Quasielastic lepton scattering and back-to-back nucleons in the short-time approximation

Understanding quasielastic electron and neutrino scattering from nuclei has taken on new urgency with current and planned neutrino oscillation experiments, and with electron scattering experiments measuring specific final states, such as those involving nucleon pairs in “back-to-back” configurations. Accurate many-body methods are available for calculating the response of light ($A\le 12$) nuclei to electromagnetic and weak probes, but they are computationally intensive and only applicable to the inclusive response. In the present work we introduce a novel approach, based on realistic models of nuclear interactions and currents, to evaluate the short-time (high-energy) inclusive and exclusive response of nuclei. The approach accounts reliably for crucial two-nucleon dynamics, including correlations and currents, and provides information on back-to-back nucleons observed in electron and neutrino scattering experiments. We demonstrate that in the quasielastic regime and at moderate momentum transfers both initial- and final-state correlations and two-nucleon currents are important for a quantitatively successful description of the inclusive response and final-state nucleons. Finally, the approach can be extended to include relativistic—kinematical and dynamical—effects, at least approximately in the two-nucleon sector, and to describe the response in the resonance-excitation region.

Figure 1

Nearest-neighbor probability density at nuclear matter density for noninteracting Boltzmann (distinguishable) particles (black circles) and free nucleons (red squares) and with realistic interactions (blue diamonds).

Figure 2

Contributions to the sum rule from single-nucleon currents in the plane-wave impulse approximation (PWIA) from low- (left panel) and higher-momentum nucleons (middle panel). The latter interfere constructively with the two-nucleon current operator as indicated in the right panel. See text for further explanations.

Figure 3

Alpha particle transverse response densities at $q=500$ MeV/$c$. The surface plots show the response densities as functions of relative energy $e$ and center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$. Results prior to (left panel) and after (right panel) the shift (see text).

Figure 4

STA results for ${}^4\mathrm{He}(e,e^{\prime })$ inclusive scattering. Left panel: Transverse response at $q=300$ MeV/$c$, prior to shift (dashed-dotted blue line) and including shift (solid red line) compared with the exact GFMC response (solid black line). Right panel: Same but at $q=500$ MeV/$c$. See text for further explanations.

Figure 5

STA results for ${}^4\mathrm{He}(e,e^{\prime })$ inclusive scattering. Longitudinal response at $q=500$ MeV/$c$. Notation as in Fig. 4.

Figure 6

Comparison of transverse responses without (dashed lines) and with (full lines) interacting two-nucleon final states. Various contributions are shown, including one-body current diagonal terms, one-body current off-diagonal ($i\ne j$) terms, interference between one- and two-body currents, and two-body currents only. See text for further explanations. Results at $q=300$ MeV/$c$ (left panel) and $q=500$ MeV/$c$ (right panel).

Figure 7

Transverse response density at $q=500$ MeV/$c$. The three-dimensional plot shows the response density as a function of relative $e$ and center-of-mass $E_{\mathrm{c}.\mathrm{m}.}$ energies in units of MeV. The contour plot below shows the fraction of the response coming from terms including two-nucleon currents.

Figure 8

Contributions to response densities at $q=500$ MeV/$c$ and final center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}=q^2/\left(4m\right)$. Contributions of one-body diagonal terms are shown (cyan solid line), along with the total one-body currents given by diagonal plus off-diagonal contributions (magenta solid line). Full (one- plus two-body currents) results are also shown for both total (solid black line) and contributions from $nn$ (blue dashed line) and $pp$ pairs (red dashed line). Left panel: Longitudinal. Right panel: Transverse.

Figure 9

Same as Fig. 8 but at $q=700$ MeV/$c$.

Figure 10

Contributions to response densities at $q=500$ MeV/$c$ and final center of mass energy $E_{\mathrm{c}.\mathrm{m}.}=q^2/\left(4m\right)$. Total contributions from all pairs (black solid line) and from from $nn$ (blue solid line) and $pp$ (red solid line) pairs are shown along with the percentage of $pp$ (dashed orange line) and $nn$ (dashed dark green line) contributions over the all pairs contribution. Left: Longitudinal. Here, the magenta dotted line represents the $pp$ pair percentage from two-nucleon momentum distributions—MD in the figure—from Ref. [63]. Right: Transverse.

Figure 11

Transverse response functions calculated with the STA at $q=300–600\mathrm{MeV}/c$ are compared with those obtained from analysis of the world data [10].