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>

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