The Scattering of Strings in a Black-Hole Background

Home , Black hole

HEP-TH-9405052 t o-dimensional w CERN-TH.7240/94 wn to b e consisten a 23, Switzerland ons in the t y h kground Bac in a Abstract Ulf H. Danielsson k-Hole Blac kground are studied. The results are sho The Scattering of Strings Theory Division, CERN, CH-1211 Genev k-hole bac In this pap er, correlation functions of tac y 1994 blac with the prediction of the deformed matrix mo del up to external leg factors. CERN-TH.7240/94 Ma

1 INTRODUCTION

In this letter, I will give some results on tachyon scattering amplitudes in

the background of a two-dimensional black hole [1, 2 ]. For this purp ose, I

will use the techniques develop ed in [3] based on Ward identities. The Ward

identities will provide p owerful constraints on the correlation functions that

are otherwise so hard to calculate. The black hole will b e constructed as a

p erturbation around the ordinary linear dilaton vacuum. Indeed, in [4] it was

shown that the action of the SL(2;R)=U (1) coset mo del [2], which describ es

atwo-dimensional black hole, can b e written as

2R + M B ); (1) (@X @X + @@

where

2 2

B(3i@ X + @)(3i@X + @)e (2)

is the `black-hole screener'. For further discussion, see [5]. In the brave

spirit of [6], I will consider tachyon correlation functions in the presence

of an integer numb er of such black-hole screeners. The hop e is that this

will teach us something ab out the true tachyon correlation functions in the

black-hole background. However, this nave p erturbative approach has severe

limitations. This will b e apparent in the form of the leg factors. I will return

to this imp ortant issue in the last section.

The main purp ose of this letter is to show that the correlation functions of

the deformed matrix mo del [7{13] are, in a highly non-trivial way, consistent

with the black-hole Ward identities. More details will b e given in a future

publication.

2 WARD IDENTITIES IN TWO-DIMENSIONAL

STRING THEORY

2.1 Without Black-Hole Screeners

In [3]Ward identities were derived, which related tachyon scattering ampli-

tudes with di erentnumb ers of tachyons. These Ward identities were shown 1

to pro duce recursion relations that determined all the amplitudes. Com-

pared to more standard calculations, see e.g. [14 ], this was a remarkable

simpli cation. Below I will brie y summarize the results of [3].

The Ward identities are derived by using the charges

I I

dz dz

Q = W c X ; (3)

m;m J;m m+1;m m;m

2i 2i

where

W = (z )O (z); (4)

J;m J+1;m J;m

rst constructed by Witten in [15]. The elds corresp ond to the gravita-

tionally dressed primary elds

2(1+J )(z )

= (z )e : (5)

J;m J;m

The O 's are elements of the ground ring. The X - elds have ghost number

1, but their precise form need not concern us.

In [3] the action of the currents on tachyons was calculated. In terms of

the rescaled tachyons, T , where

T = (1 2 p)T (6)

p p

((x)= (x)=(1 x)), it was found that

~ ~

cc( W (z )ccT (0) = 2 p +2m)T (0) + ::: (7)

m;m

pm 2

refers to chirality. A p ositive-momentum rightly dressed tachyon is de ned

to have p ositivechirality. I will follow [3] and write this, in an obvious

notation, as

~ ~

Q jT i =( 2p+2m)jT i: (8)

m;m

pm 2

As is clear from ab ove, the rst term in (3) has ghost numb ers (0; 0), while

the second term has (1; 1). Since the tachyon (if xed) has ghost numb ers

(1; 1), and has no (2; 0) part, the second term cannot contribute to the right-

hand side (apart from a BRST-exact state). It is only for rightly dressed

non-tachyonic states that one gets a non-trivial contribution. These are the

new mo duli discussed in [16 ]. 2

The same current acting on tachyons of the opp osite chirality givesavery

di erent result. In general

2m+1

+ + +

~ ~ ~

Q jT ; :::; T i =(2m+ 1)! 2 p 2m jT i: (9)

2m+1

m;m i

p p

1 2m+1

p 2m

i=1

It is hence capable of changing the numb er of tachyons. This is the reason

why it is p ossible to construct p owerful recursion relations. For instance, an

insertion of the W current leads to the relation

1=2;1=2

(N 2)A(N )=(N 2)(N 3)A(N 1) (10)

among the tachyon amplitudes A(N ), whichhave 1 negative and N 1

p ositivechirality tachyons. This, then, implies the well-known result

A(N )=(N 3)!; (11)

given that A(3) = 1. Note that as compared to [3], I have rescaled also the

negativechirality tachyon. This would really give zero since it sits on discrete

momenta (the bulk amplitudes of these renormalized tachyons are zero), but

the zero is comp ensated by including the volume factor leading to a nite

result [17]. This is the convention of collective eld theory.

2.2 With Black-Hole Screeners

Let me now generalize the ab ove construction to correlation functions involv-

ing black-hole screeners. I will denote such correlation functions by A(N; m),

where N is the total (including the negativechirality one) numb er of tachyons

and m is the numb er of black-hole screeners. Momentum conservation im-

plies

N +2m2

p = : (12)

I will use the charges Q , with n 2m 1, to derive the Ward

n=2;n=2

identities. From momentum conservation it follows that Q transform

n=2;n=2

n +12k p ositivechirality tachyons and k screeners into one tachyon. I will

assume, as an ansatz, that the precise form is

+ + k

~ ~

Q j T ::: T B i

n=2;n=2

p p

n+12k 3

n+12k

+ k

j T =2 k!(n +12k)! 2a p + b i: (13)

n;k i n;k n+12k

p n= 2

i=1

The a and b are co ecients dep ending on n and k only. The prefactor is

n;k n;k

for latter convenience. This ansatz is based on the assumption of factorization

into leg factors. If this assumption is true, it is clear that factors of must

app ear, in the same way as b efore when the black-hole screeners are involved.

Furthermore, the residual dep endence on momenta is clearly symmetric in

the di erent momenta, but must also b e at most linear in order for the

resulting recursion relations, for all N , not to dep end on individual momenta

when all contributions are summed up. This is a prerequisite for factorization

into leg factors. For each black-hole screener there will app ear a regulated

zero 1= log M (see [14] for a discussion on zero es and p oles of correlation

functions with dicrete states), which, symb olically, I will write as (1). The

1= log M will b e absorb ed into B , or rather M . This is no di erent from

standard c = 1 where = log ! .

Equations (8) and (13) are the only p ossible non-trivial contributions.

This can b e seen by using momentum conservation and examining the sin-

gularities of the contractions. All other p ossibilities give at b est discrete

states at non-discrete momenta. For n<2m1, however, discrete states at

discrete momenta would app ear, giving more complicated Ward identities.

It is straightforward to write down the generalization of the c =1 Ward

identities using the ab ove results. It follows that

(N +2m2)A(N; m)=(N2):::(N n1)(N +2(m1)(n+1)+n)A(N n; m)

+ m(m 1):::(m k + 1)(N 2):::(N n 1+2k)

k =1

(a (n +1 2k)(N +2m2) + b (N 1))A(N n +2k; m k); (14)

n;k n;k

when all p ossible non-vanishing cancelled propagators are taken into account.

The a and b could in principle b e calculated directly using the metho ds

n;k n;k

of [3]. I will, however, not attempt this in this pap er since a careful study of

regularization is rst needed.

Let me rep eat the logic of the present discussion. If the correlation func-

tions factorize as in standard c =1,then they must satisfy the Ward identities

(14) for some values of a and b . I stress that even without explicit val-

n;k n;k

ues for the co ecients a and b these are very p owerful Ward identities.

n;k n;k 4

Below I will show that the deformed matrix mo del results are compatible

with (14), while some other prop osals are not. But rst I need to explain the

prediction of the deformed matrix mo del. Rememb er that this mo del has

b een argued [7{13] to b e the matrix mo del realization of the black hole.

3 SOLUTIONS OF THE WARD IDENTI-

TIES

3.1 The Deformed Matrix Mo del Prediction

The deformed matrix mo del is obtained by adding a piece M=2x to the

matrix mo del p otential. The p otential b ecomes

1 M

x + ; (15)

0 2

2 2x

M will b e p ositive. The p osition of the Fermi level is, as usual, measured in

terms of its deviation from zero, i.e. .However, it is now p ossible to de ne

1=2

a double scaling limit, even when = 0. One then needs to keeph= M

xed, which will b e the string coupling constant.

Sp ecial cases of tachyon correlation functions have b een calculated in

several pap ers, [7{9,11{13]. In [12] the general formula (in the case with

N 1 tachyons of the same chirality) was given up to genus one. The genus-

zero piece, at =2,is

p p p

+ + + p= 2N=2+1

~ ~ ~

hT :::T T i =(N3)!!( 2 p 2)( 2 p 4):::( 2p (N 4))M ;

p p p

N 1

(16)

when normalized to collective eld theory. This result is valid only for even

N +2m2

N : for o dd N the matrix mo del gives zero. Put p = and take m

derivatives of (16) to get

A(N; m)= 2 (N 3)!!(N +2m4)!!: (17)

This, then, is the deformed matrix mo del suggestion for the correlation func-

tions with black hole screeners. 5

3.2 A Check of the Ward Identities

Let us nowcheck whether the matrix mo del prediction can b e made to satisfy

the Ward identity (14). As I will discuss in the last section, the vanishing

of the matrix mo del o dd-p oint functions is a consequence of a di erence in

the leg factors coming from a di erentchoice of vacuum. It is natural to

assume that when the p erturbativevacuum is used, the o dd-p oint functions

are given by a direct `analytic continuation' of the even-p oint result. This I

will use b elow. In the next subsection I will provide some evidence for this

by showing that it is true for the three-p oint function.

It is straightforward to substitute (17) into the Ward identities and de-

termine the necessary a and b one by one in k ; m = 1 provides k =1,

n;k n;k

m= 2 provides k = 2 (given the k = 1), etc. The answer is

a = (n 2)(n 4):::(n 2k +2) (18)

n;k

k !

and

b = na : (19)

n;k n;k

I stress that it is a highly non-trivial property of the deformed matrix model

correlation functions that coecients can be found such that the Ward identi-

ties areobeyed. For instance, it can b e checked that other prop osals, suchas

A(N; m)= (N +2m 3)! (whichwould corresp ond to c = 1 correlation

(2m)!

functions with ! M ) fail the test at m =2.

3.3 Some Explicit Examples

Even if a direct calculation of the scattering amplitudes is in general very

dicult (which is one reason to consider Ward identities), there are some

cases where the calculation can b e done and hence used as a check of the

ab ove results. This is so when m = 1. A convenient representation of the

black-hole screener is

iaX (z )+b(z )+iaX (y )+c(y )

lim @ : e : (20)

y !z

where b + c = 2 2, and the same for the anti-holomorphic piece. With

a =6 2 and b = 0 eq. (2) is repro duced. I will, for the moment, keep

the p olarization tensor, i.e. a and c, arbitrary. This can b e taken as a 6

useful check on the calculation. As explained in [19 ], one of the graviton

p olarizations is pure gauge, i.e. it can b e written as L of something. In

general, by using the gauge freedom, any sp ecial state can b e written in a

form where it contains no @ . Hence, the nal answer should not dep end

on the p olarization. However, the resulting integrals simplify only for certain

N 2c

p eculiar values of the momenta. Let me pick p = , , p = ::: = p =

1 N 2

N 2c

and consequently p = (N 2) . The integral to b e p erformed,

N 1

with the black-hole screener at 1,is

N2 N 2

Y Y

1 N

2 2 2 2

d z j z j j 1 z j j z z j ; (21) (a+c) 2c

i i i i j

i=1 i>j

where

2c 2c

= 1 N + 2; (22)

c a c a

(23) = N 2 (N 3)

c a

and

= 2: (24)

c a

The integral can b e evaluated by using the results of [20] and the result is,

with the volume factor,

p p

N 2

2(N 2)!(1 2 p) (1 2p )(1 N )(1); (25)

N 1

with no dep endence left on the p olarization tensor as promised. We recognize

the familiar leg factors and see that the remaining non-factorizable piece

agrees with (17) at m = 1 up to the normalization of B .

Let me now consider the three-p oint in a little more detail. According to

(17) it is given by

(2m 1)!!

m m

2 M : (26)

It is now necessary to comment on [21 ], where the three p oint is calculated

for general m using continuum metho ds. It was found that the three-p oint

is the same as for c = 1 up to a factor

x+4

) (m +

; (27)

x+4

( (m +1) )

8 7

where x is an unknown parameter intro duced by the regularization. In [21] x

was xed so that the factor (27) was one. However, one can argue in favour of

another prescription. Let me demand that m and the Liouville momentum of

the negativechirality tachyon are untouched by the regularization. After all,

it is only for integer m that we can p erform the calculation. As is clear from

[21], this means that x = 0. The three-p oint then b ecomes precisely (26).

This is the only way to remain consistent with (25), where this regularization

prescription has b een used implicitly. One can also check that the deformed

matrix mo del expression for the two-p oint, M , is in agreement with the

continuum calculation.

4 CONCLUSIONS

In this pap er wehave seen how the black-hole tachyon correlation functions

compare with those of the deformed matrix mo del. Wehave seen that the

latter ob ey highly non-trivial Ward identities derived for the black-hole back-

ground. I have also provided some examples of explicit calculations, where

the agreement can b e veri ed. There are therefore reasons to b elieve that

the deformed matrix mo del really is a black hole. It should b e p ossible to

complete a rigorous pro of along the lines of this pap er.

A di erence b etween the results is the form of the external leg factors or

rather the p osition of the p oles in these factors. After all, it is only the lat-

ter that are universal and can b e con dently predicted by the matrix mo del

without further physical input. In the continuum calculation we are implic-

itly using the same vacuum as in the linear dilaton theory and we are b ound

to obtain the same external leg factors in our p erturbative treatment. It

is easy to see, however, that this vacuum is an unfortunate choice. In the

black hole we often demand p erio dicity in Euclidean time. This is just a

choice, but a sensible one. It means that wehavechosen a vacuum corre-

sp onding to an equilibrium eternal black hole at the Hawking temp erature.

The particular value of the Euclidean compacti cation radius R, i.e. the

inverse temp erature, is obtained if one insists on no conical singularityat

the origin of Euclidean space, i.e. the horizon. In the two-dimensional black

hole, the compacti cation radius di ers b etween the semiclassical case [1,2]

p p

and the `exact' metric of [18 ], b eing 1= 2 and 3= 2, resp ectively. Recall

that I use conventions such that =2.For comparison, the self dual ra- 8

dius is R = 2 and the Kosterlitz{Thouless phase transition takes place at

R =2 2 . (Since the ab ovevalues are b elow R , this might b e a problem.)

T T

In standard c = 1 the p oles o ccur at p = for all integers n. If Euclidean

. Clearly time is compacti ed the allowed momenta must b e of the form

all states are allowed at R = R . The situation is di erent when we pick

one of the black-hole radia suggested ab ove. In b oth cases the o dd p oles

are not allowed! This is the reason whywe get double spacing of correlation

function p oles. The matrix mo del is clever enough to sp ot this problem. In

[8] it was shown that matrix eigenvalue wave functions that might give rise

to p oles other than the ones ab ovewere in general non-normalizable. The

p erturbative continuum calculation, however, is to o crude to take this into

account.

To conclude, there is strong evidence that the deformed matrix mo del is

describing a two-dimensional black hole. If this is indeed true, wehavea

powerful to ol at our disp osal to explore stringy quantum black-hole phenom-

ena.

Acknowledgements

Iwould like to thank K. Becker, M. Becker and, esp ecially, R. Brustein for

discussions.

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