Volume 199, number I PHYSICS LETTERS B 10 December 1987

A SIMPLE EXPRESSION FOR THE MULTILOOP AMPLITUDE IN THE BOSONIC STRING

P D1 VECCHIA ~, M FRAU b A LERDA ~ 1 and S SCIUTO d b Z

' Nordtta, Blegdam~vej 1 "~, 2100 Copenhagen O, Denma~l~ h Sezlonedt Tottno de/l'lNFA 1-10125 Turm, Itah' • Institute [or Theoretltal Physic ~, State ~ nt~er~ttl' o! New } orl~ at Stotn Btoo~, Alonl Broo]~, ~)" 11794-3840, US4 '~ Dtpatttmento dt I't~tca deg Um~erstta' dt Yapoh, 1-80125 Naples Itall

Received 18 September 1987

Using the preciously construcled new &'-string ~ertex and BRST m~arlant twisted propagator '~e obtain a simple and exphcn expression for the mululoop pamtlon function of the bosonlc string We construct also the two-loop amphtude with ~,~ external Iach2cons and gl~e the recipe for wrnmg the amphtude at an3 perlurbat~e order

A basic quant i ty in a string theory is the N-string (reggeon) vertex [ 1 ] ~1 It allows one not only to compute on-shell scattering amphtudes at the tree level reproducing the known results obtained with vertex operators,

but also to construct in a simple way mult l loop amphtudes The N-string vertex was the starting point in the old days for the construction of the multl loop diagrams [ 3 ]

In this way it was recognized that the mathematical structure underlying them was the theory of R lemann sur- faces In the last few years there has been a lot of activity for constructing multaloop diagrams starting from the path integral formulat ion of the string theory ~2 Multi loop diagrams were also computed by Mandels tam using the functional integration technique in the light-cone gauge [5,6]

A problem, that, however, was not solved in the old days, was the e l iminat ion of the unphyslcal states in an arbitrary multi loop diagram This was achieved [ 7] only for one-loop diagrams by means of the Brink-Olive projection operator

Recently the N-string vertex has been generalized to include also the contr ibut ion of the ghost degrees of freedom [ 8] This original form of the vertex has, however, the unpleasant feature of treating the external states in a non-symmetr ic way This creates problems when we use it for computing multfloop diagrams

In order to overcome th~s difficulty two new N-string vertices have been constructed [2,9 ] by sewing together three string verhces, that treat the external states symmetrlcall~ The exact connection between the vertices an refs [2,9] Is at the moment not fully understood The previous constructed vertices were used to compute one- loop diagrams reproducing correctly the already known results The authors of ref [9] wrote also a formula for the mult l loop diagrams

In this letter starting from the new N-string vertex and the BRST invarmnt propagator discussed m ref [2] we compute an arbitrary mult i loop dmgram for the bosonlc string obtaining a simple and explicit expression We start our discussion referring to the open string, or equivalently to one of the two sectors of the closed string

The basic ingredients of our construction are the BRST invar iant twisted propagator [2]

' A Della Rlc~a fellow -~ Work partially supported by the Italian Mmlstero della Pubbhca lstruzlone ~ Additional old references on the N-stung vertex can be found m refs [ 1 ] of ref [2] "-~ Seerefs [1-15] m ref [4]

0370-2693/87/$ 03 50 © Elsevier Science Publishers B V (North-Hol land Physics Publishing Divis ion)

49

Volume 199, number l PHYSICS LETTERS B 10 December 1987

><i I 4 F~g 1

1

dx T = ( b o - b ~ ) I x ( 1 - x )

0

- - P ( x ) , P(X)=xL"~(1 --X) u , (1)

and the new N-string vertex [2]

V'~"rl 2, , N ) = (dz , O(z,-z,+~ H.~IdA 1 /', \ ~ 2

,=, < ( z , + , - z , )

where

,~o , (2)

14:°~ m= dD x [ ( :< Q., q= 3 ] ] l ~ e x p ~ . ~ . .... U, I , . . . . . t = l t : ~ l ~ ? = 2 I ? I ~ - - 1

l (,~ I : I n O ) × l - [ e x p - ~ a,, D,,,,,(U,.::_,,, ~ E.,,,(V,)b}/,' , (3) m = O n = - - I / ~ 1 - 1

is the mtegrand of the "o ld" N-reggeon vertex of ref. [8] and dV.:~, is the volume of the proJecnve group

dz . dz:,d& (4) dC,t,, = ( z . - z : , ) ( z t , - z , ) ( z ~ - z . ) '

A~ ~ ~, is the product of ( N - 3 ) f e rmmmc 6 functmns

4- -3 s + l 1

A,2 ~= I] 2 Y~ bl,"e,,(U~,~;) (5) s = l t = l n = - - I

The project ive t r ans fo rmatmn U~, ~s the one mapping z,+~, z~+> z~ into 0, oo, 1, respecnvely, and

e , ,=Eo . -E_ l , , (6)

Finally, co is the following project ive t r ans fo rmatmn

r .o=(z3,z ._ ,z ,~,z l ) -" ' f i ( z , , z l . z , _ l . z , _ 2 ) - " ' ~ (7) t = 4

In ref. [2] we have shown that the new vertex has the race proper t ies of being BRST mvarmnt after the mtegra tmn over the K o b a - N l e l s e n varmbles is pe r formed and of having a project ive mvarmnt mtegrand

An unpleasant feature of I . . . . ,~ ~s that the new factors A~ e ~ and co seem to break cychc symmetry and then duahty

Now we will prove that Feynman-hke graphs bui l t sewing together couples of legs of I%, ~'~ with the propagator T in (1) (or, that ~s the same. by combining m all possible ways BRST m v a n a n t three-reggeon vertices and propagators T) and saturat ing the external legs w~th physical states are actually dual m spite of the factors A, _. ,4, and (o Namely we will prove that the s u b s m u n o n o f fig 1 can be per formed in any subgraph of any per turbat lve Feynman-hke dmgram In fact when the four-reggeon vertex V] c'~ ~s inserted reside a larger graph.

50

Volume 199. number 1 PHYSICS LETTERS B 10 December 1987

each of its four legs is saturated either with the he rmman propagator T or w~th a physical state ] q = l ) l t r a n s ) = ( b o - b ~ ) b ~ l q = 3 ) l t r a n s ) Because of the ldentmes

W I q = 1) I t rans) = 0 , W T = T W + = O , (8)

the factor ~o can always be replaced by 1 ~3 Moreover also the factor A j ~ 34 becomes cychcally symmetric once the multlphcatlve factors (b ~ - b~ ) ( t = 1,

2, 3, 4) are taken into account In fact one gets

4 4

z~2;41 H ( b ~ , - b ~ ) = - A l . ~ 4 H ( b ' o - b l ) + , (9) t = l I - - I

where the dots stand for terms that vanish when multlphed w~th the three fermlonlc delta functions already present m the old vertex (3) The minus sign on the right-hand side o f eq (9) is welcome because it cancels the one coming from the cyclic permutat ion o f the four ant lcommuting vectors , ( q = 3 [ (t = 1, 2, 3, 4)

Then cychc symmetry of V~ ~W follows from factorlzation and repeated use of the duality property we have just proved It is worthwhile to note that while BRST and conformal invariance of perturbative graphs are assured whatever BRST mvariant vertices and propagators are used, the proof of duality heavily relies on the precise form (1) o f the twisted propagator T The proof will fail f fo ther BRST mvariant propagators were used and m particular m the case of non-onentable graphs (with propagators hke TO, sg+T or bo/Lo)

The M-loop planar partition function IS obtained starting from the new 2M-string vertex and sewing a leg with the next one according to the cyclic ordering provided by the vertex after having inserted the BRST in- variant twisted propagator (1) Since the 2M-reggeon vertex depends on 2 M Koba-Nlelsen varmbles and each of the M propagators (1) is expressed as an integral over x, the mtegrand of the M-loop partition funcuon will be a functmn of 3 M - 3 varmbles (three varmbles can be fixed because of projective mvarlance) Those pa- rameters provide a natural parametnzat lon o f the moduh space

The M-loop planar pa r tmon function (for M>~ 2) is given by ~4

Z , , _ - 1 ! ] Tr,2,,_,2/~,{1"+(1,2, . 2 M ) ILl Tp} , (10) t~= I p = I

where jr, is the twisted propagator o f e q (1) and V + is obtained from (2) by changing the oscillators and the vacua of the even legs as follows

p , , - . - p , , , a , , - . - a , + , c , , - ~ - c , + , b,,--*b, + , ( O . , x , q = 3 [ - , l q = 3 , x . O . ) (11)

Since the contributions o f the orbital and ghost degrees of freedom appear in a factorized form, Z~t can be written as a product of the measure, of the contrlbuUon O~t of the orbital modes and of the contribution G~ of the ghost modes

Z ~ l = j d I ' O ~ t G ~ t , (12)

where

1 ~2~( dz, )11 ] ( d r . "] d~ '= dV./,---~,=i z~+~-z, .=, \ x , , ( ~ x . ) / (13)

We have neglected here the factors O(: . , - z ,+ t ) because the integrataon region deserves a special discussion

~ The propertv TIf + =0 is also needed because in building perturbauve graphs some legs of the vertices must be hermman conjugated Moreo~ er we observe that the equations If T= TIf + = 0 can be safelv applied because o) does not depend on the integration varmble of the propagator T

~4 In ( 10 ) we can set o) = 1 as alread~ explained

51

Volume 199. number 1 PHYSICS LETTERS B 10 December 1987

The contribution of the orbital degrees of freedom has been explicitly computed in ref [3] Here we give simply the result

O , , = H ' f i ( 1 - k . . ) D ( d e t I m r ) z)/2 (14)

where the k~ are the multlphers o f the projective transformations that are elements of the Schottky group gen- erated by

S , ,=~%P(x , , )U2 , ,_ , , / z = l . 2 , 3 . . m (15)

Remember that any projective transformation 7(z) can be parametrlzed in terms of the multlpher k and the fixed points q and ~ implicitly defined by

~(z) - ~ z - ~ l ~;(z) - ~ =kz-~-- (16)

]-[L ~s the product over all primitive elements o f the Schottky group, 1 e those that cannot be written as powers of other elements, with each conjugacy class counted only once

Finally Im r is the imaginary part of the period matrix, that ~s obtained by integrating over the momenta circulating in the M loops

The ghost contribution is given by

- " ( ' ) ~ ~ b~(J)m G , I = Tr(_~,_~2~,~ ~_/,-1(q=31 [ I e x p l-I c,, E . , , , (U,~,) ~.lt ~ ] I # l ~ n t H = 2

×,, , , I , , , = - , E " " ' ( ~ ' ) b " ' _ \ , = , e ' ( U t f f~g ~'~ p=,~ [(bG~p-L~-b(_21'- '~) lq=3)2p], (17)

where

ES,'~=b~,',, ~,',=c5, '~ , ( r ,=P(x , , )u , , ~,=v,P(x,,), (18) If t = 2 p - 1 and

bS,')=b~ ' ' ' , g ~ / ' = - c + ' ' ' , U , = U , . 9 ,=1>; . (19)

if t is even It is easy to see that the contributions of the zero and non-zero modes can be computed separately In fact

the amount of fermlonlc 6 functions, containing the oscillators bo, b~, b_ ~, corresponds to a total ghost number equal to - 3 M , that is just what we need in order to get a non-vanishing result for the M matrix elements be- tween the I q = 3 ) states Terms coming from the non-zero modes will upset the balance and they will give a vanishing contribution

The contribution of the non-zero modes ~s easy to compute and we get

G, , (non-zero modes) = H ' [~ (1-kT,)-" , (20) o¢ n = 2

in agreement with the result obtained in refs [8,9,10] The contribution of the ghost zero modes is given by the determinant of the 3 M × 3M matrix obtained with

the coefficients of the 3M terms bS, 2~'- ~ with n = 0, + 1 (identified with bS, 2~) ) and/1 = 1, 2, , M, that appear in the 2M fermlonic ~ functaons contained m the 2M-reggeon vertex and in the addlnonal M ferm~onlc 6 func- tions (b~ 2~'- ~ - b ~ i ' - ' ) present in the propagators (1)

The determination can be computed in a closed form and ~s given by

52

Volume 199. number I PHXt SICS LETTERS B 10 December 1987

G~ t ( ze romodes )= ~ ( ( 1 - k . ) I t : 1

with

-'¢,, .1 ,=l , ,~ (21)

2 2 t t - - 1 - - Z'2// 7 . - , ( 2 2 )

"~2tt + 1 - - Z21t

where k. is the multipher of the transformation S l, m (15) It is convenient to express the measure (13) m terms of the fixed points (~l,, ql,) and multipliers kl, of the

M transformanons S. in (15) It IS easy to check that they are related to the original 3M variables z, (t = 1.2, ., 2M) and x. (/~= 1. 2. . M) by the following relations

x,, ( z 2 . _ , - z e . _ e )( z2 . - z : , , + ~ ) k i t = - - {)Lit 1 - - #VIt ' O~It = - - ( z 2 t ` - I " Z 2 l , - 2 " Z 2 ' , ' Z 2 u + 1 ) ~ - - ( Z21l--1 - - Z21z + 1 ) ( Z 2 Iz - - Z21z-- 2 ) '

¢,, =>-, , . ~1,, = ze , ,+ , ( ze , ,_ , -z.,,)k,, -ze, ,_, ( z . , , + , -z: , , ) (22,,_l_z2i,)kt_(Zeu+l_Z2,,) , / t = l , 2 , , M (23)

The jacobmn of the transformation is given by

O(k, rl) f i ( _ c~,, ( 1 - k , , ) ) ( 1 _ lg[ /. },~,) (24, O(X, 2) --u=l ( l - -x , , ) 2 (l--/~,y/,): ,,=,

Using the previous jacoblan we can rewrite the measure (13) as follows

dV= IF] . , - z , + , 1 l~ ( dk,,d{,,dr/,, x , , ( ! - k , , ) ) ( 1 15] k,,7~ (25) , . , 7--77+; \ k ~ , ( ~ i , - ¢ , , ) - ( 1 - x . ) j \ . = ,

Inserting (14), (20), (21), and (25) into (12) we get the final expression for the planar M-loop parti t ion function (M>~2) of the open bosonlc string

j ¢,, ) - 1 ,t ( dk,,d{,,dr/,, ~ - ,)+2[det -,,,2 (26) Z, ,= d--~l.,~= ' \k~(r/, _~,,)2 } (~ ' ,,=fi, ( 1 - k ~ ) l m r ] \,,l~I ( l - k l . ) -~ ' ( ' - k . ) 2 .

where d I~,, takes mto account the mvanance of our expression under projecnve transformations on the fixed points and is gzven by

d V.,,, = @.do,,@. (27) (P. -P~,)(P. -P~ )(Pt,-P, ) "

where p.. Pt,, P, are three of the 2M fixed points ~,,. ~1,, Eq (26) shows that. unless the one-loop case, the contmbutlon of the ghost degrees of freedom does not just

cancel two powers of the first term m (14) This is only happening for the modes n> 1 For the mode n = 1 we get instead the addmonal last term in (26)

The previous calculatmn can be easily generahzed to the multlloop amphtude with N external tachyons In this case we must start from a (2M+N)- reggeon vertex sewing together the 2M legs as already explained and saturating the left-over states with the states [ q= 1 ), [ O, k, ),. l = 1.2. , N corresponding to tachyons

The calculation for the orbital degrees of freedom proceeds as m ref [ 3] and one gets the following result

O,,= ~ I ( ( z ' - z ' - ' ) ( z ' - : ' + " ) ~ ' ( I ( l - k : 'O-" (de t lmT) -"2 l - l . exp[k , . k ,V(z , , z , ) ] } (28) ,= i ( 2 ~ _ i - 2 ~ + l ) , , : J , ~ j

~'(z,. z,) and the permd matrix r are given exphcltly m refs [3.5.6]

53

Volume 199, number 1 PHYSICS LETTERS B 10 December 1987

The ghost con tnbu tmn is given by an expression similar to (17) where now the ( N + 2M)-strmg vertex con- tams both the legs that are sewn and the legs that are saturated with the physical states I q= 1 ),

The contribution of the non-zero modes IS again given by (20), while that of the zero modes is obtained by computing a determinant of the ( 3 M + N) × ( 3 M + N) matrix obtained as already explained in the case o f the partition funcnon The result is

G u ( z e r o m o d e s ) = 1 - x l, '~ - - z , + 2 l-[ ( 1 - k l , ) , (29)

that we have only checked for M = 2 and arbitrary N The measure d V with N external physical states has the form

1 ~ • [I \x,,(~-x,,)l (30) d V = d v ~ t . . . . 1 . . z , - , + l / l,~l

that can be rewritten after the same mampulanons as m the case of the pa r tmon function as follows

d V = f i ( x. ~"~'t (z,-z,+L) ~ (z,_,-z,+,) 1 [5[ dz, ~ (dkud~"dl/z'(l~-.k')~ ,,=, \ l - x . / ,=, \z,-z,+2 ,=, (z,~-z,~;)(z,-z,+,)' dV~.~ ,=, .=, \ k-~,(~_-~-~-). ,,t

(3~)

Putting all factors together we obtain the following final formula

A~ w(k~. k,O= dz, dV. t , , , ~ - - ; 7 - - - . ( de t lm 1~' (1 • , , , = , \k~(~l,, -~ , , ) - l . . . . ,

t < / / t = I

We can also compare our expressions (26) and (32) with the ones written by Martinec [ 10] and by Petersen and Sldemus [9] Their result consists of four factors corresponding to the contr ibunon of the orbital and ghost non-zero modes, of the determinant of the imaginary part of the period matrix and of the ghost zero modes, that are exactly the four factors that also appear m our calculation We get of course the same expression for the contribution of the orbital and ghost non-zero modes and for the determinant of the period m a m x In ad- d lnon we get a closed and explicit expression for the contribution of the ghost zero modes given in (21 ) and (29) and for the measures (25) and (31)

We must remark, however, that, as already discussed in ref [2] for the one-loop case, the integration region for the variables k,,, ~, and q~, cannot be inferred from the integration region for the variables x . and z,, but must be fixed by hand to be a fundamental region of the modular group

We have constructed the multfloop amplitude for the open bosomc stung The generallzanon to the closed string is straightforward just considering complex z and taking the modulus square of (32) except for the de- terminant o f the imaginary part of the period matrix that is unchanged, since it is coming from the integration over the momenta circulating m the M loops, that is the same for open and closed strings This procedure agrees with the conclusions m ref [ 11 ] The result obtained m this way is given by

A, ~l(kL, , k,~)= d22, d ~ ; , , . . . . , \Ik,,(~,,-¢,3 [~J .... ,

×[I,<,{lexpt2k,'k.V(z,.z,)]ll,,=,l 5 ] [ 1 - k , , 1 4 / l ~ ' . [ 1 - k o l 4 (33)

In the case of a closed string the volume element dI~/,, is given by

54

Volume 190, number 1 PHYSICS LETTERS B 10 December 1987

?_ ~ B Fig 2

d2p~d2p/,d2p~ d~/, , = ip _p/,l~_ ]p _p, 1~ IP~,-P, 12 " (34)

where p~, p/,, p~ are three o f the 2M fixed points ~,,, q. or o f the Koba-Nlelsen variables z, It is interesting to compare our result with the expression for the M-loop amphtude given by Mandelstam

[6] The comparison ~s very easy because we use the same parametrizatlon of the moduh space It is m fact straightforward to check that (33) wlthout the last term is exactly equal to eq (7 24) of [6]

However there is a new expression discussed by Mandelstam [ 12 ] m which a correction factor/~ appears Therefore our expressxon can be made to agree with Mandelstam's expression provided the following lden- nfication is made

F = [] ( 1 - k . ) 2 / H ' ( 1 - k . ) ~ (35)

Finally we wish to show that the extra factor ]~Y_~ I 1 - k . [ 4 appearing in the last term of (33) is essential to preserve modular mvarmnce

Even if the Schottky parametnzat ion ~s not statable to discuss general modular transformations, it is very easy to deal in th~s framework wzth the subgroup generated by the changes in the homology bas~s on the Rae- mann surface which only mix cycles of the same kind (see fig 2) In fact the replacement of the cycle B~, with the sum B.+B~ (with the corresponding combinat ion on the cycles .4~ and A. in order to get again a canonical basis) amounts to replace the generator S. with the product S.S.=-S ~5 The only term potentially affected by th~s subsututmn ~s the one appearing in the measure an (33)

d~, ~ d2 k"d2 ~"d2rL' I 1 -k j , 14 (35) Ik,,(4,,-q,,) 14

lnstead of calculating the jacobian from k~,. ~,. tL, to ~. ~. ~ (mult ipher and fixed points o f ~), it is easier to exploit projectwe invariance and perform the change of variables k., ~.. k ~ k . , k~, ~ keeping ~L,, ~.. tl~ fixed Using the formula for a mult tpher of a product

~'2 + ~-'/~- = [ k. + k. + ( ~., tl,,. ~,,, tl,,)(1 - L~,)(1 - k . ) ]( k,,k.) - '12 (36)

one easily gets

1 I ( 4 ~ - q . ) ( q . - ~ , , ) ( ~ , , - 4 . ) I -~

d-rl,,d-¢,.d-q. 1 ~ " • - - 3 2 =~d-k.d-k~d-~lk~,k.~l I (1 - k . ) ( 1 -L.,)(1 - ~ ) I , (37)

which ~s symmetric in '(5~. k. and Then, remembering that S . = S S y ~ and that a matrix and its reverse have the same fixed points and mul-

tiplier ~6 one can go back to the variables ~. ~. k.(q. ~ . t/~ fixed) Finally using again projective Invariance one ends up with the variables ~. ~. ~, /G, 4.. tl,,. prowng the mvarlance m form of the measure (35)

~' A very readable discussion on this point can be found m ref [ 13] We thank E C r e m m e r for ha~mg brought this reference to our a t t en t ion

~ One could quesnon i f the m u l n p h e r would not go into its inverse but an),~a~ our measure (with the factors ] 1 - L . I 4) is m v a n a n t under the subs t l tunon /~ .~ / . , 7 ~ (and m d e p e n d e n t b under ~,,~q,,)

55

Volume 199, number 1 PHYSICS LETTERS B 10 December 1987

T h i s a r g u m e n t is no t p o w e r f u l e n o u g h to p r o v e c o m p l e t e m o d u l a r m v a n a n c e o f o u r a m p h t u d e , b u t it is suf-

f icxent to ru le o u t e x p r e s s i o n s o f t he m e a s u r e w h i c h do n o t c o n t a i n the f ac to r L 1 - k . 14

O n e o f us ( A L ) w o u l d h k e to t h a n k t he I N F N , Sez lone dl T o r m o , for pa r t i a l f i n a n c i a l s u p p o r t a n d t he

D e p a r t m e n t o f T h e o r e n c a l Physxcs o f T o r l n o for ~ts k i n d hospxta l l ty

References

[1] C Lovelace Phys Lett B 32 (1970) 490 ]2] P Dl Vecchla, M Frau, A Lerda and S Scmto, Nordlta preprmt 87/36 P (1987) [3] M Kaku and L P Yu, Phys Lett B 33 (1970) 166, Ph~s Rev D 3 (1971) 2992, 3007. 3020,

V Alessandrlnl, Nuovo Clmento 2A (1971) 321, V Alessandnm and D Aman, Nuovo Clmento 4~ (1971) 793, C Lovelace. Phys Lett B 32 (1970) 703, E Cremmer, Nucl Phys B 31 (1971) 477, C Montonen, Nuovo Clmento 19A (1974) 69

]4] P D1Vecchla R Nakayama J L Petersen J Sldenlusand S Scluto Nucl Phys B 287 (1987)621 [5] S Mandelstam, Nucl Ph~s B 64 (1973) 205 [6] S Mandelstam, Unified string theories, eds M Green and D Gross (World Scientific, Singapore) p 46 [7] L BrmkandD Ohve, Nucl Phys B 58 (1973) 237 [8] P D1 Vecchla, R Nakayama J L Petersen, J S~demus and S Scluto, Phys Lett B 182 (1986l 164 [ 9 ] J L Petersen and J Sldemus. NBI preprlnt HE-87-35 ( 1987 )

[10] E Martmec, Nucl Phys B 281 (1987) 157 [ I I ] A A BelavlnandVG Knlzhmk. Ph~s Lett B 168 (1986) 201 [ 12 ] S Mandelstam, Talk Workshop on String theories ( Santa Barbara, December 1986) unpublished [13] E Cremmerand J Scherk, Nucl Ph~s B 48 (1972) 29

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