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

TL;DR

This study investigates the electromagnetic response functions of helium-4 and carbon-12 nuclei using Green's Function Monte Carlo, revealing scaling properties and their relation to nucleon-density responses, with results aligning with experimental data.

Contribution

It demonstrates that electromagnetic scaling functions can be obtained within a non-relativistic Green's Function Monte Carlo framework, showing consistency with experimental scaling behaviors.

Findings

Scaling functions exhibit characteristic asymmetry similar to experimental data.

Results are consistent with scaling of zeroth, first, and second kinds.

A direct link between scaling and nucleon-density response functions is established.

Abstract

We have studied the scaling properties of the electromagnetic response functions of $^4$He and $^{12}$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 non relativistic 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 non relativistic nature of the model. The results are consistent with scaling of zeroth, first and second kinds. Our analysis reveals a direct correspondence between the scaling and the nucleon-density response functions.

Equations

$$\frac{d^2\sigma }{d{E}_{e^{\prime }}d{\Omega }_{e^{\prime }}}=\frac{{\alpha }^2}{q^4}\frac{E_{e^{\prime }}}{E_e}{L}_{\mu \nu }{W}^{\mu \nu },$$

$$L^{\mu \nu }=2\left(k_{e^{\prime }}^{\mu }{k}_e^{\nu }+k_e^{\mu }{k}_{e^{\prime }}^{\nu }-g^{\mu \nu }{k}_{e^{\prime }}\cdot k_e\right).$$

$$W^{\mu \nu }=\sum _f⟨0|{J^{\mu }}^{†}(q)|f⟩⟨f|J^{\nu }(q)|0⟩{\delta }^{(4)}(P_0+q-P_f),$$

$$\frac{d^2\sigma }{d{E}_{e^{\prime }}d{\Omega }_{e^{\prime }}}$$

$$\displaystyle+A_{T}(|{\bf q}|,\omega,\theta_{e^{\prime}})R_{T}(|{\bf q}|,\omega)\Big{]}\ ,$$

$$\displaystyle A_{L}=\Big{(}\frac{q^{2}}{{\bf q}^{2}}\Big{)}^{2}\ \ \ ,\ \ \ A_{T}=-\frac{1}{2}\frac{q^{2}}{{\bf q}^{2}}+\tan^{2}\frac{\theta_{e}}{2}\ ,$$

$${\left(\frac{d\sigma }{d{\Omega }_{e^{\prime }}}\right)}_{\mathrm{M}}={\left[\frac{\alpha \cos ({\theta }_{e^{\prime }}/2)}{2E_{e^{\prime }}{\sin }^2({\theta }_{e^{\prime }}/2)}\right]}^2$$

$$R_L$$

$$R_T$$

$$R_L(\mathbf{q},\omega )$$

$$R_T(\mathbf{q},\omega )$$

$${\tilde{E}}_{T,L}(\mathbf{q},\tau )={\int }_{{\omega }_{\mathrm{el}}}^{\infty }d\omega e^{-\omega \tau }{R}_{T,L}(\mathbf{q},\omega ).$$

$${\tilde{E}}_L(\mathbf{q},\tau )$$

$$-|⟨0|\rho (\mathbf{q})|0⟩{|}^2{e}^{-{\omega }_{\mathrm{el}}\tau },$$

$${\tilde{E}}_T(\mathbf{q},\tau )$$

$$-|⟨0|{\mathbf{j}}_T(\mathbf{q})|0⟩{|}^2{e}^{-{\omega }_{\mathrm{el}}\tau }.$$

$$W^{\mu \nu }=$$

$$\times \theta (p_F-|\mathbf{p}|)\theta (|\mathbf{p}+\mathbf{q}|-p_F)$$

$$\times \delta (\omega +E(\mathbf{p})-E(\mathbf{p}+\mathbf{q})).$$

$$w^{\mu \nu }(p+q,p)=$$

$$\displaystyle+W_{2}(\tau)\frac{1}{m^{2}}\Big{(}p^{\mu}-\frac{p\cdot q}{q^{2}}q^{\mu}\Big{)}$$

$$\displaystyle\times\Big{(}p^{\nu}-\frac{p\cdot q}{q^{2}}q^{\nu}\Big{)}\ ,$$

$$W_1(\tau )=$$

$$W_2(\tau )=$$

$$G_E(\tau )$$

$$G_M(\tau )$$

$$F(p_F,q)=$$

$$\mathcal{F}(p_F,q,\mathbf{p})=$$

$$\times \theta (p_F-|\mathbf{p}|)\theta (|\mathbf{p}+\mathbf{q}|-p_F)$$

$$\times \delta (\omega +E(\mathbf{p})-E(\mathbf{p}+\mathbf{q}))$$

$$\mathcal{F}(p_F,q,\mathbf{p})=\frac{3\mathcal{N}{m}^2}{2p_F^3|\mathbf{q}|}\theta (E_F-\Gamma )(E_F-\Gamma ),$$

$$\Gamma$$

$$\displaystyle=\rm{Max}\Big{\{}m,E_{F}-\omega,\frac{-\omega+|{\bf q}|\sqrt{1+1/\tau}}{2}\Big{\}}\ .$$

$$\lambda =\omega /2m,$$

$$\kappa =|\mathbf{q}|/2m,$$

$${\eta }_F=p_F/m.$$

$$\displaystyle\lambda=\lambda^{0}=\frac{1}{2}\Big{[}\sqrt{(1+4\kappa^{2})}-1\Big{]}\ ,$$

$$\displaystyle\psi=sign(\lambda-\lambda^{0})\Big{[}\frac{1}{\xi_{F}}\Big{(}\frac{\Gamma_{3}}{m}-1\Big{)}\Big{]}^{1/2}\ ,$$

Full text

I Introduction

A realistic description of the electromagnetic response of atomic nuclei is a challenging many-body problem as it requires an accurate understanding of both the nuclear dynamics and of the interaction vertex. In this regard a valuable strategy consists in analyzing the scaling properties of nuclear response functions in a variety of kinematic setups West (1975); Day et al. (1987); Donnelly and Sick (1999). Scaling of the first kind is said to occur when the electron-nucleus cross section or longitudinal/transverse response functions, divided by an appropriate function describing the single-nucleon physics, do no longer depend on two variables (for example energy transfer $\omega$ and absolute value of the 3-momentum transfer ${|\bf q|}$ in the Laboratory frame), but only upon a specific function of them, which defines the scaling variable. Scaling of the second kind takes place when there is no dependence on the nuclear species. Finally, the simultaneous occurrence of both kinds of scaling is denoted as superscaling Alberico et al. (1988).

Superscaling is exactly fulfilled by the Global Relativistic Fermi gas (GRFG) model, for which a simple and symmetric scaling function can be derived in terms of the dimensionless scaling variable $\psi$ Barbaro et al. (1998) (explicit expressions are provided in Sec. II.2 below). However, contrary to the GRFG model predictions, the results extracted from experimental data reveal an asymmetric shape of the scaling function, with a tail that extends to high values of $\psi$ (and $\omega$) Amaro et al. (2005). These results represent a strong constraint for theoretical models of electron scattering reactions. Extensive studies with a large variety of models reveal the importance of a proper description of the interaction of knocked-out nucleons with the residual nucleus—final state interactions (FSI)—to obtain the tail of the scaling function Caballero et al. (2005); Caballero (2006); Caballero et al. (2007); Meucci et al. (2009); Antonov et al. (2011). The authors of Refs. Caballero et al. (2005); Caballero (2006) argue that, while this asymmetry in the scaling function is largely absent in non-relativistic mean-field models, it can be recovered within the relativistic impulse approximation, given that FSI are described using a strong relativistic mean field (RMF) potential. Asymmetric scaling functions also emerge in semi-relativistic models when FSI are described by local potentials derived from the RMF one Caballero et al. (2007). On the other hand, the comparison between semi-relativistic and relativistic results shows a breakdown of the zeroth-kind scaling, i.e. different scaling functions in the longitudinal and transverse channel, only when the fully relativistic mean field approach is employed. According to Ref. Caballero et al. (2007) this effect has been ascribed to the dynamical enhancement of the lower component of the Dirac spinors, which are not present in the semi-relativistic approach.

In this work we analyze the scaling properties exhibited by Green’s Function Monte Carlo (GFMC). GFMC is an ab initio method allowing for a very accurate description of the properties of $A\leq 12$ nuclei, in which the dynamics of constituent nucleons are fully considered Lovato et al. (2013, 2014, 2015). The longitudinal and transverse electromagnetic response functions of 12C, recently computed within GFMC turn out to be in very good agreement with experiment, when two-body currents are accounted for Lovato et al. (2016). Despite this remarkable result, GFMC is currently limited to 12C because of the exponentially growing cost of the calculation with the number of nucleons. In addition to that, the inclusion of relativistic kinematic and baryon resonance production would involve non trivial difficulties.

The study of the behavior of the scaling functions obtained from the GFMC calculations, while being interesting in its own right, is aimed at elucidating the role of initial and final state correlations in the asymmetric shape of the scaling function.

In Section II we review the derivation of the electron-nucleus cross section, as well as its expression in terms of longitudinal and transverse response functions, which are necessary to introduce the concept of scaling. In Section II.1, the main elements of the Green’s Function Monte Carlo approach are briefly outlined, while in Section II.2 we explicitly derive the expression of the longitudinal and transverse scaling functions in the context of the GRFG model, both in the relativistic and non relativistic cases. In Section III we report the results of our analysis of the scaling features of the GFMC response functions for 4He and 12C nuclei and in different kinematics. We then discuss a novel interpretation of the longitudinal and transverse scaling function in terms of the nucleon-density function. Finally, in Section IV we summarize our findings and state the conclusions.

II Scaling of the nuclear electromagnetic response within the Green’s Function Monte Carlo approach

In the one-photon-exchange approximation, the double differential electron-nucleus cross section can be written in the form

[TABLE]

where $k_{e}=(E_{e},{\bf k}_{e})$ and $k_{e^{\prime}}=(E_{e^{\prime}},{\bf k}_{e^{\prime}})$ are the laboratory four-momenta of the incoming and outgoing electrons, respectively; $\alpha\simeq 1/137$ is the fine structure constant, $d\Omega_{e^{\prime}}$, the differential solid angle in the direction of ${\bf k}_{e^{\prime}}$, and $q=k_{e}-k_{e^{\prime}}=(\omega,{\bf q})$ the four momentum transfer. The leptonic tensor is given by

[TABLE]

The hadronic tensor encompasses the electromagnetic transitions from the target nucleus to all possible final states. It is thus given by

[TABLE]

where $|0\rangle$ and $|f\rangle$ denote the initial and final hadronic states with four-momenta $P_{0}=(E_{0},{\bf p}_{0})$ and $P_{f}=(E_{f},{\bf p}_{f})$, while $J(q)$ is the electromagnetic nuclear current operator.

Equation (1) can be rewritten in terms of two response functions, denoted by $R_{L}({\bf q},\omega)$ and $R_{T}({\bf q},\omega)$, describing interactions with longitudinally (L) and transversely (T) polarized photons, respectively. The resulting expression reads

[TABLE]

where

[TABLE]

and

[TABLE]

is the Mott cross section. The L and T response functions can be readily expressed in terms of specific components of the hadronic tensor. Choosing the $z$-axis along the direction of the momentum transfer one finds

[TABLE]

II.1 The Green’s function Monte Carlo approach

GFMC provides a suitable framework to carry out accurate calculations of a variety of nuclear properties in the non relativistic regime, typically corresponding to $|{\bf q}|\buildrel<\over{{}_{\sim}}500\ {\rm MeV}$ (for a recent review of Quantum Monte Carlo methods for nuclear physics see, e.g., Ref. Carlson et al. (2015)).

The longitudinal and transverse response function are given by

[TABLE]

where $\rho({\bf q})$ and ${\bf j}_{T}({\bf q})$ denote non-relativistic reductions of the nuclear-charge and transverse-current operators, respectively Carlson et al. (2002). Valuable information on the L and T responses can be obtained from their Laplace transforms, also referred to as Euclidean responses

[TABLE]

The lower integration limit $\omega_{\rm{el}}={\bf q}^{2}/2M_{A}$, $M_{A}$ being the mass of the target nucleus, is the elastic scattering threshold—corresponding to the $|f\rangle=|0\rangle$ term in the sum of Eq. (3)—whose contribution is excluded.

Within GFMC, the Euclidean responses are evaluated from

[TABLE]

and

[TABLE]

Note that, although the states $|f\rangle\neq|0\rangle$ do not appear explicitly in Eqs. (11) and (12), the Euclidean responses include the FSI effects of the particles involved in the electromagnetic interaction, both among themselves and with the spectator nucleons.

The inversion of the Laplace transform, needed to retrieve the energy dependence of the responses, is long known to involve severe difficulties. However, maximum-entropy techniques, based on Bayesian inference arguments, have been successfully exploited to perform accurate inversions, supplemented by reliable estimates of the theoretical uncertainty. In the case of 12C, particular care has to be devoted to the subtraction of contributions arising from elastic scattering and the transitions to the low-lying $2^{+}$, $0^{+}_{2}$, and $4^{+}$ states Lovato et al. (2015).

II.2 Scaling within the relativistic Fermi gas model

The easiest, albeit quite crude approximation, to describe the hadron tensor consists on using the GRFG model. Within this approach the scattering process is assumed to take place on a single nucleon with four-momentum $p=(E({\bf p}),{\bf p})$, where $E({\bf p})=\sqrt{|{\bf p}|^{2}+m^{2}}$, $m$ being the nucleon mass. The requirement that the struck nucleon is in the target nucleus implies that $|{\bf p}|$ is smaller than the Fermi momentum $p_{F}$. Furthermore, the outgoing nucleon with four-momentum ${p^{\prime}}^{\mu}=(p+q)^{\mu}$ should lay above the Fermi surface. The expression of the hadron tensor describing the response of the target nucleus then reads

[TABLE]

Once we only discuss symmetric nuclei, $\mathcal{N}$ denotes both the number of protons and neutrons in the nucleus. The single-nucleon response tensor $w^{\mu\nu}(p+q,p)$ encodes the response of a system in which a nucleon with 4-momentum $p$ in the initial state is scattered by a (virtual) photon, leading to a final state with a nucleon carrying a 4-momentum $(p+q)$. The following general expression

[TABLE]

where $\tau=-q^{2}/4m^{2}=Q^{2}/4m^{2}\geq 0$, holds. It is well known that the nucleon structure functions $W_{1,2}$ can be written in terms of the proton and neutron electric and magnetic form factors as

[TABLE]

and

[TABLE]

where $\tau_{z,p/n}=\pm 1$.

Using the GRFG model to parametrize the nuclear amplitudes, the integral entering Eq. (13) can be analytically solved. We start by evaluating the function

[TABLE]

with

[TABLE]

resulting in Alberico et al. (1988); Donnelly et al. (1992)

[TABLE]

where we have introduced $E_{F}=\sqrt{p_{F}^{2}+m^{2}}$ and

[TABLE]

It is convenient to introduce the widespread set of dimensionless variables Alberico et al. (1988)

[TABLE]

The minimum $\Gamma_{3}/m=1$ at

[TABLE]

corresponds to the quasi elastic peak $\tau=\lambda$ Alberico et al. (1988). In the limit of large $|{\bf q}|$, the relation $\Gamma=\Gamma_{3}$ is satisfied for each value of $\omega$. Hence, a dimensionless scaling variable can be defined in terms of this quantity as Alberico et al. (1988)

[TABLE]

with $\xi_{F}=E_{F}/m-1$ and such that $\psi=0$ at the quasi elastic peak. Note that this definition of the scaling variable is equivalent to the more common expression

[TABLE]

Collecting previous results one obtains

[TABLE]

Substituting Eq. (13) and (II.2) into Eqs. (7), (8) leads to the following expressions for the L and T response functions

[TABLE]

After performing the integrations, the responses can be cast in the form

[TABLE]

where

[TABLE]

The next step consists in the definition of the longitudinal and transverse scaling functions Maieron et al. (2002)

[TABLE]

where

[TABLE]

Within the GRFG the same scaling function for the the longitudinal and transverse channel arises. This is a symmetric function centered in $\psi=0$

[TABLE]

In the non relativistic limit the L and T responses can be expressed as

[TABLE]

where $s$ and $s^{\prime}$ are the spin quantum numbers of the nucleon in the initial and final state, respectively.

In the following, non relativistic scaling variable and functions are introduced with the same non relativistic reduction of the current operator and relativistic corrections as in the GFMC calculations Carlson et al. (2002). Neglecting the very small spin-orbit relativistic correction in the definition of charge operator, the charge and current operators read

[TABLE]

As opposed to the semi relativistic model of Ref. Amaro et al. (2007), in the GFMC relativistic corrections enter only in the current definition, while the kinematics is fully non relativistic.

In the non relativistic limit, Eq. (17) reduces to

[TABLE]

with

[TABLE]

and

[TABLE]

The non relativistic Fermi energy reads $E^{nr}_{F}=m+{p_{F}^{2}}/{2m}$. We can then introduce a non relativistic scaling variable given by

[TABLE]

In the limit of large $|{\bf q}|$, Eq. (34) can be written in terms of $\psi^{nr}$ as

[TABLE]

In analogy with the relativistic case, the longitudinal and transverse responses are expressed as

[TABLE]

where

[TABLE]

We then define the non relativistic longitudinal and transverse scaling functions as

[TABLE]

where

[TABLE]

In order to compare our results with the data, we introduce the experimental scaling functions obtained from the extracted longitudinal and transverse responses for 4He and 12C

[TABLE]

It is long known that $f^{exp}_{L}$ clearly shows a scaling behavior in the limit of large momentum transfer. On the other hand, sizable scaling violations occur in the transverse channel, due to significant contributions given by two-body currents, resonance excitations and inelastic scattering. Hence, the comparison with the experimental data will be performed considering only the longitudinal contribution, $f^{exp}_{L}$.

III Results

Here we analyze the scaling features of the GFMC responses. In order to highlight the underlying nuclear dynamics we first divide them by the non relativistic prefactors $G^{nr}_{L,T}$. These have been obtained expanding the relativistic-current matrix elements in powers of $1/m$ retaining terms up to $\mathcal{O}[1/m^{2}]$ Lovato et al. (2016). Relativistic corrections appear as terms of $\mathcal{O}[1/m^{2}]$ in the longitudinal channel while they are $\mathcal{O}[1/m^{3}]$ in the transverse one, and are therefore neglected in this case. This difference plays a relevant role in the interpretation of the results presented below. For a meaningful comparison with the scaling functions extracted from experimental data, we also present the results obtained using the relativistic prefactors $G_{L,T}$. Figure 1 clearly shows the different behavior of $G^{nr}_{L,T}$ and $G_{L,T}$ for three values of the momentum transfer. Relativistic effects are particularly relevant in the transverse case; at $|{\bf q}|$= 570 MeV the ratio $G^{nr}_{T}/G_{T}$ significantly differs from 1 for $\psi\geq 0$.

In Figs. 2, 3 and 4 we show the longitudinal (blue solid lines) and transverse (red dashed lines) scaling functions extracted from the GFMC calculations of the 12C response functions. The results in the upper panels, obtained dividing the GFMC calculations by $G^{nr}_{L(T)}$, are plotted as a function of the non relativistic scaling variable $\psi^{nr}$. In the lower panels a comparison between the theoretical curves and the experimental points, in which the relativistic form of the prefactors has been adopted, is presented.

It is important to point out that the longitudinal response of 12C is known to be affected by the elastic and the low lying excited states—$J^{\pi}=2^{+},\ 0_{2}^{+},$ and $4^{+}$—contributions. In order to compare experiments —which refer only to the inclusive quasi-elastic response—with GFMC calculations, these contributions have been explicitly subtracted by using the experimental values of excitations energies and form factors. Because of the fast drop of the form factors with increasing momentum transfer, in Ref. Lovato et al. (2016) it is argued that these corrections are expected to be significant in the longitudinal channel at $|{\bf q}|=300$ MeV, but almost negligible at $|{\bf q}|=570$ MeV. On the other hand, in the transverse channel such contributions are expected to be always negligible.

The scaling functions displayed in the upper panels exhibit a clearly asymmetric shape, with a tail extending in the region $\psi^{nr}>0$, as opposed to the GRFG model predictions. The difference in magnitude between the longitudinal and transverse GFMC scaling functions, which become less evident for larger values of $|{\bf q}|$, is likely to be ascribed to small residual effects of the low lying excited state contributions. For the aforementioned reason, in the lower panels the agreement between the longitudinal GFMC scaling function and the experimental data improves with increasing momentum transfer.

The different behavior of the transverse scaling functions displayed in the upper and lower panels deserves some comments. In the lower panels, the red curves present a large non vanishing tail for $\psi>1$, although those are expected to approach zero, as shown in the upper panels. This discrepancy can be best understood considering the results of Fig.1. The relativistic and non relativistic expressions of the transverse prefactors used to extract the scaling functions are sizably different in the kinematic setups considered. In particular, for $|{\bf q}|=570$ MeV, these are very similar for $-1.5\leq\psi\leq 0$ where their ratio is almost 1, while in the region $\psi\geq 0$ their trend is significantly different and $G^{nr}_{T}/G_{T}$ increases for larger values of $\psi$.

Figure 5 shows the experimental scaling functions of 12C extracted from the experimental data of Ref. Barreau et al. (1983) for $|{\bf q}|=300,\ 380,$ and 570 MeV. Although scaling is expected to occur in the limit of large momentum transfer, within the error bars of the different data points, the longitudinal response functions scale to a universal curve over the entire quasi-elastic peak, even in the region of moderate $|{\bf q}|$.

In Fig. 6 the longitudinal GFMC scaling functions are shown as a function of $\psi^{nr}$ for $|{\bf q}|=300,\ 380,$ and 570 MeV. The theoretical results seem to indicate that first-kind scaling occurs. However, the interpretation of the differences between the three curves is obscured by the residual effect of the low-lying transitions discussed above. A more meaningful comparison can be carried out in the transverse channel, where the response functions are not affected by this effect.

Figure 7 shows the GFMC results for the transverse scaling functions. The difference between the three curves in the ${\psi}^{nr}<0$ region suggests that, for $|{\bf q}|=300,\ 380$ MeV, the requirement $\Gamma=\Gamma_{2}$ [see Eq. (36)]—which is necessary to introduce the scaling variable—is not satisfied for all the values of $\omega$. Indeed, the scaling violation in the low-energy transfer region is clearly visible.

To better elucidate the scaling properties of the GFMC calculations, it is worth to analyze the 4He nucleus, whose longitudinal response functions are not affected by low-lying transitions. In Fig. 8, the scaling functions obtained from the experimental data of the longitudinal responses of 4He at $|{\bf q}|=300,\ 400,\ 500\ ,600,$ and $700$ MeV are shown. Choosing the Fermi momentum equal to $180$ MeV, we observe that the points corresponding to different values of the momentum transfer tend to lay on top of each other, and the agreement with the 12C data at $|{\bf q}|=570$ MeV is also remarkable.

In Figs. 9-13 we show the longitudinal (solid blue) and transverse (dashed red) scaling functions extracted from the GFMC calculations of 4He at $|{\bf q}|=300,\ 400\ ,500\ ,600,$ and 700 MeV. In the upper and lower panels the same scheme followed to present the 12C scaling functions has been adopted. In the longitudinal channel, theoretical calculations and experimental data reported in the lower panels present are in very nice agreement in all the kinematic setups. Finding this agreement up to $|{\bf q}|=700$ MeV may appear surprising since the GFMC is a non relativistic approach. This can be understood because all the relativistic corrections coming from both the Dirac-spinors and the currents are kept up to $\mathcal{O}[1/m^{2}]$. However, this is not the case in the transverse channel where relativistic corrections are subleading and have been neglected. Moreover, the differences in magnitude of the transverse scaling functions, following the discussion carried out for 12C, are likely to be ascribed to relativistic effects in the prefactors.

The upper panels of Figs. 9-13 clearly show that in the 4He case the scaling of the zeroth-kind is manifest when the effects of nuclear dynamics are singled out by using the non relativistic expressions for the prefactors. The absence of low-lying transition contributions makes the scaling of the first kind apparent.

The curves of Figs 14 and 15, where we compare the longitudinal and transverse scaling functions of 4He for different values of the momentum transfer, present a remarkably good scaling behavior. The 4He results for $|{\bf q}|=600\ ,700$ MeV are almost coincident and in good agreement with the longitudinal scaling function of 12C computed at $|{\bf q}|=570$ MeV.

Figures 14 and 15 prove that the asymmetric shape of the scaling function does not depend upon the momentum transfer. Consequently, it is not likely to be ascribed to collective excitation modes, that can be accounted for within the random phase approximation.

This analysis, carried out for a variety of kinematics suggests that scaling occurs in the GFMC calculations of the longitudinal and transverse response functions of both 4He and 12C nuclei. Comparing the definition of the longitudinal response function and the one of the corresponding prefactor, see Eq. (9) and (43), while neglecting the spin-orbit contribution, one is lead to conclude that the scaling function corresponds to

[TABLE]

where $R$ is the nucleon-density response function defined as

[TABLE]

in terms of the nucleon-density operator

[TABLE]

where the $\pm$ applies to protons and neutrons, respectively. Note that Eq.(45) holds also in the relativistic case, provided that relativistic expressions for the energies are used and spinors are normalized as $\bar{u}u=\sqrt{m/E}$ to absorb the factor $m^{2}/(E({\bf p})E({\bf p+q}))$ of Eq. (18).

On the other hand, the transverse scaling function corresponds to the spin-response, which reduces to the nucleon-density response defined above in the limit of high momentum-transfer, where the impulse approximation is expected to be accurate and where $|\mathbf{q}|\gg|{\bf p}_{T}|$. We found that the longitudinal and transverse response functions obtained retaining only the one body current operator scale to the same universal scaling function: the nucleon-density response function.

The results presented in Ref. Lovato et al. (2016) show that two-body currents lead to a significant enhancement of the transverse response of 12C in the region of the quasi elastic peak. We expect that the inclusion of this contribution in the scaling analysis, while leaving the longitudinal scaling function unchanged, would contribute to the observed scaling violation of the experimental scaling function in the transverse channel for $\psi\geq 0$.

IV Conclusions

We have performed a scaling analysis of the GFMC electromagnetic response functions of 4He and 12C for a variety of kinematic setups. Despite the non relativistic nature of the calculation, all the GFMC scaling functions analyzed are strongly asymmetric, with a tail extending to the large $\psi$ region. Within the present picture, this is a consequence of nuclear correlations in both the initial and final states. This is at variance with the findings of Ref. Caballero et al. (2007), where the asymmetric shape was ascribed to relativistic effects in the treatment of the final state interactions. In this regard, it is interesting to point out that the symmetry of the scaling function is not recovered even for momentum transfer as low as $|{\bf q}|=300$ MeV in both 4He and 12C. When the nuclear dynamics is properly singled out, the 12C response function shows a fairly good scaling behavior. However, the presence of the low lying transitions which is known to affect the longitudinal channel, introduces non trivial difficulties in drawing definitive conclusions. A better understanding is given by the analysis of 4He responses which are free from the uncertainties coming from these contributions. Our results for this nucleus indicate that both the zeroth- and first-kind of scaling occur. Moreover, the 4He and 12C scaling functions fulfill scaling of the second kind once the Fermi momentum of 4He is appropriately tuned.

From our analysis, a novel interpretation of the scaling function emerges. If the spin-orbit contribution to the density-current operator is neglected, it can be easily noted that the longitudinal scaling function corresponds to the nucleon-density response. In the transverse channel, for sufficiently large momentum transfer the term proportional to the transverse momentum of the incoming nucleon can be safely neglected and the scaling function is proportional to the spin-response. In nuclei characterized by total spin $S=0$, such as 4He, 12C, 16O and 40Ca, in the impulse approximation the spin-response reduces to the nucleon-density response. Our findings on the occurrence of zeroth-kind scaling are consistent with this interpretation. In fact, within GFMC the scaling violation of the transverse response in the quasi-elastic region is likely to come from two-body currents. This was first noted by the authors of Ref. Carlson et al. (2002) in which a better agreement between the experimental data and the theoretical calculation of the Euclidean responses of 3He and 4He was found, once that this term was accounted for.

The role played by two-body current contributions in the electromagnetic responses of 12C have been recently investigated in Ref. Lovato et al. (2016), where a significant enhancement of the transverse response is observed at all momentum transfers: not only in the dip region, but in the whole quasi-elastic peak region, extending below the pion-production threshold. In the pioneering work of Ref. Fabrocini (1997), it has been shown that such enhancement is mainly due to the interference between one- and two-body currents leading to single knock out final state. In this case the kinematics would be very similar to the those analyzed in this paper, where only one-body current contributes. Hence, we expect that it would be possible to define an appropriate scaling function for these processes. The consequences of the two-body current contribution in the GFMC scaling functions as well as the study of the scaling properties of the total nuclear response —including both one- and two-body terms—will be the subject of a future work.

Acknowledgements

Research partially supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund, under contracts FIS2014-51948-C2-1-P and SEV-2014-0398, by Generalitat Valenciana under contract PROMETEOII/2014/0068, and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract DE-AC02-06CH11357 (A.L.). Under an award of computer time provided by the INCITE program, this research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357.