\section{Introduction}

The bcc phase of $^{4}$He has a pronounced quantum nature due to the
relatively open structure of the lattice. Quantum effects are manifested in
strong anharmonicity of some phonon modes and in the large zero-point
kinetic energy of the atoms \cite{glyde}. It is this large kinetic energy
which is thought to help stabilize the bcc phase over the hcp phase. In this
paper we highlight this nature of the bcc phase by proposing a new physical
model for the local atomic motion. For the sake of clarity we have
reproduced here some of the arguments and calculations already given in \cite
{niremil}. We propose that in the bcc $^{4}$He phase the local excitations
of the atoms in their potential wells, result in oscillating local electric
dipoles. The ground-state of these dipoles has the dipoles oscillating in
synchrony, thereby reducing the dipolar interaction energy between them.
Solving a mean-field Hamiltonian describing these dipoles we find that
Bosonic phase fluctuations in the (110) direction reproduce the spectrum of
the T$_{1}(110)$ phonon.

In the following we further explore the nature of the coherent ground-state
of the local-modes in the bcc $^{4}$He. We show that the bcc $^{4}$He is a
unique phase having both Diagonal Long Range Order (DLRO) of the solid
lattice and Off-Diagonal Long Range Order (ODLRO) of the local dipoles.
There is therefore a complex three-component order parameter which describes
the coherently oscillating dipoles in each of the three orthogonal
directions in the lattice. In the ground-state the local dipoles form a
Bose-Einstein condensate in the zero momentum state, and we are able to
estimate the ground-state energy reduction due to this condensation. This
estimate compares favourably with experimental results and consequently we
claim that this condensation energy stabilizes the bcc phase over the hcp
phase. In the hcp phase we expect no coherence or condensation due to the
highly isotropic lattice and the geometric frustration of the hexagonal symmetry. We also comment about the
relation of this work to previous work about the 'super-solid' concept in
quantum solids.

Additionally, we predict a high-energy optical-like mode which has fermionic
statistics. This excitation is confined to the (110) direction and involves
a local 'fliping' of a dipole with respect to the ground-state. This makes
the dipole become anti-symmetric (a $\pi $ phase difference) with respect to
the global phase of the complex order parameter and aquire Fermi-Dirac
statistics. We also give analytic expressions for the scattering intensity
of both the Bose and Fermi excitations along the (110) direction. These
predictions remain to be compared with future experimental data.

\section{Ground-state coherence and Bose excitations.}

The usual treatment of the ground-state and energy of bcc $^{4}$He employs
variational wavefunctions that aim to account for the short-range
correlations between the atoms \cite{glyde}. These correlations arise mainly
due to the hard-core repulsion between the atoms. The atoms have a high
zero-point kinetic energy which is given quite accurately by treating them
as independent particles held in place by the potential of the neighboring
(static) atoms. This type of calculation is the ''particle-in-cell''
approximation which gives surprisingly good agreement with measured
thermodynamic properties of the solid phase \cite{glyde1}. We want to focus
here on the effects of the local motion of the atoms inside this
potential-well on the nature of the ground-state. In this approach we would
like to isolate the lowest energy excited state of the atom inside its
potential well, and treat it as a local excitation of the lattice. This
local excited state consists of a local oscillatory motion of the atom along
a particular direction and produces an oscillating electric dipole,
similarly to that of the usual Van-der Waals interaction. However unlike the
case of the Van-der Waals interaction, in which the dipolar fluctuations are
random, we show that in the bcc solid there are local dipoles which are
correlated and a new ground-state of lower energy is created.

The potential well of an atom in the bcc lattice due to the standard helium
pair-potential $\upsilon (r)$ \cite{glyde}, provided one can take the other
atoms as stationary, can be maped along any direction in the lattice. We
find \cite{niremil} that in the directions normal to the unit cube's faces
(i.e. (100),(010) etc.) the confining potential well is very wide with a
pronounced double-minimum structure (Fig.1). Solving the one-dimensional
Schrodinger's equation for a $^{4}$He atom in this potential, we get a first
excited level with energy $10{\rm K}$, and a wavefunction describing atomic
motion with an amplitude of $\sim 1{\rm \AA }$ (in the (100) direction
(Fig.1)). Based on the above calculation, we shall assume that the atoms
have a local-mode that is highly directional along one of the directions
equivalent to (100). Local atomic motion along the other directions is
assumed to be severly restricted due to the higher excitation energies
(Fig.1). Experimental evidence for the existence of such a ''local mode''
comes from NMR measurements which find an activation energy of 7$\pm 1$K\cite
{schuster,allen}, and we propose to identify this local-mode with
the highly directional motion of the atoms in the normal directions.

Using this identification we can now estimate the size of the local electric
dipole moment that can be created by this local and highly directional
atomic motion. As the atom moves this instantaneous local electric-dipole is
created due to the electronic cloud and the ion being slightly displaced
relative to each other. The electric dipole moment due to mixing of the
lowest $\left| s\right\rangle $ and $\left| p\right\rangle $
electronic-levels of the $^{4}$He atom, is given from perturbation theory as
\begin{eqnarray}
\psi &=&\left| s\right\rangle +\lambda \left| p\right\rangle \Rightarrow
E_{0}\simeq \left\langle \psi \left| E\right| \psi \right\rangle
-\left\langle s\left| E\right| s\right\rangle \simeq \lambda
^{2}\left\langle p\left| E\right| p\right\rangle  \nonumber \\
\ &\Rightarrow &\lambda ^{2}\simeq 7/2.46\cdot 10^{4}\simeq 0.00284,\lambda
\simeq 0.0168  \label{lamda}
\end{eqnarray}

where $\left| s\right\rangle $ and $\left| p\right\rangle $ stand for the
ground-state and first excited-state of the $^{4}$He atom, $\lambda $ is the
mixing coefficient and $\left\langle p\left| E\right| p\right\rangle \simeq
2.46\cdot 10^{4}$ K is the excitation energy of the first atomic
excited-state \cite{white}. This small estimated mixing gives the magnitude
of the induced dipole moment as
\begin{equation}
\left| {\bf \mu }\right| =e\left\langle \psi \left| x\right| \psi
\right\rangle \simeq 2e\lambda \left\langle s\left| x\right| p\right\rangle
\simeq e\cdot 0.03{\rm \AA }  \label{mu}
\end{equation}

where $\left\langle s\left| x\right| p\right\rangle \simeq 0.9{\rm \AA }$ .
The estimation of the mixing $\lambda $ and the dipole-moment $\left| {\bf
\mu }\right| $ serves to set an upper bound on the magnitude of this effect,
since we assumed that the entire excitation energy $E_{0}$ is converted to a
local electric dipole.

It is possible to show that the lowest energy of a correlated dipolar
array
in the bcc lattice preserves the symmetry of the bcc unit cell along one
of
the symmetry axes. In such a case it can be easily shown that there will
be
no contribution to the dipolar interaction energy from oscillating dipole moments
which
are orthogonal, and
the instantaneous dipolar
interaction energy for each of the three orthogonal directions, is given by
\begin{equation}
E_{dipole}=-\left| {\bf \mu }\right| ^{2}\sum_{i\neq 0}\left[ \frac{3\cos
^{2}\left( {\bf \mu }\cdot \left( {\bf r}_{0}-{\bf r}_{i}\right) \right) -1}
\left| {\bf r}_{0}-{\bf r}_{i}\right| ^{3}}\right]  \label{edipole}
\end{equation}

where the sum is over all the atoms in the lattice, ${\bf r}_{i}$ being the
instantaneous coordinate of the $i$-th atom. For oscillating dipoles with
random phases, the average instantaneous interaction energy (\ref{edipole})
summed over the lattice would be zero. However, the energy of the dipoles
can be lowered by correlating the phases of the oscillating atoms. Since the
direction of the local dipole shows the instantaneous direction of the
motion or displacement, a state where all the dipoles point in the same
direction is just a uniform motion or translation of the entire lattice. We
therefore have to look for symmetric arrangements with respect to the number
of up/down dipoles, such as is shown in Fig.2. This is the lowest energy
'antiferroelectric' configuration with the periodicity of the bcc unit cell.
We have shown this arrangement for individual dipoles oriented along the
(001) direction, but they are similarly arranged for dipoles along the two
other orthogonal axes. The sum in (\ref{edipole}) for such a configuration
with a unit dipole is given in Fig.2. Thus, the ground state in our picture
has the atoms executing this correlated local oscillation along the three
orthogonal directions.

We therefore have, in addition to the usual (isotropic) Van-der Waals interaction,
highly directional (anisotropic) electric dipoles that become correlated so that they oscillate in synchrony.
This is a state of quantum resonance where the system oscillates between two
equivalent up/down arrangements of the ground-state of the dipoles (Fig.2).
The total interaction between the atoms is now given as the usual
second-order ($\propto 1/r^{6}$) Van-der Waals contribution that is the
result of local-dipoles which have random relative phases, and an additional
long-range (first-order, $\propto 1/r^{3}$) dipolar interaction from the correlated part.
Dipolar interactions that decay as $1/r^{3}$ occur for perfectly correlated
oscillating dipoles, such as a single electric dipole and its image in an
adjacent conducting plate. The coherently oscillating nearest-neighbor
dipoles therefore behave as perfect images of each other (Fig.2), and
oscillate with the same global phase.

The correlated oscillating dipoles do not have an average static dipole
moment, so this is not the case of an antiferroelectric structural phase
transition \cite{landau}. The array shown in Fig.2 is simultaneously
arranged along the other two orthogonal axes. Along each direction the
ground-state is given as a coherent-state of these local dipoles, i.e. has a
well-defined phase and an ill-defined occupation number.

We shall treat the dynamics of the correlated dipolar array as independent
of the other degrees of freedom of the lattice. This assumption needs
justification since there can be phonon modes that will modulate the atomic
motion, thereby coupling with the oscillating dipolar array. The oscillatory
atomic motion induced by the phonons will modulate the relative phases of
the dipoles. Let us look at the ground state of the dipoles, taking for
example dipolar
oscillations oriented along the (001) direction (Fig.2). We now need to
consider only phonons which will modulate the local motion responsible for
the oscillating dipoles in this direction. In the bcc structure, only 3
phonons fulfill this condition: L(001), T(100) and T$_{1}$(110). Let us
calculate the energy of the dipolar array when modulated along these 3
directions. For a modulation along some direction ${\bf k}$ , the dipolar
interaction energy is given by\cite{heller}:
\begin{eqnarray}
X\left( {\bf k}\right) &=&%
-\left| {\bf \mu }\right| ^{2}\sum_{i\neq 0}\left[%
\frac{3\cos ^{2}\left( {\bf \mu }\cdot \left( {\bf r}_{0}-{\bf
r}_{i}\right)%
\right) -1}{\left| {\bf r}_{0}-{\bf r}_{i}\right| ^{3}}\right] \nonumber
\\
&&\ \ \exp \left[
2\pi i{\bf k}\cdot \left( {\bf r}_{0}-{\bf r}_{i}\right) \right]
\label{xk}
\end{eqnarray}

At $k=0$ the interaction matrix $X(k)$ is just the dipolar energy (\ref
{edipole}).

In Fig.3 we plot the value of $X(k)$, the energy of the dipolar array
modulated by the relevant phonons:\ L(001), T(100), and T$_{1}$(110), for
dipole moment $\left| {\bf \mu }\right| =1$. We see that for a modulation by
L(001) and T(100) the periodicity of $X(k)$ is over a full unit-cell, that
is twice the periodicity of these phonons. Since symmetric functions of
periodicities $\pi /a$ and $2\pi /a$ are orthogonal, so are the
eigenfunctions of these particular phonons and dipole-excitations. The
dipole array cannot therefore be excited by any of these two phonon. For the
modulation produced by the T$_{1}$(110) mode, the periodicity of $X(k)$ is
the same as that of the T$_{1}$(110) phonon, which can therefore couple to
the dipole array. We conlude therefore that the coupling of the local modes
to the lattice excitations is limited to a single phonon mode, justifying
our assumption that the local modes can be treated separately to a good
approximation. We shall now calculate the dispersion relation of such an
excitation by a mean-field solution of an effective Hamiltonian. It turns
out that the only phonon mode of the bcc lattice that can couple with the
dipolar array is in fact the natural excitation of the dipolar array in the
(110) direction. Thus, the only elementary (Bose) excitations of the dipole
array would be in the (110) direction, in the form of the T$_{1}$(110)
phonon. The description of this phonon is therefore taken into account by
our treatment of the dynamics of the dipolar array, and will appear as a
solution of the mean-field treatment. This means that our assumption of an
effective decoupling between the dipolar and other degrees of freedom is
justified.

The Hamiltonian treatment of interacting local excitations was developed
originally by Hopfield \cite{hopfield} for the problem of excitons in a
dielectric material. The local excitations are treated as bosons using the
standard Holstein-Primakof procedure, and the effective Hamiltonian
describing their behavior is \cite{anderson}

\begin{eqnarray}
{H_{loc}} &=&{\sum_{k}}(E_{0}+X(k))\left( {{b_{k}}^{\dagger
}}{b_{k}}+{\frac{%
1}{2}}\right)  \nonumber \\
&&\ \ +{\sum_{k}}X(k)\left( {{b_{k}}^{\dagger }}{b_{-k}^{\dagger }}%
+b_{k}b_{-k}\right)  \label{hloc}
\end{eqnarray}
where ${{b_{k}}^{\dagger },}{b_{k}}$ are Bose creation/anihilation operators
of the local mode, $X(k)$ is given in (\ref{xk}) and $E_0$ is the energy
of exciting a local dipole out of the correlated ground-state.

The Hamiltonian ${H_{loc}}$ (\ref{hloc}) which describes the effective
interaction between localized modes can be diagonalized using the Bogoliubov
transformation ${\beta _{k}}=u(k)b_{k}+v(k)b{^{\dagger }}_{-k}$. The two
functions $u(k)$ and $v(k)$ are given by:
\begin{equation}
{u^{2}}(k)={\frac{1}{2}}\left( \frac{E_{0}{+X(k)}}{{E(k)}}+1\right) ,{v^{2}
(k)={\frac{1}{2}}\left( \frac{E_{0}{+X(k)}}{{E(k)}}-1\right)  \label{uv}
\end{equation}

The result of solving by mean-field the effective Hamiltonian for the
correlated dipolar array \cite{niremil}, is a coherent ground-state given by
\cite{huang}
\begin{equation}
\left| \Psi _{0}\right\rangle =\prod_{k}\exp \left( \frac{v_{k}}{u_{k}}{
b_{k}}^{\dagger }}{b_{-k}^{\dagger }}\right) \left| vac\right\rangle
\label{psi0}
\end{equation}

and the energy spectrum is
\begin{equation}
E(k)=\sqrt{E_0\left( E_0+2X(k)\right) }  \label{ek}
\end{equation}

In Fig.2 we see that the ground-state arrangement has the dipoles arranged
in alternating planes in the (110) direction. As we have shown the only
naturally occuring Bose excitations of this dipolar field are along this
direction and $X(k)$ is the dipolar interaction matrix element for $k$ in
the (110) direction (Fig.3). In order to calculate the energy spectrum we
now need to fix the size of the coherent dipole moment $\left| {\bf \mu
\right| $. According to our definition of the local mode the energy cost of
flipping the direction of a single dipole out of the ordered ground state is
defined to be $E_{0}$. This is equivalent to having $2\left| X(k=0)\right|
=E_{0}$, which is the condition to have a gapless mode at $k\rightarrow 0$
\ref{ek}). Using this condition, the experimental value of $E_{0}=7$K \cite{schuster,allen} determines the size
of the coherent dipole moment as: $\left| {\bf \mu }\right| \simeq e\cdot 0.01{\rm
\AA }$. This value is indeed smaller than our previous estimation, which
served as an upper bound on the size of the oscillating dipole moment (\ref{mu}).

As we have proposed, the phase modulation in the (110) direction of the
transverse atomic motion in the lattice, with energy $E(k)$ (\ref{ek})
should coincide with the T$_{1}$(110) phonon. In Fig.4 we compare the
experimental values of T$_{1}$(110) taken from neutron scattering data with
the calculated $E(k)$, and we find that the agreement is excellent for all
k $. From (\ref{ek}) and Fig.3 we see that at the edge of the Brillouin zone
the energy $E(k)$ of the phonon should be just the bare energy of the local
mode, $E_{0}$, since $X(\sqrt{2}\pi /a)=0$. We also have that at $k=\sqrt{2
\pi /a$ the dipoles have changed between the two configurations illustrated
in Fig.3, which are the two possible configurations with alternating dipoles
arranged on adjacent planes with the periodicity of the bcc unit cell.

Since the empirical value of $E_{0}$ that we used was taken from NMR data,
the agreement we find with the phonon data taken from inelastic neutron
scattering, emphasizes the self-consistency of our description. We stress
that the value of $E_{0}$ and the lattice vectors are the only empirical
inputs used in the calculation, with the functional behavior completely
given by the lattice structure and the dipolar interactions.

\section{Off-Diagonal-Long-Range-Order and condensation.}

We have found from the mean-field solution at zero temperature that the
ground-state of the bcc phase contains a coherent-state of oscillating
local-dipoles (\ref{psi0}). Since our method predicts the excitation
spectrum of the T$_{1}$(110) phonon with very good accuracy, we expect it to
be valid at the finite temperatures for which the bcc phase exists. We
therefore expect that the basic nature of the bcc phase will be well
described by our results, although the quantitative values may change due to
the finite temperature. The coherent ground-state defines a global phase and
breaks the gauge symmetry of a well-defined occupation number of local
dipoles. In the limit $k\rightarrow 0$ we find that the occupation number of
the local-modes diverges as $1/k$, signaling macroscopic Bose-Einstein
condensation in the zero-momentum state
\begin{equation}
\left\langle n_{k}\right\rangle =v^{2}(k)=\frac{1}{2}\left( \frac{E_{0}{+X(k
}}{E(k)}-1\right) \rightarrow _{k\rightarrow 0}\frac{1}{2}\frac{E_{0}/2}{E(k
}=\frac{E_{0}/2}{2\hbar kc}  \label{nk}
\end{equation}

where $c$ is the sound velocity of the T$_{1}$(110) phonon which is the
natural excitation of the dipolar array. This is identical to the result for
a Weakly Interacting Bose Gas (WIBG) problem solved by Bogoliubov \cite{bogo
, where the divergence is related to the occupation of the zero-momentum
state, i.e., the condensate fraction $n_{0}/n$
\begin{equation}
\left\langle n_{k}\right\rangle _{WIBG}=v^{2}(k)\rightarrow _{k\rightarrow 0
\frac{n_{0}}{n}\frac{mc}{2\hbar k}  \label{nwibg}
\end{equation}

where in the WIBG case we have $c$ as the $k\rightarrow 0$ sound velocity,
and $\varepsilon _{k}=\hbar ^{2}k^{2}/2m$ is the free particle energy. By
comparing (\ref{nk}) with (\ref{nwibg}) we find that in the bcc case the
role of the condensate-fraction, the WIBG order-parameter, is taken by the
parameter $E_{0}$, which is just $2\left| X(0)\right| $. This can be seen
directly from the form of the ground-state wavefunction (\ref{psi0}) where
the pair-occupation is given by:
\begin{equation}
\left\langle b_{k}^{\dagger }b_{-k}^{\dagger }\right\rangle =2u(k)v(k)=\frac
X(k)}{E(k)}  \label{pair}
\end{equation}

Equating the divergent part in (\ref{nk}) and (\ref{nwibg}) we can define an
effective condensate fraction
\begin{equation}
\frac{n_{0}}{n}=\frac{E_{0}/2}{mc^{2}}\simeq \frac{3.5}{10}=35\pm 8\%
\label{nn0}
\end{equation}

where we used for the velocity of sound $c$ the values from our calculation
\ref{ek}) ($\sim 130m/\sec $) and from elastic constants \cite{minki} ($\sim
160m/\sec $). It must be remembered that the mass $m$ in (\ref{nn0}) is not
necessarily the mass of a bare $^{4}$He atom since we are now dealing with
condensation of local dipoles. Comparing with the condensate fraction at
zero temperature in superfluid $^{4}$He \cite{sokol}, which is $\sim 10\%$,
we find that it is lower than the condensate fraction of the local modes in
the bcc phase. We again mention that our result is for T=0 which can be
depleted at finite temperature.

It is clear that it is a non-zero coherent dipole moment $\mu $ that
produces a dipolar interaction matrix $X(k)$ which in turn implies finite
pair occupation (\ref{pair}) and a coherent ground-state. This is just the
condition for the Bose-Einstein condensation of the dipoles in the bcc
ground-state (\ref{nn0}). We therefore have a broken gauge symmetry and a
complex order-parameter in the form of the pair-occupation (\ref{pair}).
This function can be complex since the conditions on $u(k)$ and $v(k)$ allow
for a relative complex phase between them, just as in the WIBG case. A
similar condensation of local dipoles in all three orthogonal axes of local
motion means that there are three independent phases at each lattice site,
since orthogonal dipoles do not interact. The order-parameter in our case
can therefore be described as a vector of three complex functions of
independent magnitude and phase:
\begin{equation}
\Phi ({\bf r})=\left(
\begin{array}{l}
\left| \mu _{x}\right| e^{i\theta _{x}({\bf r})} \\
\left| \mu _{y}\right| e^{i\theta _{y}({\bf r})} \\
\left| \mu _{z}\right| e^{i\theta _{z}({\bf r})}
\end{array}
\right)  \label{order}
\end{equation}

If the cubic symmetry is not broken by external stresses, the magnitude
of the coherent dipole moment in the three orthogonal directions should
be the same: $\left| \mu _{x}\right|=\left| \mu _{y}\right|=\left| \mu _{z}\right|$.
In the ground-state the phases are spatially uniform, while the excited
state is described through a periodic phase oscillation, i.e. the T$_{1}
(110) phonon. The order parameter (\ref{order}) is to be contrasted with the
order parameter of superfluid $^{4}$He, which also exhibits ODLRO and which
has a single complex component.

In the hcp phase we do not expect the dipoles to order in a coherent state
since the hexagonal geometry frustrates antiferroelectric-type
configurations. Also the nearly isotropic potential of the hcp lattice
does not allow the highly directional dipole moments as in the bcc case. Indeed there is good agreement between experiments and the
harmonic calculation of the phonons in the hcp phase \cite{minki2}, indicating no strong
quantum corrections, as in the bcc phase.

The bcc $^{4}$He is therefore a unique crystallographic phase having both
Diagonal Long Range Order (DLRO) of the solid lattice and Off-Diagonal Long
Range Order (ODLRO) of the local dipoles. It is not a 'super-solid' \cite
{andreev,leggett,widom,stoof} in that it does not contain both a superfluid and
a solid, but is more similar to the superconductors which have a DLRO of the
atoms in the lattice and ODLRO\ of the superconducting electrons \cite{kohn
. This system is also distinct from the case of Bose-Einstein condensation
of a phonon mode which results in a static deformation of the lattice and a
structural phase-transition \cite{kohn}.

Bose-Einstein condensation of local defects (vacancies) was previously
considered for solid $^{4}$He \cite{andreev,leggett,widom}, and is similar
to our treatment. The main difference is that in our case the physical
picture of the condensed local modes is not a local distortion of the
lattice like a vacancy, and that the condensation is unique to the bcc
phase. The estimate in these works \cite{andreev,leggett,widom} is that in the
ground-state (T=0) the density of vacancies is $\sim 10^{-4}$ per site (at
molar volume of 21cm$^{3}$), and a condensate density of $\sim 10^{-7}$ per
site. In contrast we expect in the bcc phase a sizable fraction (10-30\%) of
condensed local-modes per site (\ref{nn0}).

The fact that the region of existence of the proposed supersolid phase in
the phase diagram of solid $^{4}$He should closely coincide with the region
occupied by the bcc phase, was shown in \cite{stoof}. In this work it was
further shown that in the supersolid there should be a second-sound-like
mode, which is an oscillation in the density of the local-defects. In our
description of the bcc phase this suggests the possibility of an oscillation
in the amplitude of the order-parameter $\Phi ({\bf r})$ (\ref{order}), that
is in the amplitude of the coherently oscillating dipole moment. This is in
contrast to the T$_{1}$(110) phonon mode which is an oscillation in the
phase of the order-parameter. This mode may be produced by modulating the
density of the atoms so that the local excitation energies change and with
them the amplitude of local motion and local electric dipole moment.
Unfortunately we do not expect such a mode to have measurable consequences
which are different from the effects produced by usual longitudinal phonons.

In concluding this section, we would like to mention the recent experiments
on the behaviour of implanted metalic ions (Cs) in solid $^{4}$He \cite
{kanorsky}. These experiments are designed to look for evidence of
time-reversal symmetry breaking which is equivalent to having a static
electric dipole moment. In our description of the bcc phase we do not find a
static but a coherent-dynamic electric dipole moment. We point out that in
these experiments a marked difference between the hcp and bcc phases has
been found. In the bcc phase the electronic-spin relaxation of the Cs atom
is extremely slow and this effect could be a result of the coherence and
long-range order of the dipolar fields. The coherently oscillating $^{4}$He
electrons in the bcc phase will produce a very uniform electromagnetic
interaction with the electronic spin in the Cs atom. By comparison, in the
hcp phase the spin polarization is extremely short lived, indicating a more
random field environment. This result is in accord with our expectation that
the coherent dipoles are unique to the bcc phase. Similar experiments in the
future may allow a probe that will show directly the coherently oscillating
dipoles in the bcc ground-state.

In these experiments \cite{kanorsky} the hyperfine transition in the Cs atom
was also measured. The energy shift of this transition is sensitive to the shape
of the confining cavity of the Cs atom inside $^{4}$He lattice. The width
of the transition is a measure of the fluctuations in this cavity size \cite
{kanorsky2}, and the data show a much smaller spread in the bcc compared
with the hcp phase. Uncorrelated atomic motions of the $^{4}$He atoms will
increase the spread in instantaneous cavity sizes due mainly to breating-like motion of the cavity walls (Fig.5).
This behavior is what we expect
for the hcp case. The correlated atomic motion in the bcc phase should
result in a more constant cavity shape (Fig.5) and a narrow signal, which is indeed measured \cite{kanorsky}.

\section{Ground-state energy and the stability of the bcc phase}

The question of the relative stability of the different crystal structures
in solid He has been a long standing one. The necessity for some
non-Van-der-Waals interactions has been previously proposed to explain the
occurance of fcc over hcp structure in the heavier rare-gas solids \cite
{venables}. The bcc phase is usually found to be more stable than the
close-packed hcp phase due to the large zero-point energy in the He solids
\cite{venables}. The correlations between the dipoles in the ground-state
that we have proposed, lowers the energy of the ground-state of the bcc
phase and further stabilizes it with respect to the hcp phase. The reduction
in ground-state energy acheived by the coherent state of the dipoles along
one of the three orthogonal directions, is
given by \cite{anderson}
\begin{equation}
\Delta E=\sum_{k}\frac{E(k)-(E_{0}+X(k))}{2E_{0}}<0  \label{dele0}
\end{equation}

which is negative since $X(k)<0$ and $E(k)<(E_{0}+X(k))$.

Since the energy reduction integrand (\ref{dele0}) is non-zero only in the
(110) directions it will give a small contribution to the three dimensional
phase-space integration. At zero temperature the summation in (\ref{dele0})
will be confined to one dimensional sections along the (110) direction, so
that the contribution will be zero. At the bcc temperatures ($\sim $1.4K)
the one dimensional chains in the (110) directions are broadened so that the
summation in (\ref{dele0}) is now over finite volume sections of phase
space. We can estimate the maximum width of the conical section in $k$-space
as the momentum which corresponds to a T$_{1}$(110) phonon with energy
k_{B}T$, that is $\sim 0.13$\AA $^{-1}$. The numerical integration of (\ref
{dele0}) over such volume sections gives an energy reduction of $\Delta
E\simeq -2$mK per atom. This result is in agreement with the experimentally
interpolated energy difference between the bcc and hcp phases of solid $^{4}
He \cite{balibar}, which is of the order of a few mK per atom.

This reduction is less than 0.1 percent of the potential and kinetic
energies of the solid, and is therefore very hard to calculate accurately
theoretically \cite{nosanow}. What is shown in the usual calculations is
that the correlations between the motions of the atoms are essential in
lowering the energy of the bcc phase, compared with the hcp phase. Since
part of the correlations in the atomic motion is described by our coherent
dipole model, we expect the condensation energy of the dipoles (\ref{dele0})
to be important in determining the stability of the bcc phase. At finite
temperature the stabilization of the bcc phase compared to the hcp phase is
usually attributed to the lower zero-point energy due to the lower T$_{1}
(110) phonon energy \cite{venables}. This is just the phonon which is
softened by the long-range dipolar interactions that we have described,
indicating again the importance of the coherent dipoles to the stabilization
of the bcc phase of solid $^{4}$He. Our procedure may provide a good
estimate of the small change in energy at the structural phase transition,
by isolating the degree-of-freedom which is most affected by the transition,
i.e. the correlated atomic motion along the directions normal to the unit
cell faces.

The picture we propose is that the dipole condensation mechanism of the bcc
phase competes with the lower potential energy of the hcp phase due to its
higher coordination number. If the hcp phase has a large enough volume
(through thermal expansion or introduction of $^{3}$He impurities) its
potential energy is increased until a critical point is reached where the
bcc phase has a lower total energy due to the dipolar-condensation energy
reduction (\ref{dele0}), which is absent in the hcp phase \cite{niremil}. At
this critical point the structural phase transition occurs.

By comparison, solid $^{3}$He has a stable bcc phase due to the larger
kinetic energy of this lighter isotope. This increased zero-point energy
causes the less dense bcc phase to have a lower ground-state energy than the
hcp phase even at T=0. Since the $^{3}$He atoms are fermions with a spin 1/2
nucleus, the oscillating electric dipoles are not in resonance and can not be
treated as bosons \cite{glyde1}. We therefore do not expect a coherent state
of the atomic motion as in the bcc $^{4}$He, and bcc $^{3}$He is stable due
to its large zero-point kinetic energy alone. On the other hand, at low
enough temperatures where the bcc $^{3}$He becomes an antiferromagnet, there
could be correlations involving both the nuclear spin and electric dipole
degrees of freedom.

\section{Fermionic excitations}

In addition to the fluctuations of the phase of the coherent dipole
ground-state (i.e. T$_{1}$(110) phonons), there can be a localized 'flip' of
a dipole so that it is in anti-phase (phase difference of $\pi $) relative
to the rest of the dipoles, in the ground-state configuration. Such an
excitation is naturally treated as a Fermion since such a flipped dipole is
antisymmetric with respect to the other dipoles, that is with respect to the
global phase $\theta $ (\ref{order}) in one (or more) of the orthogonal
directions of local motion ($x,y,z$). The flipped dipole is no longer a
dipolar image of the nearest-neighboring dipoles but an anti-image, and will
be treated with Fermi-Dirac statistics.

An anti-phase localized-mode (a fermion) is not part of the correlated
ground-state, but nevertheless will feel the effect of the Bose excitations
(T$_{1}$(110) phonons) of the dipolar array as they interact with it. The
effective Hamiltonian describing such a fermion should therefore contain a
term describing the creation and anihilation of pairs of fermions from the
ground-state by a phonon (Boson). This is an off-diagonal term that
describes the fluctuation caused by a T$_{1}$(110) phonon of energy $E(k)$:
it changes a fermion 'particle' into a 'hole' and vice versa. The terms
'particle' and 'hole' are with respect to the ground-state which has
occupation of pairs of localized-modes (i.e. not an 'empty' vacuum).

In addition there should be a term that describes the excitation energy of
the bare fermionic localized-mode, that is $E_{0}$. This is just the energy
to 'flip' a dipole from the ground-state so that all it's interactions with
the other dipoles of the ground-state change sign, i.e. $-2X(0)=E_{0}$.

The many-body effective Hamiltonian that we therefore propose is
\begin{equation}
H_{D}=\sum_{k}E(k)\left( c_{k}^{\dagger }c_{-k}^{\dagger
}+c_{k}c_{-k}\right) -\sum_{k}V_{k}\left( c_{k}^{\dagger
}c_{k}c_{-k}^{\dagger }c_{-k}\right)  \label{diracham}
\end{equation}

where $c_{k}^{\dagger },c_{k}$ are the creation and annihilation operators
of the anti-phase (Fermionic) localized-mode. The first term in (\ref
{diracham}) is the 'kinetic' term due to the phonon-roton branch, where the
localized-modes are created/anihilated in pairs. The energy $E(k)$ is the
energy of the T$_{1}$(110) phonon excitation (\ref{ek}). In addition there
is a finite 'potential' energy if there is a finite density of unpaired
fermions, which is $E_{0}$. In the absence of the second term we have just
the Bose ground-state written in terms of fermionic pairs.

We linearize the equations of motion that follow from (\ref{diracham}),
similar to the BCS method \cite{kittel}
\begin{equation}
i\hbar \stackrel{\cdot }{c}_{k}=-E(k)c_{-k}^{\dagger }+\Lambda
_{k}c_{k}\qquad i\hbar \stackrel{\cdot }{c}_{-k}^{\dagger
}=-E(k)c_{k}-\Lambda _{k}^{*}c_{-k}^{\dagger }  \label{eqmotion}
\end{equation}

where we used the Fermi anti-commutation relations: $\left\{
c_k,c_k^{\dagger }\right\} =1\qquad \left\{ c_k,c_{-k}^{\dagger }\right\} =0
, and we define
\begin{equation}
\Lambda _k=\Lambda _k^{*}\equiv E_0\equiv \sum_kV_k\left\langle c_k^{\dagger
}c_k\right\rangle ,\sum_k\left\langle c_k^{\dagger }c_k\right\rangle \equiv
1,V_k=E_0  \label{linear}
\end{equation}

From (\ref{linear}) we see that the symbol $E_0$ will now indicate a finite
density of fermions.

The equations of motion have the following eigenvalues:
\begin{equation}
\left|
\begin{array}{cc}
E_{f}(k)-E_{0} & -E(k) \\
-E(k) & E_{f}(k)+E_{0}
\end{array}
\right| =0\Rightarrow E_{f}(k)=\sqrt{E_{0}^{2}+E(k)^{2}}  \label{specdir}
\end{equation}

We can now solve the equations using the Bogoliubov-Valatin transformation
for superconductivity \cite{kittel}:
\begin{equation}
c_{k}=u_{k}\alpha _{k}+v_{k}\alpha _{-k}^{\dagger }\qquad c_{-k}^{\dagger
}=-v_{k}\alpha _{k}+u_{k}\alpha _{-k}^{\dagger }  \label{bogo}
\end{equation}

with the functions $u_{k},v_{k}$ given by
\begin{equation}
u_{k}^{2}=\frac{1}{2}\left( 1+\frac{E_{0}}{E_{f}(k)}\right) ,v_{k}^{2}=\frac
1}{2}\left( 1-\frac{E_{0}}{E_{f}(k)}\right)  \label{uvdir}
\end{equation}

The ground state is
\begin{eqnarray}
\alpha _{k}\left| 0\right\rangle &=&0\Rightarrow \left| 0\right\rangle
=\prod_{k}\alpha _{-k}\alpha _{k}\left| vac\right\rangle \nonumber \\
&=&\prod_{k}\left(
u_{k}+v_{k}c_{k}^{\dagger }c_{-k}^{\dagger }\right) \left|
vac\right\rangle
\label{diracgs}
\end{eqnarray}

In Fig.6 we plot the energy spectrum (\ref{specdir}) compared with the other
phonon modes in the (110) direction \cite{minki}. It is clear that this
optic-like branch should be detectable in the low momentum range where it is
not masked by the signal from the accoustic phonon modes. There is at
present no high resolution neutron-scattering data in this energy and
momentum range, and this prediction can be hopefully checked in future
experiments. This mode could also be observed by Raman scattering, as a peak
at energy $E_{0}$.

\section{Scattering intensity}

A\ neutron scattering inelastically from the solid will create/anihilate an
elementary excitation. An excitation from the effective ground-states of the
T$_{1}$(110) phonon (\ref{psi0}) and of the fermionic mode (\ref{diracgs})
involves an anihilation of a pair of local-modes, leaving an unpaired
local-mode. We therefore expect the experimentaly measured neutron
scattering intensity to be proportional to the density of local-mode
pair-occupation at each wavevector $k$. This gives us the following results
for the two modes:

T$_1$(110) phonon:
\begin{equation}
I\propto \left\langle {{b_k}^{\dagger }}{b_{-k}^{\dagger }}\right\rangle
\frac{E_0}{E(k)}\left| \left( \frac{E(k)}{{E_0}}\right) ^2-1\right|
\label{iphon}
\end{equation}

Fermionic excitation:
\begin{equation}
I\propto \left\langle c_{k}^{\dagger }c_{-k}^{\dagger }\right\rangle =\frac{
}{2}\frac{E(k)}{E_{f}(k)}=\frac{1}{2}\frac{1}{\sqrt{\left( E_{0}/E(k)\right)
^{2}+1}}  \label{idirac}
\end{equation}

Both functions (\ref{iphon}),(\ref{idirac}) are plotted with arbitrary scale
in Fig.7. We see that the intensity of the T$_{1}$(110) phonon is such that
at small $k$ it behaves as $1/k$ which is typical for phonons at low $k$ and
was seen experimentally \cite{minki2}, but goes identically to zero at the
edge of the Brillion zone where $E(k)\rightarrow E_{0}$. The fermionic
excitation has an opposite behavior by increasing linearly in intensity with
$k$, until it saturates at the edge of the Brillion zone. The expression for
the intensity of the T$_{1}$(110) phonon is similar to the expression of
the intensity of the phonon-roton excitation spectrum of superfluid $^{4}$He
\cite{nireric}, where it agrees very well with the experimental results.

These predictions for the bcc phase have yet to be checked experimentally.

\section{Conclusion}

In this work we have investigated the nature of the quantum correlations in
the bcc phase of solid $^{4}$He. We identified a three component complex
order parameter and Bose-Einstein condensation in this phase, though not a
'super-solid' \cite{kohn}, i.e. no superfluid component. There can be
further manifestations of the ODLRO of the dipoles in the bcc phase which we
have not explored yet, such as macroscopic topological defects in the
complex order-parameter. The order-parameter or condensate-fraction can also
serve as an extra thermodynamic variable, and this opens the possibilty of
more complicated internal dynamics in the bcc solid, such as the phenomenon
of second sound in superfluid $^{4}$He.

We predict that a local excitation of a dipole out of the coherent
ground-state will behave as a Fermion, and we calculate its energy spectrum.
We find it to behave as an optical-like branch in the (110) direction.
Finally we calculate the scattering intensity as a function of wavevector $k$
for both the Bose (T$_{1}(110)$ phonon) and Fermi (new optical-like branch)
excitations. All these predictions await high-resolution neutron and Raman
scattering experiments to be compared with.

{\bf Acknowledgements}

I thank Emil Polturak for useful discussions and encouragement.

This work was supported by the Israel Science Foundation and by the Technion
VPR fund for the Promotion of Research.

\newpage

\section{Appendix A: Comparison of Bose excitations with Klein-Gordon Hamiltonian.}

We would like to point out that the Hamiltonian describing
the localized dipoles (%
\ref{hloc}) is similar to the Klein-Gordon (KG) Hamiltonian
for a
single spinless boson, written in its first-order form \cite{baymq}:

Klein-Gordon:
\begin{equation}
H_{KG}=\varepsilon _k\left( \sigma _z+i\sigma _y\right) +mc^2\sigma
_z+e\Phi
\widehat{1}  \label{hamkg}
\end{equation}

Localized dipoles (\ref{hloc}):
\begin{equation}
H_{loc}=X(k)\left( \sigma _z+i\sigma _y\right) +E_0\sigma _z
\label{hamdip}
\end{equation}

where $\sigma _i$ are the Pauli matrices, $\varepsilon
_k=\widehat{p}^2/2m$, $%
e$ is the electric charge, $\Phi $ is the electrostatic potential, $m$
is
the KG-particle's mass and $c$ is the velocity of light.

We have written the dipolar hamiltonian (\ref{hamdip}) in the basis of a two
component wavefunction
\begin{equation}
\Psi _{loc}=\left(
\begin{array}{c}
c_k^{\dagger } \\
c_{-k}
\end{array}
\right)
\label{dipolewave}
\end{equation}

In this representation we see that exciting a local dipole out of the
ground-state configuration ($c_k^{\dagger }$) has bare energy $E_0$ while
destroying an excited dipole has minus this energy.
There is a freedom of choice weather to define the positive excitation to be a
flipping of an up dipole to a down dipole or vice versa. The sign of the energy of
the dipolar bosons therefore represents this freedom which corresponds to two
equivalent dipolar configurations with a $\pi$ phase difference.

The
two-component
wavefunction of the KG Hamiltonian is:
\begin{eqnarray}
\Psi _{KG}&=&\left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right) \nonumber \\
\varphi &=&\frac 12\left( \psi +\frac{i\hbar }{mc^2}\psi
^0\right)
,\chi =\frac 12\left( \psi -\frac{i\hbar }{mc^2}\psi ^0\right)
\label{kleinwave}
\end{eqnarray}

where $\psi $ is the original wavefunction of the second-order KG
equation,
and $\psi ^0=\left( \frac \partial {\partial t}+\frac{ie}\hbar \Phi
\right)
\psi $.
The Hamiltonians (\ref{hamkg},\ref{hamdip}) are the similar
except that the KG density is not normalized to 1 but to $\left\langle
\rho
\right\rangle =E/mc^2$, describing the relativistic increase in the
density
with velocity.
By comparing the two Hamiltonians (\ref{hamdip},\ref{hamkg}) we identify that:
$E_0\leftrightarrow mc^2$, $X(k)\leftrightarrow \varepsilon _k$,
which gives the equivalence of the two hamiltonians.

The peculiarities of the KG equation appear when there is a potential $%
V=e\Phi $ (the Klein paradox for example). The equation for the momentum
of
the KG prticle:
\begin{eqnarray}
2mc^2\varepsilon _k&=&\hbar ^2c^2k^2=\left( E(k)-V\right) ^2-\left(
mc^2\right)^2  \nonumber \\
\Rightarrow k&=&\frac{\sqrt{\left( E(k)-V\right) ^2-\left( mc^2\right)
^2}}{\hbar c}  \label{kgk}
\end{eqnarray}

becomes the equation for $X(k)$ in the dipolar case:
\begin{equation}
X(k)=\frac{\left( E(k)-V\right) ^2-\left( E(k) _0\right) ^2}{2E_0}
\label{xkv}
\end{equation}

We see from (\ref{xkv}) that there is a region of energies where the
interaction parameter $X(k)$ is positive and a region where it is
negative.
We saw above that the condensation of the dipoles in the bcc phase is characterized by a negative $X(k)$
which
also gives a gapless excitation
spectrum
at $k\rightarrow 0$. The excitations with $%
E(k)>V+E_0,E(k)<V-E_0$ are therefore not contributing
to
the coherent long-range order.
A fermionic excitation is a local destruction of the coherent order
, and indeed costs at least $E_0$ (for the free case with $V=0$)
to
create (\ref{specdir}).

In the case of the KG equation the sign of the enrgy indicates the charge of
the particle/antiparticle, which have oposite charges.
Charge conjugation therefore interchanges between the two.
What is the meaning of the different signs of the energy
of the dipolar bose excitations in the bcc case ? From our definition of the second-quantized
description of the dipoles, the meaning of the sign of the energy is that the field of
resonating localized-dipoles can have two global configurations shifted
by $%
\pi $ (Fig.2). These two configurations are identical with respect to the magnitude
of the energy
spectrum, but in each the operation of spin flip changes from
up$\rightarrow$down to down$\rightarrow$up. We can therefore identify two ''charges'' for the bcc
to
distinguish between the two shifted phases. Further we find that as in the KG
case the operation of charge conjugation (which reverses the signs of the
dipoles) moves us between the two solutions.

\section{Appendix B: Symmetry breaking of the fermionic excitations.}

We see from (\ref{uvdir}) that when there is no fermion present (i.e. if
we
put $E_{0}=0$ in (\ref{linear})) the ground state has equal numbers of
fermions and holes. The symmetry between particles and holes is broken
by
the free fermion quasiparticle (or quasihole), and the sign of the symmetry-breaking
parameter $%
E_{0}$ determines which of the two kinds is present. The hole/particle
are
with respect to the equilibrium occupation by pairs of fermions in the
ground-state.

A single flipped dipole described as the fermionic excitation, breaks
the
symmetry between the number of up/down dipoles and creates a residual globally
oscillating dipole moment. The parity $P$ symmetry with respect to
reflection along the axis of the global dipole (let us choose to be $z$)
is
broken. The charge $C$ symmetry is also broken since the direction of the global dipole
is
flipped under charge conjugation. The time reversal symmetry $T$ is
unbroken
since the oscillating globel dipole does not define a unique time
direction.
We therefore have that the global $CPT$ symmetry is preserved, as is the
$CP$
and $T$ symmetries individually. The symmetry-breaking parameter in
(\ref
{diracham}) is the sign given to $E_{0}$, which corresponds to choosing
an
up or down dipole to flip. In second quantization langauge this is the
choice between an unpaired particle or hole. The broken symmetry is not
of
the $U(1)$ group such as the $\Phi $ (\ref{order}) order parameter, but
has
a $Z(1)$ discrete symmetry.

We now compare this with the situation of the two-dimensional massive
Dirac
particle \cite{berry}. The Hamiltonian describing a single fermionic
excitation (\ref{diracham}) can be written as
\begin{eqnarray}
i\hbar \frac{\partial }{\partial t}\left(
\begin{array}{c}
c_{k} \\
c_{-k}^{\dagger }
\end{array}
\right) &=&
E(k)\left(
\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}
\right) \left(
\begin{array}{c}
c_{k} \\
c_{-k}^{\dagger }
\end{array}
\right)  \nonumber \\
&+&E_{0}\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \left(
\begin{array}{c}
c_{k} \\
c_{-k}^{\dagger }
\end{array}
\right)
\label{eqmotionmat}
\end{eqnarray}

while the 2D Dirac particle is described by
\begin{equation}
i\hbar \frac{\partial }{\partial t}\left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right) =c(\widehat{{\bf \sigma }}\cdot {\bf p})\left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right) +mc^{2}\widehat{\sigma }_{z}\left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right)  \label{diracmat}
\end{equation}

By assuming momentum ${\bf p}$ in the $\widehat{x}$-direction only we
write
\begin{eqnarray}
i\hbar \frac{\partial }{\partial t}\left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right) &=&c\left(
\begin{array}{cc}
0 & -i\hbar \partial _{x} \\
-i\hbar \partial _{x} & 0
\end{array}
\right) \left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right) \nonumber \\
&+&mc^{2}\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}
\right) \left(
\begin{array}{c}
\varphi \\
\chi
\end{array}
\right)
\label{diracmat2d}
\end{eqnarray}

where $\widehat{{\bf \sigma }}=\left( \widehat{\sigma
}_{x},\widehat{\sigma }%
_{y},\widehat{\sigma }_{z}\right) $, $\widehat{\sigma }_{i}$ the Pauli
matrices, and $\varphi ,\chi $ are the particle/antiparticle scalar
wavefunctions. There is now complete analogy between
(\ref{eqmotionmat})
and (\ref{diracmat2d}). The symmetry-breaking parameter $E_{0}$ is
identical
to the $mc^{2}$ parameter in the 2D Dirac equations-of-motion (\ref
{diracmat2d}). The symmetry that is broken by choosing a non-zero
$mc^{2}$
in two-dimensions is the time-reversal symmetry (TRS) \cite{berry}. As
shown
in (\ref{diracmat}), the time component of the momentum-energy vector in
two
dimensions is taken by the $z$ axis. The broken parity $P$ in the
$z$-axis
for the fermionic excitation of the bcc phase is here replaced by the
TRS
breaking of the heavy two-dimensional Dirac particle.

Similar to the Anderson \cite{andersonbcs} transformation of the BCS
problem
to a magnetic Hamiltonian we can transform (\ref{diracham}) using:
\begin{eqnarray}
n_k &=&c_k^{\dagger }c_k\qquad c_k^{\dagger }c_{-k}^{\dagger }=\sigma
_k^{-}/2\qquad c_kc_{-k}=\sigma _k^{+}/2  \nonumber \\
&\Rightarrow &n_kn_{-k}=\frac 12\left( \sigma _k^z+1\right) \qquad
c_k^{\dagger}c_{-k}^{\dagger }+c_kc_{-k}=\sigma _k^x
\label{magtrans}
\end{eqnarray}

where the $\sigma _k^i$ are Pauli spin-1/2 operators. The basis is such
that
an up-spin in the $\widehat{z}$-direction represents an empty pair,
while a
down-spin represents an occupied pair.

The resulting Hamiltonian is:
\begin{equation}
H_{mag}=\sum \varepsilon _k\sigma _k^x-\frac 12\sum V_k\left( \sigma
_k^z+1\right)  \label{hamag}
\end{equation}

This Hamiltonian describes a fictitious magnetic field acting on the
spin $%
\overrightarrow{\sigma }$:
\begin{equation}
\overrightarrow{B}=\varepsilon _k\widehat{x}-\frac 12E_0\widehat{z}
\label{mag}
\end{equation}

where we replaced the potential energy with the constant $V_k=E_0$
(\ref{linear}).
The magnetic field (\ref{mag}) can be compared with the BCS result
\cite{andersonbcs}
\begin{equation}
\overrightarrow{B}_{BCS}=\varepsilon _k\widehat{z}+\frac 12V\sum \left(
\sigma _k^x\widehat{x}+\sigma _k^y\widehat{y}\right)  \label{magbcs}
\end{equation}

The alignment of the spins in the ground-state is shown for the two
Hamiltonians in
Fig.8.
In the BCS problem the sign of the symmetry-breaking field in $V$ (\ref
{magbcs}) has to be positive so that it induces ferro-magnetic interaction
between the fictitious spins, otherwise there will not be any rotation of the spins
across the Fermi-energy. Only when the spins rotate do they go through
the
point where the spin is entirely in the $xy$-plane. At this point the
state
has no defined occupation number but a well defined phase, while on both
sides of the domain-wall there is well defined occupation and no phase.
This
superconducting-phase at the fermi energy is just the angle of the fictitious spin in the $xy$-plane,
and
there is broken U(1) symmetry.

In the Dirac case the symmetry-breaking field $E_{0}$ in (\ref{mag}) is
a
constant external field. It can have both signs, which control the
direction
along the $z$-axis that the rotated spin has in the middle of the
domain-wall. This spin describes weather a particle or a hole is
occupied,
while away from the $k=0$ point the spins are in the $xy$-plane, with a
well
defined phase. The symmetry that is broken is therefore the binary Z(1) ($\pm
$) symmetry.

In comparison with the BCS problem we see that in the fermionic excitation in
the bcc the
symmetry-breaking parameter is a finite density of unpaired fermions: $%
\left\langle c_{k}^{\dagger }c_{k}\right\rangle \neq 0$ (\ref{linear}). The
ground-state
without unpaired fermions is a 'vacuum' of pairs of particle-holes in
equal
numbers. In the BCS problem the symmetry-breaking parameter is a finite
pair-density: $\left\langle c_{k}^{\dagger }c_{-k}^{\dagger
}\right\rangle
\neq 0$. The ground-state in the absence of electron pairing is just a
finite density of electrons below the fermi-energy and zero above. In
this
respect the two problems are 'complementary'.


\section{Introduction}
The rich and high precision data  on deep inelastic $e p $ scattering
at HERA \cite{H1} \cite{ZEUS}, covering both low and high $Q^2$ regions,
lead to a theoretical problem of matching the non-perturbative (``soft")
and perturbative (``hard ") QCD  domains. This challenging problem has
been under close investigation over the past two decades, starting from
the pioneering paper of Gribov \cite{GRIBOV} ( see also \cite{GVD} ).

Based on Gribov's general approach, one can interpret two
 time sequence of the
 $\gamma^* p$ interaction (see \fig{fig1}):
\begin{enumerate}
\item\,\,\,First the $\gamma^*$ fluctuates  into  a hadron system
(quark-antiquark
  pair to  the lowest order) well before the interaction with the
target;

\item\,\,\,Then the  converted quark-antiquark pair (or hadron
system)
 interacts with the target.

\end{enumerate}
These two stages are expressed explicitly in the double dispersion relation
suggested in Ref. \cite{GRIBOV}:

\begin{equation}\label{GRIBOVgen}
\sigma(\gamma^* N)=\frac{\alpha_{em}}{3\pi}\int
  \frac{\Gamma(M^2) dM^2}{(Q^2+M^2)}
    \sigma(M^2,M'^2,s)
  \frac{\Gamma(M'^2) dM'^2}{(Q^2+M'^2)},
\end{equation}
where $M$ and $M'$ are the invariant masses of the incoming and outgoing
quark-antiquark pairs, $\sigma(M^2,M'^2,s)$ is the cross section of a
$q \bar q $ interaction with the target, and the vertices $\Gamma^2(M^2)$ and
$\Gamma^2(M'^2)$ are given by  $\Gamma^2(M^2)=R(M^2)$, where $R(M^2)$ is the
ratio:
\begin{equation}\label{Rdef}
R(M^2)=\frac{\sigma(e^+e^-\rightarrow \mbox{hadrons})}{\sigma(e^+e^-\rightarrow
\mu^+\mu^-)},
\end{equation}
which has beem measured experimentally.
 For large masses
($M,\, M' >> M_0$) we have
$\,\Gamma(M^2\,\geq\,M^2_0)\times \Gamma(M'^2\,\geq\,M^2_0 )\,\,
\longrightarrow
\,\,R( M^2
\,\geq\,\,M^2_0 ) \,=\, 2$, where we assume that that the  number of
colours $N_c = 3 $. $M_0$ is a typical mass (a separation
parameter), which is of the order
of $M_0 = 1  GeV $, determined directly from the experimental data
\cite{TABLE}.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=fig1sx.eps,width=80mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it The generalized Gribov's formula.}}
\end{center}
\label{fig1}
\end{figure}

The key problem in all approaches utilizing  \eq{GRIBOVgen} is the
 description of the cross section $\sigma(M^2,M'^2,s)$. In this paper we
follow the approach suggested  in Ref.  \cite{GLMepj98}, which is based on
the ideas
 of Badelek and Kwiecinski  \cite{BK}.  Below we list the
mains steps of this approach.
\begin{enumerate}
\item\,\,\,We introduce $M_0 \,\,\approx \,\,1\,GeV$ in the integrals over
$M$ and $M'$ which plays the role of a separation parameter.  For $M,\, M'
\,>\,M_0$ the quark - antiquark pair are produced at short distances (
$r_{\perp}\,\,\propto\,\frac{1}{M}\,\,<\,\,\frac{1}{M_0}$ ), while for $M,\,
M'\,<\,M_0$ the distance between quark and antiquark is too long
($r_{\perp}\,\,\propto\,\frac{1}{M}\,\,>\,\,\frac{1}{M_0} $), and we cannot
treat this $q \bar q $ - pair in pQCD. Actually, we cannot even describe the
produced hadron state as a $q \bar q $ - pair;

\item\,\,\,For $M,\,M'\,<\,M_0$ we use the Additive Quark Model
\cite{AQM} in which
\begin{eqnarray}
\sigma(M^2,M'^2,s) &=&\sigma^{soft}_N (M^2,s)
\,\delta(M^2\,-\,M'^2) \nonumber\\
 &=& \left(\,\sigma_{tot}(qN
)\,+\,\sigma_{tot}( \bar q  N )\,\right)\,\delta(M^2\,-\,M'^2)
\label{SOFT}
\end{eqnarray}
The above assumption allows us to simplify the Gribov formula of
\eq{GRIBOVgen}:

\begin{equation}\label{GRIBOVeq}
\sigma(\gamma^* N)=\frac{\alpha_{em}}{3\pi}\int
     \frac{R(M^2) M^2 dM^2}{\left(
Q^2+M^2\right)^2}\sigma_{N}(M^2,s)\,\,;
\end{equation}

\item\,\,\,For $M,\, M' \,>\,M_0$ we consider the system with mass $M$
and/or $M'$ as a short distance quark - antiquark pair, and describe
its interaction with the target in pQCD (see \fig{twogluons}).

The exact formulae for $\sigma^{hard}(M^2,M'^2,s)$ is given  below (see
also Ref. \cite{GLMepj98}), but one can see from \fig{twogluons},  that
this
interaction can be expressed through the gluon structure function, and it is
{\em not} diagonal with respect to the masses, contrary to the ``soft"
interaction of a hadron system with small mass.

\item\,\,\,In principle, the short distance between the quark and antiquark
  leads to short distances in the gluon-nucleon interaction. However, the
  typical distance of the gluon interaction $r_G\,\,\propto\,1/l_{\perp}$
   (see Figs 1 and 2) is larger than the size of the
  quark - antiquark pair
  and $l_{\perp} \,\,<\,\,{k_{\perp}}$. It turns out that the calculation with
  $M_0\,\, \sim\,\,1\,GeV$,   demands a new scale in the gluon-nucleon
  interaction.  We introduce it, assuming that the gluon structure function
  $xG(x,l^2_{\perp})$ behaves as
  $xG(x,\mu^2)\,\times\,\frac{l^2_{\perp}}{\mu^2}$ for
  $l^2_{\perp}\,\leq\,\mu^2$.  This means that we assume that the gluon-hadron
  total cross section is not equal to zero for long distances (in the ``soft"
  kinematic region). It should be stressed that we introduce two scales for
  soft nonperturbative interactions, distinguishing between quark and gluon
  interactions. The two scales appear naturally in our models for ``soft",
  nonperturbative interaction.  For example, in the AQM which we use for
  $\sigma^{\scrbox{soft}}$, these two scales are the size of a hadron (distance between
  the
  quark and antiquark), and the size of the constituent quark which is
  related to gluon interaction scale.

\item\,\,\, As has been discussed, we use the AQM [see \eq{SOFT}] to calculate
$\sigma^{\scrbox{soft}}$. For  the  vector meson cross section the AQM
leads to

\begin{equation}\label{RCRSEC}
\sigma_{\scrbox{R}}\,\,=\frac{1}{2}\left(\,\sigma(\pi^+ p)\,\,+\,\,\sigma(\pi^-p)\right).
\end{equation}
For the pion-proton cross section, we use the Donnachie - Landshoff
Reggeon parameterization \cite{DL}, with an energy variable which is
appropriate for the interaction of a hadronic system of mass $M$ with the
target (see \cite{GLMepj98} for details).
\begin{equation}
\label{DL}
\sigma_{M^2N}(s) = \dlformula,
\end{equation}
with
\begin{eqnarray} \label{alphapom}
{\alpha_{\pom}}  &=& 1.079 \nonumber \\
{\alpha_{\regg}} &=& 0.55 \nonumber \\
x_M        &=& \frac{M^2}{s} \nonumber \\
A          &=& 13.1\, \mbox{mb} \nonumber \\
B          &=& 41.08\, \mbox{mb}\, .
\end{eqnarray}
The values of $A$ and $B$ are chosen so that (\ref{DL}) is valid for the
$\rho$-proton interaction. Using (\ref{DL}) in Gribov formula
(\ref{GRIBOVeq}) we derive the soft transverse contribution to $\sigma(\gamma^* p)$:
\begin{equation}\label{soft2}
   \sigma^{\scrbox{soft}}_{\scrbox{T}} = \softformula{M^2}\left\{\dlformula\right\}.
\end{equation}

\end{enumerate}

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=fig2sx.eps,width=90mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it ``Hard" contribution to
$\sigma(M^2,M'^2,s)$ in perturbative QCD .}}
\end{center}
\label{twogluons}
\end{figure}

It has been shown in \cite{GLMepj98} that  even though the simple
model
discussed above reproduces the main features of the experimental
data, it has several deficiencies which we wish to address:
\begin{itemize}
\item The calculations of the ``soft'' cross section appeared
to overestimate the
data, and to overcome this,  the soft contribution was multiplied by a
factor
 $\kappa =0.6$. There is no physical justification for
such a small value.
\item The contribution of a longitudinal polarized virtual photon to $\sigma(\gamma^* p)$
      was neglected.
\item An old GRV'94 parameterization was used for the gluons distribution
      inside the proton. This gave an energy dependence of
      $\sigma(\gamma^* p)$ which is steeper than the published data.
\end{itemize}

In the present paper we  reexamine these points developing a
formalism which also takes into account $\sigma_{\scrbox{L}}$, the longitudinal part
of $\sigma(\gamma^* p)$, for both the soft and the hard components. We find that the
contribution of $\sigma_{\scrbox{L}}$ significantly improves the energy dependence of
$\sigma(\gamma^* p)$, in a good agreement with the data for $x < 10^{-2}$.

A similar approach to photon - proton interaction was also developed in
 Ref.\cite{MRS} where quite different scales have been introduced. The goal
 of such an approach  (see Refs.\cite{GLMepj98},\cite{BK} and
\cite{MRS}) is
 to find parameters that separate the long distance (non-perturbative)
 and short distance (perturbative) interactions in QCD. A similar
 philosophy to ours is used in Ref.\cite{BW}, where an attempt was made to
 describe photon - proton interactions, assuming that the main contribution
 stems from short distances, and the long distance interaction appears as
  a result of the  shadowing corrections.

The description of $\sigma( \gamma^* p )$ can be achieved in quite a
  different way using the generalized vector dominance model\cite{GVD} or
  a Regge motivated fit of the experimental
  data\cite{ALLM}\cite{ALLM97}. In these
  approaches one looses the explicit connection with the microscopic
  theory, which makes futher theoretical interpretation of the results rather
  difficult.

\section{Description of the Model \label{secMODEL}}
\subsection{The Contribution of a Transverse Polarized Photon \label{secT}}
\begin{figure}[tbp]
\begin{center}
  \epsfig{file=fig3.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it Diagrams which cntribute to the scattering
             of a $\bar{q}q$ pair off the target proton.}
             \label{fig3}}
\end{center}
\end{figure}
As we have discussed previously, the main  assumption of our model is
that
$\sigma(M^2,M'^2,s)$ in \eq{GRIBOVgen} can be calculated as \begin{equation} \label{S+H}
\sigma(M^2,M'^2,s)=\sigma^{soft}_N(M^2\,<\,M^2_0,s)\,\, \delta( M^2 - M'^2 )+
\sigma^{hard}(M^2\,>\,M^2_0,M'^2\,>\,M^2_0,s)\,\,.  \end{equation} In the previous
section we  have discussed
the calculation of  $\sigma^{\scrbox{soft}}$. Now we present,
for completeness,  our formulae for $\sigma^{\scrbox{hard}}$ which have been given in
Ref.~\cite{GLMepj98}. The pQCD contribution for $\sigma_{\scrbox{T}}$, within the framework
of the two gluon model, is illustrated in the four diagrams  shown in
Fig.~\ref{fig3}.  The production amplitude ${\cal M}_{{\lambda\lambda'}}$ of the
$\bar{q}q$, can be factorized into the wave function $\psi_{\lambda\lambda'}$ of the
$\bar{q}q$ system inside the virtual photon, and the amplitude ${\cal T}_{\lambda\lambda'}$ for
the scattering of the $\bar{q}q$ pair off the proton:
\begin{equation}\label{mlamlam1}
{\cal M}_{\lambda\lambda'} ({k_{\perp}},z) = \sqrt{N_c}\int d^2 {k_{\perp}}'\int_0^1 dz'
                         \psi_{\lambda\lambda'} ({k_{\perp}}',z'){\cal T}_{\lambda\lambda'} ({k_{\perp}}',z';{k_{\perp}},z)\, .
\end{equation}
The transition ampiltude ${\cal T}_{\lambda\lambda'}$ for the four contributions is:
\begin{eqnarray}\label{tlamlam}
\lefteqn{{\cal T}_{\lambda\lambda'} ({k_{\perp}}', {k_{\perp}}) =} & & \\
& &      i \frac{4\pi s}{2N_c} \int \frac{d^2 \ell_{\perp}}{\ell_{\perp}^4}
              \biggl [ 2 \delta (\mbox{\boldmath $k$}_{\perp}'    -
                                 \mbox{\boldmath $k$}_{\perp})    -
                         \delta (\mbox{\boldmath $k$}_{\perp}'    -
                                 \mbox{\boldmath $k$}_{\perp}     -
                                 \mbox{\boldmath $\ell$}_{\perp}) -
                         \delta (\mbox{\boldmath $k$}_{\perp}'    -
                                 \mbox{\boldmath $k$}_{\perp}     +
                                 \mbox{\boldmath $\ell$}_{\perp} )
              \biggr ] \alpha_{\rm S} (\ell^2_{\perp}) f(x, \ell^2_{\perp})\, , \nonumber
\end{eqnarray}
where,
\begin{equation}\label{ddxg}
f(x,\ell^2)=\frac{\partial xG(x,\ell^2)}{\partial\ln\ell^2}
\end{equation}
(see \cite{LMRT} for details).
Substituting (\ref{tlamlam}) in (\ref{mlamlam1}) we have,
\begin{equation}\label{mlamlam2}
{\cal M}_{\lambda\lambda'} = i\frac{2\pi^2 s}{\sqrt{N_c}}\int\frac{d\ell^2_{\perp}}{\ell_{\perp}^4}
                  \alpha_{\rm S}(\ell^2_{\perp}) f(x,\ell^2_{\perp})\Delta\psi_{\lambda\lambda'} \, ,
\end{equation}where the definition of $\Delta\psi$ is to be understood from~
\eqs{mlamlam1}{mlamlam2}:
\begin{equation}\label{Deltapsi}
\Delta\psi = 2\psi(\mbox{\boldmath $k$}_{\perp},z) -
              \psi(\mbox{\boldmath $k$}_{\perp} - \mbox{\boldmath $\ell$}_{\perp},z) -
              \psi(\mbox{\boldmath $k$}_{\perp} + \mbox{\boldmath $\ell$}_{\perp},z)\, .
\end{equation}
$\psi_{\lambda\lambda'} $ has been calculated in
\cite{LMRT} both for a transverse and for a longitudinal  polarised incoming
photon. The wave function for a transverse polarized
photon and the amplitude for producing $\bar{q}q$ with helicities $\lambda$ and
$\lambda'$ read:
\newcommand{\lint}[2]{\int\frac{d #1^2}{#1^4}
                               \alpha_{\rm S}(#2 ^2) f(x,#2 ^2)}
\newcommand{\lintarg}[3]{\left\{ \frac{1}{#3^2+#2^2} -
                                \frac{1}{\sqrt{\left( #3^2 +
                                                       #2^2 +
                                                      #1^2\right)^2-
                                                4 #2^2 #1^2}} \right\}}
\newcommand{\lintargb}[3]{\left\{ \frac{1}{#3^2+#2^2} -
                                  \frac{1}{2 #2^2} +
                            \frac{#3^2-#2^2+#1^2}{2 #2^2\sqrt{\left( #3^2 +
                                                       #2^2 +
                                                      #1^2\right)^2-
                                                4 #2^2 #1^2}} \right\}}
\newcommand{\lintargc}[3]{\left\{ \frac{#2^2 - #3^2}{#2^2 + #3^2 } +
                            \frac{#3^2-#2^2+#1^2}{\sqrt{\left( #3^2 +
                                                       #2^2 +
                                                      #1^2\right)^2-
                                                4 #2^2 #1^2}} \right\}}
\begin{eqnarray}
\psi^{\pm}_{\lambda\lambda'}\left({\mathbf {k_{\perp}}} ,z\right)
   &=& -\delta_{\lambda , -\lambda'} Z_f e
                       \left[(1-2z)\lambda\mp 1\right]
                 \frac{{2\mathbf\epsilon_{\pm}\cdot {k_{\perp}}}}{\bar{Q}^2+{$k^2_{\perp}$}} ,
                 \label{psilamlamt} \\
{\cal M}^{\pm}_{\lambda\lambda'} &=&
 -2 Z_f e \left( i\frac{2\pi^2 s}{\sqrt{N_c}}\right)
 \delta_{\lambda , -\lambda'} \left[(1-2z)\lambda\mp 1\right]
 {\mathbf\epsilon_{\pm}\cdot {k_{\perp}}}
                               \lint{\ell_{\perp}}{\ell_{\perp}}\,\times \nonumber \\
                              & & \hspace{1cm}\lintargb{\ell_{\perp}}{{k_{\perp}}}{\bar{Q}}
                               . \label{mlamlamt}
\end{eqnarray}
In (\ref{psilamlamt}) and (\ref{mlamlamt}), $Z_f$ is the charge of the quark
with flavour $f$ in units of the electron charge $-e$, $\bar{Q}^2\equiv
z(1-z)Q^2$ and $\epsilon_{\pm}$ denotes  the photon polarization
vector
presented in a circular basis,
{
\begin{equation}
\mathbf{\epsilon_{\pm}}=\frac{1}{\sqrt{2}}\left(0,0,1,\pm i\right).
\end{equation}

To evaluate the cross section one should, first, sum over the quark
helicities
$\lambda$ and $\lambda'$, and   average over the two transverse
polarisation
states $(\pm)$ of the photon
\begin{eqnarray}
|{\cal M}^T|^2 &=& \frac{32\pi^4 s^2}{N_c} Z^2_f e^2
                   \left[z^2+(1-z)^2\right]
                               \lint{\ell_{\perp}}{\ell_{\perp}}\,\times \nonumber \\
                            & & \hspace{1cm}\lintargc{\ell_{\perp}}{{k_{\perp}}}{\bar{Q}} .
\end{eqnarray}
The cross section is obtained by integration over $z$ and over ${k_{\perp}}$,
\begin{eqnarray}
\sigma_{\scrbox{T}} &=&
\frac{\alpha_{\scrbox{em}}}{4\pi^2 N_c}\int^1_0 dz\left[z^2+(1-z)^2\right]
                       \int\frac{d {$k^2_{\perp}$}}{\bar{Q}^2+{$k^2_{\perp}$}}
                       \lint{\ell_{\perp}}{\ell_{\perp}}\,\times \nonumber \\
                         & & \hspace{3cm}\lintargc{\ell_{\perp}}{{k_{\perp}}}{\bar{Q}} .
\end{eqnarray}
Following \cite{GLMepj98}, in order to perform  the integration over
$z$,
we introduce the variables $M$ and $\tilde M$:
\begin{eqnarray}
M^2 &=& \frac{{$k^2_{\perp}$}}{z(1-z)} \nonumber \\
\tilde M^2 &=& \frac{\ell^2_{\perp}}{z(1-z)} \label{mmtdef}\, ,
\end{eqnarray}
and rewrite the integrals of $\sigma_{\scrbox{T}}$ in terms the variables $M$,
$\mathbf{\tilde M}$ and
{  $\mathbf{\ell_{\perp}}$,}
\newcommand{\sqrt{1-4\frac{\lts}{\mt^2}}}{\sqrt{1-4\frac{\ell^2_{\perp}}{\tilde M^2}}}
\newcommand{1-2\frac{\lts}{\mt^2}}{1-2\frac{\ell^2_{\perp}}{\tilde M^2}}
\begin{eqnarray}
\sigma_{\scrbox{T}} &=&
\frac{\alpha_{\scrbox{em}}}{4\pi^2 N_c}\int\frac{d M^2}{Q^2+M^2}\int\frac{d \tilde M^2}{\tilde M^2}
\int\frac{d \ell^2_{\perp}}{\ell^2_{\perp}}\frac{1-2\frac{\lts}{\mt^2}}{\sqrt{1-4\frac{\lts}{\mt^2}}}
\alpha_{\rm S}(\ell^2_{\perp}) f(x,\ell^2_{\perp})\,\times \nonumber \\
                         & & \hspace{3cm}\lintargc{\tilde M}{M}{Q} .
\label{EQsight2}
\end{eqnarray}
Using a generalization
of the Gribov approach, we take a cutoff on the integration over $M^2$ and
insert the ratio (\ref{Rdef}) inside the integration sign,
\begin{eqnarray}
\sigma^{\scrbox{hard}}_{\scrbox{T}} &=&
\frac{\alpha_{\scrbox{em}}}{2\pi^2 N_c}\int^{\infty}_{M^2_0}
\frac{d M^2\,\,R(M^2)}{Q^2+M^2}\int\frac{d \tilde M^2}{\tilde M^2}
\int\frac{d \ell^2_{\perp}}{\ell^2_{\perp}}\frac{1-2\frac{\lts}{\mt^2}}{\sqrt{1-4\frac{\lts}{\mt^2}}}
\alpha_{\rm S}(\ell^2_{\perp}) f(x,\ell^2_{\perp})\,\times \nonumber \\
                         & & \hspace{1cm}\lintargc{\tilde M}{M}{Q} .
\label{EQsight3}
\end{eqnarray}
The last integral in (\ref{EQsight3}) can be integrated by parts, using
(\ref{ddxg}). In the limit $4\ell^2_{\perp}\ll\tilde M^2$, the value of the integral is
dominated by the upper limit of the integration, therefore we replace $\ell^2_{\perp}$
in $\alpha_{\rm S}(\ell^2_{\perp})$ and $xG(x,\ell^2_{\perp})$ with $\tilde M^2/4$, and obtain, for three
colors
\begin{eqnarray}
\label{hardT}
\sigma^{\scrbox{hard}}_{\scrbox{T}} &=& \frac{2\pi\, \alpha_{em}}{3}
                  \int^{\infty}_{M^2_0}
                  \frac{d M^2 \,\,R(M^2)}{Q^2\,+\,M^2}\,\,
                  \int^{\infty}_{0}\frac{d{\tilde M}^2}{{\tilde M}^4}
                  \alpha_{\rm S}(\frac{{\tilde M}^2}{4})\,x\,G(x,\frac{{\tilde M}^2}{4})
                  \times
\nonumber \\
& & \hspace{2cm}
\left\{\,\,\frac{M^2\,-\,Q^2}{M^2\,+\,Q^2}\,\,+\,\,\frac{Q^2\,+\,{\tilde
M}^2\,-\,M^2}
{\sqrt{(\,Q^2\,+\,M^2\,+\,{\tilde
M}^2\,)^2\,-\,4\,M^2\,{\tilde M}^2}}\,\,\right\}\,\,.
\end{eqnarray}
In the case of heavy quarks, we assume that the quark is heavy enough so that
 any contribution to the soft cross-section can be neglected. The heavy
 quarks contribution is then written with the replacements: $ 4\,M^2\,{\tilde
 M}^2 \rightarrow 4\,(M^2-4m_Q^2)\,{\tilde M}^2$, and $R(Q^2) \rightarrow
 R^{QQ}(M^2)$. $R^{QQ}$ is the heavy quark contribution to the ratio
 (\ref{Rdef}) and $m_Q$ is the mass of the heavy quark,
\begin{eqnarray}
\label{hardTQQ}
\sigma^{\scrbox{hard}}_{\scrbox{T},\bar{Q}Q} &=& \frac{2\pi\, \alpha_{em}}{3}
                  \int^{\infty}_{M^2_0}
                  \frac{d M^2 \,\,R^{QQ}(M^2)}{Q^2\,+\,M^2}\,\,
                  \int^{\infty}_{0}\frac{d{\tilde M}^2}{{\tilde M}^4}
                  \alpha_{\rm S}(\frac{{\tilde M}^2}{4})\,x\,G(x,\frac{{\tilde M}^2}{4})
\nonumber \\
& & \mbox{\hfill}
\left\{\,\,\frac{M^2\,-\,Q^2}{M^2\,+\,Q^2}\,\,+\,\,\frac{Q^2\,+\,{\tilde
M}^2\,-\,M^2}
{\sqrt{(\,Q^2\,+\,M^2\,+\,{\tilde
M}^2\,)^2\,-\,4\,(M^2-4m_Q^2)\,{\tilde M}^2}}\,\,\right\}\,\,.
\end{eqnarray}
In the above formulae $x=x(M^2)=(Q^2+M^2)/W^2$ with $W$ being the
 center of mass  energy of the photon-nucleon interaction.

\subsection{The Contribution of a Longitudinal Polarized Photon \label{secSIGL}}

\subsubsection{``Soft'' Contribution to $\sigma_L$ \label{subsecSIGSL}}
A priori, it is straight forward to write the formula for the ``soft''
 component of $\sigma_{\scrbox{L}}$ in AQM. The result is similar to our model for $\sigma_{\scrbox{T}}$,
 except that $M^2$ in the numerator should be replaced with the photon
 virtuality. This factor causes the longitudinal cross section to vanish for
 $Q^2 \rightarrow 0$ (see (\ref{alphapom}) for the values of the parameters).
\begin{equation}\label{softl}
   \sigma^{\scrbox{soft}}_{\scrbox{L}} = \softformula{Q^2}\left\{\dlformula\right\}.
\end{equation}

However, it turns out that the AQM over estimates the experimental data, and
 it has to be reduced with some phenomenological procedure, such as a
 numerical factor \cite{SchSpi} or a phenomenological function which
 decreases with $Q^2$~\cite{MRS}.  In our formalism the ratio between the AQM
 and pQCD contributions depends on the separation parameter $M_0$. Thus, by
 lowering the value of $M_0$ the ``soft'' component is suppressed. We have
 found that for the longitudinal part of the cross section, taking any value
 of $M_0$ which is below the resonances mass fits the experimental data, as
 opposed to the transverse cross section where we used $0.7 < M_0^2 < 0.9
 {\,\mbox{GeV}^2}$ .

\subsubsection{``Hard'' Contribution to $\sigma_L$}
We calculate the pQCD contribution for $\sigma_{\scrbox{L}}$, from the diagrams  of
Fig.~\ref{fig3}, using \eqs{mlamlam1}{Deltapsi}. For the case of a
longitudinal polarized incoming photon, we use the expression derived in
\cite{LMRT}  for the wave function
\begin{equation}
\psi^{\scrbox{L}}_{\lambda\lambda'} = -2\delta_{\lambda , -\lambda'} Z_f e Q z(1-z)
                 \frac{1}{\bar{Q}^2+{$k^2_{\perp}$}} , \label{psiL}
\end{equation}
and substitute it in (\ref{mlamlam2}) to obtain the amplitude for a
longitudinal photon to produce $\bar{q}q$ with helicities $\lambda$ and
$\lambda'$
\begin{eqnarray}
{\cal M}_{\lambda\lambda'}^{\scrbox{L}} &=& -4\delta_{\lambda , -\lambda'} Z_f e Q z(1-z)
                               i\frac{2\pi^2 s}{\sqrt{N_c}}
                               \lint{\ell_{\perp}}{\ell_{\perp}}\,\times \nonumber \\
                              & & \hspace{4cm}\lintarg{\ell_{\perp}}{{k_{\perp}}}{\bar{Q}}
                               .
\label{mlamlaml}
\end{eqnarray}
The cross section is then obtained by squaring, summing over the final
helicities and integrating over $z$ and ${k_{\perp}}$,
\begin{eqnarray}
\sigma_{\scrbox{L}} &=& \frac{16 \pi\alpha_{\scrbox{em}}}{N_c}\sum Z_f^2
                           Q^2 \int_0^1 dz \left[ z(1-z) \right]^2
                           \int \frac{d{$k^2_{\perp}$}}{\bar{Q}^2+{$k^2_{\perp}$}}
                           \lint{\ell_{\perp}}{\ell_{\perp}} \times \nonumber \\
                           & & \hspace{4cm}\lintarg{\ell_{\perp}}{{k_{\perp}}}{\bar{Q}}.
\label{EQsighl1}
\end{eqnarray}
Changing the  integration variables ${$k^2_{\perp}$}$ and $\ell^2_{\perp}$ to $M$ and
$\tilde M$,
respectively, using (\ref{mmtdef}), we have
\begin{eqnarray}
\sigma_{\scrbox{L}} &=& \frac{16 \pi\alpha_{\scrbox{em}}}{N_c}\sum Z_f^2
           Q^2 \int_0^1 dz
               \int \frac{dM^2}{Q^2+M^2}
               \lint{\tilde M}{z(1-z)\tilde M} \times \nonumber \\
      & &      \hspace{4.5cm} \lintarg{\tilde M}{M}{Q} \, .
\label{EQsighl2}
\end{eqnarray}
We now write the formula for $\sigma^{\scrbox{hard}}_{\scrbox{L}}$, in the same way that we did
in section \ref{secT}, by taking a cutoff on the integration over $M^2$, and
insert the ratio (\ref{Rdef}) inside the integration sign,
\begin{eqnarray}
\sigma^{\scrbox{hard}}_{\scrbox{L}} &=& \frac{8 \pi\alpha_{\scrbox{em}}}{N_c}Q^2\int_{M_0^2}^\infty
           \frac{R(M^2)dM^2}{Q^2+M^2}\int_0^\infty\frac{d\tilde M^2}{\tilde M^4}
           \int_0^1 dz f\left(x,z(1-z)\tilde M^2\right)
                       \alpha_{\rm S}\left(z(1-z)\tilde M^2\right) \times
           \nonumber \\
       & & \hspace{4.5cm}\lintarg{\tilde M}{M}{Q}\, .
\label{EQsighl3}
\end{eqnarray}
Recalling (\ref{ddxg}), we can integrate \eq{EQsighl3} by parts and obtain
(for $N_c=3$),
\newcommand{\sqrt{(Q^2+M^2+\mt^2)^2-4 M^2 \mt^2}}{\sqrt{(Q^2+M^2+\tilde M^2)^2-4 M^2 \tilde M^2}}
\begin{eqnarray}
\sigma^{\scrbox{hard}}_{\scrbox{L}} &=& \frac{8\pi\alpha_{\scrbox{em}}}{3}Q^2\int_{M_0^2}^\infty
           \frac{R(M^2)dM^2}{Q^2+M^2}\int_0^\infty\frac{d\tilde M^2}{\tilde M^4}
           \, \overline{xG}\left(x,\tilde M^2\right)\times
           \nonumber \\
       & & \hspace{2cm}
           \left\{\frac{1}{Q^2+M^2} - \frac{1}{\sqrt{(Q^2+M^2+\mt^2)^2-4 M^2 \mt^2}} \,- \nonumber \right.\\
       & & \hspace{5.5cm}\left.
           \frac{\tilde M^2\left(Q^2+\tilde M^2-M^2\right)}
                {\left(\sqrt{(Q^2+M^2+\mt^2)^2-4 M^2 \mt^2}\right)^3}\right\},
\end{eqnarray}
where,
\begin{equation}
\overline{xG}\left(x,\tilde M^2\right) \equiv \int_0^1
                                        xG\left(x,z(1-z)\tilde M^2\right)
                                        \alpha_{\rm S}\left(z(1-z)\tilde M^2\right) dz\, .
\end{equation}
For $\sigma^{\scrbox{hard}}_{\scrbox{L},\bar{Q}Q}$, the contribution of heavy quarks to the longitudinal
component of $\sigma(\gamma^* p)$, we replace
$ 4M^2{\tilde M}^2 $ by $4(M^2-4m_Q^2){\tilde M}^2$ and $R(Q^2)$ with
$R^{QQ}(M^2)$, as we did for the transverse part:
\newcommand{\sqrt{(Q^2+M^2+\mt^2)^2-4(M^2-4m_Q^2) \mt^2}}{\sqrt{(Q^2+M^2+\tilde M^2)^2-4(M^2-4m_Q^2) \tilde M^2}}
\begin{eqnarray}
\sigma^{\scrbox{hard}}_{\scrbox{L},\bar{Q}Q} &=& \frac{8\pi\alpha_{\scrbox{em}}}{3}Q^2\int_{M_0^2}^\infty
           \frac{R^{QQ}(M^2)dM^2}{Q^2+M^2}\int_0^\infty\frac{d\tilde M^2}{\tilde M^4}
           \, \overline{xG}\left(x,\tilde M^2\right)\times
           \nonumber \\
       & & \hspace{2cm}
           \left\{\frac{1}{Q^2+M^2} - \frac{1}{\sqrt{(Q^2+M^2+\mt^2)^2-4(M^2-4m_Q^2) \mt^2}} \,- \nonumber \right.\\
       & & \hspace{4cm}\left.
           \frac{\tilde M^2\left(Q^2+\tilde M^2-M^2\right)}
                {\left(\sqrt{(Q^2+M^2+\mt^2)^2-4(M^2-4m_Q^2) \mt^2}\right)^3}\right\}\, .
\label{hardLQQ}
\end{eqnarray}

\section{Comparison With The Experimental Data}
We now present the results for our calculation of $\sigma(\gamma^* p)$ from the
 master equation
\begin{equation}\label{sigmatotal}
\sigma(\gamma^* p) = \sigma^{\scrbox{soft}}_{\scrbox{T}} + \sigma^{\scrbox{hard}}_{\scrbox{T}} + \sigma^{\scrbox{hard}}_{\scrbox{T},\bar{Q}Q} + \sigma^{\scrbox{soft}}_{\scrbox{L}} + \sigma^{\scrbox{hard}}_{\scrbox{L}} + \sigma^{\scrbox{hard}}_{\scrbox{L},\bar{Q}Q} \, .
\end{equation}
As stated, our model has three parameters, namely, $M_{0,L}, M_{0,T}$ and
 $\mu$. In \cite{GLMepj98} the relatively high value of $M_0=\sqrt{5}$ GeV
 was taken, in order to give an energy dependence that was in a reasonable
 agreement with the published experimental results. We found that if the hard
 longitudinal contributions are not neglected, the transverse separation
 parameter can be reduced to the value of $0.7 < M_{0,T}^2 < 0.9 \,{\,\mbox{GeV}^2}$,
 while the longitudinal one can be taken to be $M_{0,L}^2 \;\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}\; 0.4
 \,{\,\mbox{GeV}^2}$. The results of the calculation are stable if the separation
 parameters are chosen within these bounds.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=grvmrst.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it Comparison of  the gluon
             distribution
             parametrizaions which were used in Eq.\ \mbox{(\ref{sigmatotal})}
             to calculate $\sigma(\gamma^* p)$.  The full triangles correspond to data
             points
             which have been extracted in \cite{ZEUS98} from experimental
              data.}
             \label{comparexg}}
\end{center}
\end{figure}

In the calculation of the hard components of $\sigma(\gamma^* p)$, we need to specify the
 input gluon distribution $xG(x,Q^2)$ which appears in the formulae. We have
 tried several options: GRV'94 \cite{GRV94}, GRV'98 \cite{GRV98} and MRST'98
 \cite{MRST}. We compare the results of each input distribution for the
 calculation of $\sigma(\gamma p)$ in \fig{comparexg}, where it is obvious
 that the best parameterization for our purpose is MRST'98 which yields,
 together with our formalism, a good description of the data.

A difficulty in our calculation comes from the integration at very low
 $M^2$, where the published parameterizations for $xG$ are not valid.
 Naturally one has two possible options: the first is to impose a low cutoff
 on the integration below $M^2=\mu^2$. \,   The second option is to use
the general
 property of the gluon structure function which is linear in $Q^2$ at very
 small values of $Q^2$, and to rely on this property for the low mass region:
\begin{equation}\label{mu}
xG(x,l^2<\mu^2) = \frac{l^2}{\mu^2}xG(x,\mu^2)\, .
\end{equation}

In our calculation for the low mass integration we used~(\ref{mu}) for the
 gluon distribution. The coupling $\alpha_{\rm S}(Q^2)$ was also kept fixed for
 $Q^2<\mu^2$. The value of $\mu^2$ was taken as the minimal value which is
 allowed in the input parameterization that had been used:
 $\mu^2=0.4,0.8,1.25\,{\,\mbox{GeV}^2}$ for GRV'94, GRV'98 and MRST'98
 respectively. Notice that for the MRST parameterization $\mu^2=1.25{\,\mbox{GeV}^2} >
 M_0^2$. This means that the transition from soft to hard is not sharp, and
 we still have (\ref{mu}) as a soft signature inside our hard formalism at
 $M_0^2<M^2<\mu^2$.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=sigmatotw2.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it $\sigma(\gamma^* p)$ as a function of $W^2$ together
              with the
             experimental data. The full triangles correspond to data points
             which have been extracted in \cite{ZEUS98} from experimental data.}
             \label{stotw2}}
\end{center}
\end{figure}
In \fig{stotw2} we show  the results of our calculation for $\sigma(\gamma^* p)$ as
 a function of
 $W^2$ for fixed values of $Q^2$, together with the published experimental
 results. The  calculated results are shown as a band which
 corresponds to
 the two limits on $M_{0,T}$. It is clear from the figure that the width of
 the band decreases with increasing $Q^2$. As for the longitudinal separation
 parameter, the results are not sensitive to the choice of $M_{0,L}$, and in
 all our calculations we simply used $M_{0,L} = 2 m_\pi$.  The thick line
 marks the region of small $x$: to the left of it,  $x$ is not small
enough to
 justify using our model.  One can see that for $x < 10^{-2}$, our results
 reproduce the experimental results both in value and in energy dependence.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=sigmatotfixedw.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it $\sigma(\gamma^* p)$ as a function of $Q^2$ together
              with the
             experimental data.}
             \label{stotq2}}
\end{center}
\end{figure}

 In \fig{stotq2} we show our calculations for $\sigma(\gamma^* p)$ for fixed values
of $W^2$,
as a
 function of $Q^2$, scaled by factors of $1-128$. The
small
 vertical thick lines delineate  the boundary $x = 10^{-2}$, to the
right of
 the marks our results are not reliable.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=longtranfixedw.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it The ratio of $\displaystyle{\frac{\sigma_{\scrbox{L}}}{\sigma_{\scrbox{T}}}}$ as
             a function of $x$.}
             \label{longtran}}
\end{center}
\end{figure}

The longitudinal component of $\sigma(\gamma^* p)$, according to our calculation, reaches a
 maximum value of 25{\%\ } of the total cross section at $x\approx
 10^{-2}$. For small $x$, this result is closer to the published data
 \cite{NMC} than the  results of Ref.\ \cite{MRS}, where they
 obtained $\sigma_{\scrbox{L}}$ for
 $x\approx 10^{-2}$ to be 15{\%\ } of the total cross section. The ratio
 of  the longitudinal cross section to the transverse cross section, as a
 function of $x$, is shown in \fig{longtran} together with the (small $x$)
 data of Ref.\ \cite{NMC}.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=heavylight.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it The ratio of
             $\frac{\sigma^{\scrbox{heavy}}}{\sigma^{\scrbox{light}}}$ at fixed
             $Q^2$.  }
             \label{heavylight}}
\end{center}
\end{figure}
\fig{heavylight} illustrates the importance of the heavy quarks contributions
 $\sigma^{\scrbox{hard}}_{\scrbox{T},\bar{Q}Q}$ and $\sigma^{\scrbox{hard}}_{\scrbox{L},\bar{Q}Q}$ [see Eqs.\ (\ref{hardTQQ}) and
 (\ref{hardLQQ})]. The contribution has a mild $W^2$-dependence but it is
 $Q^2$-dependent, and for large $Q^2$, the ratio
 $\sigma^{\scrbox{heavy}}/\sigma^{\scrbox{light}}$ gets as high as $0.3 -
 0.4$ . Another ratio which we present is the ratio, of the hard to the
 soft contributions. Since $M_{0,L}$ is smaller than the lightest resonance
 mass, the AQM contribution to the longitudinal component is supressed, and
 therefore what we call ``soft'' contribution is actually the transverse AQM
 calculation.

\begin{figure}[tbp]
\begin{center}
  \epsfig{file=softhard.ps,width=140mm}
  \caption[]{\parbox[t]{
             0.80\textwidth}{\small \it The ratio of
             $\frac{\sigma^{\scrbox{soft}}}{\sigma^{\scrbox{hard}}}$ at fixed
             $Q^2$.  }
             \label{softhard}}
\end{center}
\end{figure}

In \fig{softhard} we present the ratio
 $\sigma^{\scrbox{soft}}/\sigma^{\scrbox{hard}}$ at fixed values of $Q^2$,
 where a clear power law behaviour as a function of $W^2$ can be seen. In
 soft processes, the hard contribution is minor, but is still present even for
 relatively small values of $Q^2$. The soft signature in hard processes is
 also present, but is smaller. For high virtualities it decreases from
 a few precent at low energy,  to less than 1{\%\ } at high energies.

\section{Conclusions}

In this paper, we have shown that an approach based on Gribov's proposal,
 provides a successful description of the experimental data on
photon-proton interaction, over a  wide range of photon virtualities $0\le
Q^2\le 100\,{\,\mbox{GeV}^2}$, and energies $3\le\sqrt{s}\, (= W) \le 300$ GeV\@.  The
key assumption on which our aproach is based, is that the non-perturbative
and the perturbative QCD conributions in the Gribov formula can be separated
by the parameter $M_0$.  Oue successful reproduction of the experimental data
(see Figs.\ \ref{stotw2} and~\ref{stotq2}) shows that this assumption appears
to be valid. It further lends credence to using the additive quark model
(AQM) to describe the non-perturbative contribution.  The successful use of
the AQM leads to an improved result presented  in this paper, compared to
our previous
result\cite{GLMepj98}, where we found it necessary to introduce an damping
factor for  the AQM.

The second important by-product of our calculation is the simple method
we have used to separate
between non-perturbative (``soft") and perturbative (``hard")
contributions. We have used three separation parameters:
$M_{0,T}$, $M_{0,L}$ and $\mu$. The values of these parameters, $M^2_{0,T} =
0.7 - 0.9 \,{\,\mbox{GeV}^2}$, $M^2_{0,L}\leq 0.4\,{\,\mbox{GeV}^2} $ and $\mu^2\approx 1\,{\,\mbox{GeV}^2}$,
were determined by fitting to the experimental data.
We
believe that these values may be useful in the future,  for more
theoretical
description of the matching between ``soft" and ``hard" processes in QCD.

The third result which we find interesting, is that the GRV parameterizations
of the structure functions cannnot successfully describe the experimental
energy dependence of the total cross section at small values of $Q^2$ (see
\fig{comparexg}),  while the MRST parameterizations can.  This result
shows the
interdependence of deep inelastic scattering data, and the theoretical
description of the mathching of the ``soft" and ``hard" contributions.

In addition  we obtain:

\begin{enumerate}
\item
 A description of the ratio $\sigma_{\scrbox{L}}/\sigma_{\scrbox{T}}$ (see \fig{longtran})
 which is in good agreement with the experimental data and other
 approaches. The fact that $\sigma_L$ appears to have only a ``hard"
 contribution should be stressed, as this could be a possible window for
 particular features of non-pertutbative QCD;

\item
 A considerable contribution of the ``soft" processes at rather large
 values of photon virtualities $Q^2$. For example at $Q^2 = 15\,{\,\mbox{GeV}^2}$, $\sigma^{\scrbox{soft}}/\sigma^{\scrbox{hard}}
 \,\approx\,\,4\%$ at $W= 10 \,GeV$ (see \fig{softhard}).
  We also find, at low $Q^2$,
 a contamination of the ``soft" processes by the ``hard" ones. For high
 energy real photoproduction this contamination amounts to
 about 5{\%\ }.
 We believe that this fact should be taken into account when
 interpreting experimental data, especially as far as their
 energy dependance is concerned;

\item
 The prediction of the cross section for the heavy quark production (see
 \fig{heavylight}) which is almost $W$ independent, while showing a steep $Q^2$
 decrease.

\end{enumerate}

We propose a simple and successful
phenomenological  model for the photon-proton interaction, which provides a
method for matching the ``soft" and ``hard" interactions at high
energy, and which can be a guide for future theoretical approaches.
It is worthwhile mentioning, that even in the present form
our model can be useful for determining the initial parton distributions at
leading twist, for the DGLAP evolution equations.

{\bf Acknowledgments:}

We thank A. Martin and A.  Stasto for useful discussions of the results of
this paper, and for providing us the kumac file for Fig.6. This research
was supporteed in part by the Israel Science Foundation, founded by the
Israel Academy of Science and Humanities.