Scaling within the spectral function approach

Scaling features of the nuclear electromagnetic response functions unveil aspects of nuclear dynamics that are crucial for interpreting neutrino- and electron-scattering data. In the large momentum-transfer regime, the nucleon-density response function defines a universal scaling function, which is independent of the nature of the probe. In this work, we analyze the nucleon-density response function of ${}^{12}\mathrm{C}$, neglecting collective excitations. We employ particle and hole spectral functions obtained within two distinct many-body methods, both widely used to describe electroweak reactions in nuclei. We show that the two approaches provide compatible nucleon-density scaling functions that for large momentum transfers satisfy first-kind scaling. Both methods yield scaling functions characterized by an asymmetric shape, although less pronounced than that of experimental scaling functions. This asymmetry, only mildly affected by final state interactions, is mostly due to nucleon-nucleon correlations, encoded in the continuum component of the hole spectral function.

Figure 1

Ladder sum of diagrams contributing to the nucleon self-energy in nuclear matter. Dashed lines represent the in-medium $NN$ interaction.

Figure 2

Feynman diagrams contributing to the polarization of the $NN$ interaction in the medium.

Figure 3

Transverse (dotted red), longitudinal (dashed blue), and nucleon-density (solid black) scaling functions of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=1.0$ GeV obtained from the CBF-SF approach including FSI corrections.

Figure 4

Longitudinal scaling functions in ${}^{12}\mathrm{C}$ computed using the hole SF [Eq. (52)] of Ref. [16] for $\left|\mathbf{q}\right|=0.57$, 0.8, 0.9, 1.0, and 1.2 GeV. Results obtained within the IA scheme are shown in the upper panel, while those including FSI effects are displayed in the bottom one. The standard definition of the longitudinal prefactor given in Eq. (30) of Ref. [10] has been used to get both the theoretical curves and the experimental points obtained from the $\left|\mathbf{q}\right|=0.57$ GeV data of Ref. [64].

Figure 5

Nucleon-density scaling functions for ${}^{12}\mathrm{C}$ computed for $\left|\mathbf{q}\right|=0.57$, 0.8, 0.9, 1.0, and 1.2 GeV. In the upper (lower) panel, results obtained using CBF (relativized LDA) SFs are shown.

Figure 6

Scaling functions for ${}^{12}\mathrm{C}$ computed for $\left|\mathbf{q}\right|=0.57$, and 0.9 GeV. In the upper (lower) panel results are obtained within the FSI (IA) and LDA (IA LDA) approaches.

Figure 7

Nonrelativistic PWIA scaling responses, using the momentum distribution of ${}^{12}\mathrm{C}$ derived in Ref. [68] for $\left|\mathbf{q}\right|=0.5$, 0.7, 1, and 1.2 GeV. The Fermi momentum was fixed to $p_F=225$ MeV.

Figure 8

Nonrelativistic scaling responses obtained within PWIA [Eq. (73)] as a function of ${\psi }^{\text{nr}}$ for $\left|\mathbf{q}\right|=0.5$, 0.7, and 1 GeV. The momentum distribution of ${}^{12}\mathrm{C}$ derived in Ref. [68] was used, and the energy of the hole state was extracted from the calculations of the nuclear matter energy spectrum of Ref. [69] and implemented in the energy conservation [see Eq. (78)]. The Fermi momentum, $p_F$, was fixed to 225 MeV.

Figure 9

Same as in Fig. 8, but nuclear potentials were used to determine both the hole and particle state energies [see Eq. (80)].

Figure 10

Scaling functions for ${}^{12}\mathrm{C}$ obtained in the IA from Eq. (70) using the CBF hole SF for $\left|\mathbf{q}\right|=0.5$, 0.7, 1.0, and 1.2 GeV.

Figure 11

Top: Breakdown of the scaling response of ${}^{12}\mathrm{C}$ at $\left|\mathbf{q}\right|=1.2$ GeV shown in Fig. 10 into the total, hole, and background contributions. Bottom: Dashed (black) and solid (green) lines correspond to the scaling function calculated with and without the inclusion of the FSI effect at $\left|\mathbf{q}\right|=1$ GeV in ${}^{12}\mathrm{C}$. The IA curve corresponds to that displayed in Fig. 10, and it is used in the convolution detailed in Eq. (57) to incorporate the FSI effects.