Off-shell string physics from conformal field theory

Nuclear Physics B308 (1988) 317-360 North-Holland, Amsterdam

OFF-SHELL STRING PHYSICS FROM CONFORMAL FIELD THEORY Stuart SAMUEL

Physics Department, City College of New York, New York, NY 10031, USA Received 8 February 1988

The subject of off-shell conformal field theory is developed and is used to produce the off-shell amplitudes of string field theory. The method is also applied to the reparametrization ghosts.

1. Introduction 1.1. INTRODUCTORY REMARKS

If string theory has physical relevance one thing is certain; the vacuum cannot be in its perturbative state. Indeed, the usual perturbative string vacuums describe flat 26- or 10-dimensional universes with large unbroken gauge groups such as 0(32) and E s × E s [1,2]. The dilaton expectation value, an important quantity in determining the strength of the coupling constant [3], must be analyzed non-perturbatively in supersymmetric string theories [4]. In short, it is important to develop non-perturbative methods so that one can see whether the true vacuum leads to spontaneous compactification, gauge-group symmetry breaking and a realistic spectrum of particle masses [5, 6]. One approach to non-perturbative physics is through field theory. During the 1970's and 1980's a plethora of non-perturbative techniques and phenomena have arisen via functional integrals: solitons, monopoles, instantons, the lattice approach with its strong coupling and numerical methods, etc. With a string field theory, one might be able to address questions concerning vacuum structure. A light-cone string field theory was achieved many years ago [7-10], but for purposes of vacuum structure a covariant approach is preferable. Covariant string field theory was developed by Witten [11] using elegant mathematical and geometrical principles. The free action is of the form f(x/', Qq,) [11,12], where g' is the string field and Q is the first-quantized BRST charge. Motivated by the fact that Q2= 0 resembles d 2= 0 where d is the exterior derivative, Witten analogized that string field theory was similar to the theory of differential forms. 0550-3213/88/$03.50©E1sevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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The field, if', is like a Lie-algebra valued one-form and Q is like d, and the analogy is made complete when the string f and * are introduced; they are counterparts of the integral and (matrix-) wedge product. The degree of a form is its b-c ghost number. In conformal field theory language, the ghost numbers of Q, q', f and * are respectively, 1, 1, - 3 and 0. To introduce interactions, the free action, fxO.Q~, is extended to the Chern-Simons form, f ( g ' * Q ~ + ~gq" * g' * g'), and the gauge transformation of the free action, 6 q = QA, is modified to 8 g ' = QA + g ( q ' * A - A * g') [11]. The basic axioms, i.e. the nilpotency of Q, the associativity of *, the graded distributive property of Q across *, the integral of an exact form being zero and the graded commutativity of * under the integral sign, are sufficient to ensure the gauge invariance of f(g" * Q q + 2xr"* g' * q). The non-commutative * product describes how strings interact. Strings are 1 The term, f(q" * q * g'), is the divided into two halves at the midpoint, o = ur. delta-function overlap of the first half of one string with the second half of the next, for each of the three strings. For more on covariant string field theory see refs. [11,131. To begin perturbation theory, the gauge must be fixed [11,13-17]. A useful choice is the Siegel-Feynman gauge [18], bog"= 0, where b0 is the zero mode of the conformal antighost, b(z). The propagator is a'bo/Lo=a'bof~drexp(-rLo), where L 0 is the zero-mode Virasoro generator. A Feynman graph can then be associated with a first-quantized world sheet [14,15]. External strings are represented as semi-infinite rectangular strips of width ~r. The propagators are internal strips of length T (also of width 7r) since this is the effect of the operator e x p ( - TLo). Finally, the interaction glues the ends of strips together in the manner described in the previous paragraph. An example, fig. la of ref. [14], is redisplayed in fig. 1. Using the above, Giddings computed the four-tachyon scattering ampfitude on-shell [14] and found agreement with the Veneziano formula [19]. Off-shell three-point couplings were obtained in refs. [20-25] by computing the three-point vertex function. It describes, in a precise mathematical way, how three strings overlap. When a mode expansion of a string is performed, the particle content is revealed. Since, in nature, particles, as opposed to strings, are observed, it is important to express string physics in terms of these degrees of freedom. Vertex functions do this since they embody in one object the interactions in the mode representation. Recently, some results, not previously obtained from a first-quantized approach, have been derived from the Witten string field theory. Ref. [26] computed off-shell four-point tachyon and vector amplitudes as well as the four-point vertex function. The scattering of second-quantized ghosts and auxiliary fields was achieved in ref. [27]. Using ref. [26], ref. [28] numerically calculated with a computer the four-point contribution to the static tachyon potential in an effort to investigate the vacuum

S. Samuel / Stringphysics "'--.....

319

"--._. _ "'--. __

--- ___,

"" " "'"" ---.

~ l ............

(

T

Fig. 1. The string configurationregion for the scatteringof four states in the Witten field theory.

structure of the open bosonic string. An off-shell approach is essential since the momentum squared of the tachyons is evaluated at 0 instead of 1/a'. Let us now turn to the first-quantized approach. It begins with the Nambu or Polyakov action. They are both reparametrization invariant and can be gauge-fixed to the following

1 fdod.r(O,X~O,X~(o,T)+OoX, A = 4~ra'

OoX,(o,.:)}

1 fdod,:(b(o,~)(O,-iOo)~(o,.:)+b(o,,:)(a,+iao)c(o,r)}

+2--~

(1.1)

which, using z = exp(T + io) and £ = exp(~ - ia) and field redefinitions, is

1 A = 8~ra' 1 + 4~

fdzd~O=X.O~X.(z,~)

fdzd~(b(z,~)O=a(z,~)+b(z,~)O~c(z,~)}

(1.2)

The ghosts, c and g, and the antighosts, b and b, arise in the Faddeev-Popov gauge-fixing procedure [29, 30]. The action in eq. (1.2) is invariant under conformal transformations, z ~ f ( z ) . Since the conformal group in two-dimensions is infinite-dimensional, conformal symmetry is particularly powerful. It and its supersymmetric extension have been developed [31-42] and used extensively in string theory [38-45]. Exemplifying its

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power are the computation of the six-point fermion amplitude in covariant form [43] and the general m-boson 2n-fermion tree amplitudes [45] in the Cartan-Weyl basis of the Lorentz group. Conformal field theory is the principal calculational tool of the first-quantized approach. Open string boundary conditions eliminate half the matter and ghost degrees of freedom. It is sufficient to drop ~ and b and use only c and b. Aside from zero-mode effects in X~'(o, ¢), dosed string amplitudes factorize into left-moving and right-moving contributions, each of which resembles an open string result. For this reason, it is sufficient to discuss open strings only and to save space, we do so. After quantization, the fields can be taken to be functions of z only. Finally, we work in the critical dimension D = 26 to avoid the conformal anomaly. The reparametrization ghosts, b and c, and their partners, fl and ~, (for supersymmetric string theories) play an important role in both the first-quantized and second-quantized formalisms. The string field is a functional of first-quantized degrees of freedom and thus a functional of the ghosts as well as matter fields. The additional degrees of freedom in q due to ghosts lead to auxiliary fields [12, 46-48] when ~/" is expanded in terms of particle fields. For the first-quantized theory, the ghosts generate the right measure factors in amplitudes. Although they do not couple to the X~'(z) in eq. (1.2), they are sensitive to the shape of the world sheet. They assure that the measure transforms properly in going from one world-sheet description to another (say from the half-plane to the disk). Anticommuting fields in two dimensions can be bosonized. For c(z) and b(z)

c(z) = e x p ( o ) ( z ) ,

b(z) = e x p ( - o ) < z ) ,

(1.3)

where o(z) is an ordinary scalar field. In the supersymmetric case, bosonization has been particularly useful [38-45,49-59] when applied to the fields g'~(z), fl(z) and ~,(z) and in the search [60-61] for a covariant analogue [38,39,45] of the Green-Schwarz light-cone spin field, S~ [62-64]. Due to the power of conformal field theory, the first-quantized approach is better adapted for computing amplitudes. The disadvantage of it is that it is an on-shell formalism. Since the vacuum structure of string theory may require an off-shell approach, one is naturally lead to investigate whether it is possible for conformal field theory to compute the off-shell quantities of a string field theory. One is searching for the best of both worlds, an off-shell formalism and a powerful means of computation. This is the subject of this paper. This concludes the introductory remarks. The reader may wish to supplement these brief discussions with the following. For string field theory, there are the introductions of refs. [11,20], sects. 1 and 2 of refs. [13,16], and the talks and reviews in refs. [65-67]. For conformal field theory, sects. 1 and 2 of ref. [40] as well as the reviews [68-70] are useful. An introduction of bosonization and spin fields can be found in sect. 1 of ref. [45].

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1.2. OFF-SHELL CONFORMAL FIELD THEORY

In the past, conformal field theory has been used to obtain on-shell results. We use the term off-shell conformal field theory to mean the application to the situation in which states do not necessarily satisfy k 2= - m 2. Before making more precise the notion off-shell conformal field theory, let us discuss some aspects of the on-shell case. To be specific, consider the computation of a tree-level diagram. One calculates a correlation function of vertex operators on a region, R, which can be taken to be the half-plane. The vertex operators are conformally i n v a r i a n t - for example, the integral of a weight-one operator. This imposes the mass-shell condition. To see this, note that a vertex operator contains exp(k. X ) ( z ) , where k is the momentum of the state. Since the contribution of e x p ( k - X ) ( z ) to the conformal weight is ct'k 2, only for a particular value of ct'k 2 does the operator have weight one. From the functional-integral viewpoint, the computation of an amplitude involves a conformally invariant measure (for D = 26) and integrand (the action and the vertex operators). As a consequence, conformal transformations leave the result unchanged. There are two types of conformal transformations: (a) those that map R into itself and (b) those that map R into a new region. For the half-plane, type (a) are the SL(2, R) transformations. The ghost and matter correlation functions change under SL(2,R) but the two appropriately compensate for each other. Type (a) are symmetries of the calculation. Modular transformations in higher genus closed string theories are further examples of type (a) transformations. The invariance under type (b) means that the domain used to compute an amplitude is immaterial. The infinite rectangular strip, the disk, the half-plane, etc. can all be used to calculate open string tree amplitudes. When the mass-shell conditions are not imposed, the vertex operators change under conformal transformations. Amplitudes are no longer necessarily invariant. Actually, they turn out to be invariant under type (a) transformations*, but not so under type (b). Consequently a surface, R, must be chosen. If R comes from a string field theory, the result from off-shell conformal field theory coincides with the field-theoretic result. Of course, on-shell computations on all conformally equivalent R agree. Summarizing, there are two ingredients in off-shell conformal field theory: the calculational method and the selection of R. Sect. 6 presents a further discussion of the latter topic. 1.3. METHODOLOGY

Let us describe the basic computational ideas. The surface, R, must be specified in some way. We do this by assuming there exists a map, p(z), from a standard * This follows from the SL(2, R) invariance of the vertex function (which requires only momentum conservation) and the equivalence of off-shell conformal field theory methods with the vertex-function approach (which is established in sect. 4 and appendix C).

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region, taken to be the half-plane, H, onto R. As z varies over H, p ( z ) varies over R. All the information about R is encoded in p ( z ) . The action in eq. (1.2) is quadratic in fields, so one is presented with a free field theory. Normally, in such a situation, it is straightforward to compute a correlation function. The calculational difficulties reside in R which may be of complicated shape (see fig. 1, for example). If the propagators on R are known the problem is solved. Instead of computing directly on R, perform the inverse transformation back to H. The correlation function changes, since it is no longer conformally invariant, but the Jacobian factors are deducible from p ( z ) . Since H is a simple surface, the propagators on H are known and the calculation is performable. Since mapping techniques have been developed [8], the above program is easily implemented, except for two tricky points. Vertex operators involve products of fields at the same point. For example, in the massless vector vertex operator, e . O X e x p ( k . X ) ( z ) (e ~ is the polarization vector and k s is the momentum), O X ~ multiplies e x p ( k . X), and exp(k. X), when expanded, exp(k. X) = 1 + k . X + ½ k . X k . X + • • • , involves products. Such operators are singular and require regularization. The on-shell tree-level formalism normal orders them. For an exponential, this is equivalent to introducing a cut-off, regularizing the operator and taking a limit as the cut-off is removed. For ordinary products, normal ordering is carried out by point splitting and subtraction [40]. The first subtle point is that, on an arbitrary R, the same regularization prescription must be used*. Consequently, all self-contractions are not eliminated. If :O: denotes the standard regularization procedure for an operator O and o° O o° denotes normal ordering on R, i.e., no self-contractions are permitted, :O: :g ~ O o° . This topic is discussed in more detail in subsect. 3.3. The second subtlety is as follows. Under a change of variable, z ~ z', a conformal operator, O, transforms. If O is a vertex operator on-shell, this is unimportant, because the calculational scheme is conformally invariant. For vertex operators off-shell, the value of an amplitude depends on whether O is a function of z or z'. This means a choice of argument variables must be made. Naively, one might use O ( p ) , where p is the above-mentioned map variable; this is incorrect. The argument of O is the exponential of what is called a string variable [8] and which string variable to use depends on the location of the operator. This is discussed in subsect. 2.1. 1.4. CONTENTS

The present work strives to provide explicit off-shell calculational methods. Throughout this paper, the abstract rules and descriptions are supplemented with concrete examples. Sect. 2 presents the procedural rules and sect. 3 provides the computational techniques along with examples. Sect. 4 shows that, when R is a * This was pointed out, for example, in ref. [57] in computations of higher loop correlation functions on exotic topologies such as RP 2, the M/Sbius strip, the Klein bottle, etc.

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surface generated from a string field theory, the results from off-shell conformal field theory and the string field theory agree. The detailed proof is relegated to appendix C. In carrying this out, we need the correspondence between string states and vertex operators. This is given in subsect. 4.1. Subsect. 4.3. recasts the string field-theoretic vertex function in conformal field theory notation. Sect. 5 calculates the b-c ghost contribution and the off-shell scattering of second-quantized Faddeev-Popov ghosts and auxiliary fields. The results are checked against those obtained in ref. [22]. The matter and ghost contributions in sects. 3 and 5 are assembled in subsect. 6.2 to obtain off-shell amplitudes. They agree with refs. [26, 27]. Although only up to 4-point processes are considered, the methods are generic and applicable to higher point amplitudes. Appendix A presents the Neumann-function components through level four for an arbitrary tree-level diagram and appendix B provides series expansions, useful for the examples in sects. 3-5. Sect. 7 is the conclusion. 1.5. NOTATION, CONVENTIONS AND NECESSARY BACKGROUND We follow the notation and conventions of refs. [14,26,45,57,71]. The slope parameter, a', is 1 so that the zero mode of OX~(z), ~ , is p". The SL(2, R) invariant vacuum [40] is denoted by 1~2>. The q-vacuums, where q is the b-c ghost number, are indicated by I + q>, the number, q, always being exhibited to avoid confusion with the ghost vacuums I - ) and l + > [29]. Hence, lI2)= I + 0 ) , ] - ) = c1112) = I + 1), and ] + ) = c0c11~2) = I + 2>. The indices r, s, t, etc. denote string labels and m, n, etc. denote mode numbers. Variables such as ~0, ~0', ~0i, etc. are the arguments of operators on R and correspond, via the map, O, to points, z, z', z i, etc., on H. The reader should distinguish the following two terms. A "vertex operator" produces a state in the first-quantized theory. A "vertex function" is a Fock-space representation of an amplitude. It appears in both the first- and second-quantized approaches. For overall reviews of string theory, see, for example, refs. [6, 64, 72-74]. To fully understand this paper, it is useful to know the material in the following works: ref. [40] for conformal field theory (operator product methods, amplitude calculations, b-c ghosts and bosonization), ref. [11] for covariant string field theory, refs. [14, 26] for string field-theoretic amplitude calculations, refs. [8, 9] for vertex functions and ref. [8] for mapping methods. 2. The rules

The goal is to compute the functional integral over the action in eq. (1.2) for a region, R, which we call the string configuration. When appropriate vertex operators are inserted, the result is a particular value of the integrand of a string scattering amplitude. For physical on-shell vertex operators, the result will coincide with the Koba-Nielsen integrand, after the transformation to the half-plane is carried out. Off-shell, the two differ.

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•.f2:i:::

9]

~f~f ~ S ~ ~ S ~

iii~iiiiii~iiiii~i~ii~ii~i'~-~ii~i' Iiiii~if ~-~iiiiiiiiiiiiiiill

iiiii i !ll i Zp

.....

Z3

Z2

Z1

(b)

Fig. 2. (a) The generic string configuration. (b) The upper-half z-plane.

We restrict R so that each state in the amplitude is identifiable with a rectangular strip, one end of which goes to infinity. Fig. 2a is an example of such a region. For the r t h state, this strip and the "point at infinity" are respectively called the rth leg and asymptotic position. Each strip terminates at a point which is called its interaction time, labelled in fig. 2a as 11, 12 . . . . . Ip for a p-point amplitude. An arrow is assigned across each leg in a clockwise direction. Except for the external legs, let the shape of R be arbitrary. Examples of suitable R appearing in the literature are fig. 1 of ref. [8], figs. 1 and 5 of ref. [9], fig. 1 of ref. [14] and fig. 2 of ref. [75]. 2.1. T H E P R O C E D U R E

(1) Specify the region by a map, p(z), from the half-plane to the string configuration plane. In other words, R is the image of p(z), as z ranges over the half-plane, H. Let z = Z r in H be the asymptotic position of the rth leg. These Zr are located on the real axis of H (see fig. 2b). (2) Define string variables, ~r(z), on the rth leg. The variable ~r(z) is of the form a p ( z ) + b where a and b are constants adjusted so that (fi) the real part of ~r(z) is

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zero at the interaction point and (b) the imaginary part of ~r(z), Im(~'r(z)), is 0 at one edge and ~r at the other edge. The arrows in fig. 2a indicate the direction of increasing Im(~r(z)). (3) Determine the asymptotic behavior of the string variables. Because ~r(z) jumps by i~r at the asymptotic position, it is necessarily of the form, f'(z)-~ ln(z - Z,) + . . . , for z near Z,. Compute the Taylor series of ~r(z) -- ln(z -- Z,) about Zr ~r(z) = l n ( z -- Zr) - N ~ r - ~,r(z), (2.1) oo

Y~'(z) = Y', y ~ ( z - Z , ) ' ,

(2.2)

m=l

where N0~~ and 7r are constants that can be determined from the map, p(z). (4) Insert operators, c(co)W(co), at asymptotic positions. The vertex operator, V~(co), which is a functional of X~(z), produces the asymptotic state. The role of c(co) is discussed in subsect. 4.1. Note: it is important to use the same vertex operator as in the usually on-shell approach, including the same normal-ordering

prescription. (5) Use for the variable, co, in step 4 co( z ) = exp(~ r( z )),

if the operator is on the r th leg.

(2.3)

Comments on (1)-(5). From eq. (2.1), ~ r ( z ) ~ - - O 0 as z ~ Z,. Eq. (2.3) then implies co(z) ---, 0,

as z ~ Z,.

(2.4)

Step (2) is exemplified in many places in the literature [8, 9, 22, 76]. Examples of V~(z) are the tachyon and vector vertex operators, exp(U.X(z)) and e'. X(z)exp(k ~. X(z)), where k ~ is the momentum and e~ is the polarization vector. The convention of ref. [40] is adopted in that products of operators at the same point imply the standard normal-ordering prescription; hence, e x p ( U . X(z)) is really :exp(k r. X(z)):. Examples of standard normal-ordering prescriptions are

OX~OX(zl) = z-~z,fim OX~(z) OX°(zO - (z

bC(ZI) = Z"~zllim b(z)c(z~)

{

(z

-

z1) 2 J'

z~) '

bcc(zl) = z'-~z,limz-~z,lim b ( z ' ) c ( z ) c ( z 0

c(zl)

(z'-z-~--~ + ( ~ ' ~ z , )

}. (2.5)

The normal ordering of exp(U-X(co)) is discussed in subsect. 3.3. The crucial ingredient in making off-shell conformal field theory work is step (5).

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2.2. T H E S T R I N G F I E L D T H E O R Y V E R T E X F U N C T I O N

It is possible to define a vertex function associated with the string configuration, R, using the quantities Z~ and ~r(z) defined above. When R corresponds to a string field theory diagram, that is, a Feynman graph generated by vertices and propagators, the vertex function is the one for the string field theory. For the X~(z)= 32,a~z -"-1, Fock spaces are introduced for each string along with their vacuums ~
x
(n

r< k r

~r=l

,)exp /

amr N.~.a. ,

½ \

r=l s=l

(2.6)

n=O

m=O

with the momentum constraint Er, k r~ = O. The Neumann-function components are [8, 9]

N~=ln[Z~-Z,[,

r4:s,

rnN'~°= ~ dz (~ dz'

!

z2~ri2"z2~ri ( z

rnnN,~, rs = ~ d z ~

dz'

z') 1

zr2~ri z2~ri ( z - - z ' ~

(2.7)

d~S(Z')exp(_m~r(z)) ' dz'

(2.8)

e x p ( - m ~ r ( z ) - n~S(z'))

(2.9)

and the diagonal zero-zero components are determined from eq. (2.1). Since eqs. (2.8) and (2.9) are contour integrals, they can be easily evaluated using Cauchy's theorem to yield an algebraic solution. For example, eq. (2.8) is

mNrS°

mN%-

exp(mN0or) Ore-1 { exp(my"r(z)) } z=zr (m - 1)! Oz "-1 z----~ , exp (mN°°r) om

rn!

Ozm ( e x p ( m Z ' ( z ) ) )

z=zr"

r 4: s,

(2 .lO)

Algebraic solutions for the N£',, for m >/1 and n >~ 1, are given in eq. (4.10). Examples of this procedure to obtain the vertex functions of a string field theory are given in refs. [8, 9, 22, 26]. The analog of eq. (2.6) for the conformal ghost and antighost, c(z)= 32,cnz-n+a and b( z ) = ~,, bnz -~- 2, is

(

b¢
r=l

.... c,,X~;,b~ n=O

s=i m=l

(2.11)

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where /~ and xr~ are R-dependent constants. Formulae for /~ and X ~ presented in sect. 5.

are

3. The Evaluation Procedure

Since the action in eq. (1.2) is quadratic in X~(z), free field theory methods may be used. One exploits conformal invariance by transforming to the half-plane where propagators are known. The computation is carried out by calculating the conformal transformation factors, by expanding about the asymptotic positions, and by using Wick's method to evaluate correlation functions. 3.1. THE EVALUATION PROCEDURE

Implementing the above leads to the following three-step procedure. (i) Perform a conformal transformation from R in fig. 2a to H in fig. 2b. Under this, primary conformal fields, O, transform as [33, 40] O(co) ~

O(z),

(3.1)

where h is the conformal dimension of O. An ancestor field transforms in a way determined by its parent primary field. Examples of primary conformal fields and their dimensions are field

dimension

OX ~ exp(k- X) b

1 ½k2 2

c

--1

exp(qo)

½q( q + Q ) ,

(with Q = - 3).

(3.2)

Examples of ancestor fields are derivatives of primary fields such as 02X(co), which transforms as 0 2 X ( c o ) ~ ( O z / O c o ) ( O / O z ) [ ( O z / O c o ) ( O / O z ) X ( z ) ] = ( OZ/OCO)2 0 2 X ( z ) q- ( 0 / 0 z ( 0 z / 0 c o ) ) ( 0 z / 0 c o ) OX(z). When a field is the product of two other operators, its transformation law is deduced by regularizing the product, i.e., one point-splits the two operators and subtracts off the singular parts (see eq. (2.5)). After the two are transformed separately, the limit in which their arguments coincide is taken. (ii) The operators appearing in a correlation are functions of the variable, co, defined in eq. (2.3). The Jacobian factor, Oz/Oco, in eq. (3.1) is

Oz ( O~d = k

) -1 Oz

exp(-fr(z))'

(3.3)

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328

and the derivative, a/0o~, is 0

az _

0~o

O exp(-~"(z))/

O~o Oz

~

a~r(z))-10q Oz

(3.4)

Oz'

when ~ is on the rth leg. It is useful to series expand eqs. (3.3) and (3.4) in terms of z - Z ~ via eqs. (3.1) and (3.2) since the operators eventually are evaluated at the asymptotic positions, Z~. The low-order expansion of eq. (3.3) in (z - Z~) is given in eq. (B.4). (iii) Evaluate the transformed correlation function using Wick's method and the following propagators

X q z ' ) = ~ " l n ( z - z'),

for Izl > Iz'l,

(3.5)

b(z)

c(z') - ( z - z') '

for Izl > Iz'l,

(3.6)

o(z)

o(z') = l n ( z - z ' ) ,

for

Izl > Iz'l.

(3.7)

3.2. EXAMPLES Here are some illustrative examples lim

lim

(OX"(o~)OX"(o~'))R

= lim

-a,,,' -

lim

OX~(z) OX"(z'))H

= exp(No~r + N ~ ) ( Z ~ - Zs) 2'

(3.8)

where ~0 = exp(~'r(z)) and ~o' = exp(~S(z')). The last equality follows from eq. (3.5) and

Oz

lim ff-dw= exp(U~r)' z z, which follows from eq. (B.4).

(3.9)

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A slightly more complicated example is lim

lim (O2X"(w) OX~(w'))R

z--.z~z'-,z,

.[ Oz

1 (z - Z s ) 2 }

= Oliraz.08 ~ '-~ e U-, ;&o ; /~- - ) () O )z{ (

23,f

= ~t~ve2N~+N~

2

1"

(3.10)

(z-zs) eq. (B.4) is useful in evaluating eq. (3.10). For a p-point amplitude there are p exp(k. X)(z) factors in the correlation function. Omnipresent as a term in an amplitude is lim

(z,--,zi)

( e x p ( k 1 . X ( ~ , ) ) e x p ( k 2. X(w2))...exp(k p. X(wp))) R

tim

{zi'*Zi}

i=l

~

I-IIz i - zsl kjk'

)

k'. N~k s . (3.11)

= exp

i
r~l

s= l

The first equality follows from eqs. (3.1), (3.2) and (3.5) and the second equality is true because of (3.9) and because Ndd in eq. (2.7) is the propagator (eq. (3.5)) when z and z' are evaluated at Z~ and Z,. 3.3. S E L F - E N E R G Y EFFECTS

The N0~r factors in eq. (3.11) are attributable to normal-ordering effects; the standard normal-ordering procedure does not eliminate all self-contractions. This is because the propagator on H is different from that on R. To see how the N~or arise in eq. (3.11), adopt the following regularization procedure for e x p ( k - X ) ( z ) Finj= 1 oo exp((

k/n )" X( zj) ) o exp(k. X(z))= ,,--,~lim(z,I-.zlimexp(½(k2/n2)~]~ilnlzi_ zjl)' o

(3.12)

where o° o° indicates strong normal ordering, that is, no self-contractions are permitted for operators within the "~". Since the same regularization procedure must be used on the string configuration lim exp(k.X(w))= lim lim rI']=~°exp((k/n)'X(*°J))°° . (3.13) z--. zr n--,+ {zi}--' zr exp(½( kZ/n 2)y'.]. ilnl ~o,- wjl )

330

S. S a m u e l /

String physics

N e x t , use

(X(~j)X(toi)}=lnlzj-zi[

= ln(exp(N0~)[ ~0j - ~oi[),

(3.14)

the latter approximation being valid in the limit in which all z, -~ zr, since via eqs. (2.1)-(2.3)

( z i - Z~ ) = ~oiexp( N~o~)exp( E~( zi ) ) ,

(3.15)

and ~r(z) ~ 0 as z ~ Zr. Eq. (3.15) becomes 1 2 rr o o :exp(k-X)(~o): = exp(~k No~)oexp(k-X)(o~) o,

(3.16)

when ~0 is on the rth leg. Eq. (3.16) remains unchanged when other regularization procedures are used. The above techniques are applicable to any product operator. Further examples are 1 2 rr o :e. OXexp(k. X(to)): = exp(Tk N0~ )(oe" OXexp(k. X)(o~)

+e.kT[eN£exp(k . X ) ( ~ ) ° ) ,

(3.17)

ax(,o): = °°El " 0 X ( ° ) ) ~ e 2 "

OS(~)

° '1 'E'I"

%exp(2N0or)(Y£ + -1[~ 'r~212, 11 J ,

(3.18)

when ~0 is on the r th leg.

3.4. U S I N G W I C K ' S M E T H O D

As stated above, we are doing free field theory on an exotic surface. Consequently, correlation functions can be computed by Wick's method. In all possible ways, one draws contraction lines between operators, and then one sums the corresponding weights. Associated with a contraction line is a propagator (or derivatives of it if derivatives are present). In addition, there are the conformal factors in performing the map to the half-plane. Using the techniques given in sects. 3.2 and 3.3, it is straightforward to specify these weight factors. There is always a factor of eq. (3.11) in an amplitude. When other operators are present, multiple contractions with the exponents are permitted. This is well known and some examples can be found, for example, in sect. 4 of ref. [45]. From eqs. (3.5), (3.8) and (3.9), the following are associated with the indicated

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331

contractions

lim lim e. OX(to)

lim

l i m e . OX(~)

e'.OX(to')

(3.19)

exp(N or+N )

e.k e x p ( k . X)(to') ~ exp(N0~r)

(3.20)

when ~o and to' are on the rth and sth legs (r 4: s) and when to and to' are both on the same r th leg,

lim el. 0X(to)E2" 0X(to )

z Zr

[

el.e2exp(2N or)(7

+ ½(,/()2),

(3.21)

1

l i m e . OX(to) e x p ( k - X ) ( t o ) ~ 71rexp(N0~r)e • k.

(3.22)

To illustrate the method

~e1. OXexp(k 1. X)(o~l)exp(k 2- X)(to2)exp(k 3. X)(to3)e 4. OXexp(k 4. X)(b)4)) R

=

E1" E4 q-

El "

kr

~_, N14~e4. k s

× eq. (3.11),

(3.23)

s=l

where, N4~ is the factor in eq. (3.19) when r = 4 and s -- 1, and N(~) is the factor in eq. (3.20) (r 4: s) or in eq. (3.22) (r = s). The term N141 arises when e4. OX(to4) and e1- OX(tol) contract with each other, whereas, the second term arises when e40X(to4) and e1. OX(tol) contract with an exp(k. X)(to) operator. Once the basic contraction factors are determined, amplitudes are quickly calculated by Wick's theorem. It is not necessary to repeat the steps given and illustrated in subsects. 3.1-3.3. In short, this subsection provides efficient techniques for obtaining off-shell results.

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4. The equivalence of off-shell conformal field theory and string field theory In the last paragraph, we gave names, N(~ and NlrZ, to certain contraction factors, which coincide with the symbols used for the Neumann components in the vertex function of eq. (2.6). We did this on purpose. They are, in fact, the same, as one can see by comparing the definitions in the last paragraph to the computations in appendix A. Appendix A contains the N,~, as computed from the integral formulae in eqs. (2.8) and (2.9), to level 4. The level number of Nr~, is m + n. The purpose of this section is to discuss the equivalence between results from off-shell conformal field theory and results from the vertex function. If the vertex function originates from a string field theory, then a link is established between off-shell conformal field theory and string field theory. The equivalence is shown by (a) establishing the correspondence between Fockspace states and vertex operators and by (b) demonstrating that the off-shell correlation functions are equal to vertex-function matrix elements. Subsect. 4.1. treats part (a). Part (b) has been verified by explicit calculation through the second level and spot-checked at levels 3 and 4. To save space, subsect. 4.2 presents only a few examples. We have, however, rigorously proven (b) and the proof is given in appendix C.

4.1. T H E

CORRESPONDENCE

BETWEEN

STATES

The connection between vertex operators and states is well known and can be found, for instance, in ref. [40]. Generally speaking, if O(z) is a vertex operator, then limz_,00(z)10 ) produces a state in oscillator form. For example, O(z)= exp(k. X)(z) produces an eigenstate of momentum k. The standard normal-operating procedure on exp(k. X)(z), places the creating operators to the left of destruction operators (see, for example, eq. (A.7) of ref. [71]) and hence lim exp(k-X)(z)lO ) = exp(ik, x)lO ) = Ik),

(4.1)

z -"* 0

where x is the position coordinate. The presence of OmX~(z)/(m- 1)! inserts an a~- m since

omX~(z) lim (~_--i3 i I0) = o,"_,,10).

z"* 0

(4.2)

By including products of OmX"(z) along with exp(k-X)(z), an arbitrary oscillator state can be created. Sect. 5 needs states involving ghost oscillators. The same procedure is used. The SL(2,R) invariant vacuum, lI2) satisfies, emil2)= 0 for m >/2 and broil2) = 0 for m >/ - 1 [40]. The expansions for c(z) and b(z) are c(z)= E,c,z -"+1 and b(z)=

S. Samuel / String physics

333

~nbnz - n - 2, s o that

lim [12) = c ,,[12), z- 0 (m + 1)! -

(m >/ - 1 )

(4.3)

O'-2b(z) lim 112) = b m112) z--,o ( m - 2)! '

(m >/2).

(4.4)

Actually, sect. 5 uses the bosonized form of ghost states. This is obtained from eq. (1.3), i.e., by substituting exp(o)(z) for c(z) and e x p ( - o ) ( z ) for b(z). For more discussion on the relation between the two representations, see, for example, ref. [16] or appendix B of ref. [71]. The connection between the operators in a correlation function and the Fock-space states is governed by the following. Rule: If the oscillator representation on the rth string is generated by lim z ~00(z)112), then in the correlation function use lim z _~zrO(~(z))The variable to has the feature of mimicking the z ~ 0 limit since ~ ~ 0 as z ~ Z~ (eq. (2.4)). To produce the usual physical states, O(~0) is of the form c(~o)V(oJ) where V(~) is a standard vertex operator depending on the matter fields, X~(~0), but not on the ghosts. The role of c(w) is to convert the SL(2, R) invariant vacuum, 112), into the string vacuum, 1-)~-~c1112 ) =lim~oexp(o)(z)112 ) [40]. It has ghost-number + 1.

4.2. ILLUSTRATION OF THE EQUIVALENCE

The simplest vertex-function matrix element occurs when each Fock-space state is

Ikr)r

.

)

x(VPl I-I Ik')s = exp

k ~- N ~ k s . \

s=l

(4.5)

r,s=l

where x(WPl is given in eq. (2.6). Since I k r ) ~ l i m ~ z r e x p ( k r . X ) ( o ~ ( z ) ) , the corresponding conformal field-theoretic correlation function is the one given in eq. (3.11). Eqs. (3.11) and (4.5) agree. Suppose one particular (the r th) Fock-space state is er. a L l l k r ) r instead of [k r)r. Since e r. olr_llkr) r ~ lim z._. Zrer" OX(~)exp(k r. X)(w), there is one ÙX operator present in the correlation function. It can contract on any exponential exp(k s. X)(o~) giving, in addition to eq. (3.11), the factor (see eqs. (3.20) and (3.22)) ~r. k S

Y" exp(N°°~) ( Z , - Zs) + y(exp(N°°r)er" kr" sq: r

(4.6)

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334

On the other hand

x(VPld • a~_l r-I[kS)s = Y'.Nf~)d . k s x eq. (4.5), s

(4.7)

s

Eq. (A.1) establishes this equivalence. The vertex-function matrix element P

x(VP] d" ar-1 es" a*-i 1-I [0)s = er" dUff

(4.8)

s=l

should equal, according to eqs. (3.19) and (3.21),

exp(Noo~ + N ~ ) ~r aX(60r )

88. aX(60s )

=

~----Z ,2 , ( L r - Ls)

( d . dexp(2N0~r)(7~ + ½(y()2),

r--/=s, r = s.

(4 9) Eq. (A.2) shows that eq. (4.8) equals eq. (4.9). As a final example, eq. (3.10) should g equal ~( VP[ ar_g2aS_rl[0> = 8 g"2 Nf~ (the factor of two comes from [ am, a v_, ] = m 8m.8 ~" when m = n = 2). Eq. (A.3) shows this to be the case. A complete proof of the equivalence is given in Appendix C.

4.3. C O N F O R M A L R E P R E S E N T A T I O N O F T H E S T R I N G F I E L D T H E O R Y VERTEX FUNCTION

While carrying out the proof in appendix C, several useful formulae were obtained. Defining 0! = 1, the general Neumann-function component is

' ' .... ( O-~-im {ln(z(60)-z'(60'))-6"qn(60rn'ndvi""= 060] ~ 060 ]

'

,z'=z~

, (4.10)

where 60 = exp(~r(z)), 60' = exp(~S(z')), 0/060 is given in eq. (3.4), and 0/060' is eq. (3.4) with r ~ s and primes on z and 60. The vertex-function in conformal field theory form is

s=l

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335

where co is on string r and co' is on string s and z and z' are to be written in terms of ~ and o~' by inverting the relations co = exp(~r(z)) and 0~' = exp(~S(z')). Understood in eq. (4.11) is normal ordering; it sets the creation operators in the OX~(o~) to zero. In eq. (4.11), ~X"(0~)X"(w'))R is the propagator on R (X~(~0) X~(0~'))R = 8""N~(~0, co'),

(4.12)

where c0 is on the rth leg and o~' is on the sth leg. The Green function, N~S(o~,~o'), in eq. (4.12) is Nr'(c0, ~') = ln(z(~0) - z ' ( ~ ' ) ) - 8rSln(o~ - o/).

(4.13)

It encodes the Neumann-function components Nrs(6o, ¢0') -~ ~. 09mNr~SnCd'n .

(4.14)

m=0 n=0

Eqs. (4.13) and (4.14) are in agreement with eq. (4.10). Refs. [77-79] have obtained representations of the form in eq. (4.11) for the cases of the CaneschiSchwimmer-Veneziano 3-vertex and the Witten 3-vertex. This subsection shows that this can be done in general. Eq. (4.12) permits calculations to be done directly on the string configuration region, if one so desires. Eqs. (4.13) and (4.14) provide an efficient method for computing the N ~ for low levels; one expresses z and z' in terms of co and c0' and expands the r.h.s, of eq. (4.13) in a power series in ~0 and ~'. The coefficients in the series are the Nm~s. Appendix B illustrates this method.

5. The ghost contribution 5.1. INTRODUCTION

The Polyakov and N a m b u - G o t o string actions [80, 81] have a gauge invariance consisting of reparametrizations of the world sheet. The ghost fields, c(z) and b(z), arise after fixing the gauge to eq. (1.1) [29, 30]. Although, it is possible to perform calculations without them, the use of ghosts simplifies most computations, particularly in regard to the integration measure. For example, in the open-string at tree level, they produce the SL(2,R)-invariant measure correction factor, (z r - Zs)(Zr - zt)(z s - zt)/(dzrdzsdzt), and in the closed string case one gets, as expected, the absolute square of this quantity [40]. In the superstring case, the supersymmetric partners of b(z) and c(z), fl(z) and 3,(z), play an important role in processes involving fermions [38-41,43-45]. In short, it is desirable to apply off-shell conformal field theory to ghosts. This is the subject of this section.

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336

Although our techniques are widely applicable, we do not give a general formula for the ghost-correlation function because there are different ways of treating ghost insertions. For example, ref. [40] defined two kinds of vertex operators: c-type and f-type. In tree amplitudes ref. [40] used three c-type vertices and the remaining ones were of f-type. In contrast, ref. [15], motivated by string field theory, used only c-type vertices; b-ghost insertions were made interior to the string configuration region, R, to soak up the Q = - 3 background charge. The insertions appeared on the rectangular propagator strips of R. Since the number of propagators in a tree diagram is p - 3, where p is the number of external lines, the net ghost number is + p - ( p - 3 ) = 3, as it should be. Only for three-point functions is there no ambiguity in the placement of ghosts. For this case, which is discussed in subsect. 5.2, our results are general. Subsect. 5.3 treats the four-point function for the configuration region associated with Witten's way of joining strings [11]. Since we wish to produce the results of string field theory, we adopt the treatment of ghost insertions in ref. [15]. This means that a physical state is associated with c(z)V(z), where V(z) is a standard X"(z)-dependent vertex operator. In computing an amplitude for a region, R, such as in fig. 2a, there is a c(~0) at each of the asymptotic positions. Allowing for external states with ghost number different from + 1 leads one to consider more complicated ghost operators. The calculational technique is the one discussed in sect. 3. Since the ghost action in eq. (1.1) is quadratic, one has a free field theory. A conformal transformation to the half-plane, H, makes the problem solvable. The transformation laws for ghost are known: one uses eq. (3.1) with dimension, h, specified in eq. (3.2). The ghost propagators on H are given in eqs. (3.6) and (3.7). To make the b-c correlation functions resemble the X"(z) case, one can bosonize them as in eq. (1.3). Then, a correlation function of exponentials like the one in eq. (3.11) is obtained. Correlation functions for derivatives of b a n d / o r c are computed by differentiating. Let us describe how to use off-shell conformal field theory to compute the ghost vertex function in eq. (2.11). From eq. (2.11) the measure, /~, is p

tL=bc(V[ I--[I - )r,

(5.1)

r=l

where FI,[ - )r = [ -

)1l

-

)2

From conformal field theory

....

/~=

lim c ( ~ g ) { . . - } •=

,

(5.2)

zi---)Zi

where '" {... }" stands for the propagator ghost-insertion factors. For p = 3, it is one.

337

S. Samuel / String physics

Once bt is known, the )(mrs, are computed from I x X ~ =6c(Vlc*-,b~-m

("

I-I] - >r

)

(5.3)

"

\r=l

Eq. (5.3) is evaluated from conformal field theory using the correspondence between Fock-space states and operators. We evidently need c _ , b lC11~2) = c_ n 112) for n >1 O, c _ , q l l 2 ) for n >1 O, b _ ' c x l I 2 ) for m >/1, and c , b _ ' q l f a ) for n >/0 and m ~ 2. The first of these is given in eq. (5.3) and b_lcllfa ) = 1~2). The remaining three are 1

0 "+I

c-"cllfa) = z,li+mr0 (n + 1)! az " + l c ( z ) c ( O ) l ~ 2 ) '

1

(5.4)

8"-2(

1)

b _ ' q l I 2 ) = z'limo (m - 2)! 8 z , ' _ 2 b ( z ' ) c ( O ) - 7 7

I/2),

(5.5)

c _ , b _ ' q 182) = - b _ ' c _ , c t I~2)

1 =-

lim z-,0 lim (m - 2)! z'-,0

8 "-2 8Z,'_

{ ---+c(0) x b(z')c(z)c(O) z'-z

1 2

(n + 1)!

8 n+l Ozn+ 1

c(Z)}z~7__,

(5.6)

where in eqs. (5.5) and (5.6) we use eq. (2.5) with z 1 = 0. To obtain the states in bosonized form, replace c ( z ) by exp(o(z)) and b ( z ) by e x p ( - o ( z ) ) . 5.2. THREE-POINT RESULTS The generic three-point case is considered in this subsection. When p = 3, there are three legs in fig. 2a and the corresponding correlation functions are related to trilinear couplings of the theory. Via conformal field theory, this subsection computes the ghost vertex associated with an arbitrary region, R. Results for this problem have also been achieved in refs. [82-84] using different methods. For p = 3, the b-ghost insertions are absent in eq. (5.2)

/* =

lim exp(o(oa(zi)

•= z~--,Z~

( E rN(' +~i 1 E

=exp -

r

r~$

R

go~) =exp

( - E rg~)I-I Izrr

- r
zsl.

(5.7)

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338

The term e x p ( - ErN~or) arises from eqs. (3.1), (3.2) (for q = 1), (3.7) and (3.9). The r.h.s, of eq. (5.7), omnipresent in string field theory [8, 9, 84] and frequently determined by ad hoc methods, is naturally generated by the ghosts. Associated with eq. (5.7) is the picture +

+

+

'

'

Z3

Z2

'

(5.8) ZI

Aside from the Jacobian transformation term exp( - Y~rN~or),the factor ~-~r < s IZr -- Zs I is the partition function for the two-dimensional electrostatic system (at a special value of the Boltzmann-temperature constant, k T ) with + charges located at the points, Zr, as in eq. (5.8). The electrostatic analogy is well-known in string theory [8,14, 31]. The charges are, of course, the ghost numbers of the state. Let us compute X~I~ 1 in detail to illustrate the method. For simplicity, let Z 1 > Z 2 > • • • > Zp throughout the rest of sect. 5. Use eq. (5.3) with r = s = 1 and m = 1. The relevant Fock-space state for string 1 is c nb_ll - ) = c_,,b_lcxlI2) = c_,1$2 ). Via eq. (5.3) ( O ln+l -0--~] (exp(o(~o))exp(o(%))exp(o(%)))R.

1

ffX11~= zlimz, ( n + l ) !

(5.9)

The term in " ( )" is the same as in eq. (5.7) except the charge at Z 1 has been displaced to z +

+

'

'

Z3

+

Z2

*

'

Z1 z

(5.10)

Transforming to H and dividing by/* in eq. (5.7), one gets from eq. (5.9)

(

1

e x p ( - ~'(z))

Xl11= zlimzI (n-t-1)!

( )10

x{exp(fl(z))()exp(Nol)(Z1

(Z -- Z2)(Z -- Z3) Z 17 3)}.

(5.11)

The term exp(fl(z))(Ofl(z)/0z)exp(NoX~) is the ratio of the transformation factors associated with eqs. (5.7) and (5.9). The term (z - Z2)(z - Z 3 ) / ( ( Z 1 - Z2)(Z 1 - Z3)) is the ratio of the partition functions of the diagrams in eqs. (5.8) and (5.10). To compute X~12, note that c 2 , b 1 1 1 - ) 1 1 - ) 2 = 1~)1C2nC2[~)2 SO that the charge at Z 1 is displaced onto the second leg +

+

+

Za

-'Z2z

-'Zl

(5.12)

S.Samuel/ Stringphysics

339

The calculation is identical to that of X~, presented in the previous paragraph, except that, since z is located on leg 2, fl(z) ---, fZ(z)

)( O~2(Z))-I 0 ) n+l

1

Oz

X~I~ = lira exp(-~2(z) ~-.z~ (n + 1)!

~z

O~2(Z) )exp(N~) ( z - Z2)(z- Z3) }

X {exp(~2(z))

o~

(z,

~(-z-~-z23)

•

(5.13)

In general, one gets

1( X~'~= lim

z-,Z, (n + 1)!

n+x exp(~S(z)) '

(

Oz ]

Oz

sz

× exp(liS(z))(oj~O]exp(N~or)i_] (z-Z,) [prop-factor] \ az ] t*r (-Z~---~)

) . (5.14)

This equation is valid for both s ~ r and r = s. For convenience, we have inserted "[prop-factor]". For p = 3, it is one. The reason for its introduction is that the general p-point result is of the form in eq. (5.14); to compute correlation functions with p legs, one needs only to specify this factor. For n = 0, eq. (5.14) is

(zs- z,)

X~'~ = exp(No~r) lqt~-~(~-r~ ~ , ) ,

(

X~g=exp(No~r) -2,/(+ ~]

1)

(Zr--Z,)

'

s * r,

(5.15)

s=r.

(5.16)

t~r

Whem m > 1, the calculation of X ~ (5.4)-(5.6). The picture for X~2n is

is more involved. One must use eqs.

+

+

+

+

-

e Z3

- ~ -Z 2

ZIz"

+

+

+

23

Z 2

Z 1

(5.17)

whereas the picture for xnn is +

-

z

Z"

(5.18)

340

S. Samuel / String physics

The general result is

X~=

1

1

lim lim z , ~ z , z--zs (m - 2)! (n + 1)! i

.

X /exp(-~'(z'))

[~

x exp(-2~'(z'))

P ( z - Zt)

)

~z'

m-2(

exp(-'~(z))

OU( z') -2exp(~,(z)) Oz'

Oz ]

-~z

Oz ] ,,

1

X I--[ - - - [prop-factor'] + exp(~'(z')) - exp(~'(z)) ,=1 (z' - z , ) (z - z')

-exp(--~'(z'))e2N~exp('~'(z))~ (O~(z) Oz

(z - Z,) [prop-factor]) )t~.,(Z,,--Zt)

(5.19) where m >t 2 and [prop-factor] and [prop-factor'] are both one. The results for m = 2 and n = 0 are

X~'~= e oo . . . . . . .

----

3y(

Iqt.r(Z,-Zt)

(

-

(Z, Zs)

-

(Z, Z t ) '

(5.20)

( 1)

X~'~= - e 2Ng 4T~+ 5(y() 2 - 3y( ~_, ( Z , - Zt) \ t4~r

1

1

+ ,.,g (z.- z,) ~ + ,.ILl,(z,- z,)

(5.21)

Let us check eqs. (5.15), (5.16), (5.20) and (5.21) for the three-point ghost vertex [22-24] of the Witten string field theory [11] using the map of Cremmer, Schwimmer and Thorn [21]. The string variables, ~'(z), are given in eqs. (3.6)-(3.8) of ref. [26]

S. Samuel / Stringphysies

341

for arbitrary Z r when z > Z 1 > 22 > Z 3. Expanding ~l(z) about (z - Zr)

(Z 2- Z3)

]

~'l(z) = ln(z - Z,) + In[ 3~-3 ( Z 1 _ 22)( Z 1 _ 23) 1

(Z 2 + Z 3 -

2Z1)

+ ~ (~-~S(~-~)

(z- Zl)

8ZZ+7(Z2+Z2)-6Z2Z3-8Z1(Z2+Z3)

(z

Z1)2+

, (5.22)

16( Z 1 - Z2)2( Z 1 - Z 3)2 determines the 7~

(1) 3(1) ( , ) ( 1 , ) 7 ( 1 )

v;=}

z r - z, '

Zr__ Zr+ 1 -

(5.23)

Zr__ Zr_l) ) - "-~ t~=Jr( Zr__ Zt)2 , (5.24)

where three-cyclicity in the indices is assumed: r + 3 = r. Substituting eqs. (5.23) and (5.24) in eqs. (5.15), (5.16), (5.20) and (5.21), the following results, which are independent of Z 1, Z 2 and Z 3 (as should be the case), are obtained 4

x~g- 3v~ = -x?x' )(lo1=0, X~,o~ = _ 8 ,

xl~ _- ~-16

(5.25)

These coefficients agree with those in eq. (11) of ref. [22]. 5.3. FOUR-POINT RESULTS This subsection computes the ghost correlation function for a region, R, generated the Witten string field theory. It is not difficult to adapt the methods to any R. The on-shell four-point scattering amplitude has been computed by Giddings [14]. We follow his treatment of the ghost measure contribution, that is, b and c are bosonized and eq. (7) of ref. [14] is used to map H onto the cut version of the string configuration (shown in fig. 3 of ref. [26]) of fig. 1.

S. Samuel/ Stringphysics

342 The map,

O(z), from H to R is [14] p(z)=p(Zo)+N f zdz'I(z'),

(5.26)

zo

where 4

I(z')=~72~/z'2

+82 r I ( z , - z , ) -1,

(5.27)

r=l

Z 1 =/3,

Z 2 = O~,

Z3=

-/3,

Zn=

-a,

(5.28)

and N=2a

2~~ T - ~

1 / r ~ 82

(5.29)

Because of SL(2, R) invariance, results depend only on one Z r. The variables a,/3, 7 and 8 satisfy [14] aft = 1,

y8 = 1,

a ~
7 ~< 8,

(5.30)

and 7 is a function of a (see eqs. (10) and (12) of ref. [14]). For a graph of 7, see fig. 1 of ref. [28]. The nature of the map, p(z), is discussed in ref. [14] (also see subsect. 2.3 of ref. [26]). There are square-root singularities at i7 and at i8 along with the associated branch cuts. The latter are drawn as jagged lines in fig. 3.

Fig. 3. Contours and cuts in the z-plane for the four-point scattering amplitude in fig. 1.

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According to sect. 2, several quantities associated with R are needed. The string variables, ~r(z), are defined in eqs. (3.10) of ref. [26], the asymptotic positions are given in eq. (5.28) and the expansions in eq. (2.1) can be found in eqs. (3.11)-(3.13) of ref. [26]. The zero-zero Neumann functions are [26] N11 = ln(fl) + ln(x),

N~0 = N0101,

N~ 2 = ln(a) + ln(x),

N~3 = N22 ,

(5.31)

where x is a function of a, defined in eq. (3.13) of ref. [26]. The region, R, is generated from covariant string field theory after gauge-fixing [11,14-17] the second-quantized theory to the Feynman-Siegel gauge, bokO= 0 [18]. As a result, external states, Is), satisfy bols) = 0 and the ghost insertion associated with the propagator strip is b0. On H, b0 = ~ dz/(2~ri)zb(z) for a contour which goes around the origin. If a second copy, R', of R is appended to R, bo can still be expressed as a contour integral do~ t

(5.32) for some variable o~I (specified below in eq. (5.33)). The copy of R is obtained by analytically extending the map to the lower half-plane. Half the contour of eq. (5.32) is given by cg in fig. 4a of ref. [14]; the other half is the counterpart, c~,, of on R'. The preimage in H of the total contour is C shown in fig. 3. Making use of bols ) = 0, ref. [14] has shown that C can be deformed to what is denoted by ~g in fig. 3. Just like the ~r are exponentially related to the ~r o~z= exp(p ( z , ) ) ,

(5.33)

where p(z) is the Giddings map in eq. (5.26). Eq. (5.33) produces the correct string measure. Eventually, the ghost correlation functions are calculated on H, so it is useful to transform eq. (5.32). This is done [14] using eqs. (3.1) and (3.2) and noting that dw1/dz I = ~i dp/dzl

-~io~,b( ~,) ~ -a~ ~2~ri

dP(Zl)

dz I

b( z,)

(5.34)

Note that

dp(:,) -1 dzt )

1(

I-Ir(Z,- zr)

(5.35)

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Four-point ghost correlation functions are calculated as in subsect. 5.3 except there is the extra insertion of the r.h.s, of eq. (5.34). When bosonized, it represents, in terms of the electrostatic analogy, a - 1 charge at z I. Let us compute/_t. Using eq. (5.2) for p = 4 u = (exp( o( O)l))exp( o(~%)) boexp( o( ~03))exp( o( o~4 )) > R

= exp(- ~"N~°r)~ / c~Z~, d~z-~t("

dzl

-1

X (exp( o ( Z 1)) exp ( o ( Z 2))exp ( - o ( z 1)) exp( o ( Z 3))exp( o ( Z 4)) ) , .

(5.36)

In transforming to H, a normalization constant is generated. It represents the world-sheet action of the combined matter-ghost system. This constant is shown to be one for D = 26 in eq. (6) of ref. [14]. Pictorially, eq. (5.36) is

+

+

+

"z 3

•+

z2

,

zl

(5.37)

and there is the additional factor of ( d o / d z z ) - 1 . Curiously, has an electrostatic interpretation. The inverse of curly bracket term in eq. (5.35) is the Boltzmann factor for the interactions of the - charge at z z with the four + charges at Z~ as well as four - ½ charges at the branch points, i6, iT, -i3~ and - i & The constant, N, is the square root of the Boltzmann factor for the interactions of the four + charges at Z~ with the - 1 charges at i& i3,, - i3, and -i& As a result, cancels the terms associated with the interaction of the charge at z z with four + charges at Z~ and leaves the Boltzmann factor for the interactions of the - charge at z z with f o u r - ½ charges as well as N -1. From this method or by straightforward algebra

dp/dz t

(dp/dzr) -t

r

-

~e2rri z

l

~

~

"

(5.38)

From eq. (5.31), the factor exp(-F~rN0~~) is x 4. The remaining term has been evaluated in ref. [14]. Calling it 2AG, 2Ao ~.~--

X4

,

(5.39)

where [14]

(5.40) and K(3, 2) is the complete elliptic integral.

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Aside from the self-energy term, x -4, eq. (5.39) is the result of ref. [14]. Assuming the validity of the calculation in ref. [14], which could be justified a posteriori since on-shell it reproduced the Veneziano formula, the factor K-4 was deduced to be present by ref. [26] (see eqs. (4.8) of ref. [26]) and the measure in eq. (5.39) was obtained (see eq. (4.9) of ref. [26]). One sees that the conformal field theory treatment of ghosts naturally leads to the correct measure. Let us calculate the X11n L Consider eq. (5.3) for r = 1, s = 1, m = 1. The relevant Fock-space state for string 1, c_,b_ll - = c _ n b _ l C l [ Q ) = c_.152>, is given in eq. (4.3). Hence

1(

0~1(Z)]-1

n+l

0

IXXll~= zFaInz,(n + 1)! e-~(~) × (exp(o (~o))exp(o (~02)) boexp(o (~03))exp(o (¢04))).

(5.41)

Substituting eq. (5.34) into eq. (5.41), the relevant diagram is +

+ "z 3

~

ZI + '

+ "-

z 2

"

(5.42)

'

which is the picture in eq. (5.37) with the + charge at Z 1 displaced to z. Eq. (5.41) is 9~dzJ(2~i)(dp/dz,) -1 times the Boltzmann factor associated with eq. (5.42). Dividing by/~ and performing the straightforward algebra, one gets eq. (5.11) with 2rr ) ~ dzt prop-factor- 4yK(y2

( Z 1 - ZI ]

1

inserted inside the curly brackets. Making the charge of variables

(5.43)

[14]

(5.44)

zi = iyy , which implies* d z 1 [ . . . ] = 2"tRef I d y [ . . . ] " ~' i

-1

(5.45) '

eq. (5.43) becomes prop-factor

1

K ( Y 2)

f.old Y - f1f l - ~ f l - -

y2y4

2.2 q_ y2y2

t

(5.46)

where r = 1. The integrand term in "( )" is the real part of the term in "( )" in eq. * The 2 × real part arises because the contour loops around the cut in fig. 3.

S. Samuel / String physics

346

(5.43) at zi= i,/y. At z = Z 1, prop-factor is one, as expected, because Ihc pz,t~Jf,,, functions for eqs. (5.37) and (5.42), which enter as a ratio, arc equal. '1~ c, mqmt< X~2, the charge at Zt is displaced to z near Z 2 +

+

+

+

~4

*Z3

Z2~

•

(5.47) ZI

The calculation is the same as before except ~'X(z) ~ ~'2(z). This does not effect propagator factor in eq. (5.46). Hence, X~2 is eq. (5.13) with eq. (5.46) (with r inserted within the curly brackets. The general X[~ is eq. (5.14) with prop-factor given in eq. (5.46). Repeating same procedure for X,~,*, for m > 2, one gets eq. (5.19) with prop-factor as in (5.46) and

1

pr°p-fact°r' - K(y2 )

JoCldy 1 ¢l_y2¢l_y2y4

(zz'+ .g2y2)

the 1) the eq.

(5.48)

(2"2 + ]t2y2) '

Ref. [27] has computed the X ~ inductively in terms of the N~;. The above formulae for X ~ are a different representation. We have checked that the two methods agree for several level-one X ~ . From off-shell conformal field theory

,.s

S~o= e

NgI-I,.,. , (Zs- Zt) 1 [ldy 1 (ZrZ, +'y2y a) - - .,~ . _ _ _ _ I-I,.,.(Z,.-Zt) X ( v 2) Jo 1 - ~ - ~ ¢ 1 - ~ , 2 y 4 ( Z 2 + v 2 y 2) (5.49)

{

X~=e N;a - 2 y / +

1 t.rE g r _ _ g t

Zr K(~2)

Soldy 1 - - ~ ¢ 1 1 _ y2y4

1

(Zr2 + 3,2y2)

}

(5.50)

From ref. [27]

xl0"=~~ ~1 -&

fl-a

~+~

5f l +- - a

+/~v 1-

O/K

x~= - c ~ , C=I+

6~fl+a

~v

12)

,

(5.51)

(5.52) +flT~-a

+3~.

(5.53)

The computer programs of ref. [28] were adapted to numerically evaluate the

S. Samuel / String physics

347

integrals and quantities in eqs. (5.49) and (5.50) (for r = 1, s = 4 and r = 4, s = 4) and eqs. (5.51)-(5.53). The two methods agreed to machine accuracy throughout the entire region 0 ~< a ~< %. They were also checked analytically in the limits a ~ 0 and o~ ---> o~0.

This section demonstrates the utility of off-shell conformal field theory in computations of correlation functions for the reparametrization ghosts, b ( z ) and c ( z ) . These correlation functions generate the correct measure and ghost-vertex functions for any string configuration region, R, and agree with results from string field theory when R comes from a second-quantized Feynman graph. By using the electrostatic analogy, results are quickly determined. Although we have only treated the three-point case and the Witten four-point case, the methods are general and results are readily obtained once the nature of the ghost insertions is specified.

6. Off-shell string amplitudes Sects. 2 - 5 discussed the computation of correlation functions for a fixed string configuration region, R. To obtain an amplitude, a sum (or integral) over a set of R is performed. As an example, consider the four-tachyon amplitude in the Witten string field theory [11,14]. The rectangular region of fig. 1 is of width ~r and length T. To produce the Veneziano amplitude [19], T is integrated from 0 to ce and six different diagrams are included, corresponding to the six different ways of joining the four external strips to the internal one (see figs. l a - l f of ref. [26]). From the viewpoint of conformal field theory, the measure for T is not immediately obvious, although it can be determined by the methods [85-91] used to handle the Polyakov approach [81]. From string field theory, one knows it is fiat: f0~ dT. With a non-flat measure, /~(T), a result different from the Veneziano amplitude arises. For example, i f / z ( T ) = O(To - T ) , the upper range of integration is cutoff at T0. Not only does the amplitude disagree on-shell with Veneziano formula but it contains no particle poles since they are generated in the T-- ~ region. If / ~ ( T ) = e x p ( - ~ m 2 T ) the mass spectrum associated with the poles is shifted: m 2 ~ m 2 + 6m 2. This is an easy way to get rid of the tachyon but the physics is not that of the bosonic string. In order that off-shell conformal field theory reproduce on-shell string amplitudes, it is necessary to correctly specify the sum over surfaces and the corresponding measure. For covariant string field theory in the Siegel-Feynman gauge [18], this is straightforward; the lengths of the propagator strips range from 0 to ~ and the measure is flat in these lengths. In other situations, one must be careful as the examples in the last paragraph illustrate. Subsect. 6.1 discusses sufficient conditions on the sum and measure so that on-shell agreement of string amplitudes is achieved. 6.1. R E Q U I R E M E N T S ON T H E SET OF S T R I N G C O N F I G U R A T I O N S

The Polyakov approach sums over all surfaces. Because the theory is gauge invariant, the volume of the gauge group must be divided out. Alternatively, a slice

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S. Samuel / Stringphysics

3-point tree

4-point tree

5-point tree

vacuum l-loop

Fig. 4. Pictorial representations of the phase space of surfaces. Sx and S2 are two typical slices. If they come from the same string field theory then they intersect the 3-point coupling at the same point because the 3-point couplings are independent of the gauge choice.

is selected that intersects each inequivalent surface once. One then integrates over the gauge slice, S, instead of over the whole space. The measure is fixed by Faddeev-Popov methods. All this is well known [81, 85-91]. We are thinking of the space of all surfaces from a very general viewpoint: tree as well as higher genus surfaces are included. Hence S is disconnected and various components have various dimensions. Even for simply-connected surfaces, the character of S changes according to the number of operator insertions (they can be thought of as surface punctures). The dimension of S is locally the number of integration parameters. Since tree-level p-point amplitudes involve p - 3 K o b a - N i e l s e n variables, the slice has dimension p - 3 for this case. Fig. 4 symbolically illustrates the situation. Pick an S. Integrate over it using the Faddeev-Popov generated measure. For each R ~ S, use off-shell conformal field theory to compute the correlation function of vertex operators. On-shell, the result coincides with the standard string model. Different slices, however, lead of different off-shell amplitudes. In other words, there is an infinite number of off-shell extensions of the Veneziano formula, one for each S. This does not imply that there is an infinite number of string field theories. For a given S, there does not necessarily exist a second-quantized action, which, when expanded perturbatively, produces a sum over surfaces corresponding to S. The number of acceptable string field theories is not known [92]. This subject is worthy of further investigation. The Witten field theory generates many S through field redefinitions a n d / o r through the selection of a second-quantized gauge. If ~ is replaced by f ( ~ ) for some function, the action and consequently the character of the perturbation series are changed. If f contains terms with two or more powers of '/', the new action

S. Samuel / Stringphysics

349

contains higher than trilinear couplings. When perturbatively expanded, the transformed theory generates a different S. This happens also through the gauge choice. The amplitudes in refs. [14, 26, 27] were computed in the Siegel-Feynman gauge*. This gauge led to the b0 insertion and the propagator strip of width 7r and length T. Another gauge choice leads to a different propagator, a different set of R, and consequently a different S**. Although the Witten string field theory generates many S, it cannot generate all of them. Consider the following construction. Take two gauges, G 1 and G 2. Call the corresponding slices, S 1 and S2. Fig. 4 shows a portion of the space of all surfaces along with the slices S 1 and S2. Choose a new slice, S3, which is S1 on the four-point tree surfaces and S2 on the five-point tree surfaces. Since the propagators for G 1 and G 2 are different, there is no second-quantized gauge choice which yields S3. For a more extreme example, consider the four-point tachyon amplitude as obtained from $1 and S2. On-shell, there is a map from S 1 (and $2) onto the integration range - ~ < x < + oo, where x is the Koba-Nielsen variable [96, 97]. For example, in the Siegel-Feynman gauge, the R associated with fig. 1 cover the range x ~ [1,1]. The other five diagrams in fig. 1 of ref. [26] cover the rest of the real axis, ~ , after an additional SL(2, R) transformation is performed. Hence each point, x ~ ~ , has a counterpart in S 1 (and $2). Break up ~ into two arbitrary disjoint sets. The preimage of the first (second) set is a set in S a ($2). Choose S3 to be the union of these sets in S 1 and S2. The off-shell four-point amplitude associated with S3 agrees on-shell with the Veneziano formula but off-shell is unlikely to even satisfy bose symmetry. Under interchange of external lines, different Koba-Nielsen regions of are mapped into each other. For example, figs. l a and lb of ref. [26] cover respectively [½,1] and [0, ½]. If these regions come from different second-quantized gauges, the amplitude will not be bose symmetric. This proves, by the way, that S3 cannot come from a field theory since bose symmetry is automatically a symmetry in field theory. One may wish to impose additional conditions on the slice, S. Certain properties [73] of on-shell string amplitudes are desirable to preserve off-shell. An example is the analyticity behavior. The on-shell tree string amplitudes are meromorphic functions of the ½p(p - 1) invariants, k i- k j, i < j and contain no new poles. Such poles would correspond to states not in the original theory. It would be attractive to have off-shell amplitudes factorize on the poles of the free particle spectrum with residues related to the off-shell couplings. Another example is symmetries. On-shell the amplitudes are crossing and bose symmetric but off-shell this need not be the * It has been shown in ref. [93] that this (second-quantized) gauge selects a slice which intersects (first-quantized) gauge-invariant surfaces once. The covariantized light-cone field theory [12, 84, 94, 95] seems to overcount surfaces by a factor proportional to f d a , where a is the string length parameter. * * The ambiguities in the treatment of the perturbation theory are expected not to affect physical results. The computation of a physical quantity should be independent of field redefinitions and gauge choices.

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S. Samuel / Stringphysics

case, as illustrated above. A gauge slice associated with a string field theory possesses the above properties. Perhaps if enough conditions are imposed, the gauge slices associated with the Witten string field theory are singled out. 6.2. FOUR-POINT COVARIANTAMPLITUDES This section demonstrates that conformal field theory reproduces the off-shell results in ref. [26] for the four-point amplitudes in the Witten string field theory. As described above, an integral over T, the length of the strip in fig. 1, is performed. The measure is [14] (6.1) dT) 2~" -d--aa = K(y2)~/1 + a z 3 , 2 ~ y 2

(6.2)

Consider the four-tachyon case. The amplitude is the product of eqs. (3.11) (for p = 4), (5.39) and (6.1). After transforming to the Koba-Nielsen variable, x, via [14]

x=

l+a------:

a=

+v%- '

eq. (4.12) of ref. [26] is obtained. The two-tachyon two-vector amplitude is the product of eqs. (3.23) (for p = 4), (5.39) and (6.1), which, after using eq. (6.3), becomes eq. (4.15) of ref. [26]. These two examples demonstrate what is true in general; that off-shell conformal field theory generates off-shell field theory amplitudes. The same holds for amplitudes involving auxiliary fields and second-quantized Faddeev-Popov ghosts [27]. The reason is that conformal field theory reproduces correctly the matter and ghost vertex functions associated with fig. 1. 7. Conclusion

This paper has developed conformal field theory into a working tool for off-shell results. The complicated direct calculations of string field theory, which involve manipulating infinite matrices (see, for example, refs. [9, 20] and sect. 4.1 of ref. [26]), can be avoided. Instead, the results are obtained by off-shell conformal field theory in an efficient manner. It applies to both matter fields and ghosts. The non-trivial off-shell extension [88,98-100] of conformal field theory has been accomplished. This is particularly important given our lack of understanding of vacuum structure. In unravelling the nature of the string vacuum, no longer is it necessary to

S. Samuel / Stringphysics

351

restrict oneself to on-shell probes. Quantities, which require an off-shell formalism, such as the static tachyon effective potential, are computable [28]. Such potentials should help to decide whether non-perturbative stable string vacua exist with the desired phenomenological properties; that is, whether the string world resembles our world. I thank A. Kosteleck~ and O. Lechtenfeld for discussions. This work was supported in part by the DOE grant DE-AC02-83ER40107 and by a NATO Collaborative Research Grant 0763/87.

AppendixA NEUMANN-FUNCTION

COMPONENTS

THROUGH

LEVEL FOUR

This appendix computes the Neumann-function components associated with fig. 2a. The expansions of the f~(z) in eq. (2.1) are substituted into the integral formulas in eqs. (2.8) and (2.9) and the latter are evaluated. In eqs. (A.1)-(A.4), if r and s appear together, r ~ s.

Level 1: eNg

N ~ - Z~- Z~ '

Niro~= eNVy(.

(A.I)

Level 2: 2 N ~ = Z r - Z~ 2T?

Zr-- Zs

2N~%~= 2e2U~(y~ + (y1~)2), ,

e N ~ + N~ "=

: :=( :eZU66,Y: " N-: + -'iv 2,

(Z_Z~)2,

1"

(A.2)

Level 3." e 3N£ [ 3N3'° = ( L - Z') 13Y~+ 9tyrO2 2\ 11

1

3y( -

-

zr-zs

"~-

(zr-

2

rr_ 3 r 3 3Ns0 - 3 e3N~(y~ + 3y~y; + :(Y:) ),

e2N~+U~ ( 2N~'~= 2 ( Z r _ Zs)2 7[

1) Zr_ Z--------~ ,

2Nf( = 2 eSNg(y3r + 2y~y( + 2( ylr)3).

(A.3)

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S. Samuel / Stringphysics

Level 4." e4N~ ( 4N4~ - ( Z ~ - Z~) 43'~+ 163'£y[ + ~(3'[)3 _ 4(3'£+ 2(V[) 2)

Z~-Z~

+

43'[ (Z~-Z,):

1_ ) (Z~ Z y '

4N,~~= 4enN~(y,[ + 43';7( + 2(y~) 2 + 8y~(y;) 2 + 8(3'1r)4) , eSN~+N~ ( r

rs_

3N~1- 3 (-~-~)2

3'2+ 3_[3"~2 2 \ 11

23'1r Z- r _- _ Z s

+ ( z,

,)

z~) ~ '

3N3~r=3e4U~(3',~ + 33';3'[+3(3'~)2+~ 93"r[3"r] 2y 1.1 2 + 9(3'1r)4), 1 e 2N~+2N~ (

2(3'; - 3';) z ~ - zs

3 (z~

] zs) 2 ] '

N¢'[ = e4Ng(3'; + 33'f3'~ + (3'~)2 + 4y£(3'[)2 + (]/lr)4) .

(A.4)

Appendix B

SOME SERIESEXPANSIONS Firstly, this appendix computes some Laurent expansions in the variable (z - Zr). They are useful for the computations in sects. 3-5. Secondly, it computes the Neumann-function components to level four by expanding eq. (4.13).

B.1. USEFUL STRING-VARIABLEEXPANSIONS The expansion for exp(-~r(z)) in terms of exp(-U(z))

(z -

Zr)

is

exp ( N0~r )

-~zZ~) E c r ( z - Z,)",

(B.1)

.=o

where c~ are the following constants C~ =

E

partitionsof n Pl,P2,...,Pm +mpm=n

pl+2p2 + -..

( (3"()Pl (,~f)P2. . (y~)Pm) . . P 1! P2 ! Pm !

(B.2)

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S. Samuel / String physics

The first few are c; - 1,

c3' =

3 +

c~ = 7~,

+ T.,.

c; = ~'~ + ½(~'()z,

,

¢4__.}t4..J_~tl~t3.+.l(~;)2nt_ . . . . . i1[\ 7 1 1r~ 2r 2r+ ~1( r l ) ,

,

(B.3)

"

The expansion for exp(~'(z)) is the same as in eq. (B.1) except (a) the prefactor on the r.h.s, is inverted and (b) 3'~ ---' _ ~,,r in the definition of the c~. On the r t h leg, the factor, (az/Ow), entering in eqs. (3.3) and (3.4) is

(0_~V -1 exp(-~'"(z))~ Oz ) = exp(No~)(1 + 23,{(z - Z,) + (3,/; + ~(3,()2)(z - Z,) 2 +(43'; + 8"/(3'; + ~(r()3)(z -

Zr) 3 -q-(5~/z~"-[-11y(3'3r +

65 r ))(Z --1- ~(~lr)2"~/; -]- ~('~I

~(y;)Z

Zr) 4 -[-"'"

).

(B.4)

B.2. THE NEUMANN-FUNCTION EXPANSION The Neumann-function components can be computed by expanding eq. (4.13). To do this, z is expressed as a function of w, that is, the relation o~= exp(gr(z)) is inverted. Using eq. (B.1) and an iterative procedure, one finds ( z - Z r ) = ~ ( 1 + c ; ~ + (c~+ ( c ; ) 2 ) ~ 2 + (c; + 3c;c~ + (c;)3)~ 3

+(c~ + 3c;c; + 2(c~) 2 + 6(c;)2c~ + (c{)4)~ 4 + ---),

(B.5)

when w is on the rth leg. In eq. (B.5) = w exp( No~") .

(B.6)

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S. Samuel / String physics

When a~ and o~' are both of the rth leg, eq. (4.13) is

ln(z - z') - ln(~, - ,~') = c ; ( ~ + ~') + ( 4 + 3(c;)2)(~2 + ~,2) + 4 ~ ' r 2• "1- ~i ( C lr) 3 ) ( ~ 3 Jr- ~,3 ) q-(C; %" 2ClC

r r -2--, ~,2)

+ ( C3 "-k ClC2)( ~ O) "-k

+ (c~ + 3c;c~ + 3(c~)2 + 3(c{)2c~ + (c~)4)(~ ` + ~,4)

+ (c~ + 2c;~; + ( 4 ) ~ + (c;)~4)(~'~' + ~ ' ~ ) r 3-tr 2-[¢2) 1' r'2 + (C{)2C~) ~2~,2 "q- " "" " "t-(C~'t- 2ClC

(B.7)

where ~' = ~0'exp(No~). From eqs. (B.7) and (B.3), the diagonal Neumann-function components are read off and are seen to agree with the results in appendix A. For the off-diagonal case, when ~0' is on a different (sth) leg

~

ln(z-z')=ln(Z•-Z,)+

~], +

(

1 )~2 +

7Zr

Zrs

(

C;+3C[C;+(C;) 3

C(--

Z~s) ~2~t

~+(cf) 2_

c2

+~

Zrs

Zrs Zrs

C2+2tc1! + Cl Z•s

~' +(r,~,~--~')

(

c ; - 2Z~, ~Tr, +

c~ +

-I- ¢~ -~ (C~) 2 +

(

+-~f +

Z,.2,

4Z 3

Z,.s

/ ~- ~ 3 ~ ~,2~,~ lcf~i+ Z~---7 2Z 2,}za~--4~ + " ' " , (B.8)

where ~ ' - ~ 0 ' e x p ( N ~ ) , Z ~ , - ( Z ~ - Z , ) and we have assumed Z~> Z,. The offdiagonal components as determined from eqs. (B.3) and (B.8) are seen to agree with those obtained in appendix A.

Appendix C

PROOFOF THEEQUIVALENCE This appendix proves that the correlation functions computed from off-shell conformal field theory are equal to the matrix elements of the vertex function. This is done for any tree-level surface, R.

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The proof has two parts. We first show that a matrix element of a vertex function is computable by a Wick contraction method of the same structure as for correlation functions. Secondly, we prove the equality of the Wick factors. C.1. WICK'S METHOD FOR THE VERTEX FUNCTION The matrix elements are of the form x ( V e l s ) where Is) is a state consisting of g . a r m acting on l--Islk*). To compute a matrix element, g . aL,, is pushed to the left until it annihilates the left vacuum. In the process, ~ ..... e r • Nr,,a,, r, s is produced. If n = 0, a~ = k*; if n > 0, a~ can be pushed to the right until it annihilates the right v a c u u m and a non-zero result occurs only if it meets e* .a_,.s In the n = 0 case, the factor g - k*mNffo is generated; in the n > 0 case, e r. e*mnNr*n is generated. In short, the aLm either contract pairwise or they contract against Ik*): ...

.

E4 " O l r _ m . . .

. E.s _ . olS n " " . $ r .

[

Ik s)

, E r. kSmNffo,

r OL--m """

) Es

• e r m n N rrs, , .

(C.1)

(C.2)

1

N o t e that several a~_,, can contract against the same [k~). The correspondence between oscillators and conformal operators is Ik ~) ~ e x p ( k ~- X)(~o~), 1

er" o~r~m, . _(m _- 1)!

: . amx( ,: )

(c.3) (c.4)

In a correlation function g . o mx(~ r) either contracts against another e~- 3nX(~ ~) and the two are removed or e r- 0 " X ( o : ) contracts against e x p ( k ~ . X ) ( o : ) and it is removed. Several g . o m x ( ~ ) can contract with the same exponential, e x p ( k s. X ) ( ~ ) . This shows that the g - Omx(~o~) contract in precisely the same way as the ~ t r- - m " C.2. EQUALITY OF WICK CONTRACTION FACTORS In the absence of any g . a~_,,, the matrix element of the vertex function is exp(½Vr.sk ~. N ~ k S ) , which is the conformal field theory result in eq. (3.11). This factor is present both in matrix elements and in correlation functions. The final answer is eq. (3.11) times the Wick contraction factors. If it is shown that the Wick factors for the vertex function are the same as those in conformal field theory, then

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356

the proof is completed. From eqs. (C.1) and (C.2), it is necessary to show 9

e'. O"X(o~')...exp( k~" X(o~)) " e" k~rnNffo, ( m - 11! I I

(a)

?

1

gs. onx(o)s)...

(b)

(C.5)

( n - 11------~.

1

( m -1l ) ,) e'.

OmX(o~") = e"ermnN,~ s, ]

(C.6)

Case (a) for r :~ s. The 1.h.s. of eq. (C.5) is

1.h.s.

(mZ1-)! e-V(~) ~

{ I n ( z - Zs) )

Oz

z~z

1 )

r

(C.7)

where in the last equality, eq. (3.4) has been used and the inner most differentiation has been carried out. The r.h.s, of eq. (C.5) is eq. (2.8). Make the change of variables from z to o~ in eq. (2.3) to arrive at

.do( z) 1 r.h.s.=e'.k J)o~i ~

1

(c.8)

~om ( z - Z~) "

Cauchy's theorem X do~ f(o~) 2~ri ~'~

f(m 1)(0) ( m - 1)! '

(C.9)

shows that eqs. (C.7) and (C.8) are equal. Case (b) for r ~ s. The 1.h.s. of eq. (C.6) is

1.h.s.

=

(m_~)i(e__~

1),(~)m-l(_O_0/n-l/(

~z ( a z ' ]

1 z~Zr

'

(c.lo) where eq. (3.4) has been used for o~ and z and for ~' and z', and one differentiation,

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357

O/O~, and one differentiation, a/O~o', have been performed. The r.h.s, of eq. (C.6) is eq. (2.9). Making the changes of variables, z ~ ~o and z ~ ~', leads to

r.h.s.

= g . e ~o

~o - ~ [ - ~ )[ 00)-"--7 o')m £otn (Z -- Z t )

(C.11)

2"

Eq. (C.11) is eq. (C.10) after eq. (C.9) is used twice. Case (a) r = s. The operators, er. a mX(o)r) and exp(k r- X)(~0~), are multiplied at the same point which implies the normal-ordering procedure

e". O"x exp( k~. x )( ~o~) ~ = lim ~

m

~---.z, 3~

[er. O"X(o~)exp(k ~- X)(O)r)

-

-

8r. kqn(o~ - o~,)] (C.12)

We write ~or even though o~r= 0 to remind the reader that we are on the r th leg. Also z ~ Z r is equivalent to ~ ~ 0. After carrying out one differentiation and using eqs. (2.1), (2.3), (3.1) and (3.5), the 1.h.s. of eq. (C.5) becomes

1.h.s.= ~,--,olim(m--T)!

~

~_b___dw]e oo

-1

,

(c.13)

which, using Cauchy's theorem (eq. (C.9)) in reverse, is ~,d~o

1

1 t oo ]

-

1))

(C.14)

After the change of variables, z---} ~0, the r.h.s, of eq. (2.8) is equal to eq. (C.14) because, for m > 0, the " - 1 " terms in " ( . ) " in eq. (C.14) are zero by Cauchy's theorem (they cannot be neglected in eq. (C.13), however!). Case (b) r = s. The 1.h.s. of eq. (C.6) is

,hs:.m~-~o ( r n - ~ ) i ( - n - 1 ) !

~

~

(ln(z-z')-ln(o~-oY)).

(C.15)

o)p ---~0

After carrying out the innermost differentiations with respect to ~ and ~0' and using Cauchy's theorem in eq. (C.9) in reverse

r

do

'((

1.h.s.=g-e~--1 o 2~ri ~o 2~ri ~0m (aOtn

azl(Oz' I 1 -3-~:~OoY/(z-z') 2

)

1 (o~

(C.16)

~o')2

For m >/1 and n >/1, the last term in "(-)" is zero by Cauchy's theorem and after

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making the change from R variables to H variables, the r.h.s, of eq. (2.9) becomes eq. (C.16). This completes the proof that the Wick contraction factors are equal for cases (a) and (b) in eqs. (C.5) and (C.6) for r ~ s and r = s. Off-shell conformal field theory methods give the same results as string field theory when R is associated with a string field theory diagram.

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