Global superscaling analysis of quasielastic electron scattering with relativistic effective mass

We present a global analysis of the inclusive quasielastic electron scattering data with a superscaling approach with relativistic effective mass. The SuSAM* model exploits the approximation of factorization of the scaling function $f^{*}\left({\psi }^{*}\right)$ out of the cross section under quasifree conditions. Our approach is based on the relativistic mean field theory of nuclear matter where a relativistic effective mass for the nucleon encodes the dynamics of nucleons moving in presence of scalar and vector potentials. Both the scaling variable ${\psi }^{*}$ and the single nucleon cross sections include the effective mass as a parameter to be fitted to the data alongside the Fermi momentum $k_F$. Several methods to extract the scaling function and its uncertainty from the data are proposed and compared. The model predictions for the quasielastic cross section and the theoretical error bands are presented and discussed for nuclei along the periodic table from $A=2$ to $A=238$: ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, ${}^3\mathrm{He}$, ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^6\mathrm{Li}$, ${}^9\mathrm{Be}$, ${}^{24}\mathrm{Mg}$, ${}^{59}\mathrm{Ni}$, ${}^{89}\mathrm{Y}$, ${}^{119}\mathrm{Sn}$, ${}^{181}\mathrm{Ta}$, ${}^{186}\mathrm{W}$, ${}^{197}\mathrm{Au}$, ${}^{16}\mathrm{O}$, ${}^{27}\mathrm{Al}$, ${}^{40}\mathrm{Ca}$, ${}^{48}\mathrm{Ca}$, ${}^{56}\mathrm{Fe}$, ${}^{208}\mathrm{Pb}$, and ${}^{238}\mathrm{U}$. We find that more than 9 000 of the total $\approx 20000$ data fall within the quasielastic theoretical bands. Predictions for ${}^{48}\mathrm{Ti}$ and ${}^{40}\mathrm{Ar}$ are also provided for the kinematics of interest to neutrino experiments.

Figure 1

Phenomenological scaling function bands compared to the $(e,e^{\prime })$ data scaled with the best parameters of the global fit and selected with the density criterion. The corresponding scaling function band C (in pink) is compared to the band B of Ref. [44] (in green). The data are selected from Refs. [49, 50, 51].

Figure 2

Color maps of the number, $N$, of QE data inside the phenomenological band divided by $N_{\max }$, as a function of the effective mass $M^{*}$ and Fermi momentum $k_F$ for different nuclei. The $\sigma \text{−}\omega$ model of Ref. [47] is shown as comparison.

Figure 3

Color maps of the ${\chi }^2$ of the QE data with the phenomenological band, divided by ${\chi }_{\min }^2$, as a function of the effective mass $M^{*}$ and Fermi momentum $k_F$ for different nuclei. The region with ${\chi }^2$ above 10% of the minimum is shown in white. The region with ${\chi }^2$ below 10% of the minimum is fitted by a ellipse. The $\sigma \text{−}\omega$ model of Ref. [47] is shown as comparison.

Figure 4

Correlation plot of the ${\chi }^2$ values versus the number of points inside the band $N_{\mathrm{band}}$ for different values of $k_F$ and $M^{*}$ around the region where ${\chi }^2$ is minimum and $N_{\mathrm{band}}$ reaches its maximum, for six nuclei.

Figure 5

Top panel: Data used in the global fit of the superscaling function. The fit is made by maximizing the number of data inside a band centered around a sample scaling function of width 0.1. In black the 4754 points falling inside the band after the fit. The central function (in red-dashed lines) is the sum of two Gaussians (shown in the figure in dashed and double-dashed lines) modified by a Fermi function (we show the denominator of the Fermi function in dash-double dotted line). Middle panel: Global set of $(e,e^{\prime })$ data scaled with the best parameters after the global fit. The scaling function appears as a dark shadow. Data are from Refs. [49, 50, 51]. Bottom panel: the same data compared to the parameterized band C.

Figure 6

Inclusive $(e,e^{\prime })$ cross-section data for ${}^2\mathrm{H}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 7

Inclusive $(e,e^{\prime })$ cross-section data for ${}^3\mathrm{H}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 8

Inclusive $(e,e^{\prime })$ cross-section data for ${}^3\mathrm{He}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 9

Inclusive $(e,e^{\prime })$ cross-section data for ${}^4\mathrm{He}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 10

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{12}\mathrm{C}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 11

Inclusive $(e,e^{\prime })$ cross-section data for nine different nuclei for different kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 12

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{27}\mathrm{Al}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 13

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{40}\mathrm{Ca}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 14

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{48}\mathrm{Ca}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 15

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{56}\mathrm{Fe}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 16

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{208}\mathrm{Pb}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 17

Inclusive $(e,e^{\prime })$ cross-section data for ${}^{238}\mathrm{U}$ for selected kinematics compared to the SuSAM* model as a function of energy transfer. Data are from Refs. [49, 50, 51].

Figure 18

Relativistic effective mass $M^{*}$ computed from the QE peak position of the selected experimental data sets (violet squares) as a function of the energy transfer $\omega$. The experimental bands for $M^{*}$ have been obtained from the SuSAM* bands for the same experimental kinematics (circles) by computing the lower and upper limits for the effective mass allowed by the band.

Figure 19

Relativistic effective mass $M^{*}$ computed from the QE peak position of the selected experimental data sets (squares) as a function of the energy transfer $\omega$. The experimental bands for $M^{*}$ have been obtained from the SuSAM* bands for the same experimental kinematics (blue circles) by computing the lower and upper limits for the effective mass allowed by the band.

Figure 20

Inclusive $(e,e^{\prime })$ cross sections of ${}^{12}\mathrm{C}$, ${}^{48}\mathrm{Ti}$, and ${}^{40}\mathrm{Ar}$ for the kinematics $\epsilon =2222\mathrm{MeV}$, $\theta =15.{541}^{\mathrm{o}}$, of the recent JLab experiment [52], compared to the SuSAM* model as a function of energy transfer.