BRST invariant tadpole operator in the bosonic string

Volume 200, number 3 PHYSICS LETTERS B 14 January 1988

BRST INVARIANT T A D P O L E OPERATOR IN T H E B O S O N I C STRING

G. CRISTOFANO, F. NICODEMI and R. PETTORINO D~parttmento dz Ftstca, Umverstt?~ dl Napoh, and lNFN Sezzone dt Napoh, 1-80125 Naples, Italy

Received 16 October 1987

Using a BRST mvarlant three-reggeon vertex we construct an exphclt expressmn for the BRST mvarsant planar tadpole operator for the bosonlc stung Joined to an (N+ I )-reggeon vertex it yields the correct expressmn for the one-loop N-tachyon amphtude

A very useful quantity in perturbative string theory is the tadpole operator, which joined with a propagator and suitably sewn to other legs allows to construct amplitudes wnh an arbitrary number of loops. Indeed this operator was constructed an the early days of dual theory [ I ] but there were difficulties in properly taking into account the contribution of the spurious states circulating in the loop. The requirement of BRST Invariance has made it clear how to cure this problem with the inclusion of Faddeev-Popov ghosts. While the final goal is the construction of amplitudes and of the partition functions for the supersymmetric and heterotic case, for the bosonic string a BRST-invariant N-string vertex operator has been recently obtained [ 2 ], allowing for the construction of multlloop amplitudes [ 3 ].

Using this vertex we construct in this letter the explicit expression of the BRST invarlant planar tadpole operator. We start from the following expression for the symmetric covariant three-reggeon vertex ~1

3 3

V3= ~ , ( 0 a , q=31 ~I exp[-~(a'ID(U, Vj)Ia~)-(c'IE(U, Vj)IbJ)] t = I t : ~ J = 1

× Z E.,,,(V,)b~,, ,5 p, . (1) t / ~ - - I I ~ l t n ~ - - I I 1

Here a ~, b" and c% are the usual operators relative to orbital and ghost excitations (wnh a ~ =p ' ) ; (0a, q = 31 are vacua of the orbital string oscilations and states of ghost number ng = 3, and

(a'lD(U,V,)laJ)= ~ a'.D.,.(U, Vj)aJ,,,, (2) n m = O

(c'IE(U, Vj)IbJ)= ~ c~E.,.(U, Vj)bJ, n. (3) n , m = 2

D( U, Vj) and E( U, Vj) are infinite dimensional representations of the projective group, U, and Vj being elements of this group corresponding to the transformations

U, z, O~ z, z,+ ~ and = (~3 l Zt -- 1 Zt Zt + 1

The general expression for D.m and E.,,, can be found e.g. in ref. [2]. In the three-reggeon case, since V3 is projective invariant, D and E become independent of the z's and are given by:

#t Covanant three-reggeon vertices have been d~scussed also m ref [4]

292 0370-2693/88/$ 03.50 © Elsevier Science Pubhshers B.V. (North-Holland Physics Publishing Division )


D"'"(U'V'+~)=D"'"(U'V'-J)=(-1)"x~n(n)'m

, [ n + l ~ 1 ) . ( m - 2 ~ E,,,,(U,V,+,)=(-1) ~ m + l ) ' E~,,(U,V,_~)=(- \ n - 2 ] "

Using these formulas it is immediately seen that the expression of the orbital part of V3 coincides with the vertex constructed by Caneschl, Schwimmer and Veneziano m 1969 [ 5 ].

Writing eq. (3) we have let both ghost and anughost operators vary only over non-zero modes while the authors of ref. [ 2 ] include a contribution from antighost zero modes. However, m any sensible calculation one has to take the scalar product of the vertex either w~th a physical state having I q= 1 ) or w~th a corresponding leg of a conjugate vertex after the insertion of a propagator which is linear in the antlghost zero modes. There- fore the inclusion of these zero modes in eq. (3) does not give any contribution due to the presence in V3 of the fermionic 0-functions.

To construct the tadpole operator one has to join two legs of the three-reggeon vertex, inserting a suitable propagator and taking the trace. Being interested in the planar tadpole, starting with the symmetric vertex we have to use a twisted propagator [6].

We will use the twisted propagator given by [ 3 ]: I

f P(x) (4) dx

D=(bo-bl) x(1-x----~ ' 0

where P(x) =xL°eL-' ( -- 1 )Lo-p2/2(1 --X) " and W=Lo-LI. As usual the L,,'s are the generators of the Virasoro algebra and can be expressed as L , = {b,, Q}, Q being

the BRST charge. As discussed m ref. [ 3] D ~s BRST invariant and also preserves duahty explicitly. To proceed to sew leg 2 with leg 3 we have to take first the conjugate of one leg's Fock space following the

rules

p.--.-p., a . ~ - a _ . , c,,~-c_n, b.-ob_~, for the oscdlators and changing the corresponding vacuum as

3(0a, q = 3 l ~ 10~, q = 3 ) 3 •

Moreover the effect of the insertion of P(x) present in the propagator amounts to making the replacements

Vz--+V2=V2P(x) and Uz--I.Lf2=P(x)U2, ( 5 )

where P(x)= (~ s~) is the 2 × 2 matrix corresponding to the conformal transformanon P(x). Finally one has to identify the operators relanve to the legs 2 and 3 and take the sum over a complete set

of states, including the integration over momentum. Using the variable k=x/(1 -x), the tadpole operator can be written as

(T . = (0~, q=3 , f d~-ffO, (6)

where the state (0. , q=31 at zero momentum refers to the leg which ~s left free, while O ~s the result of the evaluation of the trace.

The trace over the orbital degree of freedom and the integration over the loop momentum yield the result obtained by Gross and Schwarz in 1970 [ 1 ]

( ' ) ° Oo~=(lnk) -°/2 f i ~ exp[-(al(1-k)A(1-k)la)]. (7)

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Following their notat ion we have defined

(a l (1 -k )A( l - k ) la ) =- ~ ar(1-k)rAr~(1-k)'a,. r , s = 1

The matrix A is given by

2 l n k + - Mr+ M;; +MJ k---L-" - ~ s , 1 -k"M"+" '

where

M,,,. = and MT.. = ( - 1)" n

For the trace over the ghost variables we use the following definiuon [ 3]:

Trzl3F(c3, b3; ¢2, ben) = __ d2fln d2y. exp + y,*,y. , n = - - I n = 2 n n = 2

(8)

In our case

{q=0)2 I q = 3 ) 3

F(c,3, b3;c2, b2) = I] exp[-(c'lE(U, Vj)lbQ] I-I Z E,,,,,(V~)b'm , t ~ J = l n = - - I t 1 m = - - I

provided one makes the substitutions indxcated m eq. (5). The result of the trace turns out to be

1 f i (1-k")Z(-bo+~---~k)exp[(cl(1-k)l ' (1-k)[b)] ,

where, as in eq. (3)

(cl (1 - k) / ' (1 - k ) lb) = c, , ,(1-k)"Fm.(1-k) 'b. ,

and

k r k r F,.,, =-N,+.r ~ NT. + P7.~ l-k----; Pr+~ "

The matrices N +- and P+- are given by

NT. .=(_l ). (m+ l ) , ( r e + n - l ) \ n + l N'+"=\ n + l

and satisfy the following useful relations:

N- (x)N- = (1 -x )N- (1---~x) ,

P - ( x ) P - = ( 1 - x ) P - ( 1 - f - x ) ,

n - 2 ' P + " = \ n + l

N + (x)N- - (1 - x ~ N+

(9)

(10)

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where

N(x)Nl.,. =-N.rXrNr .... ( 1 - x ) N -= (1 -x)"Nn,. \-(~-xJ " n t n

Putting these results together we arrive at the following expression for the planar tadpole:

dk k) -D/2 .=, exp[ - (a (1 - k ) A ( 1 -k)la)] (TI = ( 0 . , q=31 ~-5 (ln f i 0

×(-bo+~_~k) exp[(c l ( l -k)F(1-k) lb)] . (11)

With regard to the integration hmlts we refer to the discussions given in refs. [ 1,3]. Being constructed with BRST lnvariant quantities one expects also T to be invarmnt. Indeed it can be proved that (TI Q=0.

An interesting check of this result and of the dual properties of the formalism we are using, is obtained con- structing the N-tachyon planar one-loop amphtude by sewing together the tadpole given by eq. (11 ) and an N + 1 vertex operator, with the insertion of the twisted propagator given by eq. (4).

Starting with the "asymmetric" form of the vertex of ref. [ 3 ], and saturating VN+t with the tachyonic states I q, = 1, p2 = 2 ) for l = 1 and N, and ) q, = 0, p2 = 2 ) for t = 2 ..... N - 1, we obtain:

I W ) = V'~(0)V)(0) ,=2[I dz'O(z,-z,+l)8 p , (z,-zj)P'P:exp ~)aL 1 t < J = I h

X ( ~ [Eo-I ( UI VN)E_In( UI VN+I)--E-I-I( UI VN)Eo,,(UI VN+l)]b-n~ 'p:O, Oa, q=3). (12) \ 1l~ -- [ /

Oscillators and states without index refer here to the leg N + 1 and dealing w~th the antighost zero modes of this leg we used the property that the contribution of the fermmmc 8-functmn is invanant under multiplication of all matrices V, by a common element of the projective group and E.,.( U, V,)=--8n+m.O. The one-loop N- tachyons planar amphtude is now given by the scalar product ( TI D J W). The presence of one antighost zero mode in each T, D and W ensures the correct ghost number counting to get a non-vanishing answer.

The effect of the propagator at zero momentum on W can be expressed through the relation

D(p=O) IW(VN+I))=(bo-b') x(1Vx) [ m ( ~ v + l ) ) '

where VN+~ = VN+,P(x). Inserting the expressions (11) and (12) into the scalar product, after some calculation one obtains

f dk _k~)_o+~ (z,--Zs)2 f ax ( T J D I W ) = - ~- f ( l nk ) -D/2 f i (1 .=1 1 - k x z

× I-[ dz, O(z,-z,+~) (z , -z~) p' ~ p, exp p, In 1 - ( 1 - k ) x t = 2 t < y \ t = l t = l

× I5 ~=o ,:=, i ~ ( 1 - k ) x ( 1 - o ~ : ) / ( 1 - x ~ , ) / ' (13)

where ~x, xs the anharmonic ratio (ZN--Z,)(Zx+ j--Zl )/(Zx--Zt )(ZN+ ~--Z,). Since I W) is independent of the choice of three Koba-Nielsen variables z~, ZN and zx+ ~ it is possible to put

m eq. (13)

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ZN-t- 1 Z N - - Z ! 1 - - = X , z l = l , ZNW[ = ~ ,

ZN Z N + 1 - - Z I

that allow to make the identif ication

1 (1-k)x 11-a' 1 - - XOl t Z~

which yields the well-known expression for the planar one-loop amplitude, as given for instance m ref. [ 3 ]. The most general N-loop planar diagram can be obtained by jo in ing the tadpole operator, given by eq. (I 1 ),

to N legs of a suitable (N+M) vertex operator. This problem as well as the constructions of non-planar am-

plitudes will be discussed elsewhere.

We are grateful to P. Di Vecchia and S. Sciuto for many fruitful discussions.

References

[ 1 ] D J Gross and J H. Schwarz, Nucl Phys. B 23 (1970) 333, E Cremmer, Nucl Phys. B 31 (1971) 477.

[2] P. DI Vecchla, R. Nakayama, J L. Petersen, J. Sidenlus and S Scluto, Phys. Lett B 182 (1986) 164; Nucl. Phys. B 287 (1987) 621, and references therem

[3] P D1 Vecchxa, M. Frau, A. Lerda and S Sciuto, NORDITA preprmt 87/36-P (1987), NORDITA preprmt 87/50-P (1987), J L. Petersen and J.R. Sldemus, NBI preprmt HE-87-35 (1987)

[4] A. Neveu and P. West, Phys Lett. B 168 (1986) 192, H. Hata, K. Itoh, T Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186.

[5] L Caneschi, A Schwlmmer and G. Veneziano, Phys. Lett B 30 (1969) 356. [6] D. Amatl, M Le Bellac and D. Olive, Nuovo Clmemo 66 A (1970) 831,

V Alessandrlnl, D. Amati, M. Le Bellac and D Olive, Phys Rep 1 (1971) 269

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BRST invariant tadpole operator in the bosonic string