Effective Action and All-Order Gravitational

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CERN-TH.6791/93

S.I.S.S.A. 11/93/EP

EFFECTIVE ACTION AND ALL-ORDER

GRAVITATIONAL EIKONAL AT PLANCKIAN ENERGIES

D. Amati

International Scho ol for Advanced Studies { Trieste

and INFN { Sezione di Trieste, Italy

M. Ciafaloni

Dipartimento di Fisica, Universita di Firenze

and INFN { Sezione di Firenze, Florence, Italy

and

G. Veneziano

CERN { Geneva { Switzerland

Abstract

Building on previous work by us and by Lipatov, we present an e ective action ap-

proach to the resummation of all semiclassical (i.e. O (h )) contributions to the scattering

phase arising in high energy gravitational collisions. By using an infrared safe expression

for Lipatov's e ective action, we deriveaneikonal form of the scattering matrix and check

that the sup erstring amplitude result is repro duced at rst order in the expansion parame-

2 2

ter R =b , where R, b are the gravitational radius and the impact parameter, resp ectively.

If rescattering of pro duced gravitons is neglected, the longitudinal co ordinate dep endence

can b e explicitly factored out and exhibits the characteristics of a sho ckwave metric while

the transverse dynamics is describ ed by a reduced two-dimensional e ective action. Sin-

gular b ehaviours in the latter, signalling black hole formation, can b e lo oked for.

CERN-TH.6791/93

S.I.S.S.A. 11/93/EP

January 1993

1. Intro duction

(1;2)

In a series of pap ers wehave advo cated that, as a consistent theory of quantum

gravity, string theory is able not only to repro duce Einstein's classical General Relativity

at large distances, but also to provide systematic, calculable corrections to the leading

order approximation.

The Gedanken exp erimentwe conceived in order to substantiate this idea consists of

the collision of light (or massless) particles at energies (much) larger than the Planck scale,

i.e. in an environment in which gravity b ecomes a strongly interacting theory.

At suciently large distances (impact parameters) the dominant lo op diagrams can

be evaluated and resummed in closed form and simply yield an elastic eikonal amplitude

corresp onding to the leading order gravitational de ection predicted by General Relativity.

(3;4)

This result can b e (and has b een) also obtained by other metho ds .

At extremely high energies the most imp ortant corrections to the leading term have

still the form of semiclassical contributions (i.e. of terms O (h ) in the phase) and inter-

vene at distances that are still much larger than the intrinsic (quantum) scale of string

(2)

theory. So far, they have only b een computed to the rst non-leading order .

The nature of these corrections suggests that, after a regime of large de ection angles

and copious particle pro duction at intermediate impact parameters, a regime of gravita-

tional collapse will set in b elow a critical angular momentum (for given centre-of-mass

energy). This ts well with results obtained in the study of the gravitational collapse of

(5)

large, continuous systems (such as a rotating star). In particular, numerical studies

show that, b elow a critical value of J =GM of order 1, a nite fraction of the original mass

collapses to a black hole, while the remaining fraction escap es at in nity.

(6)

General relativity studies closer to ours have b een pursued by D'Eath and Payne

in their approach to black-hole collisions at small impact parameters. So far their results

are also in agreement with the ab ove picture.

In order to check the conjecture that gravitational collapse o ccurs also in the scat-

tering of two light particles, a calculation of semiclassical terms to all orders is necessary

(and hop efully sucient). This is already a formidable task. Indeed, b esides the general

complexity of higher lo op diagrams in string or eld theory, the o ccurrence of infrared

divergences complicates the analysis further. Although a Blo ck-Nordsiek reinterpretation

(2)

is p ossible , a delicate treatment is necessary b efore b eing able to extract nite answers

for infrared-safe quantities, such as the real part of the phase shift at non-leading order.

In this pap er we prop ose an alternative approach based on an e ective action tech- 1

nique. The explicit computation of Ref. (2) allows us to identify the relevant mo des that

must enter the e ective action. This means b oth the (momentum-dep endent) pro duced-

(7;8)

graviton p olarization with its emission vertex and the mo des resp onsible for its rescat-

tering in the asymptotic regime in question.

The e ective Lagrangian generated in this way is analogous to the one recently pro-

(9)

p osed by Lipatov .We shall argue here that, with a prop er infrared regularization, this

Lagrangian is indeed suitable for a direct evaluation of the semiclassical scattering matrix:

by solving the equations of motion that follow from such e ective action one generates the

generalized gravitational eikonal.

In this new language, the correct treatment of infrared (IR) singularities determines

the correct form of the e ective Lagrangian by xing some total derivative, IR-sensitive

(b oundary) terms.

After having discussed the eikonal expansion in section 2 and the e ective action in

section 3, we shall tackle, in section 4, the simpler case in which rescattering is neglected.

We will then b e able to actually solve for the longitudinal dep endence of all classical elds

thus reducing the problem to a transverse dynamical e ective action that is automatically

IR safe.

The general form of the solution explicitly displays two longitudinal sho ck-wave met-

rics, with non-trivial transverse comp onents in b etween the twowave fronts. The trans-

verse eld, as well as the pro le function of the sho ckwaves, is determined by a reduced

e ective action in the two transverse co ordinates. We are then able to check that the

p erturbative solution of the equations of motion gives rise to a rst non-leading eikonal

contribution to the scattering amplitude that coincides with the one computed in Ref. (2)

in a diagrammatic manner.

Of course, the nal ob jective of the present approach is not the p erturbative develop-

ment it resums, but the study of non-p erturbative solutions that may uncover new classical

phenomena generated by a consistent treatment of quantum gravity; a study that we are

now ready to tackle.

2. The eikonal expansion

(1)

Let us start recalling the gravitational scattering regime that weinvestigate , and

the related length scales. In string-gravity, the fundamental scale is the string length

h, in terms of which the Planck length (and thus the Newton constant G)is

s p 2

expressed in four uncompacti ed space-time dimensions as

Gh = = g ; (2:1)

p s

where g is the string lo op expansion parameter, assumed to b e small.

Avery imp ortant scale in high-energy gravitational scattering turns out to b e the

classical (Schwarzschild) radius asso ciated with the centre-of-mass energy of the collision:

R 4GE =2G s: (2:2)

At (sup er) Planckian energies such a radius is always (much) larger than and of

the Compton wavelength of the colliding particles:

GE > (>>)h; ) R>(>>)h=E ; : (2:3)

In the case of string-gravity,physics also dep ends crucially on the ratio:

R GE

(2:4) g

which, b ecause of the smallness of g , can b e smaller or greater than 1 even at high energy.

(1)

For R< , string e ects soften gravity at short distances and con rm the validity

(10)

of a generalized uncertainty relation for the distance x actually explored at a given

momentum transfer p:

x> + p> : (2:5)

(2)

By contrast, for R>, new semiclassical phenomena that can extend much

beyond the string length do app ear. It is to this interesting regime that we shall restrict

our attention in this pap er.

In the impact parameter representation, the amplitude is given, for b R,by the

leading eikonal approximation, corresp onding to the resumation of all p owers of GE due

(1;3;4)

to multigraviton exchanges. It has the form

S (b; E ) = exp 2i (b; E );

(b; E )= log b; s =4E ; (2:6)

where is an infrared cut-o related to the well-known in nite Coulomb phase. The

Einstein de ection angle is given by the stationarity condition on the phase of the scattering

wave:

2h @ 2R 8GE

= = = : (2:7)

E @b b b 3

We can interpret the leading eikonal expression (2.6) as a multiple scattering series, in

which the (small) de ection angle {corresp onding to a p ossibly large momentum transfer

t{ is built up by many small momentum transfer graviton-exchange pro cesses (cf. Fig. 1,

t h= ).

s s

For b approaching R, the subleading terms, suppressed bypowers of R=b, b ecome

sizeable. Wehave shown in Ref. (2) that {at least up to two lo ops{ the S -matrix admits

a generalized unitary eikonal representation. In the elastic channel

S = exp 2i( + + + ) ; (2:8)

el 0 1 2

(2)

where is given by Eq. (2.6) and, for pure gravity ,

G s 6

(b; E )= log s;

3 2

s 2i 2G

[1 + log s( log b + 1)]: (2:9) (b; E )=

Notice that is purely real. The imaginary part of is exp ected from unitarity, b ecause

1 2

of graviton bremsstrahlung predicted by the inelastic amplitudes at the corresp onding

order. Therefore, the app earance of the infrared cut-o in Im is due to soft graviton

emission and thus admits a Blo ch-Nordsieckinterpretation consistent with a nite value

for Re . This extra contribution to the phase mo di es the Einstein de ection angle:

R R

= (1 + + :::): (2:10) sin

2 b b

The one-lo op correction in Eq. (2.9) is a typical quantum e ect (of order h with resp ect

to ) and is irrelevant in our regime R > .Thus we shall concentrate on and

0 s p 2

on similar semiclassical corrections, i.e. those contributions to the eikonal that {as the

2 2

leading one{ are of order h at xed "charge" Gs, and contain arbitrary p owers of R =b .

The result for was obtained in Ref. (2) by computing the two-lo op diagram of Fig.

2, where the dashed lines represent string-gravitons or, in an equivalent language, Regge-

Grib ov gravitons. In terms of graviton elds (wavy lines) the hybrid vertex o ccurring in

Fig. 2 corresp onds to a sum of Feynman diagrams (Fig. 3) where gravitons are emitted

o internal and external legs.

(7;8)

The explicit expression for the emission amplitude has the form

2 2

A = J ; =8G ; (2:11)

2 2

q q

1 2 4

in terms of the conserved and traceless vertex function J , given by

2 2 2

J = (jq j jq j q q )+c:c: (2:12)

1 2

k i

Here wehaveintro duced the complex transverse comp onents

1 2 1 2

q = q + iq ; q = q iq (2:13)

for the momentum transfers q (i =1;2) of Fig. 3, and the complex p olarization vector

2 2

= (" (k )+i" (k )); " k = " k =0; " =" =1; (2:14)

L T

in which " (" ) are 4-vectors parallel (p erp endicular) to the incoming b eam direction.

L T

(3; 4) The emission amplitude A (1; 2) was sewn up to its absorption analogue A

in Ref. (2) in order to compute Im (see Fig. 2); Re was then computed by using

2 2

analyticity and unitarity arguments for the full S -matrix, and by actually witnessing the

ensuing cancellation of infrared divergences. The extension of this rigorous pro cedure to

higher lo ops in order to compute Re , etc. lo oks to o cumb ersome to b e tractable. We

shall then prop ose a shortcut pro cedure, bychecking it later at the level of Re .

The idea is simple: infrared divergences represent a trouble related, of course, to

the physical problem of soft bremsstrahlung. Their elimination should not disturb the

computation of the real part of the eikonal, whose structure is still provided by the diagram

in Fig. 2. Thus we shall de ne our e ective eikonal function by sewing up the vertices A

by an infrared-safe prescription. At the two-lo op level, this amounts to de ning, from Eq.

(2.9), a regularized amplitude

1 2i

= hA (1; 2)A (3; 4)i =Re (1 + log s);

2 2 reg 2

1 Gs R

Re = ; R =2G s; (2:15)

2 h b

with a subtraction pro cedure to b e describ ed in detail in Section 4.

With this regularization, one can de ne an e ective S -matrix that exp onentiates the

real part of the regularized H -diagram (Fig. 2) in addition to the single-graviton exchange

diagram (Fig. 1):

S (b; E ) = exp 2i( (b; E )+Re (b; E )+:::): (2:16)

ef f 0 2

The resulting expression admits again the multiple scattering interpretation given b efore,

where, however, two kinds of irreducible interactions now take place. 5

3. The e ective action

Wehave identi ed in Ref. (1) the irreducible diagrams that generate all relevant

semiclassical corrections to the eikonal. They are subleading by one p ower of s p er lo op pair

and come from the exchange of n upp er and n lower Regge{Grib ov gravitons, interacting

at tree level as describ ed in Fig. 4. Their order of magnitude is given by

n+1 n

Gs Gh Gs R

/ (3:1)

2 2

h b h b

so that they represent the irreducible semiclassical contributions mentioned b efore, which

2 2 n

are down by(R =b ) with resp ect to the leading eikonal.

The semiclassical S -matrix of Eq. (2.16) is exp ected to come from exp onentiating

the sum of such prop erly regularized irreducible diagrams, which de nes the generalized

eikonal function we are lo oking for.

In order to implement explicitly this exp onentiation, let us rst notice that the gravi-

ton couplings to the external particle lines in Fig. 4 can b e replaced by the purely eikonal

ones (/ p p ), which, by de nition, disregard recoil e ects. Such e ects, involving the mo-

mentum transfers of the exchanged gravitons, yield subleading corrections to the eikonal

vertices (cf. Fig. 3c) whichhavealready been included in the emission vertex J of Eq.

(2.12) and are thus taken into account in the tree amplitude of Fig. 4.

The ab ove remark implies that the external particle lines can b e represented as clas-

sical sources of the graviton eld h (x)

T h = T (x)h (x)+T (x)h (x) (3:2)

through their energy momentum tensor T whose non-zero comp onents are:

2 + 2

T = E (x ) (x); T = E (x ) (x b);

0 3 1 2

(x = x x ; x =(x ;x )): (3:3)

Calling L (h ) the e ective Lagrangian generating the irreducible tree amplitudes

ef f

of Fig. 4, the scattering matrix emb o dying all iterations of such irreducible diagrams

(leading and subleading) will b e given by

i i

S (b; E )=hexp A(h )i = exp[ A(h )]; (3:4)

ef f tr ee

h h 6

where

A(h )= d x(L (h )+T h ) (3:5)

ef f

and h represents the solution of the classical equations of motion that minimize A with

the sources T provided by Eq. (3.3). Thus A(h ) represents the generalized eikonal

function.

(9)

Lipatov has prop osed an explicit form for L justi ed in the high-energy regime

ef f

in which all sub energies among initial and emitted gravitons are large. Despite the present

lack of complete derivation of an e ective action from rst principles, we think that for

the computation of the semiclassical S -matrix in Eq. (3.4) the Lipatov form can safely

b e used. The reason for this lies in the very fact that the diagrams in Fig. 4 and their

iterations never allowinternal graviton lo ops, whose regularization would probably require

again string gravity.Acheck of our conjecture will b e the agreement of the e ective action

metho d with our previous result for Re , at rst subleading level.

Lipatov's e ective Lagrangian contains not only the emission vertex J illustrated

b efore (Eq. 2.12), but also a rescattering of the emitted graviton with external particles

through eikonal vertices (/ k k ). In the diagrammatic representation of Fig. 3, this

implies the twovertices of Fig. 5.

Note that J in Eq. (2.12) emits a graviton with a well-de ned, although momentum

dep endent, p olarization =Im( q^ q^ ). This p olarization {in the high-energy regime

in question{ will b e preserved in the subsequentinteractions of that graviton with the

external particles. This is why there is a single rescattering vertex in Fig. 5.

The ab ove discussion indicates that the relevant comp onents of the gravitational eld

that will enter the e ective Lagrangian are the longitudinal ones h ;h (represented

by dashed lines in Figs. 4 and 5, emitted eachby one of the external particles) and the

intermediate (complex) comp onent

h = h =2 (z; z ;x ;x );

1 2 0 3

(z = x + ix ;x = x x ) (3:6)

represented bywavy lines in Figs. 4 and 5.

2 1

The presence of the transverse propagator (k ) in J of Eq. (2.12) shows the non-

lo cality of the e ective Lagrangian in terms of the graviton eld h . This non-lo calityis

avoided by expressing h in terms of the complex scalar eld as in Eq. (3.6), at the

price, of course, of higher derivatives in the "kinetic" term. 7

Indeed Lipatov's expression of L in terms of h ;h and reads

ef f

L = L + L + L ; (3:7)

ef f 0 e r

where

++ 2

L = 2@h @ h + @ @ @ @ ;

0 +

@ @ @

(3:8) @ = ;@ = ;@ =

@z @z @x

is the kinetic term,

L = J (x) (x)+J (x)(x);

++ 2 ++

J(x)=4[@ h @ h @ @h @ @h ]; (3:9)

is the graviton emission{absorption term corresp onding to the momentum-dep endentver-

tex J of Eq. (2.12), and

2 2

2 2 ++

(3:10) @ @ @ + h @ @ @ @ L =2 h @

+ + r

is the rescattering term, where the eikonal vertices k app ear.

We note here that, following Lipatov, the kinetic term for h contains only the

longitudinal derivatives @ and not the transverse ones. We think that this simpli cation

is plausible for the calculation of the real part of the eikonal, b ecause the longitudinal

graviton momenta vary up to E and are thus on the average much larger than the transverse

ones, of order h=b. This would not b e true for the imaginary part, in which the mass-shell

condition forces longitudinal and transverse momenta to b e of the same order. By the

same token, the real part of the longitudinal propagators of will b e evaluated in the

following with a principal value prescription, as a remainder of the Feynman prescription

for the full propagator.

The identi cation of longitudinal (h ;h ) and intermediate (; ) comp onents as

the relevant degrees of freedom of the e ective Lagrangian recalls the attempt by E. and

(11)

H. Verlinde' at separating longitudinal and transverse degrees of freedom by a gauge

choice. The non-trivial di erence lies in the fact that the de nition of the intermediate

comp onent h is momentum-dep endent, thus not purely transverse to the incoming b eam

and that, as said b efore, in terms of h the e ectivevertices are non-lo cal. Furthermore,

only in the non-interacting limit ( ! 0) that gives the leading eikonal is the decoupling

of longitudinal and transverse comp onents p ossible, as advo cated in Ref. (11).

Let us stress again that the aforementioned non-lo cality has a very physical origin. It is

indeed due to the recoil e ects of the external particles that induce a transverse propagator 8

2 1

factor (k ) in the vertex J , as discussed b efore. Thus L incorp orates graviton

ef f

mo di cations to the external particle propagation, in addition to the purely gravitational

interactions of intermediate gravitons (cf. again Fig. 3).

4. Solutions without rescattering

From the e ective action in Eqs. (3.5) and (3.8){(3.10) it is p ossible to obtain clas-

sical solutions and then compute the semiclassical S -matrix from the expression in Eq.

(3.4). We shall address, however, a simpli ed problem, which is obtained by neglecting

the rescattering term L in the e ective Lagrangian. Let us note that this simpli cation

(2)

do es not a ect our rst goal, whichistocheck the result for Re obtained previously ,

b ecause the H -diagram in Fig. 2 do es not contain rescattering vertices. Furthermore, this

simpli cation allows us to identify a general class of solutions for which the infrared-safe

de nition of the action {whichisanyway needed according to the discussion in Section 2{

can b e p erformed in an explicit and simple way.

The general equations of motion derived from Eqs. (3.5) to (3.10) are

2 2

2 ++ 2 2 ++ ++ 2 ++ 2

(@ h ) j@j [( + )j@ j (h )] h )+@ j@ j h +2[@ ( @

2 2 2

+ ( @ @ ); (4:1) (x ) (x)=@

++ 2 2 ++ 2 4

h @ h j@j h j@ j h )= @ @ j@ j 2(@

2 2

++ 2 2 2 2

); (4:2) (h @ @ ) @ (h @ @ =@

with similar ones for h and .

By dropping the rescattering terms, the r.h.s. of Eqs. (4.1), (4.2) are set to zero, and

we realize that the longitudinal variable dep endence can b e factored out in the following

solution: *.

++ +

h = E (x )a(z; z ); h = E (x ) a (z; z )

* The inversion of the @ @ op erator in Eq. (4.2) for the eld is here p erformed

by a principal value prescription that is appropriate for our S -matrix evaluation, as dis-

cussed in Section 3. Using a retarded propagator would amount to replacing (x x )

by (x ) (x ). In any case, we use a de nition of the Heaviside function with (0) = .

With this proviso, the form of Eq. (4.4) is uniquely determined. Wehave also checked that

reintro ducing transverse derivatives in the propagator do es not mo dify the nal result. 9

2 +

= (E ) (x x )'(z; z ); (4:3)

2 2

a(bz; b z ); a (z; z )=a(z; z )=

' (z; z )='(bz; b z ) ;

a ) and a which represents a sho ck-wave longitudinal metric (with pro le functions a and

non-trivial transverse metric {related to ' by Eq (3.6){ in b etween the twowavefronts.

Here a; a; ' satisfy a reduced set of equations of motion in the transverse variables z =

1 2

x + ix and z given by

2 2

(@ @ ) ' =4(@ a@ a @ @a@ @a) j(x);

2 2

2 2 2

@ @a +(R) [@ (' @ a)+@ ('@ a) @ @ [(' + ' )@ @a)] = (x) (4:4)

a(x)=a(bx);' (x)='(bx);

where we use also the x notation, if more convenient.

In the reduced equation (4.4) the gravitational radius R =4GE of the system app ears

explicitly and plays the role of coupling constant of the problem. For R = 0 eq. (4.4)

reduces to the free- eld solution

1 zz

(0) (0) (0) (0) (0)

a = log ; =0; (4:5) a (x)=a (b x);' = '

2 L

where wehaveintro duced the large-distance scale L = playing the role of infrared cut-

(0) (0)

o , which xes the scale of a by the condition a (L)=0.Thus at the leading order the

pro le function of the longitudinal sho ckwave {and the absence of transverse comp onents{

(12)

corresp onds to an Aicheburg{Sexl sho ck-wave metric. Intro ducing (4.5) into Eq. (3.4)

(13)

we obtain the leading eikonal S -matrix already derived in a varietyofways.

Togobeyond this leading expression, let us go back to our solution (4.3) of the

equations of motion. By replacing it into Eqs. (3.8){(3.10) we obtain a reduced form of

the action that involves transverse variables only and that we write in the following form

A(a; a; ')= 2Gs a(0) + a(b)+ d x a@ @a ++a@ @ a+ (4:6)

R 1 1

2 2

+ '(@ @ ) ' ' (@ @ ) ' + ' j + j ' :

2 2 2

This form will now b e argued to b e infrared-safe.

In fact, the reduced action (4.6) di ers from a nave rewriting of the various terms in

Eqs. (3.8) to (3.10) by total divergences, i.e. by b oundary terms that leave the equations 10

of motion invariant but could intro duce infrared divergences due to the long-distance con-

tributions. The particular form in Eq. (4.6) is characterized by the fact that all derivatives

act on a single eld and turns out to b e infrared-safe b ecause the b ehaviour of @ @a and

(@ @ ) ', found from the equations of motion, is go o d enough to make all the integrals

converge, as we shall see b elow.

In order to compute the S -matrix, wehave to nd the classical solutions of the reduced

equations of motion (4.4) and then evaluate the classical action (4.6) so as to obtain, by

(3.4),

S = exp A(a ; a ;' ) : (4:7)

ef f cl cl cl

For instance the leading eikonal function is recovered by setting R = 0 in Eq. (4.6) and

using the free- eld solution (4.5)

1 Gs i b

(0)

S = exp 2i (a (0) + a(b)) = exp(2 Gs log ); (4:8)

h 2 h L

in agreement with the value of quoted in Eq. (2.6).

To pro ceed further in the expansion of Eqs. (4.4) to (4.6) we need a more precise

de nition of the relevant solutions of the reduced equations of motion which, b eing a set of

fourth-order di erential equations, p ossess a lot of arbitrariness. To b e de nite, consider

the rst non-trivial iteration

(0) 2 (1) (1)

a = a + R a + ; ' = ' + (4:9)

and replace Eq. (4.9) in Eq. (4.4) to get

(0) 2 (0) 2 (1)

a @ a (4:10) (@ @ ) ' =4@

( 2

(0) (1) 2 2 1)

a )+c:c: (@ @ )a = @ (' @

(0)

where wehave used the prop erty that a , given in Eq. (4.5), is harmonic for z 6=0

(0) 2

(@ @a = (x)).

(1) (1)

To start with, a and ' are de ned up to solutions of the homogeneous equations

(1) 2 (1)

@ @a =0; (@ @) ' =0 ; (4:11)

yielding

(1)

a = f (z )+c:c:; (4:12)

(1)

' = g (z )+g(bz )+z h(z)+(b z)h(b z ); 11

where f; g and h are arbitrary analytic functions. We shall determine them by the following

requirements:

(1) (1)

(a) a and ' should b e single-valued outside the singularity p oints at z = 0 and z = b.

(1) (1)

(b) For z ! 0;b;a and ' should b e as regular as p ossible, so as to avoid ultraviolet

divergences in the action.

infrared scale of the problem.

Sub ject to the ab ove conditions, we nd the solutions

1 1 z z 1 1 1 L

(1) 2 2

; (4:13) a = j1 j log j1 j + + log

2 2 2

4 jz j b b bz bz b b

(1) (1)

( a (z; z )=a (b z; b z ));

z z z z z 1 z z z

(1)

log + log 1 + f + f 1 ]; ' = log 1 log + 1

b b b b b b b b

(4:14)

1 x

f (z ) 1 + dx log x:

2 6 x 1

(1) (1)

Note rst that a and ' are single-valued for z 6=0;b thanks to a judicious use of

the additive arbitrary functions (4.12). For instance, the basic solution of Eq. (4.10) for

z z

(1)

' , given by log log(1 ) (a pro duct of analytic and anti-analytic functions) is not

b b

(1)

single-valued, but ' in Eq. (4.14) instead is.

A similar remark holds for z ! 0;b, where the rather singular ultraviolet b ehaviour

of the basic solutions is cancelled by the additive arbitrary functions of the form (4.12),

which are thus fully determined. For instance, for z ! 0wehave the b ehaviour

1 z z 1 z z jzj jzj

(1) (1)

a const + + ; ' + log jz j ; (4:15) +0 +0

2 3 2 2

b z z b b b b

(1)

in particular ' (0) = 0, and similarly for z ! b,

jz bj 1 z z jb zj

(1) (1)

a const +0 ; ' 1 log jb z j ; (4:16) +1 +0

3 2 2

b b b b

while we also have, by Eq. (4.4),

1 1 z z

(1)

@ @a = log(1 )+ +c:c: ; (4:17)

jz j b b

1 1

2 (1)

(@ @ ) ' = : (4:18)

z (b z ) 12

It is easy to check from Eqs. (4.15){(4.18) that the reduced action (4.6) is ultraviolet-

nite close to z =0, b, thanks also to azimuthal averaging. This result may seem at rst

sight surprising, since one exp ects a quadratic divergence bypower counting, due to the

app earance of R in the action (4.6). The cancellation of such terms is due to the choice of

solution, leading to the b ehaviour in Eqs. (4.15), (4.16) such that a(a ) is regular close to

2 2

z = b (z = 0) and ' is regular around b oth. It follows that the scale of R is set by b ,so

that the only p ossible divergence is a logarithmic one, which in turn cancels by azimuthal

(1) (1)

averaging. Finally, due to the prop erty ' (0) = ' (b) = 0, the action is not sensitive

to p ossible -function singularities of the fourth derivative (4.18) close to z =0 or z =b.

On the other hand, for large jz jb,wehave

1 jz j 1

(1) (1)

a ' a ' ; (4:19) log +0

2b L bjzj

2 (z z ) jz j jz j

(1) 2

' ' log + 0((log ) ): (4:20)

b b b

(1)

Thus, the L-dep endent constant in (4.13) is needed to make a (L) = 0 (up to terms

b=L,vanishing in the large-L limit). The rather bad infrared b ehaviour (4.19) ((4.20))

is forced by the short-distance regularity conditions, but is anyway comp ensated by the

corresp onding derivatives (4.17) ((4.18)) in the reduced action. In fact, the integrand of

Eq. (4.6) b ehaves at large distances as jz j log jz j and is thus convergent, as anticipated.

(1) (1)

Finally,byintro ducing a and ' in the reduced action, we can calculate the

2 2

correction of order R =b to the leading eikonal. We nd, by Eq. (4.6)

(1) (1) (1) 2 (0) (1) (0) (1)

( A =2Gs a (0) + a (b)) + d x(a @ @ a a @ @a ) (4:21)

R 1

2 2 (1)

(1)

+ d x(' (@ @ ) ' + c:c:) :

2b 2

(0) (1)

Noting that the a a interference term is a total derivativewe nd that the b oundary

terms cancel altogether, so that the rst line of Eq. (4.21) adds up to zero. The second

line is explicitly computed by complex integration of the expression resulting from Eqs.

(4.14) and (4.18). By using the integrals

Z Z

dxdy log z log(1 z ) dxdy log z

= =0; (4:22) ;

2 2

(1 z ) (1 z ) 2 z 6 2 z

dxdy f (1 z ) 1

= f (1) = 1 ; (4:23)

2 z (1 z ) 2 6 13

we obtain

2 2

1 Gs R R

(1)

; Re = ; (4:24) A = Gs

b 2 h b

in agreement with Eq. (2.10).

The ab ove derivation of Re is not trivial, but is anyway easier than a full two-lo op

calculation and lends itself to a straightforward generalization to higher orders. In fact

2 2

further iterations of Eq. (4.4) byapower series in R =b keep, order by order, the structure

of the homogeneous equations (4.11) and thus of the additive arbitrary functions (4.12).

The latter can thus b e determined at any given order by the single-valuedness and short-

distance conditions (a) and (b) still keeping the infrared b ehaviour in Eqs. (4.17){(4.20),

apart from an -dep endent co ecient. It follows that the reduced action in Eq. (4.6) is

infrared-safe to all orders.

As a further step, one may lo ok for exact solutions of Eqs. (4.4), sub ject to the

conditions (a)to(c)above, and having an infrared b ehaviour of the typ e in Eqs. (4.19)

and (4.20), which is p erturbatively stable. Although Eqs. (4.4) are not conformal invariant,

the solutions are exp ected to b e sums of pro ducts of analytic and anti-analytic functions as

(1) (1)

for a and ' in Eqs. (4.13), (4.14). Unfortunately, the b-dep endence implies a lackof

spherical symmetry even in the large-jz j limit, as one can see from Eq. (4.20). Thus nding

such exact solutions is exp ected to b e a non-trivial problem, whose detailed analysis is left

for future work.

5. Discussion

Wehave prop osed a generalized eikonal S -matrix for high-energy gravitational scatter-

ing in 3 + 1 dimensions, based on a regularized e ective action that resums all semiclassical

terms generated from the sup erstring approach.

By semi-classical we mean here all terms that are of orderh in the eikonal phase

2 2

but contain arbitrary p owers of R =b at some xed value of Gs (which plays the role of

gravitational charge).

(9)

The action itself is the one recently prop osed by Lipatov , apart from surface terms

(total derivatives in the e ective Lagrangian) that are essential for the (Blo ch{Nordsieck)

subtraction of infrared divergences.

Wehave actually checked, by solving p erturbatively the classical equations of motion,

2 2

that the resulting S -matrix agrees up to rst subleading order (i.e. to order R =b ) with

(2)

the results previously obtained in the diagrammatic approach. 14

As compared to the latter, the e ective action metho d allows a much deep er insight

into the structure of the (generalized) eikonal phase. For instance, in a simpli ed treat-

ment in which rescattering of emitted gravitons is neglected, the longitudinal co ordinate

dep endence can b e explicitly solved for in terms of a generalized sho ck-wave. Furthermore,

the relevant comp onents of the metric tensor satisfy non-linear equations determined bya

reduced action which only involves transverse co ordinates.

This new metho d thus provides directly the dynamical ly generated metric, and in

particular the pro le function in the transverse plane. In this sense, it also provides an

"o -shell" description of the scattering pro cess that may help interpreting any kind of

singularities p ossibly emerging in the S -matrix.

As a matter of fact the classical action de nes p erturbatively a pure phase, due to the

hermiticity prop erties of the Lagrangian, which can b e interpreted as the nite real part

of the generalized eikonal, of the form

Gs R

Re = :

h b

It is conceivable that when R=b approaches some critical value of order unity, the function

builds up some non-p erturbative singularity signalled by in nities and/or unitarity and

hermiticity loss.

In order to understand such phenomenon, each one of the ab ovementioned results

will b e imp ortant. Infrared regularization allows us to disentangle new kinds of inelastic

channels from the trivial ones related to bremsstrahlung. Knowledge of the asso ciated

metric yields a con guration space picture of the singularity to b e compared with the

single-parameter b ehaviour of the S -matrix. In this sense we are ready, as claimed, to

tackle the description of black-hole formation at Planckian energies.

We are aware, of course that even if such a description were made p ossible by the

present metho d, it would still b e of semiclassical nature. It would suce, however, to

reveal novel e ects that extend to distances much larger than the string length where

small-distance uctuations are smeared out. Small-distance details and b ehaviours will

indeed b e controlled by quantum e ects generated byinternal lo ops for which the string

size regularization plays an imp ortant role.

Thus the study of non-p erturbative solutions of the e ective action now app ears as an

attainable goal to investigate novel "macroscopic" features of a consistent quantum theory

of gravity. 15

REFERENCES

(1) D.Amati, M.Ciafaloni and G.Veneziano, Phys. Lett. 197B (1987) 81; Int. J. Mo d.

Phys. A3 (1988) 1615; Phys. Lett. B 216 (1989) 41.

(2) D.Amati, M.Ciafaloni and G.Veneziano, Nucl. Phys. B347 (1990) 550.

(3) G.'t Ho oft, Phys. Lett. 198B (1987) 61.

(4) I.Muzinich and M.Soldate, Phys. Rev. D37 (1988) 353.

(5) See, e.g. R.F.Stark and T.Piran, in Pro c. of the 14th Yamada Conference on Gravi-

tational Collapse and Relativity,Kyoto 1986 (H. Sato and T. Nakamura eds., WSPC,

Singap ore 1986) p.249.

(6) P.D.D'Eath and P.N.Payne, Phys. Rev. D46 (1992)658, 675, 694.

(7) L.N.Lipatov, Sov. Phys. JETP 55 (1982) 582; Nucl. Phys. B307 (1988) 705.

(8) M.Ademollo, A.Bellini and M.Ciafaloni, Nucl. Phys. B338 (1990) 114.

(9) L.N.Lipatov, Nucl. Phys. B365 (1991) 614.

(10) G.Veneziano, Europhys. Lett. 2 (1986) 133; talk presented at the

Italian Physical So ciety Meeting (Naples, 1987);

D.Gross, Pro c. Int. Conf. in High Energy Physics, Munich, 1988.

D.Amati, M.Ciafaloni and G.Veneziano, Phys. Lett. B216 (1989) 41;

G.Veneziano, Pro c. of Texas Sup erstring Workshop, 1989;

(11) E.Verlinde and H.Verlinde, Princeton preprint PUPT-1279 (1991).

(12) P.C.Aichelburg and R.U.Sexl, Gen. Rel. Grav. 2 (1971) 303.

(13) Refs. (1), (3), (4) and (9); see also D.Amati, M.Ciafaloni and G.Veneziano, Phys.

Lett. B289 (1992) 87. 16

FIGURE CAPTIONS

Fig. 1: S -matrix diagram for the multiple scattering series of Eq. (2.6). Crosses denote

on-shell lines.

Fig. 2: Regge{Grib ov H -diagram, emb o dying the rst subleading two-lo op corrections;

dashed lines denote string gravitons and the vertex is de ned in Fig. 3.

Fig. 3: Feynman diagram de nition of the Regge{Grib ovvertex of Fig. 2, including all ex-

ternal line insertions.

Fig. 4: Structure of 2n-lo op diagrams contributing irreducible semiclassical terms to the ef-

fective eikonal.

Fig. 5: (a) Emission and (b,c) rescattering vertices of intermediate gravitons (wavy lines) by

Regge{Grib ov gravitons (dashed lines) 17 ts

b . . .

Fig. 1

p 1 q q 1 k 4

q2 q3 p2

Fig. 2 q 1 k, η =++.... q2

a) b) c)

Fig. 3

12. . . n TREE

Fig. 4

− −

q 1 k, η

q2 ++

a) b) c)

Fig. 5