Relativistic corrections for lepton-nucleus scattering in the short-time approximation

Abstract

We present an approach for including relativistic corrections in lepton-nucleus scattering calculations within the Short-Time Approximation (STA). Previous ab-initio studies employed electromagnetic currents expanded in powers of $q/m$, where $q$ is the momentum transfer and $m$ is the nucleon mass, restricting their validity to low-$q$ kinematics. We adopt an expansion scheme that treats the initial nucleon momentum perturbatively while allowing for arbitrary momentum transfer, thereby extending the applicability of the STA to high-$q$ regimes. Additionally, we incorporate a relativistic treatment of the two-nucleon final-state energies. Calculations for $^3$He and $^4$He inclusive electron scattering cross sections show a substantial improvement over previous results, achieving good agreement with experimental data in the quasi-elastic region for both low- and high-momentum transfer.

Figures (9)

• Figure 1: Ratio of the longitudinal responses obtained using: i) the high- and low-$q$ expansion schemes for the one-body charge orator at LO (blue solid line); and ii) the Darwin-Foldy correction to the charge operator and the low-$q$ expansion at LO (orange dashed line). See text for explanation.
• Figure 2: Ratio of the transverse responses obtained using the LO high-$q$ and LO low-$q$ expansion schemes for the one-body current operator. See text for explanations.
• Figure 3: Left panel: Transverse response density for $^4$He at $q=700$ MeV/$c$, as a function of relative and center of mass energies, $e$ and $E$. Results are from Ref. Pastore:2019urn and include the correlated-propagator along with one- and two-nucleon currents. Right panel: Contour plot of the figure in the left panel. See text for explanations.
• Figure 4: Left panel: Transverse response density for $^4$He at $q=700$ MeV/$c$, as a function of single nucleon momenta, $p_1$ and $p_2$. Results based on the free-propagator and single-nucleon currents up to $(p/m)^1$ in the high-$q$ expansion -- see Eqs. \ref{['eq:current_p0']} and \ref{['eq:current_p1']}. Right panel: Contour plot of the figure in the left panel. Solid blue (dashed red) lines indicate the energy-conserving delta function with relativistic (nonrelativistic) energies for the active nucleons in the final state for two given values of energy transfer $\omega$. See text for explanations.
• Figure 5: Longitudinal (left) and transverse (right) response functions for $^3$He at $q=700$ MeV/$c$. Solid blue and dashed red lines show STA and STA(0) results, respectively. Green dashed lines correspond to STA(0)NR$\delta$ results. Experimental data from Ref. Carlson:2001mp are given by the black symbols.