Off-shell conformal methods for the superstring

Nuclear Physics B310 (1988) 254-290 North-Holland, Amsterdam

OFF-SHELL CONFORMAL M E T H O D S FOR THE S U P E R S T R I N G

Olaf LECHTENFELD and Stuart SAMUEL Physics Department, City College of New York, New York, N Y 10031, USA

Received 16 May 1988

Superconformal field theory is used to compute off-shell amplitudes for the superstring. Off-shell three- and four-point functions in Witten's superstring field theory are obtained. The problem of gauge invariance due to the associativityanomaly in the bosonic sector is resolved to order g2.

1. Introduction

1.1. INTRODUCTORY REMARKS Although string theories may be the first examples of a consistent, quantum theory of gravity in which all fundamental forces are unified [1-5], it is not yet clear how they describe the real world. Given that they are most naturally formulated in 26 or 10 dimensions, a dramatic metamorphosis must take place to achieve a space-time with four flat dimensions. This may happen because the perturbative string vacuum is unstable. The study of string vacuum physics is thus of utmost importance. Vacuum physics is efficiently investigated using off-shell probes. On-shell quantities are usually of little use because the on-shell conditions in the perturbative vacuum are different from those in the true vacuum; the spectrum generically changes in going from one vacuum to another. Field theory provides a framework in which off-shell quantities can be computed. Before the development of covariant string field theory [6, 7], almost all string results were obtained on-shell. The mathematical understanding of covariant string field theory [8-21] has evolved to a point where off-shell string physics can be extracted [22-26]. Conformal field theory [27-38] on the world sheet is another approach to strings. Because the conformal group in two dimensions is infinite dimensional, it is particularly powerful. It is the principal method for obtaining string amplitudes. The marriage of these two approaches, string field theory and conformal field theory, provides an off-shell formalism with powerful calculational methods. This 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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has been the subject of much research [9, 39-53]. These efforts have resulted in off-shell conformal field theory [26], a powerful means of obtaining off-shell results.

1.2. TECHNIQUES OF OFF-SHELL CONFORMAL FIELD THEORY Let us review the basics of off-shell conformal field theory. To be specific, consider the computation of a scattering amplitude. There are two main ingredients. (a) The specification of a set of surfaces and a measure on that set, so that one can integrate over the surfaces. (b) The evaluation of correlation functions of vertex operators on each surface. The number of parameters needed to characterize the set in (a) is the number of integration variables in an amplitude. At tree level for a process involving n open string states, it is the number of K o b a - N i e l s e n variables, that is, n - 3. At the one-loop level an additional parameter is needed. For closed strings the number of parameters is doubled. For more discussion of this topic see sect. 6 of ref. {26]. The above can be contrasted with the Polyakov approach [54-56], in which one sums over all surfaces moded out by the conformal group. On-shell, the computation of a correlation function is the same for two conformally equivalent surfaces because the action, the measure and vertex operators are conformally invariant. Since the conformal weight of a vertex creating a particle with m o m e n t u m k is a ~ ' k . k + integer, vertex operators not satisfying the mass-shell condition do not have conformal dimension 0. Consequently, off-shell vertex operators are not conformally invariant. The value of an off-shell correlation function changes in going from one conformally equivalent surface to another. It is necessary to specify the subset of surfaces. This explains the raison d'Stre for (a). The basic idea in calculating step (b) is as follows. Although the action is quadratic in fields, the computations are not entirely straightforward because of the non-trivial nature of the surface, R. Instead of computing directly on R, a conforreal transformation is carried out from R to H, the half plane (in the open case) or from R to C, the full plane (in the closed case). Since propagators for H and C are known, correlation functions are evaluated using free-field theory methods, i.e., Wick's theorem. Of course, in transforming from R to H, the vertex operators pick up conformal factors. The method is not quite so simple because of two subtle effects: normal ordering and the choice of the variable on R. These are discussed in ref. [26]. We shall be concerned with R of a particular shape in which each state is identifiable with a rectangular strip, one end of which goes to infinity (see fig. 2 of ref. [26]). At this end states are created. They then evolve inwardly until they interact. The rectangular strips mean that the incoming states propagate as free strings. Although more general R can be considered, the above is relevant for covariant string field theory. R is called the string configuration. Here is the procedure for evaluating step (b).

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(1) States are created by vertex operators located at the "infinite" ends of the rectangular strips. These locations are called the asymptotic positions. The interaction positions are specified as 11, 12..... Ip, the points where the rth string joins the rest of the diagram (see fig. 2 of ref. [26]). (2) Determine the map, p(z), from the half-plane to R. Let z = Z~ be the asymptotic positions in H of the r th operator. (3) Define string variables, ~(z), for the rth state. The variable ~r(z) is of the form a~p(z) + b ~ where a" and b r are constants adjusted so that (a) the real part of ~'(z) is zero at /~ and (b) the imaginary part of U(z), Im(~(z)), goes from 0 to 7r as one crosses the width of the r th string. (4) Determine the asymptotic behavior of the string variables by computing the taylor series of ~ ( z ) - ln(z - Z~) about Z~ t ' ( z ) = ln(z - Zr) - N~o'- ~ y~,(z - Z~)m.

(1.1)

m=l

The N0~r and y~ are constants determined from the map, O(z). The N~0r are actually the diagonal zero-zero components of the Neumann function on R [23]. (5) Insert operators, V~(~), at the asymptotic positions. (6) For the variable, ~, in step 5, use oa(z) --- exp(~'r(z)),

for the rth operator.

(1.2)

(7) Perform the conformal transformation to H. In doing so, a primary conformal field, O, transforms as [36] dz) h

0(,.,)+

~

O(z),

(1.3)

where h is the conformal dimension of O. The transformation rule for other operators is determined by the transformation property of its parent primary field. The Jacobian factor, dz/dca, in eq. (1.3) is

dz (d~ r z,)l e x p ( - U ( z ) ) .

d--~ =

(1.4)

(8) Evaluate the transformed correlation function using Wick's method and the following propagators

XS(z)X"(z ')=s~"ln(z-z'),

for

[ z i > ] z ' l,

(1.5)

ep(z)q~(z') = e l n ( z - z ' ) ,

for

[z] > ]z'[.

(1.6)

I--I

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TABLE 1 The bosons, their statistics, E, their background charges, Q, and their contributions to the conforrnal anomaly, c, in the superstring. boson

e

Q

X"

1

0

o

1

3

1 -1 1

0 2 -1

qot in ~/,~' X

c

-

D 26 ~D 13 2

In eq. (1.6), q~ is a boson associated with a bosonization procedure and e = _+1 is a statistics parameter. These parameters are given in table 1.

1.3. B O S O N I Z A T I O N A N D THE EXTENSION TO THE SUPERSTRING

To extend the above procedure to the superstring is one goal of this paper. Before describing how this is done, some background material concerning chiral bosonization in the superstring must be introduced. Anticommuting fields in two dimensions can be expressed in terms of the exponentials of bosons. One boson suffices for two fermions. As an example, the reparametrization ghosts, c(z) and b(z), are [36]

c(z)

=exp(o)(z),

b(z)=exp(-o)(z),

(1.7)

for a chiral scalar field, o. For the superstring, bosonization is almost indispensable [36] because the fermion emission operator involves the covariant spin field, S(~ 1/2) [34, 35, 57]. It is expressible in terms of the five chiral bosons, epJ(z), used in bosonizing the N e v e u - S c h w a r z - R a m o n d field, q~"(z), in the helicity basis,

~'-(~2j-l~

gi+2J)(z)=exp(+_~pj)(z),

for

j=l

. . . . . 5,

and the boson, ~(z), in the bosonization of the superconformal ghosts,

(1.S)

B(z)

and

v(z) y ( z ) = e % / = e% -×

fl(z)=a~e e~=OxeXe-,, S (~- 1/2) = e a r e - ~ / 2 ,

(1.9) (1.10)

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where & is expressed in the helicity basis (see ref. [57] for notation). The exponentials in eqs. (1.7)-(1.10) should be accompanied by a cocycle operator. We adopt the convention of ref. [57], that such factors are always present even if not explicitly displayed. We use the particular representation of cocycle operators in ref. [57]. The concept of a statistically correct operator is important [57]. Without exp(-4~/2) in eq. (1.10), S a has conformal dimension ~. The ghost spin field, e x p ( - ~/2), contributes ~, so that S(s_ 1/2} has integral conformal weight [36, 58]. In addition, Sca 1/2} is an anticommuting object only when e x p ( - ~ / 2 ) is present. By combining the superconformal ghost boson, q~, with the bosons in the NSR field, a field with integral conformal weight and correct statistics is obtained. Ref. [57] showed that the GSO projection makes this happen with all operators in the superstring, so that only statistically correct operators of integer weight appear. As another example, the anticommuting conformal weight-~ NSR field, ~ ( z ) , combines with e x p ( - ~ ) , to form ~ _ l ) ( z ) , a commuting field with unit conformal weight. The space-time spin-statistics is rectified: operators with tensor indices commute and operators with a spinor index anticommute. The use of statistically correct operators is important in making off-shell conformal field theory applicable to the superstring. Operators with integer conformal weight transform nicely in going from R to H and amplitudes automatically possess the correct Bose or Fermi symmetry. This paper obtains off-shell superstring amplitudes using statistically correct operators. For the massless vector and spinor states of the open superstring the vertex operators are V{_l)=ce.+(

,}e k ' x ,

V(_ 1/2} = cu aS~- 1/2}e * x ,

(1.11) (1.12)

where e" is the polarization of the vector and u a is the wave function of the spinor. 1.4. COVARIANTSTRING FIELD THEORY Covariant open string field theory as developed in ref. [6] is based on the analogy of differential forms. Two operations, * and f, are introduced. The string field, '/', is like a matrix-valued one-form, Q, the BRST charge, is like the exterior derivative, • is like a wedge-product and f is like integration over a manifold without boundary times a trace over the matrix space. These constructs satisfy the string axioms: (i) nilpotency of Q, (ii) associativity of *, (iii) graded distributive property of Q across *, (iv) vanishing of the integral of an exact form and (v) graded commutativity of * under the integral. These axioms have been proven to be realized [10,14, 23, 24, 43]. The action is f(q" * Qq" + 2g,/,, q , , ,/,) and is invariant under the gauge transformations, 8q" = Q A + g( q" * A - A * q'). The interaction,

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f(g" * g" * g'), is the delta-function overlap of the first half of one string with the second half of the next, for each of the three strings, where o = ½~r marks the spot where strings are divided. We refer the reader to refs. [6, 7] for more details. The open superstring field theory proceeds analogously, except that there are two sectors (Neveu-Schwarz and Ramond) and the * and f operations are modified [7]. The integral is replaced by ~ = fY(i) and A * B is replaced by A * B which is equal to A * B if both A and B are in the Ramond sector and equal to X ( i ) A * B otherwise. Here f and * are the integral and star-product of the bosonic theory appropriately generalized and X(z) and Y(z) are respectively the picture-changing and inverse picture-changing operators [7, 36]

~ / ~ x = { o , ~ } = c O ~ + l q j2 (+1) "OX-¼(Orle2*b+O[~e2e~b]),

(1.13)

f 8 - Y = c 0 ~ e -2~ .

(1.14)

The factors of v~ are put in for convenience so that the free action has the standard normalization [21]. Note also that, since z = e x p ( r + io), z= i corresponds to 1 o = 5~r, the string midpoint. The action and gauge transformation laws are [7]

S= fg'NS* Qg'NS + fY(i)g"R*Qg"R

+~gfx(i)g'NS* ~'tNS* g'NS q- 2Efg"NS* % *

g'.,

(1.15)

3g'NS = QANs + g [ X ( i ) ( g"NS* ANS -- ANS * g'NS) + (g'~ * AR -- AR * g'R)], (1.16) 3 g " R = Q A R + g , X ( i ) [ ( g " R * A N s - - A N s * g ' R ) + ( g " N s * A R - A R *g"Ns)],

(1.17)

where the physical Yang-Mills coupling, g, is related to ~ by [21] g = ~-g.

(1.18)

To begin perturbation theory, the gauge invariances of eq. (1.15) must be fixed [8, 9, 59, 60]. This is done by imposing b0q" = 0, the Siegel-Feynman gauge [61]. The propagator in the Neveu-Schwarz sector is

c~'bo/Lo = edbofo ~ dTexp( - TL o),

(1.19)

where L 0 is the zero mode of the full Virasoro generator. The propagator in the Ramond sector is more complicated and can be found in ref. [62].

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Superstring

Refs. [8, 9] have shown that, after the gauge fixing, a Feynman graph is associated with a first-quantized world sheet. It is constructed as follows. External strings are semi-infinite rectangular strips of width 7r. The effect of the operator e x p ( - TLo) is to create an internal strip of finite length T and width ~r. The N S - R - R interaction glues together, in a pairwise fashion, three rectangular strips, the first half of one joining to the second half of the next. The N S - N S - N S interaction is like the N S - R - R except a picture-changing operator, X, appears at the midpoint, as one can see from eq. (1.15). A string field is expandable in terms of particle fields, +s, the coefficients of which are first-quantized states, Ds ):

= EIs>¢,.

(1.20)

s

The Is) are created by first-quantized vertex operators, V(z) (without the exp(k. X) piece which is incorporated in ~,), acting on the SL(2, R)-invariant vacuum, 1~) [36] at z = 0. The expansions for the ~Ns and g'R begin as vi! S ) b~ = cq'~-I)(O)I~2)A.m ( x ) ( X m (N ) ~ b + O~ e-Z*Occ(O)l~2)q)m(x)(X") b + " " ,

(1.21) (

=

+ ...,

(1.22)

where all massless fields are displayed, qffx) being an auxiliary field. The ( Xm) ~b are matrices associated with Paton-Chan factors [1, 63]. 1.5. T H E MAPS FOR S T R I N G F I E L D T H E O R Y

Although off-shell superconformal field theory can be developed on its own, the most interesting appfication is to string field theory. We apply it to obtain superstring three-point couplings [19-21] and four-point amplitudes. In step (2) of the procedure, the map, p(z), from the half-plane to the string configuration is needed. Via the SL(2, R) invariance of the off-shell formalism, three of the Z, can be fixed to arbitrary values. For trilinear couplings, the relevant map is given in refs. [11,12] for certain fixed values of the Z r

Z1

= 1,

z =ff,

Z 2 = O,

Z 3 --+ - m ,

z o = ~1 + i 3X/~2" ,

(1.23)

Zo=i,

(1.24)

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where z 0 is the special point, o -- ur, common to all three strings. The generic map is given in Eqs. (3.6) and (3.7) of ref. [23]. The string variables in eq. (1.1) are [23] 1

y2(z) = i v - p ( z ) ,

Yl(z) = o ( z ) ,

Y3(z) = o ( z ) .

(1.25)

The constants No~r in eq. (1.1) are

[ 4 lZr-zr+lllZr-l-zr' [Zr+l-Zr_l]

]

N°°~ = In 3 ~ -

(1.26) "

The map for the four-point amplitude is [8]

o(z) = O(~o) + N fZdz'l(z'),

(1.27)

go

where 4

I ( z ' ) = ~

zgz~ +~2 I - I ( z ' - Z r )

1,

(1.28)

r=l Z 1 = ~,

Z 2 = a,

Z 3 =

-fl,

Z4 = --a,

(1.29)

and N = 21 ~

~B = 1,

( B 2 - ~ 2) ~ ,

v8 = a,

(1.30)

~ ~ B,

v ~ 8.

(1.31)

The variable 3' is a complicated function a (see eqs. (10) and (12) of ref. [8]). For a graph of 3', see fig. 1 of ref. [25]. For more details on the map, p(z), in eq. (1.27) see ref. [8] (or subsect. 2.3 of ref. [23]). As can be seen from eq. (1.28), there are square root singularities at _+zo = _+iT and at _+z0'= + i 6 along with their branch cuts. The string variables are [23]

~l( z ) = NJ~ dz' l( z') , [2(z)

= i~" - N foZdZ' I( z') ,

~'3(z) =

Nfodz'I(z'),

~4(g)

i~r- N f* dz'I(z'),

=

(1,32)

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262

1> (2 4

3 4

(a)

4

(b)

3

4

3

(c}

i;(i 1;(

3

4

(d)

(el

3

(f)

Fig. 1. The Feynman graphs for the scattering of four states in the Witten field theory. In the R - R - R - R amplitude the external legs are fermions; in the N S - N S - N S - N S amplitude the external legs are bosons. In both cases the internal line is a NS propagator.

and the No~r of eq. (1.1) are [23] N~ 1 = N ~ = ln(fix),

N~ 2 = N~ 3 = ln(ax).

(1.33)

The function ~(a) is expressible in terms on an integral (eq. (3.13) of ref. [23]) or in terms of theta and elliptic functions (eq. (17) of ref. [22]). A graph of ~ is provided in fig. 2 of ref. [25]. The four-point amplitude is a sum of six Feynmann diagrams (see fig. 1). All correlation functions factorize into correlators for the X ~-, ~-, bc- and By-systems. Common to all amplitudes is the following term stemming from the exponentials, exp[k. X](Z~):

ek"X(¢Or) r=l

=

exp

Ndjk'. k ~

R

ek~x(Z~ r=I

= e x p [ ½ , , ,~= l

N°~k" k~] =<''' )x'

H

(1.34)

where the subscripts R and H on ( ) indicate the string configuration and the half-plane, respectively. Using the maps given above and

N~=lnlZ,-Z,L,

r4=s,

(1.35)

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one obtains

(...

)X=] ( 3@)

t

2[ I

~,l (~'kr'k~)/2

[

X~(k~k:+k~k")(1 -- X)~/2- [TXtX)I

,

for

n=3,

for

n = 4,

(1.36)

where the Koba-Nielsen variable x is related to a by [8]

i

i ~=

l

d

l+vrx

(1.37) ,

and s = ( k l + k4) 2. Generically, the remaining correlation function contributions modify the sum over the momenta squared in the exponents in eq. (1.36) into the mass-shell condition. It is easy to see that the emerging on-shell measure is the correct one.

1.6. T H E S U P E R S T R I N G

ASSOCIATIVITY ANOMALY

Crucial to covariant string field theory are the axioms. Proofs of them have been established for Fock-space states but, for certain other states, associativity is violated [64]. Extensions of these proofs to the superstring should be straightforward with one exception: associativity is violated in the Neveu Schwarz sector even for Fock-space states as pointed out by Wendt [62]. This associativity anomaly leads to a violation of gauge invariance. Since gauge invariance is crucial in eliminating spurious degrees of freedom, the superstring field theory is in jeopardy. One surprising feature of the anomaly is that it occurs at tree level rather than at one loop. This can happen in string theory because of the infinite number of degrees of freedom of the string. Although ultraviolet properties are improved vis-a-vis particles because the string is stretched over finite distances, infinities can still arise in perturbation theory due to sums over intermediate states. The integral form of the Veneziano amplitude is ill-defined unless s and t are made sufficiently large. A similar problem arises in the computation of the four-point contribution to the static tachyon potential [25]. Only by analytic continuation is a finite result achieved for arbitrary s and t. The associativity anomaly is a result of colliding picture-changing operators [62]. Except when both states are in the Ramond sector, the superstring star product, ~r, inserts an X at the common midpoint. Thus, the associativity of ~r involves the product of two X's at the same point. Since this operator product is singular (see eq. (1.38)), an ill-defined expression ensues and a regulator must be introduced. The

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regulator cannot be removed without reintroducing an infinity, hence, associativity is explicitly violated [62]. The operator product of two X's is -4

X(z)X(w)- (z_w)2 (Q, be2 (w)) 2

( z - w ) {Q' O(beZq')(w)} + "X(z)X(w)',

(1.38)

where {Q, be 20} = ( T +

4bÙc- 2bcOeo)e2.

+~0(~+" 3Xe~)be2~'+ ~03(O~be2~)be 2~,

(1.39)

and T denotes the total stress-energy tensor. Since gauge invariance is an off-shell aspect of a field theory, an off-shell approach is required. We apply off-shell conformal field theory to resolve the difficulty to order g2. A four-point '/'Ns term is added to eq. (1.15) and the transformation law in eq. (1.16) is modified. By these means, the violation of gauge invariance at order g2 is removed. Unfortunately, the requirement of gauge invariance does not unambiguously determine the modifications to the action and transformation laws. This ambiguity is presumably resolved by analyzing the order g3 violations to gauge invariance a n d / o r by invoking other principles. We obtain a finite result for the four-point vector amplitude, but are unable to fix certain off-shell pieces. On-shell the amplitude is unique and agrees with the dual-resonance model. Similar contact terms have arisen in the light-cone approach to superstrings [65 -67]. 1.7. C O N T E N T S A N D NOTATION

This paper is organized as follows. In sect. 2 the three-point superstring couplings are computed via conformal field theory methods. Sect. 3 analyzes the associativity of the * product and repairs gauge-invariance to order g2. The massless four-spinor and four-vector superstring amplitudes are obtained in sect. 4. Sect. 5 uses picturechanging methods to establish the agreement with the dual-resonance model and to demonstrate finiteness of the general four-boson amplitude. Sect. 6 is the conclusion. Two appendices present some details of the computation of the massless four-vector amplitude and the analysis of its finiteness. We adopt the conformal field theory notation in ref. [57] and the superstring field theory notation in ref. [21]. Any undefined symbol can be found in one of these two references. We set a ' = ½ and D = 10. Among the more important papers for understanding the present work are refs. [6-8, 23, 26, 36, 62].

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In string theory several different types of fields appear. Besides the states of the original string model, there are two other sets: the auxiliary fields necessary for a local Lorentz-invariant formulation [68-71] and the second-quantized FaddeevPopov ghost fields which appear after gauge fixing. We use the term "standard states" to refer to the original string model states. When a gauge group is included, amplitudes involve a trace over P a t o n - C h a n factors, like tr(XlX2... Xn), which we frequently abbreviate as t r ( ) .

2. Three-point couplings This section computes the N S - N S - N S and R - R - N S couplings [64,65] in the Witten covariant superstring theory [6] using conformal field theory. The analogous calculations in the bosonic string field theory can be found in subsect. 5.2 of ref. [261. 2.1. THE CANCELLATIONOF THE SUPERCONFORMAL ANOMALY In transforming from the string configuration, R, to the half-plane, H, an overall normalization constant is generated. It is the ratio of the functional integrals over the two world sheets, R and H, ignoring the effect of vertex operators. For the bosonic string, Giddings has shown the ratio to be one (see eqs. (5) and (6) of ref. [8]). The calculation for the superstring is as follows. The action for a boson with statistics e and background charge Q [36], is

S,= l f d2oV~(-e, Deo+iQR,),

(2.l)

where D is the covariant derivative on the world sheet [54] (see eq. (5) of ref. [8]). For bosons not associated with a bosonization procedure, such as the X ~ ( z ) , Q = O, e=+l. The two contributions to the functional integral over the action in eq. (2.1) are (i) the classical piece, Sd, obtained by shifting ~, by its solution to the equation of motion and (ii) the quantum piece, Squ, obtained from performing the functional integral over the quadratic form

Squ=So-

1

f f d2od2ot~[gR(o)D-l(o, ot)]l~R(o'),

Sol = - 3eQZSo •

(2.2) (2.3)

A boson generates c S o, where c = 1 - 3 e Q 2 may be recognized as its contribution to the conformal anomaly. Table 1 enumerates these contributions for each of the

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1

1

(a)

{b)

Fig. 2. The F e y n m a n graphs for the three-point function, For the R - N S - N S coupling external lines 1 and 2 are the ferrnions,

bosons in the superstring. The total conformal anomaly vanishes for D = 10 and hence in transforming from R to H no normalization factor is produced. This statement applies to the arbitrary n-point amplitude, since the calculation is independent of the number of vertex operators. 2.2. THE FERMION-FERMION-BOSON COUPLINGS The two Feynman graphs which enter the R - R - N S fig. 2. For fig. 2a

couplings are displayed in

AR -R-Ns-- f2gtr()ttX2)t3)(V~(col)Vd(~o2)V3ys(CO3))R

,

(2.4)

where Xr are certain matrices [1] when a gauge group is included in the covariant string field theory formalism. The V~., V~ and V~s are vertex operators for the states. For standard states, VR and VNs respectively possess q,-ghost number 2 and - 1. Both have b-c ghost number + 1. The points, ~ , ~02 and ~03 correspond to the asymptotic positions. Let us compute explicitly the couplings for massless standard states. The relevant vertex operators are given in eqs. (1.11-1.12). Transform to the half-plane

3

x

r=l

It

glc(zr)

.

(2.5)

r=l

a The v~-g The minus sign arises because the ghost c anticommutes with S(_1/2).

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O. Lechtenfeld, S. Samuel / Superstring

factor comes from the coupling normalization in eq. (1.18) and the number of ways of joining propagator lines to produce fig. 2a. Using eqs. (1.5), (1.6), (1.23), (1.24) and (1.26) (see also eqs. (3.1) and (3.7) of ref. [26]), eq. (2.5) is 4 \ (~'k~k~)/2 -~-) ifi1¢3U2 tr( ~12t")t3),

AR-R-NS=g

(2.6)

where u 1, u 2 and e 3 are the spinor and polarization vectors of the three states and fi1~3u2= "&"~ ,,1~3,,~a~,,~ t" ~B" The contribution from fig. 2b is minus eq. (2.6) with string labels 1 and 2 interchanged. The total result is aR-R-NS = g l'a+b -~

ig~l¢3U 2 tr([ 2t1, )~2]X3)

(2.7)

Eq. (2.7) agrees with eq. (5.9) of ref. [21]. Via eq. (2.4) and the above procedure, the arbitrary off-shell R - R - N S coupling can be calculated. 2.3. THE THREE-BOSON COUPLINGS

The N S - N S - N S couplings are computed similarly, except there is an extra factor of X at the midpoint due to eq. (1.15). Denoting this point by 0:0, the contribution from fig. 2a is aNS-NS-NS = v ~ g t r ( ) k l ) t z ~ 3 ) ( V l s ( w l ) V 2 s ( w 2 ) X ( o ~ o ) V 3 s ( W 3 ) } R .

(2.8)

For the massless three-vector coupling, the relevant vertex operator is that in eq. (1.11). Due to ghost-charge counting, the only piece of X which contributes is the second term in eq. (1.13). Straightforward algebra yields N S - N S - N S

A,

__

- 2g tr( )

(3~) (~rkr'kr)/2

3

× 1-[ Zr0 r=l

Z

\ r=l Zor)

Z12Z03

Z13Z02

+ - -

Z23Zol

'

(2.9)

where z 0 is the point in the half-plane corresponding to o~0 and Zrs ~ Z r - Z s ,

Zor ~- z 0 - Z r = - Zro ,

Ers = e r" e s.

(2.10)

Substituting either set in eqs. (1.23) and (1.24) for Z r and z o, and folding in the contribution from fig. 2b, one gets lia+ bAINS-NS-NS = g

( 4 ) ~

(y'rk'kr)/2

{( El" E2)E 3 • [( k 1 - k2)tr([ X1, ~k2 ] ~k3)

_ivY-k3 tr({ ~1, ~2} ~3)] + cyclic}. (2.11)

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O. Lechtenfeld, S. Samuel / Superstring

Eq. (2.11) agrees with eq. (5.9) of ref. [21]. The fact that either set gives the same result is a verification of the SL(2, R) invariance of the formalism. The i ~ - factor in eq. (2.11) arose in ref. [21] from an infinite sum over Neumann-function components (Eq. (E.2) of ref. [21]). The power of off-shell conformal field theory should be evident; the ivY- factor is obtained with significantly less effort. Via the above methods, the arbitrary off-shell N S - N S - N S coupling can be computed. 3. Superstring gauge invariance 3.1. ASSOCIATIVITY OF THE * PRODUCT USING CONFORMAL METHODS Two axioms are needed for proving order g2 gauge invariance: associativity and graded commutativity under the integral. For the bosonic string, associativity has been analyzed in refs. [6,10,14] and proved in refs. [23, 24]. To get some insight, it is useful to present a conformal field theory treatment. The proof applies for the * product (but not the * product) in either the bosonic string or the superstring. If, for an arbitrary state, ,/,3,

f((,/,4, 'k/J1), ,~2), qZ3= f(,/,4, (,/,1, ,/,2)), ,/,3,

(3.1)

then ((qt4, g,1), I/t2)= (x/.¢4,(x/11, xi[¢2)) and associativity holds. The 1.h.s. of eq. (3.1) corresponds to the Feynman graph in fig. la at T = 0 with no b0 insertion. Likewise, the r.h.s, is associated with fig. lb. Put in states for the q,r and perform the conformal transformation from the string configuration to the half plane. First consider the case when states are created by primary conformal fields. The left-hand and right-hand sides of eq. (3.1) are, respectively

e-V(Zr)

dz ]

rzl

( dfr-i

-1 e t r

r=l

]hr ) Or(Z~)

,

l(Zr_l)lhror(Zr_l) )')

(3.2)

(3.3)

where O r is the operator that produces q,r, h r is its conformal dimension and ( d f r / d z ) -1 exp(--~r)(zr) is obtained from eq. (1.32). To relate fig. la to fig. lb at T = 0, we perform the additional conformal transformation [23] z ~ w-

l+z 1

dw Z'

dz

= I(1 + w) 2,

(3.4, 3.5)

O. Lechtenfeld, S. Samuel / Superstring

269

on eq. (3.2). The transformed points, Wr, are Wr= (1 + Z r ) / ( 1 - Z r ) = Zr_ 1 [23]. Since under a conformal transformation, O ( z ) ~ ( d w / d z ) h O ( w ) , the operators in eq. (3.2) at Z r end up at Zr_ 1. Eqs. (3.2) and (3.3) are equal if the conformal factors are equal, i.e.

(d r -1

e

)

(3.6)

when z = Z r and w = Z r_l. This is easily verified. For the case of states produced by arbitrary operators, the analysis is slightly more complicated since such operators are related to primary fields by differentiation. It turns out that it suffices for eq. (3.6) to hold when w is given in eq. (3.4). Then arbitrary derivatives of the conformal factors agree on the 1.h.s. and the r.h.s, of eq. (3.6). This completes the proof of associativity of the * product.

3.2. ASSOCIATIVITY OF THE ,k PRODUCT Since the * product inserts the picture-changing operator, X(z), at the midpoint, z = i, the verification of associativity of ~- involves the product of two X at the same point. This product is singular (see eq. (1.38)) so that the use of two or more * operations in the same expression is ill defined. The effect of this problem on order g2 gauge invariance is less severe in the Ramond sector because the superstring integral inserts the inverse picture-changing operator, Y(i), (see eq. (1.15)) thereby reducing by one the number of X in the transformed action. From eqs. (1.15)-(1.17), it is seen that zero or one X appear in the transformed R - R - N S interaction term and one or two X appear in the transformed N S - N S - N S term. An order g2 violation of gauge invariance may occur in the N S - N S - N S interaction term. The presence of one X(i) in the integrands of both the left-hand and right-hand sides of eq. (3.1) does not invalidate the equation. The proof given in sect. 1 applies because (a) X is a primary field of conformal weight zero that picks up no conformal factors when transforming to the half-plane and (b) the common interaction point, z = i in H, is a fixed point of the transformation in eq. (3.4). When two X appear, a regularization procedure must be adopted. The obvious one inserts a rectangular world-sheet strip of length 2~- (~- being small) to separate the two picture-changing operators [62]. The 1.h.s. of eq. (3.1) becomes the Feynman graph in fig. l a with T = 2~', with no b0 insertion, but with X insertions at the two interaction vertices. The r.h.s, of eq. (3.1) becomes related to the graph in fig. l b in a similar manner. The strip insertion causes an explicit violation of associativity. To analyze the problem, transform to the half plane. The tri-common points map onto z 0 = i T and z~ = i~ so that present in the correlation function is the term

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O. Lechtenfeld,S. Samuel / Superstring

X(zo)X(z~). Expand this product about the point z = i X ( i g ) X ( i S ) - (Q, x101(i) + Oreg(i) } + O(iT - - i 8 ) ,

(3.7)

where ~1 =

1 -

e +

-

4~" + 0 ( ~ ' ) ,

(3.8) (3.9)

Ol(z ) = b e2q'(z), O~eg(z) = ~( Off + icg~- 1 ) O , ( z ) + vcff:~X:,

(3.10)

and e = 1 - ~,(r) is a small parameter measuring the deviation from r = 0. The relation between e and r is determined from ref. [8] (see also eq. (B.15) in appendix B). The expansion in eq. (3.7) is performed about z = i because this point is half-way between the interaction points in the string configuration. Also, z 0 and z; coincide at this point when r --+ 0. The OPE in eq. (3.7) is separated into two terms because O x and Oreg have different transformation properties under eq. (3.4):

Ol(i ) ~ - 0 1 ( i ) ,

Oreg(i ) --+ Oreg(i ) .

(3.11), (3.12)

Eq. (3.11) is deduced by comparing the OPE of X(zo)X(z~) to the OPE of X(wo)X(wO) where w0 and w~ are the transformed points (under eq. (3.4)). A direct calculation is difficult since the transformation law for :XX(i): is, a priori, not known. When X(zo)X(zD) is replaced by its OPE in eq. (3.1) and the analysis in subsect. 3.1 is reperformed, the pieces involving [~reg on both sides of eq. (3.1) are equal due to eq. (3.12). In contrast, the piece involving O 1 changes sign according to eq. (3.11) and persists in the limit r - 0 0. This implies an order g2 violation of gauge invariance in the N S - N S - N S interaction term. This difficulty was pointed out by Wendt [62]. 3.3. REPAIRING ORDER g2 GAUGE INVARIANCE The violation of gauge invariance in the NS sector can be repaired by adding additional terms to the regularized versions of the action and transformation laws in eqs. (1.15) and (1.16). Let S = S O -}- g S 1 -}- g 2 S 2 q- . . . .

= 80 " + gal " + g2a2 " + . . . .

(3.13)

(3.14)

where + . . . indicates possible higher order (in g) effects. The subscript NS has been dropped from q'Ns since throughout this section only NS fields are treated.

O. Lechtenfeld, S. Samuel / Superstring

271

The lowest order terms in eqs. (3.13) and (3.14) are [62]

So = f q , , e2"LoQq,, S1

=

(3.15)

(3.16)

~fx(i)(q,, ~, ,t,),

6og,=QA ,

61q,=e-2"CoX(i)(g,,A-a,g,).

(3.17), (3.18)

From the results given in subsect. 3.2, the violation in gauge invariance is alS 1 =

2f(q,

• ~,) X ( i ) e - 2 " L ° X ( i ) (

gt , a - A * gt)

=4f(g~*'l~)e-'L°{Q, XlOl(i)}e-'C°('t'*A)+O('r), where

e~(i

~1 = ~1

,]2

,

(3.19)

(3.20)

for maps with y(~-)= 1 - e . The additional factor in eq. (3.20) is necessary when working on R (as opposed to H). It is 1 N

H 4 = , ( Z , . - i)

~(1- ~2)(~2 1)

(3.21)

where O(z), N and Z r are given in eqs. (1.27), (1.29) and (1.30). It is cancelled by the conformal factor of b e 2~° when transforming from R to H. Consider, for the order g2 terms in eqs. (3.13) and (3.14), $2 =

~ ) , e-'LoO(i)e-'Lo(~/, • ,/'),

(3.22)

8 2 ~ = e-2"Lo [ (e-~L°O e-~'c° (g" * '/')) * A - A * ( e - ' L ° o e-~L° ( g" * g')) +**(e

"L°Oe-TLo( ~' * A - A * g ' ) )

- ( e - ' t ° o e-'L°( g" * A - A * g'))* g'],

(3.23)

where O is an operator to be determined. After a little algebra

aoS2 + a2So = - 4 f ( ~ ,

• q,)e -'Lo {Q, 0 ( i ) } e - ' L o ( g" * A) + O ( r ) .

(3.24)

O. Leehtenfeld, S. Samuel / Superstring

272

The violating 81S1 term in eq. (3.19) is cancelled by 80S2 +/~2S0 in eq. (3.24) if (3.25)

(9 = xl(91 + 6)a ,

where 02 is any operator satisfying { Q, (9 2 } = 0. Eqs. (3.22), (3.23) and (3.25) are the rectification of gauge invariance to order g2 for the open superstring field theory. By adding contact terms and modifying the gauge transformation law, order g2 gauge invariance is satisfied. A similar solution to this problem was given by Wendt [62] for on-shell 'P: Qq" = 0. For on-shell '/', the interaction term due to the (92 piece of (9 in Eq. (3.25) vanishes. Sects. 4 and 5 show that off-shell finiteness implies a non-zero (92. The requirement of off-shell gauge invariance does not, however, uniquely fix the four-point term. This is the ambiguity discussed in sect. 1. Order g2 gauge invariance may be repaired but the solution is not unique. In terms of the four-vector scattering amplitude, we are presently unable to unambiguously determine the off-shell extension. Additional analyses or symmetry arguments are needed.

4. Four-point amplitudes The techniques for computing on-shell amplitudes from the covariant bosonic string field theory were developed by Giddings [8]. Recently, the extension to the superstring has been achieved [62, 72]. Off-shell results for the bosonic theory were obtained in refs. [22-25]. Using the conformal field theory methods of ref. [26], this section computes off-shell superstring amplitudes for four massless spinors and four massless vectors. The two-spinor two-vector amplitude involves the somewhat complicated Ramond-sector propagator [62] and we prefer to leave it to a future publication. There are six Feynman graphs which enter an amplitude. They are shown in fig. 1. For both the four-fermion and four-boson cases the propagator of the intermediate line is that in eq. (1.19). 4.1. T H E F O U R - F E R M I O N A M P L I T U D E

The amplitude for four arbibtrary Ramond-sector states is A,aF---- - 2 g 2 tr(XWX3X4)

j0

dr(V~.(,ol)V~(,o2)V3(%)boV~(~4))~,

(4.1)

where the subscript R in ( )R indicates the string configuration in fig. 3a of ref. [23] with an intermediate strip of the length T. The f d T and bo insertion come from eq. (1.19). To evaluate eq. (4.1), transform to the half plane using the map [8] in eqs. (1.27)-(1.31). The insertion of bo in eq. (4.1) is [8]

dial b° =

= -

dz (dp(z))-1 dz

b(z),

(4.2)

O. Leehtenfeld, S. Samuel / Superstring

273

where the first contour is in R and the second is in H, and (dp/dz) -1 is a conformal factor generated in transforming from R to H (see eqs. (1.3), (1.4), (1.27) and (1.28)

dz

= N

if(Z2 q- y 2 ) ( Z 2 q- 3 2 )

Using off-shell conformal field theory, it is straightforward to compute eq. (4.1). For massless fermions, use eq. (1.12) for the V~. The correlation function of four spin fields is [37, 57]

1-x[1-

4

H S~'l/2)(Zr)

r=l

H

32

L

] X

oil

p.

--

(4.4)

where a has been traded for the Koba-Nielsen variable x via eq. (1.37) [8], and the b-c correlation function is given in ref. [8] (or eq. (5.38) of ref. [26]). Putting everything together A(4~= g2 tr( ) fl I /2

dx[½lC(X)](E~Uk~)/2x(k~'k2+k"k%/2(1 X) s/2-1 -

-

x(l~x(~ll"vpU2)(~13~P'u4)--(~ll~lpU4)(~12"~lZU3)],

(4.5)

where s = (pl + p4)2. The other 5 Feynman graphs in fig. 1 are obtained from eq. (4.5) by permuting indices so that the total off-shell massless four-fermion amplitude is A4F = ¼ E

( -- ,~°A4v..(a), ~(,,°('), k °('1, X"(')).

(4.6)

o~$4

On-shell, eqs. (4.5) and (4.6) agree with the standard result [1]. Off-shell factorization serves as a check of eqs. (4.5) and (4.6). Using fl I dx f ( x ) ( 1 - x )

/2

s/2

I

a's~

2f(1)

s

(4.7)

and noting that only graphs (a) and (d) have s-channel poles [ 4 ] (y'rkr'kr)/2

g2 A 4F

%

"a (a) + (d) a's ~ 6

Permissible

m

S

Paton-Chan

to¥~J

groups

require

tr([4,

a complete

set of

(48) Xm satisfying

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O. Leehtenfeld, S. Samuel / Superstring

E,,Tr(Xphm)Tr(XmM)= Tr(~PXq) [1], so

that the trace factor is

Tr([ ~4, X1][;~2, ;~3]) = ~ Tr([ X4, X1]XI)Tr(XZ [~2, X3]).

(4.9)

The residue of eq. (4.8) associated with the Ith channel is the square of eq. (2.7). This demonstrates factorization on the massless intermediate vector state.

4.2. THE FOUR-BOSON AMPLITUDE In second-order perturbation theory in the trilinear interaction in eq. (1.15) the contribution to the amplitude for four arbitrary Neveu-Schwarz states is

A4B

=

¼ E - - A(a)l., 4 B : eo(,>, k "('), N'(r)), a~S4

A(~) = 2g 2

V2s(O~2)V~s(Co3)boV4s(O~4)X(o~;)X(o~o))p.

)

= 2g2tr( )f2~dT(IT"Ns(Z1)lT"Zs(Z2)lT"~s(Z3)b°lT"~s(Z4)X(z°)X(z°))"'

(4.10)

when the quadratic action in eq. (3.5) is used: The effect of exp(2~-L0) is to modify the lower limit in the T-integration from 0 to 2~ where ~- is the cutoff parameter. The rest of the notation is as follows: b0 stands for eq. (4.2), w0 and w6 are the two tri-common interaction points and z 0 and z6 are their half-plane counterparts. The tilde over an operator indicates its half-plane transformed version. To obtain the four-vector amplitude, we use the vertex operators in eq. (1.11) and fold in the contribution from the four-point counterterm. The calculation is somewhat lengthy but straightforward and results in

A 4B (a) + (b) = 8g 2 tr(Xlx2x3X4) fo'X°da { f12 ~__ 0£2 ( 0¢2 q- /~2 -1- ~/2 q- 3 2 ) 2 ( ' ' [ -432G 43 X ( "'" >1~ (3-- ~)4 -{- (3--'it) 3

t

)X

43(3+y-3G) R.R' (3-- ~)2(OZ2 q- /~2 nt_.,[2 q_ 32) -]- (3--'}t) 2

7jp.~ R'~ ) 1 (3-7y)3 R~" (3-~)

+i(-..

d[

da

(''")x(...

)1+(a2 + j~2 _}_2) (3 - ~)~ -{-

+ ( 0 2 - t e r m ) + perm.( 1234 2 3 4 1 ) ° n k r,e r } .

:}1 (4.11)

O. Lechtenfeld, S. Samuel / Superstring

275

The nature of the term from 192, which contributes only off-shell, is discussed in sect. 5. The term involving d [ . . . ] / d a comes from 191 in eqs. (3.22) and (3.25). Because some singular T-dependence cancels between figs. 2a and 2b, they have been combined in eq. (4.11). The limit ~-~ 0 has also been taken. The next few paragraphs discuss the notation and describe the calculation. The picture-changing operator, X, can be decomposed according to q~-ghost number: X = X 0 + X 1 + X 2 (see eq. (1.13)). Only operators with q,-ghost number of 2 in X ( z o ) X ( z O ) in eq. (4.10) contribute. Hence, X X = X o X 2 + X1X 1 + X z X o + ... and there are three types of contributions: 0-2, 1-1 and 2-0. The relevant X"-correlation function in the 0 - 2 and 2-0 terms is < . . . >x, given in eq. (1.34). The ~bg-correlation function is 4

E12634

<...>2~o = ( . . . ) ~ = ( I - [ e ~ . + ( Z ~ ) >

E13E 24

~-

r=l

1~14E23

- - + - - , Z12Z34

Z13Z24

(4.12)

Z14Z23

where Z~ = Z~ - Z~, ~s = e~. e~. For the 1-1 term, the X"-correlation function contains OX"(zo) and OX~(z6) in addition to the four exponentials. When the former two contract they generate x = - T y / ( 6 y)2. When OX"(z) contracts with the other exponentials, it generates R ~ ( z ) = E4=lk~"/(z - Z~). The presence of the vectors 4

4

R " ( z ) = ~ k ~ / ( i y - Zr) ,

R'~(z) = ~ k ' ~ / ( i 6 - Z~),

r=l

(4.131

r=l

in eq. (4.11) is indicative of such contractions. The 1-1 +"-correlation function is

< -''

> 1~1,gv

=

1-I g r - ¢ ( z , ) ¢ " ( : ; ) ¢ " ( Z o )

....<

r= 1

>*1

Z~ -- Z 0

(4.14) where o(1) 0(2) 0(3)a(4)

<...

= ½ Z

(-1°

•

(4.151

o $4

In the first term in eq. (4.14), ~b"(z;) and +~(z0) contract leaving the four-~b correlation function in eq. (4.12). The second term in eq. (4.14) involves the contractions of ~ " ( z ; ) and ~b*(z0) with the er. tp(Zr). The ratio of elliptic integrals G

2) 21

(4.161

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O. Lechtenfeld, S. Samuel / Superstring

makes its appearance through the ghost correlation function and the change of variables from T to f d T - -2. . . Jo U

f% do~2~r

= J0

/3

3

/32 _ 0/2 K ( y 2 ) . . . .

(4.17)

Appendix A contains some details of the evaluation of the ghost contribution and a table of useful integrals. We have checked by explicit calculation that the integral in eq. (4.11) converges on-shell. The details are relegated to appendix B. Here we show the cancellation of the leading singularity. Reintroduce the cutoff, r, and change variables from 0/ to ~,

fo~°d0/--) fo~(') d0/= fol- ~d'/d0/ dT

(4.18)

The relation between e and r is given in sect. 3. Using the following (see also appendix B), da d~,

~/20/o(1 - 7) G(,{2) (1 + O(1 - y ) ) ,

(0/2 _1_ /32 -I- ~2 + 32)1r=0 = (0/2 + 132 + 2 ) l r = o = 8 '

( B 2 - 0/z)lT=0 = 4v~,

(3 - 7) = 2(1 - "/)(1 + O(1 - "~)),

(4.19)

the leading singularity in A(4a~comes from the 1 / ( 8 - y)4 term in eq. (4.11)

A(4leadingsingularity= 892 tr( )f

-4

(< '

>x<...

2 = 8g2 tr( ) ~ 5 ( ( "'" )x( ... 5+1)1r=0" The leading singularity of the counterterm is - [ - . . evaluated at T = 2r

(4.20)

] in " d [ . . - ]/do/" in eq. (4.11)

1

A(4~ . . . . terterm leading singularity = - 8 g 2 tr( ) 8 ~ e 2 ( ( - . .

) x ( . . . )~)lr=0"

(4.21)

The sum of eqs. (4.20) and (4.21) is zero. Subleading singularities also cancel on-shell after combining diagrams (a) and (b). This is the subject of appendix B. Off-shell, however, we find a logarithmic divergence if the O2-counterterm is absent. It comes purely from the X"-correlation, eq. (B.17), and is due to the

O. Lechtenfeld, S. Samuel / Superstring

277

x-function in Ndor (see eq. (B.18)). For the whole amplitude this results in 4

/(a)+(b)4B

__--

f i n i t e _ 47rg2 tr( ) O n e ) [ ( . . . )x( . . . )~]lr=o ~ k ~ ' k r + (02-- term), r=l

(4.22) which demonstrates the need of the O2-counterterm to render the amplitude off-shell finite.

5. Picture-changing methods Sect. 4 demonstrated finiteness of the on-shell massless four-vector amplitude by brute force calculation. It is important to establish finiteness for the general four-boson amplitude. This section does this by using picture-changing manipulations [36]. These methods have been exploited by Wendt [62] in the context of the Witten superstring field theory [7]. The techniques are used in off-shell manner in this section. Finally, agreement with the first-quantized string model is shown for on-shell physical standard states. Such states are created by BRST-invariant vertex operators of the form [57]

Y/~-1) = c e-q'V( X ~, 4,'),

(NS sector),

(5.1)

~(-1/2) = c e-¢/2V( X~, ~bt~),

(R sector),

(5.2)

where V is a ghost-independent vertex operator. 5.1. O N - S H E L L A G R E E M E N T OF T H R E E - P O I N T C O U P L I N G S

This subsection illustrates the picture-changing methods. Agreement of N S - N S - N S on-shell physical couplings in the Witten superstring field theory with the standard dual-resonance model is achieved. The general N S - N S - N S coupling in Witten's theory is given in eq. (2.8), which in the half plane, becomes •NS-NS-NS _ v~-g tr(~lX2X3 )(V~s(Z~)V~s(Z~)X(Zo)¢~s(Z~))., -1 -2 "'(a)

(5.3)

where a tilde over a V indicates a transformed operator and z 0 is the point in H corresponding to w0. For on-shell physical states

lP(z) = V(z),

(for physical on-shell V(z)),

(5.4)

because V(z) has conformal weight zero. Such states are also BRST invariant: ( Q, V(z)} = 0,

(for physical on-shell V(z)).

(5.5)

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O. Lechtenfeld, S. Samuel / Superstring

Now let the states in eq. (5.3) be physical and on-shell so that eqs. (5.4) and (5.5) are satisfied. Picture changing is accomplished by writing X ( z o ) = Vrff( Q, ~(z0)}, going to the bigger ~-algebra in which ~0 appears and inserting a ~(z) at the right in the correlation function [36] ANS-NS-NS__ v~-g tr( )fff(IT"~s(Z1)IT"~s(Z2)(Q, ~(z0) } (a)

V3s(Z3)~(Z))big. (5.6)

Since Q annihilates the left and right vacuums and the V r are BRST invariant, Q may be commuted (or anti-commuted) past operators in the correlation function until it reaches ~(z) at which point it converts ~(z) into X ( z ) . In returning to the smaller ~-algebra, ~(z0) is removed because it must supply the zero mode of (. The net result is that the position of the operator X is shifted from z 0 to z. In short, for BRST invariant vertex operators, the picture-changing operator may be moved at will. Use standard operators (eqs. (5.1) and (5.2)) and take the limit in which z approaches any Z r. Since =:

(5.v)

one vertex operator is converted from the ( - 1 ) picture to the (0) picture [36]. For on-shell physical standard states, the N S - N S - N S coupling is ANS-NS-NS = v/~g tr( (a)

)~f((l_l)(Z1)f((21)(Z2)f((3o)(Z3))smaU,

(5.8)

for the z ~ Z 3 case. Since eq. (5.8) is the result of the first-quantized string model [36], agreement has been shown. For the R - R - N S couplings, agreement with the standard dual-resonance model is immediate. No picture-changing operations are necessary; the correlation function, after converting to the half-plane and using eqs. (5.1), (5.2) and (5.4), becomes the first-quantized result.

5.2. M A N I P U L A T I O N S FOR FOUR-POINT AMPLITUDES

The first goal of this subsection is to demonstrate on-shell and off-shell finiteness of the general four-boson amplitude when the counterterm in eq. (3.22) is included. The second purpose is to establish agreement with the first-quantized approach. The difficulty with colliding picture-changing operators requires the use of the regularized action in eqs. (3.15) and (3.16). The contribution from fig. la is given in eq. (4.10). To avoid writing many tildes, we drop them in what follows. Vertex operators in the half plane are understood to be the transformed versions unless otherwise indicated. We also drop the NS subscript since all V r in this subsection are in the Neveu-Schwarz sector.

O. Lechtenfeld,S. Samuel / Superstring

279

We carry out the picture-changing manipulations twice: once for z 0 and once for z;, but we do not assume that the V r are BRST invariant. The two tri-common points are moved to w and w'. Five types of terms appear

A(a) = 2gZtr( ) ~

~]

(-

we(Z,) w'~{zs)

f2°°dT(V1V2V3V4boX(w')X(w)>small

W~W'

-~/8 b i g [ T = 2 -

~.

dr([O, glg2g3m 4] bo~(Zo)X(Zo)~(w' ) >big

--~/8big{T~2.r

--¢-8 f2~ dT<[Q'V1V2V3V4]b°~(z°)X(w')~(W)>big) = 2g 2 tr( ){term 1+ term 2 + term 3 + term4 + t e r m s ) .

(5.9)

In eq. (5.9), the last four terms are in the big ~-algebra. An average has been performed over the locations of the picture-changing operators. The first term in eq. (5.9) is the standard first-quantized correlation function in which two of the four vertex operators are pictured changed. There are six possible ways of choosing two operators out of four, and the average over these six possibilities. It is finite in the limit ¢ ~ 0. When combined with the other 5 contributions in fig. 1, it produces the dual-resonance-model result for on-shell physical standard states because the correct integration region and measure (from the b-c ghost correlation function) are obtained [8]. The second term in eq. (5.9) is infinite in the limit ~"--* 0. The third term vanishes for on-shell physical states but, for non-BRST-invariant states, is infinite as ~"~ 0. Below, it is shown that the four-point counterterm in eq. (3.22) cancels the infinite pieces of terms two and three. Terms four and five are finite and ~" m a y be set to zero; term five vanishes for BRST-invariant vertex operators. First consider on-shell finiteness. The second term is completely cancelled by the XlO 1 piece of S 2 (eq. (3.22)) in the ~-~ 0 limit. To see this, replace ~/g~(Zo')X(zo) by K1OI(i) + Oreg(i) + order(z~ - z0) (eq. (3.7)). When ~- ~ 0, the order (z~ - z0) term vanishes, Oreg(i ) is finite and ~a is infinite. Consider ~aOl(i). Since O 1 does not contain ~0, the zero mode of ~ must come from ~(w'); in returning to the smaller ~-algebra, ~ ( w ' ) is removed and all terms in the averaging sum are equal t e r m 2[ divergent = --


(5.10)

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O. Lechtenfeld, S. Samuel / Superstring

On the other hand, the gaol term in S~ yields ~tOa - t e r m in S 2 ~ g 2 tr( ) ( V ~ V 2 V 3 V 4 ~ t O ~ ( i ) } ~ m a U [ , = 2

.

(5.11)

The factor ~ is converted into ~1 when transforming to the half plane (see eq. (3.20)). The sum of the diverging piece of term 2 and the ~O~ term in S 2 cancel. The contribution for Oreg(i ) vanishes when figs. la and b are added lim [termz,{a) + term2,(b)]l~eg~ = _ 1

~

[(V1V2V3V4Oreg(i)~(w,))big

w'~{z,}

-~-(v4vlv2V3Oreg(i)~(w'))big] T=0= 0,

(5.12)

where fig. lb has been obtained from fig. la by cycling the string indices. To show the last equality in eq. (5.12), we perform the transformation in eq. (3.4) on the second term. The points Z r are mapped into Zr_ ~, so that the V ~ are evaluated at Z~, O~eg(i) is mapped into itself according to eq. (3.12), and a minus is generated when converting V4(Z4)VI(Z1)V2(Z2)V3(Z3)into VI(z1)v2(z2)v3(z3)v4(z4) because the V r are anticommuting. Let us demonstrate the agreement of the four-boson amplitude with the firstquantized theory for standard on-shell physical states. Terms 3 and 5 may be ignored by the BRST invariance of vertex operators. It is necessary to show that term 4 is zero for V ~ in the form of eq. (5.1). The effect of a picture-changing transformation on Y/~(~) in eq. (5.9) is ~(0) =" X~(( u := ~ - e * ~ V ( X, %) + . . . .

(5.13)

where only the piece having the appropriate ghost numbers to saturate the background charges is displayed. Term 4 is term4= 1 ( ? l ? 2 v 3 p 4 ) x , + x ((eq'(ce-~)(ce-*)(ce-~')}bc~'((~lflf)nb~g+

@lflf)~

+ @lflf}~,~a)

+ ((c e - * ) ( c e *)e*( c e-*)}bc*( - ((lnl~ }~i~ - (l~nl~}~ig + ( 1 1 ~ }~i~) + ((c e - * ) ( c e - * ) ( c e-*)e*}bc*( -- ((11~()~i~ -- (l(ln(}~ig

-- (ll~(}~ig)

},

(5.14)

where, to save space, the positions of the operators are omitted. When four operators are present, their arguments are Z t = rio, Z 2 = ao, Z3 = - a o and Z 4 =

O. Lechtenfeld, S. Samuel / Superstring

281

-/3 o and, when a fifth operator, ~, is present, its argument is i. Straightforward algebra yields 1 - - 1 - - 2 - - 3 - - 4 Xff terma=x(VVVV ) , y" ( _1(_) r

rE s,t

3 I_i zr~} = ZrsZrt Zr--is(4:r)

o. (5.15)

The quantity in "{ }" at r = 0 is { - ( - 4 8 ) - 3 × 16} = 0. Hence, the term 4, which arose from the second picture-changing manipulation, is zero due to a somewhat remarkable cancellation between two contributions. The importance of averaging in eq. (5.9) enters here. To show off-shell finiteness requires finding an operator O 2 in eq. (3.25) which cancels the divergent part of term 3. The only constraint on 02 is that { Q, 02 } = 0. The singular part of term 3 is generated in the small T-integration region. The operator product in eq. (3.7) may be used

term3[divergent=~/-fff27dT~V1V2V3V4[Q, bo~;Ol(i)])sman,

(5.16)

n

where K1' is x 1 in eq. (3.8) with the fixed parameter e replaced by 1 - y(T). We choose

O2= _~f27dT e --

,r/2 .,l.o[O, bo~;Ol] e ,r/2

.,Lo.

(5.17)

h

K~ and K1, a r e related in a manner similar to K1 and Xl in eq. (3.20) except the configuration is a strip of length T instead of r. Then O 2 in S 2 cancels eq. (5.16). The conformal factor goes away when transforming 02 from R to H. There are many ways to choose the 02 in S 2 to cancel the singular piece of eq. (5.16). The upper limit, m, in eq. (5.17) may be replaced by any ~--independent number. The principles of gauge invariance and off-shell finiteness do not uniquely fix the four-point counterterm. The particular solution in eq. (5.17) involves a non-local world-sheet interaction. Further investigations are needed to establish whether a local term can be found. This completes the goals concerning the four-point bosonic amplitudes in the modified Witten superstring field theory. The amplitudes are finite on-shell and off-shell and agree with the standard dual-resonance model on-shell. All four-point fermion amplitudes also agree. No picture-changing manipulations are needed and the amplitude, when transformed to the half-plane, is the firstquantized result after eqs. (5.2) and (5.4) are used. 5.3. T H E D E C O U P L I N G OF SPURIOUS STATES FOR ON-SHELL AMPLITUDES

A crucial property of string physics is the decoupling of spurious states in on-shell physical amplitudes. In the modern formalism, spurious states are produced by

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O. Leehtenfeld, S. Samuel / Superstring

vertex operators, V, which are trivial in the BRST cohomology: V = [Q, l~] + [36]. Decoupling takes place in the Witten string field theory but only after Feynman diagrams are combined. Decoupling is closely related to associativity of the * product. The proof below works for the open bosonic string and the superstring field theories. The general on-shell physical four-point amplitude is A(a) = t r ( ) f ~ d r ( V l ( Z a ) V 2 ( Z 2 ) V 3 ( Z 3 ) b o V 4 ( Z 4 ) ) , Jo

(5.18)

where the V r are of Grassmann statistics for states in the bosonic string or in the Neveu-Schwarz sector of the superstring and are of commuting statistics in the Ramond sector. Consider the four-boson case. Assuming that V 4 is spurious V4(Z4) = [Q, 124(Z4)],

(5.19)

for some commuting 154 and that the other states satisfy, {Q, v r } = 0. Eq. (5.18) is A(a ) = tr( ) ( V I ( Z 1 ) V 2 ( Z 2 ) V 3 ( Z 3 ) I P 4 ( Z 4 ) ) I T = o ,

(5.20)

since L 0 = (b0, Q } can be written as a derivative with respect to T. The upper limit does not contribute for sufficiently large s and for other values of s it is zero by analytic continuation. The amplitude A(b ) is obtained by cycling the string indices A(b) = t r ( ) f ° ~ d T ( V 4 ( Z 1 ) V I ( Z 4 ) V 2 ( Z 3 ) b o V 3 ( Z 4 ) ) . a0

(5.21)

Using eq. (5.19) and the methods in sect. 3 A(b) = -- t r ( ) ( I ~ 4 ( Z 4 ) V I ( Z 1 ) V 2 ( Z 2 ) V 3 ( Z 3 ) ) I T = =-tr(

)(VI(Z1)V2(Zz)V3(Z3)IY'a(z4))Ir=o

o .

(5.22)

Hence A(a ) + A(b ) = 0,

(5.23)

when V 4 is spurious as in eq. (5.19). For the R - R - R - R amplitudes, the argument is similar except [Q, V r] = 0 and

v4(z.)=(Q,¢"(z4)}.

(5.24)

Eq. (5.20) remains the same but eq. (5.21) has a minus sign because the Feynman rules insert an extra minus for each interchange of a fermion line. Due to eq. (5.24), eq. (5.22) and hence eq. (5.23) are unchanged. Amplitudes for spurious states decouple from on-shell physical fermion amplitudes in the Witten string field

O. Lechtenfeld, S. Samuel / Superstring

283

theory. For R - R - R - R amplitudes we have used the fact that only term 1 in eq. (5.9) survives when states satisfy [Q, V r] = 0 and when the counterterm is included. In addition, the averaging procedure in eq. (5.9) guarantees that each combination of picture-changing operators in eq. (5.20) also appears in eq. (5.22). Without such a procedure, different operators appear in eqs. (5.20) and (5.22) and the cancellation in eq. (5.23) does not take place. 6. Conclusion

This paper has developed off-shell conformal techniques [26] for the supersymmetric string. The trilinear couplings and the massless four-fermion amplitude have been obtained for the Witten superstring field theory [7]. The trilinear couplings agree with those of the lagrangian of refs. [19-21]. Via picture-changing methods, agreement with the first-quantized approach has been proven for on-shell physical states. Finally, we have analyzed the associativity anomaly [62] in the N e v e u Schwarz sector of the superstring field theory. Explicit calculation has revealed that the four-vector amplitude is infinite in accordance with ref. [62]. By modifying the action and the gauge transformation, the order g2 violation of gauge invariance has been repaired. The modified theory has a finite massless four-vector amplitude which agrees with the standard dual-resonance model when the vectors are taken to be on-shell and physical. Its off-shell extension, however, is not uniquely determined by our methods, due to a BRST-trivial ambiguity. Further principles seem to be needed to fix the amplitude completely. There still remains an unresolved question concerning the covariant superstring field theory, namely, what happens to gauge invariance in higher orders. It seems likely that violations occur at order g3 and beyond although definitive calculations are needed to confirm this conjecture. The substitution of the order g gauge transformation in the order g2 counterterm and the order g2 gauge transformation in the order g action term (see eqs. (3.13)-(3.18), (3.22), and (3.23)) leads to a singular operator product of the picture-changing operator, X, with b e 2~' in the ~-~ 0 limit that most probably necessitates further modifications of the theory. Whether the additional interactions are uniquely determined is of utmost importance; otherwise one encounters a kind of non-renormalizability. In higher point amplitudes, colliding operators involving several X a n d / o r b e 2~ can potentially generate new infinities when the cutoff ~- is set to zero. To rigorously address this question requires the computation of five- and higher-point processes. Finally, what remains of the Chern-Simons structure? Can the modified theory be given a new geometrical interpretation? We hope the off-shell methods developed in this paper will assist those willing to tackle these challenging questions. This work was supported in part by the National Science Foundation grant NSF-PHY-82-15364 and by the Department of Energy grant DE-AC02-83ER40107.

O. Lechtenfeld, S. Samuel / Superstring

284

Appendix A THE GHOST PART OF THE FOUR-VECTOR AMPLITUDE

This appendix presents some details of the ghost contribution of eq. (4.11) in sect. 4. The matter part has already been discussed in eqs. (4.12)-(4.15). As in sect. 4, there are three types of ghost correlation functions, depending on the insertion, XoX2, X1X 1 and XzX o in the amplitude. Let us first concentrate on the contribution from fig. la, given in eq. (4.10). The ghost correlators are computed straightforwardly by using bosonization and free field theory contractions. The only complication consists of the contour integral due to the b 0 insertion. The simplest ghost correlator occurs for the 1 1 part of the amplitude

(fi

)c0

{ce e")( Zr)bo e*( iv )e e"(i3 )

r= 1

small

-

1 -

N

dz 2~ri

4

1 ~-Z2 q- V2)(Z2 q- 8 2)

-

1 H (Zr iv--iSr= 1 -

_

i v ) ( Z r - i8)

.

(A.1)

The other correlation functions are slightly more involved but straightforward, and create other contour integrals on the r.h.s, of eq. (A.1). In total, five different integrals appear ~dz

1

2

/1 ~--- c 2~ri ff(z 2 q- V2)(Z 2 q- 82)

dz 13 =

~C

1

i3- z

2 [

--8 K(V2) - - 3yE(v2)]

- - =

2vri < Z 2 q- V2)(Z 2 q- 8 2) i v -- Z

~dz

1

~c dz

1

(A.2)

= -,/7- v K { ~ 2 ) '

vr

3+

iv-z=2 v [K(v 2) _ 3 +eve (v 2)] ,

t2~= c2~ri ~(z 2 + v z ) ( z 2 + 8 2 ) i 3 - z

If=

i

2vri ~ ( z 2 - ] - V 2 ) ( z 2 - ] - 8 2) i v - z = ~

dz 1 1 I3~ = ~C 2vri ~(z 2 -I- V2)(Z 2 q- 82) i8 - - Z _

_

=

2

K(V2 )

2

1

82~v2E(v2)

~. {$2 _v2e( V2 ),

(A.3)

(a.4)

(A.5)

(A.6)

O. Lechtenfeld, S. Samuel / Superstring

285

where C denotes a contour encircling the branch cut from - i y to + i3'. The results are given in terms of the complete elliptic integrals of K and E [73]. Useful combinations are 2 12= I ] + I ~ = --[(7 + 8)K(Y 2) - S E ( v 2 ) ] ,

(A.7)

q7

13=I3 v - 1 3 ~ = -

2[

8

K ( 7 2) -- - - E ( v 2 ) ] . ~r 8-7

(A .8)

Here are some details which lead to eq. (4.11). We first concentrate on the 1-1 term; it involves 11 only. Multiplying the corresponding correlators and assembling the different pieces already yields most of eq. (4.11), after the use of eq. (4.17). The contribution of the local counterterm is

Ox-term=2(

~=a f i V~

= 8(''"

1)(Zr)~l~)l(i )

)X< , , , 51~(~2 + /~2 + 2)

(1

(3 - .,//)2 +

(A.9)

which is written as a total derivative in eq. (4.11). There still remains the (0-2 + 2-0)-part of A(4~, i.e. terms containing integrals different from 11. Adding to these the explicitly dimension-dependent term (7 try~,~ = D) in 1-1, one obtains oo

2

2g2tr(hlh2X3h4) f ° dT ~ ( . . - }x(... 1 X

(8

T) 4 [ - D l l + 5/21

(8

51~(O~2q_/~2q_]/2ff_(~2) 2

1[(1 1) (1 1) ]) .y)3

0(2 + 82 q- fi2 + 82

28IZ

O~2 _{_.y~ -}- 1~2 q_ .y~ 2VI2~ + I3

. (A.10)

With the help of eqs. (A.2)-(A.8) the curly bracket becomes

8 [sE(~ ~)

{... }~o~1o=-7[~-~74

K(y ~)

(~_~),

(v + 8)K(v ~) -sE(v ~) ] + (~- ~7 ( 7 ~ U i T r 7 7 ~) ], (A.11)

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O. Lechtenfeld, S. Samuel / Superstring

which can be recognized in the second line of eq. (4.11). Contrary to the open b o s o n i c string [8], the elliptic functions do not completely cancel between the correlation functions and the measure. T h e contribution of fig. l b is generated by permuting the indices of m o m e n t a and polarizations. This concludes the derivation of eq. (4.11).

Appendix B FINITENESS OF THE FOUR-VECTOR AMPLITUDE This appendix analyses the singularity structure of the massless four-vector amplitude, as given in eq. (4.11). Since any potential divergence comes from the integrand blowing up near T = 0, or ~, = 1, it is convenient to expand the complete integrand in eq. (4.11) in powers of e = 1 - -/ around ~ = 0. As a first step, use eq. (4.17) to transform back to dT, so that elliptic functions K and E appear. Since d T / d e = O(e), we have to study the new integrand up to O(e 2) or O(e 2 In e) to rule out divergences. With d T / d a given in ref. [8], a useful f o r m u l a in this context is

dy

2(/~ 2 q- "/2)(0/2 q" ~2)E(~/2)

2q-y2(fl2q-O¢ 2)

d~

~ v ~ ( 8 2 - v ~ ) ( , ~ - ~ ) K ( y ~)

OQt(/~ 2 - O~2 )

(B.1)

Let us start with the 1 - 1 part of eq. (4.11), which seems to contain poles of order two and three in ( ~ / - 3), and hence in e. However, a short calculation yields

R.R'=

(Or -- /~) 2 012 _}_/~2 q_ ~/2 ..}_32 ( k12 q- k34)

+

__

(~+t~) ~ (c~ - / ~ + iy + i~) 2k13 4/~ 2

(/~2 -b ~ 2 ) ( ~ 2 q_ (~2)

+

(,~+/~)~ (o~ - / ~ - iT - i3)2 k24 40t 2

k 14 __

= ~(kl 12 q_ k34) _ ~(kl 14 q_ k23)

(0/2 + ~2)(0t2 q_ (~2) -

k 23

i ( k 13 - k 24) + O(e) ,

( . . . ) 4 = ~... 50~+ o(~2),

(B.2)

(B.3)

with

( ' ' " ) 0 ~ ~ ~[EI[12E34,)_ l(E13E24) q_ ~[EI,"14~23'),

(B.4)

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O. Lechtenfeld, S. Samuel / Superstring

so that, when figs. la and lb are combined, R- R' ('")1+(3-~-)2

1234 ] 1 +perm'(2341] =0(7-) "

(B.5)

The subscript 0 indicates evaluation at T = 0. Similarly, \4~ ,qt*v= 2i O~2

<. -. , 2 , ~

=

8--'y~2

+ p7

+

~E~.81 [ 12834" __ 814E23 )

[82 812834__814823..{_0(82)

813/324

(0/,~2--+ /~)2-2 ] f l 2 (~/--8) 7 ~7 ~

,

(B.6)

which implies ( " " ) 2 + ( 8 7_/y~) 3 +perm"

1234] = O 2341 ]

- . 8

(B.7)

Being of order 1/8, the above terms are both integrable. The (0-2 + 2-0)-part of eq. (4.11) necessitates the expansion of the elliptic functions K and E [73], among others

1 2 +..,), ¥-3=28(1+½8+~8 + In

e2+

(B s)

...

(B.9)

,

(B.IO)

(,~2 +/~2 + v2+ a2)2=

--

N

=

(

4)

82 I + ~ E 2 + e 2 I n - + . . .

~e 2 1 n e

1-

+

...

,

(B.12)

'

4)(

4

K ( y 2) = ½1n - + 18 - 5 + I n -

E

8

(B.11)

+~se 2 - ~ + 3 1 n

4)

+O(e31, (B.13)

E(y2) =l+e

(

-l+ln-

4) E

-

-

~812+0(E3).

(B.14)

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O. Lechtenfeld, S. Samuel / Superstring

T o a c c o u n t for t h e m e a s u r e , we n e e d

dT de

- - ~

~ r e ( l + 3 e + T 3e2 + . . . ) .

Adding figs. l a and lb one finds (

"'"

~

>l{(a)+(b)=

(

(B.15)

4)

2 ( . . . ) o ~ 1 - ~~E 2 - - ½ e 2 1 ne _ + . . .

[

(B.16)

'

4 ( kr" kr) -~- ]

< . . . >Xl(a)+(b)= 2 ( ' ' " >X 1 -- ~6~re2 Z

0(e3)

(B.17)

,

r=l

where

( (

X=~o 1 - ~ e 2 v - l - 2 1 n e

4)

+O(e3)

]

(B.18)

has been used. Putting together the above expansions and adding both counterterms leads to A4B(a)+ (b) = 8g2 tr(hih2X3h4)(... ) x < . . . )o*

×

{Zl[16 de' -~;y+ ~;y +-~ E (kr'kr) +O(lne')

+

r=l

--

+O(1)

1/

~"

+ (O2-term) 4

= f i n i t e - 47rg 2 tr( )(ln

e)(... )x,¢ E ( kr" kr) + (O2-term),

(B.19)

r=l

which is the result in sect. 4. As argued in sect. 5, the O2-term cancels the In e divergence. Before combining all contributions there were In2 e terms present, and In e terms had rational as well as transcendental coefficients (see eq. (B.18)). In eq. (B.19) we are only left with a In e divergence with a purely transcendental coefficient. We do not know whether these surprising cancellations are particular to the four-vector amplitude or are a general feature.

References [1] J.H. Schwarz, Phys. Rep. 89 (1982) 223; Scherk, Rev. Mod. Phys. 47 (1975) 123

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