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\begin{document}
\title{A realistic approach to inclusive $e$-scattering from nuclei}
\classification{24.10.Cn,25.30.-c}
\keywords {inclusive electron scattering from nuclei}
\author{R. Schiavilla}{
address={Department of Physics, Old Dominion University, Norfolk, VA 23529\\
Theory Center, Jefferson Lab, Newport News, VA 23606}
}
\begin{abstract}
We review the current status of calculations, based on realistic nuclear interactions
and currents, of the inclusive electromagnetic response of nuclei in the quasi-elastic region.
\end{abstract}
\maketitle
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\section{Introduction}
In this talk, we review our current understanding of the inclusive electromagnetic response
of nuclei in the quasi-elastic region within a dynamical approach based on realistic
nuclear interactions and currents. These interactions and currents are discussed
in the next section. The calculation of inclusive electromagnetic (or weak) response
functions is a challenging theoretical problem. In light nuclei, two different
approaches have been pursued so far, one based on sum rules and the other
utilizing integral-transform techniques aimed at removing the need
of calculating explicitly the nuclear excitation spectrum. These approaches
are reviewed in the section after next. The prospects
for extending these methods to a calculation of neutrino
inclusive response induced by charge-changing and neutral
weak currents are briefly outlined in the last section.
\section{Realistic nuclear interactions and electromagnetic currents}
The two-nucleon ($NN$) potential consist of a long-range component induced
by one-pion exchange (OPE) and intermediate- to short-range components which are
modeled phenomenologically, as in the Argonne $v_{18}$ (AV18) potential~\cite{Wiringa95},
or by scalar and vector meson-exchanges, as in the CD-Bonn potential~\cite{Machleidt01},
or by a combination of two-pion-exchange mechanisms and contact two-nucleon terms,
such as in the chiral-effective-field-theory potentials~\cite{Entem03}. All these
potential models fit the $NN$ database for energies up to the
pion production threshold with $\chi^2 \simeq 1$. However, it is by now well established
that $NN$ potentials alone fail to predict the spectra of light nuclei~\cite{Pieper01},
cross sections and analyzing powers in $Nd$ scattering at low~\cite{Marcucci09} and
intermediate~\cite{Kalantar} energies, and the nuclear matter equilibrium properties~\cite{Akmal}.
Models of the three-nucleon ($NNN$) potential include two- and
three-pion exchange~\cite{IL7,Krebs12} as well as short-range repulsive terms.
In the Illinois model 7 (IL7), these multi-pion exchange components involve excitation
of intermediate $\Delta$ resonances. The IL7 strength is determined by four parameters
which are fixed by a best fit to the energies of about 17 low-lying states of nuclei
in the mass range $ A \leq 10$, obtained in combination with the AV18 $NN$ potential.
The AV18/IL7 Hamiltonian reproduces well the spectra of nuclei with $A$=9--12~\cite{Pieper12}---in
particular, the attraction provided by the IL7 $NNN$ potential in isospin 3/2 triplets is crucial for the $p$-shell
nuclei---and the $p$-wave resonances with $J^\pi=(3/2)^{-}$ and (1/2)$^-$ in $n$-$^4$He
scattering~\cite{Nollett07}.
Realistic models for the electromagnetic charge and current operators include
one- and two-body parts. The one-body charge and current operators
follow from a non-relativistic expansion of the single-nucleon four-current.
The two-body current operators are separated into model-independent
(MI) and model-dependent (MD) terms~\cite{Marcucci05}. The MI terms
are derived from the $NN$ potential (the AV18 in the present case), and their
longitudinal components satisfy, by construction, current conservation with it.
They contain no free parameters, and their short-range behavior
is consistent with that of the potential. The dominant terms, isovector
in character, originate from the static part of the potential, which is assumed
to be due to exchanges of effective pseudoscalar ($\pi$-like) and vector
($\rho$-like) mesons. At large inter-nucleon separations, where the $NN$ potential
is driven by the OPE mechanism, the MI currents coincide with the well known
seagull and in-flight OPE currents.
The MD currents are purely transverse, and unconstrained by current conservation.
The dominant term is associated with excitation of intermediate $\Delta$
isobars~\cite{Schiavilla92}. Additional (and numerically small) MD currents
arise from the isoscalar $\rho\pi\gamma$ and isovector $\omega\pi\gamma$
transition mechanisms. These MI and MD currents have been shown to reproduce
satisfactorily a variety of nuclear properties and reactions,
including magnetic moments of, and $M1$ transitions between, low-lying
states of light nuclei~\cite{Pastore13}, the trinucleon magnetic form factors~\cite{Marcucci05},
and radiative captures in the few-nucleon systems~\cite{Girlanda10}.
The two-body charge operators represent relativistic corrections: they vanish
at zero momentum transfer because of charge conservation; they also vanish
in the static limit. The leading term is due to OPE, and is derived from an
analysis of the virtual pion photo-production amplitude. Additional (and numerically
small) contributions arise from exchange of vector mesons and $\rho\pi\gamma$
and $\omega\pi\gamma$ mechanisms. These two-body charge operators, in particular
the OPE one, are essential for reproducing the observed charge form factors of the hydrogen
and helium isotopes (see Ref.~\cite{Carlson98} and references therein).
\section{Approaches to inclusive scattering}
\label{sec:sr}
Two response functions characterize inclusive $(e,e^\prime)$ scattering, defined
as
\[
R_\alpha(q,\omega)=\sum_{f\neq 0} \delta(\omega+E_0-E_f)
\mid\langle f\!\mid O_\alpha({\bf q})\mid\!0\rangle\mid^2 \ ,
\]
where $\mid 0\rangle$ and $\mid f \rangle$ denote, respectively, the initial
and final nuclear states of energies $E_0$ and $E_f$, $\omega$ and ${\bf q}$
are the electron energy and momentum transfers, and $O_\alpha({\bf q})$ is
either the nuclear charge ($\alpha=L$) or current ($\alpha=T$) operator.
Major complications in their calculation arise in consequence of the need
of, and technical difficulties associated with, providing an accurate description
of the initial bound- and final scattering-state wave functions, based on
realistic interactions. These complications can be avoided, at least in part,
either by studying integral properties of the response functions---i.e.,
longitudinal and transverse sum rules---or by using integral transform
techniques of the type
\[
E(q,\tau)=\int_0^\infty{\rm d}\omega\, K(\tau,\omega)\, R(q,\omega) \ ,
\]
which for a suitable choice of kernel---for example, Laplace~\cite{Carlson92} or Lorentz
(see Ref.~\cite{Leidemann12} and references therein) allows the use of closure over
$\mid f\rangle$, thus removing the need of explicitly calculating the nuclear excitation spectrum.
While in principle exact, both these approaches have drawbacks.
\subsection{Sum rules}
Longitudinal (Coulomb) and transverse sum rules can be expressed as
ground-state expectation values of the charge and current operators:
\[
S_\alpha(q)=C_\alpha \int_{\omega^+_{\rm th}}^\infty {\rm d}\omega \, \frac{R_\alpha(q,\omega)}
{G^{\, 2}_{Ep}(q,\omega)}
= C_\alpha \left[ \langle 0\! \mid O_\alpha^\dagger ({\bf q})\,
O_\alpha({\bf q}) \mid\! 0\rangle - \mid \langle 0\!\mid O_\alpha({\bf q})
\mid\! 0\rangle \mid^2 \right] \nonumber \ ,
\]
where it is understood that the charge and current operators $O_\alpha({\bf q})$
have been divided by the proton electric form factor $G_{Ep}$, and the
$C_\alpha$ are normalization constants such that, in the limit $q \rightarrow \infty$
and under the approximation that only one-body terms are retained in $O_\alpha({\bf q})$,
then $S_\alpha(q\rightarrow \infty)=1$~\cite{Carlson02}.
The longitudinal and transverse sum rules defined above (as well as energy-weighted
ones) have been calculated exactly with quantum Monte Carlo techniques in $A=2$--6
nuclei~\cite{Carlson02,Schiavilla89a,Schiavilla89}. However, direct comparison
between these and the experimentally extracted sum rules cannot be made unambiguously
for two reasons. First, the experimental determination of the sum rules requires
measuring the associated response functions over the whole energy transfer,
from threshold to $\infty$. Inclusive electron scattering only
allows access to the space-like region of the four momentum transfer $\omega < q$.
Therefore, for a meaningful comparison between theory and experiment
one needs to estimate the strength outside the region covered
by experiment, either by extrapolating the data or by parametrizing
the high energy tail and using energy-weighted sum rules to constrain it.
The second reason that makes the direct comparison between theoretical
and ``experimental'' sum rules difficult lies in the inherent inadequacy of
the present theoretical model for the nuclear electromagnetic
current, in particular, its lack of explicit pion production
mechanisms. The latter mostly affect the transverse response and
make the $\Delta$ peak outside the boundary of applicability of the
present theory. However, the charge and current operator
discussed in the previous section provide a realistic
and quantitative description of the longitudinal and transverse
response function in the quasi-elastic peak region, where nucleon
and virtual pion degrees of freedom are dominant.
Experimental Coulomb sum rules in the few-nucleon systems are
in good agreement with data (after inclusion of tail contributions),
as shown in Fig.~\ref{fig:f1}~\cite{Schiavilla89}. Contributions
from two-body terms and one-body relativistic corrections in the charge operator
play a minor role in the momentum transfer range covered by experiment.
However, the situation in reference to the Coulomb sum of
medium- and heavy-weight nuclei is still controversial, that is, the question of
whether the longitudinal response in these systems is quenched or
not is yet to be resolved satisfactorily.
\begin{figure}[h]
\includegraphics[height=7cm,width=7cm,angle=-90]{../../Coulomb.eps}
\caption{The longitudinal (Coulomb) sum rule in the $A$=2--4 nuclei: theory (solid lines)
versus experiment (solid circles with error bars); circles without error
bars do not include tail corrections.}
\label{fig:f1}
\end{figure}
The ratios of transverse to longitudinal sum rules in the $A=3$--6 nuclei
are shown in Fig.~\ref{fig:f2}. The transverse sum rule is substantially
increased by two-body current contributions. One interesting feature
of the resulting enhancement is that it increases, for fixed $q$, in going from
$A=3$ to 4, and decreases from $A=4$ to 6. It has been shown in Ref.~\cite{Carlson02}
that the excess transverse strength, defined as $\Delta S_T(q)=S_T(q)-S_T(q;{\rm 1b})$
and where the label ``1b'' means one-body, is proportional
to
\begin{figure}[bth]
\includegraphics[height=4.25cm,width=6cm]{../../ratio-tl.eps}
\includegraphics[height=4.25cm,width=6cm]{../../ratio-a.eps}
\caption{The ratio of transverse to longitudinal sum rules, obtained
with one-body only and one- plus two-body terms in the charge and
current operators, as function of the momentum transfer in $^3$He, $^4$He, and $^6$Li
(left panel) or as function of mass number for $q=300,400,$ and 500 MeV/c (right panel).}
\label{fig:f2}
\end{figure}
\[
\Delta S_{T}^{A}(q) \simeq C_T \int_0^\infty {\rm d}x\,
{\rm tr}\left[ F(x;q)\, \rho^{A}(x;pn)\right]_{\sigma\tau} \ .
\]
Here $F$ is a complicated matrix in the two-nucleon spin-isospin space
depending on the current operators, and $\rho^{ A}$ is the
$pn$ density matrix in this space as function of the relative distance,
a quantity strongly affected by central and tensor correlations
induced by the repulsive core at short range and OPE component
at long range of the $NN$ potential. Thus the transverse enhancement
is primarily due to $pn$ pairs, which can be in isospin $T=0$ and $T=1$. It is
known~\cite{Forest96} that these densities scale as
\[
\rho^{A}(x;pn,T=0) \simeq R_{ A} \, \rho^{ d}(x)
\]
and similarly for $T=1$ $pn$ pairs with $\rho^{ d} \rightarrow
\rho^{ qb}$, where the label $d$ and $qb$ denote, respectively,
the deuteron and $^1$S$_0$ quasi-bound
states. The scaling factors for $T=0$ and 1 $pn$ pairs have been calculated
in light nuclei, and have been found to be close to each other, with
$R_{ A}=2.0, 4.7$, and 6.3 in $^3$He, $^4$He, and $^6$Li, respectively.
The calculated excess transverse strength is consistent with that
expected on the basis of the scaling behavior above~\cite{Carlson02}.
Furthermore, the analysis above suggests that two-body currents
may enhance significantly the transverse response function in
the quasi-elastic region.
\subsection{Euclidean response functions}
The Euclidean response functions are defined as
\[
\widetilde{E}_\alpha(q,\tau) =\int_{\omega^+_{\rm th}}^\infty {\rm d}\omega \,
{\rm e}^{-\tau \left( \omega-E_0\right)}\, \frac{R_\alpha(q,\omega)}
{G^2_{Ep}(q,\omega)}
= \langle 0\!\mid O_\alpha^\dagger({\bf q})
{\rm e}^{-\tau\left(H-E_0\right)} O_\alpha({\bf q})\mid\! 0\rangle
-\left({\rm elastic\,\, term}\right)\nonumber
\]
and represent weighted sums of $R_\alpha(q,\omega)$. At $\tau=0$
they correspond to the sum rules discussed in the previous section,
while their derivatives evaluated at $\tau=0$ correspond to
energy-weighted sum rules. Below we present results for
the scaled Euclidean responses
$E(q,\tau)={\rm exp}\left[ q^2\tau/(2\, m)\right]\, \widetilde{E}(q,\tau)$---inclusion
of this factor removes the trivial energy dependence obtained from scattering off
an isolated (non-relativistic) nucleon. The longitudinal and transverse
Euclidean responses are, respectively, $Z$ and $Z \, \mu_p^2+(A-Z)\, \mu_n^2$
for a collection of $A$ non-interacting nucleons,
$Z$ of which are protons ($\mu_p$ and $\mu_n$ are the proton and neutron
magnetic moments).
The main advantage of formulating the Euclidean response is that it can
be calculated exactly using Green's function or path-integral Monte
Carlo techniques, including both final state interactions and two-body
components in the nuclear charge and current operators~\cite{Carlson92}.
\begin{figure}[h]
\includegraphics[height=5cm,width=7cm]{../../e4l2a.ps}
\includegraphics[height=5cm,width=7cm]{../../e4t2a.ps}
\caption{The Euclidean longitudinal and transverse response functions in $^4$He,
obtained with one-body only (curve labelled IA) and one- plus two-body (curve labelled Full)
terms in the charge and current operators, are compared to experiment.}
\label{fig:f3}
\end{figure}
The calculated longitudinal and transverse Euclidean response functions of $^4$He are displayed
in Fig.~\ref{fig:f3}~\cite{Carlson02}. The experimental ones are obtained
by Laplace-transforming the data, since a direct numerical inversion of $\widetilde{E}(q,\tau)$ is not
possible due to the ill-posed nature of such a problem. In order not to include too much of
the tail of the $\Delta$ resonance, the integration has been carried out
up to the energy loss $\omega$ where the transverse response starts to increase significantly
with $\omega$. Since for the transverse Euclidean response at very small
$\tau$ the tail of the $\Delta$ peak nevertheless plays a role, the
experimental response in this region is indicated by a dashed
line only, and should not be compared to theoretical calculations.
However, as $\tau$ increases beyond $\tau \simeq 0.01$ MeV$^{-1}$,
the Euclidean response probes strength in the quasi-elastic peak region.
At $\tau > 0.03$ MeV$^{-1}$ , contributions to $E(q,\tau)$ from this region
is strongly suppressed, and the Euclidean response is mostly
sensitive to strength at threshold, which is poorly measured.
The large enhancement of the transverse response between
$( 0.01 \leq \tau \leq 0.03)$ MeV$^{-1}$ due to two-body
terms in the current operator should be noted.
\section{Outlook}
\label{sec:out}
It should be possible to use quantum Monte Carlo methods to study
neutrino response functions, and associated sum rules, in light nuclei
(including $^{12}$C) within the same realistic dynamical framework illustrated in
this talk. In recent years, this has become a hot topic in view
of the anomaly observed in recent neutrino quasi-elastic
scattering data on $^{12}$C ~\cite{Aguilar08}, i.e., the excess, at relatively
low energy, of measured cross section relative to theoretical
calculations. Analyses based on these calculations
have led to speculations that our present understanding
of the nuclear response to charge-changing weak probes may be incomplete~\cite{Benhar10}
and, in particular, that the momentum transfer dependence of the
nucleon axial form factor~\cite{Juszczak10} may be quite different from that
inferred from analyses of pion electro-production data~\cite{Amaldi79} and measurements
of neutrino and anti-neutrino cross sections from proton and deuteron~\cite{Baker81}
However, the calculations on which these analyses are based use rather
crude models of nuclear structure---Fermi gas or local-density approximations
of nuclear matter spectral functions---as well as simplistic treatments of the
reaction mechanism, and should therefore be viewed with skepticism.
As shown in this talk, ``exact'' calculations of the electromagnetic
response functions in the $A=3$--4 nuclei~\cite{Carlson02} are
in satisfactory agreement with data. In particular, these calculations
have shown that the transverse response is significantly
increased over the quasi-elastic peak region by two-body
currents, in particular those associated with OPE.
It will be interesting to see whether this mechanism is
effective in the weak sector probed by neutrino scattering,
and possibly provide an explanation for the observed anomaly
in the ${12}$C data. Work along these lines is in progress.
\begin{theacknowledgments}
I would like thank my collaborators J.\ Carlson, S.\ Gandolfi, L.E.\ Marcucci,
S.C.\ Pieper, G.\ Shen, and R.B.\ Wiringa for their many contributions to the
research reported in this talk. I also would like to gratefully acknowledge
the support of the U.S. Department of Energy, Office of Nuclear Science,
under contract DE-AC05-06OR23177.
\end{theacknowledgments}
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\end{document}