Volume 182. number 2 PHYSICS LETTERS B 18 December 1986

BRST INVARIANT N-REGGEON VERTEX

P. Di VECCHIA

Nordita, Blegdamsuej 17, DK-2100 Copenhagen 0, Denmark

R. NAKAYAMA, J.L. PETERSEN, J. SIDENIUS

The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark

and

S. SCIUTO

Dipartimento di Fisica Teorica dell’Uniuersitii di Torino, Sezione di Torino dell’INFN, I-10125 Turin, Italy

Received 27 September 1986

An N-reggeon vertex is constructed with the inclusion of the contribution of the ghost coordinates. It is shown that

it is projective and BRST invariant.

One of the most challenging problems for those working with strings is to prove that certain superstring theo- ries do not show any infinity in perturbation theory, thus providing a consistent quantum theory for gravity.

The starting point for computing dual multiloops [ 1 ] in the old days was the N-reggeon vertex written in a very simple form by Lovelace [2] for the orbital degrees of freedom and extended by Olive [3] to a form more suitable for computing multiloop diagrams.

The elimination of unphysical states was achieved in one-loop diagrams [4] by the insertion of the Brink-Olive projection operator [5], that had the net effect of killing the contribution of the longitudinal and scalar degrees of freedom.

In the new era of string theories it became clear that the right way of covariantly quantizing the string is by introducing the ghost coordinates and by requiring BRST invariance [6]. One-loop diagrams have been com- puted again using the BRST technique and the results are found to be in agreement with previous calculations

[71* In a previous paper we have included in the three-reggeon vertex [8], written in terms of conformal fields, also

the contribution of the ghost coordinates obtaining a BRST invariant expression. We have also studied its connection with the covariant three-string vertices constructed by Neveu and West and

by Hata et al. [9]. In this letter we generalize our construction to theN-reggeon vertex, obtaining an expression that can be used

as the starting point for the calculation of multiloop diagrams on the line followed in ref. [l] including also the contribution of the ghosts.

We have constructed theN-reggeon vertex both with functional integration techniques starting from the string action and with the old operator formalism. In this letter we will use the latter technique, while we will give all the details of the derivation by using path integral techniques in a future publication [IO].

The basic ingredient for constructing the N-reggeon vertex is the three-reggeon vertex constructed in ref. [8]. It is given by

164 0370-2693/86/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Volume 182, number 2 PHYSICS LEiTTERS B

i(X=O; 0, i/J =3IWi ,

18 December 1986

(1)

where

I+‘i=:exp(t dz[-X(1 -z)X;(Z)-~(l--Z)bi(Z)fb(l-Z)ei(Z)]):

and the notations are the same as in ref. [8]. In ref. [8] it has also been shown that (1) is BRST invariant since it satisfies the equation

(2)

i~X=O;Oa ;4=31[QjtQ,Wj]=o, (3)

where Qi is the BRST charge corresponding to the coordinates of the ith reggeon and Q the one corresponding to the auxiliary coordinates.

The N-reggeon vertex can be expressed in a very simple form in terms of Wi. It is given by

s N

X(p=O;Oa;*=OlljJI ~jWj~~~11P=o;oa;4=o), (4)

where i; is an operator corresponding to the ith term in (4) and acting in the space of the auxiliary oscillators (without index i), that performs the projective transformation ri E gi_r .ji ii+r ] in the notation of ref. [2]. It can be written in terms of the generators L,, L 1 and L _1 of the projective transformatrons as follows:

pi = exp[-(ci/ai) Ll] [afl(-aidi + bici)]~’ exp{-[(ai + bj)/Qi]L_l] . (5)

The parameters aj, bj, cj and dj in (5) correspond to the transformation

Z j Vi(Z) = (QiZ + bj)/(CjZ + dj) ) (6)

where Vi s [yi_r ii t,.+r] and Vi(z)= Yi(l -z).

After some calculations one gets

Uj=Zj_1(Zj_Zj+l), bj=Zj(Zj+l -Zj_l), Cj=Zj-Zj+?, dj=Zj+l-Zj_l . (7)

The product of the e-functions in (4) insures the cyclic symmetry of theN-reggeon vertex since we are deal- ing with open strings.

Because of the projective invariance of the integrand in (4), that will be shown later, we must fix three of the integration variables z, , zb and ze and insert the factor

dvabc = dzadZbdz,/lV;(0) V;(O) V;(O)1 . (8)

In the following we will show that (4) is an N-reggeon vertex, since it gives the correct N-point amplitude for (N - 3) arbitrary highest weight states with A = 1 and three highest weight states with A = 0 in the sense of con- formal field theories [ 111.

This follows from the fact that by construction the three-reggeon vertex satisfies the following basic property:

j(X =O ;Oa;q =3lWjlCYj)j = Va’,i(l) 3 (9)

as shown in ref. [8], where V,,(l) is the vertex operator corresponding to the state I?) and that in addition

16.5

Volume 182, number 2 PHYSICS LETTERS B 18 December 1986

Vai( 1) transforms as follows under a projective transformation:

i;-V,,(l) + = [$(l)]*Vai(Zi). Cl@

Inserting (9) and (10) in (4) after saturating it with IIrE, [ laijj] and choosing the vertices corresponding to a, b, c with A = 0 and the others with A = 1, we get

N N-l N

J iq dzi II e(z~-~~+l)(p=O;Oa;q=OI ITI V~i(Zi)Ip=O;Oa;~=O). i=l i=l

(if-a, b,c)

(11)

In particular, following the suggestion of Friedan, Martinet and Shenker [ 121, if we choose the vertex operators corresponding to the external states a, b, c of the form c(z) V,“(z) and those corresponding to the other states of the form V,“(z) we get from (11) the scattering amplitude relative to arbitrary physical excitations of the string. The upper index x in the previous expression denotes that the vertex operator corresponds to a physical state and contains therefore only the oscillators appearing in x,(z).

We want now to relate (4) with the form proposed in ref. [2] and show that for the xc1 part of the vertex the two expressions exactly coincide.

In fact, using the identity

Iaidi;2b.c.l m -----L-L t C J-

n=lfi

X exp nc2 c;‘[-f n(n t l)(-Ci/di)“-‘b~‘l t (n2 - l)(-Ci/di)“b6” - f n(n - l)(-C~/d~)n+lb~‘]) (

X :exp( $ dz{-X(V,(z))X~(Z)tc(C;(Z))bi(z)/~~(Z)’b(~i(z))Ci(Z)[V;(r)l2}): )

0

(12)

it is possible to show after some calculations that

N iv

where

L&(Y) = (l/m!)(m/n>1/2P [y(z)ln/8zm lzzo , n,m Z 0,

&o(Y) = (hm~/~)n , &&) = (l/4)(-CD)” > n + 0 >

Doo($ = -loglD/(AD - xp I ,

(13)

(14)

166

Volume 182. number 2 PHYSICS LETTERS B I8 December 1986

and

E,&) = [i/(m + I)!] am+1 { [r(z)]“+lir’(z)}lazm+llz=O , n,m=-1,0,1,2 ,..., (15)

with

z -+ y(z) = (AZ +B)/(Cz +D), ryz) = l/Z) 06" =p(') . (16)

D,, (7) [Em ,, (y)] is a representation of the projective group with J + 0 [J = l] corresponding to the coordinate xc1 (z) [c(z)] as explained in ref. [2].

In the derivation of (13) we have used the following mode expansions:

c(z)= 5 c,z-n+l , b(z)= 5 b,z-“-2. n=-CC *=-CC

(17)

It is interesting to notice that the two exponentials (13) can be written in the following very suggestive way

(18)

where Vii = Vi-’ Vi and the vacuum of the orbital modes is also the vacuum of the zero mode [a0 IO,) = 0 and ~0 here should not be confused with ug 5 pp used in (13)]. The suffix E means that xz (z) is a conformal field with dimension & E [E -+ 0] and the normal ordering ; ; implies that we must write a, on the right of 0,’ for n = 0, 1, . . . . Formula (18) will be shown in detail in ref. [lo].

The expression (13) provides a natural generalization of the Lovelace N-reggeon vertex to include also the con- tribution of the ghost coordinates. For N = 3 (13) reproduces the vertex of Canes&i, Schwimmer and Veneziano [ 131 with the inclusion of the contribution of the ghost coordinates [8,9].

In the last part of this letter we show two very important properties of the N-reggeon vertex, namely its invar- iance under BRST transformations and the invariance of the integrand in (4) under projective transformations of the zi.

If we perform a projective transformation such that

Zi + A(zi) = (oXi + fl)/(rZi t 6) , (19)

it is easy to see that Vi + Avi and therefore the argument I’k’:’ Vi of the representation matrices in the two ex- ponents of the right-hand side of (13) is left invariant under the transformation (19). On the other hand also the three fermionic: 6-functions are left invariant by transformation (19) since

ifn,m = -l,O, 1,and

det[~[Al,ml,,m=_,,o,l =1 for an arbitrary A.

(21)

167

Volume 182, number 2 PHYSICS LETTERS B 18 December 1986

In conclusion, the lefthand side in (13) is projective invariant. Finally it is easy to see that also the integration measure in (4) is projective invariant. This implies that the full integrand in (4) is projective invariant.

We want now to show that the N-reggeon vertex is BRST invariant or in other words that

(22)

If we remember that each Q, commutes with all the factors in the product I$!, i;IViTlr’ except one, and that

the relation (3) holds, we can rewrite (22) as follows:

+=O;O,;y=O,~fil i(x=O;0,;q=31 [

N

Q,~~ri;w,+ 1 Ip=O;O,;q=O), (23)

this is trivially zero because

Ql~=O;O,;q=o)=o. (24)

This shows the BRST invariance of the N-reggeon vertex. It is easy to generalize our construction to the N-reggeon vertex constructed by Olive [3 1.

One of us (PD.V.) wishes to thank D. Amati for very useful discussions on theN-reggeon vertex. RN. is grate- ful to the Danish Research Council for financial support.

References

[II

I21

[31 [41 [51 [61

171

[81 [91

1101 [Ill [=I [I31

168

V. Alessandrini, Nuovo Cimento A 2 (1971) 321;

V. Alessandrini and D. Amati, Nuovo Cimento A 4 (1971) 793;

C. Lovelace, Phys. Lett. B 32 (1970) 703;

C. Montonen, Nuovo Cimento A 19 (1974) 69.

I. Drummond, Nuovo Cimento A 67 (1970) 71;

G. Carbone and S. Sciuto, Lett. Nuovo Cimento 3 (1970) 246;

J. Kosterlitz and D. Wray, Lett. Nuovo Cimento 3 (1970) 491;

D. Collop, Nuovo Cimento A 1 (1971) 217;

L.P. Yu, Phys. Rev. D 2 (1970) 101,0,2256;

C. Lovelace, Phys. Lett. B 32 (1970) 490;

E. Corrigan and C. Montonen, Nucl. Phys. B 36 (1972) 58; see also V. Alessandrini and D. Amati, Proc. Intern. School of Physics (1971) (Academic Press, New York, 1972) p. 58.

D. Olive, Nuovo Cimento A 3 (1971) 399.

L. Brink and D. Olive, Nucl. Phys. B 58 (1973) 237. L. Brink and D. Olive, Nucl. Phys. B 56 (1973) 253.

M. Kato and K. Ogawa, Nucl. Phys. B 212 (1983) 443;

S. Hwang, Phys. Rev. D 28 (1983) 2614.

I. Bengtsson, Class. Quant. Grav. 3 (1986) L 31; J.H. Schwarz, Caltech preprint CALT-68-1304;

M.D. Freeman and D. Olive, Phys. Lett. B 175 (1986) 155. P. Di Vecchia, R. Nakayama, J.L. Petersen and S. Sciuto, Nordita preprint-86/8 (1986), Nucl. Phys., to be published.

A. Neveu and P. West, Phys. Lett. B 168 (1986) 192;CERN preprint TH-4358/86 (1986); H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B 172 (1986) 186; preprints RIFP637 (1985);

RIFP656 (1986).

P. Di Vecchia, R. Nakayama, J.L. Petersen, J. Sidenius and S. Sciuto, to be published.

A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. D. Friedan, E. Martinet and S. Shenker, NucI. Phys. B 271 (1986) 93.

L. Caneschi, A. Schwimmer and G. Veneziano, Phys. Lett. B 30 (1969) 356.