Exclusive electrodisintegration of ${}^3\mathrm{He}$ at high $Q^2$. II. Decay function formalism

Based on the theoretical framework of a generalized eikonal approximation, we study the two-nucleon emission reactions in high $Q^2$ electrodisintegration of ${}^3\mathrm{He}$. Our main aim is to investigate those features of the reaction which can be unambiguously identified with the short-range properties of the ground state nuclear wave function. To evaluate the differential cross section, we work in the formalism of the decay function which characterizes the property of the ground state wave function as well as the decay properties of the final two-nucleon spectator system. Our main motivation here is to explore the accessibility of two- and three-nucleon short-range correlations in ${}^3\mathrm{He}$ as well as to isolate unambiguously single and double rescattering processes in the reaction dynamics. Our analysis also allowed us to identify new approaches for investigating the role of the practically unknown three-nucleon forces in the ground state wave function of ${}^3\mathrm{He}$.

Figure 1

(Color online) definition of type $2N$-I (a) and type $2N$-II (b) correlations.

Figure 2

(Color online) definition of type $3N$-I (a) and type $3N$-II (b) correlations.

Figure 3

(Color online) spectral function for ${}^3\mathrm{He}(e,e^{\text{'}}N)X$ reaction for a struck proton (a) and a neutron (b), calculated within the PWIA and IA. The lines at the sides of the histograms show the effects due to pair distortion.

Figure 4

Missing energy distribution of the spectral function for the removal of the proton (a) and neutron (b) at different values of missing momenta. Curves counted from the above at $E_m\hspace{1em}=\hspace{1em}0$ correspond to the missing momenta 0 to 700 MeV/c with 100 MeV/c increments. Solid and dotted lines correspond to PWIA and IA predictions, respectively. Arrows correspond to the prediction of Eq. (13)—scattering off a quasi-free stationary two-nucleon correlation.

Figure 5

Missing momentum dependence of the ratio of the spectral function calculated within the IA and PWIA approximations. Solid lines correspond to $E_m=0$. For the dashed lines, $E_m$ is defined according to Eq. (13). For the dashed-dotted lines, $E_m$ is defined according to Eq. (16), and for dotted lines, $E_m=350$ MeV.

Figure 6

Missing momentum dependence of the spectral function at different values of missing momenta. The curves counted from the upper left to the lower left of the figures correspond to the missing momenta $100,200,400$, and 700 MeV/c. Dotted lines show the PWIA prediction with only NN potential considered; dashed lines show the PWIA with additional $3N$ forces included; solid lines show the IA with the same $3N$ forces included.

Figure 7

Angular dependence of transparency T as defined in Eq. (17) for the reaction with a struck proton at different values of missing momenta. The missing energy here is defined according to the type $2N$-I correlation condition of Eq. (13). For solid lines, only the single rescattering amplitude is included in FSI; for the dashed lines, both single and double rescattering amplitudes are included in FSI.

Figure 8

Missing energy dependence of transparencies for the knocked-out proton reaction at parallel, ${\theta }_m=0^{°}$ (a), and antiparallel, ${\theta }_m={180}^{°}$ (b), kinematics. The solid, dashed, and dotted curves correspond to the calculation with missing momenta equal to $400,500$, and 600 MeV/c, respectively.

Figure 9

Missing energy dependence of the spectral function for the type $2N$-II correlation condition at missing momentum $p_m=0$. Dotted, dashed-dotted, dashed, and solid lines correspond to PWIA, IA, IA+FSI1, and full (IA+FSI) calculations, respectively.

Figure 10

Missing energy dependence of the spectral function for the knocked-out proton reaction in parallel [(a)] and antiparallel [(b)] kinematics at different values of missing momenta. The upper, middle, and lower curves at the left corner of the figure correspond to the missing momenta $400,500$, and 600 MeV/c, respectively. The dashed lines show the IA prediction; the solid lines show the IA+FSI prediction.

Figure 11

Dependence of the decay function on the momenta of the two recoil protons $p_{r2}$ and $p_{r3}$ at different cuts of missing momentum of the knocked-out neutron. (a) $p_m>150$ MeV/c; (b) $p_m>300$ MeV/c; (c) $p_m>400$ MeV/c; and (d) $p_m>700$ MeV/c.

Figure 12

The same as in Fig. 11 for proton and neutron recoil with momenta $p_{r2}$ and $p_{r3}$, respectively.

Figure 13

The ($E_m,p_{r2}$) distribution of the decay function for the neutron knockout reaction. The missing momentum is restricted to $p_m<50$ MeV/c.

Figure 14

The dependence of the decay function of the reaction ${}^3\mathrm{He}(e,e^{\text{'}}\mathrm{pp})n$ on the relative angle between the recoil pn nucleons ${\theta }_{23}$ and the missing energy $E_m$, for different cuts of missing momentum $p_m$. (a) no cuts on $p_m$; (b) $p_m>300$ MeV/c; (c) $p_m>500$ MeV/c; (d) $p_m>700$ MeV/c.

Figure 15

Reproduction of Fig. 14d from different viewpoints to emphasize the signatures of type $3N$-I (a) and type $3N$-II (b) correlations.

Figure 16

Dependence of the decay function on the momentum of the registered recoil nucleon $p_{r2}$, in ${}^3\mathrm{He}(e,e^{\text{'}}{N}_f{N}_{r2})N_{r3}$ reactions. (a), (b), (c), and (d) correspond to types $2N$-I, $2N$-II, $3N$-I, and $3N$-II kinematics, respectively. Dashed lines show the PWIA prediction; solid lines show the IA predictions. Two pairs of curves in each figure correspond to different compositions of detected nucleons ($N_f,N_{r2}$).

Figure 17

Dependence of the decay function on the angle of the detected recoil nucleon ${\theta }_{r2}$ with respect to $q$ for different and fixed values of $p_{r2}$. The decay function is calculated for type $2N$-I SRC kinematics. (a) $(N_f,N_{r2})\equiv (p,n)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA; dashed lines show the IA+FSI1 (single rescattering only); solid lines show the IA+FSI. Note that in (b) the IA contribution at $p_{r2}=400$ MeV/c is not shown since it is negligible due to the node in the pp momentum distribution.

Figure 18

Dependence of the decay function on the momentum of the recoil nucleon $p_{r2}$ calculated for the kinematics of type $2N$-I correlations (a) $(N_f,N_{r2})\equiv (p,n)$ and (b) $(N_f,N_{r2})\equiv (p,p)$. Doted lines show the IA prediction; dashed and solid lines show the IA+FSI1 and IA+FSI predictions, respectively, for parallel ${\theta }_m=0^{°}$ kinematics. Curves with squares correspond to IA+FSI1 and IA+FSI predictions for antiparallel ${\theta }_m={180}^{°}$ kinematics.

Figure 19

Dependence of the decay function on the angle of the detected recoil nucleon ${\theta }_{r2}$ with respect to $q$ for different and fixed values of $p_{r2}$. The decay function is calculated for type $2N$-II SRC kinematics with missing momentum fixed at $p_m=100$ MeV/c and ${\theta }_m=0^{°}$. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI.

Figure 20

Dependence of the decay function on the momentum of recoil nucleon $p_{r2}$ calculated for the kinematics of type $2N$-II correlations with missing momentum fixed at $p_m=100$ MeV/c and ${\theta }_m=0^{°}$. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA prediction; dashed and solid lines show the IA+FSI1 and IA+FSI predictions, respectively, for ${\theta }_{r2}=0^{°}$ kinematics. Curves with squares corresponds to the IA+FSI1 and IA+FSI predictions for antiparallel ${\theta }_{r2}={180}^{°}$ kinematics.

Figure 21

Dependence of the decay function on the angle of the detected recoil nucleon ${\theta }_{r2}$ with respect to $q$ for different and fixed values of $p_m$. The decay function is calculated for type $3N$-I SRC kinematics such that the recoil momentum of the detected nucleon is opposite to the missing momentum direction and defined as $p_{r2}=\frac{p_m}{2}+50$ MeV/c. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA; dashed lines show the IA+FSI1; solid line show the IA+FSI.

Figure 22

Dependence of the decay function on the momentum of recoil nucleon $p_{r2}$ calculated for the kinematics of type $3N$-I correlations. The relation between missing and recoil momenta are the same as in Fig. 21. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\hspace{1em}\equiv (p,p)$. Dotted lines show the IA prediction; dashed and solid lines show the IA+FSI1 and IA+FSI predictions, respectively, for ${\theta }_{r2}={180}^{°}$ kinematics. Curves with squares correspond to IA+FSI1 and IA+FSI predictions for antiparallel ${\theta }_{r2}=0^{°}$ kinematics.

Figure 23

Dependence of the decay function on the angle of the detected recoil nucleon ${\theta }_{r2}$ with respect to $q$ for different and fixed values of $p_{r2}$. The decay function is calculated for type $3N$-II SRC kinematics such that $p_m=p_{r2}=p_{r3}$, and relative angles between these momentum vectors are ${120}^{°}$. Here, ${\theta }_{r2}={120}^{°}$ corresponds to ${\theta }_m=0^{°}$. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI.

Figure 24

Dependence of the decay function on the momentum of the recoil nucleon $p_{r2}$ calculated for the kinematics of type $3N$-II correlations. The relation between missing and recoil momenta is the same as in Fig. 23. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the IA prediction; dashed and solid lines show the IA+FSI1 and IA+FSI predictions, respectively, for ${\theta }_m=0^{°}\hspace{1em}({\theta }_{r2}={120}^{°}$) kinematics. Curves with squares correspond to IA+FSI1 and IA+FSI predictions for antiparallel ${\theta }_m={180}^{°}\hspace{1em}({\theta }_{r2}={60}^{°}$) kinematics.

Figure 25

Dependence of the decay function on $p_m$ at ${\alpha }_m=1$ for type $3N$-I kinematics. The direction of the recoil nucleons’ momenta is opposite to that of the missing momentum direction with $p_{r2}=\frac{p_m}{2}+50$ MeV/c. (a) $(N_f,N_{r2})\equiv (n,p)$;(b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the PWIA; dashed-dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI predictions.

Figure 26

Dependence of the decay function on $p_m$ at ${\alpha }_m={\alpha }_{r2}={\alpha }_{r3}=1$. (a) $(N_f,N_{r2})\equiv (n,p)$; (b) $(N_f,N_{r2})\equiv (p,p)$. Dotted lines show the PWIA; dashed-dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI predictions.

Figure 27

Dependence of the decay function on the momentum of the recoil nucleon $p_{r2}$ calculated for the kinematics of type $3N$-II correlations and $(N_f,N_{r2})\equiv (n,p)$. The relation between missing and recoil momenta is the same as in Fig. 23, and the $r2$ recoil proton is detected at ${\theta }_{r2}={120}^{°}$, while the angle of missing momentum ${\theta }_m=0^{°}$. Dotted lines show the PWIA; dashed-dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI predictions. Curves with squares correspond to the same contributions with three-nucleon forces included in the calculation of the ${}^3\mathrm{He}$ wave function.

Figure 28

Dependence of the decay function on the momentum of the recoil nucleon $p_{r2}$ for type $2N$-I kinematics at ${\theta }_{r2}={180}^{°}\hspace{1em}({\theta }_m=0^{°}$). $(N_f,N_{r2})\equiv (n,p)$. Dotted lines show the PWIA; dashed-dotted lines show the IA; dashed lines show the IA+FSI1; solid lines show the IA+FSI predictions. Curves with squares correspond to the same contributions with three-nucleon forces included in the calculation of the ${}^3\mathrm{He}$ wave function.