously believed, necessitating precise determination. However, for a short-lived isotope of an odd <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>Z</d:mi></d:math> element such as <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mmultiscripts><e:mi>Al</e:mi><e:mprescripts/><e:none/><e:mrow><e:mn>26</e:mn><e:mi>m</e:mi></e:mrow></e:mmultiscripts></e:math>, radius extraction relies on challenging many-body atomic calculations. We performed the needed calculations, finding an excellent agreement with previous ones, which used a different methodology. This sets a new standard for the reliability of isotope shift factor calculations in many-electron systems. The <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi mathvariant="script">F</f:mi><f:mi>t</f:mi></f:mrow></f:math> value obtained from our analysis is lower by <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mn>2.2</h:mn><h:mspace width="0.16em"/><h:mi>σ</h:mi></h:mrow></h:math> than the corresponding value in the previous critical survey, resulting in an increase in <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mrow><j:mo>|</j:mo></j:mrow><j:msub><j:mi>V</j:mi><j:mrow><j:mi>u</j:mi><j:mi>d</j:mi></j:mrow></j:msub><j:msup><j:mrow><j:mo>|</j:mo></j:mrow><j:mn>2</j:mn></j:msup></j:mrow></j:math> by <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mn>0.9</k:mn><k:mspace width="0.16em"/><k:mi>σ</k:mi></k:mrow></k:math>. Adopting <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mo>|</m:mo></m:mrow><m:msub><m:mi>V</m:mi><m:mrow><m:mi>u</m:mi><m:mi>d</m:mi></m:mrow></m:msub><m:mrow><m:mo>|</m:mo></m:mrow></m:mrow></m:math> from this decay alone reduces the CKM unitarity deficit by one standard deviation, irrespective of the choice of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mrow><n:mo>|</n:mo></n:mrow><n:msub><n:mi>V</n:mi><n:mrow><n:mi>u</n:mi><n:mi>s</n:mi></n:mrow></n:msub><n:mrow><n:mo>|</n:mo></n:mrow></n:mrow></n:math>.