Electromagnetic scaling functions within the Green's function Monte Carlo approach

We have studied the scaling properties of the electromagnetic response functions of ${}^4\mathrm{He}$ and ${}^{12}\mathrm{C}$ nuclei computed by the Green's function Monte Carlo approach, retaining only the one-body current contribution. Longitudinal and transverse scaling functions have been obtained in the relativistic and nonrelativistic cases and compared to experiment for various kinematics. The characteristic asymmetric shape of the scaling function exhibited by data emerges in the calculations in spite of the nonrelativistic nature of the model. The results are mostly consistent with scaling of zeroth, first, and second kinds. Our analysis reveals a direct correspondence between the scaling and the nucleon-density response functions. The scaling function obtained from the proton-density response displays scaling of the first kind, even more evidently than the longitudinal and transverse scaling functions.

Figure 1

Ratio of the nonrelativistic and relativistic expressions of the prefactors entering the definition of the scaling function, plotted as a function of $\psi$ for $\left|\mathbf{q}\right|=300$, 380, 570 MeV. The blue solid and red dashed lines correspond to the longitudinal and transverse channels, respectively.

Figure 2

Longitudinal (solid blue) and transverse (dashed red) scaling functions obtained from the GFMC calculation of the longitudinal and transverse responses of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=300$ MeV. Upper panel: the responses have been divided by the nonrelativistic prefactors and the resulting curves are plotted as a function of ${\psi }^{nr}$. Lower panel: the standard definition of the prefactors given in Eq. (30) has been used to get both the theoretical curves and the experimental points obtained from the data of Ref. [23].

Figure 3

Same as in Fig. 2 but for $\left|\mathbf{q}\right|=380$ MeV.

Figure 4

Same as in Fig. 2 but for $\left|\mathbf{q}\right|=570$ MeV.

Figure 5

Experimental scaling functions of ${}^{12}\mathrm{C}$ obtained from the longitudinal responses for $\left|\mathbf{q}\right|=300,380,570$ MeV [23].

Figure 6

Longitudinal scaling functions of ${}^{12}\mathrm{C}$ obtained from GFMC calculations for $\left|\mathbf{q}\right|=300,380,570$ MeV as a function of ${\psi }^{nr}$.

Figure 7

Transverse scaling functions of ${}^{12}\mathrm{C}$ obtained from GFMC calculations for $\left|\mathbf{q}\right|=300,380,570$ MeV as a function of ${\psi }^{nr}$.

Figure 8

Experimental scaling functions obtained from the longitudinal responses of ${}^4\mathrm{He}$ for $\left|\mathbf{q}\right|=300$, 400, 500, 600, and 700 MeV [18]. The value of the Fermi momentum of ${}^4\mathrm{He}$ has been set to 180 MeV. The black dots correspond to the scaling function obtained from the experimental longitudinal response of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=570$ MeV [23].

Figure 9

Longitudinal (solid blue) and transverse (dashed red) scaling functions obtained from the GFMC calculation of the longitudinal and transverse responses of ${}^4\mathrm{He}$ at $\left|\mathbf{q}\right|=300$ MeV. Upper panel: the responses have been divided by the nonrelativistic prefactors and the resulting curves are plotted as a function of ${\psi }^{nr}$. Lower panel: the standard definition of the prefactors given in Eq. (30) has been used to get both the theoretical curves and the experimental points obtained from the data of Ref. [18].

Figure 10

Same as in Fig. 9 but for $\left|\mathbf{q}\right|=400$ MeV.

Figure 11

Same as in Fig. 9 but for $\left|\mathbf{q}\right|=500$ MeV.

Figure 12

Same as in Fig. 9 but for $\left|\mathbf{q}\right|=600$ MeV.

Figure 13

Same as in Fig. 9 but for $\left|\mathbf{q}\right|=700$ MeV.

Figure 14

Longitudinal scaling functions obtained from GFMC calculations of the longitudinal response of ${}^4\mathrm{He}$ for $\left|\mathbf{q}\right|=400,500,600,700$ MeV and of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=570$ MeV.

Figure 15

Transverse scaling functions obtained from GFMC calculations of the transverse response of ${}^4\mathrm{He}$ for $\left|\mathbf{q}\right|=400,500,600,700$ MeV and of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=570$ MeV.

Figure 16

The longitudinal (solid blue) and transverse (dashed red) scaling functions obtained within the GFMC approach compared with the scaling function obtain from the proton response function (dot dashed black). Upper panel: ${}^4\mathrm{He}$ at $\left|\mathbf{q}\right|=500$ MeV. Lower panel: ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=570$ MeV.

Figure 17

Scaling function obtained from GFMC calculations of the proton response function of ${}^4\mathrm{He}$ for $\left|\mathbf{q}\right|=400,500,600,700$ MeV.