%Paper: hep-th/9405118
%From: TERRENCE@UTAPHY.PH.UTEXAS.EDU
%Date: Wed, 18 May 1994 16:33:36 -0500 (CDT)
%Date (revised): Wed, 18 May 1994 16:50:19 -0500 (CDT)
%Date (revised): Thu, 26 May 1994 10:39:39 -0500 (CDT)


\input phyzzx
\input epsf
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\def\n{\noindent}
\def\p{\partial}
\def\ge{\varepsilon}
\def\ee{\epsilon}
\def\ga{\alpha}
\def\gb{\beta }
\def\ap{\alpha '}
\def\gl{\lambda^2}
\def\sg{\sigma }
\def\sk{\sqrt{\kappa} }
\def\IN{\relax{\rm I\kern-.18em N}}
\font\captura=cmr9
\null
\rightline {UTTG-10-94}
\rightline {May 1994}

\title{Thermal ensemble of string gas in a
magnetic field
\foot{Work supported in part by NSF grant
PHY 9009850 and R.~A.~Welch Foundation.}}

\author{Jorge G. Russo \foot{
Address after September 1, 1994: Theory Division, CERN, CH-1211
Geneva 23, Switzerland} }
\address {Theory Group, Department of Physics, University of
Texas\break
Austin, TX 78712}


\abstract
We study the thermal ensemble of a gas of free strings in presence
of a magnetic field. We find that the thermodynamical partition function
diverges when the magnetic field exceeds some critical value $B_{\rm cr}$,
which depends on the temperature. We argue that there is a first-order phase
transition with a large latent heat.
At the critical value an infinite number
of states -all states in a Regge trajectory- seem to become massless,
which may be an indication of recuperation of spontaneously
broken symmetries.

\vskip 1.5cm
% \centerline {PACS numbers: 11.17.+y}

\bigskip
\endpage


\Ref\hagedorn {R. Hagedorn, Nuovo Cim. Suppl. 3 (1965) 147.}

\Ref\nqcd{A. Polyakov, Phys. Lett. B72 (1978) 477;
L. Susskind, Phys. Rev. D20 (1979) 2610.}

\Ref\otros{B. Sathiapalan, Phys. Rev. D35 (1987) 3277;
Ya. Kogan , JETP Lett. 45 (1987) 709;
K. O'Brien and C.-I. Tan, Phys. Rev. D36 (1987) 1184.}

\Ref\atwi{J.J. Atick and E. Witten, Nucl. Phys. B310 (1988) 291.}

\Ref\amati{D. Amati, Ciafaloni and G. Veneziano,  Phys. Lett. B197 (1987)
81; Nucl. Phys. B403 (1993) 707.}

\Ref\grosmen{D. Gross and P.F. Mende, Nucl. Phys. B303 (1988) 407;
Phys. Lett. 197B (1987) 129.}

\Ref\gross{D. Gross, Phys. Rev. Lett. 60B (1988) 1229.}

\Ref\moore{G. Moore, {\it Symmetries of the bosonic string S-matrix},
Yale University preprint, YCTP-P19-93 (1993); YCTP-P1-94 (1994).}

\Ref\salam{A. Salam and J. Strathdee, Nucl. Phys. B90 (1975) 203.}

\Ref\nielsen{ N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376;
J. Ambj\o rn and P. Olesen, Nucl.Phys. B 315 (1989) 606; {\it ibid}
B330, (1990) 193.}

\Ref\porrati{ S. Ferrara and M. Porrati, Mod. Phis. Lett. A8 (1993) 2497.}

\Ref\russus{J.G. Russo and L. Susskind, {\it Asymptotic level density in
heterotic string theory and rotating black holes},
preprint UTTG-9-94 (1994).}

\Ref\ferrara{E. Del Giudice, P. Di Vecchia and S. Fubini, Ann. Phys. 70
(1972) 378; K. A. Friedman and C. Rosenzweig, Nuovo Cimento 10A (1972) 53;
S. Matsuda and T. Saido, Phys.Lett. B43 (1973) 123; M. Ademollo {\it et al},
Nuovo Cimento A21 (1974) 77;
S. Ferrara, M. Porrati and V.L. Teledgi, Phys. Rev. D46
(1992) 3529.}


Among the mysteries clouding a fundamental formulation of
string theory, the Hagedorn transition is, probably, the most arcane.
It is found that the thermodynamical partition function of a free string
gas diverges at some finite temperature [\hagedorn ]. The analogy with
large $N$ QCD [\nqcd ] suggests that the
Hagedorn temperature may not be a limiting temperature but rather an
indication of another phase of the theory, perhaps where the description
of physics in terms of strings is inadequate. This interpretation is
supported by the results of refs. [\otros ], where it is shown that at
the Hagedorn transition a certain mode  becomes massless.
By studying the effective field theory near the Hagedorn temperature,
Atick and Witten [\atwi] argued that the transition should be first order
with a large latent heat, due to a genus-zero contribution to the free energy
above the Hagedorn temperature.

In quantum chromodynamics a clear evidence of partons appears
in high-energy scattering processes. High-energy scattering also provides a
way to recognize spontaneously broken symmetries.
In the case of string theory,
diverse studies in this direction were made in refs. [\amati -\moore ].
In particular, in ref. [\gross ] it was argued that in the
high-energy limit string scattering amplitudes obey an infinite number
of linear relations that are valid order by order in perturbation
theory. This suggests the existence of an enormous symmetry which is
restored at high energy.

In superconductivity, the restoration of the U(1) symmetry is achieved
either by increasing the temperature or by increasing the magnetic field.
This effect gives rise to the well-known Meissner curve separating the
superconducting phase from the normal phase in type I superconductors.
The analog of this phenomenon in the context of particle physics was explored
in ref. [\salam ], and more recently in ref. [\nielsen ],
 where it was argued that spontaneously broken symmetries by
the Higgs mechanism can be restored in the presence of a strong magnetic field.

In this paper we will further explore the Hagedorn transition by
considering a string gas in a magnetic field. We will argue that
a phenomenon analog to the case of superconductivity
takes place in string theories.
We will also argue that there are genus zero contributions to the free
energy above the critical magnetic field, and
obtain the critical $B-T$ curve.
In addition we will consider the heterotic string theory and find that
an infinite number of physical particles
become massless for approximately the same critical
value of the magnetic field. \foot {In the context
of (zero-temperature) open string theory this was first pointed out in
ref. [\porrati ], where it was also argued that this fact indicates a
phase transition with possible restoration of symmetries.}
We will find that this value of the magnetic field is precisely
the critical value beyond which the thermodynamical partition function
diverges.
This result supports the view that there should
exist an infinite-dimensional symmetry group governing string interactions,
and it may provide a clue on the organization of multiplets.

Let us first derive an asymptotic formula for the level density of states
with mass $M$ and angular momentum $J$ in the case of the bosonic open string
theory. This calculation was done in [\russus ] and we refer to this paper
for details. Here
we will present a simple, alternative derivation. We add to the world-sheet
Hamiltonian
a term containing the angular momentum in the $z$ direction with a
Lagrange multiplier,
$$
H=\sum_{n=1}^\infty \sum_{i=1}^{D-2}
\ga_{-n}^i\ga_{n}^i+\lambda J\ ,\ \ \
J=-i\sum_{n=1}^\infty {1\over n}\big( \ga_{-n}^1\ga_{n}^2
-\ga_{-n}^2\ga_{n}^1\big)\ .
\eqn\hamm
$$
 The Hamiltonian can be diagonalized by
$\ga_n^1=\sqrt{n/2} \big(a_n+b_n\big)\ ,\
\ga_n^2=-i\sqrt{n/2} \big(a_n-b_n\big)\ $.
One obtains for the partition function,
$Z=\tr \big[ e^{-\beta H}\big] $, the following expression:
$$
Z=\prod _{n=1}^\infty \bigg[ \big( 1-w ^n\big) ^{-D+4}
\big( 1-cw^n\big) ^{-1}\big( 1-{w ^n\over c} \big)^{-1}\bigg ]\ ,
\eqn\partitio
$$
where $w \equiv e^{-\beta} $ and $c\equiv e^{\beta \lambda}$.
Let us define $G=\log Z$ and consider
$$
{\p G\over\p \log c}=\big( c-c^{-1}\big)\sum_{k=1}^\infty
{w^k\over \big( 1-cw^k\big) \big( 1-c^{-1}w^k\big)}\ .
\eqn\cinco
$$
Inserting $c=e^{\gb \lambda }$ and taking the limit $\gb \to 0 $
we obtain
$$
{\p G\over\p \lambda }=2\lambda \sum_{k=1}^\infty {1\over k^2-\lambda^2}\ .
\eqn\seis
$$
The summation in eq. \seis\ can be performed explicitly. Indeed
$$\eqalign {
{\p G\over\p \lambda }&=- \sum_{k=1}^\infty \bigg(
{1\over k+ \lambda }-{1\over k-\lambda}\bigg)=\psi(1+\lambda) -
\psi(1-\lambda) \cr
&=\int_0^1 dx {x^\lambda-x^{-\lambda}\over x-1}={1\over\lambda}-
\pi {\rm cotg} (\pi\lambda ) \ .\cr }
\eqn\siete
$$
Thus
$$
G(\lambda )=\log {\lambda\over\sin (\pi\lambda )}+ G(0)\ .
\eqn\ocho
$$
The term $G(\lambda=0)$ can be obtained by writing $ Z$ in the following
way:
$$
Z(w, c)=\exp\big[ \sum_{m=1}^\infty {1\over m}\big( c^m+c^{-m}+D-4 \big)
{w^m\over (1-w^m)}\big]\ .
\eqn\masparti
$$
and setting $c=1$. The leading term as $\gb\to 0 $ is
$Z(w,0)=e^{a^2\over\gb }, \ a\equiv \pi \sqrt {(D-2)/ 6}$.
In this process subleading, power-like factors are neglected.
Thus we obtain
$$
Z(\gb ,\lambda )\cong {\rm const. } e^{a^2\over\gb }{\lambda\over
\sin (\pi \lambda )}\ .
\eqn\zetap
$$
This estimate is in agreement with the more accurate calculation
of ref. [\russus ].

By expanding  $Z$, $Z(w,k)=\sum_{n,J} d_{n,J}w ^n e^{ik J} ,\ k
=-i\beta\lambda $, then $d_{n,J}$ can be found by
$$
d_{n,J}={1\over 2\pi i}
\oint {dw\over w^{n+1}} \int_{-\infty} ^\infty {dk\over 2\pi} e^{-ik J}
\ Z(w,k )\ .
\eqn\ddnj
$$
where the contour goes over a small circle around $w=0$.
These integrals were carried out in ref. [\russus ], with the result
$$
d_{n,J} \cong {\rm const.} e^{(n+1)\gb+ a^2/\gb }
{1\over {\rm cosh}^2(\gb J/2) } \ ,\ \gb\equiv {a\over\sqrt{n+1-|J|}}\ .
\eqn\uno
$$
The constant can be chosen so that $d_{n,J}=1$ on the Regge trajectories
$J= \pm n$.

Let us consider a canonical ensemble of free string gas in the
presence of a magnetic field in the $z$ direction. We have
$$
Z(T,B)=\int_0^\infty dn \int_{-n}^n dJ d_{n,J} e^{-\gb E}\ ,
\eqn\therpar
$$
where
$$
E^2=m^2+2qB(l+1/2)-2JqB+O(B^2)\ ,\ \ \ap m^2=n-1
\eqn\energi
$$
$l$ represent the Landau level, and $q$ is the total Chan-Paton charge
of the open string. A sum over $l$ is also understood in
eq. \therpar . In eq. \energi\ we have used the fact that all
physical states in open string theory have gyromagnetic factor equal 2
[\ferrara , \russus ].
The terms of $O(B^2)$ have various origins. In particular,
a magnetic field generates non-trivial corrections to sigma-model
backgrounds starting from $O(B^2)$. Let us disregard for the moment $O(B^2)$
terms. From eq. \energi\ there seems to be a critical magnetic field at which
some states become tachyonic, as noted in ref. [\porrati ].
The first states to become tachyonic
are those with maximum spin, $J=n$, at a magnetic field
$$
qB_{\rm cr}\cong {1\over 2\ap-{1\over m^2}}\ .
\eqn\becriti
$$
The state on the Regge trajectory with $\ap m^2=1$ will become tachyonic
at $qB\cong 1/\ap $.
For large $m^2$ all the states on the first Regge trajectory will become
tachyonic at $qB_{\rm cr}\cong 1/2\ap $.
Massless states with spin 1 will be tachyonic for an infinitesimal
value of the magnetic field. This last effect may be separated,
as will be done in the case of the heterotic string theory.

The emergence of tachyons is a clear sign of instability.
It is energetically
more favorable to produce pairs with maximum spin aligned with the
magnetic field. As a result,  the partition function \therpar \ will
diverge for $B$ exceeding the critical value. This can be combined
with another well-known effect.
At zero magnetic field, the partition function diverges
if the temperature is above the Hagedorn temperature, which
for the bosonic open string theory is $T_H=1/4\pi\sqrt{\ap }$.
A critical curve $B-T$, analogous to the Meissner curve in superconductivity,
 can be obtained by inserting  eq. \uno\
into eq. \therpar \ and analysing the convergence properties.
For this purpose  power-like dependence can be neglected relative
to the exponential factors.
Let us write $J=\ga n$ and analyse the integrand of \therpar\ in the
region $-1<\ga <1 \ ,\ n\to\infty $. By using eqs. \energi\ and \uno\
we obtain the conditions
$$
\ga=0:\ \ d_{n,J} e^{-\gb E}\to 0 \iff T<T_H\ ,\ \ T_H=1/4\pi\sqrt{\ap }
$$
$$
\ga=1:\ \ d_{n,J} e^{-\gb E}\to 0 \iff B<B_{\rm cr}\ ,\ \ B_{\rm cr}=1/2\ap
$$
One can verify that these are necessary and sufficient conditions for
convergence in all the region $-1<\ga <1 \ ,\ n\to\infty $.

The phase diagram is displayed in Fig. 1.
Our analysis  only provides
a rough estimate. It is plausible that a more careful and systematic
inclusion of $O(B^2)$ and other effects will smooth out the critical curve.
In particular,
if the transition is first order one expects that the actual critical
temperature at zero field is below the Hagedorn temperature [\atwi ].
The derivative of $B$ in the coexistence curve ${dB_{\rm cr}\over dT}$
at $T=0$ should remain zero in a full treatment. There is
a thermodynamical reason for this. Indeed, on
the coexistence curve the chemical potentials of the phases must be equal,
and thus one has $S_n-S_s\prop {dB_{\rm cr}\over dT}$. But the third
law of thermodynamics implies that $S_n-S_s\to 0$ as $T\to 0$.

In any case, it is possible that the critical line is actually
not well defined,
since  the notion of temperature may cease to be valid at the Planck scale.

In superconductivity the critical magnetic field $B_{\rm cr}$ is related
thermodynamically to the difference of the Helmholtz free energy density of
the two phases. One has
$$
{B_{\rm cr} ^2\over 8\pi }={F_n(T)-F_s(T)\over V}\ .
$$
The curve $B_{\rm cr}(T)$ is quite well approximated by a parabolic law,
$B_{\rm cr}(T)\cong B_{\rm cr}(0)\big[1-\big(T/T_{\rm cr}\big)^2\big]$.


\vskip 15pt
\vbox{{\centerline{\epsfxsize=3.25truein \hskip 2cm \epsfbox{fig1.eps}}
\vskip 12pt
{\noindent{\tenrm FIGURE 1. Phase diagram $B-T$.}}
\vskip 15pt}}

By using eq. \uno ,
in the case of the heterotic string theory one gets ($n_L\cong n_R\equiv n$)
$$
d_{n,J_L,J_R} \cong
{\rm const.} { e^{(n+1)(\gb_L+\gb_R)+ a^2_L/\gb_L+a^2_R/\gb_R }
\over
\cosh^2(\gb _LJ_L/2)\cosh^2(\gb_R J_R/2) } \ ,
\gb_L\equiv {a_L\over\sqrt{n+1-J}}\ ,\ \gb_R\equiv {a_R\over\sqrt{n+1-|J|}}\ ,
\eqn\dhetero
$$
where $a_L=2\pi $ and $a_R=\sqrt{2}\pi $ and $J_{L,R}$ represent
the left and right contribution to the angular momentum. Thus
$$
d_{n,J}= \int_ {-n}^n dJ_L d_{n,J_L,J-J_L} =\int_ {-n}^n
 dJ_R d_{n,J-J_R,J_R}\ .
\eqn\dtotal
$$

The energy formula can be obtained by using the gyromagnetic coupling
derived in ref. [\russus ]. One obtains
$$
E^2=m^2+2qB(l+{1\over 2})- 2J_RqB +O(B^2)\ ,\ \ \ap m^2/2\cong 2n ,\ n>>1\ .
\eqn\enerhet
$$
Again there appears to be a limiting or critical  value for the magnetic
field.
It may be convenient to go away from the self-dual point,
so that all the particles with gyromagnetic coupling have mass.
When
$$
qB_{\rm cr}\cong {2\over \ap-{2\over m^2}}\ ,
\eqn\campob
$$
all the states on the Regge trajectory with $J_R=n$ become massless.
Let us consider the partition function
$$
Z(T,B)=\int_0^\infty dn \int _{-n}^n dJ_L \int _{-n}^n dJ_R
d_{n,J_L,J_R}e^{-\gb E}
\eqn\queseyo
$$
The peculiar coupling of the magnetic field with the right contribution
to the angular momentum is characteristic of heterotic string theories
where the gauge quantum numbers solely arise from the left sector
[\russus ].
%The integral over $J_L$ gives $\tanh (a_L\sqrt{n}/2 )$ which tends to
%1 as $n\to\infty $ and therefore it can be disregarded.
A similar analysis as  in the open string case leads to the following
convergence region:
$$
qB< 2/\ap \ ,\ \ T<T_H^{\rm het}\ ,
\eqn\hetcurv
$$
with $T_H^{\rm het}=1/\big(\sqrt{\ap }(a_L+a_R)\big)={1\over\pi\sqrt{\ap }}
\big(1-{1\over\sqrt {2}}\big)$.

A natural question is whether  the critical magnetic field is not an
artifact of having ignored $O(B^2)$ terms. An complete treatment including
all orders in the $\ap $ expansion (i.e. including all powers of $B$)
would be necessary in order to answer this question. Certainly it
would be important to verify the existence of a critical magnetic field
in some specific, exactly solvable example.
However, it should be stressed that there is no reason why higher
orders in $B^2$ should stabilize the vacuum. It does not occur in the case of
superconductivity and it does not
seem to occur in the case of the electroweak model [\salam, \nielsen ].
$O(B^2)$ corrections to
the geometry,  etc. can only lead  to further instabilities of different
nature, as e.g. gravitational collapse.

Another way to derive a $B-T$ curve  is by considering heterotic string
with a compactified (euclidean) time dimension. Let us consider
 the right-moving modes in the NS sector (the analysis at $B=0$ was
carried out in ref. [\atwi ]). In this case the energy is given by
$$
\ap E^2=-3 +{4 \ap\pi^2 n_m^2\over \gb ^2}+ {n_w^2\gb ^2\over 4\ap \pi
^2}-2J_R \ap qB +2N+2\tilde N +O(B^2)\ ,
\eqn\windi
$$
where $n_m$ and $n_w$ respectively denote quantized momentum and winding
number. At zero field the first state to become tachyonic is
that with $N=\tilde N=0$ and $n_w=\pm 1, n_m=\pm 1/2$. This has $J_R=0$
and hence it does not have gyromagnetic coupling. The curve is simply
a vertical line at the Hagedorn temperature. The first charged
state with $J_R\neq 0$ is given by $\tilde N=1$, $N=1/2$, $n_m=0$ and
$n_w=2$ ($n_w=1$ is excluded by GSO projection -for odd $n_w$ it is reversed
relative to the standard projection). One obtains the curve
$qB=1/(\ap \pi T)^2 +O(B^2)$. This curve lies entirely above
the coexistence curve \hetcurv \ and
so it is not very relevant; the phase transition already occurs
for smaller fields.
 Above this curve the thermodynamical
partition function, obtained by calculating genus $\geq 1$ contributions
in the finite temperature theory ($X^0=X^0+\gb $),
will develop another divergence in
virtue of the appearance of the new tachyon.

In superconductivity the transition at zero magnetic field at $T_{\rm cr}$
is second order, but in the presence of a magnetic field there is a
discontinuous change in the thermodynamical state
of the system with an associated latent heat, and the transition is of first
order.
By a similar analysis as in ref. [\atwi ], using the effective field theory
of a state of level $n$ on the Regge trajectory that becomes tachyonic,
 one obtains that the free energy for $B>B_{\rm cr}$ is given by
$$
F\sim- {\rm const.}{n^2\over g^2} (4/\ap  -2qB)^2\ ,
\eqn\free
$$
where $g$ is the string coupling. This represents a genus zero
contribution, just as it happens at zero field above the Hagedorn
temperature. A genus-zero contribution cannot arise on
simply connected Riemann surfaces. In large $N$ QCD,
above the deconfining transition, continuum Riemann surfaces have to be
replaced by Feynman diagrams.


In refs. [\atwi , \gross ] it was argued that
string theory  might describe the
spontaneously broken phase of a highly symmetric theory.
This view is supported by what we have found here: the partition function
diverges and an infinite number of
particles become massless at approximately the same value of the magnetic
field. This suggests that an enormous gauge symmetry is being
restored.
This symmetry would relate higher spin particles, somehow circumventing
the Coleman-Mandula theorem, which asserts that the maximum spin
of a conserved charge cannot exceed 1, but assumes that the number of
particles with masses below any given scale is  finite. This
hypothesis does not apply in the $B\to B_{\rm cr} $ limit,
where symmetries would be recuperated, since infinitely many
particles are becoming massless simultaneously.
An interesting problem, which could unravel symmetries, is deriving an
effective field theory for all
particles in the Regge trajectory in a sigma-model background $B$ near
$B_{\rm cr}$.


The author is grateful to L. Susskind for helpful discussions
and useful remarks. He also wishes to thank W. Fischler for very
valuable and stimulating conversations.

\refout
\vskip 2cm




\vfill\eject
\end


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} def
/terminate
{
currentdict Adobe_typography_AI3 eq
	{
 end
	} if
} def
/modifyEncoding
{
	/_tempEncode exch ddef

	/_pntr 0 ddef

	{
		counttomark -1 roll
		dup type dup /marktype eq
		{
			pop pop exit
		}
		{
			/nametype eq
			{
				_tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll
				put
			}
			{
				/_pntr exch ddef
			}
			ifelse
		}
		ifelse
	}
	loop

	_tempEncode
}
def
/TE
{
	StandardEncoding 256 array copy modifyEncoding
	/_nativeEncoding exch def
} def
%
/TZ
{
	dup type /arraytype eq {/_wv exch def} {/_wv 0 def} ifelse
	/_useNativeEncoding exch def
	pop pop

	findfont _wv type /arraytype eq {_wv makeblendedfont} if dup length 2 add dict

 begin

		mark exch
		{
			1 index /FID ne { def } if cleartomark mark
		}
		forall
		pop

		/FontName exch def

		counttomark 0 eq
		{
			1 _useNativeEncoding eq
			{
				/Encoding _nativeEncoding def
			}
			if
			cleartomark
		}
		{
			/Encoding load 256 array copy
			modifyEncoding /Encoding exch def
		}
		ifelse
		FontName currentdict
 end

	definefont pop
}
def
/tr
{
_ax _ay 3 2 roll
} def
/trj
{
_cx _cy _sp _ax _ay 6 5 roll
} def
/a0
{
/Tx
	{
	dup
	currentpoint 3 2 roll
	tr _psf
	newpath moveto
	tr _ctm _pss
	} ddef
/Tj
	{
	dup
	currentpoint 3 2 roll
	trj _pjsf
	newpath moveto
	trj _ctm _pjss
	} ddef

} def
/a1
{
/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
	{
	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
	{
	tr _psf
	} ddef
/Tj
	{
	trj _pjsf
	} ddef
} def
/e1
{
/Tx
	{
	dup currentpoint 4 2 roll gsave
	tr _psf
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll gsave
	trj _pjsf
	grestore 3 1 roll moveto tr sp
	} ddef
} def
/i0
{
/Tx
	{
	tr sp
	} ddef
/Tj
	{
	trj jsp
	} ddef
} def
/i1
{
W N
} def
/o0
{
/Tx
	{
	tr sw rmoveto
	} ddef
/Tj
	{
	trj swj rmoveto
	} ddef
} def
/r0
{
/Tx
	{
	tr _ctm _pss
	} ddef
/Tj
	{
	trj _ctm _pjss
	} ddef
} def
/r1
{
/Tx
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	tr _ctm _pss
	grestore 3 1 roll moveto tr sp
	} ddef
/Tj
	{
	dup currentpoint 4 2 roll currentpoint gsave newpath moveto
	trj _ctm _pjss
	grestore 3 1 roll moveto tr sp
	} ddef
} def
/To
{
	pop _ctm currentmatrix pop
} def
/TO
{
	iTe _ctm setmatrix newpath
} def
/Tp
{
	pop _tm astore pop _ctm setmatrix
	_tDict begin /W {} def /h {} def
} def
/TP
{
 end
	iTm 0 0 moveto
} def
/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
{
_ctm setmatrix _tm concat 0 _rise translate _hs 1 scale
} def
/Tm
{
_tm astore pop iTm 0 0 moveto
} def
/Td
{
_mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto
} def
/iTe
{
	_render -1 eq {} {_renderEnd _render get dup null ne {load exec} {pop} ifelse}
ifelse
	/_render -1 ddef
} def
/Ta
{
pop
} def
/Tf
{
dup 1000 div /_fScl exch ddef
exch findfont exch scalefont setfont
} def
/Tl
{
pop
0 exch _leading astore pop
} def
/Tt
{
pop
} def
/TW
{
3 npop
} def
/Tw
{
/_cx exch ddef
} def
/TC
{
3 npop
} def
/Tc
{
/_ax exch ddef
} def
/Ts
{
/_rise exch ddef
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
100 div /_hs exch ddef
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Th
{
pop pop pop pop pop
} def
/TX {pop} def
%/Tx
%/Tj
/Tk
{
exch pop _fScl mul neg 0 rmoveto
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop neg Td
} def
/T*-
{
_leading aload pop Td
} def
/T-
{
_hyphen Tx
} def
/T+
{} def
/TR
{
_ctm currentmatrix pop
_tm astore pop
iTm 0 0 moveto
} def
/TS
{
0 eq {Tx} {Tj} ifelse
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_pattern_AI3 1.0 0
%%Title: (Adobe Illustrator (R) Version 3.0 Pattern Operators)
%%Version: 1.0
%%CreationDate: (7/21/89) ()
%%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_pattern_AI3 16 dict dup begin put
/initialize
{
/definepattern where
	{
	pop
	}
	{
	Adobe_pattern_AI3 begin
	Adobe_pattern_AI3
		{
		dup xcheck
			{
			bind
			} if
		pop pop
		} forall
	mark
	cachestatus 7 1 roll pop pop pop pop exch pop exch
		{
		{
		10000 add
		dup 2 index gt
			{
			break
			} if
		dup setcachelimit
		} loop
		} stopped
	cleartomark
	} ifelse
} def
/terminate
{
currentdict Adobe_pattern_AI3 eq
	{
 end
	} if
} def
errordict
/nocurrentpoint
{
pop
stop
} put
errordict
/invalidaccess
{
pop
stop
} put
/patternencoding
256 array def
0 1 255
{
patternencoding exch ( ) 2 copy exch 0 exch put cvn put
} for
/definepattern
{
17 dict begin
/uniform exch def
/cache exch def
/key exch def
/procarray exch def
/mtx exch matrix invertmatrix def
/height exch def
/width exch def
/ctm matrix currentmatrix def
/ptm matrix def
/str 32 string def
/slice 9 dict def
slice /s 1 put
slice /q 256 procarray length div sqrt floor cvi put
slice /b 0 put
/FontBBox [0 0 0 0] def
/FontMatrix mtx matrix copy def
/Encoding patternencoding def
/FontType 3 def
/BuildChar
	{
	exch
 begin
	slice begin
	dup q dup mul mod s idiv /i exch def
	dup q dup mul mod s mod /j exch def
	q dup mul idiv procarray exch get
	/xl j width s div mul def
	/xg j 1 add width s div mul def
	/yl i height s div mul def
	/yg i 1 add height s div mul def
	uniform
		{
		1 1
		}
		{
		width 0 dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		0 height dtransform
		dup mul exch dup mul add sqrt dup 1 add exch div
		} ifelse
	width 0 cache
		{
		xl 4 index mul yl 4 index mul xg 6 index mul yg 6 index mul
		setcachedevice
		}
		{
		setcharwidth
		} ifelse
	gsave
	scale
	newpath
	xl yl moveto
	xg yl lineto
	xg yg lineto
	xl yg lineto
	closepath
	clip
	newpath
 end
 end
	exec
	grestore
	} def
key currentdict definefont
end
} def
/patterncachesize
{
gsave
newpath
0 0 moveto
width 0 lineto
width height lineto
0 height lineto
closepath
patternmatrix setmatrix
pathbbox
exch ceiling 4 -1 roll floor sub 3 1 roll
ceiling exch floor sub
mul 1 add
grestore
} def
/patterncachelimit
{
cachestatus 7 1 roll 6 npop 8 mul
} def
/patternpath
{
exch dup begin setfont
ctm setmatrix
concat
slice exch /b exch slice /q get dup mul mul put
FontMatrix concat
uniform
	{
	width 0 dtransform round width div exch round width div exch
	0 height dtransform round height div exch height div exch
	0 0 transform round exch round exch
	ptm astore setmatrix
	}
	{
	ptm currentmatrix pop
	} ifelse
{currentpoint} stopped not
	{
	2 npop
	pathbbox
	true
	4 index 3 index eq
	4 index 3 index eq
	and
		{
		pop false
			{
			{2 npop}
			{3 npop true}
			{7 npop true}
			{pop true}
			pathforall
			} stopped
			{
			5 npop true
			} if
		} if
		{
		height div ceiling height mul 4 1 roll
		width div ceiling width mul 4 1 roll
		height div floor height mul 4 1 roll
		width div floor width mul 4 1 roll
		2 index sub height div ceiling cvi exch
		3 index sub width div ceiling cvi exch
		4 2 roll moveto
		FontMatrix mtx invertmatrix
		dup dup 4 get exch 5 get rmoveto
		ptm ptm concatmatrix pop
		slice /s
		patterncachesize patterncachelimit div ceiling sqrt ceiling cvi
		dup slice /q get gt
			{
			pop slice /q get
			} if
		put
		0 1 slice /s get dup mul 1 sub
			{
			slice /b get add
			gsave
			0 1 str length 1 sub
				{
				str exch 2 index put
				} for
			pop
			dup
				{
				gsave
				ptm setmatrix
				1 index str length idiv {str show} repeat
				1 index str length mod str exch 0 exch getinterval show
				grestore
				0 height rmoveto
				} repeat
			grestore
			} for
		2 npop
		}
		{
		4 npop
		} ifelse
	} if
end
} def
/patternclip
{
clip
} def
/patternstrokepath
{
strokepath
} def
/patternmatrix
matrix def
/patternfill
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternclip
Adobe_pattern_AI3 /patternpath get exec
grestore
newpath
} def
/patternstroke
{
dup type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get
	} if
gsave
patternstrokepath
true
	{
		{
			{
			newpath
			moveto
			}
			{
			lineto
			}
			{
			curveto
			}
			{
			closepath
			3 copy
			Adobe_pattern_AI3 /patternfill get exec
			} pathforall
		3 npop
		} stopped
			{
			5 npop
			patternclip
			Adobe_pattern_AI3 /patternfill get exec
			} if
	}
	{
	patternclip
	Adobe_pattern_AI3 /patternfill get exec
	} ifelse
grestore
newpath
} def
/patternashow
{
3 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 4 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath currentpoint
	6 index 6 index 6 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
5 npop
} def
/patternawidthshow
{
6 index type /dicttype eq
	{
	Adobe_pattern_AI3 /patternmatrix get 7 1 roll
	} if
	{
	2 npop (0) exch
	2 copy 0 exch put
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	9 index 9 index 9 index
	Adobe_pattern_AI3 /patternfill get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
8 npop
} def
/patternashowstroke
{
4 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 5 1 roll
	} if
4 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	gsave
	false charpath
	currentpoint
	4 index setmatrix
	7 index 7 index 7 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
6 npop
} def
/patternawidthshowstroke
{
7 index type /dicttype eq
	{
	patternmatrix /patternmatrix get 8 1 roll
	} if
7 1 roll
	{
	2 npop (0) exch
	2 copy 0 exch put
	gsave
	_sp eq {5 index 5 index rmoveto} if
	false charpath currentpoint
	7 index setmatrix
	10 index 10 index 10 index
	Adobe_pattern_AI3 /patternstroke get exec
	grestore
	newpath moveto
	2 copy rmoveto
	} exch cshow
9 npop
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_Illustrator_AI3 1.0 3
%%Title: (Adobe Illustrator (R) Version 3.0 Full Prolog)
%%Version: 1.0
%%CreationDate: (7/22/89) ()
%%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI3 71 dict dup begin put
/initialize
{
userdict /Adobe_Illustrator_AI3_vars 67 dict dup begin put
/_lp /none def
/_pf {} def
/_ps {} def
/_psf {} def
/_pss {} def
/_pjsf {} def
/_pjss {} def
/_pola 0 def
/_doClip 0 def
/cf	currentflat def
/_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
/_ay 0 def
/_cx 0 def
/_cy 0 def
/_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
/_wv 0 def
/Tx {} def
/Tj {} def
/CRender {} def
/_AI3_savepage {} def
/_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
/_pd 1 dict def
/_ed 15 dict def
/_pm matrix def
/_fm null def
/_fd null def
/_fdd null def
/_sm null def
/_sd null def
/_sdd null def
/_i null def
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3 dup /nc get begin
	{
	dup xcheck
		{
		bind
		} if
	pop pop
	} forall
end
end
end
Adobe_Illustrator_AI3 begin
Adobe_Illustrator_AI3_vars begin
newpath
} def
/terminate
{
end
end
} def
/_
null def
/ddef
{
Adobe_Illustrator_AI3_vars 3 1 roll put
} def
/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
{
	{
	pop
	} repeat
} def
/sw
{
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll mul add
} def
/swj
{
dup 4 1 roll
dup length exch stringwidth
exch 5 -1 roll 3 index mul add
4 1 roll 3 1 roll 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
{
4 1 roll
	{
	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
{
4 1 roll
	{
	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
/sp
{
	{
	2 npop (0) exch
	2 copy 0 exch put pop
	false charpath
	2 copy rmoveto
	} exch cshow
2 npop
} def
/jsp
{
	{
	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
/pl
{
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
	{
	curveto
	} def
	/C
	/c load def
	/v
	{
	currentpoint 6 2 roll curveto
	} def
	/V
	/v load def
	/y
	{
	2 copy curveto
	} def
	/Y
	/y load def
	/l
	{
	lineto
	} def
	/L
	/l load def
	/m
	{
	moveto
	} def
	}
	{
	/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
/d
{
setdash
} def
/cf	{} def
/i
{
dup 0 eq
	{
	pop cf
	} if
setflat
} def
/j
{
setlinejoin
} def
/J
{
setlinecap
} def
/M
{
setmiterlimit
} def
/w
{
setlinewidth
} def
/H
{} def
/h
{
closepath
} def
/N
{
_pola 0 eq
	{
	_doClip 1 eq {clip /_doClip 0 ddef} if
	newpath
	}
	{
	/CRender {N} ddef
	}ifelse
} def
/n
{N} def
/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
{
closepath
F
} def
/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
{
closepath
S
} def
/B
{
_pola 0 eq
	{
	_doClip 1 eq
	gsave F grestore
		{
		gsave S grestore clip newpath /_lp /none ddef _sc
		/_doClip 0 ddef
		}
		{
		S
		}ifelse
	}
	{
	/CRender {B} ddef
	}ifelse
} def
/b
{
closepath
B
} def
/W
{
/_doClip 1 ddef
} def
/*
{
count 0 ne
	{
	dup type (stringtype) eq {pop} if
	} if
_pola 0 eq {newpath} if
} def
/u
{} def
/U
{} def
/q
{
_pola 0 eq {gsave} if
} def
/Q
{
_pola 0 eq {grestore} if
} def
/*u
{
_pola 1 add /_pola exch ddef
} def
/*U
{
_pola 1 sub /_pola exch ddef
_pola 0 eq {CRender} if
} def
/D
{pop} def
/*w
{} def
/*W
{} def
/`
{
/_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
/@
{} def
/&
{} def
/O
{
0 ne
/_of exch ddef
/_lp /none ddef
} def
/R
{
0 ne
/_os exch ddef
/_lp /none ddef
} def
/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
{
/_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
{
_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
{
_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
{
/_gf exch ddef
findcmykcustomcolor
/_if exch ddef
/_fc
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[
39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
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/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
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/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
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Adobe_cshow /terminate get exec
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%%EOF



