A finite representation for a superstring scattering amplitude and its low-energy limit

Nuclear Physics N rth-Holland

B363

A FINITE

(1991)

527-542

REPRESENTATION FOR A SUPERSTRING SCA’ITERING AMPLITUDE AND ITS LOW-ENERGY LIMIT* J. Lee

lnsrifuie

MONTAG

for Tlzeoretical

and

18 April

1991

of the one-loop graviton-graviton is resolved by separating the amplitude the graviton vertex operators. Each

kinematic region and can cuts required by unitarity.

be analytically The scalar

continued factor for

tion of a box diagram with internal lines summed are additional terms polynomial in the external intrinsic ultraviolet cut-off, and the low-energy internal states over an infinite

1. WEISBERGER

Physics, Stale University of New York al Stony Brook, Stony Brook, NY I 1794-38409 USA Received

The divergence superstring theory orders of inserting

William

are kept. One-particle set of mass levels

on

reducible an internal

each

scattering into terms term is well

amplitude in corresponding defined in an

the type-II to different appropriate

to physical regions where it develops branch term contains the proper-time representa-

over the mass levels of the superstring. There momenta. String theory gives the diagrams an field theory limit appears when only massless graphs line.

are

generated

from

duality

by summing

1. Introduction

String perturbation theory is formulated as a topological expansion. At a given order, the contribution to an S-matrix element is expressed as an integral over the (super)modular parameters of a (super)Riemann surface with g handles. The world-sheet coordinates of vertex operators inserted for the emission of physical states are integrated over a fundamental region of the Riemann surface. In point particle field theory many Feynman diagrams contribute, in general, to a scattering amplitude at a given order of perturbation theory. String theory appears to simplify the analysis by summing all of these diagrams to get a single integral representation. Superstring scattering amplitudes for massless external particles have been argued to be one-loop finite [l]. However, Amano has demonstrated that the genus-one graviton-graviton elastic scattering amplitude appears to be divergent for all non-trivial values of the external momenta [2]. This divergence is caused by the behavior of the integrand for large values of the imaginary part of the modular parameter r when the world-sheet coordinates, Im I/;, of the vertex operators are separated by distances of order Im T. *Work

supported

0550-3213/91/$03.500

in part 1991

by National - Elsevier

Science

Foundation

Science

Publishers

under B.V.

grant All

rights

PHY

89-08495.

reserved

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Superstring

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The integral representation for the amplitude is manifestly real while it must develop branch cuts and imaginary parts associated with unitarity. One might have hoped that there was some kinematic region in which the original integral representation converged and from which the answer could have been analytically continued to show the development of absorptive parts. For the elastic scattering of massless excitations there are branch cuts starting at s = t = u = O*. There exists no range of the invariants for which the whole amplitude is real. Amano proposed an ad hoc off-shell continuation of the amplitude which allows us to relax the on-shell condition s + t + u = 0 and which preserves modular invariance. He then distorted the contour of integration in moduli space to obtain a convergent integral whose analytic continuation back to the mass shell may give the correct value of the scattering amplitude. While this procedure may lead to an acceptable answer for one-loop amplitudes, it does not seem to have a straightforward generalization to higher orders. In addition, by invoking an ad hoc off-shell extension and by distorting the complex structure of Teichmiiller space, Amano’s prescription takes us quite far from the original formalism. In this paper we develop a way to make sense of the representation for the scattering amplitude which does not involve such drastic deformations of its original form. This approach is manifestly unitary and leads naturally to an expression for the ‘low-energy limit of string amplitudes in terms of a local field theory of massless particles. Furthermore, it provides an expansion to calculate the correction due to internal heavy states and seems to be generalizable to arbitrary order in the loop expansion. ,In four-graviton scattering, the integrals over the imaginary parts of the worldsheet coordinates for the vertex insertions for the scattered particles can be divided into six terms, each corresponding to a particular ordering of the variables on the torus. These six regions correspond to a double counting of the three sets of box graphs for one-loop amplitudes in a point particle theory. Each term consists of a kinematic factor multiplied by the scalar factor for a sum of box diagrams in a theory with cubic interactions. The internal lines of the boxes are summed over the infinite set of mass levels of the superstring spectrum. The integral representation for each Feynman-diagram-like term converges in a certain kinematic region and can be analytically continued to values of the kinematic variables where it develops unitarity branch cuts corresponding to the two particle intermediate states in the appropriate channels. Thus in order to extract a finite result from superstring perturbation theory, the single integral representation for the scattering amplitude at a given order can be decomposed into several pieces, each corresponding to a set of Feynman diagrams with a field theory interpretation. As a result of duality, the number of diagrams is still less than that in field theory. l

we

take

pij=

-(Pi

for

the

+Pj)*.

Mande]stam

variables

S=

-(PI

+Pz)*v

t = -(Pz+P~)‘~

’ = -(PI

+p3)z’

and

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Superstring

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The Feynman diagrams are given in a proper-time representation with the Regge slope parameter (Y’ appearing as a proper-time cut-off in exactly the same way as shown by Polchinski [3] in his calculation of the cosmological constant for the bosonic string. The slope parameter plays two other roles in the expansion. It sets the mass scale for the heavy particles, and it serves as a dimensional factor in derivative couplings to the heavy states. Therefore, the decoupling of massive states in the low-energy limit (cu’ + 0) coincides with the removal of the ultraviolet cut-off and the vanishing of certain coupling constants. Which effect is dominant depends on the ultraviolet behavior of the effective field theory, and that depends on the number of uncompactified space-time dimensions of the vacuum. In sect. 2 we examine the one-loop amplitude for graviton-graviton scattering in type-II superstring theory. We consider the theory in the critical dimension d = 10 and with 10 - d dimensions compactified. For simplicity we consider the example of compactification onto circles of radius R. After briefly reviewing Amano’s results in order to clarify the possible source of divergences, we show how the division of the integrals into different orderings of the Im vi variables gives terms which are well defined in appropriate kinematic regions. In sect. 3 we take the zero-slope ((Y’ + 0) or low-energy limit [4] and show that these terms correspond to the proper-time representation for the three sets of box graphs with massless states on the internal lines. In sect. 4 we analyze the contribution of the massive excitations by including the next-order term in the low-energy expansion of the integrand. We show explicitly how the amplitude once again corresponds to a sum of box graphs and explain how this result may be proved to all orders. We present the O(a’> correction to the low-energy amplitude and discuss the structure of the higher-order contributions. We then give an example of how box graphs with one internal line summed over the mass levels of the superstring spectrum generate an infinite sum of one-particle reducible graphs with poles in a crossed channel at masses given by the superstring spectrum. In sect. 5 we conclude by discussing the meaning of modular invariance for our result. We also present some of the problems to be solved in calculating finite amplitudes with more than four external particles and for higher-genus surfaces. 2. The one-loop amplitude The complete

four-graviton

amplitude

in the critical dimension

d = 10 is given

by El iA(s,t,u)

=m

r( 1 - $‘S)q a.4 I-( 1 + $X’.+0

1 - ~cK’t)r( 1 - &A) + &Y’t)r(l + &‘u)

(1)

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amplitude

where Ki? is a kinematical factor, and cl is a dimensionless constant which is determined uniquely by unitarity. We keep all factors of CY’ explicit and take (Y’= 1/2rrT, where T is the string tension. The one-loop contribution A ,(s, t, U) is given by d2r = q CY’ LqT2)2

A,(s,t,u)

xjj=exp[-r(y~y’)2]~e1~(~,~~T’~,

where the complex modular mental domain,

parameter

(T

=

T,

(2)

+

ids)

is integrated

The coordinates of the vertex operators (vi = xi + iy;) are integrated ~y;
1,

over the funda-

over the torus,

(4)

O
72

Due to translational invariance on the world-sheet we can fix v4 = T . For the case with 10 -d dimensions compactified onto circles, the form of the tree-level amplitude with massless external states is unchanged, and the one-loop amplitude is modified by the insertion of the factor [F2(a,~)1'0-d into the integrand [41, F,( a,

T)

= a&

C exp[ -2irrrrmn W2.n

- rrr2(m2a2 + n’/Q’>]

Y

(5)

where the dimensionless parameter a = G/R, and R is the radius of compactification. Following ref. [2], we decompose the integrand as follows in order to isolate the divergent piece:

(6) where iij

is a bounded function of

TV,

and

(7)

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531

amplitude

For @ > 0, the integrand blows up exponentially as r2 --) 03. It is not difficult to show that this divergence exists for all non-trivial values of the external momenta. The presence of the absolute value in @ suggests dividing the Im vi integration region into the six possible orderings of these variables [4]. For each of the six ordered integrals there is a kinematic region where it can be well defined. After resealing yj + 7*yi we find 1 >Yy,>Y,ZY, l>Y,>Y,zY;zo,

‘SY,(Y3-Yz) +t(Y,-Y*)(l-Y,), s(l -YI)(Yz-Y3) +ty,(y, -Y2>, sy,(Y,-Y,)+u(Y,-Y,)(l-Y,), @= s(1 -Yz)(Y, -Y3) +uY,(Y,-Y,)7

20,

l~Y,~Y,~Y*~O7 (8) l~Y2~Yl

fY2(Y,-Y3)+u(Y3-Y2)(1-Y~)?

&Y3>09

1~Yl~Y3~Y2~o~ 1 >Y,>Y,>Y,

~t(l-y,)(Y,-y,)+“y,(y2-y3),

>o.

The three kinematic regions for convergence of the one-loop amplitude can be read from eq. (8). The first two orderings give finite contributions only if Re s Q 0 and Re t =G0, the second two only if Re s G 0 and Re u Q 0, and the last two only if Re t G 0 and Re u =S0. Therefore, for s + t + u = 0, at least one ordering will give a divergent contribution to the amplitude. The expression for i(vi - vj) can now be written as a function of an ordered difference of world-sheet coordinates such that Yi - Yj > OF 03 (l-e X(v) = (1 - e2iru) n n=l

2isrm+2im~)(~ (1

_ _

e2iir7n)2

e2i7r7tz-2i7rv) ’

(9)

We find, therefore, that the integral expression for the one-loop contribution to the four-graviton scattering amplitude can be evaluated as follows. The Im vi integrations are first split up into the six possible orderings of the variables which can be grouped into three pairs. The two orderings in each pair give identical contributions to the amplitude. This can be verified by the substitutions vi -+ T + 1 - vi, which correspond to rotating the parallelogram representing the torus through 180” about its center (T + 1)/2*. Each of the orderings gives a finite contribution to the scattering amplitude when the complex external momenta are appropriately restricted so that the corresponding integrals converge. Each pair of orderings must be treated separately so that momentum conservation (S + t + u = 0) is not violated. Once the finite integrals are performed, they can be analytically *This same diffeomorphism gives rise to a factor l/2 in the overall normalization of amplitudes on the torus [3]. The necessity of these factors for unitarity has been discussed in the operator formalism by Sakai and Tanii [6].

532

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/ Superstring

continued to the physical region for scattering point particle field theory [7].

scattering

amplitude

in analogy to the procedure

for

3. The low-energy limit In this section we examine the zeroth-order contribution to the low-energy expansion of the superstring amplitude. This is equivalent to the results of Green et al. [4], and we present it in order to set up the notation for the generalization to the case when heavy particles propagate in the loop. With some caution, the low-energy expansion may be performed inside the integrand. The four-graviton amplitude is known to contain no massless poles. Therefore for values of the external momenta satisfying I&s], la’t 1, I&U I +C 1, we may take

ij

n

[iii]

-fm’Pu = 1 _ K2 Cpij~ij+f U

[

~Cpijdij I/

1

*+ . ..)

(10)

where eij = log iii is only part of the two-dimensional Green function on the torus. It is implicit that the product and sums are performed such that vi - vi is an ordered difference with yi -yj 2 0. Note that the factor err0”2@ in the decomposition (6) cannot be expanded directly in a power series in a’ for large values of r2. In order to interpret the low-energy expansion of the amplitude, we find it convenient to divide the fundamental domain into two regions. We take 9= 9, + F2, and define

The F, region has no direct interpretation in terms of a particle field theory. In this region the entire integrand may be expanded in a power series in CX’since T* is bounded from above, and the integral therefore gives only a polynomial in the external momenta to any finite order in a’. It is the .9* region which gives a sum of box graphs (with proper-time cut-off) in the low-energy limit. The resealed yi variables act as Feynman parameters, and the modular parameter r2 serves as the proper-time variable. Furthermore, due to the periodicity of the integrand for xi +xi f 1, we may shift the limits of the xi integrations given in (4) to make them independent of T and yi. In this way the xi and T, integrations act independently as projectors onto the superstring spectrum for the internal lines of the corresponding Feynman diagram. In order to make the connection with Feynman graphs in the low-energy field theory more transparent, the resealed yj can be written as linear combinations of the usual Feynman parameters pi [S]. For

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533

example, for the first ordering in (8) the substitutions YI =P,,

Yz=P,

+P2,

y3=P,+P*+P3=

1 -P4

(12)

give

Similar expressions can be obtained for each of the other five orderings. The zeroth-order term in the expansion (10) from region F2 is given by

xCe

dr*O(s.r) + esro’r2w.u) + edrz@(r.u)] IO-d

2i~r,mn-rrrz(tnzo’+n2/a2)

1

9

(14)

where the xi integrations are trivial, and @(s, t> = sp,p3 + tP2P.,. The T, integral can be easily performed. At this order of the expansion (lo), only the string solitons introduced by the compactification scheme contribute to the massive internal states. We make the cut-off dependence on (Y’ explicit by resealing 72 -+ T~/(T(Y’) to get

Xk

r#J(s.r) + eT2ws,rr)+ er*dJv,u) 1

IO-d “,rJ

[

s mi,n, 1xmp, Ce-

(r*/a’M,,lfoZ+rrf/a2)6

(15)

This result is seen to be the scalar part of a sum of box graphs in d dimensions in the proper-time representation with a proper-time cut-off given by ~TTLY’(see appendix A). These graphs include the three box graphs with only massless states on the internal lines plus graphs with a massive soliton propagating around the loop. The &-dependence in the overall factor gives the correct change in dimen-

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scattering

amplitude

sion of the scattering amplitude and multiplies the compactified tree-level contribution as well. If we are in a number of dimensions where there is no ultraviolet divergence as a’ + 0, the region 9, does not contribute, and we may set the cut-off to zero in the 9*-region. For fixed o, the zero-slope limit (a’ + 0) gives a massless theory, and we do not get any contribution from the string solitons in the loop [4],

-

iPj

j=l

I

1 -@(t,u)

4-g

1

1.

(16)

This expression is finite for 4
= ilrrn2

C m=l

e-T2m202

+

O(

a’)

.

(17)

This term is independent of the external graviton momenta since the loop has been reduced to a point interaction in the low-energy limit. The massive superstring excitations do not contribute to the expansion (10) until O(CU’~). Therefore, their contributions vanish in the zero-slope limit regardless of any ultraviolet divergence. This is quite different from the way heavy particles decouple in point particle field theory. Returning to the uncompactified theory, we set d = 10 in eq. (14) and integrate by parts twice to extract the cut-off dependence in the form of a first-order pole in (Y’. We then take (Y’ + 0 in the lower limit of the remaining convergent 72 integral

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scattering

amplitude

erz*(s.9

+

535

to get

x

[@‘(s,

t)

evw.f)

+

@‘(s,

U)

@‘(t,

u)

er2w.uq

+0(d).

The contribution

(18)

from 9, merely renormalizes lim A:0)1~, = $( d-0

the cutoff-dependent

term,

t - 1) + O((Y’).

(19)

The pole at cr’ = 0 can be absorbed by local counterterms in the field theory limit. At this order, the only effect of the F, region is to adjust the cut-off, and the entire result may be written lim A\“’ =

27~~~

d-0

Xk

s*d’(s,r)

4. The contribution

+ eTz@(s.rr)

+ erfP(r.tr)]

+ O( Q’)

.

(20)

from the massive states

In this section we examine the next non-zero term in the expansion (10). In what follows we consider only the ten-dimensional theory so that the contribution of the massive superstring states appears in a more straightforward way. The term linear in (Y’ from the expansion (10) vanishes due to the Re vi integration. Concentrating on the first pair of orderings in eq. (8) and taking d = 6(x + iffy), we write for the term quadratic in (Y’ e rrdr,[sy,(Y,-Y2)+r(Yz-ylXI

(jtv)

=

_ f

C

C

n=O

tn=l

~[e2i*um+2ininm

-Ydl

+ ,-2i7rCtn-Zi7rFnm]

(21)

536

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where we have used momentum conservation to eliminate the (vi - vi)-independent factor from i(vi - vj) as defined in eq. (9). The xi and T, integrations project out the superstring spectrum for the internal lines of the resulting Feynman diagrams in the field theory limit. This can be proved to all orders in (Y’ by expanding the factor II, Gi
n~oe-4~“~~2/d)

x

+t*(e+u2(e-

[ S*(e-4mT2P2/a’

4mrz(J34+P3+P*)/d 4mr,(P,+Pz)/d

+

e4mr2P4/a’)

+ e-4mT2P3/a’) +

e-4m7,w,+P3)/~’ )I

+ C e-4mnT2/u’ [pi + -pi] . ( n=l 1 1 This expression can be identified as the scalar part of an (s, t) channel box diagram with massive states from the superstring spectrum propagating in the loop (see appendix A). Including the remaining orderings we can write

+t*(e-

4mr2[n(PI+&+P4)+(n+

I)Pjl/a’

+e-4mrz[n(P2+P3+P4)+(n+l)~,l/a’ +u*(e+

lXP1+P2+8d+nPd/~

+ e-4mr21(n+lXP2+P1+P4)+nPll/~‘)

4m721Wl +P2)+W

e-4m72[n(P2+P~)+(n+

+ ,-4mr2[(tt+

+ ~XP3+Pdl/~

1XPl +P4Wa’

+ e-4mr2[(n+lXP~+P2)+"(P3+P4)1/a' + e- 4m7m

+ ~xPz+Pl)+~wI

+(S,t,U~S,~t)+(S,t,U~t,U,S).

+P.dl/~’

)I (23)

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By comparison to eq. (15) with d = 10, one finds that this expression gives corrections proportional to the inverse fourth power of the mass levels found in the superstring spectum. The sum over m contributes both to this overall scale and to the massive states propagating in the loop. The sum over n, however, only contributes to the massive internal states. Any two internal lines in a diagram can only differ by m mass levels, and all combinations of these diagrams are included in the above expression. The n = 0 terms in eq. (23) contribute Feynman diagrams which each have both massless and heavy internal lines. These graphs do not reduce to point interactions in the low-energy limit as do the graphs with all heavy internal states. Contraction of the heavy internal lines gives one-loop graphs with effective vertices for at least four particles. However, the leading contribution is just a polynomial in the external momenta, lim A\*‘~g,=const.X~‘(~*+t*+~*)

+O(cy’*loga’).

(24)

d-0

The low-energy limit of the F, region gives only a polynomial in the external momenta. At O(cr’) it has the same momentum dependence as the term above. By taking the low-energy limit in the expansion (lo), the single poles for the massive excitations corresponding to the one-particle reducible graphs seem to have been lost since a box diagram has no single pole. However, these poles reappear when we sum the infinite series. In order to show how the s-channel poles can be generated by summing an internal line of box graphs over the superstring spectrum, we look at the (s, t) channel box graphs contributed by the first pair of orderings in (8) with only one massive internal line corresponding to p4 = 1 -y3 and the (s, U> channel box graphs contributed by the second pair of orderings in (8) with only one massive internal line corresponding to p4 = y3. By inspection of the integrand for the superstring scattering amplitude given in the decomposition (61, it is not difficult to see that such graphs come only from the expansion of

X

e~“‘r21~Yl(Y,-Y*)+l(Y2-)IIXI

X 1+ [

C at(s) m=l

-Yx)]

e-4mi@J/o’

11 _

1 .

e2i7r(T,

-x3)-2?rrz(l

-y3)

1 -o’s/*

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The coefficient

a,,,(s) in the above expansion is

scattering

amplitude

(26) and to lowest order gives lim,, -) 0 a,,(s) = a’s/4rn. This limit reproduces the first term in eq. (23) for n = 0 in addition to the zero-slope result (15). The summed expression for A(s; t, u) in eq. (2.5) has poles for s = ~N/(Y’, where N is a positive integer. These poles arise from integrating the vertex coordinates of the third particle over the corners of the torus. The (s, t) channel graphs give contributions at the top two corners and the (s,u) channel graphs at the bottom two. Using polar coordinates r eis for the displacement of the third particle vertex from each of the corners of the torus (which are identified with the fourth particle vertex), the sum of the two integrals for small r is

x1+ c

- &Y’s

1

,n=, (m + I>!

(2irr

eie)‘”

+ terms finite as s + 4/V/a’.

(27)

The e-integral projects powers of r which generate poles in the s-channel at the mass levels of the superstring. The residue of each pole contains the proper-time representation for one-loop three-point functions with massless internal states (see fig. 1). Powers of [t/3, + u(p2 + ps)] correspond to derivative couplings of the heavy higher-spin states in the s-channel. Symmetry of the amplitude in t and u is demonstrated by the substitutions p, c, p3 and 8 -+ 8 + rr. Thus the sum over an

Fig.

1. Duality

diagrams

for s-channel

poles.

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infinite number of mass levels on the internal line of a box graph produces a sum over one-particle reducible graphs in a crossed channel as expected from duality. 5. Discussion

In order to obtain a finite answer for the one-loop graviton-graviton scattering amplitude in type-II superstring theory, we have shown that one can split the original divergent formula into several terms each corresponding to a different ordering on the world-sheet of the positions for the vertex functions of the scattered particles. In an approach where the vertex operators are replaced by punctures, this would correspond to dividing the moduli space for the punctured Riemann surface into several regions. In addition to a polynomial in the external momenta, each term has a contribution which corresponds to a set of box diagrams with a proper-time cut-off summed over the mass levels of the superstring spectrum. Each box diagram converges in one of three kinematic regions and can then be continued analytically to define a piece of the amplitude in the physical region where it develops branch cuts as demanded by unitarity. In the low-energy or zero-slope limit only zero mass internal excitations survive corresponding to an effective field theory limit [4]. Cutoff-dependent terms, if present, contribute to polynomials in the external momenta which can be represented by local counterterms. Although they are all finite and computable from the string theory, the effective low-energy field theory may be finite, renormalizable or non-renormalizable according to the number and nature of the counterterms required. The O(cw’> correction to the low-energy expansion of the scattering amplitude gives only a polynomial in the external graviton momenta,

+const.Xa’(s2+t2+u2)

+O(cL210gcu’).

(28)

At higher order in LY’ the low-energy amplitude will receive contributions from Feynman diagrams with effective vertices giving non-polynomial dependence on the external momenta. First-quantized string theory contains only cubic interactions, and the Polyakov path integral formulation does not require the explicit inclusion of ghosts. String theory, however, reproduces effective low-energy field theories which require not only cubic but also higher-order interactions plus Faddeev-Popov ghosts. The field theory formalism, of course, allows one to compute off-shell Green functions.

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.I. L. Montag,

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amplitude

In sect. 3 we noted that because of the explicit factors of cr’ multiplying the amplitude, the contribution of the massive excitations vanishes in the zero-slope limit even though the proper-time integral diverges. This is in contrast to decoupling of massive particles at low energy in field theory where both renormalization resealings and the limit of taking the ultraviolet cut-off to infinity must be carried out before we can take the heavy mass limit. Therefore, in a suitably defined second-quantized form of string theory, it might be possible to integrate out the heavy particles and derive the effective low-energy theory directly from the action. Our decomposition of the amplitude does not directly give terms corresponding to diagrams with single-particle intermediate states. However, we have seen that such terms result from duality by summing over an infinite number of mass levels on a given internal line in the box diagram. For the one-loop four-graviton amplitude no massless pole appears because of the vanishing of the two- and three-point amplitudes for the torus. The five-graviton amplitude, however, should have a massless pole in two to three particle channels. Therefore, it cannot consist simply of a kinematic factor times a scalar factor involving products of the same string propagator that appears in the four-graviton amplitude of eq. (2). The explicit ultraviolet finiteness of the amplitude, the manifest presence of unitarity branch cuts at all the expected two-particle thresholds, and the presence of one-particle poles for the massive excitations seem strong physical evidence for the validity of our analysis and interpretation of the string formula for the scattering amplitude. The division of the integral into pieces by the simple procedure of ordering the yi variables for the vertex operators depended on our particular choice of a fundamental region for moduli space. Modular invariance is maintained if the action of an element of the modular group (PSL(2,Z) for the torus) is accompanied by the corresponding transformation of the six regions of integration for the vertex insertions, thereby expressing these regions in terms of the transformed variables. This is analogous to computing a Lorentz covariant result in a convenient reference frame and then transforming it to a general inertial system. A general modular transformation for the torus, ar+b 7’ = cr+d’

a,b,c,dEZ,

corresponds to a conformal

ad-bc=l,

(

;

;

1

E PSL(2,Z),

(29)

change of coordinates, (30)

To get to the standard representation for the torus as a parallelogram of sides 1 and T’, we must also make some lattice translations. For a general modular

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541

transformation this will turn the simple orderings of the yi into a complicated expression. Consider the generating transformations for the modular group of the torus. For the generator T’ = T + 1 there is no change in the ordering conditions. However, for the generator T’ = - l/7, after resealing yi + 72yi and y: -+ ~;yj, the inequalities l=y,>y,>Y,aYy,

a0

(31)

become

and the proper-time variable is T;/ 1~~1~. At higher genus we expect that an appropriate ordering of vertex operator coordinates will decompose an amplitude into a sum of Feynman-diagram-like terms consisting of 1PI graphs. Length-twist or Fenchel-Nielsen variables seem a likely choice for the decomposition of the moduli into proper-time variables and mass level projectors. Naive divergences of the original amplitude should appear at the boundaries of moduli space where any of the handles gets infinitely long. Additional problems will be to relate the propagators expressed in terms of the period matrix to the modular parameters and to find convenient coordinates for the vertex operators on a given Riemann surface.

Appendix A

For reference we display the proper-time representation for a one-loop Feynman graph [93. The scalar factor for a one-loop graph with N vertices in d dimensions is

A=jddq_fi (2Tr)d

i=l

1

[

(q+ki)*+m;

1

where ki is a sum of external momenta, and rn; is the mass of an internal line. Using a standard representation for the gamma function, we can exponentiate the

542

J. L. Montag,

sum of propagators

W.I.

Weisberger

/

Superstring

scattering

and perform the gaussian integrations

xexp

amplitude

over the loop momenta,

-~~Pj[(4+kj~2+m~]

i

1

j=l

2

Xexp

-7 i

I

~/Ij(k~+m~)I

j=

( )I) :P,k,

.

(34)

j=l

An ultraviolet divergence for N G d/2 now appears as a singularity of the integral at T = 0 which can be regulated by inserting a proper-time cut-off at the lower limit. For the (s, t) channel box graph with four massless external particles, we have in particular

(35) j=l

References [l] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge Univ. Press, Cambridge, 1987) [2] K. Amano, Nucl. Phys. B328 (1989) 510 131 J. Polchinski, Commun. Math. Phys. 104 (1986) 37 [4] M.B. Green, J.H. Schwarz and L. Brink, Nucl. Phys. B198 (1982) 474 [5] J.H. Schwarz, Phys. Rep. 89~ (1982) 223 [6] N. Sakai and Y. Tanii, Nucl. Phys. B287 (1987) 457 [7] R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix (Cambridge Univ. Press, Cambridge, 1966) 181 G. ‘t Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365 [9] C. Itzykson and J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980) sect. 6.2