The multiloop covariant tadpole operator and amplitudes for the bosonic string

Volume 211, number 4 PHYSICS LETTERS B 8 September 1988

T H E M U L T I L O O P COVARIANT TADPOLE OPERATOR AND AMPLITUDES FOR THE BOSONIC STRING "~

G. CRISTOFANO 1 NORDITA, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark

R. MUSTO, F. NICODEMI and R. PETTORINO Dipartimento di Scienze Fisiche, Universitiz di Napoli and INFN, Sezione di Napoli, 1-80125 Naples, Italy

Received 19 May 1988

An explicit expression is given for the covariant tadpole operator with an arbitrary number of loops, including the ghost zero mode contribution. Its integrand is recast in geometrical terms and is recognized to give the g-vacuum for the bosonic string recently discussed in the literature. By joining the g-tadpole to an N-reggeon vertex the general orientable dual amplitude with g loops is obtained.

The perturbative expansion of bosonic string theory based on the covariant operator formulation, which following the principle of BRST invariance includes the contribution of ghost fields to decouple spurious states in the loops, has turned out to be particularly simple and elegant.

The beautiful results of old dual theory have been reproduced, with the advantage of obtaining the correct measure of integration in moduli space, and finding that the effect of the longitudinal and time-like oscillations is not exactly compensated by the ghost contribution beyond one loop [ 1 ].

Now, due to the crucial property of duality, a perturbative diagram of arbitrary complexity can be built out of a set of"bas ic" operators; in this framework the basic operators for the orientable loop calculations, namely the covariant orietable tadpole and self-energy diagram have been recently evaluated [ 2,3 ].

In this letter we apply such an approach to the case of a surface of arbitrary genus, constructing the g-loop tadpole operator Tg. In the framework of old dual theory this operator was discussed in ref. [ 4 ]. Properly sewing this tadpole to an N-reggeon vertex one can obtain the operator for the general g-loop orientable amplitude. We will build this operator saturating the ghost contribution so that, in the end, only standard DDF states [ 5 ] have to be used as external physical states. In particular the g-loop scattering amplitude for N external tachyons coincides with the one obtained in ref. [ 1 ], thus showing explicitly the property of duality. The measure is obtained, in the Schottky parametrization, starting from lower genus surfaces and encodes the antighost zero modes contribution. Sewing Tg to another tadpole yields instead the orientable partition function.

Following the notation of ref. [ 3 ] the one-loop tadpole operator including ghosts can be written as 1

f dk f dpexp ½p21nk) l_kn)2_Otbo_ l/k) bo_b_,) 1 (TI = (0~, q=31 k ( 1 - k ) n=J 0

× exp[ ( aUlz)p~, - ½ ( a lA°r l a ) - ( clAghl b ) ] . (i)

¢r Work supported in part by the Italian MPI. J On leave of absence from Dipartimento di Scienze Fisiche, Universityh di Napoli, 1-80125 Naples, Italy.

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

417


Here the state (0a, q= 3[ at zero momentum refers to the free leg and p is the loop momentum. The matrices A °r and A gh are defined as

A= ~ (U1)(~r2U3)n(VI)=(UI~-"'V1), (2) n:#0

where 3--= ~ +~_ooS" is the sum over all the elements of the Schottky group generated by S= [72U3, while the prime means that the identity must be omitted in the summation. We write (M) for a representation of a projective matrix M, understanding that it will be D, of conformal weight 0, or E, of conformal weight - l, when referring to orbital or ghost degrees of freedom respectively. We use the twisted propagator given by [ 6 ]

l I dx (3)

D=(bo-bl) x(1-x-------)P(x) , 0

where P(x) =xL°eL-' ( -- 1 )/-o-p2/2( 1 --X) w, with W=Lo-L1, and U = P ( x ) U, [7= VP(x); k=x/( 1 - x ) is the

S multiplier and

Z. = ( 1 - k ) " / , ~ / ~ . (4)

Using Ul =FV V ~ , it is sometimes useful to rewrite

A= Z (FST)=(FY-'c), (5) n#0

where

Sc=VII[72U3VI=VI-ISVI=(; I)

and ~chas the same meaning as 3-. Eq. ( 1 ) conveniently exhibits the automorphized structure of the tadpole, which, as we will see, generalizes

to higher loop operators. Since D is BRST invariant only after integration on x, surface terms might arise if one applies the BRST charge

to ( TI because of the truncation of the integration region in eq. ( 1 ) [6,7 ]. We will neglect this problem in the

following. The planar self-energy operator is obtained by sewing the tadpole to a symmetric three-reggeon vertex. Taking

the scalar product ( TI D I V3) in the space of leg 1 excitations, it turns out to be

f dk dx dpexp(½p21nk ) f i (l_k,,)_D+26(p2+p3) G23= k(1-k---~)x(1-x-~ n=l

× e x p ( - ~ (a;-"l(ujg~)12)p.- ~ Z" [½(a[ l(Uj~,Vs) la?)-(c;-l(Uj',~V~)lb2-)l) j=2 i,j=2 ot

(.._.~X /~3/~2 1--X 3 3 2 2 ) I~I 10a, ) ×\ l_xUO~,O - ~ ( b o - b l ) ( b o - b l ) i=2 q=3 i- (6)

Here 12)=D(P(x) ) IX) while Z~ denotes the sum over all the elements of the Schottky group generated by S= (PUI) -'Sc(PU1 ), with the identity missing only for i=j. For the orbital part all these elements, except the identity, are defined to have non-vanishing matrix elements only in the space of non-zero modes.

In ref. [ 3 ], starting with these operators, also the two-loop orientable tadpole has been evaluated. The pattern of the two-loop case can be generalized to any number of loops. Indeed not only the non-zero mode contribution can be written in exactly the same form with the Schottky group having now a generator for each loop, but also the term coming from the zero modes preserves its structure identically. However, to write the g-loop tadpole

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Volume 2 11, number 4 PHYSICS LETTERS B 8 September 1988

Fig. 1. g-loop tadpole drawn as a Feynmann like diagram. The tilde denotes the insertion of a twisted propagator.

operator, see fig. 1, it is convenient, for the sake of clarity to distinguish the variables x,, (i= 1, . . . . g), associated with the propagators directly attached to the tadpoles, from those yi (j= 1, . . . . g- 2) of the propagators, con- necting different three-vertices.

Let us now introduce the projective tranformation Vi defined as follows: starting from the “free” leg and moving along the “horizontal” line in fig. 1, i.e. without going around any tadpole, one writes from left to right a (z-independent) factor V,’ V,,, when crossing the n-th three-vertex (whose legs are labelled by CX,,, /3,, and yn= 1,2, 3 cyclically) and a factor P(y)rfor each propagator crossed, ending with the leg of the three-vertex to which the ith tadpole is jointed #’ . Hence, since there are g- 1 vertices, g- 2 “horizontal” propagators and, for i> 2, the ith tadpole is joined to the (i- 1 )th three-vertex, we can write 1 V,=V&!, v~,_~~p~Y~-~~~Bg--2T/ag-*l~~~~p~Yi-~I)u~,-~ va,_~l (7)

for i= 2, . . . . g, while f, is obtained from p1 by the substitution V,, + V,, . We recall that in terms of 2 x 2 matrices one has

uava+, = ( 1 -1 0 -1 > ’ ~olVti-,= ( 1 0 1 -1 > 9

and that for the D-representation D( U, V,+ 1 ) DT ( U, V,_ , ). The g-loop tadpole is then found to be

Xexp ,~,p,(~~((~~.r’P)la+)-f(a+I(~~~~)la+)+(c+I(~~~P)lb+) ( > (8)

Here the matrices 0 and P refer to the free leg of the tadpole and of course 0; =@y’. The quantity Y-‘= C&T AZ) (S’J= 12 r$)) denotes the sum over all the elements of the Schottky group which do not begin with TF on the left (and do not end with 7’jg) on the right, with the identity missing for i=j). The generators T jg), which depend on g, are given by

T,‘g’ = (P(~,)~~i)-‘S,(l’(x~)~~) , (9)

#I Using (YPY) = (P- ’ ) one can easily see that moving immediately backward on a piece of a path cancels the contribution of this piece.

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where S~ = (o k~ I) are the generators related to each isolated tadpole. With such a choice of the Schottky group one has V= 1 in eq. (8), but we have left it explicitly written to underline the fact that one can always make an arbitrary overall projective transformation on the generators: 7",.~ V - ~ T~ V.

The generators T~ g) as well as the various matrices appearing in the exponentials of eq. (8) have a very clear diagrammatic interpretation. Indeed T(~ g) is the projective transformation constructed applying the diagrammatic rules given above for l~,, see eq. (7), to a path which starts from the free leg, goes once around the ith tadpole and comes back to the starting point. The matrix 127 L y -o ~ corresponds to a path which starts from the three-vertex leg joined to the ith tadpole, winds an arbitrary number of times around each tadpole in either direction (giving a factor S ~ or S - ' ) and ends on the three-vertex leg joined to the jth tadpole. For i Cj one can also go directly from tadpole i to tadpole j, which is the contribution of the identity, while for i=j one has to go at least around another tadpole. Analogous meaning can be given to other matrices. Of course the projective transformation must be taken in the representation appropriate for the orbital and ghost degrees of freedom.

A very important point in eq. (8) is that in terms of the fixed points (~!g), q!g) ) of T ! g) o n e has

f~, 12,7 = I~f) - I ~ f > ,

while

(2,l 0 , = ( l / ~ f l - ( l / u f l ,

as is easily seen from eq. (9), since (Z~[ verifies the same relations in terms of the fixed points of &. The structure of the multiloop tadpole operator is even more clearly exhibited if eq. (8) is translated into a more explicitly geometrical language, in which the paths discussed diagrammatically above become paths on the associated Riemann surface.

Consider first the term in the exponent quadratic in the loop momenta

½ ~ p,[diiln k, + ()~/I ( O~Y°~ ) [)~j) ]Pj = - ½ ~ Pff2opj. (10) t,J U

It is possible to see that g2 is 2n times the imaginary part of the period matrix for the g-loop Riemann surface [ 8-10,3 ], which in terms of the fixed points (~, q) of the Schottky group generators can be written as

0 £2 o = -8o ln k, + ~ In ~i -t~(~)rli-t'~(rlJ) (11)

~, -t~(rb)rl,-t~( ~j) "

The coefficient of the term linear in p~ can be expressed instead by means of the first abelian integrals. Indeed using the definition of the D-representation and its properties, given e.g. in the appendix of ref. [ 3], one can see that

V(z)

xfna+ f OJiz= 0 ( 1 2 ) (2'[ (OJ - 'V) l a + ) = n=~'~l TOzn ZO

where in the Schottky parametrization of the surface the first abelian integrals are given by V(z)

t o f = - ~ In V(z)-&(¢i)Zo-t~(q,) (13) V(z) - t~(~/,)Zo - t~(~) "

Z0

For the term bilinear in the bosonic oscillators one has the relation

(a+I(UY-'V)Ia+)=a+(U),~ Z (Tc~)ls(V)~m a+ a~I

1 1 E[V(z), V(w)] . . . . o.~ma+~ (14) = - x / ~ a + ~ . O ~ a m In V-~ZV(w)

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a sum on n and m f rom 1 to ~ being understood. The quanti ty E(z, w), normalized to E(z, w)~( z -w) as z ~ w is given by

[z- t~(w)l[w-t , (z)] E(z, w)=(z-w) o~,~' [z-t~(z)][w-t~(w)] " (15)

In the above equat ion 1-[' indicates that T~ and T~- ~ are counted only once. E(z, w) is known as the pr ime form [ 8,11 ] and its logari thm yields the singular part of the N e u m a n n function on the surface.

Analogously using the propert ies of the E-representation, the bil inear term of the b-c system in the exponent of eq. (8) can be writ ten as

(c+I(UJ'V)Ib+) = ~ Z C+Enm(UTc~V) b+ n=2,m=-- 1 ot¢-I

1 1 - c , ( n - 2 ) ! ( m + l ) ! a " - 2 3 m + ' F [ V ( z ) ' V(w)] [ . . . . ob + (16)

Here

F[V(z), V(w)]=Sb,c[V(z), V ( w ) ] - [OzV(z)]2 Ow V(w) [ V(z) - V(w) ] (17)

is the regular part of the quant i ty

[0zV(z)] 2 1 S~.c[ V(z), V ( w ) ] = ~ Owt,[ V(w) ] V(z)- t ,[ V(w) l ' (18)

where the sum is running over the whole Schottky group. As we shall see Sb,c is the equivalent o f the Szeg6 kernel for the b-c system ~z.

In eq. (16) also the contr ibution of the antighost zero modes is included, which however is irrelevant when taking the scalar product of the tadpole operator with a physical state having q = 1, or with the leg of another tadpole or vertex opera tor after the insertion of the propagator given in eq. (3) , because of the zero modes present in the last line ofeq . (8) .

Substituting eqs. ( 10 ), ( 12 ), ( 14 ) and ( 16 ) into eq. (8) , and integrating over the loop m o m e n t a we obtain

I Tg)= f f i dk, dx, ~[g-2 dyj I ~ , ~ ' = , ( 1 - k , ~ ) -D+2 (detg2)_D/2 ,=, ki x,(1--x,)j-=q y j ( l - - y j ) l - [ ~ ( 1 - k , ) z

(, +1 1 V(z),V(w)] , +) × e x p 2x/n ~m.V V(z)-V(w) . . . . o

- a~ 070m In E l , f m ~ a , ,

V(z) V(w)

× e x p ( ½'/-nan+ i,j= ~ , llo"f°fz=o[g2-1]iSO~n? m? -~ f °Y w=oV/m½am+ ) zo W0

( l 1 ~n_2~m+l~ . . [ ] . . . . O ) × e x p c + ( n - 2 ) ! ( m + 1)( . . . . . ~V(z), V(w) b +

i=l i= '~X i ~ y j bO- f i l-x` ---Ti---(bo-b,) 10a, q = 3 ) . (19)

~2 Strictly speaking eq. (17) is exact only if the matrix V does not contain inversions, or else and extra term, which is a polynomial of second degree in w, should be included. The above expression has also been obtained by the authors of ref. [ 10] (see ref. [ 11 ] ). We thank them for useful discussions on this point.

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It is interesting to remark that correlation functions on a genus-g surface can be expressed as expectation values between the SL (2, ~) invariant vacuum 10a, q= 0) and a state (gl defined as the Bogoliubov transformation [ 13-15 ] generated by the integrand of the multitadpole operator

(gl = (0a, q=3 [exp[ (a[ ( U:~ ~i)12,) [ t2-1 ]gj(2j I (Uj y-iV) la) ]exp[ - ½ (a l ( UY" V)la) - (cl( UY-' V)Ib)] • (20)

Indeed the two-point function for the bosonic field X(z)

Gx(z, w) = (glX(z)X(w) lOa, q = 0 )

turns out to be

V(z) V(w) Gx(z, w)=lnE[V(z), V(w) ]+ ~ f oji[g2-1]i j I O)J

i , j = I zO wo

i , j = 1 zo wo

which is the holomorphic part of the Neumann function on the surface [ 8 ]. The last equality in eq. (21 ) is justified from the definitions given in eqs. ( 13 ) and ( 15 ), provided one makes a similarity transformation on the Schottky group generators and apart from terms depending separately on z and w which can be absorbed into a redefinition of the Neumann function.

In a similar way, for the propagator of the ghost field one has

Gb.c(z, w) = (glc(z)b(w)10a, q = 0 ) =Sb.c[ V(Z), V(W) ] .

Using the g-tadpole operator one can now build quantitites like partition functions and scattering amplitudes. In order to construct the partition function relative to a planar, orientable surface of arbitrary genus g, we

have to join an h-loop tadpole to a ( g -h ) - l oop one by means of a twisted propagator. To be specific we evaluate the g-loop partition function by performing the scalar product

Z g = ( T g _ 1 [BIT) . (22)

By a straightforward calculation this yields

i=1 (l--xi)2j~=l (1--yj) 2]

× I ~ f i (1--k~,)2-Dl-If=~(l--~z) (det g2) -D/2 (23) . . = t 1-[, ( 1 - k ~ )

The generator 7",. of the Schottky group is here obtained applying the rules given before eq. (7) to a closed path starting from an arbitrary point of the diagram in fig. 2 and making a tour around the/th tadpole, i.e. going once around the cycle ai of the canonical homology basis. Changing the starting point is equivalent to performing on the generators a similarity transformation, which leaves Zg invariant.

The g-loop N-reggeon vertex VN,, see fig. 2, can be obtained by joining the operator I T~) to one leg of an N + 1-reggeon vertex, i.e.:

( VN,gl = ( VN+I I DI / ' 8 ) . (24)

There are several possible forms for such a vertex (see e.g. ref. [6 ] ) but any choice will lead to the same result if one first saturates the ghost degrees of freedom making them disappear from the resultant operator. The

422


~ m

Xll

Y l - 2

1

N + I i

' . " , , j ,

Fig. 2. g-loop N-reggeon vertex obtained by joining a g-loop tadpole to the leg N+ 1 of an (N+ 1 )-reggeon vertex.

matrix element of this operator with DDF states yields then the relative scattering amplitude. Of course explicit BRST invariance is lost once such a gauge choice has been made.

By using the "asymmetric" vertex (VN+~I given in ref. [6], saturated with the state ] qt = 1 ) ] qN= l ) 1-[~=-21 I q~ = 0) , and joining leg N+ 1, with q = 1, to ] Tg), we find, following their notation

/,=, ,=, k~ ~, j = x ' ~ ' 2 ~ , 2 r , = , u , a b ~ O ( z " - z u + ' ) , yj y . = V',(O)V'w(O)V'N+I (0)

/,=~ 1-I~(1-k~) 2 [det~l-D/2.,~=1171 exp[(a, lM.~la.)16 , P . . (25)

Here y is related to the propagator in eq. (24) and

M,u = ( U~ W'l?,) 1~) [ Q - ' I,A2j I ( ~ : - J v , ) - ½ ( u~ ~-" ( V~).

Also here all transformations are build according to the diagrammatic rules discussed above, starting from leg v and ending on leg/2, the identity being omitted in : ' " only for /2= v. We recall that all the elements of the Schottky group different from the identity, have non-vanishing matrix elements only in the space of non-zero modes of the oscillators a~.

We notice that both the coefficients of the oscillators bo and b~ present in IT g) contribute to Zg, while only the last one contributes to ( VN,g[. This term is the only one corresponding to the old string formalism that ignores the ghost degrees of freedom. The non-zero ghost modes present both in ( VN+][ and [ Tg) do not contribute to the scalar products of eqs. (22) and (24), their effect reducing therefore to the (partial) compensa- tion of non-transverse orbital excitations in the loops, as shown by eqs. (23) and (25).

The above results can be rewritten in terms of the fixed points (~i, t/~) of the Schottky group generators. In the case of the g-loop N-reggeon vertex, trading y and ZN+ ~ for two fixed points of the set of generators

L = (P(Y) U,v+ l ) - 'T f (P(y) UN+ ~ )

one finds after some calculation

(Vu,gl= if[ u(0al f f i dki d~idr/i ( l _k i ) 2 [I dzu lL=' '=1 d"~'~abc O ( Z ' - z l 2 + I ) k 2 (¢i __ q i ) 2 u = l

× FI V'u(O)-ll-[~l-In~=l(1-k~)2-O[det-Q] -D/2 f i exp[(a~lM, u lau) ld ( ~ Pu) . u=, 1-[a(l-k,~) 2 u,~=l /,=1

(26)

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Us ing this express ion it is easy to see tha t the o p e r a t o r Vu, g co inc ides wi th the one g iven in ref. [ 11 ], thus

showing expl ic i t ly the dua l i ty p rope r ty o f the f o r m u l a t i o n used.

A c o r r e s p o n d i n g express ion can be wr i t t en also for the pa r t i t i on func t i on Zg.

We like to t hank P .Di Vecchia , M. F r a u and S. Sc iu to for several useful discussions. The au thors acknowledge

the k ind hosp i t a l i ty e x t e n d e d to t h e m by N O R D I T A at va r ious stages o f this work.

References

[ 1 ] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Phys. Lett. B 199 (1987) 49. [2] G. Cristofano, F. Nicodemi and R. Pettorino, Phys. Lett. B 200 (1988) 292;

T. Kobayashi, H. Konno and T. Suzuki, University ofTsukuba preprint UTHEP-175. [3 ] G. Cristofano, F. Nicodemi and R. Pettorino, Naples University preprint (February 1988 ). [ 4 ] E. Cremmer, Nucl. Phys. B 31 ( 1971 ) 477. [5] P. Di Vecchia, E. Del Giudice and S. Fubini, Ann. Phys. (NY) 70 (1972) 378. [6] P. Di Vecchia, M. Frau, A. Lerda and S. Sciuto, Nucl. Phys. B 298 (1988) 526. [ 7 ] A. Neveu and P. West, Phys. Lett. B 194 ( 1987 ) 200. [8 ] S. Mandelstam, in: Unified string theories, eds. M. Green and G. Gross (World Scientfic, Singapore, 1986). [9] E. Martinec, Nucl. Phys. B 281 (1987) 157.

[ 10] J.L. Petersen and J.R. Sidenius, Nucl. Phys. B 301 (1988) 247. [ 11 ] P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, N-string g-loop vertex for the bosonic string, NORDITA preprint

(1988). [ 12 ] P. Di Vecchia, K. Hornfeck, M. Frau, A. Lerda and S. Sciuto, private communication. [ 13 ] C. Vafa, Phys. Lett. B 190 ( 1987 ) 47. [ 14] S. Mukhi and S. Panda, Tata Insitute preprint TIFR/TH/87-51. [ 15 ] L. Alvarez-Gaum6, C. Gomez, P. Moore and C. Vafa, CERN preprint TH-4883/87.

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The multiloop covariant tadpole operator and amplitudes for the bosonic string