Meaning of the nuclear wave function

Background: The intense current experimental interest in studying the structure of the deuteron and using it to enable accurate studies of neutron structure motivate us to examine the four-dimensional space-time nature of the nuclear wave function and the various approximations used to reduce it to an object that depends only on three spatial variables.

Purpose: The aim is to determine if the ability to understand and analyze measured experimental cross sections is compromised by making the reduction from four to three dimensions.

Method: Simple, exactly calculable, covariant models of a bound-state wave-state wave function (a scalar boson made of two constituent-scalar bosons) with parameters chosen to represent a deuteron are used to investigate the accuracy of using different approximations to the nuclear wave function to compute the quasielastic scattering cross section. Four different versions of the wave function are defined (light-front-spectator, light-front, light-front with scaling, and nonrelativistic) and used to compute the cross sections as a function of how far off the mass shell (how virtual) is the struck constituent.

Results: We show that making an exact calculation of the quasielastic scattering cross section involves using the light-front-spectator wave function. All of the other approaches fail to reproduce the model exact calculation if the value of Bjorken $x$ differs from unity. The model is extended to consider an essential effect of spin to show that constituent nucleons cannot be treated as being on their mass shell even when taking the matrix element of a “good” current.

Conclusions: Developing realistic light-front-spectator wave functions to meet the needs of current and planned experiments is a worthwhile activity.

Figure 1

Scattering process of interest.

Figure 2

Virtuality as a function of $x$ and $Q^2$ for $k_{\perp }=0$.

Figure 3

Virtuality as a function of $x$ and $Q^2$ for $k_{\perp }=0.25$ GeV.

Figure 4

Virtuality as a function of $x$ and $Q^2$, lower values of $Q^2$, for $k_{\perp }=0$.

Figure 5

$\frac{d\mathrm{\Sigma }}{d^2{k}_{\perp }}$ as a function of $x$ for two different ranges of $Q^2$ with $k_{\perp }=0$. (a) $Q^2$ from 1 to 6 ${\mathrm{GeV}}^2$. (b) $Q^2$ from 10 to 60 ${\mathrm{GeV}}^2$. For each case, the lower the value of $Q^2$, the higher the cross section. For larger values of $Q^2$ the curves tend to coalesce.

Figure 6

$\frac{d\mathrm{\Sigma }}{d^2{k}_{\perp }}$ for three models at $x=1$ and $k_{\perp }=0$. Only two curves are observable easily because of the confluence of the exact and on-mass-shell light-front wave function approach.

Figure 7

Exact to on-mass-shell ratios as a function of $x$ and $k_{\perp }=0$ for different values of $Q^2$.

Figure 8

Nonrelativistic approximation. Ratios of exact to nonrelativistic cross sections are shown as a function of $x$ for different values of $Q^2$.

Figure 9

$\frac{d\mathrm{\Sigma }}{d^2{k}_{\perp }}$ as a function of $x$ for two different ranges of $Q^2$ with $k_{\perp }=0$. (a) $Q^2$ from 1 to 6 ${\mathrm{GeV}}^2$. (b) $Q^2$ from 10 to 60 ${\mathrm{GeV}}^2$. For each case, the lower the value of $Q^2$, the higher the cross section. For larger values of $Q^2$ the curves tend to coalesce. Deuteron wave function of Eq. (33).

Figure 10

Exact to on-mass-shell ratios as a function of $x$ and $k_{\perp }=0$ for different values of $Q^2$. Deuteron wave function of Eq. (33).

Figure 11

Nonrelativistic approximation. Ratios of exact to nonrelativistic cross sections are shown as a function of $x$ for different values of $Q^2$. Deuteron wave function of Eq. (33).