PHYSICAL REVIEW D VOLUME 38, NUMBER 4 15 AUGUST 1988

BRST-invariant tadpole calculation and diagrammatic formalism in the bosonic string

T. Kobayashi, H. Konno, and T. Suzuki Institute of Physics, University of Tsu kuba, Zbaraki 305, Japan

(Received 26 February 1988)

Planar and nonorientable tadpole operators are evaluated in the Becchi-Rouet-Stora-Tyutin- .vc + l / 2

(BRST-) invariant formalism. It is shown that the insertion of ( - I ) ( N , = ghost-number operator) is needed for keeping the BRST invariance. We shall also discuss the extension of the BRST-invariant diagrammatic calculation to arbitrary diagrams.

A covariant quantization of the bosonic string is given by the Becchi-Rouet-Stora-Tyutin (BRST) formalism.' It was shown that the nilpotency of the BRST charge ( e 2 = 0 ) requires the critical values D =26 and ao= 1 for the space-time dimension and the intercept of trajectory, respectively. Following this formalism, the ghost interac- tions corresponding to the Caneshi-Schwimmer- Veneziano (CSV) vertex2 and the general N-Reggeon ver- tex' have been d e r i ~ e d . ~ The BRST invariance of planar one-loop diagrams was also discussed in this f ~ r m a l i s m . ~ However, no BRST-invariant diagrammatic operator for- malism based on Fock spaces has yet been completed. In this paper we shall study a BRST-invariant calculation satisfying unitarity for the planar and nonorientable tad- pole operators and show a'possible BRST-invariant di- agrammatic operator formalism.

The three-string vertex given by Fig. 1 is described in the symmetric form as

where vX and vgh, respectively, stand for the vertices of the X part and the ghost part. Following Ref. 4, the ghost vertex is given by

where 123(8/ ~ l ( f i / ~ ( f i I 3(fi 1 , ( 8 1 is the vacuum with

( vk,h3 i = ,,,(a / exp

FIG. 1. Symmetric three-string vertex. The dot on each leg indicates the side of the line on which the external particles are to be attached.

3 3

- 2 2 (c"' U,V, 1 b ( r ) ) r f l s = I

( r f s t

nonzero norm to the SL(2binvariant vacuum 0 ) ( ( D l O ) = l ) ( R e f . I) ,

U , and V, are defined in conventional forms:,

c and b are, respectively, the ghost and the antighost os- cillators, and B'lvO) denotes the (1,O) representation of SL(2,R). (The ( n, m ) element of the ( J , P - J) representa- tion is defined by

B(J,P-JI (A),, = 1 d m + P (P +m)! !dzm + P

where

and detA= 1. This is the same as the conventional repre-

FIG. 2. Diagrams for the (a) planar and the (b) nonorientable tadpole operators, where X stands for the twist.

1150 @ 1988 The American Physical Society

38 - BRST-INVARIANT TADPOLE CALCULATION AND . . .

sentation except for the normalization coefficients.) The BRST invariance of ( V12, I is shown by the equation4

( V12, I ( Q " ' + Q ' ~ ' + Q ' ~ ' ) = O . (3)

Since the BRST charge commutes with all Virasoro operators L, ( n = - C C , . . . , cc 1, operators written in terms of L,, e.g., the propagator

1 1 I Dz-= l d x x L O - (4) Lo 0

and the twist operator

are BRST invariant, where

In order to evaluate loop amplitudes illustrated in Fig. 2, we need a reflection operator which transforms a string state ( x to I x ) in the BRST-invariant way. Such an operator is easily derived from the vertex as follows:

where 1 + ) is the physical vacuum defined by j + ) =cocl 1 0 ) (Ref. 1). Since ( V 1 and 1 0 ) 2 are BRST invariant, the BRST invariance of ; R ,, ) is trivial, that is,

(Q" '+Q'") 1 R 1 3 ) = O . (8)

It should, however, be noted that the prefactor (bbl' - b a ' ) is required to keep the BRST invariance. In the evaluation of the loops we also have to define the trace of the inner product such as O,i= ( V1,23 / 0',' / R ),

which may be defined in terms of the coherent states as

where 03i - OfiO$, the coherent states follow the usual convention^,^ and ) A and 1 B, y ) , represent the coherent states written by auxiliary oscillators aAA', bAA', and cAAJ which do not couple with any other oscillators. The contrac- tion in the auxiliary Fock spaces automatically yields the identification of aA"=aA3', bA1'=bA3', and C;"=C,!~'. It is not- ed that the minus sign of ( -B"', -7"' / must be introduced to derive the right trace over the ghost states.

(Although the identity operator of the coherent states for one fermionic operator d, is give by J d a d a ePea 1 a ) ( a / where / a ) = e - a * t l ~ ) a n d ~ =(O1eE*with ( 0 / 0 ) = 1 , t h e t r a c e o f O m u s t b e d o n e J d ~ d a e - " ~ ( - a ~ a ) b e - cause of the antiperiodic boundary condition for 4. The consistency of the definition is easily checked by setting 0 = 1, i.e., l d ~ d a e - ' ~ ( -a 1 a ) = 2 gives the right answer, while l d E d a e u ( 1 a ) = O is wrong.)

Following the definition (91, we can easily derive the well-known result

=Tr :exp [y t ( A,-l)y + y t ~ , + ~ , y ] : :ox(a ta )::Ogh(bt,c;ct,b): , (10) [y=a.b.c i 1

where / 6 ) , and 3 ( 6 1 should be chosen as ( 6 1 6 ) = 1, i.e., ( 0 / 0 ) or ( + j - ) for the ghost oscillators, the creation and annihilation operators and the normal-ordered product have to be defined on the vacuum / 6 ) and the coefficient ma- trices A, and also the coefficient vectors B, and C, do not contain any oscillator variables.

A. Planar tadpole operator ( T )

The simplest formula for computing the planar tadpole operator is given by

1152 T. KOBAYASHI, H. KONNO, AND T. SUZUKI

where the ghost-number operator1 is defined by

and bb3' has to be inserted because the expectation values between 3 ( ~ and x ) , should have no ghost number. The Nc+1/2

insertion of ( - 1 ) may be understood as the change of the antiperiodic boundary condition to the periodic one for Ne+1/2

the ghost.6 It is noted that ( - 1 ) anticommutes with Q ' ~ ' , b '3 ' , and c ' ~ ' , and commutes with all operators con- structed from LA3' and the oscillators except b'3' and c ' ~ ' , whereas bb3' does not anticommute with Q ' ~ ' . This indicates that the bo insertion is not BRST invariant. We can, however, show that in the tadpoles it is effectively BRST invariant by using Feynman's tree theorem. Following the argument of Freeman and olive,' we can write ( T 2 1 Q"' as

where the Eqs. (3) and (8) are used in the derivation of the first term and the last term with L F ' derived from (Q'",bi3 '] = ~ h ~ ' vanishes because of SL(3, By making use of the definition of the trace ( 9 ) we can change Q'" to

0 '0'

- Q ' ~ ' as

Then we have

( T2 Q ( ~ ' = O

~; , '+1/2 That is, the tadpole operator defined by (1 1 ) is BRST invariant. It should be stressed that the insertion of ( - 1 ) is essential to keep the BRST invariance.

The explicit expression of ( T2 1 is given by

where ( T? I and ( Tgh stand for the contributions of the X part and ghost part, respectively, and are evaluated as fol- lows:

with

A ( x ) ~ ~ = ( x ) +? [ M *M + M ~ M + ]

2 v ' G lnx 2 + 1 - ( X I 1 - ( x ) nm

fo rn ,m 2 1 and

with ! ( x ) G ( x ) , , = M --- + M + ? L + 1 - ( x ) 1 - ( x ) nm

M- +MT- 1

BRST-INVARIANT TADPOLE CALCULATION AND . . . 1153

for n ,m 2 2. The notation in (16) follows that of Ref. 7 as

where differs from a)'030' only by the normalization, while, in (17),

In both equations ( ( x ) / [ l - ( x ) ] ) , , = [ x " / ( 1 - ~ " l ] 6 , ~ is used.

In the above calculation we can derive the following useful formulas. The multiple law of an operator 0 writ- ten in terms of L o and L., to ( V ) is simply described as

where 0, = U, and = V, for t#i and 0 , = O"jt U , and - v,o"' for t =i. The trace for the t - Nc+ l / 2

operator with ( - 1 1 can be reduced to a simple form

Nc+1/2 T r [ O ( - 1 ) 0 ' 1

-B,,Y, -P P =I n (d8,dy.r ) n ( d p m d B m e " " 1 n m

x (Bp 10'0 I B y ) (19)

by inserting the identity operator between 0 and Nc+l/2

( - 1 ) . This equation shows that the BRST- invariant trace over the ghosts with the insertion of

N,+1/2 ( - 1 ) may be represented with the trace formula defined by the coherent states with the periodic boundary condition. It is well known in the statistical mechanics of covariant gauge theory.6

B. Nonorientable tadpole operator ( NT)

The nonorientable tadpole operator shown in Fig. 2 is evaluated as

(lVT2 I =Tr13[( V1,23 I Q ( 3 ) t ~ ( 3 ) b a ) Q ( 3 1

The BRST invariance of N~ is easily proved in terms of Feynman's tree theorem as was done in A. The operator N~ is also described as

where

with (22)

and

( '"Tfh 1 = 1 n [ i - c - ~ ) ~ ] ~

x ( l + x 1 , = I

x 2 ( + i e ~ ~ [ - ( c ' ~ ' 1 J ( x ) 1 b ' 2 ' ) - ( ~ ( 2 ) 1 B o ) b i 2 ) - ( c ( ~ ' I B~ )b \2 '+bi2) ( l+x)c ( ,2 ) ] (231

with

J ( x ) , , =( l + x ) " F ( - ~ ) , , ( l + x ) ~ , m I I 1 - x - x 2

o n = M , M I n m I*] [ x - m x m + ' - m =2 l + x

+ ( l + x ) " M ? I l - ( - x ) M C I n m [e I n ( r n + x ) }

- 2 (M+) , , xm m x + 1 - x - x 2 [ m ; l ] ] 1 [ 1+x f

1154 T. KOBAYASHI, H. KONNO, AND T. SUZUKI

and the same matrices as were used in (17) should be tak- en for M* and M T .

It is noticeable that attaching an N-tachyon state con- structed on the vacuum / - ) to the tadpole operators derives the well-known one-loop N-point formulas as il- lustrated in Fig. 3. Since the details of the derivation were given in the paper by Gross and ~chwarz , ' we do not give them here. It is noted, however, that in their derivation the definitions of the variables should be changed as follows:

<.=I--(1-alpj-<,=I-pj I for the planar l o o p , (24)

f , = 1 - ( 1 - w 2 ) p , - f j = 1 - ( 1 + x , p j

for the nonorientable loop .

It is noted that the factor ( 1 + x ) ' in "Tgh cancels out the Jacobian factor ( 1 + x ) associated with the above change of variables in the nonorientable loop. Both loops, of course, have the right measures.

In the extension of the BRST-invariant calculation to general diagrams a difficulty arises from the bo insertion, because bo effectively anticommutes with Q only in one- loop amplitudes where LjLo,, in the Feynman tree theorem

saves it. Remembering the relations

and

FIG. 3. Diagrams where an N-tachyon state is joined to the (a) planar and the (b) nonorientable tadpoles. They coincide with the usual one-loop N-point formulas.

L where P ( x ) = x O C ~ ( ~ - X ) ~ with W = L o - L , , we find out one possibility of the BRST-invariant bo insertion by using

instead of Db,, which was used in the recent paper by DiVecchia, Frau, Lerda, and ~ c i u t o . ~ This operator D T is nothing but the multiple of the ( bo - b , ) factor to the twisted propagator. In this choice the tadpole operators are expressed as

NC + 1 / 2 ( T , / = ~ r ' ~ [ ( Vi '23 D T ' 3 ) ( ~ ) ( - 1 ) ! R I I , ) ] ,

(25) + l / 2

("T, 1 = ~ r ' ~ [ ( V1,23 1 D ~ ' ~ ' ( x ) S ~ ' ~ ' ( - 1 ) 1 R I I , ) ] . (26)

Equivalence of (25) and (26) to our ( 1 1 ) and (20), may be shown in the following consideration: We have an opera- tor identity in SL( 2, R )

Lo - L - , where z =x /( 1 -x ). Since S1= ( - 1 ) e , we rewrite (27) as

Considering that the factor e P L - ' may be removed be- cause it represents a conformal transformation continu- ously connected to the identity, we may consider that D is effectively equivalent to

which is nothing but the operator used in our formulas. Note that the replacement of D with fltDbo makes cal- culations very simple as was done in our evaluations.

Now we can calculate arbitrary diagrams in terms of T + l / 2

the operators ( V I 2 , , D T, S1, ( - 1 ) , and 1 R I 3 ) . The details for the BRST-invariant diagrammatic oscilla- tor formalism will be discussed in a future paper.

After the completion of our evaluations, we found the paper by DiVecchia, Frau, Lerda, and ~ c i u t o . ~ In their work and also that by ~ e ~ l a i r , ~ however, the necessity of

Nc + 1 / 2 . the ( - 1 ) insertion is not discussed and the BRST

t invariance of the definition of the reflection, i.e., b, -+b,,

3 8 - BRST-INVARIANT TADPOLE CALCULATION AND . . . 1155

t c, - -c,, and ( 0 / -+ 1 ), in Ref. 7 and the b, insertion to general diagrams by LeClair are not clear.

Recently Cristofano, Nicodemi, and Pettorino derived the planar tadpole formula9 by using the method given in Ref. 7. There is some difference between their formula and our Eq. (171, especially in the expression for the ghost modes. The difference due to that between the technical terms used there is not essential, e.g., the difference be- tween the propagators and also that between the vacuum (10 1 and ( + . It should, however, be noted that the

BRST invariance of their formula is only recovered when their formula is attached to physical states, whereas ours is manifestly BRST invariant as was shown in (14). The main difference for the expression of b, and b , modes in the exponent arises from this point. Of course, both for- mulas derive the same results for arbitrary physical quan- tities. An example has already been shown by the deriva- tion of the one-loop N-point formula, where the difference between the definition of the variables noted in (24) has to be taken into account.

'M. Kato and K. Ogawa, Nucl. Phys. B212, 443 (1983); S. Hwang, Phys. Rev. D 28,2614 (1983).

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B297, 603 (1988); H. Konno (in preparation). 5M. D. Freeman and D. Olive, Phys. Lett. B 175, 151 (1986);

175, 155 i 1986). 6C. W. Bernard, Phys. Rev. D 9, 3312 (1974); H. Hata and T.

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B298, 527 (1988); Phys. Lett. B 199, 49 (1987). 8 ~ . J. Gross and J. H. Schwarz, Nucl. Phys. B23, 333 (1970). 9G. Cristofano, F. Nicodemi, and R. Pettorino, Phys. Lett. B

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