%Paper: hep-th/9310029
%From: dij@s-a.amtp.liv.ac.uk
%Date: Wed, 6 Oct 1993 14:24:58 +0000 (BST)




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%   Figures are appended as a standard PostScript file
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\input harvmac
\def \x{\xi}
\def \pd{\partial}
\def \s{\sigma}
\def \b{\beta}
\def \a{\alpha}
\def \g{\gamma}
\def \d{\delta}
\def \e{\epsilon}
\def \l{\lambda}
\def \th{\theta}
\def \ph{\phi}
\def \Th{\Theta}
\def \Ph{\Phi}
\def \p{\pi}
\def \P{\Pi}
\def \D{\Delta}
\def \G{\Gamma}
\def \z {\zeta}
\def \m{\mu}
\def \n{\nu}
\def \o{\omega}
\def \O{\Omega}
\def \D{\Delta}
\def \r{\rho}
\def \k{\kappa}
\def\app{{Acta Phys.\ Pol.\ }{\bf B}}
\def\apny{Ann.\ Phys.\ (New York)\ }
\def\cmp{Comm.\ Math.\ Phys.\ }
\def\fortphys{{Fort.\ Phys.\ }{\bf A}}
\def\ijmpa{{Int.\ J.\ Mod.\ Phys.\ }{\bf A}}
\def\jetp{JETP\ }
\def\jetpl{JETP Lett.\ }
\def\jmp{J.\ Math.\ Phys.\ }
\def\mpla{{Mod.\ Phys.\ Lett.\ }{\bf A}}
\def\nc{Nuovo Cimento\ }
\def\npb{{Nucl.\ Phys.\ }{\bf B}}
\def\physrep{Phys.\ Reports\ }
\def\plb{{Phys.\ Lett.\ }{\bf B}}
\def\pnas{Proc.\ Natl.\ Acad.\ Sci.\ (U.S.)\ }
\def\pr{Phys.\ Rev.\ }
\def\prd{{Phys.\ Rev.\ }{\bf D}}
\def\prl{Phys.\ Rev.\ Lett.\ }
\def\ptp{Prog.\ Th.\ Phys.\ }
\def\pw{Part.\ World\ }
\def\zpc{Z.\ Phys.\ {\bf C}}
\def\lmp{Lett.\ Math.\ Phys.\ }
\def\ua{\underline{\a}}
\def\ud{\underline{\d}}
\def\uH{\underline{H}}
\def\ur{\underline{\r}}
\def \cH{{\cal H}}
\def \pp{\prime\prime}
\def \pri{\prime}
\def \tD{\tilde\D}
\def \tP{\tilde\Ph^+}
\def \trho {\tilde \r}
\def \tPm{\tilde\Ph^-}
\def \ba{\bar\a}
\def \ta{\tilde\a}
\def \tb{\tilde \b}
\def \del {\partial}
\def \bdel{\bar \del}
\def \bz {\bar z}
\def \cG {{\cal G}}
\def \tG {\tilde  G}
\def \tcG {\tilde{\cal G}}\def \ti {\tilde \imath}
\def \tj {\tilde \jmath}
\def \bi {\bar \imath}
\def \bj {\bar \jmath}
\def \tra {\tilde a}
\def \trb {\tilde b}
\def \bJ{\bar J}
\def \bT{\bar T}
\def \bA{\bar A}
\def \bP{\bar \P}
\def \cJ{{\cal J}}
\def \ad{{\rm ad}}
\def \cC {{\cal C}}
\def \rank{{\rm rank}}
\def \bs{\setminus}
\def \wr{\tilde r}
\def \tg{\tilde g}
\def \bg{\bar \g}
\def \tB{\tilde B}
\nopagenumbers
\line {\hfil LTH 317}
\vskip .5in
\centerline{\titlefont The Exact Tachyon Beta-Function}
\vskip 5pt
\centerline {\titlefont for the }
\vskip 5pt
\centerline {\titlefont Wess-Zumino-Witten
Model}
\vskip 10pt
\centerline{I. Jack and D. R. T. Jones}
\bigskip
\centerline{\it DAMTP, University of Liverpool, Liverpool L69 3BX, U.K.}
\vskip .3in
We derive an exact expression for the tachyon $\b$-function for the
Wess-Zumino-Witten model. We check our result up to three loops
by calculating the three-loop
tachyon $\b$-function for a general non-linear $\s$-model with torsion,
and then specialising to the case of the WZW model.
\Date{October 1993}

\newsec{Introduction}
The Wess-Zumino-Witten (WZW) model\ref\ew{E. Witten, \cmp92 (1984) 483.}\
 is a particularly interesting
example of a conformal field theory. Indeed it is currently believed
that all rational conformal field theories may be derived from the WZW
model by the Goddard-Kent-Olive (GKO) construction\ref\GKO{P. Goddard, A. Kent
and D. Olive, \plb152 (1985) 88.}
 (or equivalently
by gauging\ref\Gaug{K. Gawedzki and A. Kupiainen, \plb215 (1988) 119;
\npb320 (1989) 625\semi
D. Karabali and H. J. Schnitzer, \npb329 (1990) 649.}).
The WZW model on a Lie group manifold $\cG$ is parametrised
by the level (which is constrained to be an integer for a compact
$\cG$). The properties which characterise the conformal field
theory--the central charge and the conformal dimensions of the primary
fields--have exact non-perturbative expressions in terms of the level
 and the Casimirs for the Lie algebra of $\cG$\ref\kz{V. G. Knizhnik and
A. B.Zamolodchikov, \npb247 (1984) 83.}.

In general, we may consider perturbing the WZW model by adding a
potential term to the action. This potential term is usually taken to be
a primary field of the WZW model; however we would like to consider the
case where this restriction is not applied. Our motivation for this
comes from non-abelian Toda field theories\ref\ls{A. N. Leznov and
M. V. Saveliev, \lmp6 (1982) 505; \cmp89 (1983) 59.}, which have recently
been receiving some attention\ref\natta{M. V. Saveliev,
\mpla5 (1990) 2223.}
\ref\gs{J.-L. Gervais and M. V.
Saveliev, \plb286 (1992) 271\semi
F. Delduc, J.-L. Gervais and M. V. Saveliev, \plb292 (1992) 295\semi
L. O'Raifeartaigh and A. Wipf, \plb251 (1990) 361\semi
L. O'Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, \cmp143 (1992)
333\semi
L. Feh\'er,
L. O'Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf,
\apny213 (1992) 1\semi
G. Papadopoulos and B. Spence, ``The Space of Solutions of Toda Field
Theory'', preprint UM-P-93/38, KCL-93-7, hep-th/9306088.}.
The action for these models
consists precisely of a WZW model coupled to a potential term. In a
recent paper\ref\natt{I. Jack and D. R. T. Jones,
``Quantum non-abelian Toda field theories'',
Liverpool preprint LTH 315.}\
 we discussed the conformal properties of the non-abelian
Toda field theory at the quantum level. The conformal behaviour of the
potential term is described by the tachyon $\b$-function (in terminology
derived from string theory) and therefore we needed an exact expression
for this $\b$-function. Our aim in this paper is to derive this. In the
particular case where the potential is simply $\tr(g)$, where $g\in\cG$,
the tachyon $\b$-function is simply related to the conformal dimension
of $\tr(g)$, and we shall exploit this to deduce the general form of the
tachyon $\b$-function. We shall then check this result by performing an
explicit perturbative calculation up to three-loop order. We first
obtain the result for a general non-linear $\s$-model, and then specialise
to the case of the WZW model. In the WZW case, our perturbative
calculation is similar to a three-loop calculation of the conformal
dimension of $\tr(g)$ carried out some time ago\ref\bos{M. Bos, \apny181 (1988)
177}. However, we use a
different prescription\ref\ht{
C. M. Hull and P. K. Townsend, \plb191 (1987) 115\semi
R. R. Metsaev and A. A. Tseytlin, \plb191 (1987) 354;\npb293 (1987) 385\semi
D. Zanon, \plb191 (1987) 363\semi
D. R. T. Jones, \plb192 (1987) 391.}
\ref\hugh{H. Osborn, \apny200 (1990) 1.}
 for continuing the two-dimensional alternating
symbol to $d$ dimensions in the context of dimensional regularisation.
This prescription has the conceptual advantage of avoiding the need for
introducing extra evanescent terms into the action, and also preserves
explicit $O(d)$ covariance. It now appears to have been accepted as the
standard prescription for calculations of this type.
\newsec{Exact result for tachyon $\b$-function}
 In this Section we derive our principal result, namely the exact
tachyon $\b$-function for the WZW model. We first write down the action for
the Wess-Zumino-Witten (WZW) model defined on a group
manifold $M_{\cG}$:
\eqn\eaa{
kS_{WZW}(g)=-{k\over{8\p}}\int_S d^2x \tr(g^{-1}\del_{\m} gg^{-1}\del^{\m}g)
+{ik\over{12\p}}\int_B d^3x \e^{\m\n\r}\tr(g^{-1}\del_{\m} gg^{-1}\del_{\n}g
g^{-1}\del_{\r}g)  }
where $g$ is a group element in the defining representation of $\cG$,
whose generators satisfy
\eqn\eab{
[T_a,T_b]=if_{abc}T_c.}
%
$B$ is a 3-dimensional ball whose
surface is the two-dimensional worldsheet $S$. We assume that the group
generators satisfy $\tr(T_aT_b)=\d_{ab}$.
We are using here the conventions of Ref. \ref\go{P. Goddard and D. Olive,
\ijmpa1 (1986) 303.}. The level $x$ is defined in terms of $k$ by
$x={2k\over{\psi^2}}$, where $\psi$ is the highest root in
$\cG$.
The central charge is given by
\eqn\eac{
c={k{\rm dim}\cG\over{k+{1\over2}c^{\cG}}},  }
%
and $g$ is a primary field with conformal dimensions given by\kz
\eqn\ead{
h=\bar h={c^R\over{k+{1\over2}c^{\cG}}},  }
%
where $c^{\cG}$ and $c^R$ are the eigenvalues of the quadratic Casimir in
the adjoint and defining representations respectively, so that
\eqn\eaf{
f_{acd}f_{bcd}=c^{\cG}\d_{ab}, \qquad T_aT_a=c^R1  .}
%
The WZW model is a particular example of a two-dimensional non-linear
$\s$-model, whose action is given in general by
\eqn\eag{
S(\ph)={\l\over{8\pi}}\int d^2x\sqrt{\g}
\{\g^{\m\n}G_{ij}(\ph)\del_{\m}\ph^i\del^{\n}\ph^j
+\e^{\m\n}B_{ij}(\ph)\del_{\m}\ph^i\del_{\n}\ph^j
 +{1\over{\l}}D(\ph)R^{(2)}+V(\ph)\}  }
%
where $\g_{\m\n}$ is the two-dimensional metric, $\g$ is its determinant and
$\e^{\m\n}$ is the two-dimensional alternating symbol.
$\{\ph^i\}$ represent co-ordinates on some target manifold with
metric $G_{ij}$ and antisymmetric tensor field $B_{ij}$ defined on it,
$D$ is the dilaton field coupling to the two-dimensional scalar curvature
$R^{(2)}$, and $V$ is the tachyon field. (The terminology derives from string
theory; our conventions are equivalent to taking $\a^{\pri}={2\over{\l}}$
in Ref. \hugh, where $\a^{\pri}$ is the string coupling.)
The conformal invariance conditions for the $\s$-model Eq. \eag\ may
be written\hugh
\eqna\eh
{$$\eqalignno{
B^G_{ij}&\equiv\b_{ij}^G+{2\over{\l}}\nabla_i\del_jD+2\del_{(i}W_{j)}=0
&\eh a\cr
B_{ij}^B&\equiv\b_{ij}^B+{2\over{\l}}H^k{}_{ij}\del_kD+2H^k{}_{ij}W_k=0
&\eh b\cr
B^V&\equiv\b^V-2V+{1\over{\l}}\del^kD\del_kV+W^k\del_kV=0&\eh c\cr}$$   }
%
where $\b^G_{ij}$, $\b^B_{ij}$ and $\b^V$ are the standard renormalisation
group $\b$-functions for $G_{ij}$, $B_{ij}$ and $V$, and $H_{ijk}$ is the
torsion, defined by $H_{ijk}=3\nabla_{[i}B_{jk]}$. $W_i$ is a vector field
which can be determined perturbatively within a given renormalisation
scheme. We shall be using
a renormalisation scheme in which $W_i$ vanishes for the WZW model.

When $B^G_{ij}$ and $B^B_{ij}$ both vanish, the quantity $B^D$ given by
\eqn\eai{
B^D\equiv\b^D+{1\over{\l}}\del^kD\del_kD+W^k\del_kD, }
where $\b^D$ is the dilaton $\b$-function, becomes constant
\ref\CP{G. Curci and G. Paffuti, \npb286 (1987) 399.}\hugh\ and is then
related to the central charge $c$ for the conformal field theory by
\eqn\eaj{
c=3B^D. }
%
The tachyon $\b$-function is of the form
\eqn\eal  {
\b^V=\O V  }
%
where $\O$ is a differential operator, in general of arbitrary order.
We may write $\O$ in the form
\eqn\eam  {
\O=\sum X^{(n)k_1\ldots k_n}\nabla_{k_1}\ldots\nabla_{k_n} }
%
where $X^{(n)k_1\ldots k_n}$ is an $n$th rank tensor constructed from
the Riemann tensor, the torsion and their derivatives contracted together.

In the case of the WZW model Eq. \eaa, the target manifold is the group
$M_{\cG}$.
There is no tachyon field $V$, and the
metric $G_{ij}$ and $B_{ij}$ may be read off by comparing Eqs. \eaa\ and
\eag. We have\ref\braat{E. Braaten, T. L. Curtright and C. Zachos,
\npb260 (1985) 630.}
\eqn\eak{
G_{ij}=e_{ai}e_{aj}, \qquad \l=k, }
%
where the vielbein $e_{ai}$ is defined by
\eqn\eaka{
ig^{-1}\del_ig=e_{ia}T_a }
%
and satisfies
\eqn\eakab{
e_{a}{}^ie_{bi}=\d_{ab} .}
%
In fact $B_{ij}$ can only be defined locally, but we have a globally defined
expression for $H_{ijk}$\braat,
\eqn\eakb{
H_{ijk}={1\over2}f_{abc}e_{ai}e_{bj}e_{ck}. }
%
We also have
\eqn\eakc{
\nabla_ie_{aj}=f_{abc}e_{bi}e_{cj}  }
%
from which it follows that the Riemann tensor is given by\braat
\eqn\eakd{
R_{ijkl}={1\over4}f_{abe}f_{cde}e_{ai}e_{bj}e_{ck}e_{dl} }
%
and also that
\eqn\eake{
\nabla_lH_{ijk}=0  .}
%

We will denote quantities
evaluated for the particular case of the WZW model
by subscripts $WZW$. $\O_{WZW}$ has an
expansion similar to Eq. \eam, but in terms of $X_{WZW}^{(n)k_1\ldots k_n}$
obtained from $X^{(n)k_1\ldots k_n}$ by
substituting the expressions in Eqs. \eakb\ and \eakd\
 for the Riemann tensor and
torsion. $X_{WZW}^{(n)k_1\ldots k_n}$ is constructed purely from the structure
constants and the vielbeins.

If $V$ is a primary field, it will be an eigenfunction of $\O_{WZW}$ whose
eigenvalue is the conformal dimension of $V$. In particular, for $V=\tr(g)$,
we must have
\eqn\ean  {
\O_{WZW}\tr(g)={c^R\over{k+{1\over2}c^{\cG}}}\tr(g). }
%
This information is sufficient to determine $\O_{WZW}$. We can assume without
loss of generality that $X_{WZW}^{(n)k_1\ldots k_n}$ is symmetric in
$k_1 \ldots k_n$ (indeed, $X_{WZW}^{(n)k_1\ldots k_n}$ is naturally given in
this form by the explicit calculation--see later). So we have, using Eqs.
\eaka\
and \eakc,
\eqn\eao{
X_{WZW}^{(n)k_1\ldots k_n}\nabla_{k_1}\ldots\nabla_{k_n}\tr(g)=
i^nX_{WZW}^{(n)k_1\ldots k_n}\tr(gT_{a_1}\ldots T_{a_n})e_{a_1k_1}\ldots
e_{a_nk_n}
,  }
%
since the structure constant terms in Eq.~\eakc\ vanish by symmetry here.
Eq. \ean\ can be expanded in powers of $1\over k$ (or
equivalently in powers of $c^{\cG}$), and this expansion must
correspond to the perturbation expansion for $\O$.
Hence we see that
$X_{WZW}^{(n)k_1\ldots k_n}T_{a_1}\ldots T_{a_n}e_{a_1k_1}\ldots e_{a_nk_n}$
must reduce to a function of $c^R$ and $c^{\cG}$ linear in $c^R$.
The only way in which this can happen is if
\eqn\eap{
X_{WZW}^{(n)k_1\ldots k_n}=0,\qquad i\ne2  }
% and if
\eqn\eaq{
X_{WZW}^{(2)k_1k_2}T_{a_1}T_{a_2}e_{a_1k_1}e_{a_2k_2}
=-{c^R\over{k+{1\over2}c^{\cG}}}.}
%
We then must have
\eqn\ear{
X_{WZW}^{(2)k_1k_2}e_{a_1k_1}e_{a_2k_2}=-{c^R\over{k+{1\over2}c^{\cG}}}
\d_{a_1a_2}, }
%
and hence, from Eq. \eak,
\eqn\eas{
X_{WZW}^{(2)k_1k_2}=-{c^R\over{k+{1\over2}c^{\cG}}}g^{k_1k_2}.}
%
So we finally have the desired result
\eqn\eat{
\O_{WZW}=-{c^R\over{k+{1\over2}c^{\cG}}}\nabla^2, }
%
and hence
\eqn\eau{
\b^V_{WZW}=-{c^R\over{k+{1\over2}c^{\cG}}}\nabla^2V.  }
\newsec{Three-loop perturbative calculation}
In this Section we go some way towards corroborating our exact result for the
tachyon $\b$-function for the WZW model, derived in the previous Section,
by performing a perturbative calculation up to three-loop order. We
start by doing the computation for the general non-linear $\s$-model of Eq.
\eag, before specialising to the WZW model. It is most convenient to perform
these computations using dimensional regularisation; however, the crucial
issue which then arises for a $\s$-model of the form Eq. \eag, one with a term
involving the antisymmetric tensor $B_{ij}$, is how to extend the
two-dimensional alternating symbol, $\e^{\m\n}$, away from two dimensions.
In two dimensions one has the relation
\eqn\eba{
\e^{\m\n}\e^{\r\s}=\g^{\m\s}\g^{\n\r}-\g^{\m\r}\g^{\n\s} .}
%
However, if one tries to apply this relation for $d\ne2$, one encounters
inconsistencies. One solution is to regard $\e^{\m\n}$ as strictly
two-dimensional even within dimensional regularisation\bos. The drawback of
this approach is that the tangent-space group is reduced from $O(d)$ to
$O(d-2)\times O(2)$, and as a consequence one is obliged to introduce
additional, ``evanescent'' couplings which were not present in the original
two-dimensional action. Physical results independent of these evanescent
couplings can be obtained, but only at the expense of additional
calculation. This is the method used in Ref. \bos\ to compute the anomalous
dimension of $\tr(g)$ up to three loops.

An alternative approach, which has been fairly widely used\ht\hugh, is to
abandon Eq. \eba\ away from two dimensions, but to assume the existence
of a tensor
$\e^{\m\n}$ in general $d$ dimensions with the property
\eqn\ebb{
\e^{\m\r}\e_{\r}{}^{\n}=-\g^{\m\n}   .}
%
We should stress that the $\g^{\m\n}$ which appears on the RHS of Eq. \ebb\
is the $d$-dimensional metric. Evanescent couplings are required in this case
also for the rigorous discussion of renormalisability\hugh; however,
the important
difference as compared to the previous prescription is that these evanescent
couplings completely decouple from physical results. Moreover it turns out
that a relation of the form Eq. \ebb\ is sufficient for perturbative
calculations.

The most convenient means for discussing the quantisation of the non-linear
$\s$-model is the use of the background field method\ref\afm{L.
Alvarez-Gaum\'e, D. Z. Freedman and S. Mukhi, \apny134 (1981) 85.}
. The field $\ph^i$ is
expanded around a classical background configuration as
\eqn\ebc{
\ph^i=\ph_0^i+\p^i  . }
%
However, the field $\p^i$ is not very convenient for use as the quantum
field variable, since it does not transform as a vector. It is customary to
write $\p^i$ in terms of the field $\x^i$, the tangent vector to the
geodesic linking $\ph_0^i$ to $\ph_0^i+\p^i$, and to use $\x^i$ as the
quantum field\afm\ref\muk{S. Mukhi, \npb264 (1986) 640.}
. This guarantees a manifestly covariant perturbation
expansion written in terms of tensor quantities such as the Riemann tensor,
the torsion and their covariant derivatives. The expansion of the action
for the non-linear $\s$-model in Eq. \eag\ in terms of $\x$ takes the
following form (setting the dilaton term, which will not concern us, to zero)
\afm\ref\fv{B. E. Fridling and A. E. M. van de Ven, \npb268 (1986) 719.}\muk:
\eqn\ebd{
S(\ph+\p)=S(\ph)+\sum_{i=1}S^{(i)}(\ph,\x) , }
%
where
\eqna\ebe{$$\eqalignno{
S^{(1)}&=\int d^2x(g_{ij}\del_{\m}\ph^iD^{\m}\x^j+\e^{\m\n}H_{ijk}\del_{\m}
\ph^i\del_{\n}\ph^j\x^k+\nabla_iV\x^i), &\ebe a\cr
S^{(2)}&={1\over2}\int d^2x\{g_{ij}D_{\m}\x^iD^{\m}\x^j
+(\g^{\m\n}R_{iklj}+\e^{\m\n}\nabla_lH_{ijk})\del_{\m}
\ph^i\del_{\n}\ph^j\x^k\x^l
\cr&\quad+2\e^{\m\n}H_{ijk}\del_{\m}\ph^iD^{\m}\x^j\x^k+\nabla_i\nabla_jV
\x^i\x^j\},
&\ebe b \cr
S^{(3)}&=\int d^2x({1\over3}\e^{\m\n}H_{ijk}D_{\m}\x^i
D_{\n}\x^j\x^k+{1\over6}\nabla_i\nabla_j\nabla_kV\x^i\x^j\x^k)
+\ldots,  &\ebe c \cr
S^{(4)}&=\int
d^2x({1\over6}\g^{\m\n}R_{iklj}+{1\over4}\e^{\m\n}\nabla_lH_{ijk})
D_{\m}\x^iD_{\n}\x^j\x^k\x^l\cr&\quad
+{1\over{24}}\nabla_i\nabla_j\nabla_k\nabla_l
V\x^i\x^j\x^k\x^l
)+\ldots  ,&\ebe d\cr}$$  }
%
We omit the subscript $0$ on $\ph^i_0$ here and henceforth. The operator
$D_{\m}$ is defined by
\eqn\ebf{
D_{\m}\x^i=\del_{\m}\x^i-\G^i{}_{jk}\del_{\m}\ph^j\x^k .}
%
   In Eqs. \ebe{c}\ and \ebe{d}\
we have omitted terms involving $\del_{\m}\ph^i$, since these
are irrelevant for our calculation. Feynman diagrams are constructed using
2-, 3- and 4-point vertices corresponding to the quadratic,
cubic and quartic terms in $S^{(2)}$,
$S^{(3)}$ and $S^{(4)}$ (except for the first term in $S^{(2)}$, which
supplies the propagators used to link the vertices). The 1- and 2-loop
diagrams contributing to the tachyon $\b$-function are shown in Figs. 1
and 2. The corresponding contributions to $\O$ are given by\ref\df{D. Friedan,
\prl45 (1980) 1057;\apny163 (1985) 318.}\ref\aatb{A. A. Tseytlin, \plb178
(1986) 34.}\hugh
\eqn\ebfa{\eqalign{
\O^{(1)}&=-{1\over{\l}}\nabla^2 \cr
\O^{(2)}&={2\over{\l^2}}H^{kmn}H^l{}_{mn}\nabla_k\nabla_l\cr}  }
%
The 3-loop diagrams
contributing to the tachyon $\b$-function are depicted in Fig. 3. We
subtract subdivergences on a diagram-by-diagram basis. The results for the
divergent contributions $W_{(a)}^{(3)}$--$W_{(g)}^{(3)}$ of the diagrams
Figs. 3(a)--(f) respectively, incorporating the corresponding subtractions,
are:
\eqna\ebg{$$\eqalignno{
W_{(a)}^{(3)}&={4\over3}{1\over{\l^3}}\bigl({1\over{\e^2}}-{5\over4}
{1\over{\e}}\bigr)H^{kmn}H^{lpq}H_{mpr}H_{nq}{}^r\nabla_k\nabla_lV, &\ebg a\cr
W_{(b)}^{(3)}&=-{4\over3}{1\over{\l^3}}\bigl({1\over{\e^3}}-{2\over{\e^2}}
+{7\over4}{1\over{\e}}\bigr)H^{prs}H^q{}_{rs}H_{pm}{}^kH_q{}^{ml}
\nabla_k\nabla_lV, &\ebg b \cr
W_{(c)}^{(3)}&=-{4\over3}{1\over{\l^3}}\bigl({1\over{\e^3}}-{1\over{\e^2}}
\bigr)H^{kmn}H^p{}_{mn}H^{lrs}H_{prs}\nabla_k\nabla_lV &\ebg c \cr
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
W_{(d)}^{(3)}&=-{2\over3}{1\over{\l^3}}\bigl({1\over{\e^3}}-{1\over{\e^2}}\bigr)
R^{mpqn}H^k{}_{mp}H^l{}_{nq}\nabla_k\nabla_lV &\ebg d \cr
W_{(e)}^{(3)}&=-{4\over3}{1\over{\l^3}}\bigl({1\over{\e^3}}-2{1\over{\e^2}}
+{7\over4}{1\over{\e}}\bigr)R^{kpqm}H^{nl}{}_pH_{nmq}\nabla_k\nabla_lV &\ebg e
 \cr
W_{(f)}^{(3)}&=-{1\over{\l^3}}\bigl({2\over9}{1\over{\e^3}}
-{1\over9}{1\over{\e^2}}+{1\over2}{1\over{\e}}\bigr)R^{kmnp}R^l{}_{mnp}\nabla_k
\nabla_lV &\ebg f\cr
W_{(g)}^{(3)}&={1\over{\l^3}}\bigl({2\over9}{1\over{\e^2}}-{5\over18}
{1\over{\e}}\bigr)
\nabla^kH_{mnp}\nabla^lH^{mnp}\nabla_k\nabla_lV \cr&\quad
+{1\over{\l^3}}\bigl({4\over3}{1\over{\e^3}}-{2\over{\e^2}}+{1\over2}
{1\over{\e}}\bigr)
\nabla^pH^{kmn}\nabla_pH^l{}_{mn}\nabla_k\nabla_lV &\ebg g\cr
W_{(h)}^{(3)}&={1\over{\l^3}}\bigl({4\over3}{1\over{\e^3}}-{8\over3}
{1\over{\e^2}}
+{4\over3}{1\over{\e}}\bigr)\nabla^kH^{lnp}H^m{}_{np}\nabla_{(k}\nabla_l
\nabla_{m)}
&\ebg h \cr
}$$ }
This leads to the following contribution to $\O$ at 3 loops:
\eqn\ebh{\eqalign{
\O^{(3)}&={1\over{\l^3}}\bigl(-{3\over2}R^{kmnp}R^l{}_{mnp}
-7R^{kpqm}H^{nl}{}_pH_{nmq}\cr
&\quad-5H^{kmn}H^{lpq}H_{mpr}H_{nq}{}^r
        -7H^{prs}H^q{}_{rs}H_{pm}{}^kH_q{}^{ml}
       +{3\over2}\nabla^pH^{kmn}\nabla_pH^l{}_{mn}\cr
&\quad        -{5\over6}\nabla^kH_{mnp}\nabla^lH^{mnp}\bigr)
\nabla_k\nabla_l
+{4\over{\l^3}}\nabla^kH^{lnp}H^m{}_{np}\nabla_{(k}\nabla_l\nabla_{m)}  \cr}  }
%
Specialising to the case of the WZW model, we readily find, using Eqs. \eakb,
 \eakd\ and \eake,
\eqna\ebi{$$\eqalignno{
\O_{WZW(1)}&=-{1\over k}\nabla^2   &\ebi a \cr
\O_{WZW(2)}&={1\over{2k^2}}c^{\cG}\nabla^2 &\ebi b \cr
\O_{WZW(3)}&=-{1\over{4k^3}}c^{\cG2}\nabla^2 &\ebi c \cr }$$  }
%
which is consistent with the expansion of Eq. \eat\ up to $O({1\over{k^3}})$.
\newsec{Conclusions}
Our central result is the exact expression for the tachyon $\b$-function
in the WZW model, given by Eq. \eau. This quantity played a crucial role
in a recent paper deriving the exact conformally invariant action for the
non-abelian Toda field theory. We checked this result up to 3rd order in
perturbation theory by calculating the tachyon $\b$-function at this order
for a general non-linear $\s$-model, and then specialising to the case of
the WZW model. The result for the general case seems to us to be of interest
since it is the first full, direct computation of a renormalisation group
quantity at three loops for the general non-linear $\s$-model with torsion.
(The three-loop contribution to the dilaton $\b$-function was, however,
calculated indirectly in Refs. \ref\jj{I. Jack and D. R. T. Jones, \plb200
(1988) 453.}, \ref\htb{C. M. Hull and P. K. Townsend, \npb301 (1988) 197.},
\hugh.)

A more stringent check on the validity of Eq. \eat\ would be provided by a
four-loop calculation of $\b^V$. For instance, Eq. \eap\ is trivially satisfied
at three loops since the term involving three derivatives in $\O^{(3)}$ in
Eq. \ebh\ manifestly vanishes upon specialisation to the WZW model. On the
other
hand, at four loops there are possible four-derivative terms in the general
result for $\O^{(4)}$ (see Ref. \ref\jjm{I. Jack, D. R. T. Jones and
N. Mohammedi, \npb334 (1990) 333} for a calculation of $\O^{(4)}$ in the
torsion free case), and these do not obviously vanish immediately upon
specialisation to the WZW case. A four-loop calculation would be formidably
difficult, however.
\vskip 12pt
\line{\bf Acknowledgements}
\vskip 5pt
The Feynman diagrams were produced using the package FeynDiagram written by
Bill Dimm. One of us (I. J.) thanks the S.E.R.C. for financial support.
\listrefs
\line{\bf Figure Captions \hfil}
\item {Fig. 1.} One-loop diagram contributing to tachyon $\b$-function (with
tachyon insertion denoted by cross).
\item {Fig. 2.} Two-loop diagram contributing to tachyon $\b$-function.
\item {Fig. 3.} Three-loop diagrams contributing to tachyon $\b$-function.

\bye

%!
%%Creator: FeynDiagram 1.17  by Bill Dimm
%%BoundingBox: 82.8 79.92 463.824 758.16
%%LanguageLevel: 1
%%Pages: 2
%%EndComments
%%BeginProlog
% @(#) abbrev.ps 1.9@(#)

/CP /charpath load def
/CF /currentflat load def
/CPT /currentpoint load def
/C2 /curveto load def
/FP /flattenpath load def
/L2 /lineto load def
/M2 /moveto load def
/NP /newpath load def
/PBX /pathbbox load def
/RM2 /rmoveto load def
/SD /setdash load def
/SLC /setlinecap load def
/SLW /setlinewidth load def
/S /show load def
/ST /stroke load def

% @(#) vertex.ps 1.9@(#)

/vtx_dict 20 dict def
vtx_dict /vtx_mtrxstor matrix put

/vtx_create
  {
  vtx_dict begin

  /ang exch def
  /rad exch def
  /y exch def
  /x exch def
  /vtx_proc exch def

  /vtx_oldmatrix vtx_mtrxstor currentmatrix def

  x y translate
  ang rotate

  vtx_proc

  vtx_oldmatrix setmatrix
  end
  } bind def


/vtx_dot
  {
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  0 setgray
  fill
  } bind def


/vtx_box
  {
  NP
  rad rad M2
  rad rad neg L2
  rad neg rad neg L2
  rad neg rad L2
  closepath
  0 setgray
  fill
  } def

/vtx_cross
  {
  NP
  -1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  L2

  -1 2 sqrt div rad mul
  1 2 sqrt div rad mul
  M2
  1 2 sqrt div rad mul
  -1 2 sqrt div rad mul
  L2

  ST
  } def

/vtx_circlecross
  {
  /xshrfct .63 def

    % blank out whatever is underneath
  NP
  rad 0 M2
  0 0 rad 0 360 arc
  1 setgray
  fill

  NP
  -1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  L2

  -1 2 sqrt div rad mul xshrfct mul
  1 2 sqrt div rad mul xshrfct mul
  M2
  1 2 sqrt div rad mul xshrfct mul
  -1 2 sqrt div rad mul xshrfct mul
  L2

  rad 0 M2
  0 0 rad 0 360 arc

  0 setgray
  ST
  } def

% @(#) arrow.ps 1.9@(#)

/tan
  {
  dup
  sin exch cos div
  } bind def

/arrow_dict 12 dict def
arrow_dict /arrow_mtrxstor matrix put

/arrow
  {
  arrow_dict
    begin

    /alpha exch def
    /y2 exch def
    /x2 exch def
    /y1 exch def
    /x1 exch def

    /arrow_oldmtrx arrow_mtrxstor currentmatrix def

    /dx x2 x1 sub def
    /dy y2 y1 sub def

    /len dx dx mul dy dy mul add sqrt def

    /theta 1 alpha tan atan def
    /rad len alpha tan mul
      theta sin alpha tan theta cos -1 add mul add div def

    x1 y1 translate
    dy dx atan rotate

    len 0 M2
    rad neg 0 rad theta 360 theta sub arcn
    closepath

    arrow_oldmtrx setmatrix
    end
  } bind def

% @(#) fill.ps 1.9@(#)

/fillbox_dict 20 dict def
fillbox_dict /fillbox_mtrxstor matrix put

% fillbox_create is to be used for filling a box which is AT LEAST AS LARGE
% as box given by xmin,ymin xmax,ymax - use clipping to strip off what
% you don't want

/fillbox_create
  {
  fillbox_dict begin

  /ang exch def
  /ymax exch def
  /xmax exch def
  /ymin exch def
  /xmin exch def
  /fillbox_proc exch def  % the fillbox_proc fetches rest of stack

  /fillbox_oldmtrx fillbox_mtrxstor currentmatrix def

  xmin xmax add 2 div
  ymin ymax add 2 div
  translate

  ang rotate

  % rad tells how big a circle which encompasses the box must be
  /rad
    xmax xmin sub xmax xmin sub mul
    ymax ymin sub ymax ymin sub mul
    add sqrt 2 div
    def

  fillbox_proc

  fillbox_oldmtrx setmatrix
  end
  } def


/fillbox_lines
  {
  /incr exch def

  rad neg incr rad
    {
    /yv exch def
    rad neg yv M2
    rad yv L2
    ST
    } for
  } bind def


/fillbox_dots
  {
  /dtrad exch def
  /incr exch def

  /shx 0 def

  rad neg incr rad
    {
    /yval exch def
    rad neg shx sub incr rad
      {
      /xval exch def
      xval dtrad add yval M2
      xval yval dtrad 0 360 arc
      fill
      } for
    shx 0 eq { /shx incr 2 div def } { /shx 0 def } ifelse
    } for
  } bind def

% @(#) ellipse.ps 1.9@(#)

/ellipse_dict 6 dict def
ellipse_dict /ellipse_mtrxstor matrix put

/ellipse
  {
  ellipse_dict begin

  /yrad exch def
  /xrad exch def
  /y exch def
  /x exch def

  /ellipse_oldmtrx ellipse_mtrxstor currentmatrix def

  x y translate
  xrad yrad scale
  0 0 1 0 360 arc

  ellipse_oldmtrx setmatrix

  end
  } def

% @(#) text.ps 1.9@(#)

/text_dict 20 dict def
text_dict /text_mtrxstor matrix put

/SF* {  exch findfont exch scalefont setfont } bind def

systemdict /selectfont known
  { /SF /selectfont load def }
  { /SF /SF* load def }
  ifelse

/txtsta
  {
  text_dict begin

  /angle exch def
  /y exch def
  /x exch def

  /text_oldmtrx text_mtrxstor currentmatrix def

  x y translate
  angle rotate
  } bind def


/txtend
  {
  text_oldmtrx setmatrix
  end
  } bind def

% @(#) max.ps 1.9@(#)

/max { 2 copy lt {exch} if pop } bind def

% @(#) grid.ps 1.9@(#)

/grid_dict 40 dict def
grid_dict /grid_mtrxstor matrix put

/grid
  {
  grid_dict
    begin

    /ysteps    exch def
    /xsteps    exch def
    /delpts  exch def
    /yminpts   exch def
    /xminpts   exch def
    /delta     exch def
    /ymin      exch def
    /xmin      exch def

    0 SLW

    /xmaxpts xminpts xsteps delpts mul add def
    /ymaxpts yminpts ysteps delpts mul add def

    /fnthgt 8 def
    /Times-Roman findfont fnthgt scalefont setfont
    /labsep fnthgt 2 mul def
    /extndfr 1 2 div def
    /graylevel 0.2 def

    /labseptrk -1 def
    0 1 xsteps
      {
      /xn exch def
      /xpts xminpts delpts xn mul add def
      /x    xmin    delta    xn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      xpts
      dolabel { yminpts delpts extndfr mul sub } { yminpts } ifelse
      M2
      xpts ymaxpts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xpts yminpts M2
        x 20 string cvs false CP
        PBX
        pop exch pop exch sub /txtwid exch def
        NP
        xpts txtwid 2 div sub
          yminpts fnthgt sub delpts extndfr mul sub M2
        x 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    /labseptrk -1 def
    0 1 ysteps
      {
      /yn exch def
      /ypts yminpts delpts yn mul add def
      /y    ymin    delta    yn mul add def

      /dolabel labseptrk 0 lt def

      graylevel setgray
      dolabel { xminpts delpts extndfr mul sub } { xminpts } ifelse
      ypts M2
      xmaxpts ypts L2
      ST

      dolabel
        {
        /labseptrk labsep def
        0 setgray
        xminpts ypts M2
        y 20 string cvs false CP
        PBX
        exch 4 -1 roll sub /txtwid exch def
        sub neg /txthgt exch def
        NP
        xminpts txtwid sub fnthgt sub delpts extndfr mul sub
          ypts txthgt 2 div sub M2
        y 20 string cvs S
        } if

      /labseptrk labseptrk delpts sub def
      } for

    0 setgray

    end
  } def

%%EndProlog
%%Page: 1 1
%%PageBoundingBox: 82.8 414.82 264.758 758.16
%%PageOrientation: Portrait
save

NP
0.756 SLW
1 SLC
180 648 54 180 -180 arcn
ST
1.134 SLW
{vtx_circlecross} 126 648 6.1152 0 vtx_create
82.8 745.2 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(Fig. 1. One-loop Diagram) S
txtend
NP
0.756 SLW
1 SLC
180 468 54 180 -180 arcn
ST
NP
0.756 SLW
1 SLC
126 468 M2
234 468 L2
ST
1.134 SLW
{vtx_dot} 234 468 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 126 468 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 180 522 6.1152 0 vtx_create
236.16 463.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
82.8 565.2 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(Fig. 2. Two-loop Diagram) S
txtend
116.28 463.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
restore
showpage
%%Page: 2 2
%%PageBoundingBox: 82.8 79.92 463.824 758.16
%%PageOrientation: Portrait
save

NP
0.756 SLW
1 SLC
180 648 M2
180 702 L2
ST
NP
0.756 SLW
1 SLC
180 648 M2
226.764 621 L2
ST
NP
0.756 SLW
1 SLC
180 648 M2
133.236 621 L2
ST
NP
0.756 SLW
1 SLC
179.999 648 54 449.999 330 arcn
ST
NP
0.756 SLW
1 SLC
180.001 648 54 90.0015 210 arc
ST
NP
0.756 SLW
1 SLC
180 647.998 53.9976 330.001 209.999 arcn
ST
1.134 SLW
{vtx_dot} 180 648 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 180 702 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 226.764 621 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 133.236 621 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 180 594 6.1152 0 vtx_create
182.16 648 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
226.764 610.2 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
124.92 610.2 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
175.68 704.16 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
172.44 561.6 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(a\)) S
txtend
82.8 745.2 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(Fig. 3. Three-Loop Diagrams) S
txtend
NP
0.756 SLW
1 SLC
126 414 M2
234 414 L2
ST
NP
0.756 SLW
1 SLC
255.6 468 140.4 202.62 157.38 arcn
ST
NP
0.756 SLW
1 SLC
-3.6 468 140.4 337.38 382.62 arc
ST
NP
0.756 SLW
1 SLC
104.4 468 140.4 337.38 382.62 arc
ST
NP
0.756 SLW
1 SLC
363.6 468 140.4 202.62 157.38 arcn
ST
NP
0.756 SLW
1 SLC
126 522 M2
234 522 L2
ST
1.134 SLW
{vtx_dot} 126 414 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 234 414 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 234 522 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 126 522 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 180 522 6.1152 0 vtx_create
235.08 523.08 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
116.28 523.08 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
116.28 404.28 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
235.08 404.28 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
172.44 392.4 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(c\)) S
txtend
NP
0.756 SLW
1 SLC
342 594 M2
450 594 L2
ST
NP
0.756 SLW
1 SLC
471.6 648 140.4 202.62 157.38 arcn
ST
NP
0.756 SLW
1 SLC
212.4 648 140.4 337.38 382.62 arc
ST
NP
0.756 SLW
1 SLC
320.4 648 140.4 337.38 382.62 arc
ST
NP
0.756 SLW
1 SLC
579.6 648 140.4 202.62 157.38 arcn
ST
NP
0.756 SLW
1 SLC
342 702 M2
450 702 L2
ST
1.134 SLW
{vtx_dot} 342 594 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 450 594 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 450 702 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 342 702 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 331.2 648 6.1152 0 vtx_create
451.08 703.08 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
332.28 703.08 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
332.28 584.28 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
451.08 584.28 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
388.44 572.4 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(b\)) S
txtend
NP
0.756 SLW
1 SLC
342 432 M2
450 432 L2
ST
NP
0.756 SLW
1 SLC
431.1 396.9 95.7644 158.499 111.501 arcn
ST
NP
0.756 SLW
1 SLC
306.9 521.1 95.7644 291.501 338.499 arc
ST
NP
0.756 SLW
1 SLC
360.9 396.9 95.7644 21.5014 68.4986 arc
ST
NP
0.756 SLW
1 SLC
485.1 521.1 95.7644 248.499 201.501 arcn
ST
NP
0.756 SLW
1 SLC
396 486 M2
396 486 L2
ST
1.134 SLW
{vtx_dot} 342 432 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 450 432 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 396 486 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 396 432 6.1152 0 vtx_create
391.68 488.16 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(R) S
txtend
332.28 427.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
452.16 427.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
388.44 399.6 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(d\)) S
txtend
NP
0.756 SLW
1 SLC
126 252 M2
234 252 L2
ST
NP
0.756 SLW
1 SLC
215.1 216.9 95.7644 158.499 111.501 arcn
ST
NP
0.756 SLW
1 SLC
90.9 341.1 95.7644 291.501 338.499 arc
ST
NP
0.756 SLW
1 SLC
144.9 216.9 95.7644 21.5014 68.4986 arc
ST
NP
0.756 SLW
1 SLC
269.1 341.1 95.7644 248.499 201.501 arcn
ST
NP
0.756 SLW
1 SLC
180 306 M2
180 306 L2
ST
1.134 SLW
{vtx_dot} 126 252 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 234 252 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 180 306 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 212.4 284.4 6.1152 0 vtx_create
175.68 308.16 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(R) S
txtend
116.28 247.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
236.16 247.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
172.44 230.4 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(e\)) S
txtend
NP
0.756 SLW
1 SLC
396 206.1 98.1 123.398 56.6015 arcn
ST
NP
0.756 SLW
1 SLC
396 259.2 61.2 151.928 28.0725 arcn
ST
NP
0.756 SLW
1 SLC
396 369.9 98.1 236.602 303.398 arc
ST
NP
0.756 SLW
1 SLC
396 316.8 61.2 208.072 331.928 arc
ST
1.134 SLW
{vtx_dot} 342 288 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 450 288 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 396 320.4 6.1152 0 vtx_create
331.2 283.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(R) S
txtend
452.16 283.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(R) S
txtend
388.44 234 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(f\)) S
txtend
NP
0.756 SLW
1 SLC
180 62.1 98.1 123.398 56.6015 arcn
ST
NP
0.756 SLW
1 SLC
180 115.2 61.2 151.928 28.0725 arcn
ST
NP
0.756 SLW
1 SLC
180 225.9 98.1 236.602 303.398 arc
ST
NP
0.756 SLW
1 SLC
180 172.8 61.2 208.072 331.928 arc
ST
1.134 SLW
{vtx_dot} 126 144 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 234 144 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 180 176.4 6.1152 0 vtx_create
105.48 139.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
() S
/Symbol 12.96 SF
(\321) S
/Times-Roman 12.96 SF
() S
txtend
235.08 139.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
() S
/Symbol 12.96 SF
(\321) S
/Times-Roman 12.96 SF
() S
txtend
115.2 139.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
244.8 139.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
172.44 90 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(g\)) S
txtend
NP
0.756 SLW
1 SLC
342 108 M2
450 108 L2
ST
NP
0.756 SLW
1 SLC
431.1 72.9 95.7644 158.499 111.501 arcn
ST
NP
0.756 SLW
1 SLC
306.9 197.1 95.7644 291.501 338.499 arc
ST
NP
0.756 SLW
1 SLC
360.9 72.9 95.7644 21.5014 68.4986 arc
ST
NP
0.756 SLW
1 SLC
485.1 197.1 95.7644 248.499 201.501 arcn
ST
NP
0.756 SLW
1 SLC
396 162 M2
396 162 L2
ST
1.134 SLW
{vtx_dot} 342 108 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 450 108 1.092 0 vtx_create
1.134 SLW
{vtx_dot} 396 162 1.092 0 vtx_create
1.134 SLW
{vtx_circlecross} 450 108 6.1152 0 vtx_create
386.28 164.16 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
() S
/Symbol 12.96 SF
(\321) S
/Times-Roman 12.96 SF
() S
txtend
396 164.16 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
332.28 103.68 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(H) S
txtend
388.44 86.4 0 txtsta
0 0 M2
/Times-Roman 12.96 SF
(\(h\)) S
txtend
restore
showpage
%%EOF
\\

