Eikonal resummation and topological expansion coefficients in superstring theory

Volume 244, number 3,4

PHYSICS LETTERS B

26 July 1990

Eikonal resummation and topological expansion coefficients in superstring theory Gerardo Cristofano a,b, Marco Fabbrichesi a,c,1 and Kaj Roland c,2 a NORD1TA, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark b Dipartimento di Scienze Fisiche, Universit~ di Napoli and INFN, Sezione di Napoli, 1-80125 Naples, Italy

Received 15 March 1990

We explicitly show that in the Regge regime of large s, fixed t the leading part of the g-loop amplitude for the scattering of two gravitons is the product of g+ 1 tree amplitudes times the expectation value of a factorized operator. This result proves that the superstring loop series is summable in the eikonal approximation. We determine the normalization dictated by unitarity of all terms in the topological expansion as a function of the two independent constants in the theory, the string tension 1~or' and the coupling constant g.

1. Unitarity is violated by the tree-level superstring amplitudes in the Regge regime of large center-of-mass energy x/~ and fixed transferred m o m e n t u m x/cL-t [ 1 ]. It has been argued [2,3] that this problem is solved by an eikonal resummation - very much alike the one of field theory [ 4 ] - of the perturbative series. In ref. [ 3 ], by using Regge-Gribov techniques [ 5 ] to approximate the string amplitudes in the Regge regime, such a resummation was shown to take place in the impact-parameter space. Nevertheless, a formal proof of the eikonal resummation in string theory requires starting from the actual superstring amplitudes and by taking the appropriate limit in moduli space recovering the same results. In a previous letter [ 6 ], as a preliminary step in this direction, we have proved that the expected Regge behavior upon which Regge-Gribov techniques rest - is indeed a feature of the asymptotic limit of the superstring amplitudes at any loop order. In this letter we complete the analysis by explicitly proving that the part leading in powers o f s of the superstring amplitude at the g-loop order does factorize in the Regge regime. The perturbative series so obtained can be readily summed by going into the impact-parameter space to give an exponential function. This exponentiation manifestly restores unitarity. At the same time, for the exponentiation to occur, the numerical coefficient in front of the amplitude at each loop order must take particular values. These coefficients are nothing but the constants Ng giving the relative weights of different topologies in the loop expansion. They are fixed by unitarity. We therefore obtain, as a byproduct of the proof o f exponentiation in the Regge regime, the absolute normalization of the superstring amplitudes at any loop order. Until now, such normalization was known only up to the one-loop level [ 7 ]. 2. For the sake of definiteness, and also because it is the most relevant case, we restrict ourselves to the scattering of two gravitons (see fig. 1 ). The graviton state is given by i : D X U D X ~ e x p ( i p : X ) : 10) [Ei'Pi) -~-~u*'

(1)

for i = 1, 2, b and c. The superstring amplitude for this process at g-loop order can be written by using the covariant loop calculus [ 8 ]. By singling out the kinematical piece dominant as s ~ , we obtain A. Della Riccia Foundation Fellow. 2 Supported in part by the Danish National Science Research Council. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

397

Volume 244, number 3,4

PHYSICS LETTERS B

26 July 1990

I

2

k2

j CZa o

Ze z

0

'")

b Fig. 1. The leading diagram in the two-graviton scattering amplitude at the g-loop order (the wiggling lines representing the exchange of the leading trajectory ).

ag4oop(S, t)=Ng~l "~2*b'~c

[

× (Ik~/~\l_k.) lt=l

(I

#=1

Zc~.--Za~

dZ, dO~l-I'[I(l,_k~ I ~r) .=2,,

(

FI'I] 1 1-k~ ~) .=t

]

XzcDz, Dz~G( Z~, Zz)DzbDz~ G( Zb, Zc)exp ( - ½a' sV~ - ½a' tV¢)] × {c.c} × [det(2z~Im

T)]-5[F(ku, Z,~., Zp.)] ~°-° ,

(2)

where Z = (z, 0) and D z = 0o+ 00z. We have already fixed super-projective invariance by choosing zc = ~ , zz = O, zb = 1 a n d 0c = 02 = 0. In eq. ( 2 ) appears the super Green function on the genus-g Riemann surface, given by

I[Z- Ty( Y) ] [ Y - Tr(Z) ] G(Z, Y)=loglZ-Yl+logI~I' [ Z - T y ( Z ) ] [ Y - T ~ ( Y ) ] Z

+½

~

Re

(3/

Y

.(2u ( 2 z t I m T ) ; u ' Re

12.

,

(4)

u,V=I

with Z2 f Zj

£2u = -

Y' log

(,u)

[ z~ - y~( z p . ) ] [ z , - T~( Z . ~ ) ] TT(Zfl..) ] "

[Z 2 - Ty(Zau) ] [Z 1 -

(5)

and

l(

Tu" = ~ni log kuOuv+

<~log [Zp,-T~(Zp.)][Z.,-T~(Z~..)]] [Zp~- Ty( Z,~.) I [Zc~ - Ty( Zp~) ],] '

(6)

where Ty are elements of the super-Schottky group. More details on the notation can be found in refs. [6,8 ]. The function Vs and V, are defined by combinations o f these Green functions:

Vs = ½[ G ( Z I , Zb) -'1-G(Z2, Zc) --G(Z,, Zc) -G(Z2, Zb) ], Vt = ½ [ G ( Z l , Z 2 ) + G ( Z b , Zc)-G(Z,, Zc)-G(Z:, Zb)] • 398

(7)

Volume 244, number 3,4

PHYSICS LETTERS B

26 July 1990

We have also compactified 10 - D space-time dimensions to tori of equal radii and F( k u, Z , , , Zp, ) takes this into account. Eq. (2) contains only the Neveu-Schwarz sector of the superstring. As it was shown in ref. [ 6 ], this is the only sector relevant to the Regge regime. G (Z, Y) can be decomposed into a sum of two terms: one purely bosonic and one containing the 2 (2g+ 2) grassmannian variables. Accordingly, the Vs and the Vt in eq. (2) split into a bosonic part, Vsat, and a fermionic one, V s,! F• In eq. (2) no approximation has been made. We just chose the part of the amplitude proportional to ~'~2 %" ec. Considering now the ~ ~ limit, a number of significant simplifications occur. First, as argued in ref. [ 6 ], to bring down as many powers of s as possible, all grassmannian integrations must be saturated from the factor exp[ ( - a ' s / 2 ) V[ ] 5< (c.c.) in eq. (2). This means we can take all grassmannian variables equal to zero everywhere else. Second, the leading term in powers ofs in the amplitude (2) comes from the configuration in which the two gravitons exchange the leading trajectory g + 1 times (see fig. 1 ). This corresponds to the corner in moduli space where k~, zp, and z~ are all close to 0. Third, as we consider the limit ku-~0, we can neglect o(k) corrections everywhere except in the V~, where we keep terms of o(ku), and in the V[, where we should keep terms involving 1, x/~, ~ .... , ~ . . . k g on an equal footing since they are equally important after summing over the spin structures. In particular, det(27c lm T) can be replaced by A - d e t ( 2 z t l m z), where ~ is the purely bosonic part of the period matrix. Also, the compactification factor simplifies to give [F(ku, Z~,, Z~, ) ] =A ~/2 [ 1 +o(k~) ]. Taking all this into account, we can re-write eq. (2) as ag.loop(S ,

t)=NgE1 "¢2eb'~c zo,

x

In principle, the integration over the 2g variables 0,~ and 0a, seems difficult. However, they enter only through the prime form part ( 3 ) and the abelian integral part (4) of the Green function. A close inspection of V v reveals that any attempt to saturate the integration over one of the pairs from the abelian integral part would invariably bring down powers of A - ~ suppressing the amplitude by a factor of one divided by the logarithm of a pinched modular parameter. Therefore, all 0~, and 0p~ must come from the grassmannian part of the prime form. This is an enormous simplification. In the prime form part, the super-fixed points enter only through the non-trivial elements of the Schottky group. This means that each pair 0,. 0B, will always carry with it (at least) one power of x/~u. We have to pull down g pairs of 0,,0p~, that is (at least) g powers of x/r~. Since we have g ku's, in order not to get a sub-leading contribution we can allow each 0,,0p, to carry one and only one x/~u. In general a block element of the Schottky group like S + ~ S + J ... S~ ~ will give rise to a grassmannian contribution to the prime form proportional to ~ . This has been explicitly checked at the two-loop level [ 6 ]. This means that we can afford to consider l-block elements only, the various Schottky generators decouple completely and each pair 0,~0p~ is saturated by considering the contribution from S u only to the prime form part of VsF, that is

zo z, z~-zb

~=~

)

(1-z~,)zp,(z~-zp,)

"

Therefore the integration over the grassmannian variables and the sum over the spin structures gives us /

\2g+2

2[OL'S~] ag_loop(S,t)=Ng2g-~l\

,[~',A--D/2 ~l'E2~b'EceXp(--oCsVs--Ot'o,t,~

g

2

r7 d ~ u k 2d2pud2trud2Zl12 ,=, I ~1 Iz, ' 11

(10)

399

Volume 244, number 3,4

PHYSICS LETTERS B

26 July 1990

where we have introduced the geometrical variables [ 6 ]

pu=;Wu=--log(ZP~Z'--za¢]

kZo,,, Zl - zp,, / '

ZI

tr. = i o)u = -- log ( I - u ~ - ~ \ 1 -- Z.e,,.]

(11)

Zb

(oJu denotes the bosonic part of ~2u in eq. (5), i.e. the ordinary holomorphic one-forms of a genus-g Riemann surface). Notice that our convention for a complex integration measure is d2z = d Re z ^ d Im z. The integral over the 2g variables Re pu and Re a u in (10) is dominated by a stationary phase at Repu = Re au = 0 - as it is described in detail in ref. [ 6 ] - and can be easily performed. Next, we can integrate out the phases of the g + 1 pinched moduli zi, ku to obtain / , \g+2

ag.loop(S, t)--- N g 2 g ( 2 7 t )2~g +4l i ) I ~ ' S

e, "'Z'b''c

2n

× f ~,~,l~ld klku u ]l f -~ d lz l l u= ,fid lm pu d lm truJ° ( a ' s-~-Iz, Xexp(--½oYtlog,zll--½oYtlog~- 1

I ) ~,=tfiJ°~2i-l-~l[a'sku~2

log z ~ ( 2 r t l m r ) u ) )

,

- (D-2)/2

(12)

where 2zr Im

zu~ =

- log ]k u I

for/t = v, ~uu

=-log

Izt I - l o g ~ u ~ ,

for/tCv,

(13)

and q~u=4sin

~__~.

Imau ~ ,

sm

q~u,=4sin

Impu-Imp~ 2

sin

Imau-Ima, 2

(14)

We integrate the modulus of the g + 1 pinched moduli between zero and some small positive number, the dependence on which drops out in the end. To perform these remaining integrations it is necessary to expand in (12) the Bessel functions. The form of the integral suggests that the integration is easier if we introduce the new variables xu=-log

xg+,

Ik~l +log [z, 1 - 2 1 o g Iq~l ,

=-log

IZll •

(15)

When written in terms of these variables, eq. (12) is just the the Schwinger parametrization of an amplitude depending on g + 1 momenta qu: /

,\ g(D-2)/2/

ag.looo(S,l)=Ng2g(27Q2g+l(~} \Z~/ X

dx. l u=l

t \g+2

(~7.s} \ c41 /

fi dlmpudlmtr u .u=l

~

~q.~.2{b.~.c ( - 1 ) '' +'-'+ ....

rtl ,...,ng+ 1 = 0

{0[ s~2nl+'"+2ng+l{

×~1) Xexp

400

~, g~Il exp(--2nUXj') ~ dD--2qu)~D--2( q- Y, q~ I,=l F 2 ( n u + 1) /t=l

-- ~o~'

g+l

~ ~=1

2 + quXu

~.,

1 ~<,tt< v~
ot qu'qvl°g

/

I qbu~ I ,

(16)

Volume 244, number 3,4

PHYSICSLETTERSB

26 July 1990

where ~u g+ ~--- q~u for # = 1, ..., g. q is the transferred momentum in the center-of-mass system, q 2 t. The x-integration can now be carried out. Next, the infinite sums are performed by means of the SommerfeldWatson formula [ 5 ]. At this point the dependence on the lower limit of integration I of the x-integrations explicitly drops out. The final result is =

a ,oo.(S, ,)= ' u g

\2n}

\ 4i ,]

~"

__

~, "~2~b'~,:

(

,+'

"~+'

V(aq~/4)

( a s ' ~ -<~''v~ ..... <"="+'/=

X f u=i fi dlmpudlmauSf-fu=, f dD--2qur~D--2\q - ~ lqla) f-I--t F ( i Z a'--~u/4) \ 4i } 0

×exp

(

~

a qu q~log I q ~ l

1 ~
)

(17)

Eel. (17) above amends in the two-loop case eel. (16) in ref. [6] that contains a mistake. The completely factorized form of the amplitude can be written by introducing a new set of (redundant) variables a~u,a [ 3 ] to be associated to the world-sheet parametrization of the two, up and down, strings:

Imau=trdu--adg+l,

Impu = a ~ , - a ~ + l ,

(18)

for/z = 1, ..., g and by recalling that [ 7 ] /

\2+ct't/2

2 tot's~ atree(s, t)=-2got-~l )

F( - or' t/4 P(l+a't/4)"

(19)

Accordingly, by choosing Ng as given by eq. (23) below, the amplitude can be written as

ag.ioo,(S, t) = ( ig)~ s ~, "~2eb'~< f~g-=l;+ ,(2d°-2 n)g(qg_+2') b ~ 2( q - g~'u=,'#)\ g +I'J' atree(S,/Jx) 2n

x <

o i ! "I*I' .=t

do .

(27Q 2 :exp

{iqu" [XU(a~) -.~'~(adu)]}: 0) ,

(20)

where ~ ' i ( a ) = i ~ _l [a~,exp ( i a n ) + ~ e x p

(-ian)]

n~O n

and tu= - q2. Eq. (20) agrees with ref. [ 3 ] and is an explicit proof that the leading part of the amplitude (2) factorizes in the product of g+ 1 tree amplitudes times the expectation value of a factorized operator. In this form it can be diagonalized by going into the impact-parameter space where the series is readily summed into the exponential form of ref. [ 3 ]

a(s' t) ='~ "~2'b "c4S ~ d°-2b exp (iq'b ) ( Ol(exp[2i~(s' b ) ] - l) 10 '

(21)

where d°-2q atree(S, t) ~ do'Uda d 4s _~ :exp {iq" [ b + £ U ( a ~) - - x d ( a d) ]}:.

3(s, b ) = f (2n)D_ 2

(22)

In order to obtain eel. (20) and thereby the exponentiation required to restore unitarity, the value of the numerical constant Ng that multiplies the amplitude at the g-loop order had to be chosen as follows: 401

Volume 244, number 3,4

( |

(g+ 1)! \2-~]

PHYSICS LETTERS B

\~l

(Ol')g(O--4)/2.

26 July 1990

(23)

O u r result agrees with the one previously found [ 7 ] for the case g = I. 3. U n i t a r i t y is a very stringent constraint on the string p e r t u r b a t i v e series. It fixes all numerical coefficients in the theory in terms o f only two free parameters, the string tensions 1 / a ' and the coupling constant g. A particularly challenging kinematical region from the p o i n t o f view o f unitarity is the Regge regime, where unitarity is restored only after s u m m i n g the entire series. It is r e m a r k a b l e that in this regime the requirement o f s u m m a b i l i t y can be used to d e t e r m i n e all numerical coefficients without having to actually enforce order-by-order unitarity and p e r f o r m lengthy calculations. F u r t h e r m o r e , this m e t h o d , since it relies only on the gravitational interaction to fix the topological expansion coefficients, is in principle applicable to any string theory.

Note added. We c o m m e n t on two aspects o f eq. ( 2 3 ) that might seem to contradict some general results o f Gross a n d Periwal [ 9 ] on the string p e r t u r b a t i o n series. ( 1 ) The factor 1 / ( g + 1 )! appearing in eq. (23) merely reflects the fact that in eq. ( 2 ) we do not integrate over just one f u n d a m e n t a l d o m a i n o f the m o d u l a r group. Rather, we integrate over the g! copies o f the fundamental d o m a i n o b t a i n e d by arbitrarily p e r m u t i n g the g sets o f l o o p - p a r a m e t e r s ( k u, z~, zp~ ). (This overcounting is what allows us to consider the integrals over the x u in eq. (16) to be i n d e p e n d e n t ) . (2) The factor ( - 1 )g+ ~is an artifact; the a m p l i t u d e ag (s, t) contains (s/i)2g+ z, that is, another factor ( - 1 )g+ l, leaving a p e r t u r b a t i o n series o f manifestly positive terms.

References [ 1] M. Soldate, Phys. Lett. B 186 (1987) 321, and references cited therein. [2 ] I.J. Muzinich and M. Soldate, Phys. Rev. D 37 ( 1988 ) 359. [ 3 ] D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 197 (1987) 129; Intern. J. Mod. Phys, A 3 ( 1988 ) 1615. [4] H.D. Abarbanel and C. Itzykson, Phys. Rev. Lett. 23 (1969) 53; M. Levy and J. Sucher, Phys. Rev. 186 (1969) 1656; G. Tiktopoulos and S.B. Treiman, Phys. Rev. D 3 (1970) 1037. [ 5 ] See, e.g., P.D.B. Collins, Regge theory and high energy physics (Cambridge U.P., Cambridge, 1977 ), and references cited therein. 16 ] G. Cristofano, M. Fabbrichesi and K. Roland, Phys. Lett. B 236 (1990) 159. [ 7 ] N. Sakai and Y. Tanii, Nucl. Phys. B 287 ( 1987 ) 457; J. Polchinski, Commun. Math. Phys. 104 (1986) 37. [8] See, e.g., P. Di Vecchia, M. Frau, K. Hornfeck, A. Lerda and S. Sciuto, Phys. Lett. B 211 (1988) 301; J.L. Petersen, J.R. Sidenius and A.K. Tollsten, Nucl. Phys. B 317 ( 1989 ) 109; G. Cristofano, R. Musto, F. Nicodemi and R. Pettorino, Phys. Lett. B 217 (1989) 59. [9] D.J. Gross and V. Periwal, Phys. Rev. Lett 60 (1988) 2105.

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