text:
\section{Introduction}

The problem of studying nuclear matter with chiral Lagrangians is not
trivial; for instance models based on the linear $\sigma$-model fail
to describe nuclear matter already at $\rho \sim \rho_0$ because the
normal solution in which chiral symmetry is broken becomes unstable
respect to the Lee-Wick phase. In Ref.~\cite{R.J.Furnstahl1993}
Furnstahl and Serot conclude that the failure of the $\sigma$-model is due to the
restrictions on the scalar field dynamics imposed by the Mexican hat
potential. A possible solution to this problem is still to use a linear realization, but
with a new potential, which includes terms not present in the Mexican
hat potential.  A guideline in building such a potential is scale
invariance~\cite{E.K.Heide1994, G.W.Carter1998, G.W.Carter1997, G.W.Carter1996}.

In QCD the invariance under dilatation is spontaneously broken due to the presence of
the parameter $\Lambda _{QCD}$ coming from the renormalization
process.  Formally, the non conservation of the dilatation current is
strictly connected to a not vanishing gluon
condensate:
\begin{equation}\label{current}
\langle \partial_{\mu}j^{\mu}_{QCD} \rangle =\dfrac{\beta(g)}{2g}\langle F^{a}_{\mu \nu}(x)F^{a\mu \nu}(x)\rangle .
\end{equation}

In the approach of Schechter~\cite{J.Schechter1980}, and of Migdal and Shifman~\cite{Migdal:1982jp}
 a scalar field representing the gluon
condensate is introduced and its dynamics is regulated by a potential
chosen so that it reproduces (at Mean-Field level) the divergence of
the scale current that in QCD is given by Eq.~(\ref{current}). The
potential of the dilaton field is therefore determined by the
equation:
\begin{equation}\label{dil}
\theta_\mu ^\mu = 4 V(\phi)-\phi \dfrac{\partial V}{\partial \phi}=4
\epsilon_{vac} \left (\dfrac{\phi}{\phi_0}\right )^4
\end{equation}
 where the parameter $ \epsilon_{vac}$ represents the vacuum energy.
To take into account massless quarks a generalization was proposed in
Ref.~\cite{E.K.Heide1992}, so that also chiral fields contribute to
the trace anomaly. In this way the single scalar field of
Eq.~(\ref{dil}) is replaced by a set of scalar and pseudoscalar fields $\lbrace \sigma,
\boldsymbol{\pi}, \phi \rbrace$.

It has already been shown that an hadronic model based on this
dynamics provides a good description of nuclear physics at densities
about $\rho_0$ and it describes the gradual restoration of chiral
symmetry at higher densities~\cite{Bonanno:2008tt}. In the same work
the authors have shown a phase diagram, where the interplay between chiral
and scale invariance restoration lead to a scenario similar to that
proposed by McLerran and Pisarski in~\cite{L.McLerran2007}.
This is not too surprising since the large $N_c$ limit explored in
\cite{L.McLerran2007} should be well represented in chiral models as the
one discussed in \cite{Bonanno:2008tt}.
It is therefore tempting to explore the scenario presented in
~\cite{Bonanno:2008tt} at a more microscopic level.

\textit{The new idea we develop in this work is to interpret the
fermions as quarks, to build the hadrons as solitonic solutions of the
fields equations as in~\cite{Birse:1986qc, Kahana:1984be} and, finally,
to explore the properties of the soliton at finite density using the
Wigner-Seitz approximation.}

Similar approaches to a finite density system have been investigated
in the past~\cite{Achtzehnter:1985ur, N.K.Glendenning1986, D.Hahn1987,
Birse:1987pb, Birse:1991cx, U.Weber1998, P.Amore2000}.  A problem of
those works is that the solitonic solutions are unstable and disappear
already at moderate densities when e.g. the linear $\sigma$-model is
adopted~\cite{U.Weber1998}.  We are therefore facing an instability
similar to the one discussed and solved when studying nuclear matter
with hadronic chiral Lagrangians.
The first aim of our work is \textit{to check whether the inclusion of the
logarithmic potential allows the soliton crystal to reach higher
densities.} The second and more important aim is \textit{to check
whether the inclusion of vector mesons in the dynamics of the quarks
can provide saturation for chiral matter.}

We should remark that the version of this model without
vector mesons has already been studied
at zero density in~\cite{AbuShady:2010zza}, but using a different
technique to describe the single nucleon. Instead no calculation
exists with the vector mesons. Moreover, both versions of the model
are unexplored at finite density.

The structure of the paper is as follows. In Sec.~\ref{model} we
describe the model we are using, in Sec.~\ref{hedgehog} we review the
Mean-Field approximation and the hedgehog ansatz while in Sec.~\ref{proiezione}
we present the technique adopted for projection.
Later in Sec.~\ref{hedgedensity} we describe the Wigner-Seitz approximation used to mimic
a system at finite density.
 Next in
Sec.~\ref{resultsvac} and in Sec.~\ref{hedgefindensity} we present our results, firstly for the soliton in vacuum
both at
Mean-Field level and by adopting a projection technique and then
 we show the results for the
Wigner-Seitz lattice of solitons. Finally, in Sec.~\ref{conclusions}
we present our conclusions and future outlooks.

\section{The model}\label{model}

In a chiral quark-soliton model quarks are coupled to mesons in a chirally invariant way.
Following Refs.~\cite{E.K.Heide1994,G.W.Carter1998,G.W.Carter1997,G.W.Carter1996,Bonanno:2008tt}
we consider the Lagrangian:
\begin{align}\label{lagr}
\mathcal{L}_0  =& \bar{\psi}  \big(i \gamma ^\mu\partial _{\mu}-
g_\pi(\sigma + i \boldsymbol{\pi}\cdot \boldsymbol{\tau}\gamma _{5}))\psi \nonumber\\
  &+\dfrac{1}{2}(\partial_{\mu}\phi \partial^{\mu}\phi +
\partial_{\mu}\sigma \partial^{\mu}\sigma + \partial_{\mu}\boldsymbol{\pi}\cdot \partial^{\mu}\boldsymbol{\pi})\nonumber\\
  &-V(\phi,\sigma,\mathbf{\pi}).
\end{align}
Here $\psi$ is the quark field, $\sigma$ and $\pi$ are the chiral fields and $\phi$ is
the dilaton field which, in the present calculation, is kept frozen at its vacuum value $\phi _0$.

An extension of this model,
already discussed in Refs.~\cite{E.K.Heide1994,G.W.Carter1998,G.W.Carter1997,G.W.Carter1996,Bonanno:2008tt},
 is to add the dynamics of vector mesons
and to incorporate the idea of universal coupling~\cite{Sakurai1966}.
This can be achieved by considering the vector mesons as massive gauge fields,
following also the scheme of Ref. \cite{W.Broniowski1986}.
The new Lagrangian is given by:
\begin{align} \label{lagrVM}
\mathcal{L}_{VM} & =\bar{\psi}  \big(i \gamma ^\mu\partial _{\mu}-g_{\pi}(\sigma + i \boldsymbol{\pi}\cdot
\boldsymbol{\tau}\gamma _{5}) +g_{\rho}\gamma ^\mu \dfrac{\boldsymbol{\tau}}{2}\cdot
(\boldsymbol{\rho} _{\mu}+\gamma _{5} \boldsymbol{A} _{\mu})\nonumber \\
& -\dfrac{g_{\omega}}{3} \gamma ^\mu \omega _{\mu}\big)\psi
  +\dfrac{\beta}{2}(D_{\mu}\sigma D^{\mu}\sigma + D_{\mu}
  \boldsymbol{\pi}\cdot D^{\mu}\boldsymbol{\pi})\nonumber \\
& -\dfrac{1}{4}(\boldsymbol{\rho}_{\mu \nu}\cdot \boldsymbol{\rho}^{\mu \nu}+
\boldsymbol{A}_{\mu \nu}\cdot \boldsymbol{A}^{\mu \nu} + \omega_{\mu \nu} \omega ^{\mu \nu})\nonumber \\
&   +\dfrac{1}{2}m_{\rho}^2(\boldsymbol{\rho}_{\mu} \cdot \boldsymbol{\rho}^{\mu}
+\boldsymbol{A}_{\mu} \cdot \boldsymbol{A}^{\mu})+\dfrac{1}{2}m_\omega ^2 \omega _\mu \omega ^\mu \nonumber\\
& -V(\phi_0,\sigma,\mathbf{\pi})
\end{align}
where $\omega_\mu$ is a vector-isoscalar coupled to baryon current,
 $\boldsymbol{\rho}_\mu$ and $\boldsymbol{A}_\mu$ are respectively a vector-isovector
  and an axial-vector- isovector fields coupled to isospin and axial-vector current.
The covariant derivatives for the chiral fields and the field tensors for vector mesons read:
\begin{align}
&D_\mu \sigma =\partial_\mu \sigma + g_\rho \boldsymbol{A}_\mu \cdot \boldsymbol{\pi}\nonumber ,\\
&D_\mu \boldsymbol{\pi} =\partial_\mu \boldsymbol{\pi} +
g_\rho (\boldsymbol{\rho}_\mu \wedge \boldsymbol{\pi} - \boldsymbol{A}_\mu \sigma)\nonumber ,\\
 &\omega_{\mu\nu} = \partial_\mu \omega_\nu-\partial_\nu \omega_\mu \nonumber,\\ &\boldsymbol{\rho}_{\mu\nu}=\partial_\mu \boldsymbol{\rho}_\nu-\partial_\nu \boldsymbol{\rho}_\mu+
 g_\rho(\boldsymbol{\rho}_\mu \wedge \boldsymbol{\rho}_\nu + \boldsymbol{A}_\mu \wedge \boldsymbol{A}_\nu)\nonumber ,\\
& \boldsymbol{A}_{\mu\nu}=\partial_\mu \boldsymbol{A}_\nu-
\partial_\nu \boldsymbol{A}_\mu + g_\rho(\boldsymbol{\rho}_\mu \wedge \boldsymbol{A}_\nu + \boldsymbol{A}_\mu \wedge \boldsymbol{\rho}_\nu).
\end{align}
The pion field mixes with the longitudinal component of the $a_1$~\cite{Gasiorowicz:1969kn, Ko:1994en}
 and so to ensure that the pion gets its physical mass once the two fields are decoupled, we need to introduce the constant:
\begin{equation}\label{beta}
\beta= \dfrac{m_\rho ^2}{m_\rho ^2 -g_\rho ^2 f_\pi ^2}.
\end{equation}

The potential is given by:
\begin{align} \label{pot}
V(\phi,&\sigma,\mathbf{\pi}) =  B\phi_0 ^4\left( \ln \dfrac{\phi}{\phi_0} -\dfrac{1}{4}\right ) -
\dfrac{1}{2}B \delta \phi^4 \ln \dfrac{\sigma^2 +
\boldsymbol{\pi}^2}{\sigma_{0} ^2}\nonumber\\
  +& \dfrac{1}{2}B\delta \phi^2\dfrac{\phi_0 ^2 }{\sigma^2_0}
\left (\sigma^2 + \boldsymbol{\pi}^2-\phi^2 \dfrac{\sigma^2_0}{2 \phi_0 ^2}\right ) \\
  -&  \dfrac{1}{4} \epsilon_{1} \left (\dfrac{\phi}{\phi_0}\right)^2 \left[\dfrac{4\sigma}{\sigma _{0}}
 -2 \left (\dfrac{\sigma^2 + \boldsymbol{\pi}^2}{\sigma_0^2}\right)-
\left(\dfrac{\phi}{\phi_0}\right)^2\right]-V_0\nonumber
\end{align}
where the logarithmic term generates from~(\ref{dil}). The first two
terms of the potential are responsible for the breaking of scale
invariance, while the second line is needed to ensure that in the
vacuum $\phi=\phi_0$, $\sigma=\sigma_0$ and $\boldsymbol{\pi}=0$, i.e
it provides spontaneous symmetry breaking. The last line
explicitly breaks the chiral invariance of the lagrangian, giving mass to the pion.

In this work we assign the following values to the masses of
 bare fields: $m_\pi =139$ MeV, $m_\rho=m_A=776$ MeV and $m_\omega=782$ MeV.
 For the sigma field, since there are no experimental constraints,
 we use $m_\sigma =550$ MeV and $m_\sigma=1200$ MeV which are typical values used in nuclear physics
if the sigma has to provide the intermediate range attraction.
We relate the quark-omega and the quark-rho couplings to the corresponding
couplings with the nucleons. We keep fixed $g_\rho=4$ and we vary $g_\omega$
between $10$ and $13$. The pion-quark coupling constant $g_\pi$
will vary from $3.9$ to $5$.
By varying the coupling constants in the ranges indicated above we will
be able to tune the attractive and the repulsive interactions, in order to
obtain saturation when studying the total energy of the soliton at finite
density.
The constants $B$ and $\phi_0$ can be fixed by choosing a value for the
mass of the glueball and for the vacuum energy $\epsilon_{vac}$, while
$\delta=4/33$ is provided by the QCD beta function and it corresponds
to the relative weight of the fermionic and of the gluonic degrees of
freedom.  Finally the constant $V_0$ ensures that the potential energy
is vanishing in the vacuum.

As anticipated, we choose to keep the dilaton field frozen.
This decision (which obviously
simplifies the dynamics of the system) can be justified by the results
obtained in~\cite{Bonanno:2008tt, G.W.Carter1998}, where it has been
shown that at low temperatures the dilaton remains close to its vacuum
value even at large densities.  Therefore the degrees of freedom of
our model are limited to quarks, chiral fields and vector mesons, similarly to the
linear $\sigma$-model whose solitonic solutions have been obtained
in~\cite{Birse:1986qc,W.Broniowski1986}. The potential can be written
in a simpler form and it reads:
\begin{align} \label{potfro}
& V(\sigma,\mathbf{\pi}) =  \lambda_1 ^2 (\sigma^2 +\boldsymbol{\pi}^2)-\lambda_2 ^2\,\ln(\sigma^2 +
\boldsymbol{\pi}^2)-\sigma_0 m_\pi ^2 \sigma
\end{align}
where:
\begin{align}
& \lambda_1 ^2=\dfrac{1}{2}\dfrac{B \delta \phi_0 ^4+\epsilon_1}{\sigma_0 ^2}=\dfrac{1}{4}(m_\sigma ^2+m_\pi ^2)\\
& \lambda_2 ^2=\dfrac{1}{2}B \delta \phi_0 ^4= \dfrac{\sigma_0 ^2}{4}(m_\sigma ^2-m_\pi ^2)\\
& \epsilon_1 =m_\pi ^2 \sigma_0 ^2.
\end{align}
Here the vacuum value $\sigma_0$ is fixed to be equal to $f_\pi =93$
MeV.

It is interesting to compare the logarithmic with the Mexican hat
potential. In Fig.~\ref{comppot} it can be seen that in the case of the
Mexican hat potential it is relatively easy to restore chiral symmetry
by climbing the maximum located at the center.  This is not
possible in the case of the logarithmic potential as long as the
dilaton field remains frozen. Since only at large temperatures the
dilaton field changes significantly~\cite{Bonanno:2008tt}
we can expect that at large
densities and moderate temperatures this model provides more stable
solitonic solutions.
This is a crucial question which will be investigated in our paper.

An important point in our approach is that we aim at describing
the dynamics of nuclear matter by incorporating all the interactions
already at a quark level. This is at variance with e.g. the
approach of Ref.\cite{Smith:2003hu, PhysRevC.70.065205} where the vector field $\omega$
was introduced only at the nucleon level, but was not
present in the dynamics of the quarks.

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{potential.eps}
\caption{Comparison between the logarithmic (solid line) and the Mexican hat potential (dashed line).}\label{comppot}
\end{figure}

\section{Mean-Field and hedgehog ansatz}\label{hedgehog}

The Euler-Lagrangian equations that follow from the Lagrangian
given in eq.~(\ref{lagrVM}) read:

\begin{align}
& [i\slashed\partial-g_{\pi}(\sigma + i \boldsymbol{\pi}\cdot \boldsymbol{\tau}\gamma _{5})
+g_{\rho} \dfrac{\boldsymbol{\tau}}{2}\cdot (\boldsymbol{\slashed\rho}
+\gamma _{5} \boldsymbol{\slashed A} )-\dfrac{g_{\omega}}{3} \slashed\omega ]\psi = 0 \nonumber,\\
& \beta \partial _\mu D^\mu \sigma = -\beta g_\rho \boldsymbol{A} _{\mu}\cdot \boldsymbol{\pi}
-g\bar{\psi}\psi -\dfrac{\partial V}{\partial \sigma}\nonumber,\\
& \beta \partial _\mu D^\mu \boldsymbol{\pi} = \beta g_\rho (-\boldsymbol{\rho} _{\mu} \times D^\mu \boldsymbol{\pi}
+\boldsymbol{A} _{\mu}D^\mu \sigma)-i g \bar{\psi}\boldsymbol{\tau}\gamma_5
\psi-\dfrac{\partial V}{\partial \boldsymbol{\pi}}\nonumber,\\
& -\partial ^\mu \boldsymbol{\rho}_{\mu \nu}=g_\rho \boldsymbol{\upsilon}_\nu
+m_\rho ^2\boldsymbol{\rho}_{\nu}\nonumber ,\\
& -\partial ^\mu \boldsymbol{A}_{\mu \nu}=g_\rho \boldsymbol{a}_\nu
+m_\rho ^2\boldsymbol{A}_{\nu}\nonumber ,\\
& -\partial ^\mu\omega_{\mu \nu}=-\dfrac{1}{3}g_\omega \bar{\psi}\gamma_\nu\psi +m_\omega ^2\omega_{\nu}.
\end{align}
Here  $\boldsymbol{\upsilon}_\nu$ and $\boldsymbol{a}_\nu$ are the vector and the axial-vector currents:
\begin{align}
&\boldsymbol{\upsilon}_\nu = \boldsymbol{\rho}_{\mu}\times \boldsymbol{\rho}_{\mu \nu}
+\boldsymbol{A}_{\mu}\times \boldsymbol{A}_{\mu \nu}+
\beta \boldsymbol{\pi}\times D^\nu \boldsymbol{\pi}+\bar{\psi}\gamma_\nu \dfrac{\boldsymbol{\tau}}{2}\psi,
\end{align}
\begin{align}
\boldsymbol{a}_\nu = \boldsymbol{\rho}_{\mu}\times & \boldsymbol{A}_{\mu \nu}
+\boldsymbol{A}_{\mu}\times \boldsymbol{\rho}_{\mu \nu}+\beta \boldsymbol{\pi}\times D^\nu \sigma-\beta\sigma D_\nu \boldsymbol{\pi}\nonumber\\
& +\bar{\psi}\gamma_5\gamma_\nu \dfrac{\boldsymbol{\tau}}{2}\psi.
\end{align}

The fields equations for the model without vector mesons, provided by the Lagrangian in eq.~(\ref{lagr}),
 can be obtained by the previous ones by fixing $g_\rho=g_\omega=0$.\\

The previous equations are relations between quantum fields.
The starting point of our calculation is the Mean-Field approximation,
where mesons are described by time-independent, classical fields and
where powers and products of these fields are replaced by powers and
products of their expectation values.  The quark spinor in the
spin-isospin space is:
\begin{equation}
\psi=\dfrac{1}{\sqrt{4 \pi}}\left (\begin{array}{c}
u(r)\\
iv(r)\boldsymbol{\sigma} \cdot \boldsymbol{\hat r}\end{array}\right) \chi_h
\end{equation}
where the spinor $\chi_h$,
defined as:
\begin{equation}
\chi_h= \dfrac{1}{\sqrt 2}(|u_\downarrow\rangle-|d_\uparrow\rangle)
\end{equation}
 satisfies the condition for the Grand Spin $\boldsymbol{G}=\boldsymbol{I}+\boldsymbol{J}$:
\begin{equation}\label{grandspin}
\boldsymbol{G}\chi_h=0.
\end{equation}
The hedgehog baryon $\vert B\rangle =\vert N_c q\rangle \vert \sigma
\rangle \vert \pi \rangle\vert \omega\rangle \vert \rho\rangle \vert A\rangle$ is given by the product of quarks and
coherent states of mesons fields and it corresponds to a linear
combination of states with $I=1/2$ and $I=3/2$:
\begin{equation}\label{hedgestate}
\vert B\rangle = \sum_{JMM_I}(-)^{J+M}C_J\delta_{M,-M_I}\vert J=I,M=-I_3,M_I\rangle.
\end{equation}
An explicit expression for the coefficients $C_J$ will be given in Sec.~\ref{proiezione}.

It can be shown that the hedgehog state is one element
of an infinite class of degenerate solutions of field equations at Mean-Field level~\cite{Fiolhais:1985bt}.
The profiles of the chiral and the vector mesons fields
in the hedgehog state are given by the relations:
\begin{align}
& \dfrac{\langle B\vert \sigma \vert B\rangle}{\langle B\vert B\rangle} =
\sigma_h (r)\nonumber ,\quad \dfrac{\langle B\vert\boldsymbol{\pi}_a \vert B\rangle}
{\langle B\vert B\rangle} =\widehat{r}_a h (r)\nonumber\\
& \dfrac{\langle B\vert \omega_\mu \vert B\rangle}{\langle B\vert B\rangle}
\omega_0 (r)=\omega (r)\nonumber\nonumber\\
& \dfrac{\langle B\vert\boldsymbol{\rho}_a \vert B\rangle}
{\langle B\vert B\rangle} =\rho_i ^a(r)=\rho (r)\epsilon^{ika} \widehat{r}^k\nonumber\\
& \dfrac{\langle B\vert\boldsymbol{A}_a \vert B\rangle}
{\langle B\vert B\rangle} =A_i ^a(r)=A_S (r)\delta ^{ai}
+A_T (r)(\widehat{r}^a \widehat{r}^i-\dfrac{1}{3}\delta^{ai})\nonumber
\end{align}
At Mean-Field level the mesons fields are classical and the
differential equations governing their dynamics have to be
supplemented by the appropriate boundary conditions.  For the single
nucleon case we impose the following boundary conditions to the
fields:
\begin{eqnarray}\label{bcvac}
& u'(0)=v(0)=0\nonumber ,\\
&\sigma_h '(0)=h (0)=0,\\
&\rho (0)=\omega '(0)=A_S '(0)=A_T (0)=0, \nonumber
\end{eqnarray}
while at infinity the boundary conditions read:
\begin{align}
& \sigma_h (\infty)=\sigma_0 ,\,  h (\infty)=0,\nonumber\\
& \dfrac{v(\infty)}{u(\infty)}=\sqrt{\dfrac{-g \sigma_0+\varepsilon}{-g \sigma_0-\varepsilon}} , \,\\
& \omega'(\infty)=\rho '(\infty)=A_S '(\infty)=A_T '(\infty)= 0,\nonumber
\end{align}
where $\varepsilon$ is the quark eigenvalue.\\
The total energy of the soliton at Mean-Field level is given by:
\begin{align}\label{enmfa}
 E_{MFA}= 4\pi \int &r^2 dr (E_{int}+E_{kin,Q}+E_{\sigma}+E_\pi \nonumber\\
 & +E_\omega +E_\rho +E_A+ E_{pot})
\end{align}
where explicit expressions for each term in the energy density can be found in Appendix I.
A test of the numerical accuracy of the solution originates from
another way of expressing the energy, obtained by
Rafelski~\cite{Rafelski1977} by integrating out the fermionic fields:
 \begin{align}
E_{Raf.}&=\int  d^3r\bigg[4\left(V-\sigma\dfrac{\partial V}{\partial \sigma}-\pi \dfrac{\partial V}{\partial \pi}\right)-m_\omega ^2\omega ^2 \nonumber\\
& +m_\rho ^2\left(2\rho ^2+3A_S ^2+\dfrac{2}{3}A_T ^2\right)\bigg]
\end{align}
Our solutions satisfy this consistency test up to a precision of the order of $10^{-3}$.

\section{Projection}\label{proiezione}

The hedgehog baryon defined in section~\ref{hedgehog} is not an
eigenstate of isospin and angular momentum. Moreover this
semi-classical solution also breaks the translational symmetry of the
Lagrangian, since the localized soliton is not an eigenstate of the
linear momentum, either. In this work we restore the
invariance under rotations by using the projection technique developed
in~\cite{Ruiz1995,Birse:1986qc}.  Instead, we will not restore the
translational invariance of our soliton. The spin-isospin eigenstates
are defined as follows:
\begin{equation}\label{projstate}
\vert JMM_I\rangle = N_{JM_I} \int d^3\Omega D_{M,-M_I}^{J}(\Omega)^* \widehat{R}(\Omega)\vert B\rangle
\end{equation}
where the weight functions $D$ are the Wigner functions,
$\widehat{R}(\Omega)$ is a spatial rotation through Euler angles
$\Omega \equiv (\alpha,\beta,\gamma)$ and $ N_{JM_I}$ is a
normalization factor.\\
Since the hedgehog states are eigenstates of the Grand Spin $\boldsymbol{G}$,
it is equivalent to rotate either in spin or in isospin space.
Moreover, when studying diagonal matrix elements of nucleon states, it
is customary to work with states where the third component of the
angular momentum $M$ is equal to $-M_I$ since in this case the expression of
the Wigner function is particularly simple.  In this way the
projection operator becomes:
\begin{equation}\label{projoperator}
P_{JM}=\dfrac{2J+1}{8\pi^2}\int d^3 \Omega D_{M,M}^{J}(\Omega)^* \widehat{R}(\Omega).
\end{equation}
The normalization factor has been determined by using~(\ref{projstate})-(\ref{projoperator}):
\begin{align}
N^2 _{J, -M} & = \left(\dfrac{2J+1}{8\pi^2}\right)^2 (\langle B \vert P_{JM}\vert B\rangle)^{-1}.
\end{align}
Finally, the coefficients $C_J$ in eq.~(\ref{hedgestate}) are given by the expression:
\begin{align}
C_J ^2 & = \langle B \vert P_{JM}\vert B\rangle\nonumber \\
& = \dfrac{2J+1}{8\pi^2}\int d^3\Omega  D_{M,M}^{J}(\Omega) \langle B\vert\widehat{R}(\Omega)\vert B\rangle.
\end{align}
Once we obtain the projected state, we proceed to evaluate the corresponding energy.
Basically we need to calculate the expectation value of the Hamiltonian on the projected state given by~(\ref{projstate}).
The projected energy can be written as:
\begin{align}\label{projen}
& E_{J} & = &\langle JM-M\vert :H:\vert JM-M\rangle \nonumber\\
& & = &4\pi \int r^2 dr (E_{int}+E_{kin,Q}+E_{J,\sigma}+E_{J,\pi}\nonumber\\
& & &+E_{\omega}+E_{J,\rho}+E_{J,A}+E_{J,pot})
\end{align}
More details about the projection technique and the evaluation of the each term in the energy can be found in~\cite{Birse:1986qc, Ruiz1995}.\\

\subsection{Projected observables}

The formalism needed to compute most of the observables can be found
in~\cite{Birse:1986qc,Ruiz1995}.  The only quantity for which we need
to provide a new explicit expression is the potential energy, since in our
potential there is a logarithmic term, not present in the $\sigma$
model.\\
The matrix element for which we need to develop a full calculation is:
\begin{equation}\label{potproj}
E_{J,pot}= \langle JM-M\vert :\int d^3r V(\sigma_h,h):\vert JM-M\rangle
\end{equation}
 More details about the calculation of this term are given in Appendix II.\\
To compute the static
observables we have used the explicit formulae given in the Appendix
of Ref.~\cite{Ruiz1995}; in addition to radii and magnetic moments we also
show the results for the average number of pions in the projected
state, given by:
\begin{equation}
\langle N_\pi\rangle _J=\overline{N}_\pi C_0 (J,\overline{N}_\pi),
\end{equation}
where $C_0(J,\overline{N}_\pi)$~\cite{Birse:1986qc} is a projection coefficient depending
on the spin and on the average number of pions $\overline{N}_\pi$ in
the unprojected state.
As already mentioned before, we do not perform a
projection on the linear momentum, but we adopt an easier
approach~\cite{Dethier:1982ax} which provides a rough estimate of the
center-of-mass corrections to the baryon total energy.  The masses for
the $N$ $(J=1/2)$ and for the $\Delta$ $(J=3/2)$ are then given by:
\begin{equation}\label{spurious}
M_J = (E_J-\boldsymbol{P}^2)^{1/2} \, .
\end{equation}

\section{Wigner-Seitz approximation to nuclear matter}\label{hedgedensity}

In order to describe a soliton system at finite density we use the
Wigner-Seitz approximation. This approach is very common
and it has already been widely applied to both
non-linear~\cite{D.Hahn1987,N.K.Glendenning1986,P.Amore2000} and
linear-$\sigma$ models~\cite{U.Weber1998}. Specifically, the
Wigner-Seitz approximation consists of replacing the cubic lattice by
a spherical symmetric one where each soliton sits on a spherical cell
of radius R with specific boundary conditions imposed on fields at the
surface of the sphere.  The configuration of the meson fields,
centered at each lattice point, generates a periodic potential in
which the quarks move.

The spinor of quark fields
must satisfy the Bloch theorem:
\begin{equation}
\psi _{\boldsymbol{k}} (r)=e^{i \boldsymbol{k}\cdot \boldsymbol{r}}\Phi_{ \boldsymbol{k}}(r),
\end{equation}
where $\boldsymbol{k}$ is the crystal momentum (which for the ground state
is equal to zero) and $\Phi_{ \boldsymbol{k}}(r)$ is a spinor that has
the same periodicity of the lattice.

\subsection{Boundary conditions}
In the literature various sets of possible boundary conditions have
been discussed~\cite{U.Weber1998,P.Amore2000}.  In our work we adopt
the choice of Ref.~\cite{U.Weber1998} which relates the boundary conditions
at $R$ to the parity operation, $\boldsymbol{r}\rightarrow -\boldsymbol{r}$.
Respect to this symmetry the lower component $v(r)$ of quark spinor,
the pion $h (r)$ and the rho $\rho (r)$ are odd, and therefore they have to vanish at $R$:
\begin{equation}
v(R)=h (R)=\rho (R)=0.
\end{equation}
Similarly, for the $\sigma$ field, the upper Dirac component, the $\omega$ and the $A$ fields the
argument based on parity provides the conditions:
\begin{equation}
u'(R)=\sigma_h '(R)=\omega '(R)=A_S '(R)=A_T '(R)=0.
\end{equation}

The boundary conditions at $r=0$ remain the ones given in eq.~(\ref{bcvac}).
Basically the calculation consists in solving the set of coupled field
equations in a self-consistent way for a given value $R$; practically we
start from $R=4$ fm, for which the periodic solutions are
indistinguishable from the vacuum ones, and we slowly decrease the
cell radius down to the smallest radius for which self-consistent
solutions can be obtained. The parameter set is the same used for
vacuum calculations.

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{campi_densfinita.eps}
\caption{Upper and lower components of the Dirac spinor ((a), (b)), $\sigma$ and
pion fields ((c), (d)), in the model without vector mesons, as functions of the cell radius $R$.}\label{campi}
\end{figure}

In Fig.~\ref{campi} we plot the Dirac and the chiral fields in the model without vector mesons
for different values of $R$; down to $R=2$ fm, the solutions do not change
significantly, but as the cell radius shrinks to lower values, we see
that all the fields are deeply modified by the finite density.

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{campi_densfinitaVM.eps}
\caption{Vector mesons fields, as functions of the cell radius $R$.
In (a) we plot the $\omega$ field, in (b) the $\rho$ field, in (c) the $A_S$ field and in (d) the $A_T$ field.}\label{campiVM}
\end{figure}

For the model including vector mesons, we present in Fig.~\ref{campiVM} the trend of the vector
mesons fields.
To better clarify the difference between the models without and with vector mesons, in Fig.~\ref{densityvm} we show the baryon density profiles in the two cases.
The relevant feature is that in the model without vector mesons the shape of the soliton becomes significantly more flat, at large densities, than in the case with vector mesons.
This effect is due to the repulsion between the two solitons provided by the $\omega$ field,
which prevents the baryon density to become large in the inter-nucleon region.
This feature will have an important consequence on the dependence of the radii on the density,
as discussed in Sec~\ref{properties}.
 \begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{Densita_LOGVMNOVM.eps}
\caption{Color online. Baryon density as a function of the ratio $r/R$ for the model without (dashed line) and with vector mesons (solid line). Two values of the cell radius $R$ are shown, namely $R=2$ fm and $R=1.2$ fm.
}\label{densityvm}
\end{figure}

\subsection{Band width}
We are solving a problem in which quarks are moving in a
periodic potential and therefore Bloch theorem tells us that a band
should form. How to define the width of the band is highly not
trivial.  The most sophisticated technique is the one provided
in~\cite{U.Weber1998}.\\
Here we adopt two much simpler procedures. The first method is taken
from~\cite{D.Hahn1987}, where the authors estimates the band
width as:
\begin{eqnarray}\label{bandglend}
\Delta & =& \sqrt{\epsilon _0 ^2 + \left(\dfrac{\pi}{2R}\right)^2}-|\epsilon_0| , \\
\epsilon_{top} &=&\epsilon_0 +\Delta ,
\end{eqnarray}
where $\epsilon _0$ is the eigenvalue of the ground state.\\
An alternative approximation to the band width is obtained, following~\cite{Birse:1987pb},
by imposing that the lower Dirac component vanishes at the  boundary:
\begin{equation}\label{bandmike}
u(R)=0.
\end{equation}

A more accurate evaluation of the band effects can be obtained by
solving in a self-consistent way the field equations,which depend also
on $\boldsymbol{k}$, as discussed in~\cite{U.Weber1998}.  There it is
shown that $\epsilon_{top}$ obtained by imposing the condition~(\ref{bandmike}) is an upper limit to the top of the
band and that the true top lies about half way between this upper
limit and the bottom of the band. In conclusion, the band width turns
out to be considerably smaller than the one estimated using the first
method which leads to the upper limit $\epsilon_{top}$ in eq.~(\ref{bandglend}).

Concerning the filling of the band, when working with chiral solitons
at Mean-Field level the relevant quantum number is the grand-spin $G$
and the lower band corresponds to $G=0$. The only degeneracy remaining
is color and therefore the three quarks per soliton are completely
filling the band. The total energy of the cell is estimated by assuming a uniform
filling and by averaging the energy within the band.

\section{Results: properties of the nucleon in vacuum}\label{resultsvac}

We start by showing how the
solitonic solutions for the
single nucleon can be built and we check
that it is possible to provide a reasonable description of the single
nucleon properties with the chosen parameter set.
In particular we present here results for the lagrangian without
vector mesons $\mathcal{L}_0$ for the set $m_\sigma=550$ MeV and the
typical value $g_\pi=5$~\cite{E.K.Heide1992, Birse:1986qc}. Instead
the parameters for the model with vector mesons are given by $m_\sigma=1200$ MeV, $g_\pi=3.9$,
$g_\omega =12$ and $g_\rho =4$.
This set has been chosen in order to both
get saturation at finite density and a reasonable description of the
nucleon in vacuum. Anyway, since the calculation of nuclear matter properties
done by using the Wigner-Seitz approximation is affected by large uncertainties
we also present a second set of results with parameters better fitted to
single nucleon properties.

In Tables~\ref{res} and~\ref{resVM} we present the static properties of the hedgehog
baryon at Mean-Field level and we compare them with results obtained
in the linear $\sigma$-model~\cite{Birse1985, W.Broniowski1986}.

In Tables~\ref{contrib} and~\ref{contribVM}
we show the decomposition of the soliton total energy in its various
contributions and again we compare with the linear $\sigma$-model~\cite{Birse1985, W.Broniowski1986}.
We see that the contributions to the total energy coming from the chiral fields
and from the vector mesons are comparable: as expected vector mesons play an important role in the dynamics of the soliton.
It is also interesting to notice that the results obtained with the logarithmic model are very similar
to the ones obtained with the Mexican hat potential, at zero density, specially when vector mesons are included.

\begingroup
\begin{table}[t]
\caption{Various nucleon properties at Mean-Field level in the present work without vector mesons and in the $\sigma$-model~\cite{Birse1985}.}\label{res}
\centering
\begin{tabular}{|c | c  c  c|}
\hline
 & & & \\
 Quantity & Log. Model & $\sigma$-Model & Exp.  \\[7pt]
 \hline
 & & & \\
 $M\,(MeV)$ & $1176$ & $1136$ & $1085$\\[5pt]
 $\langle r_e ^2\rangle_{I=0}$& $(0.76\, fm)^2$ & $(0.78\, fm)^2$  & $(0.72\, fm)^2$ \\ [5pt]
 $\mu_{I=1}$ $(\mu_N)$& $3.83$ & $3.63$ & $4.70$ \\[5pt]
 $\langle r_m ^2\rangle_{I=1}$& $(1.12\, fm)^2$ & $(1.14\, fm)^2$ & $(0.80\, fm)^2$ \\[5pt]
 $g_A$ & $1.27$ & $1.22$  & $1.26$ \\ [5pt]
 $\overline{N}_\pi$& $2.4$& $1.9$ & $\diagup$\\[10pt]
 \hline
\end{tabular}
\end{table}

\begingroup
\begin{table}[b]
\caption{Various nucleon properties at Mean-Field level in the present model and in the $\sigma$-model~\cite{W.Broniowski1986} with vector mesons.}\label{resVM}
\centering
\begin{tabular}{|c | c  c  c|}
\hline
 & & & \\
 Quantity & Log. Model & $\sigma$-Model & Exp.  \\[7pt]
 \hline
  & & & \\
 $M\,(MeV)$ & $1329.5$ & $1331.7$ & $1085$\\[5pt]
 $\langle r_e ^2\rangle_{I=0}$& $(0.78\, fm)^2$ & $(0.76\, fm)^2$  & $(0.72\, fm)^2$ \\ [5pt]
 $\mu_{I=1}$ $(\mu_N)$& $4.49$ & $4.51$ & $4.70$ \\[5pt]
 $\langle r_m ^2\rangle_{I=1}$& $(0.99\, fm)^2$ & $(1.01\, fm)^2$ & $(0.80\, fm)^2$ \\[5pt]
 $g_A$ & $1.34$ & $1.35$  & $1.26$ \\ [5pt]
 $\overline{N}_\pi$& $2.62$& $2.66$ & $\diagup$\\[10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

\begingroup
\begin{table}[t]
\caption{Contributions to the soliton total energy at Mean-Field
level in the Logarithmic model and in the Linear $\sigma$-model~\cite{Birse1985}. All quantities are in MeV.}\label{contrib}
\centering
\begin{tabular}{|c c c|c|c|}
\hline
 & & & & \\
Quantity &  &  & Log. Model & Linear $\sigma$-Model  \\[7pt]
\hline
 & & & & \\
Quark eigenvalue &  &  & $83.1$ & $107.4$\\[5pt]
Quark kinetic energy &  &  & $1138.0$& $1056.9$ \\ [5pt]
$E_\sigma$ (mass+kin.)&  &  & $334.5$& $320.3$ \\[5pt]
$E_\pi$ (mass+kin.)&   &  & $486.0$ & $373.1$\\ [5pt]
Potential energy $\sigma -\pi$& & & $105.7$& $120.7$ \\ [5pt]
$E_{q\sigma}$&   &   & $-101.4$ & $-62.3$\\ [5pt]
$E_{q\pi}$ &   &  & $-787.0$ & $-673.2$\\ [5pt]
Total energy  &   &  & $1175.6$ & $1136.2$\\ [10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

\begingroup
\begin{table}[b]
\caption{Contributions to the soliton total energy at Mean-Field
level in the Logarithmic model and in the Linear $\sigma$-model with vector mesons~\cite{W.Broniowski1986}. All quantities are in MeV.}\label{contribVM}
\centering
\begin{tabular}{|c c c|c|c|}
\hline
 & & & & \\
Quantity &  &  & Log. Model & Linear $\sigma$-Model  \\[7pt]
\hline
 & & & & \\
Quark eigenvalue &  &  & $114.5$ & $112.9$\\[5pt]
Quark kinetic energy &  &  & $1075.8$& $1080.6$ \\ [5pt]
$E_\sigma$ (mass+kin.)&  &  & $213.8$& $212.2$ \\[5pt]
$E_\pi$ (mass+kin.)&   &  & $393.2$ & $397.3$\\ [5pt]
Potential energy $\sigma -\pi$& & & $81.2$& $80.4$ \\ [5pt]
$E_\omega$ (mass+kin.)&   &  & $-194.4$ & $-196.5$\\ [5pt]
$E_\rho$ (mass+kin.)&   &  & $162.6$ & $165.4$\\ [5pt]
$E_A$ (mass+kin.)&   &  & $329.5$ & $334.1$\\ [5pt]
$E_{q\sigma}$&   &   & $6.54$ & $4.74$\\ [5pt]
$E_{q\pi}$ &   &  & $-621.9$ & $-627.1$\\ [5pt]
$E_{q\omega}$ &   &  & $388.9$ & $393.0$\\ [5pt]
$E_{q\rho}$ &   &  & $-163.8$ & $-165.9$\\ [5pt]
$E_{qA}$ &   &  & $-341.8$ & $-346.4$\\ [5pt]
Total energy  &   &  & $1329.5$ & $1331.7$\\ [10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

\begingroup
\begin{table}[t]
\caption{Projected nucleon properties in the present work
without vector mesons and in the linear
$\sigma$-model and comparison with experimental values.}\label{resproj}
\centering
\begin{tabular}{|c | c   c  c|}
\hline
 & & & \\
 Quantity & Log. Model & $\sigma$-Model & Exp.  \\[7pt]
 \hline
  & & & \\
 $E_{1/2}\, (MeV)$ & $1075$ & $1002$ &  \\[5pt]
 $M_N \,(MeV)$ & $960$&  $ 894$& $938$ \\[7pt]
 $E_{3/2}\, (MeV)$ & $1140$ & $1075$ &  \\[5pt]
 $M_{\Delta}\, (MeV)$ & $1032$& $975$& $1232$\\[7pt]
  $\langle r_E ^2\rangle_p\, (fm^2)$ & $0.55$ & $0.61$ & $0.74$\\[5pt]
  $\langle r_E ^2\rangle_n\, (fm^2)$ & $-0.02$ & $-0.02$ & $-0.12$\\[5pt]
  $\langle r_M ^2\rangle_p\, (fm^2)$ & $0.7$ & $0.72$ & $0.74$\\[5pt]
 $\langle r_M ^2\rangle_n\, (fm^2)$ & $0.72$ & $0.75$ & $0.77$\\[5pt]
  $ \mu_p$ $(\mu_N)$ & $2.25$ & $2.27$ & $2.79$\\[5pt]
 $\mu_n$ $(\mu_N)$ & $-1.97$ & $-1.92$ & $-1.91$\\[5pt]
 $g_a$ & $1.52$ & $1.10$ & $1.26$\\[5pt]
                    & $1.6$ $(J=1/2)$& $1.2$ $(J=1/2)$ &  \\
  $\langle N_\pi \rangle_J$&                  &                  & $\diagup$\\
                    & $2.$ $(J=3/2)$& $1.6$ $(J=3/2)$& 	\\[10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

\begingroup
\begin{table}[b]
\caption{Projected nucleon properties  in the present work and in the linear
$\sigma$-model with vector mesons and comparison with experimental values.}\label{resprojVM}
\centering
\begin{tabular}{|c | c   c  c|}
\hline
 & & &  \\
 Quantity & Log. Model & $\sigma$-Model & Exp.  \\[7pt]
 \hline
  & & & \\
 $E_{1/2}\, (MeV)$ & $892$ & $882$ &  \\[5pt]
 $M_N \,(MeV)$ & $763$&  $ 750$& $938$ \\[7pt]
 $E_{3/2}\, (MeV)$ & $1030$ & $1029$ &  \\[5pt]
 $M_{\Delta}\, (MeV)$ & $918$& $917$& $1232$\\[7pt]
 $\langle r_E ^2\rangle_p\, (fm^2)$ & $0.59$ & $0.58$ & $0.74$\\[5pt]
 $\langle r_E ^2\rangle_n\, (fm^2)$ & $-0.03$ & $-0.02$ & $-0.12$\\[5pt]
  $\langle r_M ^2\rangle_p\, (fm^2)$ & $0.69$ & $0.69$ & $0.74$\\[5pt]
 $\langle r_M ^2\rangle_n\, (fm^2)$ & $0.70$ & $0.71$ & $0.77$\\[5pt]
  $ \mu_p$ $(\mu_N)$ & $2.72$ & $2.71$ & $2.79$\\[5pt]
 $\mu_n$ $(\mu_N)$ & $-2.49$ & $-2.5$ & $-1.91$\\[5pt]
 $g_a$ & $1.6$ & $1.48$ & $1.26$\\[5pt]
                    & $1.1$ $(J=1/2)$& $1.8$ $(J=1/2)$ &  \\
  $\langle N_\pi \rangle_J$&                  &                  & $\diagup$\\
                    & $1.3$ $(J=3/2)$& $2.2$ $(J=3/2)$& 	\\[10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

\begingroup
\begin{table}[t]
\caption{Projected nucleon properties  in the present work and in the linear
$\sigma$-model with vector mesons and comparison with experimental values for the parameter set: $g=3.6$, $g_\omega=13$, $g_\rho=4$ and $m_\sigma=1200$ MeV.}\label{resprojVMa}
\centering
\begin{tabular}{|c | c   c  c|}
\hline
 & & & \\
 Quantity & Log. Model & $\sigma$-Model & Exp.  \\[7pt]
 \hline
  & & & \\
   $E_{1/2}\, (MeV)$ & $1020$ & $1008$ &  \\[5pt]
 $M_N \,(MeV)$ & $926$&  $912$& $938$ \\[7pt]
 $E_{3/2}\, (MeV)$ & $1148$ & $1147$ &  \\[5pt]
 $M_{\Delta}\, (MeV)$ & $1066$& $1063$& $1232$\\[7pt]
 $\langle r_E ^2\rangle_p\, (fm^2)$ & $0.67$ & $0.66$ & $0.74$\\[5pt]
 $\langle r_E ^2\rangle_n\, (fm^2)$ & $-0.05$ & $-0.05$ & $-0.12$\\[5pt]
   $\langle r_M ^2\rangle_p\, (fm^2)$ & $0.77$ & $0.76$ & $0.74$\\[5pt]
 $\langle r_M ^2\rangle_n\, (fm^2)$ & $0.78$ & $0.77$ & $0.77$\\[5pt]
  $ \mu_p$ $(\mu_N)$ & $2.63$ & $2.64$ & $2.79$\\[5pt]
 $\mu_n$ $(\mu_N)$ & $-2.37$ & $-2.38$ & $-1.91$\\[5pt]
 $g_a$ & $1.58$ & $1.46$ & $1.26$\\[10pt]
 \hline
\end{tabular}
\end{table}
\endgroup

In Tables~\ref{resproj} and~\ref{resprojVM} we present the results after projection in
both models, with and without vector mesons.
 Moreover,
in Table~\ref{resprojVMa} we present the results obtained
using a second parameter set, better fitted to the single nucleon properties.
It is important to remark that our results in the logarithmic model with only chiral fields
are consistent with the
ones obtained in~\cite{AbuShady:2010zza}. There, a different approach
based on the coherent pair approximation was used.
Their results are similar
to ours when the coherence length parameter $x$ is taken to be of the
order of one, as suggested in~\cite{PhysRevD.60.114022}.

The results obtained both without and with the vector mesons
in general overestimate the experimental values, particularly
for the magnetic observables,
 once the parameters are chosen so that
the projected mass of the nucleon is close to its physical value. One has anyway
to recall that for the mass an approximate correction for the spurious center
of mass motion has been taken into account (see eq.(~\ref{spurious})), while no
center of mass correction has been done for the other observables.
When this further corrections are taken into account the value of some
 observables typically reduces.

\section{Results: solitons at finite density}\label{hedgefindensity}
In this section we present the results obtained by studying a Wigner-Seitz
lattice of solitons.
We first discuss the energy of the system at finite density and we then present
the effect of the density on the single nucleon properties.

\subsection{Energy of the lattice}
In Figs.~\ref{Etot} and~\ref{EtotVM}  we present the results of the total energy per unit cell in the present model and in the
linear $\sigma$-model with and without vector mesons.
  For each given value of $m_\sigma$, the
logarithmic model allows the system to reach higher densities.
Notice that as
$m_\sigma$ increases, the system remains stable down to lower values of $R$ because the
chiral fields are more and more restricted to lay on the chiral
circle. Moreover the introduction of vector mesons stabilises even more
the solution, hence in comparison to the model with only $\sigma$ and $\pi$ we
can reach higher densities.

\begin{figure}[b]
\centering
\includegraphics*[width=\columnwidth]{Etot.eps}
\caption{Total energy of the soliton as a function of cell radius $R$ for the linear
$\sigma$-model \cite{U.Weber1998} and for the present model without vector mesons.
Different values of $m_\sigma$ are considered, $g_\pi=5$, $g_\omega=12$ and $g_\rho=4$.}\label{Etot}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{EtotVM.eps}
\caption{Total energy of the soliton as a function of cell radius $R$ for the linear
$\sigma$-model \cite{U.Weber1998} and for the present model with vector mesons.
Different values of $m_\sigma$ are considered.}\label{EtotVM}
\end{figure}

In Fig.~\ref{band} we plot the quark eigenvalue for the model without vector mesons
 as a function of the
cell radius $R$. The line labeled $\epsilon_{top}^{(a)}$ corresponds to the estimate
of the top of the band given by eq.~(\ref{bandglend}), while
$\epsilon_{top}^{(b)}$ follows from eq.~(\ref{bandmike}).
It is clear that in absence of vector mesons we never obtain saturation. Moreover
to change the value of $m_\sigma$ does not modify this result.
  In Fig.~\ref{bandVM} we show the analogous results in the case with vector mesons.
The band structure is similar to the one in Fig.~\ref{band}, but here the main difference
is given by a significant increase of the top of the band
at high densities which allows us to obtain saturation.\\

\begin{figure}[b]
\centering
\includegraphics*[width=\columnwidth]{FigEnTildateLOGVMNOVM.eps}
\caption{Total energy of the soliton at Mean-Field level in the logarithmic model without vector mesons (dashed line) and with vector mesons (solid line). The parameter values are: $m_\sigma=1200$ MeV, $g_\pi=5$, $g_\omega=12$ and $g_\rho =4$.}\label{FigEnTildateLOGVMNOVM}
\end{figure}

To better understand our result which indicates the possibility of getting
saturation, in Fig.~\ref{FigEnTildateLOGVMNOVM} we compare the total energy of the soliton at Mean-Field without
the contribution associated with the band. In order to emphasize the effect of the density
on the energy we subtracted the mass of the nucleon in vacuum.
It is clear that the exchange of vector mesons plays a crucial role,
by contributing $\sim100$ MeV at $R=1$ fm, but it is not sufficient to get saturation.
To determine which ingredients of the model are actually providing the repulsion at high densities, in Fig.~\ref{entildate} we plot the interaction energies
for each term contributing to the total energy at finite density. The plotted quantities are
 defined as the value of the chosen energy contribution
at $R$ minus the corresponding vacuum value:
\begin{equation}
\tilde{E}_i (R) = E_i(R)-E_i (\infty).
\end{equation}
From the figure it is clear that the band effect strongly influences the total energy of the soliton, by providing the largest contribution to repulsion at high densities.
This is not surprising, because the band is associated with the sharing of quarks between nucleons.
It is well known~\cite{Shimizu:1989ye,Faessler:1996wt} that in calculations of the $N-N$ potential based on
quark models the short-range repulsion is associated with the formation of a six-quark bag.
In our calculation the exchange of vector mesons is the dominant effect at densities up to
$\rho_0$, but at very high densities the band effect dominates.
The total amount of these contributions leads to the repulsive mechanism responsible for the steep rising of the total energy at high densities, as it can
be seen in the upper panel of Fig.~\ref{bandVM}.\\

In Figs.~\ref{scomposizenVM} and~\ref{scomposizenVM1} we show in details all the contributions
to the total energy (without the band effect).
It is clear that the $\omega$ meson provides short-range repulsion, partially compensated
by the interaction of the quarks with the $A$ meson.

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{bandprova2.eps}
\caption{Quark eigenvalue as a function of the cell radius $R$,
in the model without vector mesons.
The shaded areas represent the band as estimated in eq.(\ref{bandglend}) and in eq.~(\ref{bandmike}).
The first excited state $1^+$ is also shown. The quark mass in vacuum, here $465$ MeV,
is indicated by the dashed line.}\label{band}
\end{figure}

\begin{figure}[b]
\centering
\includegraphics*[width=\columnwidth]{Fig_EnergiaSat_ms1200g39gw12gr40_2.eps}
\caption{Panel (a): total energy of the soliton with band effects (solid line) and without band effects (dashed line) as a function of the cell radius $R$ for the model with vector mesons. Panel (b): the quark eigenvalue as a function of the cell radius $R$ for the model with vector mesons.
The shaded areas represent the band as estimated in~ eq.(\ref{bandglend}) and in eq.~(\ref{bandmike}).
The quark mass, here $362.7$ MeV,
is indicated by the dashed line. }\label{bandVM}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{Figura_EnergieTildate_ms1200g39gw12gr4.eps}
\caption{Color online. Interaction energies $\tilde{E}_i$, as defined in text,  as a function of the cell radius $R$ in the model with vector mesons. Parameters as in Fig.~\ref{bandVM}.}\label{entildate}
\end{figure}

\begin{figure}[b]
\centering
\includegraphics*[width=\columnwidth]{mesonsenergy_NEW.eps}
\caption{Color online. Panel (a): contributions of the chiral fields to the total energy of the soliton as a function of the cell radius $R$ in the model with vector mesons. Panel (b): contributions of the vector meson fields to the total energy of the soliton as a function of the cell radius $R$. Parameters as in Fig.~\ref{bandVM}.}\label{scomposizenVM}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{QuarkmesonsEnergy_NEW.eps}
\caption{Color online. Panel (a): interaction energies of the chiral fields with quarks as a function of the cell radius $R$ in the model with vector mesons. Panel (b): interaction energies with the vector mesons as a function of the cell radius $R$. Parameters as in Fig.~\ref{bandVM}.}\label{scomposizenVM1}
\end{figure}
It has been discussed in the literature how to interpret the results obtained
using the Wigner-Seitz lattice and in particular which should be the
indications of quark deconfinement. In~\cite{D.Hahn1987} it has been
suggested that deconfinement takes place when the upper band, which
corresponds to Grand Spin $G=1$, merges with the lower
band.
In the case with vector mesons this occurs roughly at densities
slightly larger than the saturation density.
We should keep in mind that the estimate of the width of the band is affected by large uncertainties
and it is well possible that in a more refined calculation saturation density and
deconfinement density turn out to be well separated.

Which conclusions can be drawn from our analysis?
First and foremost we have a scenario in which it is possible, for the first time,
to obtain saturation by making use of the rather common idea of getting attraction
from the chiral fields and repulsion from the vector fields. In this game the
logarithmic potential plays a crucial role, by allowing the solitons to remain
stable at densities large enough that the vector mesons can play a role.
It is important at this point to clarify whether we can get this result only
in a special and tiny parameters' range or if the mechanism leading to saturation
is rather stable respect to the choice of the parameters' values.
This analysis is shown in Fig.~\ref{coester} where we plot the value of the total energy,
including the band effects, at the "saturation density" point minus the energy in
vacuum for different values of the parameters, at fixed $m_\sigma=1200$ MeV.
Here "saturation density" means the density at which a local minimum in the total energy
appears, even if that minimum is not the global one. The minimum is global when the
energy plotted in Fig.~\ref{coester} is negative and in that situation we are getting
real saturation. Instead, when the plotted energy is positive the local minimum corresponds
to a sort of metastable state. Since our calculation is affected by large uncertainties
we think it is interesting to show also the parameters leading to this "false" minimum,
because in a more sophisticated calculation (based for instance on a better estimate of the
band) those energies could easily become negative. For parameters' values significantly
outside the indicated range no local minimum exists. For instance for values of $g_\omega > 12$
the local minimum of the energy disappears because the energy keeps raising as the density increases, while for small
values of $g_\omega$ the repulsion cannot contrast the attraction and the energy gets smaller and smaller
at large densities.
It is important to notice that the range of parameters' values providing "saturation" is
at least in part overlapping
with the range of parameters' values for which a reasonable description of the single soliton in vacuum can
be obtained. Finally, we have to recall that we are not really studying nuclear matter, but rather
$G=0$ matter, which is composed of degenerate nucleons and deltas. Obviously, no experimental data
exist for that type of theoretical matter, but we can expect it to be saturating, probably with a larger
saturation energy than that of nuclear matter. The corresponding saturation density is also obviously
unknown.

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{CoesterLinesplusgw_NEW.eps}
\caption{Minimum of the total energy as a function of the density. Different sets of couplings are shown, for which the model
admit saturation. See the text for more details.} \label{coester}
\end{figure}

\subsection{Nucleon properties at finite density}\label{properties}
The question about the
modification of the nucleon properties at finite density have been
investigated in many analysis, both experimental~\cite{Aubert:1983xm}
and theoretical~\cite{PhysRevLett.53.892,Ericson:1986ud,Mulders:1990xw,Lu:1998tn,PhysRevC.70.065205}.
In Figs.~\ref{obs} and~\ref{obsVM} we show how the values of
a few observables evolve as a function of the spherical cell density
$\rho_C=(4 \Pi R^3/3)^{-1}$.
The Mean-Field observables evaluated as a function of the density cell are the isoscalar electric and the isovector magnetic radius and the number of pions.
The formulae for these quantities for the model without and with vector mesons read~\cite{Cohen:1986va,Ruiz1995}:
\begin{align}\label{obsmeanf}
& \langle r^2_E\rangle_{I=0}=\int r^4 (u^2+v^2)dr\nonumber\\
& \langle r^2 _M\rangle_{I=1}=\frac{1}{\mu_{I=1}}\int r^5 \frac{2\pi}{9}(G_{M} ^Q+G_{M} ^\pi +G_{M} ^\rho +G_{M} ^A) \,,
\end{align}
and $\mu_{I=1}$ and the radial functions $G_m ^{field}$ are given in Appendix III.
The problem with our evaluation is that in the Wigner-Seitz approach
we are forcing a unit of baryon number in every cell of the lattice.
Therefore, at densities large enough that the fields start occupying most of the cell and their
value is no more strongly varying inside the cell, the various radii all simply scale
with the size of the cell. The results we are obtaining are therefore indicative
only at densities low enough that the fields are still relatively well
contained inside each cell.
As shown in Figs.~\ref{campi} and ~\ref{campiVM} all the fields are well confined up to values of density close to $\rho_0$.
This implies that the behaviour
of the observables, evaluated from these fields,  have no physical relevance at densities
of the order or above nuclear matter saturation density $\rho_0$.

The problem we are facing is deeply associated with the Wigner-Seitz approximation
in which the effect of the finite density to the various observables is due only to Hartree contributions. In the real case two neighboring nucleons interact also via the Fock term.
Notice that at densities large enough that the field fluctuations are suppressed, the Hartree
contributions associated e.g with the pion field vanishes.
On the contrary the Fock term becomes relevant at those densities and it provides
contributions to the electromagnetic
observables which in nuclear physics are sometimes called pion-in-flight.
These terms, of course, cannot be evaluated in the Wigner-Seitz approach.

 Another problem with the Wigner-Seitz approach is that
it imposes spherical boundary conditions on the fields. This is particularly
dangerous in the case of the chiral fields, since at Mean-Field level directions
in ordinary space are connected with directions in isospin space, a situation
which is certainly quite far from reality. Due to these problems a work is in
progress~\cite{Sarti:2012un} in which a real lattice will be studied, with
boundary conditions which can change depending on the direction.
We think it is nevertheless worthy to present our results in Figs.~\ref{obs}-\ref{obsVM}
so to compare them with future more precise estimates.

For the model containing $\sigma$ and $\pi$ only, our results can be meaningful at low densities where
the dynamics is dominated by chiral fields.
The introduction of vector mesons affects in quite a interesting way the isoscalar radius: as shown in Fig.~\ref{densityvm}, the repulsion provided by the $\omega$ field, for densities smaller than $\rho_0$, prevents the swelling of the nucleons.
The qualitative effect of the inclusion of the vector mesons is to stabilize the shape of the solitons respect to compression.
This can be seen also in the case of the magnetic radius where the reduction of this quantity as a function of the density is less marked than in the case without vector mesons.
It is also possible to evaluate the so-called "super-ratio" defined as $(G_E/G_M)^\rho/(G_E/G_M)^{vac}$,
where $G_{E,M}$ are the electric and magnetic form factors.
In our calculation we obtain a reduction of this quantity as a function of the density, similarly to what obtained in other works, although the effect here is much larger. Let us stress again that
although the model with vector mesons allows to reach much larger densities, we do not
attribute too much significance to the behaviour of the observables at $\rho\gtrsim \rho_0$ for the reasons explained above.
In the lowest panel of Figs.~\ref{obs} and~\ref{obsVM}
we also show the behaviour of the number of
pions $\overline{N}_\pi$ per unit cell at finite density. The possibility
of an enhancement of the pion cloud, when the nucleon is not isolated, has been
discussed in the literature~\cite{Ericson:1986ud}.  In the present approach we obtain instead a
decrease of $\overline{N}_\pi$.
This result stems from the behaviour of the pionic field on the Wigner-Seitz lattice as shown in Fig.~\ref{campi}. The strong reduction of the pionic field is due to the boundary conditions
requested by the spherical symmetry.
It will be interesting to see if this result survives when a real lattice is used in the calculation.
\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{OBS_FINDEN_LOGNOVMsqrtNEW_23Febb.eps}
\caption{Isoscalar electric radius (panel (a)), isovector magnetic radius (panel (b))
and average number of pions (panel (c)) as a function of cell density $\rho_C$ for the model without
vector mesons.
}\label{obs}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics*[width=\columnwidth]{OBS_FINDEN_sqrt_LOGVM_NEW_23Febb.eps}
\caption{Isoscalar electric radius (panel (a)), isovector magnetic radius (panel (b))
and average number of pions (panel (c)) as a function of cell density $\rho_C$ for the model with vector mesons.}\label{obsVM}
\end{figure}

\section{Conclusions}\label{conclusions}
We used a Lagrangian with quarks degrees of freedom based on chiral and scale
invariance to study how the soliton behaves in vacuum and at finite density.
We presented results for the simple model with just chiral fields and also for the
model including vector mesons.
To describe the single nucleon properties in vacuum we have
used a projection technique.
The values of the static observables in the
logarithmic model at zero density are comparable to the ones
obtained using the linear $\sigma$-model.

For the description of
the soliton at finite density we have employed the Wigner-Seitz
approximation.
We have shown that the new potential, which includes
a logarithmic term originating from the breaking of scale invariance,
allows the system, for each given $m_{\sigma}$, to reach
densities larger than the ones obtained with the $\sigma$-model.
Moreover, as expected, the addition of the vector mesons plays a
double role in the study at finite density: on one hand it stabilizes the solution
and allows to reach even higher densities, on the other hand it partially provides the
repulsion necessary to obtain saturation.
The remaining repulsive contribution originates from the band effect.

The possibility to obtain saturation seems to be a firm result of the model, at the level of the Wigner-Seitz approximation.
In fact, by exploring the space of parameters, we have shown that the interplay between
attraction, provided by the chiral fields, and repulsion, given by the omega field and the band effect,
allows to get saturation for a rather extended range of parameters' values.

At sub-nuclear densities the dynamics should be dominated by
the chiral fields and the modifications
of the nucleon observables obtained in our work can
therefore be physically relevant in the low-density range.
In particular the
isoscalar radius presents a slight swelling, of the order of $5\%$
in the model without vector mesons and at even smaller dependence on the density in the complete
model.
This trend is in agreement with previous calculations~\cite{PhysRevLett.53.892,Lu:1998tn,Smith:2004dn}.

The present work will be extended in several directions. First
 a more precise and accurate
calculation of the band in the soliton crystal will be
done following Ref.~\cite{U.Weber1998}.
Work is in progress in order to study this same model by using the
technique developed in Ref.~\cite{Park:2008xm}, which provides
a more precise description of a multi-soliton system.
Finally, the
model can also be studied at finite temperature, including the
dynamics of the dilaton field.
We can expect that the effect of the finite temperature
on the soliton lattice will be to reduce the stability, by lowering
the value of the dilaton field and therefore making it more easy
for the chiral fields to fluctuate. It will be interesting
to compare the obtained phase diagram
with the one proposed by McLerran and Pisarski~\cite{L.McLerran2007}.

\begin{acknowledgements}
It is a pleasure to thank B.Y.Park and V. Vento for many stimulating discussions,
M.Birse and J.McGovern for useful comments and tips on calculations.\\
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


Google search relies on an algorithm known as PageRank. It's fairly easy to understand in principle, but implementing it is another matter. Figuring out how to search through all of the internet's pages is a hard problem, generally. Evaluating websites for the purposes of search means evaluating them within the context of graph data structures, where entities like webpages are represented by nodes, and the relationships between those webpages are represented by edges connecting them.
Some "naive" (read: crappy) graph exploration algorithms rise to the level of NP-hard, a class of computational complexity encompassing problems that are likely unsolvable by even the most powerful computers that exist or will exist. Given some arbitrary unexplored graph, it's just plain difficult to know the most efficient way to visit all of the nodes and links. As it turns out, we might as well explore the graph randomly, in the form of what's known as a "random walk."
The random walk method lets the search algorithm account for unusual geometric features of the graph, which might trip up a more directed search. Now, physicists have come up with a method that exploits quantum physics to improve on the class random walk strategy, as described in a recent paper published in the journal Physical Review A. By enlisting subatomic particles as the "searchers," the researchers made it possible to explore the graph in a way that is both properly random and that exploits interference effects between the particles. In other words, the searchers can be both random, and be aware of the larger search context.
Exploiting quantum effects in the interest of randomization is pretty obvious, since the quantum world is all about randomness. That is, it's all about uncertainty, or having incomplete information. It turns out there's a problem in using quantum effects for graph search, however. This is called unitarity, a principle that sets a limit on how quantum systems can evolve over time. Basically, this evolution needs to happen symmetrically, which limits the sorts of graphs that can be explored using quantum particles. If the links between nodes are bidirectional, we say that the graph is undirected; otherwise, it's a directed graph. Directed graphs are the problem.
The physicists behind the paper were able to come up with a clever fudge that allowed them to maintain this unitarity restriction by exploiting a different kind of symmetry having do with how the parity of a quantum system evolves over time. The upshot is that they could exploit quantum effects in the service of searching directed graphs. In experiments, they were able to match the performance of PageRank in evaluating the "centrality" of nodes in a graph, which can be thought of as a measure of how likely a random walker is to encounter those nodes as it explores the graph.
In some metrics, the quantum method was able to beat PageRank, such as in identifying equivalent sets of nodes within a graph. That said, there's more work to be done in even understanding how the quantum walker operates.


text:
NAGU SOO BIIR
XUQUUQDA SHAQAALAHA
Statement on the International Day for the Elimination of Violence against Women
The Federation of Somali Trade Unions (FESTU), umbrella organization of eleven independent and representative trade unions of Somalia, joins hands with all progressive organizations and trade union movement to support the global campaign to eliminate violence against women.
FESTU is appalled by the rising levels of violence and the increasingly heinous acts that are unleashed on women in our society. This scourge has been concentrated in the working class communities; but we note how all classes are affected by gender-based violence.
In Somalia, women are regularly and brutally assaulted, raped and even murdered. As well as the physical impact this has a massive mental and psychological impact on those affected. Sexual harassment and the violence against Somali women that is inherent in it is not taken seriously – not by the employers (bosses), not by the law enforcement, not by society in general.
A more widespread problem is that the majority of women lack economic independence. The high levels of unemployment which deprive millions of women of the chance of a job and an income, or at best restrict them to low-paid, insecure jobs make this problem even worse. Women have quality jobs and their salaries and working conditions are raised to a level that allows them to live independent lives.
This federation and its 11 member unions are committed to fight these injustices tooth-and-nail. We insist that violence against women is always illegal, always wrong and always a violation of the fundamental human right to live in safety and security.
We will never be able to claim victory in the struggle for social justice and respect for human rights so long as Somali women are being battered, harassed, intimidated and denied their constitution rights. Somali nation cannot claim to be civilized while such daily assaults on our most vulnerable fellow human beings continue to happen in our workplaces, houses and streets. All other human rights are insignificant if women are denied the most basic right to personal safety.
Patriarchy and unscrupulous entrepreneurship are bed fellows. Inequality and discrimination of certain groups and women are the order of the day. FESTU calls for a speedier shift to an egalitarian society where all individuals and groups will have rights and be respected and we can live in peace.
We will fight for an end to poverty and economic oppression of women. We will fight for an end to violence and abuses against women. We will fight for economic and social justice across the country. We call on our trade unions, businesses and governments (both federal and State levels) to ensure that women live in a violence-free society.
All of us need to educate our members about the rights of women as well as how to deal with these incidents and how to ultimately eradicate them from our society. This ugly situation needs to be reversed and this kind of violence must be stamped out of our society.
FESTU joins the International Trade Unions Confederation (ITUC) in campaigning for a new international labour Convention to tackle the various forms of gender-based violence that occur in the world of work.
Down with the oppression of women!
Forward to a peaceful and egalitarian society!
Ahmed Osman Said Omar Faruk Osman
FESTU President FESTU General Secretary
COP27 Must Deliver Climate Justice if Somalia is to Survive Devastating Impact of Climate Change
Somali Workers Deserve Decent Pay – World Day for Decent Work (WDDW)
UN High-Level Political Forum on Sustainable Development
Economic justice for working women's empowerment is a key to achieving a fair world of work in Somalia
Solidifying action against child labour in Somalia: World Day Against Child Labour 2022
Tweets by festusomalia
ABOUT FESTU
FESTU is the potent and authentic voice that fights for workers' rights by striving to guarantee a fair standard of living, improve working conditions and decent work for all, while improving the visibility of Somali unions and guaranteeing the participation of Somali workers in the struggle for peace, democracy and social justice.
FESTU ADDRESS
KM5, Bula Hubey,
Wadajir District,
Mogadishu, Somalia
info@festu.org
Copyright © 2023 FESTU

redpajama_set_name: RedPajamaCommonCrawl


text:
VIDEO / Dassault Aviation announces the Falcon 10X
Today, Dassault Aviation announced an all-new Falcon jet. Featuring a range of 7,500 nautical miles, the Falcon 10X will fly nonstop from New York to Shanghai, Los Angeles to Sydney, Hong Kong to New York or Paris to Santiago. Top speed will be Mach 0.925. The Falcon 10X will enter service at the end of 2025.
"Today we are introducing a new benchmark in business aviation," said Dassault Chairman and CEO Eric Trappier. "The Falcon 10X will offer an unrivalled passenger experience over both short- and long-duration flights, along with breakthrough safety features from Dassault's frontline fighter technology. We have optimized every aspect of the aircraft with the passenger in mind and established a new level of capability for ultra-long-range aircraft."
OFFICIAL / Dassault Falcon announces the Falcon 10X, which will take flight in a few years, powered by Rolls-Royce plc…
Posted by Aeronews on Thursday, May 6, 2021
The 10X will have the biggest and most comfortable cabin on the market and offer greater modularity than any other aircraft in its class, with a selection of multiple interior configurations. The 10X is large enough to accommodate 4 cabin zones of equal length but owners can configure their cabin to create a truly customized interior, including for example, an expanded dining / conference area, a dedicated entertainment area with a large-screen monitor, a private stateroom with a queen-size bed or an enlarged master suite with a private stand-up shower.
The 10X will have a cabin cross section larger than some regional jets. Its cabin will be 6-feet, 8-inches (2.03 m) tall and 9 feet, 1 inch wide (2.77 m). That will make it almost 8 inches (20 cm) wider and 2 inches (5 cm) taller than the widest and tallest purpose-built business jet flying today. Pressurization will also be the best on the market, with passengers experiencing a 3,000-foot cabin pressure altitude while flying at 41,000 feet. A next-generation filtration system will provide 100-percent pure air. The aircraft will be at least as quiet as the Falcon 8X, currently the quietest business jet in service.
FALCON 10X / The Rolls-Royce plc Pearl X engines can be controlled via a single thrust lever, a technology borrowed from…
Fuselage and engines
The 10X will feature an entirely new fuselage with extra-large windows—nearly 50 percent larger than those on the Falcon 8X. 38 windows will line the fuselage making for the brightest cabin in business aviation. The high-speed wing will be made of carbon fiber composites for maximum strength, reduced weight and minimum drag. Tailored for speed and efficiency, the very-high aspect ratio wing will be equipped with advanced, retractable high-lift devices offering superior maneuverability at low approach speeds. The twin-engine aircraft will be powered by business aviation's most advanced and efficient engine, the in-development Rolls Royce Pearl 10X. The 10X is the latest, largest and most powerful version of the Pearl series, delivering more than 18,000 pounds of thrust.
FALCON 10X will enter service at the end of 2025. Featuring a range of 7,500 nautical miles, the #Falcon10X will fly…
The Falcon 10X's flight deck will set a new standard in intuitive design, with touch screens throughout the cockpit. A next-generation Digital Flight Control System, derived directly from Dassault's latest military technology, will provide an unprecedented level of flying precision and protection, including a revolutionary, new single-button recovery mode.
A single smart throttle will serve as the primary power control, connecting both engines to the Digital Flight Control System which will automatically manage the power of each engine as needed in different flight scenarios. Thanks to Dassault's breakthrough FalconEye combined vision system—the first to offer both enhanced and synthetic vision capabilities—combined with dual HUDs able to serve as primary flight displays, the 10X will be capable of operating in essentially zero ceiling/visibility conditions.
Get a fresh program presentation on the @RollsRoyce #Pearl10X engine: https://t.co/vAqXrBc0vF pic.twitter.com/Xie01Pvt5K
— Dassault Falcon (@DassaultFalcon) May 6, 2021
"The #Falcon10X cabin is the biggest and most comfortable on the market and offers greater modularity with a selection of multiple interior configurations." Carlos Brana pic.twitter.com/0ecSpvDkPY
"With it's 7,500 nm range, the #Falcon10X connects all major international business destinations nonstop. Fly New York to Shanghai, Los Angeles to Sydney, Paris to Santiago." Carlos Brana pic.twitter.com/5zNSbyafQw
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Born September 6, 1822 in Virginia.
Parents were ISAAC BOOCO and BETSEY HOPPES.
Married ANGELINE GARRINGER on February 27, 1848 in Fayette County, Ohio.
Abraham apparently could not write, which was not uncommon for men of his generation. He had applied for the marriage license for Samantha Garringer (Angeline's sister) and Arthemus Hendrixon, and he signed with his mark (X).
Both Abraham and Angeline are buried in Fairview Cemetery, Jefferson Township, Fayette County, Ohio.
IRA D. BOOCO, born in 1849; died in 1910.
ISAAC BOOCO, born November 20, 1851 in Fayette County, Ohio; died April 8, 1925 in Jeffersonville, Fayette County, Ohio. Married HARRIET S. SMITH on July 26, 1875 in Fayette County, Ohio.
ALBERT B. BOOCO. Married SARAH E. STRALEY on December 2, 1879 in Fayette County, Ohio, by D. Mann, M.G.
CYRENA BOOCO, born in 1858 in Fayette County, Ohio; died December 23, 1909 in Los Angeles, Los Angeles County, California. Married SAMUEL ALEXANDER HILL on July 28, 1880 in Fayette County, Ohio.