Momentum distributions for ${}^2\mathrm{H}(e,e^{\prime }p)$

Background: A primary goal of deuteron electrodisintegration is the possibility of extracting the deuteron momentum distribution. This extraction is inherently fraught with difficulty, as the momentum distribution is not an observable and the extraction relies on theoretical models dependent on other models as input.

Purpose: We present a new method for extracting the momentum distribution which takes into account a wide variety of model inputs thus providing a theoretical uncertainty due to the various model constituents. To test the extraction, pseudodata were generated, and the extracted “experimental” distribution, which has theoretical uncertainty accounted by this extraction method, can be compared to the theoretical distribution. This procedure can provide an upper bound on the deuteron momentum distribution.

Method: The calculations presented here are using a Bethe-Salpeter-like formalism with a wide variety of bound state wave functions, form factors, and final state interactions.

Results: In the examples we compared, the original distribution was typically within the error band of the extracted distribution. The input wave functions do contain some outliers which are discussed in the text. Due to the reliance on the theoretical calculation to obtain this quantity, any extraction method should account for the theoretical error inherent in these calculations due to model inputs.

Conclusions: The extraction method works well and provides a systematic way to investigate the deuteron momentum distribution, while accounting for theoretical uncertainty and providing a theoretical error band.

Figure 1

Feynman diagrams representing the impulse approximation to deuteron electrodisintegration. In all diagrams particle 1 is a proton and particle 2 is a neutron.

Figure 2

Momentum density distributions for the eight wave functions used in the calculations presented in this work.

Figure 3

A comparison of three different parametrizations of the electric and magnetic form factors of the proton and neutron, divided by the corresponding dipole form factors. In each case the vertical line corresponds to $Q^2=4.25{\mathrm{GeV}}^2$ which is the value for the chosen kinematics.

Figure 4

PWIA cross sections calculated for the E121003 kinematics with the IIB wave functions and the three form factor parametrizations. Cross sections are plotted as a function of the missing momentum $p_m$.

Figure 5

Center of momentum $pn$ elastic cross sections calculated using both SAID and Regge methods. Data close to the chosen value of $s$ are also displayed.

Figure 6

The envelope containing all 24 calculations of the cross section in PWIA is shown in panel (a). The corresponding envelope containing all 48 of the complete calculations represented by the diagrams of Fig. 1 is shown in panel (b).

Figure 7

The ratio of FSI to PWIA cross sections for wave function IIB, the GKex05 electromagnetic form factors at $\varepsilon =11\mathrm{GeV},Q^2=4.25{\mathrm{GeV}}^2$, and a range of values for $x$.

Figure 8

The envelope containing the ratio FSI to PWIA cross sections for all 48 calculations is shown by the shaded band.

Figure 9

Ratio of FSI to PWIA cross sections for wave function IIB and the AMT electromagnetic form factors at the E1210003 kinematics for both SAID and Regge FSI. Curves labeled with (a)+(c) contain only the proton contributions to the IA [diagrams (a) and (c) of Fig. 1], while those without the label also contain contributions of from the neutron given by diagrams (b) and (d) of Fig. 1.

Figure 10

This figure uses the calculation of the cross section with the WJC 2 wave functions, the GKex05 electromagnetic form factor, and the Regge FSI as pseudodata. The dotted line is the reduced cross section using the AMT form factor in ${\sigma }_{ep}$, and the dashed line is the extracted momentum distribution using the procedure described above with a shaded band representing the theoretical error. The solid line is the momentum distribution for the WJC 2 wave functions.

Figure 11

Same as Fig. 10 but using the AV18 wave function.

Figure 12

Same as Fig. 10 but using the CD Bonn wave function.

Figure 13

Same as Fig. 10 but using the WJC 1 wave function.

Figure 14

Diagram representing the half-on-shell deuteron vertex function.