The analytic renormalization setting for the string effective action

Volume 240, number 1,2

PHYSICS LETTERS B

19 April 1990

THE ANALYTIC RENORMALIZATION SETTING FOR THE STRING EFFECTIVE ACTION J. G A I T E Grupo de Fisica Te6rica, Universidad de Salamanca, E-3 7008 Salamanca, Spain Received 3 January 1990

The analytic renormalization procedure for 2D field theory is studied. The consequences for the string effective action are investigated, comparing the R-operation with the Legendre transform. To show where and why they differ, the renormalization is explicitly carried out to third order.

1. Introduction

We are now at a good stage of development of string theory. We have 2D conformal field theories ( C F T ) as good candidates for compactified space-time [ 1 ]. Besides, some m i n i m u m principles which should allow to establish the actual one are being constructed [2,3]. Although the starting point was the study of 2D a-models [4], where the celebrated gradient flow relation was conjectured, there are also more general approaches within the renormalization group ( R G ) philosophy. The Lovelace-Polyakov approach [ 5,6 ] indicates a closer relation between the process which leads from the dual model amplitudes to the string effective action (the Legendre transform) and the 2D perturbative renormalization. It has inspired an evergrowing literature [ 7 ]. Moreover, the inclusion of all string modes allows to connect string field theory with the nonperturbative Wilson R G [8]. Regarding the first issue, in ref. [9 ] (paper I) we considered the relation between bare and renormalized couplings and identified it with the integral equation of motion. Since this does not turn out to work for next to leading logarithm renormalization (or, equivalently, higher poles in analytic renormalization), we obtained in the second part, including irrelevant couplings, the exact second order gradient flow. In this paper we will formulate the 2D Bogoliubov R-operation and compare it with the Legendre transform, deducing the general reasons that prevent them from coinciding. The departure of renormalization from obtaining the equation of motion is explicitly shown to third order, the first, nontrivial, by the appearance of the renormalization double pole (paper I). To this order, it will be highlighted why the R-operation produces this pole.

2. Renormalization and RG

Let us start considering the Callan-Symanzik equations for any field theory:

(°

XZ ~ox

i°)

+ Ak + fl ~ 5

(A,~(x,)'"AzN(XN) ~ = 0 .

(1)

They imply the breaking of scale invariance by a logarithmic dependence of correlators on an overall scale factor that we call x (x may be, e.g. defined as x 2 = Ykx 2 ). The way it comes into play can be shown explicitly from the renormalization procedure: Using dimensional regularization, any divergent integral is of the form

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87

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PHYSICS LETTERS B

P 1= J ddv(y2) -C~ac_F(oe_d/2) [R-2
a 2(o'--d/2)] ,

19 April 1990 (2)

with a < lYl 0), the gamma function has the 1/{ pole. For { < 0 the integral is UV convergent and we can safely perform the zero cutofflimit a--,0. For {> 0 the integral for a=O is defined by analytic continuation from { < 0. It still has the factor R - L I ~ R - V e + finite = 1 / e - l o g R + finite.

(3)

Therefore, the finite part has an ambiguity proportional to log R and, if the integral includes the overall scale x, it appears in the form log(R/x), as can be easily checked in the usual 044 calculations. A useful remark can be made when we consider the RG equation for marginal fields 4,. A simple way to obtain it is by the derivation from the trivial RG equation for the renormalized functional integral F dF ( O a r Orr +fl' O~~) V = 0

O ') V = ; 0~ + fl ' r q + Ofl dV i = O ~O f~O~z + fl ,l~ii

(4,5)

Comparing with ( 1 ) we easily deduce that in this case the anomalous dimension operator acts as

at}j= off' ~2T o,.

(6)

The equivalence of RG logs to poles of the analytically continued correlators is of fundamental importance in string theory. Any solution of the equations of motion (on-shell string fields) must be a 2D CFT, therefore, dilation invariant. The breaking of this invariance caused by renormalization, eq. ( 1 ), can be translated into the analytic structure of off-shell dual amplitudes. This is the basis of the R-operation procedure used in paper I. This procedure has also been considered in ref. [ 10], where it is called "stringy regularization". Similar techniques have been applied by the russian school in the pure 2D context. Zamolodchikov has studied the renormalization flow between next minimal models in the region where they are closely packed together and nearly gaussian, taking { = l - A = 2 / p + l as the splitting parameter. This recalls the Wilson { expansion method. The technique is founded in that, since we have in 2D complete information on the critical behavior of minimal models, it is better to change the dimension of the fields, that leads also to regularized theory, than moving outside d = 2. This procedure has been followed by Dotscnko in a study of the off-critical Ising model (the first minimal model with p = 3) [ 11 ], giving the explicit equivalence between logs and poles 1/{, eq. (2.20). The relation to string theory is as follows. When p-+oe, the minimal model Mp goes to the gaussian model or the free string in one dimension, while the number of quasimarginal fields goes to infinity and they become exactly marginal [ 12,2 ]: they are all described by the graviton vertex, V(k, z) = 0X 0Xexp (ikX), for different k. The corresponding perturbation theory gives the dual amplitudes [ 13,14 ]. The dimension of V(k) is A = k 2+ 2 and the regularization by analytic continuation in its dimension is performed simply by taking the graviton offshell. The concrete way to do this in perturbation theory is the subject of the next section.

3. Analytic 2D renormalization as the Legendre transform

In order to formulate the R-operation for string theory, let us look at it as an elaboration on the 2D gaussian model, or Coulomb gas. The perturbation theory for a Green function of the gaussian model is

88

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19 April 1990

G1, .lE(z'l ..... z'E) = J D X VI~ (z'l)... VIE(Z~) e x p ( - S o -Sint)

=

DXVl,(z~)...ViE(z~)N~=o~.CflilVil(kl)...c, oiNViN(kN) exp(--So),

(7)

with the external vertices Vs,,..., VIE, relevant or marginal, fixed, while the internal variables ZI,...,ZN, are integrated over and the corresponding vertices are marginal. It is known since long [ 15 ] that this Green function to the order N adopts the form t ~t G'I...IE(Z''''"~E)=

l~

i , j = I,...,E

t t IZi--zJl "j t d22'"'d2Zu ")

I~

i = I,...,E

]Z~--ZJlpij ~

I z , - z J l ~° ,

(8)

i , j = 1 ,...,N

J = 1,...,N

with a~j, rio, 7,j = (ks + k s) 2 + n~j, being nonpositive integers. The divergences in the integrations appear for values such that some internal points approach either an external point or another internal point. The elementary divergence is shown by the OPE Vj(k2, z ) V , ( k ~ , 0 ) = IzlZ'-aJ-ZiVm(k, O) ,

k=k2 +kl ,

(9)

with, for graviton vertices, Am-Aj-A,=k2+2-k~-2-k~-2=k2k,

-2.

(10)

As we know from dual model theory, the resulting 1/k 2 pole means the existence of an off-shell propagating particle, a graviton in this case. The integral is automatically regularized while it stays off-shell. So the analytic continuation of the dimension of vertices needed for regularization amounts to take them off-shell. The actual way it manifests itself in (8) is in assigning a different complex parameter to the set of propagators joining each couple of vertices. Noticeably, this procedure has also been used in 4D (ref. [ 16 ], see the appendix of ref. [ 5 ] ). The expression (8) can be holomorphically factorized in 2 functions of z and f variables. The holomorphic and antiholomorphic sectors are equivalent for the bosonic string and it is necessary to consider only one; therefore, one dimensional integrals. The Green function (8) depends on a large number of complex parameters but it may be split into several divergence structures. The systematic way to do that constitutes in the BPHZ R-operation. Each structure, a family of nonoverlapping divergences, is called a Zimmermann forest. The R-operation is a recursive procedure. It proceeds from the renormalized N order integrals to the N + 1 divergent ones. Once the subdivergences are subtracted up to order N, the overall divergence is removed by the N + 1 order counterterm. In the usual notation, J ( G N + I ) is the divergent integrand for the graph GN+ 1, '~ (GN+ I ) is the integrand without subdivergences and !Y (GN+ ~) is the renormalized one, ~(GN+a) =~(GN+,)+

C~(Gu+l ) ,

(11)

with g ( O u + l ) the global counterterm. Zimmermann gave the explicit form of ~ (GN+~) from the fact that nonoverlapping divergences renormalize independently: ~(Gu)=J(Gx)+

Z l~I c#(7),

(12)

U 7~U

where U= (~, ..., ~,~) is a forest. We shall use these formulae in the next section. For analytic renormalization in the MS scheme ~(7) = - K ~ ( 7 ) ,

(13)

K being the operation that picks up the pole part of ~. The remarkable fact of the R-operation is that the counterterms are local and of the same kind of the already interacting vertices, and can be generated simply by changing the coupling constants. 89

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AS=d(t'~,

PHYSICS LETTERS B

19 April 1990

(14,15)

S r = ( t ~ V , = S + AS=((t'~ +&o'r)V, .

Then, the infinities in the expansion in (t[, are cancelled in the expansion in (t'r by similar infinities appearing in

@'r. In perturbation theory, the analytic renormalization in the MS scheme which we use, &~'r has a particular form generalizing the known one for dimensional renormalization [17,5 ]. Graphically, it is the pole part of the V, tadpole: (t~are(k)= e

j,-2( (ttren(k)+ ~ ' ~N ..~(1)i t,~Jl [b "J ---/NW ren\r'l)'''~OjN(kN)'~

~k2(

, )n--, ~N aJt.-..JN N)i (t jl ""(t ]v ) '

N

k = y, k;.

(16)

s=l

Let us consider now the Green functions that indeed appear in string theory. They are the ones without external vertices, or rather with only three due to M6bius gauge fixing. Their expansion in the bare coupling constants, which are now the space-time fields (graviton, ...), gives the dual S-matrix elements when they are onshell. This is the field theory to which dual models reduce in the low energy limit. The functional of off-shell fields Zb (~0{,) =log j" D X e x p [Sb(X ) ]

( 17 )

is, therefore, the S-matrix generator [ 13,14 ], and according to old conventions we name their arguments incoming fields, (tio = (t~,. Its perturbative expansion yields the truncated vertices, which have poles corresponding to resonances. The low energy effective action f is obtained by the standard method of the functional Legendre transform, and its arguments named classical fields (t~, are the solutions of the equations of motion 8F k , 2~0c,i a(t~. = k2(t{n " F[(t~(k)] =Z[(t{~(k)] + f (tin

(18,19)

The Legendre transform is realized perturbatively as a pole subtraction process: The effective action generates the proper vertices which indeed do not have poles corresponding to propagating virtual particles. If we write eq. (19) perturbatively, separating the kinetic term from the interaction/~, "2 , ~" k2(tln =k: (to q - ~ - = k 2 ~ o i +

1 , ~ !~. Cu,..jN(tac'...(t~' ,

(20)

The Fderivatives Fu~jN are the proper vertices. Let us put them in the form

1

(t{. =(t~+ ~

1 ~ ~..FU,..#,(t~' ...(t~N.

(21)

Eq. (18) can be written to order N in (to as ~,' .

1

s* Jx__ - -1 Fu~s~'(tc'''(t~ - ~ L!

g

a)j l CnjL f i 2 i (tin k (tic, u,...,L~c."r~ +

(22)

where (21 ) is substituted for (tin in order to achieve the required N power of(tc on the left-hand side. This makes F,v intervene in the second term, which is cancelled by the first term for L = 2. The remaining terms have proper vertices up to order N - 1. Therefore, we can give a recursive interpretation to (22). It serves to calculate the proper vertex of order N, which has no poles in k 2 (k is the m o m e n t u m in any intermediate channel), from the truncated vertices up to order N and the proper vertices up to order N - 1. The poles in the truncated vertex of 90

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19 April ! 990

order N are subtracted by terms with truncated vertices of lower order whose legs end on lower order proper vertices. This is done through the substitution (Pin((Pc). Recalling that the q~i, are the bare couplings and the cited poles are the UV divergences of the underlying gaussian 2D FT, we are forced to conclude that we are considering nothing but the R-operation in the MS scheme and the ~0care the renormalized couplings, q~c= ~0ren. In spite of this wonderful realization, comparison of eqs. (21) and (16) shows that the latter has the higher pole terms, which are not present in the other one. This relation between string theory and 2D renormalization, in fact suggested by Polyakov's method [ 6 ], may be written as ~0~,= e x p ( -- ~ k 2 ) / k 2/~t,

(23)

with z the logarithmic renormalization scale, z = l o g R. It is stronger than the usual gradient flow, which is deduced from it. (We reproduce the deduction in paper I):

0= d;dr =exp(-rk

)

1

dq~{\

(24,25)

Here F 0 must be a double covariant derivative in order to have a covariant equation (see ref. [ 18] also ref. [8] ), and it is proportional to the metric in the space of metrics [2]. This has been checked in sigma model calculations [ 19,14 ]. More precisely, it is the term proportional to r in it. Taking into account the equivalence between r and 1/k 2 expansions already commented on, 1 ~ Fij=G,] .

(26)

The problem is due to the explicit dependence of Zr and its derivatives on r, which is checked with the RG equation (4), (5), and comes from the elimination of the M6bius group volume [ 18,20 ] while the string effective action and its derivatives are independent of any 2D concept. As those references propose, we must define these in terms of the others factorizing out the dependence on r. However, a quick look at (4), (5) shows that it cannot be only linear, as they say. In paper 1, we proposed to take simply Felt= Zr (r= 0). Other prescriptions are probably equivalent. Once that is done, ( 16 ) should be (21 ), and the deduction of the gradient flow holds. We understand now the difference between 2D renormalization and the Legendre transform. The first operation is designed as to give a finite fl function, which is indeed obtained only from the first order pole a [ 17 ]. This fact implies that the higher poles are determined from it by the RG pole equations and none can be null. This is also shown by the Klebanov-Susskind method [21 ] of obtaining the relation between bare and renormalized couplings by perturbatively integrating the equation dW _ y(~o) = 3j~0J+ff(~0), dr

(27)

and taking the limit r-~ov to enter the region where the bare couplings are defined. The ensuing series substantiates (16) and shows explicitly how the higher poles are constructed from the simple one.

4. Applying the R-operation. The third order counterterm This third order counterterm appears for the first time when the renormalize the five-vertex correlator. It cures the divergence originating from three coalescing vertices in the remaining double integration after M6bius gauge fixing, as well as in any high order correlator. 91

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The counterterms for (0~are obtained from the tadpole with the off:shell vertex V,. This is easily seen from the factorization properties of dual amplitudes. The general M6bius gauge fixed N point amplitude for vector particles can be written as R

z3 dZN_2..,

0

dZ2(ZN_I --Zl) 0

=RkN-lkl+lfdzzv_2.., R

0

(Zi--Za) I'&' [ I

[I I~
Z,+ I

__Zi)

z3

N-- 2

f dz2 0

2

i=1,3 .... N

3

17 ( & - z J ) k ° n ( R - z / ) ku.... [ I ( z , + l - z , ) IGi
2,

(28)

generalizing the gauge fixing to z N_ 1= R instead of 1, for it will act as a renormalizatlon scale, log R = r. It has an overall logarithmic divergence when zN_2, ..., z2~0, which is found after integration by parts over zN 2 R

ZN--2

f dZ,v-2(ZiV-2--ZN 0

3 )k

' 3--1.~D . ,k,v klk--Z'N--2]

23

'-2 f dZN 3... f 0 0

R

kN--I'N--2

-- kN 2,N--3 -- 1

I

ZN

dZN-2(ZlV_2- z x - 3 j "~kN2"-'--I(R--ZN '" "

2)k ............ I f

0

2

dzx_3 ....

(29)

0

It has also "dual divergences" when some z , - , R . They may be segregated dividing the Z~: 2 integration range, say by R / 2 , though this is not necessary because the one we need is easily picked up. The change 6( v in the vector field, eqs. ( 15 ), ( 16 ), due to second order renormalization, is straightforwardly found [6,9]: R k2 i 6¢"(k) = ~ 5 - ~ (k2 - k , )~[¢(k2)¢(k: ) ] .

(30)

Let us find the three on-shell one off-shell tadpole, and the third order counterterm from it. Again, we shall follow closely paper I. The five-point vector amplitude is considered,

, 4 5 = R k4:+l

dz2(R-z3)k43-1(R-z2)ka2(z3

dz3 0

---72) 17u'rs~3-2 2 .

(31)

0

It is selected because it has logarithmic overall divergence and we save one integration by parts. Let us also convert the subdivergence in the logarith as in (29), ; dz2 -2 .... 2,_ w 3 - ~ 2_ , , t ,tD" - ~ a_J ,s42 - s - 1I

; dz2z~-l[t(z3-z2)

0

0

* L(R-z2)~42+s42(.73-22)'(R-7.2) s42-1 ] ,

(32)

the second summand, proportional to 5'42, does not contribute to the overall divergence. From the first the second order counterterm is to be subtracted. Let us name the Feynman graphs involved, G=

( Z l , z2, z 3 ) ,

y=(zi,z2)

(33)

.

The integrand without subdivergences is obtained by ,~(G)=,Y(G) +~J(6/7) [-K)(7)]

.

Since G / 7 = (Zl, z3) and, obviously, .2(7) =,~: (7): 92

(34)

Volume 240, n u m b e r 1,2

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19 April 1990

~ ( O ) = d (O) + d (O/7) (-KS(?') ) z3

R

~3.

(35)

It is easy to check that there is no subdivergence dz2~(G)=tz~

dzzz~-l(z3-zz)~-l(R-zz)'~z-tz~-

o

zt3-1R "42

0 =tR

( l z3 nU+1S 4 12 R s

Z3 + ~Sl n 2Z3 -R ~- +... ) +finites=o.

(36)

The distribution zt3+u- ~ l o g ( z a / R ) is well defined and shows the first RG logarithm, according to the CallanSymanzik equation ( 1 ). The global counterterm ~6(G) is proportional to 6(z3) and consists of a simple pole 1/S, S:--s + t + u, whose residue yields the four-vector irreducible vertex, and a double pole R

double pole t

f

dz 3

(~

RS) -t+u-l(R-Z3)S43-1 s z3

0

= tRS. 3- ~R ~

1 ( t-~u)s

= _ tRY.3- l _ _ S( S-s)

(37)

"

To understand the origin of this pole we must go deeper in the renormalization procedure. At first sight, (37) shows that the counterterm RS/s, being defined at R, does not subtract all the second order tadpoles, but only the pole. This causes at the same time the RG logarithm in (36) and the double pole. The analysis of Collins (see ref. [17] p. 98) (made in momentum space) teaches us that the region of the double integration which produces it, occurs when z3, z2-~0 but z2 approaches to 0 faster, Z2/Z3--~0, and for this reason it is properly called subdivergence. One might think that in this region the second order tadpole should be renormalized at a larger scale. As a matter of fact, it is not difficult to see that different values of the double pole are obtained by choosing different renormalization scales. A better understanding is achieved by introducing Chan variables, Z2

Z3

X=--

y= ~ ,

Z3

(38,39)

dz3dz2=R2ydydx,

and writing the integral to renormalize

(40)

As=idz3idz2z~Z'(z3-z2)C~3Zz~3'(R-z3)C~43(R-z2)~42

o

o

in the more symmetrical form I

(41)

A s = f &rdyxb~-l(1--X)h2-1yP~-l(1--y)#2-1(1--xy) - a , o

where, in the case selected, a ~ - o L 4 2 = - s 4 2 , b l = o L 2 1 + l = s ,

b2 ~ OL32-~'-1 =t,

fll = OL32"~-OL31 ~- OL21 -~- 2 = S,

,~2 = OL43"[- l =$43.

The three regions of integration of ref. [ 17 ] are 93

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x finite, y ~ 0 ,

third order overall divergence,

y finite, x ~ 0 ,

second order divergence,

x~0, y~0,

subdivergence.

19 April 1990

The second one has not yet been considered because we have taken the 1/S pole part from the beginning. Nevertheless, in the whole five-point amplitude renormalization, it has the same role as the 1/S pole, because of inversion symmetry, realized in (41 ) as x ~ y . Then, there exists a symmetrical double pole 1/s ( s - S ) , coming from G' = (zs, z4, z3). Curiously, adding both, 1 / s ( s - S ) + 1I S ( S - s )

(42)

= 1 / s - S ( 1 / s - 1I S ) = - 1 / s S ,

we get the normal double pole in the amplitude. This suggests an overcounting of it by the R-operation. Let us see that this is indeed the case, dividing the integration region in order to track the appearance of the poles [22 ]. Symbolically,

0

0

0

il(lf i/' f

1/2

0

1/2

'Si

0

0

I/2

i If ii

1/2

= 1/sS+ 1/s+ 1 / S + finite + dual poles,

0

I/2

1/2

(43)

corresponding to the above mentioned regions. The subtraction is realized independently for the second order divergence and the overall divergence, by inserting 1 / s d ( x ) and 1/solo,) in (41) instead o f x '-~ and y'-~, respectively. We get two integrals over one variable: I

--l--f dyvS s o = -2/sS-

I

l(1--y)s43-1-- I f

dxxs-l(l-x)ss2-1_o

1 / s - 1 / S + finite+dual poles.

_

_

1

1 B(S, s a 3 ) - ~ B ( s , s32) S

(44)

Clearly, the pole 1/sS is subtracted twice. Let us conclude by discussing how the results of this paper should be interpreted. In spite of the tbund departure of renormalization and eq. (16) from the equation of motion (20), the gradient flow seems to hold, as is shown by a-model calculations. Even a fairly rigorous proof, related to the methods in this paper, is available [ 18 ]. So. one may turn to another philosophy. In paper I it was proposed to consider renormalization flow (RF) in theory space, where the gradient flow holds, as a dynamical system. The projection on the manifold described by quasimarginal couplings (the center manifold) is called, in the stochastic language, the elimination of fast variables [23 ], and must produce the restricted gradient flow. In fact, this type of flow has appeared already in that context [24]. Another interesting point, though rather speculative, concerns the Zamolodchikov program [2 ] on establishing the RF between known 2D CFT and its utility in the string ground state problem. This comes from its lagrangian formulation in terms of bosonic fields: They can be interpreted as Landau-Ginzburg mean-field models [ 12,25]. For c> 1 there is no thorough classification of 2D CFT. To this aim, the most powerful tool is catastrophe theory [26], which is also applicable in the case of off-criticality and in any dimension. Having a bosonic interpretation and, therefore, being space-time-like, which extends off-shell, the generalization of Zamolodchikov's method would materialize the compactification process.

Acknowledgement

The author is grateful to J.L. Miramontes for useful conversations, to D. Kutasov for correspondence and 94

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A.A. T s e y t l i n f o r m a k i n g h i s w o r k k n o w n to h i m p r i o r to p u b l i c a t i o n . H e is also i n d e b t e d to C. G 6 m e z for a d v i c e o n t h e m a n u s c r i p t a n d to t h e C I C Y T f o r s u p p o r t .

References [ 1 ] D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. [2] A.B. Zamolodchikov, JETP Lett. 43 (1986) 730; Sov. J. Nucl. Phys. 46 (1987) 1090. [ 3 ] G. Curci and G. Paffuti, Nucl. Phys. B 312 (1989) 227; N.E. Mavromatos and J.L. Miramontes, CERN preprint CERN-TH.5408/89; H. Osborn, Phys. Lett. B 214 (1988) 555. [4] C.G. Callan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593. [5] C. Lovelace, Nucl. Phys. B 273 (1986) 413. [ 6 ] A.M. Polyakov, Gauge fields and strings, Contemporary Concepts in Physics, Vol. 3 (Harwood Academic, New York, 1987 ). [7] B. Sathiapalan, Nucl. Phys. B 294 (1987) 747; R. Akhoury and Y. Okada, Phys. Lett. B 183 (1987) 65; H. Ooguri and N. Sakai, Phys. Lett. B 197 (1987) 109; T. Kubota and G. Veneziano, Phys. Lett. B 207 ( 1988 ) 419. [8] T. Banks and E. Martinec, Nucl. Phys. B 294 (1987) 733; J. Hughes, J. Liu and J. Polchinski, Nucl. Phys. B 316 (1989) 15. [9] J. Gaite, Mod. Phys. Lett. A 4 (1989) 941. [10] D. Kutasov, Phys. Lett. B 227 (1989) 68. [ 11 ] VI.S. Dotsenko, Nucl. Phys. B 314 ( 1989 ) 687. [ 12 ] A.W.W. Ludwig and J.L. Cardy, Nucl. Phys. B 285 ( 1987 ) 687. [13] B.E. Fridling and A. Jevicki, Phys. Lett. B 174 (1986) 75. [ 14] A.A. Tseytlin, Nucl. Phys. B 276 (1986) 391. [ 15 ] S. Coleman, Phys. Rev. D 11 ( 1975 ) 2088. [ 16 ] K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Nucl. Phys. B 174 (1980) 345. [ 17 ] J.C. Collins, Renormalization (Cambridge, U.P., Cambridge, 1984). [ 18] D. Kutasov, Phys. Lett. B 220 (1989) 153. [ 19 ] N.E. Mavromatos and J.L. Miramontes, Phys. Lett. B 201 (1988) 433. [ 20] A.A. Tseytlin, On the renormalization group approach to string equations of motion (Lebedev Institute, Moscow, 1989 ). [21 ] I. Klebanov and L. Susskind, Phys. Len. B 200 (1988) 446. [22] P.H. Frampton and K.C. Wali, Phys. Rev. D 8 (1973) 1879. [23] N.G. Van Kampen, Phys. Rep. 124 (1985) 69. [24] H. Haken, Synergetics (Springer, Berlin, 1977 ). [25] A.B. Zamolodchikov, Soy. J. Nucl. Phys. 44 (1987) 529; P. Howe and P. West, Multicritical points in 2 dimensions, the renormalization group and the epsilon expansion, CERN preprint CERN-TH 5299/89 (February 1989 ). [26] C. Vafa and N. Warner, Phys. Lett. B 218 (1989) 51; J. Gaite, Phase transitions as catastrophes: the tricritical point, University de Salamanca preprint FTUS 07/89 (July 1989), Phys. Rev. A, to appear.

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