Quasi-elastic electron scattering with KIDS nuclear energy density functional

Electrons in the process we consider are highly relativistic, so the available code is constructed in the relativistic formalism in which both electrons and nucleons are treated relativistically. Many and well-known nuclear structure models such as Skyrme force models and Gogny force models are formulated on the non-relativistic ground. KIDS model is also based on the non-relativistic phenomenology, so it is necessary to bridge the two approaches.

In the relativistic formalism, single particle wave function with angular momentum $\mathbf{J}^{2}$ and $J_{z}$, and in good quantum state of parity and time reversal takes the form

$\displaystyle\Psi(\mathbf{r})=\left(\begin{array}[]{c}f(r)\chi^{\mu}_{\kappa}(\hat{\mathbf{r}})\\ ig(r)\chi^{\mu}_{-\kappa}(\hat{\mathbf{r}})\end{array}\right).$

(3)

Orbital and spin states are represented by

$$\chi^{\mu}_{\kappa}(\hat{\mathbf{r}})=\sum_{m,s}\left<lm\left.\frac{1}{2}s\right|j\mu\right>Y_{lm}(\hat{\mathbf{r}})\chi_{s},$$

(4)

where $\chi_{s}$ and $Y_{lm}$ are the Pauli spinor and the spherical harmonics, respectively. $\kappa$ is the eigenvalue of the operator $(\mathbf{\sigma}\cdot\mathbf{L}+1)$ given by

$$\kappa=\left(\begin{array}[]{c l}l&;\,j=l-\frac{1}{2},\\ -l-1&;\,j=l+\frac{1}{2}.\end{array}\right.$$

(5)

Solving the non-relativistic wave equation

$$\frac{d^{2}F}{dr^{2}}+\frac{2}{r}\frac{dF}{dr}-\frac{\kappa(\kappa+1)}{r^{2}}F-2M\left[V_{\rm cen}(r)+V_{\rm SO}(r)(-\kappa-1)\right]F+p^{2}F=0,$$

(6)

one can determine $F(r)$. Radial functions $f(r)$ and $g(r)$ can be calculated from the relation

$$f(r)=D^{1/2}(r)F(r),\,\,\,g(r)=D^{-1/2}(r)G(r),$$

(7)

where Darwin factor $D(r)$ is defined as

$$D(r)=\exp\left[-2\int^{\infty}_{r}drMrV_{\rm SO}(r)\right].$$

(8)

Lower component function $G(r)$ is obtained from the relation with $F(r)$

$$G(r)=\frac{1}{E+M}\left[\frac{dF}{dr}+\left(\frac{\kappa+1}{r}-T(r)\right)F\right],$$

(9)

where

$$T(r)=MrV_{\rm SO}(r).$$

(10)

With non-relativistic nuclear potentials $V_{\rm cen}$ and $V_{\rm SO}$ given from a model, Hartree-Fock equation is solved and $F(r)$ is obtained. $G(r)$ is calculated from Eq. (9), and finally the relativistic wave functions $f(r)$ and $g(r)$ are obtained by calculating Eq. (7).

One purpose of the electron scattering is to get better knowledge about the structure of nuclei. At the energy of quasi-elastic scattering, the distribution of nucleons in the interior and the gradient of density around the surface could have effects on the cross section. In terms of the Skyrme force, derivative terms account for the contribution of density gradient. These terms also contribute to determining the effective mass of the nucleon, so the structural uncertainty around the surface as well as the core could be explored by probing the dependence on the effective mass.

Role of the effective mass could be singled out when other conditions (e.g. binding energy, charge radius) are unchanged. Independent control of the nuclear properties is accessible with the KIDS energy density functional kidsnm . In the KIDS framework several rules are assumed to develop a nuclear model. At first energy per particle in homogeneous nuclear matter is expanded in powers of Fermi momentum (equally $\rho^{1/3}$) as

$\displaystyle{\cal E}(\rho,\,\delta)={\cal T}+\sum_{i=0}(\alpha_{i}+\beta_{i}\delta^{2})\rho^{1+i/3}.$

(11)

${\cal T}$ is the kinetic energy, and $\delta=(\rho_{n}-\rho_{p})/\rho$. Model parameters $\alpha_{i}$ and $\beta_{i}$ are fixed or fit to symmetric and asymmetric nuclear matter properties. In this work we fix $\alpha_{0}$, $\alpha_{1}$ and $\alpha_{2}$ to three saturation properties: saturation density $\rho_{0}=0.16$ fm^-3, bindind energy per particle $E_{B}=16.0$ MeV, and incompressibility $K_{0}=240$ MeV kidsnuclei2 . Several ways have been tried to fix the $\beta_{i}$ values. It has been proved that four $\beta_{i}$’s are good and enough to reproduce the neutron matter equation of state obtained from microscopic calculations kidsnuclei1 , static properties of neutron rich nuclei kidsnd , and neutron star observations kidsk0 ; kidssymene . In this work we use the values of $\beta_{i}$ fit to the pure neutron matter of equation of state calculated by Akmal, Pandharipande and Ravenhall apr . The model thus fixed is labeled KIDS0.

When nuclei are described in the KIDS framework, energy density functional is transformed to the form of Skyrme force kidsnpsm2017 . Terms accounting for the density gradient and spin-orbit interactions are added In the notation of Skyrme force, they correspond to parameters $t_{1}$, $t_{2}$, $x_{1}$, $x_{2}$ and $W_{0}$. We assume $t_{1}=t_{2}$, $x_{1}=x_{2}=0$, and $t_{1}$ and $W_{0}$ are fit to the data of binding energy and charge radius of ⁴⁰Ca, ⁴⁸Ca and ²⁰⁸Pb. In the KIDS0 model, $x_{1}$ and $x_{2}$ are assumed to be 0, so the effective masses are obtained as results of determining $t_{1}$. By adjusting $x_{1}$ and $x_{2}$, one can produce specific values of isoscalar and isovector effective masses $\mu_{S}$ and $\mu_{V}$ defined by

	$\displaystyle\mu_{S}$	$\displaystyle=$	$\displaystyle\frac{m^{*}_{S}}{M}=\left[1+\frac{M}{8\hbar^{2}}\rho(3t_{1}+5t_{2}+4y_{2})\right]^{-1},$
	$\displaystyle\mu_{V}$	$\displaystyle=$	$\displaystyle\frac{m^{*}_{V}}{M}=\left[1+\frac{M}{4\hbar^{2}}\rho(2t_{1}+2t_{2}+y_{1}+y_{2})\right]^{-1},$		(12)

where $y_{i}=t_{i}x_{i}$. In this work we consider two cases $(\mu_{S},\,\mu_{V})=(0.7,0.7)$ and $(0.9,0.9)$. Each model is labeled KIDS0-m*77 and KIDS0-m*99, respectively.

Central and spin-orbit potentials entering Eq. (6) are obtained from KIDS functional as

	$\displaystyle V_{\text{cen}}$	$\displaystyle=$	$\displaystyle U_{q}+U_{\text{coul}},$
	$\displaystyle U_{q}$	$\displaystyle=$	$\displaystyle\left(t_{0}+\frac{1}{2}y_{0}\right)\rho-\left(\frac{1}{2}t_{0}+y_{0}\right)\rho_{q}$		(13)
		$\displaystyle+$	$\displaystyle\frac{1}{8}[(2t_{1}+y_{1})+(2t_{2}+y_{2})]\tau-\frac{1}{8}[(t_{1}+2y_{1})+(t_{2}+2y_{2})]\tau_{q}$
		$\displaystyle-$	$\displaystyle\frac{1}{16}[(6t_{1}+y_{1})-(2t_{2}+y_{2})]\nabla^{2}\rho+\frac{1}{16}[(3t_{1}+6y_{1})+(t_{2}+2y_{2})]\nabla^{2}\rho_{q}$
		$\displaystyle+$	$\displaystyle\frac{1}{12}\sum_{k=1}^{3}\rho^{k/3}\left[\left(2+\frac{k}{3}\right)\left(t_{3k}+\frac{1}{2}y_{3k}\right)\rho-(t_{3k}+2y_{3k})\rho_{q}-\frac{k}{3}\left(\frac{1}{2}t_{3k}+y_{3k}\right)\frac{\rho^{2}_{p}+\rho^{2}_{n}}{\rho^{2}}\right]$
		$\displaystyle-$	$\displaystyle\frac{W_{0}}{2}\left(\frac{\partial}{\partial r}+\frac{2}{r}\right)(J+J_{q}),$
	$\displaystyle V_{\text{SO}}$	$\displaystyle=$	$\displaystyle\frac{1}{2}W_{0}\left(\frac{\partial}{\partial r}+\frac{2}{r}\right)(\rho+\rho_{q})+\frac{1}{8}(t_{1}-t_{2})J_{q}-\frac{1}{8}(y_{1}+y_{2})J,$		(14)

where $\tau$ is the non-relativistic kinetic energy, $J_{q}$ denotes the spin current of the neutron and the proton, and $J=J_{n}+J_{p}$. Parameters in the potential of each model are summarized in Tab. 1. We consider the SLy4 model for a comparison with a standard Skyrme force model. With these potentials, the transformed relativistic wave functions are generated and applied into the exclusive $(e,e^{\prime}p)$ reaction in quasi-elastic region.

Charge and neutron distributions in ¹⁶O, ⁴⁰Ca and ²⁰⁸Pb are displayed in Figs. 1, 2, and 3, respectively. Measured values are denoted with gray bands.

In the result of ¹⁶O, KIDS models show weak dependence on the effective mass, and the results of models agree to each other in both charge and neutron distributions. SLy4 model agrees well with KIDS model at $r>2$ fm, but in the interior region ($r<2$ fm) densities are slightly suppressed compared to the KIDS model. All the models reproduce the data of charge distribution o16density well over $r>$ 2.5 fm. Visual discrepancy is found in $0.5<r<2.5$ fm, but the difference from experiment is less than 10%. Data for the neutron are not available so only the theory results are presented.

For ⁴⁰Ca, model dependence is weak again, and the four models predict similar distributions of both protons and neutrons. In the comparison with experiment sick1981 ; ray1979 , distribution of the proton is reproduced well by the theory. In case of the neutron, theoretical results are within the errors of experiment in the core region ($r<2$ fm), but on the surface ($2<r<4$ fm) where the density drops rapidly, theory exceeds experiment.

We have seen that the distribution of the proton and the neutron are insensitive to the effective mass in the light nuclei ¹⁶O and ⁴⁰Ca. In the result of ²⁰⁸Pb, on the other hand, the effect of effective mass appears clear. For the proton, dependence on the effective mass is negligible at $r>1$ fm. In this region model dependence is weak and theory agrees well with experiment pb208density . At $r<1$ fm, theories are divided into two groups: one with KIDS0, KIDS0-m*99 (Group1), and the other with KIDS0-m*77, SLy4 (Group2). Group2 shows charge density large than Group1 as $r\rightarrow 0$. Models in each group have similar isoscalar effective mass, $\mu_{S}\sim 1.0$ for Group1 and $\mu_{S}=0.7$ for Group2. Therefore different behavior of charge density at $r<1$ fm could be originated from the value of isoscalar effective mass.

Neutron distribution shows pattern of agreement and disagreement to experiment saito2007 similar to the neutron distribution of ⁴⁰Ca. Dependence on the effective mass is divided into three categories: no dependence at $r>2.5$ fm, models are classified into two sets Group1 and Group2 at $1<r<2.5$ fm, and four models behave independently at $r<1$ fm. In the inner core $r<2$ fm, theory results are within the range of experimental uncertainty. In the outer core region $2<r<5$ fm, all the models obtain neutron densities less than experiment. The discrepancy does not exceed 10%. In the surface region $5<r<8$ fm, theory overwhelms experiment slightly, and it is reversed in the tail $r>8$ fm. There seems to be a pattern in the discrepancy of the neutron distribution. On the other hand, distribution of the proton which is relevant to the scattering with electrons agrees well with experiment.

Electron-nucleus scattering data provide cross sections from protons at specific states. Table 2 collects the single particle levels of the proton for which cross sections calculated from theory will be compared with experiment. For light nuclei ¹⁶O and ⁴⁰Ca, models with smaller isoscalar effective mass (Group2) obtain results closer to experiment than the models in Group1. For ¹⁶O, Group2 models differ from experiment by 9.4–13.2%, and Group1 models by 16.6–18.5%. For ⁴⁰Ca, Group1 models give deviations from experiment 16.6–18.5% and 12.8–16.6% in the 2s_1/2 and 1d_3/2 states, respectively. With the models in Group2, we have 10.1–10.2% and 2–3% for 2s_1/2 and 1d_3/2 states, respectively.

For ²⁰⁸Pb Group1 models show better agreement to data. In the 3s_1/2 state Group1 models give differences 2.5–5.7% and Group2 models 7.9–9.7%. In the 2d_3/2 state, Group1 and Group2 give differences 7.2–10.5% and 14.5–15.3%, respectively. A similar pattern is reported in the UNEDF model unedf0 . Isoscalar effective mass is $0.9M$ in the UNEDF model, and the model shows agreement better for heavy nuclei than light ones.

As far as single particle levels are concerned, light nuclei favor small isoscalar effective mass, but heavy nuclei support effective masses close to 1.

In the present work, we calculate the reduced cross section $\rho(p_{m})$ at one particular shell, which is related to the probability that a bound nucleon at a given orbit with the missing momentum $\mathbf{p}_{m}$ can be knocked out of the nucleus with asymptotic momentum $\mathbf{p}$. The reduced cross section as a function of $p_{m}$ is commonly defined by

$\displaystyle\rho(p_{m})={\frac{1}{pE\sigma_{ep}}}{\frac{d^{3}\sigma}{dE_{f}d\Omega_{f}d\Omega_{p}}},$

(15)

where the missing momentum is determined by the kinematics $\mathbf{p}_{m}=\mathbf{p}-\mathbf{q}$ with $\mathbf{q}$ the momentum of virtual photon mediating EM interactions between electron and proton. The off-shell electron-proton cross section, $\sigma_{ep}$ is not uniquely defined. We use the form CC1 $\sigma_{ep}$ given by Ref. npa1983 . In all the calculations SFs are calibrated with the KIDS0 model.

Figure 4 shows the cross section of ⁴⁰Ca in the parallel kinematics where $\mathbf{p}\parallel\mathbf{q}$. The incident electron energy is $E_{i}=412$ MeV and the energy transfer to proton is $\omega=100$ MeV. Theory results are represented with lines, and data from NIKHEF kramer are noted with red circles. In the 2s_1/2 state, theory results are similar to each other. Spectroscopic factors are adjusted to reproduce the cross section data around peak at $p_{m}=0$ MeV/$c$ with KIDS0 model. Theory agrees well with data not only at $p\simeq 0$ MeV/$c$, but over $p_{m}<200$ MeV/$c$. In the 1d_3/2 state, experiment at peaks around $p_{m}\simeq-150$ MeV/$c$ and 100 MeV/$c$ are reproduced with good accracy, and agreement is extended over the range $p_{m}<200$ MeV/$c$. Model dependence is weak, but a small gap between Group1 and Group2 is seen at $p_{m}<0$. Group2 models predict the cross sections slightly larger than the Group1 models.

Figure 5 presents the result of ²⁰⁸Pb in the parallel kinematics. The incident electron energy is $E_{i}=412$ MeV and the kinetic energy of knocked-out proton is $T_{p}=100$ MeV. Predictions are divided into two groups around the peaks in both 3s_1/2 and 2d_3/2 states. Since the SFs are calibrated with KIDS0 model, models in Group1 are in good agreement with the NIKHEF data bobeldijk . Models in Group2 that have $\mu_{S}=0.7$ give cross sections larger than the Group1 models by 15–30%. Isovector effective mass does not follow the grouping of isoscalar effective mass: KIDS0 and SLy4 models have $\mu_{V}=0.8$, KIDS0-m*77 has $\mu_{V}=0.7$, and $\mu_{V}=0.9$ for the KIDS0-m*99 model. Therefore it is likely that the model dependence of the cross section is dominated by the isoscalar effective mass, and the effect of the isovector effective mass is, if ever, quite limited. Result of $\mu_{S}=0.9$ (KIDS0-m*99) can hardly be differentiated from that of $\mu_{S}=1.0$ (KIDS0) in the 3s_1/2 state, but in the 2d_3/2 state, result of the former is slightly enhanced over the latter. This difference is, though small, in accordance with the correpondence of small effective mass to large cross section.

Figure 6 depicts the result of ¹⁶O in the perpendicular kinematics. In the perpendicular kinematics $|\mathbf{p}|=|\mathbf{q}|$, so $\mathbf{p}_{m}$ is almost perpendicular to $\mathbf{q}$. Upper panels compare the result with data from Saclay chinitz , and the lower panels with data from JLab gao . Energies of the incident electrons are 580 MeV in Saclay and 2441 MeV in JLab. The kinetic energies of the knocked-out protons are $T_{p}=159$ MeV and $T_{p}=427$ MeV, respectively. As a result, even though the ranges of $p_{m}$ are similar, both theoretical results and experimental data are different in the upper and lower panels. For the same reason, SFs are different depending on the incident electron energies. Results obtained from the models are not sensitive to the effective mass, so they agree well with experiment at both energies. Looking into the details around peaks, cross sections are increasing in the order of KIDS0, KIDS0-m*99 and KIDS0-m*77. Results of SLy4 are indistinguishable from those of KIDS0-m*77. Again the isoscalar effective masses are decreasing in the order of increasing cross sections. This behavior is consistent with what has been observed in the result of ²⁰⁸Pb. The result supports that the correlation between the isoscalar effective mass and the scattering cross section is not limited to specific nuclei, but valid over a wide range of mass number.

Figure 7 reports the cross section of ⁴⁰Ca in the perpendicular kinematics with the incident electron energy and the energy transfer to proton the same with those in Fig. 4. Similar to the parallel kinematics, theory reproduces the data with accracy, and the dependence on the effective mass is marginal. Spectroscopic factors are the same with that of the parallel kinematics in the 2s_1/2 state, but we have a reduced value in the 1d_3/2 state. In the result of ¹⁶O, we saw that the kinematic conditions have effect to the value of SF. Different SF values in the 1d_3/2 state could be attributed to the kinematic conditions.

Spectroscopic factors provide understanding of the structural details of nuclei, and allow the estimation of the contribution of many-body correlations that could be missed in the mean field approximation. Therefore SFs is a measure to figure out the validity, accuracy and limit of shell description of a model. Table III collects SFs from literature, and compares them with the result of this work. Calculations based on relativistic formalism jin1992 ; udias1993 obtain results smaller than the values evaluated from experiment volya2007 and a phonon-coupling calculation gnez2014 . Results of the latter two volya2007 ; gnez2014 agree to each other. Result of ²⁰⁸Pb in this work is consistent with Refs. volya2007 ; gnez2014 , but those of ⁴⁰Ca agree with relativistic calculations jin1992 ; udias1993 . Consequently our work predicts SFs of ⁴⁰Ca smaller than those of ²⁰⁸Pb. It could be interpreted that the mean field approximation works better in heavy nuclei. Result of ¹⁶O is consistent with this tendency because the SFs are 0.62 – 0.75 in 1p_1/2 and 0.61 – 0.70 in 1p_3/2.

In the laboratory frame, the quasi-elastic cross section for the $(e,e^{\prime}p)$ reaction is simply written as

	$\displaystyle{\frac{d^{3}\sigma}{dE_{f}\,d\Omega_{f}\,d\Omega_{p}}}$	$\displaystyle=$	$\displaystyle K\Big{[}v_{L}R_{L}+v_{T}R_{T}+\cos 2\phi_{p}\,v_{TT}R_{TT}$		(16)
			$\displaystyle\qquad\qquad\qquad+\cos\phi_{p}\,v_{LT}R_{LT}+h\sin\phi_{p}\,v_{LT^{\prime}}R_{LT^{\prime}}\Big{]}~{},$

where the kinematic factor $K$ is given by $K={\frac{p\,E}{(2\pi)^{3}}}\,\sigma_{M}$ with the Mott cross section $\sigma_{M}={\frac{\alpha^{2}}{4E^{2}_{i}}}\,{\frac{\cos^{2}({\theta_{e}}/2)}{\sin^{4}({\theta_{e}}/2)}}$. In Eq. (16), $R_{L}$, $R_{T}$, $R_{TT}$, and $R_{LT}$ are referred to as the longitudinal, transverse, longitudinal-transverse, and transverse-transverse interferences response functions, respectively. The fifth one, $R_{LT^{\prime}}$, indicates the polarized longitudinal-transverse interference, which is directly proportional to the electron beam asymmetry. The four-momenta of the incoming and outgoing electrons are labeled $p_{i}^{\mu}=(E_{i},{\bf p}_{i})$ and $p_{f}^{\mu}=(E_{f},{\bf p}_{f})$. In the parallel kinematics, three interference terms disappear, but they can be extracted in the perpendicular kinematics. The detailed discussions are in Ref. kimepja01 . The electron kinematic factors in Eq. (16) are given in terms of the four-momentum transfer, $q=(\omega,\mathbf{q})$, and the electron scattering angle $\theta_{e}$:

	$\displaystyle v_{L}=\frac{q^{4}}{\mathbf{q}^{4}}~{},\qquad v_{T}=\tan^{2}\frac{\theta_{e}}{2}-\frac{q^{2}}{2\mathbf{q}^{2}}~{},\qquad v_{TT}=-\frac{q^{2}}{2\mathbf{q}^{2}}~{},$
	$\displaystyle v_{LT}=-\frac{q^{2}}{\mathbf{q}^{2}}\,\left(\tan^{2}\frac{\theta_{e}}{2}-\frac{q^{2}}{2\mathbf{q}^{2}}\right)~{},\qquad v_{LT^{\prime}}=-\frac{q^{2}}{\mathbf{q}^{2}}\,\tan^{2}\frac{\theta_{e}}{2}.$		(17)

In Eq. (16), the fourth structure function could be obtained by subtracting the cross sections at azimuthal angles of the outgoing proton $\phi_{p}=0$ and $\phi_{p}=\pi$ and keeping the other electron and outgoing proton kinematics variables fixed. The fourth structure function is a function of the missing momentum given by

$$R_{LT}={\frac{\sigma^{R}-\sigma^{L}}{2Kv_{LT}}},$$

(18)

where $L$ (left) and $R$ (right) indicate the left side at $\phi_{p}=0$ and the right side at $\phi_{p}=\pi$ of the cross section in Eq. (16), respectively.

Figure 8 shows the fourth response functions from ¹⁶O with the same kinematics as the JLab experiment. The explanations of the curves are the same as the previous results. The difference between the effective masses is at most about 45 % and 10 % around the position of peak at 1p_1/2 and 1p_3/2 orbits, respectively. The sensitivity of the effective masses is clearly exhibited on the fourth response function. We learn that the values of the SF for the fourth response function and for the reduced cross section may be different for a good agreement with the data.

Finally, we also calculate another left-right asymmetry, $A_{LT}$, defined as

$$A_{LT}={\frac{\sigma^{R}-\sigma^{L}}{\sigma^{R}+\sigma^{L}}}.$$

(19)

The kinematics in Fig. 9 are the same as in Fig. 8. Since this asymmetry does not require any SF, it is possible to compare the theoretical result with experimental data directly. The role of the effective masses is not large compared to the fourth response function but our theoretical results describe the experimental data relatively well except one point at $p_{m}\simeq 350$ MeV/$c$ in the 1p_3/2 orbit.