On loop corrections to string theory effective actions

Nuclear Physics B298 (1988) I09-132 North-Holland, Amsterdam

ON LOOP CORRECTIONS TO STRING THEORY EFFECTIVE ACTIONS R.R. METSAEV and A.A. TSEYTLIN Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky pr. 53, Moscow 117924, USSR Received 19 June 1987

We make some comments concerning the structure of loop corrections to the effective action (EA) for massless fields of a string. The singular part of the a' ~ 0 limit of the one-loop EA in the open (super)string theory is studied and is shown to be in correspondence with the ultraviolet divergent part of the one-loop effective action in the (super) Yang-Mills theory. In particular, we reproduce the known result about the absence of AD-4F.2~one-loop infinities in D = 26 Yang-Mills theory starting from the open Bose string theory. We also discuss the path integral representation for EA (EA = generalized partition function for the o-model) and use it to compute the open string theory EA on the disc and the annulus.

I. Introduction T h e useful concept of the low-energy tree-level effective action (EA) for the massless string modes is well-known at present (see e.g. refs. [1-4]). By definition, the tree-level E A is a field-theory action that reproduces the massless sector of the tree-level string S-matrix. Thus it represents an off-shell extension of the tree amplitudes with massless exchanges subtracted out. It is natural to define the loop-corrected E A as an action with the tree S-matrix equal to the massless sector of the loop-corrected string S-matrix [4, 5]. Then the EA corresponds to the diagrams with all kinds of loops (massless, massive and mixed) but only with massive " t r e e " exchanges (i.e. it is "one-particle irreducible" only with respect to a finite n u m b e r of massless particles). N o t e that thus defined EA is different from a standard q u a n t u m effective action corresponding to a field theory with an action equal to the tree-level EA, which does not include certain diagrams, e.g., purely massive loops. Thus the study of q u a n t u m properties of a tree-level effective field theory does not give correct i n f o r m a t i o n about the full loop-corrected EA. Given a string loop amplitude we are to subtract all its singularities due to massless tree exchanges in order to obtain the corresponding term in the EA. Since 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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R.R. Metsaec; A.A. Tseytlin / String theo~

the modular infinities can also be interpreted as being due to the massless exchanges* [6] the resulting EA must be finite. This is consistent with the expectation that the stationary points of EA should give us stable vacua with the corresponding scattering amplitudes being free of modular infinities [7, 8]. The subtraction of the modular infinities which are present on an arbitrary background may be carried out directly in the EA by means of the renormalization of the fields**. The resulting "fl-functions" (corresponding to the full-"local" and "modular"-renormalization) should be the same as in the a-model approach [7-10] and hence should be equivalent to the effective equations of motion. A contact between the S-matrix and the a-model approaches can be established through the concept of the generating functional for string amplitudes [11]. It is very plausible that the total (summed over genera) renormalized partition function for the o-model Z = F~Zg coincides with the full loop-corrected EA [4, 5] (see also below). The tree-level EA has a natural low-energy expansion in powers of a' (or, equivalently, in powers of derivatives of the fields). One should be careful in attempting to use analogous expansion in loops since the loop amplitudes are non-analytic in a' and momenta even after the subtraction of the massless exchanges. The reason why string loop amplitudes are, in general, singular in the a'---, 0 limit is that a ' ~ 0 corresponds to decoupling of the massive modes and hence leaves us with loops of the "limiting" a' = 0 field theory which are ultraviolet divergent. The relation lim~,~0(loops of string theory)= (loops of the limiting a' = 0 field theory) was checked in ref. [12] using the examples of the one-loop open and closed superstring 4-point amplitudes. It was assumed in ref. [12] that a number (n > 2) of D = 10 dimensions are compactified on a torus of radius R and the limit R ~ 0 was taken simultaneously with a ' ~ 0. One aim of the present paper is to provide additional checks of the above relation in the case of the Bose string and superstring one-loop amplitudes in uncompactified space. We shall first present the results for one-loop ultraviolet infinities in D-dimensional Yang-Mills (YM) theory and D -- 10 super-Yang-Mills (SYM) theory (sect. 2) and then will rederive them by taking the a' --* 0 limit of the open Bose string and superstring one-loop amplitudes (sect. 3). In particular, we shall give a string-theory derivation of the curious fact of the absence of the A D 4F~, one-loop infinities in YM theory in D = 26 [13]. Our procedure of taking the a ' - ~ 0 limit of the open string loop amplitudes is based on the introduction of an explicit ultraviolet cut-off A in the amplitudes. * In the Bose string theory there are additional singularities due to the tachyon exchanges which, in principle, can be eliminated using a "dimensionless" regulator (accounting for logarithmic infinities only). A more consistent procedure is to include the tachyon field in the arguments of the EA and to subtract also the tachyon exchanges (i.e. to renormalize the tachyon field). A more serious problem is how to deal with the teachyonic loop (" handle) (IR) infinities which apparently have no sigma-model interpretation and formally propagate into the EA. We shall ignore this difficulty in what follows (see also ref. [5]). * * This is analogous to the "local" renormalization subtracting the massless exchanges, which is needed if the massless poles of the tree amplitudes are traded for an explicit 2d cutoff dependence [4].

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With A fixed the expansion in powers of a' is regular. What we determine are the leading A-dependent terms in the a' expansion of the one-loop EA. Open string theory is UV finite only for G = SO(2D/2). The situation is different in closed string theory where there is a built-in UV cutoff A - 1 / ~ (see sect. 3). The ~ ' ~ 0 singularities of the closed string EA thus correspond to UV divergences of the limiting field theory. The S-matrix method based on expansion in powers of the fields may not be the best one for computing loop corrections to the string EA. In fact, most of a limited number of exact results for the one-loop EA in an ordinary quantum field theory are most easily found using the background field method and heat kernel ("proper time") techniques. An expansion of the (one-loop) EA in powers of the fields in certain cases is even singular. It is easy, of course, to write down a string-theory counterpart of the r.h.s, in the formula

logdet(-~2(A)+X)= -f0 T f ~x(~)exp

+

A remarkable fact is that the resulting Polyakov path integral is a generating functional for the (one-loop) string amplitudes [11]. The basic conjecture is that after renormalizations, subtracting all kinds of massless exchanges [4, 5], this path integral (the partition function for a 2d o-model) coincides with the EA as defined in the S-matrix approach. We would like to stress that while a rigorous proof of this conjecture at the tree level is complicated by the issue of non-compact M~Sbius invariance, no such complication exists in loops. In sect. 4 we shall present the expression for the one-loop EA in the open Bose string theory computed using the partition function ansatz. We shall also study the a' ~ 0 limit of the EA, reproducing the results for infinities of the limiting YM theory. An important formula we use in the analysis of the a' --->0 limit is proved in appendix A. In appendix B we relate the two types of traces which appear in the string-theory and field-theory expressions.

2. One-loop infinities in Yang-Mills and super-Yang-Mills theories in D dimensions Below we shall summarize several (old and new) results concerning the one-loop infinities in a ' - - 0 "limiting" field theories- Yang-Mills and super-Yang-Mills theories. We shall use proper time cutoff to account for power infinities. The one-loop effective action depending on the background YM field strength F~, is given by )2saslogdetA~, where As ( s = 0 , 1 , ~ ) are second order (scalar, vector, spinor) operators, A = - ~ 2 ( A ) + X(F) (for details see ref. [13] and references

R.R. Metsaev, A.A. Tseytlin / String theory

112

therein)• I n general, F 1 ~_ ½log det A -__ - 5l f~ 1 ~1~ =

o~ dt -~-Sp e -tz~ ,

[1

(4qr)O/2

~ ADbo +

1 D - 2

Flo~=f d°x~l~ ,

A e - 2b2 +

1 D - 4

(2.1)

AD-4b4 + • . .

+½1og(AZ/#2)bDI,

A =e

1/2~ ~,

(2.2)

where bp are the DeWitt-Seeley coefficients• In ref. [13] several leading bp coefficients were c o m p u t e d for (super) Y M and (super) gravity in arbitrary n u m b e r of dimensions• In particular, it was found that for Y M theory in D dimensions* bo = / 3 o T r l , 66 =

--

b 4 = ~t'~ P l T r F ~2,

b 2 = 0,

~ / 3 2 T r ( ~ F ~ , ) 2 - ~/33Tr( F~F~xFx~ ) ,

(2•3)

where 13o = 133 = D - 2,

131 = D - 26,

/32 = D - 42.

(2.4)

A remarkable observation that follows from (2.4) is that there are n o F 2 infinities in D = 26 Y M theory. We shall rederive this fact from string theory in sects• 3, 4. In the case of the combination of a non-abelian vector and a Majorana-Weyl spinor corresponding to the N = 1 SYM theory it was found [13] that (2.4) is replaced by /30 ~--- / 3 3

~--"

D - 2 - ¼n,

/32 = D - 42 + n,

/31 = D - 26 + ½n,

n =- 2 D/2 .

(2.5)

Hence/3o . . . . . /33 = 0 if D = 10, i.e. N = 1, D = 10 SYM is free of A 1°. . . . , A 4 infinities• Yet there are A 2 and log A infinities in this theory, i.e. b 8 and bl0 v~ 0. In ref. [13], b 8 4 : 0 was found for a constant abelian 4-dimensional background. In fact, it is possible to give the general expressions for b 8 and blo corresponding to A = _ ~ 2 + X using the results of ref. [14]• In the case of the operators A 0, A1/2, A 1 in non-abelian background 1

b8 = Tr[vlF .Fo.F

F.

+ v2F..F. Fo F

+ y3F~F~,~FxpFxp + "{4F~FxoF~,~Fxo] + (terms with ~ F ~ ) , 1

(2.6)

9

blo = ~.wTr E xiKi + ( F5 terms and terms with ~ F ~ , ) , •

i=1

* Here Tr is the trace in the adjoint representation (see appendix B).

(2.7)

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113

TABLE 1 Coefficients in eq. (2.6) (n = 2 D/z)

A0

Yl

3'2

2

1

35 26n

A1/2 A~

"/3 17

105 4n

35 (2D + 336)/35

74

21 (D + 672)/105

1

42~

21o 19n

16n

10~ (17D - 672)/210

105 (D - 336)/420

where gl

D -

2

eJ.9.FpoF

K3_

.,

K== e.F.,eFooF/o, K,=(e F oFoo)

2

K6 =

,

K8 = F,,~Fpo~xF~,.~xFp~,

K 7 = (..@~,F,,p)2F2 ,

F¢-

K9 =

(2.8)

The coefficients 7n and ~, we have found are presented in tables 1 and 2. Using them it is straightforward to compute the total be coefficients in YM (b~°t = bp(A1) 2bp(Ao) ) and in SYM (b~°t= bp(A1) - 2bp(Ao) - ¼bp(A1/2)) theories. We shall list only the results which we shall use for comparison with the string theory. In the case of the abelian background in YM theory (see appendix B) -

b8

=

(N

-

5) [ 3~o(D + 238)F 4 + ~ x( D

_ 50)(FZ)Z]+o(O~F..)

(2.9)

The A 2 and log A counterterms in SYM theory in D = 10 in a general non-abelian

TABLE 2

Coefficients in eq. (2.7) (n = 2 °/2) N1

A° A1/2 A1

g2

4

4

189 7n

18"-'-9 148n

2-~ 4(D+468) 189

189 4(D+288) 189

K3

2

6-3 10n 63 2(D-48) 63

g4

2

~5

~6

~7

6~ 5n

2

g9

0 19n

0 2n

84 4

3 16

21

3

4

63 lln

189 187n

0 n

252 2(D-60)

189 2D

7 8

189

7

63

g8

63 4(D-18) 63

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114

background are

b s = }Tr( F..Fp~Ff,xFox + ½F~Fo~FpxF~x

I r..F~.r~oF~ . - ~F~.F~oZ~F~. ) + o( ~ . ~ ) , 2 blo = ~((4K1 +

(2.10)

K4-4Ks-Ks)+O(FS,~F~)

= -

(2.11)

~ oF~.F~9oF.~F~-

~oF~F~9oF~oF~

~

~oF.~F~oF~F~.)

+ O(F~,~.C~).

(2.12)

-

To pass from (2.11) to (2.12) we have made "integration by parts". Note that the structures in (2.10) and (2.12) are very similar: they differ only by the two contracted derivatives in the first and the third F factors in (2.12). The F 4 terms in (2.10) are just the same as the a '2 correction in the open superstring tree-level effective action [4]. These observations will be understood in sect. 3 starting from the superstring one-loop 4-vector amplitude. In sect. 4 we shall derive (2.9) from the one-loop EA in the open Bose string theory. It is useful to remark that in order to compute the F 4 terms in b 8 and (DF)ZF 2 (and F 5) terms in bl0 in D = 10 SYM theory we only need to know the following (easily established) terms in the general expressions for bp'S for A = _ ~ 2 + X: 1 b 8= ~ . . S p X 4 + - " ,

1

bl0=-~Sp(-Xs+2X3~2X+X

2

(~,X) 2 +..'). (2.13)

The reason is that all other possible terms cancel out in the total 1 "sum rules" [13] (As= _ ~ 2 + X,, s = 0,5,1) Z

Sp(X(-2X~-

1 k ~X~/2) =0,

k=0,...,3.

bp because

of the

(2.14)

s ~ 0 , 2,1 1

Finally, we recall that according to ref. [13] b0 = b2 =

b4 = b6 = 0

(2.15)

is true also for D = 11 supergravity and hence for all its reductions, e.g. for N = 2, D = 10 (non-chiral) supergravity. As we shall find in sect. 3, type II superstring theory predicts the presence of the A2R4 infinity in N = 2, D = 10 supergravity.

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3. a' ~ 0 limits of string theory one-loop amplitudes The aim of this section is to discuss the a' ~ 0 correspondence between the string theory and the limiting (a' = 0) field theory at the one-loop level. We shall mainly consider the open string theory case. To make the a' ---, 0 limit of the string one-loop amplitudes regular we shall introduce the explicit UV cutoff A in the open string theory. The A-regularized one-loop string theory EA will go smoothly in the a' ~ 0 limit into the A-regularized one-loop quantum effective action for the a ' = 0 field theory. Since we used the proper-time regularization in the field theory case (2.1), (2.2) we shall introduce its analog in the string theory. To this end let us recall the general expression for the one-loop open string amplitudes in the operator formalism (see e.g. ref. [15]) aC~v f d ° p S p t a -A=L o-l=a'p

'V~ 1 V . . . - A - 1 V l, 2+ " .

,

~-1

=

- - X foldX ~, ,

X

(3.1)

~=a'A.

The proper times t i ( k l = f ~ d t e - t a ) with the canonical dimension cm 2 are t i = - - a'ln x v As usual for the one-loop graphs in a field theory, a cutoff is needed only for the integral over the total proper time (which, in fact, is the one that appears in the background field method expression (2.1)): t = E ~ t i = - a ' l n w , w = X x . . . x N. Introducing the standard modular parameter for the annulus q (the ratio of its radii) we get q = e 2 ~ / 1 ~ = e -2~2'~'/' ,

q = e -2"2"'x ,

0 ~ X ~ A2 ,

(3.2)

where we have introduced the parameter 2, = t 1. It is now possible to consider the two limits: (i) String-theory U V limit: a' =fixed, A --+ ~ , i.e. q ~ O. This limit corresponds to shrinking of the hole of the annulus (or the M~Sbius strip). If q-+ 0 infinities cancel out (as they do in the case of non-orientable SO(N) open strings for N = 2 °/2 [16,17]) then the string theory is independent of A. (ii) Field-theory limit: ~' ~ O, A =fixed, i.e. q ~ 1. In this limit the annulus degenerates to a thin circular "wire". Since all massive states* of the string decouple in the a' ---, 0 limit we are left with the one-loop graph in the a' = 0 field theory. The subsequent limit A --+ ~ corresponds to shrinking the "wire" to a point, i.e. is just the usual short distance limit in field theory. It is the second limit that we shall study below. It is important to note that the Y M coupling constant gD = g 0 t ' ( ° - 4 ) / 4 ( g is the dimensionless open string theory coupling) should be kept fixed in the field-theory limit, Then the general structure * Except the tachyon (see below).

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116

of the loop-corrected T=S+I’,+

...

+ ld”x

open string-theory EA is

- ljdDx(F2+&‘3+a’2F4+ SZJ

..e)

[a,AD+a,AD-4F2+a3AD-6F3+

...

+a,vlogAFD/2

+ finite non-local terms + 0( a’)] .

(3.3)

We shall check that the A-dependent terms in (3.3) coincide with the infinite part of the one-loop effective action of the (super) YM theory. In this section we shall use the S-matrix approach, i.e. we shall take the a + O-limit directly in the one-loop string amplitudes. The open Bose string one-loop vector scattering amplitudes are, in general, given by (we omit the group-theory factors) (see e.g. refs. [15,4,18]) .~$.,--g;a’-~‘~

MN= 1

/

[dq MN,

MN=(vl...v~)’

1

---a’ $

doi...

cyclic order

( GP41P.i’,

fJ+4

-2idJp.k,+

d2 Gpq= YG dd,dt$

GPY k P

p4’

where V = j,j?VdOlPipeeikx and Gp4 = G(6,, B4) is the restriction of the Neumann function on the annulus to the boundaries. It has the following useful representation ]19,41 G(W’)

= ;

E

&osn(B-80,

A,,=

1 + q2” ___ I- q2” ’

B,=

___

n-1

2Y”

1-p

’

(3.6)

(3.7)

(G” corresponds to the case when B and 8’ belong to different boundaries). We shall formally extend the modular measure [dq] to arbitrary D

Tq-(D-2)/12ex’(D-26)/121ny[~(qZ)]Z-D,

= fi (1 -Xn). ,,= 1

(3-g)

For D = 26, eq. (3.8) reduces to the standard expression (dq/q3)[J(q2)]p24.

This

[dq] =

f(x)

R.R. Metsaev, A.A. Tseytlin / String theol.

117

particular measure follows from the operator formalism blindly extended to D dimensions. We shall see that the a' ~ 0 limit of (3.4) with (3.8) indeed reproduces the infinities of the YM theory in arbitrary D. As was noted above, the field theory a ' ~ 0 limit corresponds to q---, 1. The analysis of the q ~ 1 limit of the integral in (3.4) is most easily done in terms of the variable w, In w = 2rr2/lnq (q ---, 1 corresponds to w ~ 0). The measure (3.8) rewritten in terms of w is

1

[dw] = -- d i n w ( - l n w ) - l - o / 2 f ( w ) 2-0 ,

[dq] = - ½(2~r)l+D/2[dw],

(3.9)

w

where we have used the standard Jacobi transformation formula 2~r )-1/2 _ ~ w l / 2 4 e ~ra/61nWf(w) "

f(q2) =

(3.10)

The explicit 1 / w factor in (3.9) is a consequence of the presence of the tachyon. It produces essential singularity in the w --+ 0 limit. If w --+ 0

f ( w ) = l - w + O(wZ).

(3.11)

Suppose that m N in (3.4) has the following w ~ 0 behaviour

MNWSO~,(P(N k) + WQ(Nk))(ln w) k + O ( w 2 ) ,

(3.12)

k

where PN and QN depend only on momenta and polarization vectors. According to (3.2), In w = - (a'?t) -1, so that we get, using (3.11), (3.12),

ag y - gN foa2d)~ )t-' + D/2 y"

{[ p(Nk)el/~,'X + R~)l(a,)t)-k + O(e_l/~,x) } , (3.13)

k

R ~ ) = ( D - 2)PN~k) + Q{Nk) .

(3.14)

The term with the factor e 1/~'x= 1 / w (or e "2t, m 2 = - 1 / a ' ) is the tachyonic contribution which does not have a sensible a' expansion. It should be simply dropped in order to have the a' ---, 0 correpsondence with the one-loop amplitude of the massless field theory. This complication is absent in the superstring theory where [15] [dql . . dq. . q

2.7/'2o/'d X

(3.15)

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118

and hence the a' ~ 0 limit of A N is determined simply by the first term in (3.12), i.e. R~ ) = Pff'.

(3.16)

Thus we finish with (both in Bose and superstring cases) AD

lim

a'~0

~N

--

,,N~'~(k)

OD A.~:'N

k

D

-

2k 2k

,

R~)=

lira a' kR(~/ "

~'-~o

(3.17)

Let us now compute the explicit expressions for R N in (3.17). It is trivial to see that all the two-point one-loop vector amplitudes (the annulus, the cylinder and the MObius strip) are identically vanishing so that the vector field remains massless*. Because of the kinematical r e a s o n s ( ~ i . k j = 0, k 1 = - k 2, k ~ = 0) the 2-point amplitude is equal simply to ~l "~2 times a constant which can be computed for zero momentum. But the vector vertex operator vanishes at zero momentum ( f d 0.¢ = 0). The vanishing of the 2-point amplitude is also obvious from (3.4), (3.5) ( f dO G = 0). The vanishing of the one-loop 2-vector amplitude was recently interpreted as the absence of the mixing between the abelian A~ and the antisymmetric tensor B~ of the closed string sector of the theory of oriented open and closed strings. It is certainly true that the particular four-dimensional mixing e,""XPF~,,Bxp [22] which is gauge-invariant with respect to both A, and B~ is ruled out. However, the D-dimensional mixing F~,,,B~,,, [23] which breaks down the B,~-gauge invariance cannot be excluded (if .£a_ (0[,B~x])2 + F~B,~+ . . . . Bu~[3B~ + BII~E.~, Bu~ is decoupled from F,,). In fact, such a term is present in the one-loop E A fdDxtdet(6,~+ B ~ + 2~ra'F~,) that was found in ref. [9] using the fl-function approach. In sect. 4 we shall give a very simple derivation of this action using the path integral approach. The 3-vector amplitude corresponding to the orientable planar diagram (i.e. to the case when all the particles couple to the same boundary of the annulus) easily follows from (3.4), (3.5) (k i . kj = O) •~¢3 -- g3Da'2

D/2 [ C 1 ( ~ 1 " ~2~3"

k~ + two transpositions)

+ a"C2~ 1 • k 2 ~ 2 • k 3 ~ 3 • k l ] ,

(3.18)

c~- f[dq]f~yclic order d01d02d03G12(e32-

d 631),

Gpq= dOp Gpq

C 2 - f[dq]fcyc1~corderdOldO~dO3(G32-G31)(G21-G23)(Gle-G13 ) .

(3.19)

* This is true to all orders in the loop expansion. As is obvious from the form of the generating functional for the vector scattering amplitudes, the Y M symmetry is an exact symmetry on manifolds of arbitrary genus. This is not so in the case of the antisymmetric tensor gauge invariance in the theory of oriented open and closed strings [21] (see also sect. 4).

R.R. Metsaev, A.A. Tseytlin / String theo~

119

T h e first b r a c k e t in (3.18) is the standard Y M vertex while the a ' term corresponds (just as at the tree level [1,4]) to the F 3 term in the EA. To c o m p u t e C 1 we put 0 i in cyclic order, fix the rotational s y m m e t r y by taking 03 = 0 and use (3.6)

f[dq]f0

27r

Ca-

02

d0afo d01 ~

mAmcosm(Ol-02) ~ A,(sinnO2-sinn01).

m=l

n=l

(3.20) Computing finish with

the 0-integrals (fo2'~d02 fo02 d01cos m(O a - 02)sin nO2 = (qr/n)Smn) we

C 1 -

1+

f[dqlI(q),

q2n )2. (3.21)

n=l

I can be rewritten as oe I=

-½+4

q2. (3.22)

5 =1 ( l - - -q-2- n ) 2 '

oo where we defined ~,=1 1 as ~'(0) -- - ~1 ~ . Using the fact that E,,=lnx we can also transform (3.22) into the form

I-

21

n

d 2q ~ q l n f ( q 2 ).

= x/(1 - x) 2

(3.23)

where f is the partition function defined in (3.8). Making the Jacobi transformation (3.10) we get lnw (lnw) 2 (lnw) 2 d . . . . I _ 1 + 2rr 2 + 24~r 2 + rr 2 d l n w In f ( w ) , w-*O

I ~

lnw (lnw) 2 - 1 + -2~r - - Z + 24~r 2

w

7ri(lnw) 2+ O(w2).

(3.24)

C o m p a r i n g (3.24) with (3.14), (3.17) we conclude that only the last two terms in (3.24) contribute to the a ' --* 0 limit of C 1. We find: AD 4 lim C 1 - R(32) ,~'--,o D - 4 '

D-2

e(32)

24

D-26

1

24

* The result for the a' --, 0 limit of ~3 is in fact independent of the value of ~2,~_11.

(3.25)

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R.R. Metsaev, A.A. Tseytlin / String theory

Thus we have reproduced the field-theory result (2.2), (2.4) for the coefficient of the

AD-4F 2 infinities: they are absent in D = 26 and hence C 1 has a regular a'--* 0 limit (modulo the tachyonic singularity). Similar analysis can be carried out also for the C 2 coefficient in (3.19) with the a' + 0 result - ((D - 2 ) / ( D - 6))A ° 6 being again in agreement with (2.2), (2.4). Let us now turn to the case of the open superstring theory. As is well-known [24] the first non-vanishing one-loop massless superstring amplitude is the 4-point one. The one-loop corrected 4-vector amplitude is given by [24,12] d4(tot)- g2oK [ 1 F ( 1 - a ' s ) F ( 1 - a ' t )

-F-(-~+-dTu3

where K is the standard kinematic factor K = -

f ( 1 ) - g20 f[dq] Off

f[du]

+ fO)( s, t ) + ""

]

,

(3.26)

¼st~1 . ~2~3" ~4 + " " " ,

f[du] Of""~; °'',

(3.27)

3

= .o f l I - I dviO(v'+l- vi)'

ui=Oi/2~r'

i=1

~12~34 ~ 1 - ~13~24

~2

U4= 1,

1~23@14 1~13@24

27r °e [1-- 2q2ncos2~uji + q 4n ] ~ij=~(w",,,w)= -- lnq sinTrujiII -(1- - - - -q2n) 2 , n=a -

Pji=uj-Pi

-

(3.281 and [dq] is given by (3.15). Expanding the integrand of (3.27) in powers of a', I2~-~"~22~'t= 1 - o's In ~21 - a't In ~22 + O(a'2),

(3.29)

and using the asymptotic formula (see e.g. ref. [12]) ~p(w ~, w) WS°exp(- ½v(1 - v)ln w),

(3.30)

we find

f~'-

g~ofA2dXrid.

s

,

1 + ~U32/-'1 q- ~1"21"43-{- O(off

,]

.

(3.31)

Computing the v-integrals

ftdul =

~,

1

f[dv]u32~,l= f[dv]v21v,3 = 57'

(3.32)

R.R. Metsaev, A.A. Tseytlin / String theory

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we finish with f ( x ) _ g2o(A 2 _ ~ u In A 2) + O ( a ' ) .

(3.33)

Using the fact that r(1 - a's)r(1

- ~'t)

r(1 + a.u)

= 1 - ~ 2 a ' 2 S t + "'"

it is possible to check that the effective lagrangian corresponding to the tree-level part of (3.26) is given by [4, 3] 1

2

1

t 2

.LPo = 4~12otr[ F~ + ~(2~ra ) (F~,~Fo~F~xFox+13 r , . r o . F,x r,~ - ~1r,~r,.r~or~o

~F..FxoF~.Fxo ) + O ( a ' 3 ) ] ,

(3.34)

where we rescaled A~ so that F = dA + A 2. Hence (3.33) corresponds to the AZF4+ log A 2 ~ F F ~ F F terms in the one-loop EA which are exactly the same as found above in the SYM theory (2.12), (2.13) (note that u = - 2k 1 • k3). Once again the structure of the one-loop corrected superstring EA is thus

r-fdl°x

1

--~- ( F 2 + gt0

a'2F 4

+".)+A2F4+logA2~FF~FF+

. . . }.

(3.35)

It should be noted that the contribution of the M/Sbius strip diagram present in the case of the SO(N) type I superstrings has the a' ~ 0 limit which is different from the a ' ~ 0-limit of the annulus contribution only by an overall coefficient. Both contributions sum up with the contribution of non-planar (cycinder) diagram to reproduce the SO(N) SYM theory result (see appendix B)*. We finish this section with a discussion of the a ' ~ 0 limit of the 4-graviton one-loop amplitude in type II superstring theory in flat D = 10 space. The amplitude is given by [24,12]

4 r ( 1 - ~ ' , ) r ( 1 - ¼~',)r(1 - ~'u)

d 2 °" - ~ 0 ~

s,u r(17 ~',)r(l+ ~',)r07¼~'.) +ClOts't)+ t'

K?0 , . d 2 , r ,

v4=r,

3

(d2pi

0~< Imv;~< r:,

)

H

([ ,im i,21 I01(vj;Ir)[ kkj ,

exp

IRe v;I ~< ½,

~ = ~1 + i~2.

(3.36)

* The cancellation of the q --, 0 infinities in the case of N = 32 is irrelevant for our discussion of the a ' -~ 0 (i.e. q --, 1) limit (see also ref. [12]).

122

R.R. Metsaee, A.A. Tseytlin / String theoO,

As in the case of the open strings we are to establish the relation between the m o d u l a r p a r a m e t e r r and the field theory " p r o p e r - t i m e " . Recalling the operator f o r m a l i s m derivation of (3.36) we conclude that (see also ref. [25]): r2 = t/2~ra' = i / 2 ~ r a ' ~ , 0 <~~ <~A 2 (note that r is dimensionless). Suppose that a ' = fixed, and consider A ~ oo, i.e. r 2 ~ 0. This limit is, in fact, excluded since the m o d u l a r invariance restricts the r-integration to the fundamental domain [26]: Irl >/1, ]rll ~< 1. This explains why the U V infinities are automatically absent in m o d u l a r invariant closed string theories: the largest value that ~ can take is A 2 = (~r~/3-a') 1. In other words, the closed string theory has a built-in U V cut off A - ( a ' ) 1/2. Note that the restriction imposed by modular invariance disappears in the a' ~ 0 limit: A 2 - 1 / a ' ---* ~ . Thus, in contrast to the open string case, here the limits a ' --* 0 and A -* oo are not independent. Taking the a' ~ 0 limit of (3.36) keeping Kxo fixed we obtain lim

~(1-1oop)

~

g~Oo,~A 2 q_ O(ot'),

(3.37)

A 2 ~ 1/a'.

Since the kinematic factor off in (3.36) corresponds to the R 4 term in the EA [3] we conclude that the general structure of the one-loop corrected EA in type II superstring theory is*

Eq. (3.37) implies the presence of the A2R 4 one-loop infinity in N = 2, D = 10 supergravity. In contrast to the SYM case it is easy to see that the total s y m m e t r y of the a m p l i t u d e (3.36) and s + t + u = 0 imply the absence of the ~ R ~ R R 2 1 o g A term in (3.37), (3.38). However, this does not exclude possible presence of RSlog A terms in (3.38). In general, the local part of F in a closed string theory has the following structure

1

~C,¢0+~9°1+g2~CP2+g4~9°3+ . . .

]f~-

(3.39)

where ~ are dimensionless local functions of R and its derivatives (we ignore other massless string modes), £,0i = a~°) + a}l>a'R + a}Z)(a'R) 2 + - . . , and hence contain only positive powers of a'. a ' - 0 / 2 comes from normalization of the constant m o d e of the string coordinate, g is the dimensionless string coupling. Rewriting (3.39) in terms of ~D = g a'(D-2)/4 we get negative powers of a ' in the loop corrections, reproducing in the a' --, 0 limit the U V infinities of the limiting field theory. Note * The presence of the R4 one-loop correction was recently mentioned in ref. [27]. However, the necessity of keeping ~10 fixed in the a' ~ 0 limit, i.e. the singular (A2) nature of this R 4 t e r m (and the a' -~ 0 limit in general) was not properly understood there.

R.R. Metsaev, A.A. Tseytlin / String theory

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that while in the tree approximation the expansion in a' is equivalent to the low-energy (derivative) expansion, this is not true after the account of loop contributions since now the EA nontrivially depends on two dimensional parameters: a' and x D.

4. Path integral representation for string loop corrected effective action The plan of this section is the following. We shall first summarize several qualitative arguments supporting the path integral (o-model partition function) ansatz for the loop EA. Then we shall demonstrate how it works, deriving the generalization of the Born-Infeld action [19] to the case of the presence of the constant antisymmetric tensor of the oriented closed string sector. The same action was recently found using the r-function method [9]. The advantage of our approach is that it is very simple and reveals the connection of the resulting action with the generating functional for string amplitudes. Finally, we shall compute the one-loop correction to the open string EA in the case of the abelian and constant F~. We shall study the a' - , 0 limit of the result and demonstrate its correspondence with the one-loop infinities (2.2), (2.4), (2.9) in the YM theory. Consider the generating functional for "massless" scattering amplitudes in the theory of oriented closed and open Bose strings [11,19, 21]*

Z[G..eO,B..A.I=

g,h=O

e-"xfidgj[dx"le -',

(4.1)

I = 12 + 11 ,

I2-

1 fd2z[V Oax. Oax.%(x)+i bOax. Obx.

47ra'

(4.2)

Ii= f dO[iS:"A.(x)+

~----~eK~p(x)].

(4.3)

Here X = 2 - 2 g - h is the Euler number of a 2-manifold with a topology of a sphere with g = 0,1 .... handles and h = 0,1 . . . . holes, go = e°/2 is the open string interaction constant. The special case of the closed Bose string theory corresponds * Here for simplicity we take the vector field to be abelian, ignore tr 1 = N factors coming from boundaries (exp(-11) should in fact be tr P exp(-i~A~2e'+ . . . ) ) and disregard the possibility of different A~,coupled to different boundaries (see ref. [4] for details).

R.R. Metsaev, A.A. Tseytlin / String theory

124

to h = 0 in (4.1) (then, of course, there is no boundary term (4.3)). K in (4.3) is the trace of the second fundamental form of the boundary introduced [21] in order to have possibility to absorb o into the constant part of the dilaton ~ (X = (1/4~r) f d 2 z g r g R ( Z ) + ( 1 / 2 ~ r ) f d O e K ) . Let us represent I in (4.1) as I 0 + V-9~ where I o = (1/4~ra')fOx ~ Ox ~, V are the vertex operators and q0 are the fields. Expanding e v.~ in powers of V and equating ¢p to their "plane-wave .... in"-values we obtain a sum of on-shell scattering amplitudes. We assume that the path-integral measure in (4.1) includes division by the factor of the (regularized) volume of the M~3bius group (see e.g. ref. [28]) which is non-trivial in the case of the sphere and the disc. This complication is absent for the annulus, torus and higher genus surfaces. It is usually claimed [28] that for D = 26 the integral over g,b in the amplitudes can be reduced to a finite-dimensional integral over the moduli { rp ), i.e. that the integral over the conformal factor p (gab eZPgab(~')) decouples. In fact, this seems not to be true because of the modular infinities present in the Bose string amplitudes. These infinities need to be regularized in a reparametrization-invariant way (e.g. by cutting the lengths of geodesics on a surface) and this should produce additional anomalous dependence on p (see also refs. [8, 29, 9]). The dependence on p should disappear only in the amplitudes computed by expanding near a true vacuum. To deal with a "critical" string theory we, in any case, must not integrate over p (postulating that a "critical" string theory must be Weyl-invariant in a true vacuum). A natural suggestion is thus simply to drop the integral over p choosing g,b = gab as a Weyl gauge [4]. The amplitudes near a true vacuum will be independent of this choice. Adopting this prescription let us consider (4.1) for arbitrary off-shell values of the fields =

oo

Z=

E f[d'rl[dx~'] e-1'x'~]= Zsphere q- Zdisc -]- Zann -~- Ztor-~

"'"

"

(4.4)

g h= 0

This is the generalized partition function for the o-model (4.2), (4.3). It suffers from "local" as well as "modular" infinities. Suppose that the same (e.g. geodesic distance) cutoff e is introduced for both types of infinities. We know that the o-model is renormalizable with respect to the "local" infinities on each type of surface. The basic point is that the total model (4.4) is, very likely, renormalizable also with respect to the modular infinities, i.e. that all dependence of Z on e can be absorbed in the fields G~, ~, B,~, A~*. It is this statement (and not, e.g., the finiteness of the string amplitudes on a trivial background) that should be proved in order to demonstrate consistency of the string theory. Its validity is supported by observations [6-10] that a modular infinity in (V1... VN) on a higher genus surface * Strictly speaking, to absorb the quadratic infinities we need also to renormalize the tachyon field. For simplicity we shall disregard quadratic infinities assuming, e.g., that they have been canceled by the bare tachyon field which was then set equal to zero.

R.R. Metsaev, A.A. Tseytlin / String theory

125

can be represented as log e(V0) times (I11... VN) on a lower genus ("degenerate") surface. Here Vo is a dimension-two (the dilaton in the case of a trivial background) vertex operator. To renormalize the modular infinities, e.g., in the closed string amplitudes on a disc, or Zdisc, we need to add the corresponding bare terms to the fields cp in Zsphere. Note that the exponentiation of the counterterm e x p ( f d 2 z l o g e (Vo)ai~c Ox Ox) properly accounts for renormalizations of similar multiple infinities on higher genus surfaces (see also ref. [5]). Another crucial observation is that the renormalization of the infinities seems to correspond to subtraction of all types of the massless exchanges in the amplitudes [7,4]. Then a natural conjecture [4, 5] is that the renormalized partition function (4.4) represents just the full loop-corrected EA as defined from the S-matrix, F = z(ren)[{p]. Note that thus obtained EA is finite, i.e. is free of all kinds of infinities which may be present in the amplitudes computed on particular nonvacuum backgrounds. In this sense the EA for the superstring theory is no "better" than the EA for the Bose string theory, e.g., both are free of modular infinities*. The only difference is in their extremas: while a flat "empty" D = 10 space seems to be a stable perturbative vacuum of the superstring theory, a flat empty D = 26 space is not a vacuum of the Bose string theory already at the one-loop level [8]. A natural consistency requirement is that the effective equations of motion ~F/6~p = 0 determining the true vacuum, should be equivalent to the full Weyl invariance conditions, i.e. to the vanishing of the full "/}-functions" [7-9]. Then the string theory vacuum amplitudes will be p (or Weyl gauge) independent and, in particular, finite (since a covariant cutoff can always be absorbed in 0)- Note that according to (4.4) ~F/6~ = 0 is equivalent to the vanishing of the full massless tadpole in a background and hence is a natural equation of motion. The general structure of F that follows from (4.4) is

r =

,-D,2f

-

2,2+ 4(0,) 2) + .

+d,e-*[~det(G..+ B,~+ 2~a'F,,) +(d2+d3+

...)+d4e~+

..-},

]

+ O(0¢p)] (4.5)

where d o = D - 26, dl, d2, d3, d4,.., are, naturally, the values of the finite parts of the dilaton tadpoles on the sphere, the disc, the annulus, the torus, etc. The loop corrections in (4.5) contain, of course, non-local pieces. Let us now derive the expression for the disc contribution given in (4.5). The important advantage of the generating functional (4.1) is that it reveals the gauge symmetries of the massless sector of the theory. We conclude that the antisymmetric * This is true only modulo the tachyonic loop infinity problem mentioned in the introduction (see also

[5]).

R.R. Metsaev, A.A. Tseytlin / String theory

126

tensor gauge symmetry 8B¢. = 0 . ~ - 0 ~ . is broken down in the presence of boundaries. On the world sheets with boundaries it is replaced by the generalized symmetry

8B~,~= 0 ~ -

0.~,

8~ = -~,

£ = 2~ra'A~.

(4.6)

At the same time, 6A. = 0.e is always unbroken. As a consequence of (4.6) the EA should depend on B.~ and F~ only through the gauge-invariant combination ~ = B.. + F.., F= 2~ra'F. Hence to obtain the part of Fd_i~c which depends on B.~ = const it is sufficient simply to replace F~ by B.. + F~. in the Born-Infeld action of ref. [19]. In more detail, expanding B..(x) Ox ~ Ox ~ for x = y + ~, y = const, we see that the first term B..(y) O~~ O( ~ reduces to the boundary, fdO~"~"B.~. The term of the same structure fdO~ ~ F.. comes from the expansion of fdO2"A.(x). Computing the resulting gaussian integral on the disc [19] we get

+O(Oa,O.,Oe)].(4.7) The first Born-Infeld term in (4.7) is free of "modular" infinities (corresponding to shrinking the boundary of the disc so that to make a sphere) even without renormalizations. The derivative-dependent terms in Zdisc (those of them which depend on Ha. ~ = 30ExB~.1, R(G) and 0~) contain modular infinities which should cancel in Fdisc against the corresponding counter-terms reserved in the Zsphere. Note that this cancellation implies that Aqoi6FspherJ8qOi, Aq0i - flqn e, is proportional to the infinite part of Zdisc. One interesting consequence of this is that Zdisc must be finite when evaluated on a classical solution (SF~ph~rJ& p = 0) of the Bose string theory (e.g., on the solution, corresponding to the group manifold with parallelizing torsion). Let us now consider the dependence of Za~. on F~. = const. In general, there are two types of diagrams with the topology of the annulus: the planar and the non-planar one. Hence Z.~. should contain two pieces, one corresponding to the case when all vectors are coupled to the same boundary and the other - to the case when vectors are distributed between the two boundaries. In ref. [4] we computed Za~. in the general case when two different constant gauge fields (F1. . and F2~.) are coupled to the two boundaries

Z, (fl, FO=vf[dql2(F,,f ),

(4.8)

D/2

Z= H

f i [detl2.,p1-1,

p=l n=l

~n,p =

1 + fl2 A2 + flpf2pB 2

f2p(flp + fzp)A. B.

fl~ ~

1 + f,~A:+ flpfzpB~

(4.9)

R.R. Metsaev, A.A. Tseytlin / String theory

127

A., B. and [dq] were defined in (3.6)-(3.8) and fl.2p are the "skew eigenmodes" of FL2... The case when all the vectors sit on one boundary corresponds to F 1 = F, F 2 = 0. Then [4] 21~2(F,0)

= f i [det(8~.+A.F~.)] -1

(4.10)

n=l

The expression (4.8) for F 1 = F 2 was derived in [19]. As was pointed out in [9], in the case of the neutral strings (with opposite charges at the ends) we are to set F x = - F 2 = F. Then using that A2, - B 2 = 1 and Fl,,=lc = c -1/2 we get, in agree~ ment with [9],

22 =

=det(8,, +

(4.11)

Thus the full Z~.,(F) can be represented as ( t r l = N )

z <'°t'- vf[dql(2N21 + 22).

(4.12)

The modular infinities present in (4.12) can be absorbed in renormalization of the fields in Zai~c (in particular, in the Born-Infeld term in (4.7)). Let us compare Fa~ = z(r~ ) with the one-loop EA of the mass YM field theory. For simplicity, we shall compare only the planar diagram contributions which are proportional to N (see Appendix B). Let us study the a ' - + 0 limit of Z(la~)n= Nvf[dq]Z 1 introducing the cut-off A as in (3.2). According to the analysis carried out in sect. 3 we need to know the q -~ 1 (i.e. w --+ 0) expansion of 2a*. Recall that In w = - 1/a'X and if= 2~ra'F. Hence to get a non-vanishing a' ~ 0 limit we need to extract the following terms in 21 (see also (3.12)) D/2 Z l w~-~O E k=l

(ek+hkw)(lnw) kpk+

"",

(4.13)

where e k and h k are constants (here for simplicity we ignore the possibility of different contractions of F,~ which have the same order in F, see below). We then obtain (computing the integral over X as in (3.10)-(3.12) and dropping the singular tachyonic contribution) D/2 Z(1) ex'-.*O s,r t-D~2 jfu*D x f d ( l n w ) - lva

E (lnw) k-x-D/2pk

(4.14)

k=l

D/2 A D - 2 k

- Nfd'x

E D - 2k v*rk'

"/k = hk + (D - 2)e k .

k=l

Note that all a'-dependent factors have cancelled out. * The renormalization of the q -0 0 infinity does not affect the analysis of the q ~ 1 limit.

(4.15)

R.R. Metsaev, A.A. Tseytlin / String theory

128

To determine e k and h k in (4.13) we rewrite 2~1 as

I-[ exp [ - tr ln(6~ + A.ff.~)] =

Zl =

n=l

exp n=l

ex

A2nmff2m

E I_m=l

(4.16)

%m----~ A2m= ~ {l+q2n 2,~

2rn

n=l

I_m=l

(4.17)

n =1 I ~1 - - " ~ " ~

It is possible to prove the following relation (see appendix A) w---,0

(~m -- ( Km "}- Pmw )(ln w )2m -l- O( (ln w )X2m-1, W 2 ) ,

Xm

[(2m) (2~r2)2m ,

(4.18)

(-1) m Pm= (2m - 1)9r 2m"

(4.19)

Substituting (4.18) into (4.16) it is straightforward to compute the coefficients ek and h k in (4.13) by expanding the exponent. Let us determine the coefficients of the F 2 and F 4 terms: exp[l(K1

q-

plw )(ln w )2 ff 2 +

¼(l¢ 2 +

P2w )(ln w )4 ff 4 + . . . ]

= 1 + ~(1¢ 1 1 q_ &w)(lnw)2ff2+ ¼(I¢2..kP2w)(lnw)4ff4 + [(K , l2+ 2 K l p l w ) ( l n w ) 4 ( f f 2 ) 2 +

...,

Fk

=_

F~F~x. Fp~ . .

.

(4.20)

M a k i n g use of (4.]3)-(4.15) and of ~'(2)= !,;7-2 6 ~ ~'(4)= ~ r 4 w e finish with

c~

=

lira 7(1)

~p__+ 0 - a l a r l

× {I D - 2 6

= - ~(4~r)-D/2NfdDx

\ -57a--4 A

d

t~ 4 2

-r

Ao8 +-D - - 8 [ ~ ( D + 238)F 4+ ~ ( D -

} 50)(F2) 2] + "-"

,

(4.21) This expression is in complete agreement (see appendix B) with the one-loop result in YM theory (2.2), (2.4), (2.9)*. It is interesting to note that our D-dimensional * Note that (4.14), (4.18) thus make possible to compute all the bp coefficients in (2.2) in the case of abelian F~, = const background in YM theory. Analogous F~, = const formulas can of course be derived using heat kernel methods directly in YM theory. It is interesting to note that while in YM theory we have non-minimal term - F in the A1 operator, string theory manages to reproduce the same effective action by only minimal interaction f2~'A~(x) on the world sheet.

R.R. Metsaev, A.A. Tseytlin / String theory

129

expression for the measure (3.8) reproduces correctly the YM theory results in arbitrary number of dimensions. Hence it seems that the string EA has a natural extension to arbitrary D. In fact, this should be so since D = 26 is simply one of equations of motion (for the dilaton) corresponding to the flat (tree-level) vacuum. But EA is a more general object, that contains information about all possible vacua which may have D ~ 26. Appendix

A

PROOF OF EQ. (4.18) Let us start with

1 + q2n 12m A2/"=l~ ] = [sinh(n lnq)] - 2rn + O((1-

q2")1-2~).

(A.1)

It is not difficult to prove that

n=l

22m ~p~ n(na-1)"'(n2-(m-1)2)q2. (sinh(n In q))-am _ (2m--- 1)[ .=t 1 -- q2. 22m

q22

(2m---1)r n~= l 1 - q2,

n2m 1 0 ( ( l n q ) l +

2m). (A.2)

Hence 22m

n2m-1

°m-n=l

" n=l

Now let us make use of the following relation 2£sin(2Trn,) =

~ln~(x(,),w)-~rcot~ru

,

(A.4)

tl=X

where x0') = w" and 1-x

( ln2x ] f i

+ ( x , w ) = --~-x exp 21nwJ.=~ Differentiating (A.4) over ~ (2m n2m-1

.=1 1--q 2nq - - - - -

2n__

- 1)

(1-w"/x)(1-w"x)

(a.5)

0-w°) ~

times and putting

p = 0

we get

(-1)m+l d2m-1 [ d 1 2(2rr) 2m d--~-q d u l n + ( x ( u ) ' w ) - ~ r c ° t ~ r u J ~ o "

(A.6)

R.R. Metsaev, A.A. Tseytlin / String theory

130

Since we are interested only in the terms in om which go as (ln W) 2m times a constant or the first power of w we have n2rn-1 1) m+l d e , , 1 [ ~ - T ~ q2~ = ( [ in w ,=1 2(2~r) 2" dl, 2"-1 1 - e -~"w

2w In w sinh(u In w)

-½1n w + u ln w - ~-l + O(w2)],~o"

(A.7)

Making use of the formulas t e t-

~ 1

--

k=0

Bk k ~.~t,

~(2k)

"

22k-17r2k (2k)! B2k'

(A.8)

where B~ are the Bernoulli numbers and ~(s) is the Riemann zeta-function we finally obtain for o" in (A.3)

(-1)"w ]

~(2m)

w~0

om -

(2~r2)2"+ ( 2 m _ l ) ! ~ r 2 "

\2m-1

( l n w ) 2 " + O ( ( l n w)

w2).

(A.9)

Appendix B The group traces which appear in the field-theory expressions (2.3), (2.6), (2.7), (2.10), (2.12) are taken in adjoint representation. However, the group traces which appear in the open string theory amplitudes are taken in the fundamental representation (see e.g. ref. [15]). Hence in order to be able to compare the field-theory and string-theory results we need a relation between the traces in adjoint (Tr) and fundamental (tr) representations of a gauge group. Let M be a matrix in a gauge algebra. Then (see e.g. ref. [16]) T r M ~ = (N + 2l)tr M 2 ,

TrM

4 =

(N + 8/)tr M 4 -I- 3(tr M2) 2,

(B.1)

where l = - 1 for SO(N) and l = 0 for U(N). Let M = Ma~ a, where )t" are the group generators in a corresponding representation. Then from (B.1) we have T r ( ) k (al . . . ]ka4)) = ( N +

8 l ) t r ( X (al . . . )t a ' ) ) + 3tr()k(al~:)tr(2ta3~a4)),

(B.2)

where ( a l . . . a 4 ) = ( 1 / 4 ! ) ( a l . . . a 4 + -.- ). Using the formula al

a2

a3

a3 - -

Fo F xFpx +

4 ~ p . ~ a t~ v ~t XlO~t X p ] o l J \

1Fl vFovFo F

h _ xF.vF 1

~Sp Y,

.....

FxoFxo -

1Fp.. f x#f

Fxo '

(B.3)

R.R. Metsaev, A.A. Tseytlin / String theory

131

where Sp denotes any kind of trace, we get Tr Y = ( N + 8l)tr Y + 3 tr(F~xF~x)tr(F~pF~o) + ~2tr(F~Foo)tr (F~oFo.) - ¼tr(F,,Foo)tr (F""Foo) - 3 ( t r F 2 ) 2 .

(B.4)

This f o r m u l a has a simple interpretation. The 1.h.s. corresponds to the A 2 infinity in the S Y M theory while the r.h.s, corresponds to a ' ~ 0 limit of the full 4-vector superstring scattering amplitude. N tr Y term is the contribution of the planar annulus diagram, - 8 tr Y is the contribution of the M~Sbius strip and (tr) 2 terms are the contributions of the non-planar (cylinder) diagram. It is only the first contribution that we explicitly discussed in the text (see sect. 3). The n u m b e r - 8 naturally a p p e a r s if the annulus and the M/Sbius strip are added with the unit weight:

a n d if the integral over q is cut off in the same way for the annulus and the MiSbius strip (see ref. [16]). A relation analogous to (B.4) is true also for the D F F D F F terms in (2.12). Let us also c o m m e n t on the derivation of the eq. (2.9). F r o m (2.6) and table 1 we get f8 - Tr( F4 + " " " ). Using (B.1) it is possible to prove a generalization of (B.2) (here and in (2.9) we consider the SO(N) group) Tr(M'...

Xa,) = ( N - 4 ) t r ( M , . . . X~,) - 2tr(~kal~ka2~a3~ka4)

--2tr(~ta,~a4Xa2~a3)

+

3 tr()t(~ M~)tr(M~X a,)).

(B.5)

T h e n f8 can be rewritten in the t r ( F 4) + ( t r F 2 ) 2 form. In the case of the abelian b a c k g r o u n d (with normalization t r F 2 = F 2) we get (2.9). In order to c o m p a r e with the planar d i a g r a m contribution to the string EA we are to consider only the terms in (2.9) which are proportional to N. References [1] A. Neveu and J. Scherk, Nucl. Phys. B36 (1972) 155; J. Scherk and J.H. Schwarz, Nucl. Phys. B81 (1974) 118 [2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46 [3] D. Gross and E. Witten, Nucl. Phys. B277 (1986) 1 [4] A.A. Tseytlin, Nucl. Phys. B276 (1986) 391; Phys. Lett. B176 (1986) 92 [5] A.A. Tseytlin, Lebedev Inst. Preprint N290 (1987), to appear in Int. J. Mod. Phys. A [6] J. Shapiro, Phys. Rev. Dll (1975) 2937; M. Ademollo et al., Nucl. Phys. B94 (1975) 221; D. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B267 (1986) 75; E. Gava, R. Iengo, T. Jayaraman and R. Ramachandran, Phys. Lett. B168 (1986) 207; A. A. Belavin and V.G. Knizhnik, Zh. Exp. Theor. Phys. 91 (1986) 364

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