Toward resolving the discrepancy in helium-3 and helium-4 nuclear charge radii

The discrepancy in the squared nuclear charge radius difference, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi mathvariant="normal">Δ</a:mi><a:msup><a:mi>R</a:mi><a:mn>2</a:mn></a:msup></a:mrow></a:math>, between <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mmultiscripts><c:mi mathvariant="normal">He</c:mi><c:mprescripts/><c:none/><c:mrow><c:mn>3</c:mn></c:mrow></c:mmultiscripts></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mmultiscripts><e:mi mathvariant="normal">He</e:mi><e:mprescripts/><e:none/><e:mrow><e:mn>4</e:mn></e:mrow></e:mmultiscripts></e:math>, as determined from electronic and muonic atom energy levels, presents an unresolved puzzle. This paper shows that accounting for off-diagonal hyperfine mixing effects can substantially reduce this discrepancy. We find that hyperfine mixing with the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>n</g:mi><g:msup><g:mspace width="0.16em"/><g:mn>3</g:mn></g:msup><g:mi>S</g:mi></g:mrow></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>n</i:mi><i:msup><i:mspace width="0.16em"/><i:mn>1</i:mn></i:msup><i:mi>S</i:mi></i:mrow></i:math> states (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>n</k:mi><k:mo>&gt;</k:mo><k:mn>2</k:mn></k:mrow></k:math>) in <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mmultiscripts><l:mi mathvariant="normal">He</l:mi><l:mprescripts/><l:none/><l:mrow><l:mn>3</l:mn></l:mrow></l:mmultiscripts></l:math> introduces a correction of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mo>−</n:mo><n:mn>1.37</n:mn></n:mrow></n:math> kHz to the isotope shift of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mrow><o:mn>2</o:mn><o:msup><o:mspace width="0.16em"/><o:mn>1</o:mn></o:msup><o:mi>S</o:mi><o:mo>−</o:mo><o:mn>2</o:mn><o:msup><o:mspace width="0.16em"/><o:mn>3</o:mn></o:msup><o:mi>S</o:mi></o:mrow></o:math> transition, a factor of seven times larger than the current uncertainty. This correction modifies <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mrow><r:mi mathvariant="normal">Δ</r:mi><r:msup><r:mi>R</r:mi><r:mn>2</r:mn></r:msup></r:mrow></r:math> by <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mrow><t:mo>−</t:mo><t:mn>0.0064</t:mn><t:mspace width="3.33333pt"/><t:msup><t:mrow><t:mi>fm</t:mi></t:mrow><t:mn>2</t:mn></t:msup></t:mrow></t:math>, shifting it from <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:mrow><v:mn>1.0757</v:mn><v:mrow><v:mo>(</v:mo><v:mn>15</v:mn><v:mo>)</v:mo></v:mrow><v:mspace width="3.33333pt"/><v:msup><v:mi>fm</v:mi><v:mn>2</v:mn></v:msup></v:mrow></v:math> to <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"><x:mrow><x:mn>1.0693</x:mn><x:mrow><x:mo>(</x:mo><x:mn>15</x:mn><x:mo>)</x:mo></x:mrow><x:mspace width="3.33333pt"/><x:msup><x:mi>fm</x:mi><x:mn>2</x:mn></x:msup></x:mrow></x:math>, as initially reported by Werf []. This brings <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:mrow><z:mi mathvariant="normal">Δ</z:mi><z:msup><z:mi>R</z:mi><z:mn>2</z:mn></z:msup></z:mrow></z:math> closer to the value of <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"><bb:mrow><bb:mn>1.0636</bb:mn><bb:mrow><bb:mo>(</bb:mo><bb:mn>31</bb:mn><bb:mo>)</bb:mo></bb:mrow><bb:mspace width="3.33333pt"/><bb:msup><bb:mi>fm</bb:mi><bb:mn>2</bb:mn></bb:msup></bb:mrow></bb:math> obtained from muonic helium <db:math xmlns:db="http://www.w3.org/1998/Math/MathML"><db:mrow><db:mi>μ</db:mi><db:msup><db:mrow><db:mi>He</db:mi></db:mrow><db:mo>+</db:mo></db:msup></db:mrow></db:math> by Schuhmann [], narrowing the existing discrepancy from <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML"><eb:mrow><eb:mn>3.6</eb:mn><eb:mi>σ</eb:mi></eb:mrow></eb:math> to <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML"><fb:mrow><fb:mn>1.7</fb:mn><fb:mi>σ</fb:mi></fb:mrow></fb:math>. The adjusted value <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"><gb:mrow><gb:mi mathvariant="normal">Δ</gb:mi><gb:msup><gb:mi>R</gb:mi><gb:mn>2</gb:mn></gb:msup></gb:mrow></gb:math> also agrees well with the result of 1.069(3) <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML"><ib:msup><ib:mi>fm</ib:mi><ib:mn>2</ib:mn></ib:msup></ib:math> derived from the helium <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML"><jb:mrow><jb:mn>2</jb:mn><jb:msup><jb:mspace width="0.16em"/><jb:mn>3</jb:mn></jb:msup><jb:mi>S</jb:mi><jb:mo>−</jb:mo><jb:mn>2</jb:mn><jb:msup><jb:mspace width="0.16em"/><jb:mn>3</jb:mn></jb:msup><jb:mi>P</jb:mi></jb:mrow></jb:math> transitions, as analyzed by Patkóš []. Our results provide key insights for resolving the discrepancy in <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML"><mb:mrow><mb:mi mathvariant="normal">Δ</mb:mi><mb:msup><mb:mi>R</mb:mi><mb:mn>2</mb:mn></mb:msup></mb:mrow></mb:math>.