# Comparing proton momentum distributions in $A=2$ and 3 nuclei via $^2$H $^3$H and $^3$He $(e, e'p)$ measurements

## TL;DR

This study measures and compares proton momentum distributions in $A=2$ and 3 nuclei using electron scattering, revealing discrepancies at high momenta that could inform short-range nuclear force models.

## Contribution

First measurement of $(e,e'p)$ cross-section ratios for $^3$He, $^3$H, and deuterium across a wide momentum range, providing new insights into nuclear interactions at short distances.

## Key findings

- Ratios extend above Fermi momentum, showing differences up to 50%.
- $^3$He/$^3$H ratios agree within 3% at lower momenta.
- Discrepancies at high momenta suggest sensitivity to short-range NN interactions.

## Abstract

We report the first measurement of the $(e,e'p)$ reaction cross-section ratios for Helium-3 ($^3$He), Tritium ($^3$H), and Deuterium ($d$). The measurement covered a missing momentum range of $40 \le p_{miss} \le 550$ MeV$/c$, at large momentum transfer ($\langle Q^2 \rangle \approx 1.9$ (GeV$/c$)$^2$) and $x_B>1$, which minimized contributions from non quasi-elastic (QE) reaction mechanisms. The data is compared with plane-wave impulse approximation (PWIA) calculations using realistic spectral functions and momentum distributions. The measured and PWIA-calculated cross-section ratios for $^3$He$/d$ and $^3$H$/d$ extend to just above the typical nucleon Fermi-momentum ($k_F \approx 250$ MeV$/c$) and differ from each other by $\sim 20\%$, while for $^3$He/$^3$H they agree within the measurement accuracy of about 3\%. At momenta above $k_F$, the measured $^3$He/$^3$H ratios differ from the calculation by $20\% - 50\%$. Final state interaction (FSI) calculations using the generalized Eikonal Approximation indicate that FSI should change the $^3$He/$^3$H cross-section ratio for this measurement by less than 5\%. If these calculations are correct, then the differences at large missing momenta between the $^3$He/$^3$H experimental and calculated ratios could be due to the underlying $NN$ interaction, and thus could provide new constraints on the previously loosely-constrained short-distance parts of the $NN$ interaction.