Scale dependence of deuteron electrodisintegration

Background: Isolating nuclear structure properties from knock-out reactions in a process-independent manner requires a controlled factorization, which is always to some degree scale and scheme dependent. Understanding this dependence is important for robust extractions from experiment, to correctly use the structure information in other processes, and to understand the impact of approximations for both.

Purpose: We seek insight into scale dependence by exploring a model calculation of deuteron electrodisintegration, which provides a simple and clean theoretical laboratory.

Methods: By considering various kinematic regions of the longitudinal structure function, we can examine how the components—the initial deuteron wave function, the current operator, and the final-state interactions (FSIs)—combine at different scales. We use the similarity renormalization group to evolve each component.

Results: When evolved to different resolutions, the ingredients are all modified, but how they combine depends strongly on the kinematic region. In some regions, for example, the FSIs are largely unaffected by evolution, while elsewhere FSIs are greatly reduced. For certain kinematics, the impulse approximation at a high renormalization group resolution gives an intuitive picture in terms of a one-body current breaking up a short-range correlated neutron-proton pair, although FSIs distort this simple picture. With evolution to low resolution, however, the cross section is unchanged but a very different and arguably simpler intuitive picture emerges, with the evolved current efficiently represented at low momentum through derivative expansions or low-rank singular value decompositions.

Conclusions: The underlying physics of deuteron electrodisintegration is scale dependent and not just kinematics dependent. As a result, intuition about physics such as the role of short-range correlations or $D$-state mixing in particular kinematic regimes can be strongly scale dependent. Understanding this dependence is crucial in making use of extracted properties.

Figure 1

SRG running of the AV18 potential in the ${}^{3}S_1–{}^{3}S_1$ channel.

Figure 2

SRG running of the AV18 potential in the ${}^{3}S_1{–}^3{D}_1$ channel.

Figure 3

Deuteron wave functions in position space for the unevolved ($\lambda =\infty$) and evolved (to $\lambda =2{\mathrm{fm}}^{-1}$) AV18 potential.

Figure 4

Deuteron wave functions in momentum space for the unevolved ($\lambda =\infty$) and evolved (to $\lambda =2{\mathrm{fm}}^{-1}$) AV18 potential.

Figure 5

$|\mathrm{\Delta }{\psi }^{\lambda }(p^{\prime };k)|$ in the ${}^{3}S_1$ channel for various $p^{\prime }$ and SRG $\lambda$ values. $\mathrm{\Delta }{\psi }^{\lambda }(p^{\prime };k)$ in the evolved case is suppressed for large momentum. For large $p^{\prime }$, the values of $\mathrm{\Delta }{\psi }^{\lambda }(p^{\prime };k)$ even at small momenta differ for different SRG $\lambda \mathrm{s}$.

Figure 6

$|{\psi }^{\lambda }(p^{\prime };r)|$ in the ${}^{3}S_1$ channel for various $p^{\prime }$ and SRG $\lambda$ values. ${\psi }^{\infty }\left(r\right)$ has a correlation wound at short distances that is absent for the evolved wave functions.

Figure 7

Heat map plot for the unevolved ($\lambda =\infty$) and the evolved current ($\lambda =1.5{\mathrm{fm}}^{-1}$) for $q^2=36{\mathrm{fm}}^{-2}$ in the ${}^{3}S_1–{}^{3}S_1$ block for $m_{J_d}=0$. The heat map plots for $m_{J_d}=1$ look very similar. The unevolved current is peaked at $(0,q/2)$ and $(q/2,0)$. SRG evolution induces smooth two-body currents in the low-momentum region.

Figure 8

Heat map plot for the evolved current ($\lambda =1.5{\mathrm{fm}}^{-1}$) for $q^2=36{\mathrm{fm}}^{-2}$ in the ${}^{3}P_1–{}^{3}S_1$ channel for $m_{J_d}=1$. While the changes due to evolution are distributed, only the low-momentum region of the evolved current is relevant for our examples (see the boxed region). In this region, the evolved current is smooth and linear in $k_1$ [cf. Eq. (15)].

Figure 9

Local decoupling in the final state of the ${}^{3}S_1$ partial wave at large momentum $p^{\prime }=1.5$ fm.

Figure 10

Comparing the unevolved IA and the full calculations of $f_L$ at $p^{\prime }=1.5{\mathrm{fm}}^{-1}$ (or $E^{\prime }=100$ MeV) and $q^2=49{\mathrm{fm}}^{-2}$ with the evolved IA result at $\lambda =1.5{\mathrm{fm}}^{-1}$ and $\lambda =1.2{\mathrm{fm}}^{-1}$. We find that the contribution of FSI is minimal in the evolved picture at this kinematics corresponding to $x_d=1.64$ and $Q^2=1.78{\mathrm{GeV}}^2$.

Figure 11

Same as Fig. 10 but at $q^2=36{\mathrm{fm}}^{-2}$. Here $x_d=1.55$ and $Q^2=1.34{\mathrm{GeV}}^2$.

Figure 12

Comparing the unevolved IA and the full calculations of $f_L$ at $E^{\prime }=100$ MeV and $q^2=10{\mathrm{fm}}^{-2}$ with the evolved IA result at $\lambda =1.5{\mathrm{fm}}^{-1}$ and $\lambda =1.2{\mathrm{fm}}^{-1}$. This kinematics with $x_d=0.99$ and $Q^2=0.39{\mathrm{GeV}}^2$ corresponds to the quasifree ridge. At the quasifree ridge, the contributions from the FSI are small for all SRG scales.

Figure 13

Comparing the unevolved IA and the full calculations of $f_L$ at $p^{\prime }=1.5{\mathrm{fm}}^{-1}$ (or $E^{\prime }=100$ MeV) and $q^2=1{\mathrm{fm}}^{-2}$ with the evolved IA result at $\lambda =1.5{\mathrm{fm}}^{-1}$ and $\lambda =1.2{\mathrm{fm}}^{-1}$. In the region $p^{\prime }>q/2, {\psi }_f^{\lambda }$ picks the contribution from the nonsmooth high-momentum region of $J_0^{\lambda }(k,k^{\prime })$. Here $x_d=0.14, Q^2=0.03{\mathrm{GeV}}^2$.

Figure 14

Exact $f_L$ (solid line) for $E_{np}=20$ MeV and $q^2=36{\mathrm{fm}}^{-2}$ compared to $f_L$ obtained using the derivative expansion for the evolved current ($\lambda =1.5{\mathrm{fm}}^{-1}$). For this kinematics $x_d=1.88, Q^2=1.3{\mathrm{GeV}}^2$.

Figure 15

Convergence of the derivative expansion in the $S$ channel. Here $x_d=1.88, Q^2=1.3{\mathrm{GeV}}^2$, and $\lambda =1.5{\mathrm{fm}}^{-1}$.

Figure 16

Exact $f_L$ (solid line) compared to $f_L$ obtained using the SVD expansion for the evolved current at successive orders for $\lambda =1.5{\mathrm{fm}}^{-1}$ and $k_{\max }=1.6\lambda$. S+P+D indicates that we have gone up to the $D$ channel in the partial wave expansion of the final state.

Figure 17

Convergence of the SVD expansion in the $S$ channel for $\lambda =1.5{\mathrm{fm}}^{-1}$ and $k_{\max }=1.6\lambda$. Here $x_d=1.88, Q^2=1.3{\mathrm{GeV}}^2$.

Figure 18

Comparison of singular vectors for different $q^2$ for two different channels for $\lambda =1.5{\mathrm{fm}}^{-1}$ and $k_{\max }=1.6\lambda$.

Figure 19

Demonstration that for $p^{\prime }\ll q$ the $q$ dependence of $f_L$ factorizes.

Figure 20

Demonstration that $f_L$ is a strong function of $q$. $Y$ axis is the denominator of the ratio plotted in Fig. 19. For the region in which the ratio of $f_L$ plateaus, the denominator by itself varies by over three orders of magnitude.

Figure 21

Contribution to $f_L$ from the deuteron $S$ state in the evolved and unevolved case. Here $x_d=1.88, Q^2=1.3{\mathrm{GeV}}^2$.

Figure 22

Contribution to $f_L$ from the deuteron $S$ and $D$ states as a function of SRG $\lambda$ for the same $x_d$ and $Q^2$ as in Fig. 21 but with fixed ${\theta }^{\prime }=0^{\circ }$.

Figure 23

Schematic pictures of the most important incoming and outgoing proton and neutron momenta for ${\theta }^{\prime }=0^{\circ }$ scattering in the impulse approximation with quasifree kinematics at any $\lambda$, and analogous pictures for scattering with large momentum transfer near threshold at high and low resolution.

Figure 24

$Q^2$ values as a function of $E^{\prime }$ and ${\mathbf{q}}_{\mathrm{c}.\mathrm{m}.}^2$.

Figure 25

$x_d$ values as a function of $E^{\prime }$ and ${\mathbf{q}}_{\mathrm{c}.\mathrm{m}.}^2$.