Loop amplitudes in covariant string field theory

PHYSICAL REVIEW D VOLUME 35, NUMBER 4 15 FEBRUARY 1987

Loop amplitudes in covariant string field theory

Hiroyuki Hata* Research Institute for Fundamental Physics, Kyoto Uni~lersity. Kyoto 606, Japan

Katsumi Itoh, Taichiro Kugo, Hiroshi Kunitomo, and Kaku Ogawa Department of Pl~ys~cs . Kyoto Uniuersity. Kyoto 606, Japan

(Received 3 October 1986)

The string-length-parameter (a ) independence of the on-shell physical A-string amplitudes in the covariant string field theory presented by the authors is shown to all loop orders in perturbation theory and for IV l;ld =26. The proof is based on the new invariance of the string action under the field transformation P Q = p ' ' ( a / a a ) Q + ( 0 2 , Q3 terms) with pf' being the center-of-mass momentum carried by Q. By making use of this u independence, the calculations of the tree and one-loop amplitudes are greatly simplified and shown to reduce directly to the operator formalism expressions. Most remarkably, we find that our pure open-string field theory contains the closed string as a bound state.

I. INTRODUCTION

In a series of papers'-5 we have constructed the interacting bosonic string field theory in a manifestly covariant manner for both the open- and closed-string sys- tems. It is described by the action

which has a gauge invariance (when d=26) under the stringy local gauge transformation

where, both in (1.1) and (1.2), terms enclosed by the square brackets are necessary only for the open-string system. We refer the reader, in particular, to Refs. 4 and 5, where the full details of the formulation are spelled out for the open and closed cases, respectively. In this paper we follow the notations and conventions of these two references and quote them as I and 11, respectively. There we have confirmed the full gauge invariance, and have shown that the on-shell physical amplitudes correctly give the usual dual amplitudes at the tree level. Furthermore, the zero-slope limit of our covariant open-string field theory were calculated in an off-shell manner to reproduce the Yang-Mills theory.4

However, there remains an important problem to be clarified before we can say that our string field theory is a satisfactory quantum field theory. It is the problem of a . Our string field @ is a functional of bosonic string coordinates X Y u ) and Grassmann Faddeev-Popov (FP) ghost coordinates c ( o ) and ?(a). In addition, @ also depends on another (real) variable a ( - cz <a < cc which we called the string length parameter:132

The role of a is as follows. The variable a is something

like a momentum and each term in (1.1) is constructed so as to conserve a . In the open-string case the cubic interaction term a3 is the product of three strings as depicted in Fig. 1, and describes the joining of two strings into one or, reversely, splitting of one string into two. In Fig. 1 the strings are connected in such a way that the string r has length n- a, . Let the string 3 be the "longest" and

then the splitting point u1 of the string 3 in its parameter space 0 < a3 < T is specified by a, as o1 = I a2/a3 1 T .

If the field @ does not have a as its argument, the splitting point oI must be given by using a probability function j i(ur ). However as was explained in detail in Sec. V D of I, there exists no &(uI which is consistent with the gauge invariance of the action at 0 (g2) .

Then the problem arises. Since the field @ contains a as its argument, the corresponding string state also has the a degree of freedom besides the mass, momentum, etc., and the unitarity holds by regarding a as a physical quantum number. However, of course, we know no such new physical quantum number in nature, and there should exist some mechanism which ensures that we may consistently discard a as an unobservable variable.

A possible answer to this question has been proposed in Refs. 1, 2, and 4. Suppose that the a dependence of the on-shell physical N-string scattering amplitude .7$;s 1s only through the conservation 6 function 6( 2, =, a, 1,

TA, (a independent (1.4)

with completely a-independent amplitude T. .. Then, if we go to the Fourier-conjugate space, say d space, of a variable, the amplitude (1.4) takes the form

1356 @ 1987 The American Physical Society

35 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY 1357

FIG. 1. The @' interaction term in the action (1.1).

S-I

T,v(a independent) . (1.5)

This implies that (physical) scattering takes place only among string states possessing the same d. Therefore, we have a unitary theory by choosing a physical state vector space with some fixed (but an arbitrary) value of d. (Physical state vector spaces with different d are equivalent to one another but completely disconnected.) Namely, unitarity in the usual sense without an a degree of freedom holds for the reduced amplitude T N ( a independent).

In order for the above scenario to work, however, we have to prove (1.4). We actually had two encouraging results which support this. First is the tree scattering amplitude. From the explicit computation of the general N - string amplitude at the tree level, we proved that (1.4) indeed holds for the on-shell physical amplitudes (see Secs. VII C of I and VI C of 11). Second is the zero-slope limit of the open-string field theory. By taking the limit a' (Regge slope parameter)+O with the Yang-Mills coupling constant held fixed in the gauge-fixed Becchi- Rouet-Stora- (BRS) invariant action obtained from (1.11, we get the following local field theory action for the mass- less Yang-Mills and FP-ghost modes contained in @ (Ref. 4):

where

6~ is the BRS transformation and ( * ) is. a complicated quantity which is nonlocal in d [see Eq. (8.34) of I, hereafter denoted as (1.8.34)]. Since the nontrivial a depen-

A

dence of S (1.6) appears only in the gauge-fixing + FP- ghost term iSB( * 1, it is clear that the physical scattering amplitude calculated from $ takes the desired form (1.4) to all orders in f i . The action (1.6) further tells us that the L-loop physical amplitude contains the divergence ( f d a / 2 ~ ) ~ :

X T.$ ' O o p ( a independent) . (1 .8 )

Since the L-loop amplitude is multiplied by (gyM I L or #, the divergent factor can be absorbed into the renormalization of them.

The purpose of this paper is to give a general proof for (1.4); namely, we show to all orders in perturbation theory that the on-shell physical amplitude in our covariant field theory depends on a, of the external states only through the conservation 6 function 6( 2, a, ). [We call this property ( 1.4) "a independence" hereafter.] The proof is based on the new invariance of the gauge-invariant action ( I . I), (1.11,

under the nonlinear field transformation GLa,

where p p is the center-of-mass momentum carried by the field @ and Y[@] is a functional of @ which consists of cubic and quartic terms in @. This invariance (1.9) with ( I . 10) is equivalent to the following property of the gauge invariant action:

under

where 6B is the BRS transformation defined by [see Eqs. (1.5.71) and (11.3.111

Since

under an arbitrary variation 6@ of @ [see (I.6.19)], Eq. (1.9) with (1.10) follows from (1.11) and (1.12).

In order to perform the perturbation calculation of the scattering amplitudes, we have to fix the local gauge invariance of S ( I . 1) and introduce the corresponding FP- ghost string fields. However, this conventional gauge- fixing procedure seems to be very difficult to be carried out and has not been successful in the interacting string field theory,3 although it has been fully clarified in the free Fortunately, we know already a gauge-fixed BRS-invariant action 5, which was first constructed in Refs. 1 and 2. It is obtained from the gauge-invariant action (1.1) simply by putting the $ component of the original field @= --Tad+* equal to zero

In proving the a independence (1.4) of the amplitude obtained from this gauge-fixed action 5, the properties (1.9) or ( 1.1 1 ) of the gauge-invariant action give sufficient in- formation. In fact, by applying the well-known diagrammatic technique8 to prove the gauge-fixing independence of the on-shell physical amplitudes in a gauge theory, we can show

1358 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA - 3 5

Although (1.16) is weaker than

which surely implies (1.4), we can deduce the a independence from (1.16) at least for .9pxJ, with 11; 5 d=26.

As a by-product of the a independence, the calculation of the tree and one-loop amplitudes in our covariant string field theory are greatly simplified. Namely, if we suitably put the external string lengths a, equal to zero, the calculation reduces to the well-known operator formalism'^.^ From the one-loop amplitudes in the open- string system we find a remarkable fact: Closed-string states are dynamically generated as a bound state in the pure open-string field theory. In this respect covariant string field theory is very different from the light-cone gauge ~ n e ' ' - ' ~ (where the pure open-string system is in- consistent and the closed string field must be introduced as an elementary field from the start1') and seems to contain further attractive features.

The rest of this paper is organized as follows. In Sec. I1 we prove our basic formula (1.9) or (1.11). The proof is

I .v 6ai is arbitrary but x 6ai =0

i = l

similar to the nilpotency proof of the BRS transformation SB (1.13) given in I and 11. However, it is rather complicated in the present case and the impatient reader may skip this section in the first reading. In Sec. I11 we derive (1.16) from (1.9) and show the a independence for N ( 526)-string on-shell physical amplitudes. In Sec. IV, as a bonus of the a independence the tree and one-loop amplitudes are found to coincide with those in the operator formalism expressions. Generation of closed-string bound state in the pure open-string field theory is also dis- cussed there. The final section (Sec. V) is devoted to the discussion. In Appendices A-F we give various formulas and technical details needed in the text.

, (1.17)

11. PROOF OF EQ. ( 1 . 1 1)

In this section we present a proof of the basic formula (1.1 1) for the gauge-invariant action. Because of the presence of quartic string interaction term in addition to the cubic one, the proof for the open-string system is much more involved than for the closed-string system. There- fore, for completeness, we shall mainly concentrate on the open-string system, and the proof for the closed-string case will be briefly sketched at the final part of this section.

First, let us recall the definitions of the 3-string and 4- string interaction terms, @3 and in the open string action (1.1) (Ref. 4):

where

r = ( p , , ~ b " , a r ), d r = d p , d T ~ ) d a , / ( 2 ~ ) ~ ~ ' ( r = 1, . . . , 4 ) (2.3)

and tr implies the trace operation for the matrix index of the fields ( @ . The vertices v'") and 1 v'") are given as

where ,u, G (oI 1, 6, E, F, and w"' are expressed commonly for both the 3- and 4-string vertices as

35 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY

G (a,) =flralfi.rr['(cr~') ( r :arbitrary) (aYJ=interaction point of the r string) ,

In the above formulas the Neumann functions Rrm are understood to be the 3- (4-) string one when they appear in the quantities of the 3- (4-) string vertex. Here, we are taking a gauge for the projective transformation which fixes Z , , Z 2 , and Z 3 to some constants for both the 3- and 4-string vertices. For the 4-string vertex (2.6) we have put Z, =x, and x ix - < x + ) correspond to a* in (1.4.7).

Equations (2.4)-(2.12) are found in Secs. I I IF , IV B, and V A of I. However, we have made a small change of notations from those of I. (i) We use the superscripts (3) and (4) to distinguish between the 3- and 4-string quantities. The equations with no superscript hold for both the 3- and 4-string vertices. [However, we shall often omit the superscript (3) for the 3-string quantities when there is no confusion.] (i) F",' and w","~ ' in (2.6) were denoted in I as F:~' and u"', respectively [see (1.6.17~) and (1.6.3011.

iiii) Equation (2.7) for ,u '~ ' gives a projective invariant ex-

, U " , ~ ~ X in (2.5) with the previous f i a o ) d a o in I is understood from (I.5.lb) and (1.4.29) with ( a,b,c, ) = (1,2,3) and z', =X.

In order to evaluate the variation %S of the action under the transformation 6: (1.12) we first calculate z r p l ( a / a a , ) 1 V ) . (From now on we often omit the Lorentz index p . ) Then we rewrite 6,s as a sum of the term possessing the form of the BRS transform 6B Y and the rest. Finally, the latter unwanted terms are shown to vanish. The calculations are rather complicated although we need no new techniques and formulas than those contained in I. Therefore we proceed by dividing the proof into several steps.

Let A, be a differential operator pression A , = z d a r i a / a a r ) [ ) : S a r = 0 ] . (2.13)

r

r = i We computed in Sec. VII C of I [Eq. (1.7.4911 the response of our vertex function p4 10) under the operation of

which coincides with p =exp( -70 x:= ( 1 /ar 1) given by Aa. There the following formula (1.7.48) for the Neu- (1.3.57). The coincidence of the 4-string vertex measure mann function was essential:

(2.14)

where N:m is related to N rm in (2.10) by - -

N:~ ==Nymexpi -n~ & - - m ~ & ) . (2.15)

Here we need to make a similar calculation including also the ghost oscillator terms. By making use of the formulas

(1) (11 ( r i - ( r i ) nu'", = - [ L , a _ , ] i a~"=&' ,y n~ y n . 1 (2.16)

hay''', =A,( - ina ,c '~ ; ) = i ~ a , / a , ) ~ ' ~ \ , (2.17a)

Aay '?\ = A ~ ( F ':\ /ar ) = - ( Sar /al )7 '?\ , (2.17b)

1 2 S a i 2 NZOco~ik~ :" ) (2.18)

k > l

[Eq. (2.18) is Eq. (I.1.9)], we obtain

1360 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA

where

Now we add and subtract

2 w'.' 2 nR Y0p ?',Sa, r n > l

s

to rewrite (2.19) into the form

where

and in deriving (2.251 use has been made of the relation

The complicated quantity RF (2.26) has in fact a very simple expression. Let us define Rya.p and R y p 2 by

3 5 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY 1361

QB v '~ ' ( 1,2,3 ) ) (3-string case) ( l - ~ ~ ' w " ' ) p e ~ I O)G(Q.,p)

QB 1 vi4'( 1,2,3,4) ) (4-string case) I other types of terms)]peF1 0 ) . 6 ( a , p ) ,

where QB - pr Q:', and in the last line we have made a prescription of (1.3.18) to rewrite the quantities in front of peFl 0 ) solely in terms of creation and zero-mode operators. Rya.p and Ryp2 represent the terms of the form ytut.p and ytp.p, respectively. Then we can show that RF (2.26) and Rya.p have a simple relation (see Appendix A):

(11 RF=Rya .p (a -n -7 '",,p r+aar 1 . (2.29)

Equation (2.29) tells us that RF vanishes in the 3-string case since we have [(I.3.7)]

QB / ~ ' ~ ' t l , 2 , 3 ) ) = 0 . (2.30)

In the 4-string case let us recall the following relation which is obtained by combining (1.4.19) and (1.4.20).

By making use of the formulas

Eq. (2.3 1) is rewritten as

In the last term b stands for the creation oscillator expression of G a r c ' r ' ( a ~ ' ) ( r is arbitrary): - -

b = V'7iarc"'(a:") =O(n 1) + [ d 7 i a , c ' " ( a : " ) , ~ ' ~ ' ]

I

where originally arc"!( a has an expansion [see From (2.29) and (2.39) we finally obtain (1.2.311

By comparing Eqs. (2.28) and (2.36) we get, in the 4-string case, Before going to the next step we further rewrite (2.25)

into the form

(2.40) where


(2.44) Our next task is to calculate with wf r ' given by (2.12). This is accomplished by the formula (2.40). In

(2.45) Appendix A, RYpZ defined by (2.28) is shown to be equal to13

1 ! n - l , = 2 T P r 2 w 1 r . - j 2 z - 6 r s L ~ a+ 2 ( n -m)m - RYP- (2.46)

r , s . tn 2 I a r ,=, 2n

in both the 3- and 4-string cases. By equating the right-hand sides (RHS's) of (2.40) and (2.46) [in the 3-string case we have R ' ~ ' ? = O instead of (2.4011 and taking the a variation, we reach after a tedious but straightforward calculation using

YP - (2.14) and (2.18) the equation

where in the 3-string case the terms containing ( d x / d a o ) b should be omitted. Since p r2 ( r = 1-3) in the 3-string case, and p r 2 ( r = 1-41 and zr,, ( d /dx)N &, .pS in the 4-string case are independent Lorentz invariants, we conclude from (2.47) the formulas

Finally for W"' defined by

wlrl= 1 -T;lwlrl

we get, from (2.48)

h a w i r t = [ jQB,GI. wlr)l+ p , d l ( r l l -c:ri~lrl

It should be noted that every term on the RHS of (2.52) contains no annihilation oscillators.

C. A, I v ! ~ ' ) and A, / v ' ~ ' )

Now from the results of the preceding subsections, (2.42) and (2.52), we obtain

where use has been made of (2.411, (2.49), (2.431, and the property [see (2.4411

{ g , d l r l j = [ ~ , M ' ~ I ~ ; ~ + [ ~ , L ( ~ I I ~ ! ~ ) = i g,lll(rll wir l .

3 5 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY

In the 3-string case, (2.531, (2.301, and

A,S(a,p)=O

gives

A, 1 ~ ' ~ ' ( 1 , 2 , 3 ) ) = ~ , b I V1"(1,2,3)) . In the 4-string case (2.53) and (2.36) lead to

Since the formulas of the a variation of the vertices have been found [(2.56) and (2.5711, we are in a position to consider the response of the gauge-invariant action S ( 1.1) under the field transformation 6, ( 1.12). However, before proceeding to this step we make some preparations.

In the preceding subsections Sa, in A, (2.13) may be any c numbers satisfying 2, Sa, =O. However, from now on we set

Sa, = p r (2.58)

with some (fixed) p. This choice does not modify the formulas of the previous subsections. Then, various quantities A,, 8, b , etc., should carry the Lorentz index p , which is often omitted for simplicity.

With this particular choice of Sa, (2.58), the following important relation holds (see Appendix B):

where

is a generator of OSp( d / 2 ) rotation: - 1

[ J p , a z ] = i n p T n n = O , 2 , . , J , J = - L n i O ) , ( J p , c o ) = _ x p

Now let us consider 6,s. First, the free kinetic term @.QB@ is invariant:

dp da a &$(@.QB@)= J 7 P p J ~ J - - - [ t r ( @ ( p , ~ o , a ) 1 QB , @ ( p , ~ o , a ) ) ] = O . (277) 2 7 ~ aa

For the 3-string interaction term @3 (2.1), we obtain, from (2.56) and (2.59),

~ , @ ~ = J d l d 2 d 3 t r 6 , { ( @ ( 1 ) ( @ ( 2 ) (@(3) ] V(1,2,3))

= - J d l d 2 d 3 t r ( @ ( l ) ( W 2 ) ( @ ( 3 ) [ A , V(1,2 ,3)) ]

where 3

Y , = - i J d l d 2 d 3 t r ( @ ( l ) / (@(2) / ( @ ( 3 ) 2 [ ~ i r ' - ~ ~ a r - l ~ ~ ' ~ ' r ' ( o ~ ' ) ] V(1,2,3)) r=1

and 6; is the BRS transformation at the tree level [(I.2.25)]:

s;(@ 1 = ( @ 1 QB, 8; / @ ) = Q B / @ ) . The 4-string interaction term is more complicated. From (2.5) and (2.57) we get


where

The desired form of 6,S, ( 1 . 1 l ) , is a total BRS transform ?jBY of some quantity Y . Here, the total BRS transformation 6 , ( 1.13) is given by [see Secs. 111 and IV of I ]

6~ =s: +g6; +g26; , (2.68)

6; 1 @ ( 3 ) ) = J d l d 2 ( @ ( 1 ) I ( @ ( 2 ) / ; V ( 1 , 2 , 3 ) ) , (2.69)

By expressing (2.63) and (2.66) as Y , ( n = 3,4) plus the rest, we obtain the desired result 6,s =6B Y with Y given by

y = + g Y 3 + f g 2 Y 4 , (2.71)

i f the following three conditions are satisfied:

E. Proof of (2.72). I. Nonhorn diagrams

The proof of the three conditions (2.72)-(2.74) is very similar to the O ( g n ) ( n 2 2 ) nilpotency proof of the BRS transformation 6 B [see (I.3.4)]:

Similarly to the case of (2.75a1, the proof of (2.72) needs a careful treatment with respect to the horn diagrams. First, let us rewrite 6; Y 3 into a more explicit form. From (2.69) and the Hermiticity condition of the string field @

[(1.2.15)],

where fl is the twist operator (1.2.14) and

the 0 ( g ' ) BRS transformation 6; on the bra-field ( @ is given by

6 ; ( @ ( 4 ) = J d 1 d 2 ( @ ( 1 ) 1 ( @ ( 2 ) 1 J d 3 ( ~ ( 3 , 4 ) v ( 1 , 2 , 3 ) ) R t 4 ! .

1 I , 2 I 2 i i r 1 2 ) ( R (1,21 / = ( 2 ~ j ~ + ' 6 ( p ~ + p 2 ) 6 ( o l + a 2 ) 6 ( ~ b 1 ' - ~ ~ 2 ~ ( ~ exp a a i y iy,, y ,, 1

From (2.64) and (2.78), we have

, (2.77)

3 5 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY 1365

where

J ( r l = J l r ! - d ~ a r - l F g l ~ ( r ~ ( u i r ' I ) (2.80)

and use has been made of the cyclic symmetry of the vertex (1.3.58)

1 ~ ( 1 , 2 , 3 ) ) = / ~ ( 2 , 3 , 1 ) ) = I V(3,1,2)) . (2.81)

Similarly to the case of ( 6 i l2 1 @ ), (2.79) consists of various types of diagrams depicted in Figs. 8 and 10 of I which arise by combining two 3-string vertices. In the rest of this subsection let us confine ourselves to the nonhorn diagram contribution to (2.79). For the nonhorn diagrams we may use the formula

( R ( 6 , 5 ) 1 n ' s i ( J ( 5 ) + J ( 6 i ) = ~ , (2.82)

to rewrite (2.79) as

with

FIG. 2. Four types of diagrams contributing to (2.29). X and G indicate the position of the interaction point factors X(cr,)and G (a,) , respectively. In diagrams A'-2-a and b strings (1,2,3,4,7,8) were originally (4,1,2,3,6,5) in (2.9 1).

As we see now, these two terms (I) and (11) separately vanish when d=26. First, the vanishing of (I) is a direct consequence of the nilpotency condition (2.75a), which has been proved in Ref. 1

and I. In fact, from (2.75a) we obtain the formula (see Appendix C)

(ufr ' ) x S d 5 d 6 ( ~ (6 ,5) 1 R ' ~ '

S d 5 d 6 ( ~ ( 6 , 5 ) j~1"' V(1,2,6)) 1 V(5 ,3 ,4)) / .onhom+[first term with (1,2,3,4)-- ,(2,3,4,1)]=0. (2.86)

V(1,2,6)) / V(5,3,4)) i nonhorn . (2.85)

Letting J d 1-d 4 t r (@( 1) 1 . . . ( @ ( 4 ) I ( ~ ~ = l ~ ( r ' ) operate on (2.86) and making the cyclic change of variables (1,2,3,4)+(4,1,2,3) in the second term, we see the vanishing of (2.84).

Next, consider the second term (TI) (2.85). Recall that the 3-string vertex / V( 1,2,3)) is written as

I ~ ( 1 , 2 , 3 ) ) = G ( u : ~ ~ ) v 1 ~ ( 1 , 2 , 3 ) ) (2.87)

with

being the oscillator representation of the 6 functional expressing the connection of string coordinates [x" ' (u ) , c " ' (u ) ,T '~~(u ) ] ( r= 1,2,3). Since / V( 1,2,3 ) ) (2.87) satisfies the same connection condition as Vo( 1,2,3) ) with respect to X(cr) coordinates, the factor 2, ( G / a r ) F ~ ' ~ ' r ' ( u ~ ' ) in (2.85) can be rewritten into the form

where, for example, x (u :~~) stands for the common X coordinate at the interaction point of the strings 1, 2, and 6: X I ~ ~ ( ~ ~ I I ) = ~ ( Z ~ ( ~ ( ~ ' = x ~ u ~ ' . Incidentally, there arise in (2.89) the same antighost zero-mode factors

(2, (1/ar)c[') as in / Vo(1,2,6)) or 1 V0(5,3,4)) . By using

1366 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA - 35

we get

The vanishing of (2.91) at d = 2 6 can be seen in quite the same manner as in the proof of the 0 (g2) nilpotency (2.75a) for the nonhorn diagrams [or (2.8611 which has been fully described in I. For example, consider the pair of diagrams A2- 1 and A2-2 of Figs. 8 and 10 of I between which the cancellation occurred in (2.75a). Corresponding to these configurations we have in the present case the four diagrams shown in Fig. 2. The cancellations occur between A2- l -a and A - 2 - b and between A - 1 - b and A - 2 -a. [That we get the right sign factors for the cancellation can easily be understood by comparing (2.86) and (2.911.1

Other two conditions (2.73) and (2.74) can be shown quite similarly to the proof of ( 6 ; ~ ~ )nonhorn=O described above.

F. Proof of (2.72). 11. Horn diagrams

Let us now complete the proof of (1.1 1) by showing (2.72) for the horn diagrams. Namely, we have to show

As we saw in Sec. V C of I, for the horn diagram contribution to ( + 16; Y3 (2.79) we must make a kind of regularization to separate the two 3-string vertices V( 1 ,2 ,6 ) ) and I V(5,3 ,4) ) by an infinitesimal "time" T: viz.,

T L ' ~ /a , x lim J d 5 d 6 ( ~ ( 6 , 5 ) 1 fY5'e , j V 1 2 , , v i5 ,3 .4 ) ) 1 horn T-0

Because of the presence of the factor exp( TL1" /a j ) , we cannot use (2.82) to carry (2.93) into the form of (2.83). Instead, we rewrite (2.93) as

where

i 1 X t r ' ( o ) = [non-zero-mode part of ~ ( " ( g ) ] = -= 2 -ur 'cos(nu) , I1

(2.95) v'r n+O

and we have, for simplicity, omitted to write explicitly the regularization factor limT,oexpi TL1"/a5) for the two terms in the square brackets of (2.94). Equation (2.94) is owing to

T L " ' / ~ ~ = - ( R (5 ,6) I O"'e ( J ( ~ ) + J ( ~ ~ ) = o (2.96)

for - 1 J"' = (non-zero-mode part of "'1 = i 2 -aIrL n Y "' - r t ,

n g 0

and - - J ( r ) = T ( r ! - v ~ a r - l ~ ; ) ~ ~ r l ( u y l ) ,

For the second term of (2.92) we have

[second term of ( 2 . 9 2 ) ] = + J d l - d 4 t r ( @ ( l ) I . . . ( W 4 ) I

35 LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY 1367 -

where use has been made of (2.59). As the horn diagram counterpart of (2 .86) , the following formula holds (see Appen- dix C ) :

J d 5 d 6 ( ~ ( 6 , 5 ) 0" ' V ( 1 , 2 , 6 ) ) V ( 5 , 3 , 4 ) ) h,,,+[first term with ( 1 , 2 , 3 , 4 ) + ( 2 , 3 , 4 , l ) ]

From this we find that the first terms on the RHS's of (2.94) and (2.99) cancel each other because z:=, ( ~ ~ / ~ , ) T [ ' x ' ' ' ( u ~ ' ) and hence z:=, J " ' is cyclically symmetric in the horn diagram case.14 Summing the rest of terms of (2.94) and (2.99) the equation to be proved reduces to

Here we should note that the curly brackets on the LHS of (2.101) is proportional to pr ( r = 1-4) when it is expressed in terms of creation and zero-mode operators a:' ( n 5 0 , r = 1-41. This is more easily seen in (2.92) and (2.93). From ( B 1 1 ) and ( B 6 ) of Appendix B , we see that V I 3 ' ) and / v i 4 ' ) are proportional to p,, and so are A,xk because 6al=pr . Therefore, in calculating the individual terms in the curly brackets of (2.101) we may drop the terms not proportional to p, since they are ensured from the beginning to cancel one another. From this fact we immediately find that the ] Z r B " ' , b ) term in (2.1011 may be dropped because as seen from (2.37) and (2.60) it contains only the creation operators a':', ( n 2 1 ) .

Now, let us consider the first term in the curly brackets of (2.101):

where again we have made explicit the regularization factor. We have the following identity for the 3-string vertex:

lim J d 5 d 6 ( ~ ( 6 , 5 ) 0 ' 5 ' e T ~ ~ ' ~ ' / c I ~

T-0

2 a r i l r 1 ( u y ' ) 1 V ( 1 , 2 , 3 ) ) = O , (2.103) r=l

since ~ ' " ( u : " ) ( r= 1,2,3) are common and 25=, a r x r = O in front of / V ( 1,2 ,3) ). By making use of (2.1031, the quantity (2.102) is rewritten as

I I (2 .102)= lim 2 a r G f ' r ' i u y ' i T-+O r = I I

V ' G - ! r ' - , 2 y C o x "(0)")

r=5,6

where we have used the formula

~ ( 1 ~ 2 . 6 1 ) V ( 5 9 3 9 4 ) ) hen,

obtainable form the Cremmer-Gervais identity (1.5.31 ) . The vertex vi4'( 1-4) ) is given by

where p';l' and E;~ ' are given by (2.7) and (2.10), respectively, with suitable 4-string Neumann functions ' f f i ~ for a diagram like BA-l of Fig. 9 in I substituted. If we move

~ ( a ; ~ ~ ) = d ~ a ~ ~ i r ~ ' ( u ~ ' ) ( r = 1 or 2 )

and

~ ( o ~ ~ ~ j = / ~ a , . i 7 ~ ~ ' ( u y ' ) ( r = 3 or 4 )


to the right, we obtain two terms:

[ ~ ( u ~ ' ~ ~ , ( l / a ~ ) ~ ~ " + ( l / a ~ ~ ~ ~ ~ ~ ~ ( ~ ) 1 v ~ ~ ~ ) = G ( ( T : " ) v;')

and

[ ( l / a l ) ~ b 1 ' + ( l / a 2 ) ~ ~ 2 ' ] ~ ( ~ 2 6 ) ~ ( f l ) v$') .

However, the contribution of the latter term to (2.104) vanishes in the limit T-0 since we have 1 d x / d T I - T-"~,

(see Sec. V C of I and below). Therefore, (2.102) is finally written as

4

2 a r ~ G z ' r ) ( u y ' )

where use has been made of the relation

I vk4') - T ~ / ~ and G(U;~ ' )G (cT:~" I vf? ) - T I / ?

and in the second line we have retained only the p, part of

by the reason mentioned above. For a given set of values a , ( r = 1-4) with s g n ( a I , a 2 , a 3 , a 4 ) = ( + , , +, - ) or ( - , +, -, + ), one of the two horn dia-

gram configurations corresponding to x =x, and x is made by combining l V( 1 ,2 ,6) ) and l V( 5 ,3 ,4 ) ) , and another by combining V(4 ,1 ,6) ) and V(5 ,2 ,3) ) (see Figs. 12 and 13 of I). Because of the cyclic symmetry, the factor 2 in front of the first terms in the curly brackets of (2.101) can be understood as the sum of these two horn diagram configurations. Taking this fact into account, the condition (2.101) is rewritten by using (2.107) as

where al, denotes the length of the intermediate string of the horn diagram.

G. Proof of (2.109) -

First, let us consider the LHS of (2.109). By putting i-z, ({,-- cc and z-z: (interaction point of the vth string) in the defining equation of the Neumann function, (I.A7I, we get

2 ~ ~ O c ~ ~ ( n u ~ ' ) = l n ~ z ~ ~ - Z s I . (2.1 10) n 2 0

We apply this formula to the LHS of (2.109), i.e., the horn diagram configuration with time interval T (see Fig. 9 BA-1 or Fig. 11 CA-2 of I):

LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY 1369

where use has been made of (1.A20):

with rr' being the interaction time of the rth string. In (2.1 11) a ( = 1 or 3) denotes the string which belongs to the same 3-string vertex as the string 4. Note that yi4'=y'"' and y ' 2 ' = y ' a + 2 ' for y =zO and TO. When T tends to zero, two (real) interaction points zb4' and zb2' coincide:

Another factor dx /dT I is obtained by differentiating

with respect to x i = Z 4 ) (recall that we are taking the gauge Z1-3 =const):

where it should be noted that , T 1 = 1 rr'-rk2' / and that the limit T+O corresponds to x + x t . When we multiply (2.111) and (2.115), the last term of (2.1111, (p4+pa)irb4'-~b2') , does not contribute since we have

1 d x / d T 1 a zb4' -zb2' -' ar T - ~ ' ~ . [Actually, from (2.114) and dp/dz=O at z =zo, we have rb4'-rb2' cc (zb4' -zb2' 1,. ] Therefore, we get

[LHS of (2.109)] = -sgn(aIM)sgn(zb4' -zb2')

where we define the intermediate string IM as the one be-

longing to the 3-string vertex of strings 4 and a , and hence we have

On the other hand, the RHS of (2.109) is obtained as follows. First, note that x =xi is the point where dp(z)/dz=O has a double root z =z$:

By taking the a variation of (2.118) and using (2.119) we get

Therefore, from (2.116) and (2.1201, the condition (2.109) reduces to

By inspecting the z-plane diagrams, one can easily convince oneself that (2.121) actually holds. The details are given in Appendix D. [In fact, if we repeat the nilpotency proof of (2.75a) given in I by adopting the x-integration form of the 4-string vertex (2.5) in place of the uo- integration form iI.4.7), we get the same condition (2.121) as a correspondent of (1.5.491.1

H. Proof for the closed-string theory

In the closed-string case the proof of (1.1 1) is much simpler since the action (1.1) contains only the cubic interaction term @3. The explicit expression of each term in the action reads5

with

where ( I ' O ( r 1 r= (p r ,TO ,nc , a , ) , d r - d p r & ~ ' d n ~ ' r ' d a , / ( 2 n ) d + 1 ( r =1,2,3) ,

Q B = Q R + + Q R - ,

Qsi = 2QBOpen[Eq. (2.21 ) ] with a~" -a~*"" ( a =a ,c ,~ ) , ab ' " "=~ , /2 ,

I V+(1,2 ,3))= / ~ ' ~ ' ( 1 , 2 , 3 ) ) [ ~ ~ . (2.4)]/6(3'(a,p) with the same substitution as Q ~ ~ i 2 . 1 2 7 ) . (2.129)

1370 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA 35 -

For the oscillator mode expansions, see (11.2.2) and (11.2.3). In particular, we have

The closed string field @ is subject to the constraint

P@=@ [or iL+ -L- )@=O] ,

expressing the invariance under the rigid a translation. It may be either the unprimed field (11.2.5) which contains 7: or the primed field without T: having the expansion @= -Tod+ll' since the difference contributes neither to (2.122) nor (2.123).

Now, let us consider A , I V c ( 1 , 2 , 3 ) ) with A , given by (2.13). From the results for the 3-string vertex of the open string, Eq. (2.56), we know that

~ A , - + Q ~ ~ G , ) V i i 1 , 2 , 3 ) ) = 0 , (2.132)

where QBi is defined by (2.127) and ai is obtained from G (2.43) by the same substitution. Further, if we make the identification (2 .58) , 6ar=pr =2aA'"", Eq. (2.59) holds for V i ) :

[ J ~ ' - ~ % a , l ~ ~ " ( r ' ~ ~ ' ( a ~ ' ) ] / V i ( 1 , 2 , 3 ) ) = 0 ,

where

I

- (The factor 2 is due to the relation pr=2ab'"r' .) By using (2.1321, (2.133), and the relations I Q B * , O ~ ] =0 and QBi V i ) = 0 , we obtain

A,{ V , ) v - ) ] = ~ ( Q ~ + + Q B ) ~ G + + ~ - ) ~ V + ) I V - )

where

with x:" being the string coordinate of the closed string:

In deriving A , 1 V c ( 1 , 2 , 3 ) ) from (2.136), we should note the following points: (i) Because of the presence of the factor T : ' ~ ' in (2.1231, only the i r ,s , t )=i1,2 ,3) and (2,1,3) term in (2.124) contributes to (2.123). Hence we need not differentiate the a's in front of 1 V + ) V - ); (ii) The term in (2.136) proportional to T:"' does not contribute since { Q ~ , T : " ' ~ = - ~ ( L $ ' - - L : ' ) , 2 ( L + - L - )=0, and T : ~ ~ ' ~ T : ~ ~ ) ~ T : ' ~ ' T ~ ' " = o . Therefore, we obtain

Aa 1 V c ( 1 , 2 , 3 ) ) = - i Q B y 1 2 3 2 [ ~ ~ ' - / % a ~ l ~ ~ l ~ j r ' ( a ~ ' ) ] 1 F c i 1 , 2 , 3 ) ) , (2.140) r -

where I Vc ) i: / Vc ) without 9 1 2 3 ; 1 Vc ) = 9 ' 1 2 3 PC ). From (2.1401, the variation of the closed-string action S c = @.QB @ + under the transformation i 1.12) is given by


where

Equation (2.141) can be written in a desired form (1.1 11, if we can show

where the total BRS transformation 6, (1.13) for the closed string is given

The proof of (2.143) is quite similar to that of (6; ) 2 = 0 presented in Ref. 2 and Sec. IV of 11, and can be carried out following the previous proof of (6;Y3)nonhorn=0 in the open-string theory.

111. a INDEPENDENCE OF ON-SHELL PHYSICAL AMPLITUDES

In the preceding section we showed that our gauge- invariant action ( 1.1 ) has a marvelous invariance under the transformation (1.10). By making use of this invariance we prove in this section the desired a independence of the on-shell physical amplitudes, i.e., Eq. (1.4), to all loop orders in perturbation expansion.

Since our gauge-invariant action has a stringy local gauge symmetry (1.21, we have to fix the gauge and introduce the corresponding ghost fields in order to perform the loop expansion of scattering amplitudes. However, such a conventional gauge-fixing procedure, which has been fully studied in the free case,637 seems to be very difficult to apply in the interacting case3 and is not known at present.

Fortunately, however, we know a gauge-fixed action of the interacting string. It is obtained from the gauge- invariant action (1.1) by setting the $ component of the original string field @,

equal to zero,

[See (1.6.15) and (11.5.27) for the notations.] Here (b is re- laxed to carry any internal ghost number nFp contrary to the case of gauge-invariant action in which @ is restricted

A

to the npp = - 1 sector.33' This S is no longer gauge invariant but instzad has an invariance under the new BRS transformation 8,; i.e., under

we have

where the * and ( o o ) products are defined in (1.6.29) and

(11.5.37). This BRS transformation 6, is nilpotent on the mass shell [i.e., if we use the equation of motion of 2 (3.2)]:

12d = 0 (on shell) . (3.5)

Now, the a invariance (1.9) or (1.1 1) of the gauge- invariant action is inherited by the gauge-fixed action (3.2) in the following manner. Let us start from (1.11). Since 6,@=p(a/aa)@ does not mix the (b and t,h components, we have

under

By expanding Y [@I in powers of $

(3.6) is rewritten as

where use has been made of (3.3) and

[see (1.6.231 and (II.5.32)] which is a consequence of (1.14). Equation (3.9) says that 6,S consists of two terms: one is the BRS transform o f f = Y *=o and another is proportional to the equation of motion of 2. Similarly to (1.10) the latter term can be included in the fielcj transformation. Defining the nonlinear a transformation 6NLa by

we have

[This result is also obtained directly from (1.9) and (1.10). In fact we have 6NLa4 = ( 6NLa(b ]

Now, let us turn to the proof of the a independence of on-shell physical amplitudes obtained from the gauge- fixed action S^ (3.2). The argument is an application of two well-known techniques in the particle field theory used in the proofs of (i) the invariance of the S matrix under a nonlinear field redefinition, 4-4 + f ( d ) with f ( 4 ) being at least quadratic in 4 and (ii) the invariance of the physical S matrix in a gauge theory under the change of gauge-fixing condition. [See Ref. 8. In fact, (ii) is an application of (i).]

The (renormalizedl on-shell amplitude of N-string scattering is generally given by


'V

,F .v 1 i p l , . . . ,ps ;a l , . . . ,as)= n lim ~ ~ - ' / ~ i p i , a ~ ) i p i * + m i ~ ) ( d ~ ( p ~ , a l ) . . . ~ Y ( P . v , ~ . v ) ) , (3.13) i=l p,2--mi2

where ( d l . . . 4 ) is an N-string Green's function

( d l . . m,)= J a m m , . . . d \ e 2 [ $ l / ~ g d e 2 [ $ l ,

and 2; is a wave-function renormalization constant

( 2 ~ ) ~ + ~ 6 ( p +p1)6 ia+a ' )Z i ip , a )= , lim (p i2+mi2)(d i (p ,a)d i iP ' ,a ' ) ) . p---m,2

The field d l appearing in (3.13) should be understood to represent the i component with mass mi of the original string field d

d l = ( i I d ) . (3.16)

From (3.13) we have

First, let us consider the last term. By making a change of integration variable in (3.14)

d+d

with SNLa given by (3.111, we get15

Equation (3.19) together with the BRS Ward identity

leads to

Here we should note the following points: ii) Only the part of (3.2 1) which has poles at p12z-m, * in each ith leg can contribute in (3.17); (ii) h ( d ) = d 2 [ +d3] since Y [ @ ] =@'[ +a" [see (2.71) and (2.14011; (iii) s B d = e B d +d2[ +d3], but (&dlk = ( k ( eB d ) = O since we are considering the scattering of transverse physical states k ) annihilated by oB. Hence, the well-known argument8 leads to

X a \

2 ~ k - - ( d ~ . . . d ~ ) = - 2 i A k - B k ) . ( d l . . . dV)+inonpole part) , k=l ask k = 1

where Ak and Bk are the values at p k 2 = -mk2 of the diagrams shown in Fig. 3.

The change of the wave-function renormalization constant Z, can be read off immediately from the definition (3.15) and Eq. (3.22) with N = 2 (Ref. 16):

Remarkably, the first and second terms on the RHS of (3.17) just cancel each other, and we obtain the desired identity

Equation (3.24) is indeed a necessary condition for the a independence (1.4) to hold. But, is it also sufficient? The answer is "yes," at least when N 5 d= 26. The reason is as follows. The a independence is assured if we can show

where 6 a i is subject to the constraint x:=l 6 a i = 0 but otherwise arbitrary. Let us multiply (3.24) by @ = ( O , E ) with an arbitrary 25-vector E. Then 6a i corresponds to €.pi. When N 5 26, the set of momenta ip l , . . . , p ~ ~ - ~ ) are almost always independent vectors in 25 dimensions,


FIG. 3. Diagrammatic representatiotl of A k and Bk in Eq. (3.22). In both diagrams the external hk line is amputated. In Bk the dashed line shows the FP-ghost field line.

i.e., except for some zero-measure regions. Therefore, by varying 6 we can vary e a p i ( i = 1, . . . , N - 1 ) independent- ly. This fact, together with the analyticity of .Y~$:,, with respect to p, ensures the a independence. When N 2 27, p i ( i = 1, . . . , N - 1) are not independent and the above argument does not apply. However, we hope that there exists some ingenious argument to deduce the u independence for arbitrary N.

In this section we have shown the a independence of on-shell physical N-string amplitudes calculated from the gauge-fixed action $ (3.2) to any loop orders and for N <26. The basic tool was Eq. (3.12) for the present gauge-fixed action 2 (3.21, which has been derived from the special symmetry t iNLa (1.10) of the gauge-invariant action. However, Eq. (3.12) shows a typical form of the response of a gauge-fixed action under a symmetry transformation of the original gauge-invariant action: In the conventional gauge-fixing procedure the relation between the gauge-fixed action and the gauge-invariant one is generally given by"

and the choice of X corresponds to the choice of gauge. If Sin , has an invariance under a field transformation 6 , 6S,,, = O (and if [ 6 , 6 B ] =0) we have

Therefore, if the conventional gauge-fixing procedure applies to the interacting string field theory, we expect the proof presented in this section holds valid for any gauge- fixed actions.

IV. CALCULATION OF ONE-LOOP AMPLITUDES

In the previous section we proved the a independence of on-shell physical amplitudes, which implies that in

FIG. 4. Two types of diagrams contributing to the scattering with ordering (1,2,3,4).

computing the reduced amplitude T.,, of (1.4) we can take any values for the a ' s of the external states as long as they are conserved. This fact greatly facilitates the calculation of the tree and one-loop amplitudes in our covariant string field theory since by suitably setting the external a ' s equal to zero the calculation reduces to the well-known one of the operator f o r r n a l i ~ m . ~

A. Tree amplitudes

Tree amplitudes in our covariant string field theory were computed in Secs. VII of I and VI of I1 for general external a, and their a independence was shown from the explicit expression of the amplitudes. In this subsection, as a preliminary step to consider the one-loop amplitudes, let us reexamine the tree amplitudes and show that they coincide with the operator formalism's.

Consider, for example, the scattering of four open strings 1-4 with a l , a 2 > 0 and a 3 , a 4 < 0. As explained in I the complete dual amplitude with ~ h a n - P a t o n I 8 factor ordering (1,2,3,4) consists of two types of diagrams: (a) and (b) depicted in Fig. 4. (Here we assume

1 a* I > / a3 / . ) In these diagrams let us take the limit az--t + 0, a3- -0 keeping the ratio az/a, held fixed. Then, diagrams (a) and (b) of Fig. 4 reduce to ( a ' ) and (b ' ) of Fig. 5, respectively.

First, we argue that Fig. 5(b1) vanishes. By examining the a3=e+O limit of the reduced 3-string vertex I u ( 1,2,3) ) (I.6.16a) in the gauge-fixed action 2 (3.21,

FIG. 5. Two diagrams (a') and (b'), to which the diagrams (a) and (b) of Fig. 4 reduce in the limit az,a3-+0, respectively. The strings with infinitesimally small a are drawn by wavy lines.

1374 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA 3 5 -

we can show (see Appendix E)

1 " D I 2 / 2 i L 1

u (1 ,2 ,3 ) ) - a, = e - 0 ,

In Fig. 5(b1) we apply this formula for the vertex 1 u ( 7 ,4 ,1 ) ) with a7+0. If the intermediate (off-shell)

momentum p, satisfies ( p 7 2 / 2 ) - 1 > 0 , we have / u(7,4,1))-0 (a7-0). Other quantities in Fig. 5(b1) are

nonsingular in the limit a,-0. [Since the vertex (4.1) is a function of the ratios a r + l / a r , the vertex u (2 ,3 ,7) ) in

Fig. 5(b1) is invariant under the uniform rescaling of a2 and a3 with a2/a3 fixed.] Therefore, by defining the amplitude as the analytic continuation from the region satisfying ( ~ ~ ~ / 2 ) - 1 > 0 (i.e., the spacelike region beyond the tachyon mass shell), we can regard Fig. 5(b1) as giving vanishing contribution. [This is the usual procedure. For instance, recall the conventional, manipulation in the operator formalism: ( - L ) ' = Jo dx x - L - ' I

The vanishing of Fig. 4(b) in the limit u2,a3-0 can also be understood from its expression as an integral over the Koba-Nielsen variables:

[See (1.7.26). The present Fig. 5(b1) corresponds to Fig. 22(b) of I, and the integration region in (1.7.26) should be re- placed by x o <x j 1.1 Here, x o is the value of x when the two interaction times T I and T 2 given by (1.7.14) coincide. By inspecting Eqs. (1.7.101, (1.7.131, and (1.7.14) one can easily see that x,-1 as a2,a3-0, and hence the integral (4.3) vanishes in this limit. [Note that x i - 1, T z -(x, 1 ) and T2 - T I -a41nx as a2,a3-0.1

Next, let us consider diagram (a ' ) of Fig. 5. In this diagram the external lines 2 and 3 with infinitesimal string lengths are put on shell and physical. In general, we can show that the following formula holds for the DDF stateI9 D D F ) which does not contain the ghost oscillators and satisfies the physical state condition2' Q* D D F ) = O (see Appendix El:

lim (I ;DDF u(1 ,2 ,3) ) I p , onshell=v(2) 1 r ( 2 , 3 ) ) , (4.4) a,-0

where V(2) is the operator formalism vertex made of aL2' oscillators for the emission of i 1 ;DDF) state, and r ( 2 , 3 ) ) is just R ' ~ ' 1 R ( 2 , 3 ) ) with the ghost zero mode Fo removed:

In fact, we can show that V given by (4.4) satisfies9

where L,X is the Virasoro operator

L , X = + 2 : ~ , - ~ . a ~ : . (4.7) rn

Equation (4.6) is a consequence of 25 = , ( Q g' - L "'w l r ' ) 1 u ( 1 ,2 ,3) ) = O which is the To-independent part of the identity 2 j = , Q:' i V( 1,2,3 ) ) =O. For example, for the tachyon state 1 ;DDF) = 0 ) ' we have

where k = p l and x 2 = i ( a / a p 2 ) . Therefore, the calculation of diagram (a ' ) has been reduced to that in the operator formalism. [Our (inverse) propagator

L = - (p2 /2 ) - - 2 { a _ n . a n + n ( c _ , ~ + T - , c n ) ] + l n > l

in (3.2) contains the ghost oscillators. However, they do not contribute to the operator formalism calculation of the tree amplitudes if all the external physical states are chosen to contain no ghost excitations.]

If we put a l , a3 -0 in Fig. 4 instead of a2,a2-0, both diagrams (a) and (b) survive. However, as is well known in the operator formalism, the sum of these two diagrams just give the correct dual amplitude which coincides with (a ' ) of Fig. 5.

Let us next discuss the contribution of the quartic interaction term. So we consider the scattering of the same strings 1-4 with a different ordering (1,3,2,4). In this case, three diagrams (el, (0, (g) of Fig. 6 contribute. As before, we take the limit a2,a3-0. In this limit diagram (e) vanishes by the same reason as (b ' ) of Fig. 5, and diagram (fl reduces to the

A

operator formalism diagram tf') of Fig. 7. Diagram (g), which contains the 4-string vertex u i 4 ' ( 1 ,3,2,4)) of S [see (1.6.15~) and (1.6.16b)l

35 LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY -

FIG. 6. Three diagrams contributing to the scattering with (1,3,2,4) ordering. Diagram (g) contains the 4-string vertex.

also vanishes in this limit. This is because the interval of x integration of (4.91, [ x - ,x + 1, tends to zero as a2,aj-0 as seen from (2.1 18) and (2.119). (Actually, we have x + - -2, as a2,a3-0.) Therefore, in the case of ordering (1,3,2,4) also, where the 4-string vertex was indeed necessary for finite a , ( r = 1-41, we have only to calculate the operator formalism diagram (f ' ) .

In this subsection we have considered only the 4-string tree amplitudes. However, the reader can easily convince himself that any tree amplitude reduces to the operator formalism's by suitably putting the external a, equal to zero except for two a, (a l and a4 in Fig. 5 ) .

B. Planar one-loop amplitudes

FIG. 7. The operator forma!ism diagram to which diagram (D of Fig. 6 reduces in the limit a2,a3-+O.

plitude is also a independent. Hence, let us consider the limit of putting all the external a, ( r = 1-41 equal to zero uniformly (i.e., by fixing the ratios a, + /a, 1. In this limit, remarkable simplification occurs: All the diagrams except for (a) and (c) in Fig. 8 vanish. The contribution of diagram (b) is written as an integral over the loop-string length aL :

[We are considering the reduced amplitude T.\, of (1.4) which is truely a independent.] By making the rescaling a~ +€aL, we have

The contribution of diagram (a) is similarly written as

Let us consider the one-loop correction to the diagrams where we have introduced the cutoff A, which should be given in Fig. 4. There are many diagrams to be con- taken to be cc in the end. Now, if we take the limit sidered for finite a, ( r = 1-41, and typical of them are a , + O in the integrand f 'ai(aL /a, ), it reduces to the drawn in Fig. 8. (Here, we do not consider the correction one-loop amplitude of Fig. 9 in the covariant operator for- to the external lines.) r n a l i ~ r n , ~ ~ which is clearly a, independent. Therefore (by

Now, we know from the general argument given in the allowing the exchange of aL integration and the a,-0 previous section that the (on-shell physical) one-loop am- limit) we get

A JaL ~ a ) - 1 Jo "(operator formalism amplitude of Fig. 9 )

a,-0

A and the divergent factor ( So da, /2n) is factorized out. Quite similarly, the contribution of diagram (c) becomes

~ " ' ( a , - [ J :~ 2 1 x (operator formalism amplitude of Fig. 9 ) . a,-0

Diagram (d) vanishes in the limit a , + O by the same reason as diagram (b), (dl, or (el. Therefore, summing reason as diagram (b') of Fig. 5: The intermediate string (4.13) and (4.141, we get the expected result [see (1.811: 5 which is attached to the annulus of Fig. 9 is off shell and has vanishing as. Diagram (e) contains a 4-string ~f~ 2 1 x (operator formalism vertex, which vanishes in the limit a l ,a4-0 like in diagram (g) of Fig. 6. Other diagrams not drawn in Flg. 8 also vanish in the limit a , + O ( r = 1-4) by the same amplitude of Fig. 9) .


fT] C . Nonplanar one-loop amplitudes

Tu, 7

T T T j L

At the one-loop level there arises the open AV-string amplitude which makes the Chan-Paton group singlet by ( I , . . . , k ) and ( k + I , . . . , N ) strings separately. Such amplitudes do not appear at the tree level. For example, we have a 4-string amplitude which is group singlet in the (1,2) channel, whose typical diagrams are drawn in Fig. 10. As in the previous subsection, by taking the limit a,-0 ( r = 1-41, only diagrams (a) and (c1 survive and the whole amplitude is given by calculating the nonplanar operator formalism diagram of Fig. 1 1 :

. . . . . - - - - -

G-: x (operator formalism

2 " 3 - 5 : - amplitudes of Fig. 11 ) . (4.16) 1 ; : 4 This result (4.16) implies indeed a surprising fact. We

have started from a pure open-string system. However, as is well known in the operator formalism calculation (see, e.g., Sec. IV.5 of Ref. 221, the amplitude of Fig. 11 contains poles at

1 ' '4

(d) (e) which are just mass levels of the closed string. [Kecall also that (1,2) channel is a Chan-Paton group singlet.] Therefore this fact shows that the closed string is dynami-

FIG. 8. Typical diagrams contributing to the one-loop correction to the diagrams of Fig. 4. Each diagram is drawn in two manners. In the lower conventional Feynman diagram the ordering of interaction is more transparent than the upper string diagram. Note that diagrams (a)--(c) belong to the same Feyn- man diagram. Two vertical wavy lines XY in diagram (e) with a 4-string vertex should be identified to for111 a string loop.

A The divergent factor ( $-,daL/2rr) should be absorbed into the renormalization of f i or the coupling constant g.

Here we have considered only the open-string one-loop amplitudes. However, the above argument applies straightforwardly to the closed-string case, and the closed-string one-loop amplitude also reduces to the operator formalism's by letting all external a, tend to zero. [Therefore, in the closed-string case we have a divergence arising from the modular invariance2-? in addition to the a~ divergence of (4.151.1

\ , -- /' - (c) (d) - L -

1 -- 4 F I G 10 Typ~cal one-loop diagrams which contr~bute to the

FIG. 9 One-loop operator formal~sm amplitude il,2)-singlet ampl~tude


1274 FIG. 12. Open string two-loop diagram with external a set

FIG. 1 1 . Nonplanar operator formalism diagram. equal to zero.

cally generated as a bound state of the open string. In the light-cone gauge string field theory, however,

things are very different. First, in the light-cone gauge, pure open-string system is not consistent with Lorentz invariance and we have to consider an open-closed mixed system which contains closed-string field as an elementary field.'' Second, diagrams such as (a) and (c) of Fig. 10, which were the origins of generating closed string in the covariant formulation, are missing in the light-cone gauge since strings can propagate only in the positive T direction. [However, it is actually a very subtle problem whether the contributions of the "Z graphs" such as Figs. 10(a) and 10(c) really vanish in the light-cone gauge, and is not yet quite settled (see, e.g., Ref. 24). If such Z graphs have nonvanishing contributions, there is still a possibility that the light-cone gauge formulation of open string is also a consistent theory even without introducing an elementary closed-string field and produces the latter as a bound state.]

Therefore, covariant formulation seems to be much simpler and fascinating although there remain many points to be clarified as we shall discuss in the next section.

D . Higher loops

zero. For example, in the two-loop case we encounter diagrams such as the one given in Fig. 12. In this diagram the vertices to which the external wavy lines with infinitesimal a attach are the operator formalism vertices, but the two vertices where the strings with length a l and a2 join to form another string with length a l+a2 are the original 3-string ones 1 u ( 1,2,3 ) ) (4.1). In addition we have to perform (nontrivial) integrations over the loop string lengths a , and a 2 . Hence, the actual calculation seems to be very difficult.

However, one can easily E e that the L-loop amplitude contains the divergence ( J da/2771L which is factored out like (1.8). After puttin-gmall the external a equal to zero, the L-loop amplitude is written as an integration over the loop string length a - a~ :

where the integrand f depends only on the ratios a k / a l . Let us introduce the polar coordinate ( a , f l a ) for

L ( a l , a 2 , . . . , a L ) [ a = ( zk ak2)1 /2 ] . Then, since the in- In higher-loop amplitudes also, much simplifications tegrand f does not depend on the radius a , (4.18) is (naive-

occur by taking the limit of all the external a, tending to ly) rewritten as

I

A

thus showing that ( Ipmm da/277)l is factored out as was the gauge invariance and the nonlinear a invariance 6,,, conjectured in I. This divergent factor can be absorbed is necessary. intomthe renormalization of the coupling constant g or f i , ( J P m d a / 2 ~ ) . g - g , since the L-loop amplitude is multi-

V. DISCUSSION plied by gL or #, leaving the genuine L-loop amplitude ( r( L /2 / 2 d / * 1 d f l f ( a, ) which corresponds to In this paper we have proved the a independence of the T , { ' ~ O of (1.8). Of course, (4.19) contains dangerous ma- on-shell physical N-string amplitudes to all loop orders in nipulation of divergent integrals, and a careful treatment perturbation theory and for N < d = 26, by making use of with some kind of regularization which is consistent with the nonlinear a invariance (1.10) existent in our gauge-

1378 HATA, ITOH, KUGO, KUNITOMO, AND OGAWA 3 5 -

invariant action. Owing to this a independence the actual calculation of the tree and one-loop amplitudes are greatly simplified and are shown to coincide with the operator formalism expressions. Remarkably, from the one-loop amplitudes in the pure open-string field theory, we have found that the closed strings are dynamically generated as bound states of the open strings.

Now there remain two important problems to be solved in order to make the content of this paper a more satisfactory one. One is to extend the proof of the a independence to N-string amplitudes with N > 26. We know that the tree on-shell physical amplitudes are a independent for any number of the external string states. Further, the one-loop N-string amplitudes calculated by assuming the a independence reproduce the correct ones of the operator formalism also for N > 26. Therefore, it is quite natural to expect that the a independence holds for any N . This may be proved from (3.24) or by any other method. In any case, the a independence must be proved for all N in order that we can consistently discard the string length a as an unobservable degree of freedom.

Another problem is to invent a regularization method for the divergences arising from the a integration in the multiloop diagrams (and also for the divergences from other origins). As we have shown in Sec. I V D by a simple-minded manipulation, the L-loop on-shell physical amplitude is expected to contain a multiplicative divergent factor ( d a / 2 ~ ) ~ which can be absorbed into the renormalizazgn of the loop-expansion parameter g or fi. However, in order to consistently factorize out such divergences more rigorously in multiloop diagrams, we have to introduce a regularization, which should violate neither the gauge invariance nor the nonlinear a invariance S N L a .

The most striking result of this paper is the finding that the closed string is contained as a bound state in the pure open-string field theory. This fact raises some questions. First, can we construct, as in the light-cone gauge case, a covariant string field theory which contains both the open- and closed-string fields as elementary fields and describes the interactions between them? And if possible,

what is the relation between our pure open-string field theory and the open-closed mixed theory? If such a mixed theory existed, then we would have two kinds of closed strings, in particular, gravitons there: one is elementary and the other is a bound state. Therefore, we think that it is impossible to construct a open-closed mixed field theory in a covariant and consistent manner. In any case, this problem will be clarified soon if we try to construct a nilpotent BRS transformation in the mixed system which contains open-closed and open-open-closed vertices.

The second problem is concerned with the closed-string gauge invariance in the pure open-string field theory. Since our pure open-string field theory conceals a closed string as a bound state, we can expect that the open-string field theory action has in fact an additional gauge invariance which corresponds to ( 1.2) of the pure closed-string field theory. This gauge invariance with closed-string transformation parameter A should contain a part corresponding to the general coordinate transformation. To give an explicit expression of this gauge invariance is our future problem. It may also be important for constructing the type-I superstring field theoryZ5 in a covariant manner.26

Note added in proof: A. Neveu and R. West [Nucl. Phys. B278, 601 (198611 have shown that our 3-string vertex is related to the Caneschi-Schwimmer-Veneziano vertex through a conformal transformation, which also implies the a independence of the on-shell physical 3-string amplitude at the tree level.

ACKNOWLEDGMENTS

The authors would like to thank K.-I. Aoki, H. Aoya- ma, and M. Bando for warm encouragement from Cali- fornia. They also thank their colleagues at Kyoto Univer- sity for discussions. The work of one of us (T.K.) was supported in part by a Grant-in-Aid for the Scientific Research Fund from the Ministry of Education, Science and Culture, No. 61540206.

APPENDIX A

In this appendix we show Eqs. (2.29) and (2.46). [A similar calculation was done also in Appendix E of I for the 3- string vertex in the special gauge ( Z , , Z 2 , Z 3 ) = ( l,O, a 1.1 The following calculation applies to both the 3- and 4-string cases.

From the ghost zero-mode structure of QB (Ref. 20)


we have

( 1 -T~ 'w" ' ) eF 0 ) = 2 (0 g ' - ~ " ' w " ' ) e ~ 10) +(terms containing To) I The terms Ryap and RypZ are contained in the first term on the RHS of (A5), and we make the following manipulation to express it in terms of creation and zero-mode operators:

with d given by

@ = 2 ( Q g ' - W ( " ~ ( " ) ,

r

(Note that the difference between w"'L(" and ~ " ' w " ' affects neither Ryap nor RypZ . After a straightforward but tedious calculation by using the formulas

we are led to

&eFI O > = ~ ~ ( [ ~ B , F ] + ~ [ [ ~ B , F ] , F ] ) / 0 )

1 + 2 2 ( - i ) ['~"6"+--- +m,08rs- L% ~ , , , 8 ~ ~ rst n t l m t l a r a r n s

+ ( y , y a a , y y Y terms) eFIO) , I

Equations (2.29) and (2.46) are immediate consequences of (2.281, (A5), (A8), and (A9).

L" 'eF O ) = e F ( ~ ' r ' + [ ~ " l , F ] ) / O ) =

APPENDIX B

In this appendix we show Eq. (2.59). First let us consider

- + p , 2 + 1 - 2 2 nN~oa'", .p,+(aa,yy terms) (A9) s n t l

with

1 V ) = 1 ~ ' ~ ' ( 1 , 2 , 3 ) ) or 1 ~:"(1--4)) . (B2)

Recall that / V) is given in both cases by [see (2.4) and (2.6)]

( r : arbitrary) , (B4)

we have

{ J " ' , v"Gas-irF'(of') ]

= { Gar-'~~)~(r'(o~)),fias.irF'(o~') ]

-8rs / ;x(r~(~y)) - (B6)

and hence 2, [ J "'- (&/ar ) T ~ ' x " ' ( ~ ~ ' ) ] anticommutes with the ghost prefactor G (oI 1:

From (2.60), (2.61), and Then (B 1 ) is calculated from the formulas


a r l r ~ ~ ( r ' ( o : " ) e E ~ ) . b ( a , ~ ) = V ' ; (B10) X" ( o y l ) e E 1 O ) . 6 ( a , p ) Is: arbitrary) .

Equation (B10) is because X " ' ( o ) in front of e I 0 ) .6( z r p r ) satisfies the connection condition of Fig. 1. Since (2, ( l / a r )T[ ' ) '=o we get

Next we consider

From (2.221, ( B 4 ) , and (2 .18) , we have

and similarly to ( B 7 ) , G anticommutes with G (o I ) :

( G , G ( o ~ ) ] =O . (B14)

Therefore, in ( B 1 2 ) can be moved to the place in front of e E O).S(cr,p) of V ) ( B 3 ) . Let us rewrite - 2, A , ~ ' , $ F [ ' in b:

However, the last term of (B15) does not contribute to b V ) . This is because 2, 6 a s ( 2, ,oi@ ~ocos (no :" ) ) with 2, Sas=O is independent of r [see ( I .B4)] and hence 2, ( l / a , ) ~ ; ' can be factorized. From (2.22) and (B15) we finally obtain

If we set 6 a r =pr , (B11) and ( B 1 6 ) implies (2.59).

APPENDIX C

In this appendix we derive Eqs. (2.86) and (2.100). These equations are bra-ket representation of the 0 (g ' ) nilpotency condition (2.75a) of the B R S transformation. From (2.69) and (2.76) we get

3 5 - LOOP AMPLITUDES IN COVARIANT STRING FIELD THEORY

On the other hand, from (2.651, (2.701, and (2.36) we have

From (2.75a) and the above two formulas, we find that this equation holds

J d 5 d 6 ( ~ ( 6 , 5 ) I R ' ~ ' / V(1,2,6)) ~ ( 5 , 3 , 4 ) ) + J d 7 d 8 ( ~ ( 7 , 8 ) R"' / V(2 ,3 ,8)) V(7,4,1))

[Precisely speaking, this equation (C3) is stronger than the nilpotency condition (2.75a) itself. However, in Sec. V of I we have actually proved (C3) which is equivalent to (I.5.73bl.l Equations (2.86) and (2.100) are consequences of (C3) and the fact that the nonhorn diagram contributions to the first and the second terms on the LHS cancel each other and their horn diagram part cancels with the third 1 vi4' ) ( x = x ) term.

APPENDIX D

In this appendix we show that Eq. (2.121) actually holds. The proof is accomplished by drawing the z plane diagrams. Here it is convenient to consider a unit disc instead of the upper-half z Figure 13 shows a diagram corresponding to the 4-string configuration. The positions of z1-3 are held fixed and we vary Z,( =x). The increase of x on the real axis in the upper-half z- plane diagram corresponds to the anticlockwise movement of Z , on the edge of the disc in Fig. 13. Hence the positions of x (x - < x + ) are as indicated in the figure.

Now let us move Z, in the anticlockwise direction. Then Fig. 13 changes as shown in Fig. 14. This is easily

FIG. 13. The z-plane diagram corresponding to the 4-string configuration ( x - < x < x + ) with al,a3 > 0, a2,aj<0 or a,,a3<0, a2,a4>0.

understood if we notice that as we increase x, the points Z3 and Z , , and hence the strings 3 and 4, become "closer." (Figure 14 shows the case a, 1 > a3 . When

1 a, 1 < a3 the interaction point z, in Fig. 13 moves to the edge of the disc between Z1 and Z,. From Fig. 14 corresponding to x + < x < Z3 we see that (2.121) holds for x =x because we have sgn(a4aIM)= - and h' sgn(zb4 -zo I= - . [Recall -aIM=a,+a3 in (2.1 17).] When a31 > ! a 4 we have sgn(a4aIM)=sgn(zb4' - z F ) = + as can easily be seen by drawing the corresponding diagram.

The diagrams corresponding to x =x- and Z , < x < x - are drawn in Fig. 15 in the case

( a, ( (2, / ai,li In Fig. 15 we have sgn(a,aIM) = - and sgn(zo -zo ) = + , and hence (2.121) with x =x- actually holds. [When a, / < a, I , the interaction point zo in Fig. 13 moves to the edge of the disc between Z, and Z3 and we have sgn(a,aIM)= + and sgn(zb4'

( 2 ' -zo ) = - . I

APPENDIX E

In this appendix we show Eqs. (4.2), (4.41, and (4.6). For this purpose we need expression of the 3-string Neu- mann functions N:m in the limit a , + O Here we take

FIG. 14. Diagrams corresponding to x =x + (horn diagram) and x + <x < Z 3 in the case a4 1 > a3 1 .


FIG. 15. Diagrams corresponding to x = x - (horn diagram) and Z 2 < x < x in the case I a4 > a , ( .

( Z 1 , Z 2 , Z 3 ) = ( l , 0 , x ) and use the formulas of (1.3.11). For N we have

n - l 1

---[sgn(~a,)]" (a1 =t-0) , ea3

- ( E l )

1 - 3 ( - Y N: - ----- , N n - - ----- ( e =2.718 28. . . ) , n a3 " ' a 3

and for N;m(n,m 2 1)

In addition, since

we have

Equation (4.2) is an immediate consequence of (E3). From the above formulas one can easily understand

that the LHS of (4.4) can be written in the form of the RHS, and that V(2) does not contain the ghost oscillators since ( 1;DDF does not. [The vertex u ( 1,2 ,3) ) has originally the 6 function 6 ( p , +p2 +p3 1, which, however,

can be rewritten as e - i p 1 ' x 2 6(p l + p 2 as in (4.8).] Then we show that V given by (4.4) satisfies (4.61. As seen from (A5), the property ( xr Q:') V( 1 ,2 ,3) ) = 0 implies

Multiplying this by ( 1;DDF from the left and using (4.4) and

Q;' l ;DDF)=O ,

L"' 1;DDF) =O ( p , : on shell) ,

we get

where w"' ( r=2 ,3 ) here are understood to represent w"' (1.3.51) with c" ( c n > l ) put equal to zero since ( 1;DDF I does not contain the ghost excitations. The last equality in (E6) is owing to the fact that V(2) does not have ghost oscillators in it and hence [ v(2 ) ,w1" ]=0 . Now we have the formula

lim w12 ' r ( 2 , 3 ) ) = - lim w'" 1 r ( 2 , 3 ) ) a , -0 al-0

which is derived from (1.3.511, ( E l ) , (E2), and

' 2 ) [a, +(--)"a1?:] j r ( 2 , 3 ) ) = 0 ( a n = a n , y n , p n ) . (E8)

By using (E7) and

in Eq. (E6), we have

v ( z ~ ] - [ L ~ ~ ~ , v (211 2 c;21

i ! n + O 1 - v 2 , 2 2 j 2 , . E ~ O )

n # O

Equation (4.6) is obtained by substituting

QB = - 2 c-,L,X+ (purely ghost term , ( ~ 1 1) n #O

[ L C , I =ncn , ( E 12)

in (E10) and comparing the coefficient of ~ ' 2 ; .

APPENDIX F

In this appendix we show that the string field path- integral measure G d in (3.14) is invariant under the changes of variables (3.18) and


d-4 + h$,d (A: Grassmann odd) . (F1)

We consider the open-string case. The proof for the closed string is quite simgar. First, let us consider ( F l ) . The BRS transformation 8B is given by (3.3) with the * and ( 0 o ) products defined by (1.6.29). In particular we have

where r and dr ( r = 1,2,3) here denote ( p r , a r ) and

dp,dar/(2rr)d+1, respectively, and the reduced 3-string vertex 1 u ( 1,2,3)) is given by (4.1). The path-integral measure L3d is invariant under (F1) if we can show

~r,[8(^s,d(3))/84(3)1=0, (F3)

where Tr implies the trace with respect to the first- quantized string state._

The ~~d term in ijBd clearly satisfies (F3) since eB transforms one (first-quantized) string state to another with one more ghost number and hence has no diagonal element. Next, the contribution of d*$ term (F2) to (F3) is given by

where use has been made of the formula

S(d (1 ) I / 8 ! d ( 3 ) ) = ( r ( 1 9 3 ) ! (F5)

with ( r ( 1,3 ) ( given by (4.3, and the cyclic symmetry of the vertex u ( 1,2,3 ) ) . Equation (F5) is a consequence of the Hermiticity condition for d obtained from (2.76) and (3.1):

( d ( 2 ) l = s d l ( r ( l , 2 ) p # ~ ( l ) ) . (F6)

The presence in (F4) of ( r (2,3 ) which contains S ( a z + a 3 ) enforces a , =O. This implies that

in (F4) as can be seen from (E7) or more easily from the second equation of (1.3.53):

[ w ' ~ ' is nonsingular in the limit a l + O as is verified by using formulas ( E l ) and (E21.1 However, a care must be taken in concluding the vanishing of (F4) from (F7).

Equation !F7) implies more precisely w ' ~ ' + w ' ~ ' = 0 ( € 1 as a l =€--to. On the other hand in this limit a l + O the vertex 1 u(1,2,3)) behaves like (4.2) [see (E3)]. The momentum p l in (4.2) should also be put equal to zero in (F4). Therefore (F4) is in fact a rather subtle quantity. [The origin of - 1 in the exponent of (4.2), and hence the origin of the subtlety, is the presence of the tachyon state.] Here we assume that the a l + O and pl--0 limits in (F4) are taken in the following manner: first a,-0 with held positive and finite, and then p l -0. !We also assume that ( 4 ( 1) 1 is nonsingular as a l+O.) In this sense we may conclude that (F4) vanishes.

The contribution of the $o$od term in to (F3) also vanishes in the same manner. Equation (F8) holds also for the 4-string w'""" [denoted as u"' in (1.6.29b)I.

Next let us consider the change of variables (3.18). From (2.711, (2.641, (2.67), and (3.81, zNLad (3.11) is expressed in the bracket notation as

with 1 h 3 ) and 1 h ) given by

- where 2 " and X ( r ) are defined by (2.95) and (2.971, respectively, U"'=LL"~"' ' and vi") is the integrand of (4.9). For

A A -

this i?iNLnd we repeat the above argument for EBd. However, since 2, ? ' ' I is cyclically symmetric and

holds for both / u ) = u (1 ,2 ,3)) and , ui4 ' ( 1-4)) [see (2.103), which holds also for the 4-string vertex], it is clear that the measure is invariant under (3.18).

1384 HATA, ITOH, KUGO, KUNITOMO, A N D OGAWA - 3 5

'Present address: Department of Physics, Kyoto University, Kyoto 606, Japan.

'H. Hata, K . Itoh, T. Kugo, H . Kunitomo, and K . Ogawa, Phys. Lett. 172B, 186 (1986).

2H. Hata, K . Itoh, T. Kugo, H. Kunitomo, and K . Ogawa, Phys. Lett. 172B, 195 (1986).

3H. Hata, K . Itoh, T. Kugo, H . Kunitomo, and K . Ogawa, Nucl. Phys. B283, 433 (1987).

". Hata, K . Itoh, T. Kugo, H. Kunitomo, and K . Ogawa, Phys. Rev. D 34, 2360 (1986).

5H. Hata, K . Itoh, T. Kugo, H . Kunitomo, and K . Ogawa, preceding paper, Phys. Rev. D 35, 1318 (1987).

6H. Terao and S. Uehara, Phys. Lett. 173B, 134 (1986). 7T. Banks, D. Friedan, E. Martinec, M. E. Peskin, and C. R .

Preitschopf, Nucl. Phys. B274, 71 (19861. %ee, e.g., G. 't Hooft and M. Veltman, in Particle Interactions

at Very High Energies, edited by D . Speiser, F . Halzen, and J. Weyers (Plenum, New York, 1974), part B, p. 177; B. W. Lee, in Method in Field Theory, edited by R . Balian and J. Zinn- Justin (North-Holland, Amsterdam, 1976).

9J. Scherk, Rev. Mod. Phys. 47, 123 (1975); in Dual Theory, edited by M . Jacob (North-Holland, Amsterdam, 1974).

IOM. Kaku and K . Kikkawa, Phys. Rev. D 10, 11 10 (1974); 10, 1823 (1974).

"S. Mandelstam, Nucl. Phys. B64, 205 (1973); B69, 77 (1974): B83,413 (1974).

12E. Cremmer and J.-L. Gervais, Nucl. Phys. B76, 209 (1974); B90, 410 (1975).

131ncidentally we can obtain the following projective invariant expression for w'" in the 3-string vertex from 12.46)=0:

5

I4This is because the horn diagram has only one interaction

point. For the nonhorn diagrams, which have two interaction points, ~ ~ = , ( G / a , ) ~ f ' ~ " ' ( a : ' ' ) is not cyclic symmetric since precisely a?' should be written as for r= 1,2 and a:34!" for r= 3,4.

I5The path-integral measure 8 b is invariant under the change (3.18). This is also the case for the change of variables ~+-++h^6~d (see Appendix F) .

I6If we can apply the formal argument that the string length pa- rameters a, always appear only through their ratios then Z , cannot depend on a, and equation Ai = B, follows from (3.23).

]'See, e.g., T . Kugo and S. Uehara, Nucl. Phys. B197, 378 (1982).

I8J. E. Paton and H. M. Chan, Nucl. Phys. B10, 519 (1969); T. Matsuoka, K . Ninomiya, and S. Sawada, Prog. Theor. Phys. 42, 56 (1969); H. Harari, Phys. Rev. Lett. 22, 562 11969); J. Rosner, ibid. 22, 689 (1969).

I9P. Di Vecchia, E. Del Giudice, and S. Fubini, Ann. Phys. (N.Y.) 70, 378 (1972).

20M. Kato and K. Ogawa, Nucl. Phys. B212, 443 (1983). I1We know no suitable reference to cite here which gives a co-

variant operator formalism with F P ghosts. In any case, the one-loop diagram calculated from the operator formalism with the (inverse) propagator L (1.2.23) including the FP- ghost oscillators reproduces the conventional results. The role of FP-ghost oscillators is simply to produce [ f (w)]-* , which, together with the contribution of a" oscillators [ f ( w)Id, gives the correct power [f ( ul)]d-2 [see Eq. (IV.10) of Ref. 221.

Z2Scherk (Ref. 9). 23J. A. Shapiro, Phys. Rev. D 5, 1945 (1972). 24Stanley J. Brodsky, Ralph Roskies, and Roberto Suaya, Phys.

Rev. D 8, 4574 (1973). 25J. H. Schwarz, Phys. Rep. 89, 223 (1982); M. B. Green and J.

H. Schwarz, Nucl. Phys. B218,43 (1983); B243,475 (1984). Z6E. Witten, Nucl. Phys. B276, 291 (1986).

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Loop amplitudes in covariant string field theory