Deep-inelastic and quasielastic electron scattering from $A=3$ nuclei

We perform a combined analysis of inclusive electron scattering data from $A=3$ nuclei in the deep-inelastic and quasielastic scattering regions, using Monte Carlo analysis methods and the nuclear weak binding approximation to establish the range over which the data can be described within the same theoretical framework. Comparison with quasielastic ${}^3\mathrm{He}$ cross sections from SLAC and Jefferson Lab suggests that most features of the $x\gtrsim 1$ data can be reasonably well described in the impulse approximation with finite-$Q^2$ nuclear smearing functions for momentum transfers $Q^2\gtrsim 1{\mathrm{GeV}}^2$. For the DIS region, we analyze the recent ${}^3\mathrm{He}$ to deuterium cross-section ratio from the Jefferson Lab E03-103 experiment to explore the possible isospin dependence of the nuclear effects. We discuss the implications of this for the MARATHON experiment at Jefferson Lab and outline how a Bayesian analysis of ${}^3\mathrm{He}$, ${}^3\mathrm{H}$, and deuterium data can robustly determine the free neutron structure function.

Figure 1

Proton smearing functions in ${}^3\mathrm{He}$, $f_{ij}^p(y,\gamma )$, $i=2,L$, for the on-shell [(a)–(c)] and off-shell [(d)–(f)] distributions, computed from the KPSV spectral function [57] for $\gamma =1$ (red solid curves), 1.5 (green dot-dashed curves), and 4 (blue dashed curves).

Figure 2

Neutron smearing functions in ${}^3\mathrm{He}$, $f_{ij}^n(y,\gamma )$, $i=2,L$, for the on-shell [(a)–(c)] and off-shell [(d)–(f)] distributions, computed from the KPSV spectral function [57] for $\gamma =1$ (red solid curves), 1.5 (green dot-dashed curves), and 4 (blue dashed curves).

Figure 3

Ratio of deuteron pole contribution to the total smearing function for (a) the proton on-shell $f_{22}^p$ function and (b) proton off-shell ${\tilde{f}}_{22}^p$ function, for different values of $\gamma$. The ratio of the total proton smearing functions for the SS [53] and KPSV [57] spectral functions is given in panels (c) and (d) for the on-shell and off-shell functions, respectively.

Figure 4

QE electron-${}^3\mathrm{He}$ cross section as a function of $x$ for fixed incident electron energies $E$ and scattering angles $\theta$, with $Q_0^2$ the value of the momentum transfer squared at $x=1$. Data from the early SLAC experiment by Day et al. [35] [(a)–(c)] and the subsequent NE9 experiment by Meziani et al. [36] [(d)–(f)] are compared with the WBA calculation using the KPSV and SS spectral functions, and the “cc1” and “cc2” off-shell prescriptions, as well as a calculation using smearing functions at $\gamma =1$ [(a) and (d)], scaled by a factor 1/2 for clarity.

Figure 5

As in Fig. 4, but for the Jefferson Lab QE data from Fomin et al. [37] at $E=5.766$ GeV. Cross sections which include inelastic contributions are illustrated in panels (e) and (f) (black dot-dashed curves).

Figure 6

QE cross section for ${}^3\mathrm{He}$ (red solid curves) and ${}^3\mathrm{H}$ (blue dashed curves) [in units of $\mathrm{nb}/\mathrm{sr}\mathrm{GeV}$] at typical kinematics of the E12-11-112 experiment [38] with $E=4.3$ GeV, for (a) $\theta ={15}^{\circ }$ and $Q_0^2=1{\mathrm{GeV}}^2$ and (b) $\theta ={30}^{\circ }$ and $Q_0^2=3{\mathrm{GeV}}^2$. The corresponding ratios of the ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ cross sections in panels (c) and (d) illustrate the effects of the different $A=3$ spectral functions, from the KPSV (red solid curves) and SS (green dashed curves) models.

Figure 7

Ratio of the neutron magnetic form factor $G_{Mn}$, extracted from Eqs. (37) and (38), to the input form factor $G_{Mn}^{\left(0\right)}$ taken from the Kelly parametrization [63]. The extracted $G_{Mn}$ is computed using the exact on-shell calculation (dashed blue curve), giving a ratio of unity, and with $G_{Mn}$ computed from Eq. (38) but with $f^p=f^n$ (solid red curve), or for $f^p=f^n$ with the off-shell calculation using the “cc2” prescription (dot-dashed green curve).

Figure 8

Isoscalar off-shell function $\delta f^0$ from the CJ15 [15] global QCD analysis of proton and deuteron data (red solid curve) and the Kulagin-Petti (KP) [25] fit to of nuclear structure function ratios, assuming $\delta f^p=\delta f^n$. The kinematics of the Jefferson Lab E03-103 experiment [33] are indicated by the green horizontal band.

Figure 9

Ratio of ${}^3\mathrm{He}$ to deuterium cross sections, ${\sigma }^{{}^3\mathrm{He}}/{\sigma }^d$, scaled per nucleon, from the Jefferson Lab E03-103 experiment [33] compared with the full Monte Carlo fit results (red solid curves and bands) using (a) the CJ [15] and (b) the KP [25] isoscalar off-shell function $\delta f^0$, as well as with fits assuming isospin symmetric off-shell corrections, $\delta f^p=\delta f^n$ (green dashed curves and bands), and with the on-shell only fits (blue solid curves and bands). The experimental error bars include statistical and systematic uncertainties added in quadrature, with an overall 1.84% normalization uncertainty not shown [85].

Figure 10

Normalized yield of the Monte Carlo parameter distributions for the proton off-shell function $\delta f^p$, for the normalization $C$ [(a) and (c)] and intercept $x_0$ [(b) and (d)], using the isoscalar off-shell function $\delta f^0$ from the CJ [15] [(a) and (b)] and KP [25] [(c) and (d)] analyses. The colored histograms represent 10 statistically independent Monte Carlo analyses, while the black histograms are the combined result.

Figure 11

Off-shell functions for the proton, $\delta f^p$ (red solid curves and bands), and neutron, $\delta f^n$ (blue solid curves and bands), from the fit to the E03-103 data [33], for a given isoscalar off-shell function, $\delta f^0$ (green dashed), from (a) the CJ [15] global analysis and (b) the Kulagin-Petti [25] nuclear ratios fit. The functions are shown only in the range of $x$ constrained by the E03-103 data.

Figure 12

Normalized yield of Monte Carlo distributions for the data normalization factors $N_{\mathrm{dat}}$ for the CJ [15] [(a) and (b)] and KP [25] [(c) and (d)] isoscalar functions $\delta f^0$, assuming isospin symmetry ($\delta f^p=\delta f^n$) [(a) and (c)] and the isospin-dependent analysis ($\delta f^p\ne \delta f^n$) [(b) and (d)], along with the on-shell only fit [(e)]. The colored histograms represent 10 statistically independent Monte Carlo analyses, while the black histograms are for the combined result. The pile-up in some of the fits occurs at the upper boundary of the allowed 1.84% normalization uncertainty in the E03-103 experiment [33].

Figure 13

Ratios of nuclear to nucleon structure functions (a) $R\left({}^3\mathrm{He}\right)=F_2^{{}^3\mathrm{He}}/(2F_2^p+F_2^n)$ and (b) $R\left({}^3\mathrm{H}\right)=F_2^{{}^3\mathrm{H}}/(F_2^p+2F_2^n)$, for the off-shell Monte Carlo fits using the CJ [15] (solid red curves and bands) and Kulagin-Petti [25] (solid blue curves and bands) isoscalar off-shell function $\delta f^0$, compared with the ratios assuming isospin symmetric off-shell functions from CJ (dashed red curves) and KP (dashed blue curves), and the on-shell only fit (dotted green curves). The range of $x$ constrained by the Jefferson Lab E03-103 experiment [33] is indicated by the green horizontal band, and a scale of $Q^2=4{\mathrm{GeV}}^2$, which is close to the average for the E03-103 data, was used for all structure functions.

Figure 14

(a) Super-ratio $R\left({}^3\mathrm{He}\right)/R\left({}^3\mathrm{H}\right)$ of the EMC ratios in ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ and (b) the ratio $F_2^{{}^3\mathrm{He}}/F_2^{{}^3\mathrm{H}}$ of the ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$ structure functions. The off-shell Monte Carlo fits using the CJ [15] (solid red curves and bands) and Kulagin-Petti [25] (solid blue curves and bands) isoscalar off-shell functions $\delta f^0$ are compared with the results assuming isospin symmetric off-shell functions from CJ (dashed red curves) and KP (dashed blue curves), and the on-shell only fit (dotted green curves). The range of $x$ constrained by the Jefferson Lab E03-103 experiment [33] and the extended range expected by the MARATHON experiment [20] are indicated by the green horizontal bands.