Weak-charge form-factor determination at the electron-ion collider

Determining the weak charge form factor, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>F</a:mi> <a:mi>W</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:msup> <a:mi>Q</a:mi> <a:mn>2</a:mn> </a:msup> <a:mo stretchy="false">)</a:mo> </a:math> , of nuclei over a continuous range of momentum transfers, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mn>0</e:mn> <e:mo>≲</e:mo> <e:msup> <e:mi>Q</e:mi> <e:mn>2</e:mn> </e:msup> <e:mo>≲</e:mo> <e:mn>0.1</e:mn> <e:mtext> </e:mtext> <e:mtext> </e:mtext> <e:msup> <e:mrow> <e:mi>GeV</e:mi> </e:mrow> <e:mrow> <e:mn>2</e:mn> </e:mrow> </e:msup> </e:math> , is essential for mapping out the distribution of neutrons in nuclei. The neutron density distribution has significant implications for a broad range of areas, including studies of nuclear structure, neutron stars, and physics beyond the Standard Model. Currently, our knowledge of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>F</g:mi> <g:mi>W</g:mi> </g:msub> <g:mo stretchy="false">(</g:mo> <g:msup> <g:mi>Q</g:mi> <g:mn>2</g:mn> </g:msup> <g:mo stretchy="false">)</g:mo> </g:math> comes primarily from fixed target experiments that measure the parity-violating asymmetry in coherent elastic electron-ion scattering. Fixed target experiments, such as CREX and PREX-1,2, have provided high-precision weak charge form factor extractions for the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mmultiscripts> <k:mrow> <k:mi>Ca</k:mi> </k:mrow> <k:mprescripts/> <k:none/> <k:mrow> <k:mn>48</k:mn> </k:mrow> </k:mmultiscripts> </k:mrow> </k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mmultiscripts> <m:mrow> <m:mi>Pb</m:mi> </m:mrow> <m:mprescripts/> <m:none/> <m:mrow> <m:mn>208</m:mn> </m:mrow> </m:mmultiscripts> </m:mrow> </m:math> nuclei, respectively. However, a major limitation of fixed target experiments is that they each provide data only at a single value of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>Q</o:mi> <o:mn>2</o:mn> </o:msup> </o:math> . With the proposed electron-ion collider (EIC) on the horizon, we explore its potential to impact the determination of the weak charge form factor. While it cannot compete with the precision of fixed target experiments, it can provide data over a wide and continuous range of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msup> <q:mi>Q</q:mi> <q:mn>2</q:mn> </q:msup> </q:math> values, and for a wide variety of nuclei. We show that with data corresponding to an integrated luminosity of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi mathvariant="script">L</s:mi> <s:mo>∼</s:mo> <s:mn>500</s:mn> <s:mo>/</s:mo> <s:mi>A</s:mi> <s:mtext> </s:mtext> <s:mtext> </s:mtext> <s:mrow> <s:msup> <s:mrow> <s:mi>fb</s:mi> </s:mrow> <s:mrow> <s:mo>−</s:mo> <s:mn>1</s:mn> </s:mrow> </s:msup> </s:mrow> </s:math> , where <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>A</v:mi> </v:math> is the nucleus atomic weight, the EIC can significantly impact constraints by lifting degeneracies in theoretical models of the neutron density distribution. Ensuring EIC detector coverage at low <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:msup> <x:mi>Q</x:mi> <x:mn>2</x:mn> </x:msup> </x:math> and large negative pseudorapidities will be essential for such <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msub> <z:mi>F</z:mi> <z:mi>W</z:mi> </z:msub> <z:mo stretchy="false">(</z:mo> <z:msup> <z:mi>Q</z:mi> <z:mn>2</z:mn> </z:msup> <z:mo stretchy="false">)</z:mo> </z:math> measurements.