$A=3(e,e^{\prime })x_B\ge 1$ cross-section ratios and the isospin structure of short-range correlations

We study the relation between measured high-$x_B$, high-$Q^2$, helium-3 to tritium, $(e,e^{\prime })$ inclusive-scattering cross-section ratios and the relative abundance of high-momentum neutron-proton ($np$) and proton-proton ($pp$) short-range correlated nucleon pairs in three-body ($A=3$) nuclei. Analysis of these data using a simple pair-counting cross-section model suggested a much smaller $np/pp$ ratio than previously measured in heavier nuclei, questioning our understanding of $A=3$ nuclei and, by extension, all other nuclei. Here, we examine this finding using spectral-function-based cross-section calculations, with both an ab initio $A=3$ spectral function and effective generalized contact formalism spectral functions using different nucleon-nucleon interaction models. The ab initio calculation agrees with the data, showing good understanding of the structure of $A=3$ nuclei. An 8% uncertainty on the simple pair-counting model, as implied by the difference between it and the ab initio calculation, gives a factor of 5 uncertainty in the extracted $np/pp$ ratio. Thus we see no evidence for the claimed “unexpected structure in the high-momentum wave function for hydrogen-3 and helium-3.”

Figure 1

The relation between the average ${}^3\mathrm{H}/{}^3\mathrm{He}$ cross-section ratio for $1.4\le x_B\le 1.7$ and the relative number of $np$ and $pp$ SRC pairs in helium-3. The experimental results of Ref. [1] are shown by the black band, whose width equals the data uncertainty. The AV18 prediction is shown by the blue data data point using the $np/pp$ SRC-pair ratio from Ref. [11]. The simple pair-counting model is shown by the red line and its uncertainty is shown by the red band. The uncertainty of $\pm 8\%$ is determined by the difference between the model and the AV18 calculation for the same $np/pp$ ratio. The resulting $np/pp$ SRC-pair ratio uncertainty is shown by the vertical lines.

Figure 2

Tritium to helium-3 $(e,e^{\prime })$ cross-section ratio, ${\sigma }_{\text{T}}/{\sigma }_{\text{He}}$, at $Q^2\approx 1.9{\mathrm{GeV}}^2$ as a function of $x_B$ for data [1] (black points) and calculation (blue line). The factorized cross-section calculation [Eq. (5)] uses an exact ab initio spectral function calculated using the AV18 interaction [12]. The gray band shows the $1.18\%$ data normalization uncertainty ($1\sigma$).

Figure 3

Calculated contributions to the ${}^3\mathrm{He}(e,e^{\prime })$ cross section from different initial nucleon states characterized by their initial momentum $k$ (top) and separation energy $E_s$ (bottom) [12]. The dashed blue line in the bottom panel, labeled “$=5.5$,” represents the two-body $pd$ breakup contribution. Calculations are performed using Eq. (5) for the $Q^2\approx 1.9{\mathrm{GeV}}^2$ kinematics of Ref. [1], using an ab initio AV18 spectral function [12].

Figure 4

Tritium to ${}^3\mathrm{He}$ GCF cross-section ratios for different $NN$ interactions [${\mathrm{AV4}}^{\prime }$: magenta; Norfolk: red; N2LO(1.0 fm): blue; N2LO(1.2 fm): green] relative to AV18, plotted vs the corresponding $np/pp$ contact ratio relative to the AV18 ratio. The different points represent different input parameters and intensity of the points is proportional to the probability of the input parameters.