%Paper: 
%From: lreina@ulb.ac.be (Reina Laura)
%Date: Fri, 2 Jul 93 19:17:05 MDT

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\input harvmac.tex
\input psbox.tex
\psfordvips
\def\dirac{\partial \!\!\! /}
\def\LR{$SU(2)_L \otimes
SU(2)_R \otimes U(1)_{B-L}$}
\def\SM{$SU(2)_L \otimes U(1)_Y$}
\def\PRL{ {\sl Phys. Rev. Lett.} }
\def\PR { {\sl Phys. Rev.} }
\def\NP { {\sl Nucl. Phys.} }
\def\PL { {\sl Phys. Lett.} }
\def\MPL{ {\sl Mod. Phys. Lett.} }
\Title{\vbox{\baselineskip12pt\hbox{ULB-TH-08/93}
\hbox{July 93}}}
{\vbox{\centerline{Spontaneous Baryogenesis with}
	\vskip6pt\centerline{Observable CP Violation}}}

\bigskip
\centerline{L. Reina\footnote{$^\natural$}{Chercheur IISN, e-mail:
lreina@ulb.ac.be} and
M. Tytgat\footnote{$^\flat$}{Aspirant FNRS, e-mail: mtytgat@ulb.ac.be}}
\bigskip
\centerline{Service de Physique Th\'eorique}
\centerline   {Universit\'e Libre de Bruxelles, CP 225}
\centerline{Boulevard du Triomphe, B-1050 Bruxelles, Belgium}
\vskip 1in
Using a left-right symmetric model with Spontaneous CP Violation and
the hypothesis of a weakly first order electroweak phase transition we
derive a relation between the produced baryon asymmetry and the
observed parameter $\varepsilon$, describing CP violation in the $K$
system.  \vskip .3in
%\draft
\vfill
\break

\newsec{Introduction.}

The Electroweak Baryogenesis models came as a response to the
discovery of the existence in the Early Universe of rapid baryon
number violating anomalous processes\ref\manton{F.Klinkhamer and N.
Manton, \PR {\bf D30}, 2212 (1984)}, which could have erased any
pre-existing asymmetry. In this way the problem of preserving some
post-inflationary baryon asymmetry until now \ref\fuku{M. Fukugita and
T. Yanagida, \PR {\bf D42} (1990) 1285; J. Harvey and M.Turner,
\PR{\bf D42} (1990) 3344; B. Campbell, S. Davidson, J.Ellis and K.
Olive, \PL{\bf B256} (1992) 457} can be replaced by the problem of
generating the baryon asymmetry of the universe (BAU) at the latest
possible stage: the electroweak phase transition (EWPT) \ref\all{V.
Kuzmin, V.  Rubakov and M. Shaposhnikov, \PL{\bf B155}, 36 (1985)}.
This scheme has also opened the exciting possibility to further test
the standard model (SM) and probe its numerous extensions.

The task of baryogenesis is twofold: first one has to insure that
Sakharov's conditions \ref\sakh{A.D. Sakharov, JETP Lett. {\bf 6},24
(1967) } are fulfilled and then check that an efficient mechanism is
realised. It is very nice and intriguing that the SM can satisfy by
its own the former point: C and CP violation, B non-conservation and
an out-of-equilibrium stage through a first order phase transition
{\all}. Whether this model is able to generate the right asymmetry is
still an open question \ref\shapo2{G.R. Farrar and M.E. Shaposhnikov,
CERN preprint TH.6732 (1993)} and the most popular point of view is to
consider the baryon excess of the universe as an hint for something
standing beyond the SM (see
\ref\andiii{A. G. Cohen, D.B. Kaplan and A.E. Nelson, UCSD-PTH-93-02 preprint,
(1993)} and \ref\turok{N. Turok,Imperial/TP/91-92/33 preprint (1992)} and
references therein).

In this letter we investigate the left-right model (LR) based on the
gauge group $SU(2)_L
\otimes SU(2)_R \otimes U(1)_{B-L}$, already considered in
\ref\moh{R.N. Mohapatra and
X. Zhang, \PR {\bf D46}, 5331 (1992)} and \ref\moi{J.-M. Fr\`ere, L.
Houart, J.M.  Moreno, J. Orloff and M. Tytgat, CERN preprint TH.6747
(1992)}. One of the most interesting suggestions is that, if CP is
spontaneously violated at the electroweak symmetry breaking scale,
then it should be possible to relate the baryon asymmetry of the
universe (BAU) to the low energy CP violation phenomenology and in
particular to the precisely measured $\varepsilon$ parameter in the
$K^0$ system.

We think this model deserves further attention since it offers a
common source to the, until now, only two {\it well measured}
manifestations of CP violation in nature.

In {\moi} it was supposed that the electroweak phase transition was
strongly first order occurring through nucleation and expansion of
bubbles of true vacuum with {\bf thin} walls. The reflection of quarks
on the moving wall in a CP violating way creates a charge excess which
is converted into a baryon excess by the rapid B violating processes
occurring in the symmetric phase\foot{This is the so called Charge
Transport Mechanism proposed in \ref\andy1{A.G. Cohen, D.B. Kaplan and
A.E. Nelson, \NP {\bf B373}, 453 (1992)}}.

Here we will follow the opposite hypothesis, {\it i.e.} that the
expanding bubbles have a {\bf thick} wall. The appropriate mechanism
here is the so called Spontaneous Baryogenesis
\ref\andi2{A.G. Cohen, D.B. Kaplan and A.E. Nelson, \PL {\bf 245B},
561 (1990)} in which
the BAU is created within the wall. We will get a strong upper bound
for the produced baryon asymmetry in terms of the $\varepsilon$
parameter and purely dynamical quantities ($\kappa$, $m_{top}$, $M_2
\approx M_R$ and the signature of the quark mass matrices). If we
further suppose that this LR model `` saturates" $\varepsilon$, we get
the nice result that the BAU and Re $\varepsilon$ must have the same
sign.


\newsec{Spontaneous CP violation in the LR model}

The matter content of the model and the symmetry properties imposed on
the Lagrangian are given in the Appendix. The salient points are:

\item{-} the explicit LR symmetry is broken to the SM one at a scale
$M_R$ = O(TeV)
through the $vev$ of a scalar field transforming as a triplet under
$SU(2)_R$;
\item{-} the electroweak symmetry is broken through the $vev$ of a
scalar bi-doublet
$i.e.$ a field in the $(1/2, 1/2, 0)$ representation of the gauge
group.

\noindent The Yukawa couplings of quarks read:
\eqn\yuk{{\cal L}_{\hbox{{Yukawa}}} = - \Gamma_{ij}\, \bar q_{iL}\, \phi
\, q_{jR} - \Delta_{ij}\, \bar q_{iL}\tilde \phi\, q_{jR}\quad , }
where $\phi$ is the bi-doublet field, with $vev$:
\eqn\bida{\langle\phi\rangle = \left(\matrix{v & 0 \cr 0 & w \cr}\right)}
and $\tilde\phi \equiv \sigma_2 \phi^*\sigma_2$. The coupling matrices
$\Delta$ and $\Gamma$ are real and symmetric due to the assumed CP
invariance of the original Lagrangian.

Spontaneous CP violation occurs if $v$ and $w^*$ are relatively
complex.  After a $SU(2)_L$ or $SU(2)_R$ transformation it is possible
to express this by one single phase $\alpha$:
\eqn\bidb{\langle\phi\rangle = e^{i\,\alpha/2}\left(\matrix{|v| & 0
\cr 0 & |w| \cr}\right)}
\noindent As noted in \ref\branco{G.C. Branco and L. Lavoura, \PL {\bf
165B}, 327 (1985)}
the simplest model cannot give rise to a non trivial phase without
fine tuning, but the minimal extension of adding a pseudo-scalar
singlet suffices to solve the problem. This is briefly summarised in
the Appendix.

In the chosen phase convention the mass matrices of the quarks are
given by
\eqn\mass{\eqalign{M^{(u)} &= v \Gamma + w^* \Delta = |v|
e^{i\,\alpha/2}
(\Gamma + r e^{-i\,\alpha} \Delta)\cr M^{(d)} &= w \Gamma + v^* \Delta = |v|
e^{i\,\alpha/2}(r\Gamma + e^{-i\,\alpha} \Delta)\cr}} with $w/v^* =
|w/v|\, e^{i\,\alpha}= r\,e^{i\,\alpha} $. These are complex symmetric
matrices which can be diagonalized by two unitary matrices:
\eqn\rot{\eqalign{M^{(u)} &= e^{i\,\alpha/2} U D^{(u)} U^T \cr
M^{(d)} &= e^{i\,\alpha/2} V D^{(d)} V^T \cr}} At the difference of
the SM the L and R quarks are not rotated independently. This has as
important consequence that the mixing matrices $K_L$ and $K_R$ (the
generalisation of the KM matrix) are not independent and in the basis
(phase convention) we have chosen they are related by \eqn\KM{K_L =
U^{\dag}V = K_R^*} Note that the factorisation of $e^{i\,\alpha/2}$ in
{\rot} leaves {\KM} invariant. In the same way one can factorize
$e.g.$ $|v|$ without changing the rotation matrices $U$ and $V$. Note
also that, for the case of $N_f$ flavours, they are $(N_f^2 - N_f +1)$
physical phases in the mixing matrices and so it is possible to have
CP violation already with two generations. The nicety of the LR model
with spontaneous CP violation is that all the phases are functions of
$\alpha$ and that in the limit of small $r \sin \alpha$ they are
analytically calculable.  Furthermore, an enhancement of the
$\Delta$S$=2$ channel ensures a small value of
$\varepsilon^\prime/\varepsilon$ for dynamical reasons
\ref\JMJM{G. Branco, J.-M. Fr\`ere and J.-M. Gerard, \NP {\bf B221},
317 (1983)}.

The key point, as remarked by Chang \ref\chang{D. Chang, \NP {\bf
B214}, 435 (1983)}, is that in the limit where $r\sin\alpha$ goes to
zero in {\mass} there is no CP violation. With the hypothesis of small
$y = r\sin\alpha$ ($a\,posteriori$ verified) one may expand $K_L$ as
follows
\eqn\dev{K_L = e^{-i\,\alpha/2}(K_0 + i y K_1) + O(y^2)}
where the lowest order matrix $K_0$ is real and experimentally known
being equal to $\vert K_{KM}\vert$, up to signs. As shown in {\chang}
$K_1$ is calculable as function of the measured mixing angles and
quark masses.

We will now show  that it is possible to use the vacuum (at T=0)
results to describe the features of CP violation during the EWPT, provided:
\item{-}$r$ is constant during the phase transition;
\item{-}$\alpha$ is small.

\noindent Let us define from {\mass} the following matrices:
\eqn\masst{\eqalign{\tilde M^{(u)} &= \Gamma + r e^{-i\,\alpha} \Delta
=\Gamma +r\cos
\alpha \Delta - i r \sin \alpha \Delta = U \tilde D^{(u)}U^T\cr
                   \tilde M^{(d)} &= r\Gamma + e^{-i\,\alpha} \Delta
=e^{-i\alpha}(r\cos \alpha\Gamma + \Delta + i r \sin \alpha
\Gamma)=V\tilde D^{(d)}V^T\cr}} As the $\tilde D$'s are dimensionless
they can only be function of $r$ and $\alpha$. Chang's expansion of
$U$, $V$, $D^{(d)}$ and $D^{(u)}$ is given by
\eqn\matexp{\eqalign{U &= U_0 - i y U_1 + O(y^2) \cr
                     V &= e^{-i\, \alpha/2}(V_0 + i y V_1) + O(y^2)
\cr D^{(u,d)} &= D_0^{(u,d)} + i y D_1^{(u,d)} + O(y^2)\cr}} It can be
shown {\chang}, \ref\ecker{G. Ecker and W. Grimus, \NP {\bf B258}, 328
(1985)} that these matrices have the following properties:
\item{-}$U_0$ and $V_0$ are orthogonal, they diagonalize the real part
of {\masst} and
$K_0 = U_0^\dagger V_0$; \item{-}$D_1^{(u,d)}=0$;
\item{-}$U_0^\dagger U_1$ and $V_0^\dagger V_1$ are real and symmetric.

\noindent As appears from {\masst} and {\matexp} if $r$ is constant during
the EWPT and $\alpha$ is small, $U_{0}$, $V_0$ and $\tilde D_0^{(u,d)}$ are
constant to order $\alpha$. Moreover one has from {\ecker}:
\eqn\eck
{\eqalign{(1-w_{\alpha}^2)(U_0^\dagger U_1)Q{ij} & = {(K_0
D^{(d)}K_0^T)_{ij}\over m_i^{(u)}+ m_j^{(u)}} - 1/2\, w_\alpha
\delta_{ij}\approx{(K_0 \tilde D_0^{(d)}K_0^T)_{ij}\over \tilde m_i^{(u)}+
\tilde m_j^{(u)}} - 1/2\,
 r  \delta_{ij}\cr  (1-w_{\alpha}^2)(V_0^\dagger V_1)Q{ij} &
= {(K_0^T D^{(u)}K_0)_{ij}\over m_i^{(d)}+ m_j^{(d)}} - 1/2\, w_\alpha
\delta_{ij}\approx{(K_0^T \tilde D_0^{(u)}K_0)_{ij}\over \tilde m_i^{(d)}+
\tilde m_j^{(d)}} - 1/2\, r \delta_{ij} \cr}}
where $w_\alpha = r
\cos\alpha \approx r$ to the same order.
One concludes that $U_0^\dagger U_1$ and $V_0^\dagger V_1$ are also
almost constant during the EWPT. Hence the matrices $U_{(0,1)}$,
$V_{(0,1)}$ and $\tilde D^{(u,d)}$ introduced above are the same as in
the $T=0$ vacuum. All the effects of CP violation during the EWPT can
be parametrized by the sole variation of the phase $\alpha$ or more
properly of the small expansion parameter $y$. This property will be
extensively used below but we want to show first that our hypotheses
of constant $r$ and small $\alpha$ may indeed be easily satisfied.

The $vev$ of $\alpha$ and $r$ are given by the minimisation of the
most general scalar potential compatible with LR and CP symmetry (see
the Appendix) minimally extended to give rise to spontaneous CP
violation {\branco}:
\eqn\potalpha{\tan \alpha = {C_1 \eta^2 \over C_2 \eta^2 + B_4\,\sigma_R^2}}
and with $\tan s \equiv 1/r$,
\eqn\potr{\tan 2s = {(B_4 \cos\alpha)\, \sigma_R^2 + (C_1 \sin \alpha
+ C_2 \cos \alpha)\,
\eta^2 \over A_3 \sigma_R^2}}
where $\eta$ and $\sigma_R$ are the $vev$'s of the pseudoscalar
singlet and the right scalar triplet respectively. The other
parameters are combinations of dimensionless couplings in the scalar
potential. For $\eta =O(100$ GeV), corresponding to the requirement of
spontaneous CP violation at the EW scale and $\sigma_R = O$(1 TeV), a
generic value for the R scale,
\eqn\alp{\tan\alpha \approx C_1 \eta^2 / B_4 \sigma_R^2}
has a well defined sign and is monotonically varying from $0$ to some
{\bf small} finite value as $\eta$ varies from $0$ to its $vev$ during
the EWPT. Also, as $\alpha$ is small, \eqn\rrr{\tan 2s \approx B_4 /
A_3 = \hbox{constant}.} The temperature dependent corrections to the
above results can only be logarithmic and should consequently barely
change the conclusions.


\newsec{Spontaneous Baryogenesis with Spontaneous CP Violation}
In {\moi} it was argued that the phase transition in the model we consider
could be strongly first order because of the presence of trilinear couplings in
the scalar potential (cf Appendix). However, the astonishing
complexity of the potential cannot allow us to exclude the opposite
possibility, $i.e.$ that, for some values of the parameters, the EWPT
is weakly first order, occurring through nucleation and expansion of
bubbles of true vacuum with a thick wall.

The mechanism in {\moi} used the reflection and transmission of heavy
quarks on the wall in a CP violating way. If the wall is thick, the
reflection probability at threshold is exponentially suppressed and
the mechanism breaks down. A possible alternative is to create the
baryon asymmetry within the wall. For this we will suppose that all
the processes, but baryon number violating ones, are in quasi thermal
equilibrium within the wall. This means that the thermalization time
of the particles $\tau_{th}\approx (0.25-0.08\, T)^{-1}$ {\andiii} is
significantly smaller than the characteristic time of the moving wall
$\tau_{wall} = \delta_w / v_w$, where $\delta_w$ and $v_w$ are
respectively the thickness and the velocity of the wall. The
characteristic time for baryon number violation $\tau_B$ is estimated
to be of the order $(\alpha_W^4 T)^{-1}\approx 10^6/T$ in the
symmetric phase and is much longer in the broken phase. Baryon number
violating processes are thus always out of thermal equilibrium. We
further suppose that within the wall $\ddot\alpha \ll \dot \alpha$ so
that the phase is quasi-static.

The hypothesis $\tau_{th} \ll \tau_w$ with $\ddot\alpha\ll\dot\alpha$
is referred as the {\it adiabatic regime} in \andiii. Thermal
equilibrium is maintained through the bubble wall by fast
interactions, while chemical equilibrium may not be satisfied as some
processes, among which baryon number violation, are comparatively
slower. The departure from chemical equilibrium is handled by
introducing effective chemical potentials due to the interaction with
the slowly varying phase $\alpha$.

To see how this may happen we go back to the Yukawa Lagrangian for the
quarks
\yuk. In the rest frame of the plasma, the time varying phase of the bi-doublet
scalar field may be removed by a time dependent rotation of the quarks
fields.  We choose the rotation which diagonalizes the couplings with
the scalar field
\eqn\roti{\eqalign{u_L \longrightarrow u_L^\prime &=
e^{-i\,\alpha/4}U^\dagger u_L\cr
                  u_R \longrightarrow u_R^\prime &= e^{i\,\alpha/4}
U^T u_R\cr d_L \longrightarrow d_L^\prime &= e^{-i\,\alpha/4}
V^\dagger d_L\cr d_R \longrightarrow d_R^\prime &= e^{i\,\alpha/4} V^T
d_R\cr}} where we have used the convention of {\rot} and {\KM}. The
first consequence of {\roti} is that the kinetic part of the quark
Lagrangian gives rise to a new term: \eqn\kine{{\cal L}_K
\longrightarrow {\cal L}^\prime_K\, + \,{\cal L}_{eff}} where
\eqn\eff{\eqalign{{\cal L}_{eff} = &-{1 \over 4}\left \{ \bar
d^\prime_L (\dirac\alpha)
d^\prime_L \,- \,\bar d^\prime_R (\dirac\alpha) d^\prime_R \, + \,
\bar u^\prime_L (\dirac\alpha) u^\prime_L \, -\,\bar u^\prime_R
(\dirac\alpha) u^\prime_R\right \}\cr &+ \,i\,\bar
d^\prime_L(V^\dagger\dirac V)d^\prime_L \,+ \,i\,\bar
d^\prime_R(V^T\dirac V^*)d^\prime_R \cr &+
\,i\,\bar u^\prime_L(U^\dagger\dirac U)u^\prime_L \,+ \,i\,\bar
u^\prime_R(U^T\dirac U^*)u^\prime_R}} Now we use the expansions
{\matexp} of $U$ and $V$ to get:
\eqn\effii{\eqalign{{\cal L}_{eff} = &-{1 \over 4}\left \{ -\bar
d^\prime_L (\dirac\alpha)
d^\prime_L \,+ \,\bar d^\prime_R (\dirac\alpha) d^\prime_R \, + \,
\bar u^\prime_L (\dirac\alpha) u^\prime_L \, -\,\bar u^\prime_R
(\dirac\alpha) u^\prime_R\right \}\cr &-\,\bar d^\prime_L (V_0^\dagger
V_1\dirac y) d^\prime_L\, + \,
\bar d^\prime_R (V_0^\dagger V_1\dirac y)^* d^\prime_R \cr
& -\,
\bar u^\prime_L (U_0^\dagger U_1\dirac y) u^\prime_L \, +\,
\bar u^\prime_R (U_0^\dagger U_1\dirac y)^* u^\prime_R\cr}}
where we have used that $U_{(0,1)}$ and $V_{(0,1)}$ are constant
matrices. As $y$ is time dependent, this new part of the Lagrangian is
of the form
\eqn\potchem{{\cal L}_{eff} = - \sum_i \mu_i\, \bar q_i\gamma_0 q_i }
where the summation is over flavours and chiralities, and $\mu$ is
proportional to $\dot y $: the $\mu$'s look like chemical potentials
{\andiii} though being dynamical quantities rather than Lagrange
multipliers as in thermodynamics.

The quark distributions will try to minimize the effective potential,
possibly through baryon number violating processes, that could lead to
a net baryon excess at the end of the EWPT {\andiii}. However we now
show why this does not happen here. First, rapid chirality changes
occur both by emission and absorption of real scalar particles and by
interaction with the chirality non-conserving external scalar field.
Consequently baryon violating processes act equally on left-handed and
right-handed quarks. Then the baryon creation rate is
proportional to the rate for B violation times the sum of the
effective chemical potentials or, in the above formulation, to the
trace of {\it all} the matrices in {\effii} : \eqn\naivB{\eqalign{\dot
n_{B} \propto \Gamma_B\,
\biggl\{   \left(3/4\dot\alpha - \dot y Tr(V_0^\dagger V_1)\right)_{d_L} &+
\left(-3/4\dot\alpha + \dot y Tr(V_0^\dagger V_1)^*\right)_{d_R}\cr
\left(-3/4\dot\alpha + \dot y Tr(U_0^\dagger
U_1)\right)_{u_L} &+ \left(3/4\dot\alpha - \dot y Tr(U_0^\dagger
U_1)^*\right)_{u_R} \biggr\}\cr}} Finally, as $U_0^\dagger U_1$ and
$V_0^\dagger V_1$ are real matrices, the different contributions
cancel to zero. This can be understood by the fact that parity
violation is one of the ingredient required for baryogenesis {\sakh}:
the cancellation between the left-handed and the right-handed terms is
then a remnant of the P-conserving structure of our LR model. The
following figure gives an equivalent (naive) illustration of this
phenomenon: $${\psboxscaled{650}{LRfigure.1}} $$

The previous conclusions would be disastrous if not for the presence
of a second contribution. Indeed our rotation, being chiral, has a
gauge anomaly. The Lagrangian is then modified as
\eqn\anomaly{{\cal L} \longrightarrow {\cal L}+{\cal L}_\Theta\,={\cal
L}\,+\,{\Theta (x) \over 16 \pi^2} TrW_{\mu\nu}\tilde W^{\mu\nu}}
where \eqn\thet{\Theta (x)	= \hbox{Arg det} M^{(u)}M^{(d)}} and
$W_{\mu\nu} = g / 2i \sum_a \lambda_a W_{\mu\nu}^a $, $\lambda_a/2$
being the gauge group generators in the appropriate representation. A
priori such a term arises for each gauge group of the model, but the
only ones relevant for baryogenesis are $SU(2)_L$ and $SU(2)_R$
because they are chiral and have a non-trivial topological structure.
Moreover, as the $SU(2)_R$ symmetry is broken, there is a high energy
barrier (a R-sphaleron {\moi}) between the different R-vacua and
topological fluctuations are exponentially suppressed in this sector
of the model at the electroweak temperature. So, as in the SM,
``rapid" baryon number violation will occur through configurations of
$SU(2)_L$.

Note that {\anomaly} also gives a contribution in the strong sector.
Because parity is spontaneously broken, the LR model belongs to a
class of models in which the $\Theta_{strong}$ is {\it a priori}
calculable, but the final value of $\Theta$ we will obtain largely
exceeds the bound from the $e.d.m.$ of the neutron
\ref\marciano{S.M. Barr and W. J. Marciano, in CP Violation, C.
Jarlskog ed., World
Scientific, 455 (1988)}. Some fine tuning -- $i.e.$ the introduction
of a non zero $\Theta_{strong}$ bare -- is thus necessary in order to
cancel the $\Theta_{strong}$. As our mechanism of baryogenesis is only
sensitive to the {\it variation} of $\Theta$, this will not affect our
final result.

{}From the anomaly equation
\eqn\ano{\partial_{\mu} j^\mu_5\, = -\,{1\over 8 \pi^2}\, Tr W^L_{\mu\nu}\tilde
W_L^{\mu\nu}} we deduce the divergence of the baryonic current:
\eqn\barcur{\partial_{\mu} j^\mu_B\, =-{N_f \over
2}\partial_{\mu} j^\mu_5\, = {N_f\over 16 \pi^2}\, Tr
W^L_{\mu\nu}\tilde W_L^{\mu\nu}} where $N_f$ is the number of
flavours\foot{Notice that if we take into account the anomalous
contribution of the right gauge bosons, then {\barcur} becomes
\eqn\barcuri{\partial_{\mu} j^\mu_B\, = {N_f\over 16 \pi^2}\lbrace
Tr W^L_{\mu\nu}\tilde
W_L^{\mu\nu} - Tr W^R_{\mu\nu}\tilde W_R^{\mu\nu} \rbrace.} As the
$\Theta$-term acts the same for the L and R sector their contributions
would cancel as is the case for the effective chemical potentials. The
breaking of the LR symmetry at an higher scale is really crucial here
in order to satisfy Sakharov's conditions.}. Using the anomaly
equation with {\anomaly} we may get an effective potential for the
baryon number density
\eqn\potB{{\cal L}_\Theta = {\Theta \over N_f}\, \partial_\mu j_B^\nu = -\,
{\dot\Theta\over N_f}\, j^0_B.} The fermions will try to minimize the
potential {\potB} through baryon number violating processes. This, as
pointed in \andiii, is equivalent to the introduction of chemical
potentials constraining the processes in a system slightly
out-of-chemical-equilibrium.

The master equation for the baryon number density is then given by
\ref\willy{M. Dine, O. Lechtenfeld, B. Sakita, W. Fishler and J.
Polchinsky
\NP {\bf B342}, 381 (1990)} \eqn\botlz{{d \,n_B \over d\, t}\, =\, - 3 N_f
{\Gamma_B \over T}{\partial F \over \partial B}} where $F$ is the free
energy of the system.  If $\Theta$ is quasi static during the EWPT,
then ${\partial F /
\partial B}\approx {\partial V / \partial j^0_B}= \dot\Theta /N_f$.

As we do not know the details of the phase transition, hence the exact
time dependence of $\Theta$, and as little is known on the exact rate
for B violation in the intermediate regime of the phase transition, we
will make the following simplifications:
\item{-}the rate per unit volume will be taken to be constant
and the same as in the symmetric phase, $\Gamma_B = \kappa
\,\alpha_W^4 T^4$ were $\kappa$ parametrizes our ignorance of the
exact rate but has been estimated from numerical simulations to be
O(1);
\item{-}the variation of $\Theta$ during the EWPT
takes its maximal value.

\noindent These assumptions clearly give a very conservative upper
bound for the
baryon number produced during the phase transition. Using the
expression {\rot} for the mass matrices and the expansion {\matexp}
for the rotation matrices, we have the following expression for
$\Theta (x)$:
\eqn\thetaexp{\eqalign{\Theta (x) &= \hbox{Arg det}
M^{(u)}M^{(d)} = \hbox{ImTr ln}M^{(u)}M^{(d)}\cr &= \hbox{ImTr
ln}\lbrace 1 - 2\, i\, y(x) (U_0^\dagger U_1 - V_0^\dagger V_1)\rbrace
+ O(y^2)\cr &= 2\, y(x)\, Tr(V_0^\dagger V_1 - U_0^\dagger U_1)\cr}}
where we have used the small $y$ approximation and the properties of
the matrices given in the previous section. Note also that
$U_0^\dagger U_1$ and $V_0^\dagger V_1$ are explicitly given by {\eck}
and are {\it a priori} experimentally accessible quantities.

With the above assumptions we get the following upper bound on the
baryon-to-entropy ratio:
\eqn\BAU{\eqalign{{n_B \over s} &\leq {n_B \over s}\bigr\vert_{max} =
- 3\,\, {\kappa
\alpha_W^4 T^3 \over s}\, (\Theta_f - \Theta_i)\cr  &\leq - \kappa \,
{135 \over g_{*S}
\pi^2}\, \alpha_W^4\, r \sin \alpha\, Tr(V_0^\dagger V_1 - U_0^\dagger
U_1)\cr}}
where we have used the fact that $\sin \alpha$ starts from zero (see
{\alp}) and $s = 2
\pi^2 /45 g_{*S} T^3$ is the entropy density. Note that {\BAU} does
not depend of the bare value of $\Theta$.

\newsec{Phenomenological Implications:
{\bf $\varepsilon$} and the BAU} The $\varepsilon$ parameter as
predicted in the LR model with spontaneous CP violation is given by
{\ecker}, \ref\JM{J.-M. Fr\`ere, J. Galand, A. Le Yaouanc, L. Olivier,
O.  P\`ene and J.-C. Raynal \PR {\bf D46}, 337 (1992)}:
\eqn\epsi{\varepsilon =
\varepsilon_{SM} + \varepsilon_{LR}} where
\eqn\epssm{\varepsilon_{SM} = e^{i\,(\pi/4)}\, 1.34\, s_2\, s_3 \sin
\delta \left\{ 1 +
860\, S\left ( {m_t^2\over M_1^2}\right)s_2\, \hbox{Re}V_{ts}\right\}}
and
\eqn\epslr{\varepsilon_{LR} = -
e^{i\,(\pi/4)}\,0.36\sin(\delta_2-\delta_1)\left [
{M_2\over \hbox{1.4 TeV}}\right ]^2\left\{1 + 0.05
\ln\left[{M_2\over\hbox{1.4 TeV}}\right]\right\}.} Some remarks are in
order here:
\item{-}the expression for $\varepsilon_{SM}$ is the same as the one
calculated in the
standard model from the KM matrix with three generations. $M_1$ is
essentially the $W_L$ boson of the SM. The product $s_2\, s_3 \sin
\delta$ is calculable in the LR model and is linear in $y =
r\sin\alpha$ {\ecker};
\item{-}In the LR model, there is  CP violation
in the $K_0-\bar K_0$ system already with two generations. In that
case, there are three CP violating phases ($\gamma$, $\delta_1$ and
$\delta_2$) in the mixing matrices which are also calculable functions
of $r$ and $\alpha$. In the case of three generations, the LR
contribution of the third generation is usually negligible in
comparison to the one of the first two families. Moreover the LR
contribution with the first two generations usually dominates the SM
one for not too heavy $W_R
\approx W_2$ {\JM};
\item{-} There is a subtlety in the fact that in a LR model the
sign of the quark masses are observable. One can remove them from the
mass matrices by doing a rotation on the right-handed quarks to the
cost of effects in the $K_R$ mixing matrix. So there is actually a
discrete set of $2^{2 N_f -1}$ different models. This clearly weakens
the predictability of the model {\JM}: for example for some signatures
there are cancellations in $\varepsilon_{LR}$ and $\varepsilon_{SM}$
so that the third generation dominates in {\epsi}\JM.


\noindent The interesting point for us is that $\varepsilon$ is
linear in $y = r \sin\alpha$, just as is our upper bound for the BAU.
For the reasons given above, a complete analysis of the consequences
for the LR model requires some care. Nevertheless, if we suppose that
the LR model with two generations ``saturates" $\varepsilon$ in the
sense that $\varepsilon_{LR}$ dominates $\varepsilon_{SM}$, we have
the following expressions:
\eqn\domin{\eqalign{\delta_2 - \delta_1 =& {r \sin\alpha \over
1 - w_\alpha^2}\biggl [ {m_u c^2 + m_c s^2 \over m_d} - {m_u s^2 + m_c
c^2 \over m_s} \cr &+ 2{ m_d - m_s \over m_u + m_c} + 2 {m_u - m_c
\over m_d + m_s} (s^2 - c^2)\biggr ] \cr
\approx & {r \sin\alpha \over 1 - w_\alpha^2}
\biggl [ 7\, sign\left ({m_c \over m_s}\right
) + 6 \,sign \left ({m_c \over m_d}\right) \biggr ]\cr}} and
\eqn\domini{\eqalign{r \sin\alpha\, Tr(V_0^\dagger V_1 - U_0^\dagger
U_1) =& {1 \over 2}{r
\sin\alpha \over 1 - w_\alpha^2}\biggl [c^2 \left ({m_u \over m_d} - {m_d \over
m_u}\right ) + c^2 \left ({m_c \over m_s} - {m_s \over m_c}\right )\cr
&+ s^2
\left ({m_c \over m_d} - {m_d \over
m_c}\right ) + s^2 \left ({m_u \over m_s} - {m_s \over m_u}\right )
\biggr ]\cr
\approx & {1 \over 2}{r \sin\alpha
\over 1 - w_\alpha^2} \biggl [ 7\, sign\left ({m_c \over m_s}\right
) + 6 \,sign \left ({m_c \over m_d}\right) \biggr ]\cr}} where $s
\equiv \sin \theta_C$ is the Cabibbo mixing angle.

We finally get from {\BAU} and {\epslr}:

\eqn\fnal{{n_B \over s} \leq 135 \, {\sqrt{2}\over 0.72}\,
{\kappa \alpha_W^4 \over \pi^2 g_{*S}}\, \hbox{Re }
\varepsilon \, \left ({1.4\, \hbox{TeV} \over M_2 }\right )^2 \left [1
-0.05 \ln \left({
M_2 \over 1.4 \,\hbox{TeV}}\right ) \right ]}

\noindent There are only two unknowns in the upper bound $i.e.$
$\kappa$, describing
the amount of B violation and $M_2$, the mass of the charged
right-handed $W$: these are dynamical parameters independent of CP
violation. $\hbox{Re } \varepsilon$ is observable and the expression
{\fnal} predicts that it must have the same sign as the BAU. {}From the
experimental value of $\hbox{Re } \varepsilon$ we get also an upper
bound on the mass $ M_2$ \ref\kolb{E. Kolb and M. Turner, {\it The
Early Universe}, Addison--Wesley, New York (1990)}: \eqn\num{4\,
10^{-11}\leq {n_B \over s} < 0.7\,\kappa \left ({1.4\,\hbox{TeV} \over
M_2 }\right )^2 10^{-9}} or
\eqn\bound{M_2 < 6\, \sqrt{\kappa}\, \hbox{TeV}}
where $\kappa$ is usually estimated to be O(1). This upper bound is
consistent with the one needed to satisfy $\varepsilon \approx
\varepsilon_{LR}$ which is $M_2 \leq 19$ TeV
\JM.



\newsec{Conclusions}

The LR symmetry coupled to spontaneous CP violation offers the
interesting opportunity to describe in a unified framework the
generation of the baryon asymmetry and the LR phenomenology of the
$K_0-\bar K_0$ system. The existence of one unique source of CP
violation naturally permits to relate $n_B/s$ and $\varepsilon$ and
this in agreement with the observations.

The upper bound on the baryon asymmetry we have obtained rests on very
few assumptions.  The most important ones are that the phase
transition is weakly first order and that spontaneous CP violation
occurs at the electroweak scale. Their possible validity is hidden by
the complexity of the scalar potential, in which we do not think it is
useful to go much further at this stage. Our very conservative bound
still indicates that the mechanism of spontaneous baryogenesis may be
in difficulty if the R scale is too high ($M_2 >$ few TeV). However we
have only considered the ``natural" case where the LR model
``explains" CP violation. Whether the conclusion is the same for
others signatures is an open question. Linked to this is the agreement
in sign we have found between $n_B/s$ and $\hbox{Re }
\varepsilon$ which could possibly not be satisfied in all the cases. Clearly
this must be complemented by a conjoint analysis of the consequences
on $\varepsilon^\prime$ and possibly the electric dipole moment of the
neutron.

Also it could be interesting to return to the thin wall case in which
baryogenesis is more efficient {\andiii} to see if it is possible to
obtain a similar relation between the sign and magnitude of $B$ and
$\varepsilon$.



\newsec{Acknowledgements}
We would like to thank J.-M. Fr\`ere and Ph. Spindel for discussions
and encouragements.


\appendix {}{} The model is based on the gauge group \LR with quarks
and leptons in fundamental representations
\eqn\rep{\eqalign{Q_L &= \left (\matrix{u \cr d}\right )_L \hskip15pt ({1/2, 0,
1/3})
\hskip40pt Q_R = \left (\matrix{u \cr d}\right )_R \hskip15pt ({0, 1/2,
1/3})\cr Le_L &= \left (\matrix{\nu \cr e}\right )_L \hskip15pt ({1/2,
0, -1})
\hskip40pt Le_R = \left (\matrix{\nu \cr e}\right )_R \hskip15pt ({0,1/2,
-1})\cr}} The generalisation of the Gell-Mann-Nishijima relation is $Q
= T_{3L} + T_{3R} + {B-L \over 2}$.  The two $SU(2)$ coupling
constants $g_L$ and $g_R$ are set to be equal by imposing the discrete
symmetry L $\longleftrightarrow$ R.


To break the symmetries down to $U(1)_Q$ different Higgs mutliplets
are introduced
\ref\desh{N.G. Deshpande, J.F. Gunion, B. Kayser and F. Olness,
 \PR{\bf D44}, 837 (1991)}:

\eqn\higgs{\eqalign{\Delta_L =(1,0,
2) \hskip15pt & \hskip15pt \Delta_R =(0,1, 2)\cr
\phi=(1/2,&1 / 2,0)\cr}}
The introduction of the scalar bi-doublet imposed by the LR structure
of the model is fundamental in the discussion of spontaneous CP
violation and makes the difference with respect to models using two
doublets for baryogenesis. The breaking of {\LR} occurs in two steps:
firstly through the vacuum expectation value of the triplet $\Delta_R$
at some O(TeV) scale; secondly \SM is broken to $U(1)_Q$ at the
electroweak scale O(100 GeV) through the $vev$ of the bi-doublet
field.

 For completeness we also sketch here the results obtained in
{\branco}. Their objective is to know if spontaneous CP violation is
possible with the matter content given above. For this the following
transformation properties under P and CP are imposed on the scalar
fields:
\eqn\P{\hbox{P} : \phi_i \longrightarrow \phi_i^\dagger\hskip15pt;
\hskip15pt\Delta_L \leftrightarrow \Delta_R}
and
\eqn\CP{\hbox{CP}: \phi_i \rightarrow \phi_i^* \hskip15pt;
\hskip15pt\Delta_L \rightarrow \Delta_L^*\hskip15pt;
\hskip15pt\Delta_R \rightarrow \Delta_R^*}
where $\phi_1 \equiv \phi$ and $\phi_2 \equiv \tilde \phi=\tau_2
\phi^* \tau_2$.  Imposing P and CP to be symmetries of the scalar
potential
\eqn\scapot{V = V_{\phi} + V_\Delta + V_{\Delta\phi}} give constraints on the
couplings \branco. The $vev$ of the scalar mutltiplets are set to
\eqn\vevs{\langle\phi\rangle = e^{i\,\alpha/2}\left(\matrix{|v| & 0 \cr 0 &
|w| \cr}\right)} and
\eqn\vevss{\langle\Delta_L\rangle = \left(\matrix{0 & 0 \cr \sigma_L
e^{i \delta} &
0 \cr}\right)\hskip30pt\langle\Delta_R\rangle = \left(\matrix{0 & 0
\cr \sigma_R & 0 \cr}\right)} It is shown in {\branco} that, without
fine tuning, $\alpha$ must be $0$ or $\pi$. In order to have
spontaneous CP violation a simple extension is proposed: a
pseudoscalar singlet is introduced which transforms under P and CP as
follows:
\eqn\PCP{\hbox{P}: \eta \rightarrow -\eta \hskip50pt \hbox{CP}: \eta
\rightarrow -\eta
} This allows one to introduce in the scalar potential the following
additional terms
\eqn\poten{V_{\eta\phi} = i\, C_1 \,\langle \eta\rangle \,\eta\,
Tr(\phi_2^\dagger
\phi_1 - \phi_1^\dagger \phi_2)\,+\, C_2\, \eta^2 Tr(\phi_2^\dagger \phi_1 +
\phi_1^\dagger \phi_2)}
which contain a term odd in $\alpha$. Then a non zero $vev$ is
obtained for $\alpha$ by minimizing the potential:
\eqn\tanalpha{\tan \alpha = {C_1 \eta^2 \over C_2 \eta^2 + B_4\,\sigma_R^2}}
where $B_4$ is simply a combination of couplings from
$V_{\Delta\phi}$.  As this is used in the body of the paper we also
give the expression for the ratio $|v/w| =1/r = \tan s$:
\eqn\tandeuxs{\tan 2s = {(B_4 \cos\alpha)\, \sigma_R^2 + (C_1 \sin
\alpha + C_2 \cos
\alpha)\, \eta^2 \over A_3 \sigma_R^2}}
where as above $A_3$ comes from the dominant terms in
$V_{\Delta\phi}$.

Other interesting possibilities to have spontaneous CP violation are
given in {\branco}, but we have only considered the simplest one.





\listrefs
\end



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Adobe_cmykcolor_vars /_setrgbcolor get exec
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Adobe_cmykcolor_vars /_currentrgbcolor get exec
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{
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aload pop pop
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currentpacking true setpacking
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% modifyEncoding ==> [ modified array ]
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	/_pntr 0 ddef

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		% get bottom object
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		{
			% exit
			pop pop exit
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% re-define font
% expected arguments
% for 'normal fonts :
% [ /_Helvetica-Bold/Helvetica-Bold direction fontScript defaultEncoding TZ
%
% for cartographic, pictographic, and expert fonts :
% [ ... number value stream ... /_Symbol/Symbol
%	direction fontScript defaultEncoding TZ
% for blended fonts w/ default encoding :
% [ /_AdobeSans_20ULig1XCond-Bold/AdobeSans
%	direction fontScript defaultEncoding [ w0 w1 ... wn ] TZ
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% [ ... number value stream ... /_AdobeSans_20ULig1XCond/AdobeSans
%	direction fontScript defaultEncoding [ w0 w1 ... wn ] TZ
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{
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def
% text painting operators
/tr					% string tr ax ay string
{
_ax _ay 3 2 roll
} def
/trj				% string trj cx cy fillchar ax ay string
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx	% text							%
   	% textString Tx -
	{
	dup
	currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	} ddef
/Tj	% justified text				% textString Tj -
	{
	dup
	currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	} ddef

} def
/a1
{
/Tx	% text
% textString Tx -
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj	% justified text				% textString Tj -
	{
	dup currentpoint 4 2 roll gsave
	dup currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef

} def
/e0
{
/Tx	% text
% textString Tx -
	{
	tr _psf
	} ddef
/Tj	% justified text				% textString Tj -
	{
	trj _pjsf
	} ddef
} def
/e1
{
/Tx	% text
% textString Tx -
	{
	dup currentpoint 4 2 roll gsave
	tr _psf
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj	% justified text				% textString Tj -
	{
	dup currentpoint 4 2 roll gsave
	trj _pjsf
	grestore 3 1 roll moveto tr sp
	} ddef
} def
/i0
{
/Tx	% text
% textString Tx -
	{
	tr sp
	} ddef
/Tj	% justified text				% textString Tj -
	{
	trj jsp
	} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx	% text
% textString Tx -
	{
	tr sw rmoveto
	} ddef
/Tj	% justified text				% textString Tj -
	{
	trj swj rmoveto
	} ddef
} def
/r0
{
/Tx	% text
% textString Tx -
	{
	tr _ctm _pss
	} ddef
/Tj	% justified text				% textString Tj -
	{
	trj _ctm _pjss
	} ddef
} def
/r1
{
/Tx	% text
% textString Tx -
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj	% justified text				% textString Tj -
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef
} def
% font operators
% Binding
/To	% begin text 					% bindType To -
{
	pop _ctm currentmatrix pop
} def
/TO	% end text					% TO -
{
	Te _ctm setmatrix newpath
} def
% Text paths
/Tp	% begin text path
% a b c d tx ty startPt Tp -
{
	pop _tm astore pop _ctm setmatrix
	_tDict begin /W {} def /h {} def
} def
/TP	% end text path					% TP -
{
	end
	iTm 0 0 moveto
} def
% Render mode & matrix operators
/Tr	% begin render					% render Tr -
{
	_render 3 le {currentpoint newpath moveto} if
	dup 8 eq {pop 0} {dup 9 eq {pop 1} if} ifelse
	dup /_render exch ddef
	_renderStart exch get load exec
} def
/iTm % internal set text matrix
% - iTm -	(uses _tm as implicit argument)
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm % set text matrix				% a b c d tx ty Tm -
{
_tm astore pop iTm 0 0 moveto
} def
/Td % translate text matrix 		% tx ty Td -
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/Te	% end render					% - Te -
{
	_render -1 eq {} {_renderEnd _render get dup null ne {load
exec} {pop} ifelse} ifelse
	/_render -1 ddef
} def
% Attributes
/Ta	% set alignment					% alignment Ta -
{
pop
} def
/Tf	% set font name and size		% fontname size Tf -
{
dup 1000 div /_fScl exch ddef
exch findfont exch scalefont setfont
} def
/Tl	% set leading
% leading paragraphLeading Tl -
{
pop
0 exch _leading astore pop
} def
/Tt	% set user tracking				% userTracking Tt -
{
pop
} def
/TW % set word spacing
% minSpace optSpace maxSpace TW -
{
3 npop
} def
/Tw	% set computed word spacing		% wordSpace Tw
{
/_cx exch ddef
} def
/TC % set character spacing
% minSpace optSpace maxSpace TC -
{
3 npop
} def
/Tc	% set computed char spacing 	% charSpace Tc -
{
/_ax exch ddef
} def
/Ts % set super/subscripting (rise)	% rise Ts -
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti	% set indentation
% firstStartIndent otherStartIndent stopIndent Ti -
{
3 npop
} def
/Tz % set horizontal scaling		% scalePercent Tz -
{
100 div /_hs exch ddef
iTm
} def
/TA % set pairwise kerning			% autoKern TA -

%	autoKern = 0 -> no pair kerning

%			 = 1 -> automatic pair kerning
{
pop
} def
/Tq % set hanging quotes			% hangingQuotes Tq -

%	hangingQuotes 	= 0 -> no hanging quotes

%			 		= 1 -> hanging quotes
{
pop
} def
% Text Bodies
/TX {pop} def
%/Tx	% non-justified text			% textString Tx -
%/Tj	% justified text				% textString Tj -
/Tk	% kern
% autoKern kernValue Tk -

%  	autoKern = 0 -> manual kern, = 1 -> auto kern

%	kernValue = kern value in em/1000 space
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK	% non-printing kern
% autoKern kernValue TK -
{
2 npop
} def
/T* % carriage return & line feed	% - T* -
{
_leading aload pop neg Td
} def
/T*- % carriage return & negative line feed	% - T*- -
{
_leading aload pop Td
} def
/T-	% print a discretionary hyphen	% - T- -
{
_hyphen Tx
} def
/T+	% discretionary hyphen hyphen	% - T+ -
{} def
/TR	% reset pattern matrix 			% a b c d tx ty TR -
{
_ctm currentmatrix pop
_tm astore pop
iTm 0 0 moveto
} def
/TS	% special chars
% textString justified TS -
{
0 eq {Tx} {Tj} ifelse
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_IllustratorA_AI3 1.0 2
%%Title: (Adobe Illustrator (R) Version 3.0 Abbreviated Prolog)
%%Version: 1.0
%%CreationDate: (7/22/89) ()
%%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_IllustratorA_AI3 61 dict dup begin put
% initialization
/initialize				% - initialize -
{
% 47 vars, but leave slack of 10 entries for custom Postscript fragments
userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put
% paint operands
/_lp /none def
/_pf {} def
/_ps {} def
/_psf {} def
/_pss {} def
/_pjsf {} def
/_pjss {} def
/_pola 0 def
/_doClip 0 def
% paint operators
/cf	currentflat def	% - cf flatness
% typography operands
/_tm matrix def
/_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def
/_renderEnd [null null null null /i1 /i1 /i1 /i1] def
/_render -1 def
/_rise 0 def
/_ax 0 def			% x character spacing	(_ax, _ay,

% _cx, _cy follows awidthshow naming convention)
/_ay 0 def			% y character spacing
/_cx 0 def			% x word spacing
/_cy 0 def			% y word spacing
/_leading [0 0] def
/_ctm matrix def
/_mtx matrix def
/_sp 16#020 def
/_hyphen (-) def
/_fScl 0 def
/_cnt 0 def
/_hs 1 def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
% typography operators
/Tx {} def
/Tj {} def
% compound path operators
/CRender {} def
% printing
/_AI3_savepage {} def
% color operands
/_gf null def
/_cf 4 array def
/_if null def
/_of false def
/_fc {} def
/_gs null def
/_cs 4 array def
/_is null def
/_os false def
/_sc {} def
/_i null def
Adobe_IllustratorA_AI3 begin
Adobe_IllustratorA_AI3
	{
	dup xcheck
		{
		bind
		} if
	pop pop
	} forall
end
end
Adobe_IllustratorA_AI3 begin
Adobe_IllustratorA_AI3_vars begin
newpath
} def
/terminate				% - terminate -
{
end
end
} def
% definition operators
/_					% - _ null
null def
/ddef				% key value ddef -
{
Adobe_IllustratorA_AI3_vars 3 1 roll put
} def
/xput				% key value literal xput -
{
dup load dup length exch maxlength eq
	{
	dup dup load dup
	length 2 mul dict copy def
	} if
load begin def end
} def
/npop				% integer npop -
{
	{
	pop
	} repeat
} def
% marking operators
/sw					% ax ay string sw x y
{
dup length exch stringwidth
exch 5 -1 roll 3 index 1 sub mul add
4 1 roll 3 1 roll 1 sub mul add
} def
/swj				% cx cy fillchar ax ay string swj x y
{
dup 4 1 roll
dup length exch stringwidth
exch 5 -1 roll 3 index 1 sub mul add
4 1 roll 3 1 roll 1 sub mul add
6 2 roll /_cnt 0 ddef
{1 index eq {/_cnt _cnt 1 add ddef} if} forall pop
exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop
} def
/ss					% ax ay string matrix ss -
{
4 1 roll
	{				% matrix ax ay char 0 0 {proc} -
	2 npop
	(0) exch 2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	4 index setmatrix
	stroke
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
3 npop
} def
/jss				% cx cy fillchar ax ay string matrix jss -
{
4 1 roll
	{
       % cx cy fillchar matrix ax ay char 0 0 {proc} -
	2 npop
	(0) exch 2 copy 0 exch put
	gsave
	_sp eq
		{
		exch 6 index 6 index 6 index 5 -1 roll widthshow
		currentpoint
		}
		{
		false charpath currentpoint
		4 index setmatrix stroke
		}ifelse
	grestore
	moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
% path operators
/sp					% ax ay string sp -
{
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	false charpath
	2 copy rmoveto
	} exch cshow
2 npop
} def
/jsp					% cx cy fillchar ax ay string jsp -
{
	{
% cx cy fillchar ax ay char 0 0 {proc} -
	2 npop
	(0) exch 2 copy 0 exch put
	_sp eq
		{
		exch 5 index 5 index 5 index 5 -1 roll widthshow
		}
		{
		false charpath
		}ifelse
	2 copy rmoveto
	} exch cshow
5 npop
} def
% path construction operators
/pl				% x y pl x y
{
transform
0.25 sub round 0.25 add exch
0.25 sub round 0.25 add exch
itransform
} def
/setstrokeadjust where
	{
	pop true setstrokeadjust
	/c				% x1 y1 x2 y2 x3 y3 c -
	{
	curveto
	} def
	/C
	/c load def
	/v				% x2 y2 x3 y3 v -
	{
	currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y				% x1 y1 x2 y2 y -
	{
	2 copy curveto
	} def
	/Y
	/y load def
	/l				% x y l -
	{
	lineto
	} def
	/L
	/l load def
	/m				% x y m -
	{
	moveto
	} def
	}
	{%else
	/c
	{
	pl curveto
	} def
	/C
	/c load def
	/v
	{
	currentpoint 6 2 roll pl curveto
	} def
	/V
	/v load def
	/y
	{
	pl 2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
	pl lineto
	} def
	/L
	/l load def
	/m
	{
	pl moveto
	} def
	}ifelse
% graphic state operators
/d					% array phase d -
{
setdash
} def
/cf	{} def			% - cf flatness
/i					% flatness i -
{
dup 0 eq
	{
	pop cf
	} if
setflat
} def
/j					% linejoin j -
{
setlinejoin
} def
/J					% linecap J -
{
setlinecap
} def
/M					% miterlimit M -
{
setmiterlimit
} def
/w					% linewidth w -
{
setlinewidth
} def
% path painting operators
/H					% - H -
{} def
/h					% - h -
{
closepath
} def
/N					% - N -
{
_pola 0 eq
	{
	_doClip 1 eq {clip /_doClip 0 ddef} if
	newpath
	}
	{
	/CRender {N} ddef
	}ifelse
} def
/n					% - n -
{N} def
/F					% - F -
{
_pola 0 eq
	{
	_doClip 1 eq
		{
		gsave _pf grestore clip newpath /_lp /none ddef _fc
		/_doClip 0 ddef
		}
		{
		_pf
		}ifelse
	}
	{
	/CRender {F} ddef
	}ifelse
} def
/f					% - f -
{
closepath
F
} def
/S					% - S -
{
_pola 0 eq
	{
	_doClip 1 eq
		{
		gsave _ps grestore clip newpath /_lp /none ddef _sc
		/_doClip 0 ddef
		}
		{
		_ps
		}ifelse
	}
	{
	/CRender {S} ddef
	}ifelse
} def
/s					% - s -
{
closepath
S
} def
/B					% - B -
{
_pola 0 eq
	{
	_doClip 1 eq 	% F clears _doClip
	gsave F grestore
		{
		gsave S grestore clip newpath /_lp /none ddef _sc
		/_doClip 0 ddef
		}
		{
		S
		}ifelse
	}
	{
	/CRender {B} ddef
	}ifelse
} def
/b					% - b -
{
closepath
B
} def
/W					% - W -
{
/_doClip 1 ddef
} def
/*					% - [string] * -
{
count 0 ne
	{
	dup type (stringtype) eq {pop} if
	} if
_pola 0 eq {newpath} if
} def
% group operators
/u					% - u -
{} def
/U					% - U -
{} def
/q					% - q -
{
_pola 0 eq {gsave} if
} def
/Q					% - Q -
{
_pola 0 eq {grestore} if
} def
/*u					% - *u -
{
_pola 1 add /_pola exch ddef
} def
/*U					% - *U -
{
_pola 1 sub /_pola exch ddef
_pola 0 eq {CRender} if
} def
/D					% polarized D -
{pop} def
/*w					% - *w -
{} def
/*W					% - *W -
{} def
% place operators
/`					% matrix llx lly urx ury string ` -
{
/_i save ddef
6 1 roll 4 npop
concat pop
userdict begin
/showpage {} def
0 setgray
0 setlinecap
1 setlinewidth
0 setlinejoin
10 setmiterlimit
[] 0 setdash
newpath
0 setgray
false setoverprint
} def
/~					% - ~ -
{
end
_i restore
} def
% color operators
/O					% flag O -
{
0 ne
/_of exch ddef
/_lp /none ddef
} def
/R					% flag R -
{
0 ne
/_os exch ddef
/_lp /none ddef
} def
/g					% gray g -
{
/_gf exch ddef
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_gf setgray
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/G					% gray G -
{
/_gs exch ddef
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_gs setgray
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/k					% cyan magenta yellow black k -
{
_cf astore pop
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_cf aload pop setcmykcolor
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/K					% cyan magenta yellow black K -
{
_cs astore pop
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_cs aload pop setcmykcolor
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/x
% cyan magenta yellow black name gray x -
{
/_gf exch ddef
findcmykcustomcolor
/_if exch ddef
/_fc
{
_lp /fill ne
	{
	_of setoverprint
	_if _gf 1 exch sub setcustomcolor
	/_lp /fill ddef
	} if
} ddef
/_pf
{
_fc
fill
} ddef
/_psf
{
_fc
ashow
} ddef
/_pjsf
{
_fc
awidthshow
} ddef
/_lp /none ddef
} def
/X
% cyan magenta yellow black name gray X -
{
/_gs exch ddef
findcmykcustomcolor
/_is exch ddef
/_sc
{
_lp /stroke ne
	{
	_os setoverprint
	_is _gs 1 exch sub setcustomcolor
	/_lp /stroke ddef
	} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
% locked object operator
/A					% value A -
{
pop
} def
currentdict readonly pop end
setpacking
% annotate page operator
/annotatepage
{
} def
%%EndResource
%%EndProlog
%%BeginSetup
%%IncludeFont: CMR10
Adobe_cmykcolor /initialize get exec
Adobe_cshow /initialize get exec
Adobe_customcolor /initialize get exec
Adobe_typography_AI3 /initialize get exec
Adobe_IllustratorA_AI3 /initialize get exec
[
39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
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/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
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/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
/guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe
/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
/hungarumlaut/ogonek/caron
TE
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299.2598 521.7502 299.0878 524.4369 297.3247 525.9878 c
295.5616 527.5387 292.875 527.3667 291.3241 525.6037 c
S
297.6974 519.9471 m
296.1296 518.199 296.2758 515.5108 298.0239 513.943 c
299.772 512.3752 302.4601 512.5214 304.0279 514.2695 c
S
310.2865 508.5173 m
311.8374 510.2804 311.6654 512.967 309.9023 514.5179 c
308.1393 516.0688 305.4526 515.8969 303.9017 514.1338 c
S
310.3531 508.5966 m
308.7853 506.8485 308.9314 504.1604 310.6795 502.5926 c
312.4276 501.0248 315.1158 501.1709 316.6836 502.919 c
S
322.9004 497.1203 m
324.4513 498.8834 324.2793 501.57 322.5163 503.1209 c
S
322.5163 503.1209 m
320.7532 504.6719 318.0665 504.4999 316.5156 502.7368 c
S
323.0457 497.381 m
321.4779 495.6329 321.624 492.9447 323.3721 491.3769 c
325.1202 489.8091 327.8084 489.9553 329.3762 491.7033 c
S
335.2961 485.6672 m
336.847 487.4303 336.675 490.1169 334.9119 491.6678 c
S
334.9119 491.6678 m
333.1489 493.2187 330.4622 493.0468 328.9113 491.2837 c
S
335.5714 485.9792 m
334.0036 484.2311 334.1497 481.5429 335.8978 479.9751 c
S
335.8978 479.9751 m
338.55 477.5964 343.1722 479.4744 347.5174 475.5171 c
343.7577 483.4032 l
S
347.5174 475.5171 m
338.06 476.6758 l
S
284.9625 531.2011 m
286.5134 532.9642 286.3414 535.6508 284.5783 537.2017 c
S
284.5783 537.2017 m
282.8152 538.7527 280.1286 538.5807 278.5777 536.8176 c
S
272.4242 542.6956 m
270.8564 540.9475 271.0026 538.2593 272.7506 536.6915 c
S
272.7506 536.6915 m
274.4987 535.1237 277.1869 535.2699 278.7547 537.0179 c
S
272.3103 542.5597 m
273.8613 544.3228 273.6893 547.0094 271.9261 548.5603 c
S
271.9261 548.5603 m
270.1631 550.1113 267.4765 549.9393 265.9255 548.1762 c
S
259.7312 554.0069 m
258.1633 552.2588 258.3095 549.5707 260.0576 548.0029 c
S
260.0576 548.0029 m
261.8057 546.4351 264.4938 546.5812 266.0616 548.3293 c
S
322.2892 528.253 m
320.7603 530.0353 318.0761 530.2406 316.2938 528.7118 c
314.5116 527.1829 314.3062 524.4986 315.8351 522.7164 c
S
309.4838 517.1088 m
311.0298 515.3414 313.7159 515.1618 315.4833 516.7077 c
317.2508 518.2537 317.4303 520.9398 315.8844 522.7073 c
S
309.4455 517.1254 m
307.9167 518.9076 305.2324 519.113 303.4502 517.5841 c
301.6679 516.0553 301.4626 513.371 302.9914 511.5888 c
S
296.4728 506.1332 m
298.0187 504.3657 300.7048 504.1861 302.4723 505.7321 c
304.2397 507.278 304.4193 509.9642 302.8734 511.7316 c
S
296.5428 506.0568 m
295.0139 507.839 292.3296 508.0444 290.5474 506.5155 c
288.7652 504.9866 288.5598 502.3024 290.0886 500.5201 c
S
283.5293 495.112 m
285.0752 493.3445 287.7613 493.165 289.5288 494.7109 c
S
289.5288 494.7109 m
291.2963 496.2569 291.4758 498.943 289.9299 500.7104 c
S
283.7688 494.9339 m
282.2399 496.7162 279.5557 496.9215 277.7734 495.3927 c
275.9912 493.8638 275.7858 491.1796 277.3147 489.3973 c
S
270.5586 484.3145 m
272.1045 482.547 274.7907 482.3675 276.5581 483.9134 c
S
276.5581 483.9134 m
278.3255 485.4594 278.5051 488.1455 276.9592 489.9129 c
S
270.832 484.0009 m
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S
264.8366 484.4596 m
262.1326 482.14 263.3923 477.3125 258.9025 473.5201 c
267.2113 476.22 l
S
258.9025 473.5201 m
261.2838 482.7458 l
S
322.2631 528.2852 m
323.809 526.5178 326.4951 526.3382 328.2626 527.8842 c
S
328.2626 527.8842 m
330.03 529.4301 330.2096 532.1162 328.6637 533.8837 c
S
335.2934 539.2187 m
333.7645 541.0009 331.0802 541.2063 329.298 539.6774 c
S
329.298 539.6774 m
327.5158 538.1486 327.3104 535.4643 328.8393 533.6821 c
S
335.1735 539.3493 m
336.7194 537.5818 339.4056 537.4023 341.173 538.9483 c
S
341.173 538.9483 m
342.9404 540.4942 343.12 543.1803 341.5741 544.9478 c
S
348.1623 550.3295 m
346.6334 552.1117 343.9491 552.3171 342.1669 550.7882 c
S
342.1669 550.7882 m
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S
U
0 O
0 g
198.0773 499.1401 m
215.1601 512.8171 L
213.5901 513.4412 L
196.5073 499.7642 L
198.0773 499.1401 L
b
205.8337 506.2906 m
S
0 O
0 g
1 w
196.9039 510.1663 m
213.9867 523.8434 L
212.4167 524.4088 L
195.3339 510.7317 L
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f
0 R
0 G
0.5 w
204.6603 517.3169 m
S
u
u
0 To
1 0 0 1 180.6603 509.3169 0 Tp
TP
0 Tr
0 O
0 g
1 w
/_CMR10 12 Tf
14.5 0 Tl
(B) Tx
(\r) TX
TO
U
U
177.1603 520.8169 m
192.1603 505.3169 l
F
177.1603 504.8169 m
192.1603 521.8169 l
F
u
u
0 To
1 0 0 1 419.4108 508.3169 0 Tp
TP
0 Tr
(B) Tx
(\r) TX
TO
U
U
415.9108 519.8169 m
430.9108 504.3169 l
F
415.9108 503.8169 m
430.9108 520.8169 l
F
u
0 R
0 G
0.5 w
141.1051 518.3859 m
153.7111 529.7914 l
S
158.1603 533.8169 m
170.7663 545.2224 l
S
175.2155 549.2479 m
188.5631 561.3244 l
185.1733 550.166 l
S
188.5631 561.3244 m
176.8442 558.8128 l
S
U
u
141.7659 507.5219 m
153.0159 494.7769 l
S
156.9865 490.2787 m
168.2365 477.5337 l
S
172.2071 473.0354 m
184.1189 459.5407 l
173.003 463.0672 l
S
184.1189 459.5407 m
181.7513 471.2895 l
S
U
u
415.2905 459.4482 m
427.2615 471.5185 l
S
431.4865 475.7787 m
443.4575 487.849 l
S
447.6825 492.1091 m
460.3577 504.8895 l
457.576 493.5642 l
S
460.3577 504.8895 m
448.7917 501.7481 l
S
U
u
417.0353 561.3158 m
428.4558 548.7231 l
S
432.4865 544.2787 m
443.907 531.686 l
S
447.9377 527.2415 m
460.0299 513.9081 l
448.8676 517.2849 l
S
460.0299 513.9081 m
457.5046 525.6241 l
S
U
u
1 w
38.1603 416.8169 m
38.1603 602.8169 l
42.0353 594.6919 l
S
38.1603 603.3169 m
34.1603 594.4419 l
S
U
u
0 To
1 0 0 1 56.6603 593.8169 0 Tp
TP
0 Tr
0 O
0 g
/_CMR10 10 Tf
12 0 Tl
(Energy) Tx
(\r) TX
TO
U
u
0 R
0 G
239.1595 400.6065 m
386.6621 400.5798 l
378.7719 396.3579 l
S
386.6621 400.5798 m
378.9255 404.6093 l
S
U
u
0 To
1 0 0 1 286.1603 413.3169 0 Tp
TP
0 Tr
0 O
0 g
(Time arrow) Tx
(\r) TX
TO
U
u
0 R
0 G
0.5 w
391.918 552.0853 m
389.5705 552.1428 387.6208 550.2864 387.5633 547.939 c
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S
391.6228 535.1123 m
393.9706 535.0773 395.9024 536.9524 395.9373 539.3002 c
395.9723 541.6481 394.0973 543.5799 391.7494 543.6148 c
S
391.5848 535.095 m
389.2373 535.1525 387.2877 533.2961 387.2301 530.9486 c
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391.0646 518.0993 m
393.4125 518.0643 395.3442 519.9394 395.3792 522.2873 c
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S
391.1682 518.1 m
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S
390.5855 501.106 m
392.9358 501.0709 394.8629 502.7896 394.9001 505.294 c
S
394.9001 505.294 m
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S
390.8781 501.1647 m
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S
389.9221 484.2423 m
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S
394.2367 488.4302 m
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S
390.3381 484.2355 m
387.9906 484.2929 386.0409 482.4365 385.9834 480.0892 c
S
385.9834 480.0892 m
385.8961 476.5276 390.3253 474.2312 390.1364 468.3572 c
393.6937 476.3366 l
S
390.1364 468.3572 m
384.8778 476.3027 l
S
0 O
0 g
383.5803 498.915 m
400.6631 512.592 L
399.0931 513.2161 L
382.0103 499.5391 L
383.5803 498.915 L
b
391.3367 506.0655 m
S
0 O
0 g
1 w
382.4069 509.9412 m
399.4897 523.6183 L
397.9197 524.1837 L
380.8369 510.5066 L
382.4069 509.9412 L
f
0 R
0 G
0.5 w
390.1633 517.0918 m
S
U
%%PageTrailer
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%%Trailer
Adobe_IllustratorA_AI3 /terminate get exec
Adobe_typography_AI3 /terminate get exec
Adobe_customcolor /terminate get exec
Adobe_cshow /terminate get exec
Adobe_cmykcolor /terminate get exec
Adobe_packedarray /terminate get exec
%%EOF

