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UTTG-01-96

Collective Co ordinates for D-branes



Willy Fischler, Sonia Paban and Moshe Rozali

Theory Group, Department of Physics

University of Texas, Austin, Texas 78712

Abstract

We develop a formalism for the scattering o D-branes that includes collective co-

ordinates. This allows a systematic expansion in the string coupling constant for such

pro cesses, including a worldsheet calculation for the D-brane's mass.

1 Intro duction

In the recent developments of string theory, solitons play a central role. For example,

in the case of duality among various string theories, solitons of a weakly coupled string

theory b ecome elementary excitations of the dual theory [1]. Another example, among

many others, is the role of solitons in resolving singularities in compacti ed geometries,

as they b ecome light[2].

In order to gain more insight in the various asp ects of these recent developments, it

is imp ortant to understand the dynamics of solitons in the context of string theory.In

this quest one of the to ols at our disp osal is the use of scattering involving these solitons.

This includes the scattering of elementary string states o these solitons as well as the

scattering among solitons. An imp ortant class of solitons that have emerged are D-branes

[4]. In an in uential pap er [3]itwas shown that these D-branes carry R R charges,

and are therefore a central ingredient in the aforementioned dualities. These ob jects are

describ ed by simple conformal eld theories, which makes them particularly suited for

explicit calculations.

In this pap er we use some of the preliminary ideas ab out collective co ordinates de-

velop ed in a previous pap er [5], to describ e the scattering of elementary string states

o D-branes. We clarify and expand these ideas, providing a complete analysis of this

pro cess.



Research supp orted in part by the Rob ert A. WelchFoundation and NSF Grant PHY 9511632

The pap er b egins by reviewing the presence of non-lo cal violations of conformal in-

variance that are a consequence of the existence of translational zero mo des. It is then

shown how the world sheet theory removes these anomalies through the intro duction of

collective co ordinates (which app ear as \wormhole parameters"), and the inclusion of the

contribution of disconnected worldsheets to the S-matrix elements. As a bypro duct of

this analysis we calculate the mass of the soliton by a purely worldsheet approach (that is

amenable to higher order corrections). We thereby recover a result obtained in [4] using

e ectivelow energy considerations.

These ideas are then applied to the example of a closed string state scattering o a

0-brane. We showhow to include the recoil of the brane, and discover the interpretation

of the wormhole parameters as collective co ordinates.

The conclusion contains some comments on higher order corrections to scattering and

to mass renormalization as well as p ossible extensions of this work in di erent directions.

2 Conformal anomalies.

D-branes placed at di erent lo cations in space have the same energy. This implies

the existence of spacetime zero mo des, one for each spatial direction transverse to the

D-brane (i.e. one for each Dirichlet direction). Because of these zero mo des, the attempts

to calculate scattering matrix elements in a weak coupling expansion encounter spacetime

infra-red divergences. In p oint-like eld theory these divergences arise from zero mo des

propagating in Feynman diagrams (for example diagrams as in Fig. 1 , where  is a zero

0

mo de).

Fig. 1: Diagram containing the propagation of a zero-mo de. 2

Such b ehavior can also b e found in the canonical formalism, where it manifests itself by

the presence of vanishing energy denominators, characteristic of degenerate p erturbation

theory. This is resolved in quantum eld theory by the intro duction of collective co ordi-

nates. The right basis of states, in which the degeneracies are lifted, are the eigenstates

of momenta conjugate to these co ordinates.

In string theory,however, we do not havea workable second quantized formalism.

Instead, wehave to rely on a rst quantized description. The weak coupling expansion

consists of histories spanning worldsheets of successively higher top ologies. Yet, the afore-

mentioned divergences are still present, and are found in a sp eci c corner of mo duli space.

The divergences app ear as violations of conformal invariance, as describ ed b elow. To the

lowest order, these singularities are found when one consideres worldsheets as depicted in

Fig. 2.

P

Fig. 2: The surface .

P

This surface, , is obtained by attaching a very long and thin strip to the b oundary of

the disc. This is equivalent to removing two small segments of the b oundary of the disc

and identifying them, thus creating a surface with the top ology of an annulus. In the

degeneration limit, when the strip b ecomes in nitely long and thin, the only contribution

comes from massless op en string states propagating along the strip. In the case of Dirichlet

b oundary conditions, the only such states are the translational zero mo des.

In the case of closed string solitons, the divergence app ears for degenerate surfaces

containing long thin handles, along which the zero mo de, a closed string state in that 3

case, propagates. In b oth cases these limiting world-histories are reminiscent of the eld

theory Feynman diagrams, as in Fig. 1 .

We are now ready to prop erly extract the divergences. As discussed in [7] for the

closed b osonic string theory, the matrix element in the degeneration limit can b e obtained

by a conformal eld theory calculation on the lower genus surface, with op erators inserted

on the degenerating segments (see Fig. 3 ).

P

Fig. 3: The degenerations limit of the surface .

This can b e expressed formally as follows:

Z

X

dq

2h

a

q h (s )  (s ) V :::V i (2.1) hV :::V i =

a 1 a 2 1 n Disc 1 n 

q

a

where f g is a complete set of eigenstates of L (which generates translations along the

a 0

strip), with weights h . The limit q ! 0, therefore, pro jects the sum into the lowest

a

weight states.

As an illustration, it is useful to rst consider the same limit for the usual op en

b osonic string (with Neumann b oundary conditions). The lowest weight states (excluding

the tachyon) are:

!

i

p dX

i ik X

 = N g e (2.2)

k

k

ds

2

k

with h = and N a normalization factor. Using the ab ove formula (2.1), one gets:

k k

2

Z Z

d D E

X

d k 2

(1+k ) i i

hV :::V i = dq q  (s ) (s ) V :::V : (2.3)

1 n  1 2 1 n

k k

d

Disc

(2 )

i

This expression exhibits the exchange of a photon, where log q is the time parametrizing

the photon worldline. The requirement of factorization xes the normalization of the

op erators  .

a 4

Wenow turn to the analysis for the D-branes, which is rather similar. The Dirichlet

N 0 p

p-brane is de ned by imp osing Neumann b oundary conditions on X = X :::X , and

D

Dirichlet b oundary conditions on the remaining co ordinates, X . The lightest states

(excluding the tachyon) are the translational zero mo des of the D-brane:

D i! X

0

 = N@ X e i = p +1;:::;d 1 : (2.4)

i n

i

where X is target time, @ is the derivative normal to the b oundary, and N is a normal-

0 n

ization constant. The existence of these zero mo des re ects the degeneracy of di erent

~

D-branes con gurations, obtained by rigid translations (k = 0) in the Dirichlet direc-

N

tions.

In order to allow a general p-brane (p> 0) to recoil one needs to compactify the spatial

Neumann directions and wrap the p-brane around them. In that case the mass of the p-

brane is nite and prop ortional to the spatial Neumann volume. For the sake of clarity

we fo cus on the scattering o a 0-brane, where no such compacti cation is necessary.

The normalization constant N that app ears in (2.4) is determined by factorization;

consider the S-matrix elementinvolving two closed string tachyons whose vertex op erators

are:

Z

D D

~ ~

2 i! X (z )ik
X (z )

1 0 1 1

1

V = N d z e

1 t 1

Z

D D

~

~

2 i! X (z )ik
X (z )

2 0 2 2

2

V = N d z e : (2.5)

2 t 2

The factorization on op en string p oles is obtained by letting one of the tachyons

approach the b oundary (see Fig. 4).

Fig. 4: Factorization on the Disc.

In the factorization one recognizes the various op en string p oles, in particular the

1

2 2

tachyon at ! = , and the zero mo de at ! = 0. This yields the normalization factor

2 5

1

for the zero mo de:

p

g

: (2.6) N =

4

Since these states don't carry Dirichlet momenta, they are discrete quantum mechan-

ical states.

We are now ready to extract the violations of conformal invariance for S-matrix ele-

ments calculated on the annulus. As mentioned ab ove, these divergences app ear in the

limit of mo duli space describ ed in Fig. 2 . In this limit the divergent contribution to an

S-matrix elementinvolving n elementary string states is:

   

Z Z Z Z

g

~ ~

= (2.7) V ::: V X (s ) V ::: V X (s )
@ log  @

1 n D 2 1 n D 1 n n

16T

 Disc

where T is a large target time cut-o , and  isaworld-sheet cut-o . The app earance of T is

necessary in order to prop erly extract the divergent contributions due to the translational

zero mo des.

3 Wormhole Parameters

In order to restore conformal invariance we app eal to the Fischler-Susskind mechanism

[6]. The idea is to mo dify the two-dimensional action such that the sum of the various

world-sheet contributions to the S-matrix element is conformally invariant.

The Fischler-Susskind mechanism requires in this case the addition of a non-lo cal

op erator O to the world-sheet action:

Z Z

g

2

~ ~

O = log  ds @ X (s ) ds @ X (s ) : (3.1)

1 n D 1 2 n D 2

8T

This op erator is a bilo cal involving the translational zero mo de vertex op erators. Notice

the additional logarithmic divergence b eyond the one that is already present in (2.7). This

additional divergence accounts for the volume of the residual Mobius subgroup that xes

the lo cations of twovertex op erators:

1

In our notation the op en string tachyon is

1 p

i! X

0

ge

2 6

 

Z Z Z Z

1

ds @ X (s ) ds X (s ) V ::: V =

1 n 1 2 2

V

Mobius

Disc

 

Z Z

1

@ X (s ) @ X (s ) V ::: V (3.2)

n 1 n 2

V

Residual

Disc

where V = 2 log  is the residual Mobius volume as de ned ab ove [11]. In the

Residual

case of the Fischler-Susskind mechanism involving a lo cal op erator this would corresp ond

instead to a nite factor.

In order to rewrite the sum of histories as an integral weighted by a lo cal action we

intro duce wormhole parameters , one for each Dirichlet direction. As will b ecome clear

i

these parameters are prop ortional to the momenta conjugate to the center of mass p osition

of the 0-brane.

The S-matrix element then takes the form:

Z Z Z Z

2

~ =2 I

S = d~ e DX e V ::: V (3.3)

1 n

p

Z Z

~ g

1

2 0 0

~ ~ ~ ~

p

I = d z (@ X
@ X + @X
@X )+ log  ds @ X

n

4

2 T

Z

1

~

+ ds ~a(X )@ X (3.4)

0 n

8

As will b e shown b elow, the inclusion of the last term in the action is imp ortant in restoring

conformal invariance. At the lowest order in the string coupling constant g :

~a(X )=~a

0 0

where ~a is a constant (indep endentofX ), and represents the xed lo cation of the brane.

0 0

4 Scattering o a 0-brane

In organizing various contributions that violate conformal invariance we b egin by con-

sidering the \vacuum p ersistence amplitude":

(

p

Z Z Z

~ g

1

~;~a

 ~

p

@X @X + log  @ X + h0 j 0i = DX exp

n

T=2

T=2

4

2 T

)

Z

1

~

+ ~a(X )@ X : (4.1)

0 n

8 7

This expression di ers from the vacuum p ersistence amplitude in the absence of ;~ ~a,

as can b e seen in the following expression:

! !

2 2

a_

~;~a

h0 j 0i = exp log  exp log 

T=2

2

T=2

T  g

 

1

~=0;~a=~a

0

p

exp ~a
~ h0 j 0i (4.2)

0

T=2

T=2

2 gT

In order to explain equation (4.2), we note rst that this contribution to the S-matrix ele-

ment shows up diagrammatically as the sum of disconnected diagrams (see Fig. 5). These

disconnected diagrams have to b e added to the more conventional connected diagrams in

2

order to restore conformal invariance

...

Fig. 5: The sum over disconnected diagrams.

The various terms app earing in the exp onent in eq. (4.2) are calculated on the disk

0

as the two-p oint function of the op erator O :

p

 

Z

~ g

1

0

~ ~

p

O = ds log @ X + ~a(X )@ X :

n 0 n

8

2 T

Care has to b e exercised to carefully account for the Mobius volume asso ciated to the

disc. The S-matrix element then b ecomes:

0 1

!

2

Z Z

2

~

a_ ~

2

~ =2

@ A

exp log  log  S = d~ e DX exp

2

T  g

 

Z Z

1

I

p

exp ~a
~ e V ::: V : (4.3)

0 1 n

2 gT

For simplicitywe analyze the scattering of a closed string tachyon o the D-brane. The

~ ~ ~ ~

~

initial and nal momenta are resp ectively k and k with excess momentum P = k + k .

1 2 1 2

The S-matrix element (4.3) is then, after some straightforward algebra:

2

Disconnected diagrams have previously b een considered in the case of the D-instanton [9]. 8

  

Z

2 ~ ~a

0

~ =2

~

p

S = d~ e exp
iP

2  gT

( )

 

2

p ~

~

p

exp i g P log 

T

9 8

1 0

2

= <

~

a_

2 2

~

A @

 g P log  exp

2

; :

 g

( )

 

2

1 1

~ ~

~

~ ~

exp ! + ! + a_
P log  F ( ;~ k ; k ; a_ ) (4.4)

1 2 1 2

2 2

0

where F is the nite part of the S-matrix element, up to order g ,evaluated using the

action (3.4).

Conformal invariance is restored only if:

p

~

~ = i P gT (4.5)

2

~ ~

a_ =  g P (4.6)

1

~

~

a_
P =0 (4.7) ! + ! +

1 2

2

equation (4.5) identi es ~ as b eing prop ortional to the recoil momentum; equation (4.6)

~

identi es a_ as the recoil velo city (interpreting the co ecient as the inverse mass); equation

(4.7) expresses the conservation of energy; using (4.6) it can b e written as:

2

~

P

! + ! + =0: (4.8)

1 2

2M

This can also b e obtained from (4.4) byintegrating over the constant part of X . The

0

S-matrix element then b ecomes:

 

1 1

2 2 2 2

~ ~

~ ~

 g P T  (! + ! +  g P ) G(k ; k ) (4.9) S = exp

1 2 1 2

2 2

~ ~

~

where G(k ; k ) is obtained from F ab oveby substituting the ab ovevalues for ;~ a_ .

1 2

Note that relation (4.6), that enforces momentum conservation, eliminates the dep en-

3

dence on ~a . We therefore conclude that (4.10) represents the transition from initial

0

~

D-brane at rest, to a nal state in which the D-brane has recoil momentum P . This

allows us to identify:

1

M =

2

 g

which is consistent with the results of [4].

3

~

Similarly, the conformal violations prop ortional to a cancel, at this order, when adding connected and

disconnected diagrams. 9

5 Conclusion

In conclusion, wehave applied the Fischler-Susskind mechanism to the case of D-brane

recoil. This approach can b e systematically extended to higher orders in string p ertur-

bation theory. Contributions from disconnected diagrams have to b e included in order

to restore conformal invariance and translational invariance. A purely two-dimensional

calculation of the soliton mass is obtained.

In this pap er we consider scattering of elementary string state o the D-brane. This

still leaves the question is the scattering among D-branes [12]. Since D-branes are BPS

saturated one might hop e to nd the metric on the mo duli space describing such scattering

( in analogy to the work of [8] on BPS monop oles). This would entail nding the space

on which the wormholes move.

Another interesting question is the treatment of fermionic collective co ordinates, from a

two-dimensional viewp oint, and the app earance of a manifestly sup ersymmetric collective

co ordinate description.

In summary, a b etter understanding of the dynamics of D-branes can shed more light

on the relations among string theories.

Acknowledgments

We thank D. Berenstein, S. Chaudhuri, M. Dine, and J. Distler for discussions. This

work is supp orted in part by the Rob ert A. WelchFoundation, and NSF grant PHY9511632.

After the completion of this work wehave received a pap er dealing with similar issues

[10]. We disagree with the conclusions of that pap er.

References

[1] E. Witten, Nucl. Phys. B443, (1995) .

[2] A. Strominger, hepth/ 9504090 (1995).

[3] J. Polchinski, hepth/9510017 (1995).

[4] J. Dai, R.G. Leigh, and J. Polchinski, Mod. Phys. Lett. A4, 2073 (1989) .

[5] W. Fischler, S. Paban, and M. Rozali, Phys. Lett. B352, 298 (1995) . 10

[6] W. Fischler and L. Susskind, Phys. Lett. B173, 262 (1986) .

[7] J. Polchinski, Nucl. Phys. B307, 61 (1988) .

[8] M.F. Atiyah and N.J. Hitchin, \The Geometry and Dynamics of Magnetic Monop oles,

Princeton, 1988."

[9] J. Polchinski, Phys. Rev. D50, 6041 (1994) .

M. Green, Phys. Lett. B354, 271 (1995) .

[10] V. Periwal and O. Tafjord, PUPT-1607.

[11] J. Liu and J. Polchinski, Phys. Lett. B203, 39 (1988) .

[12] C. Bachas, hepth/9511043;

U. Danielsson, and G. Ferretti, hepth/9603081;

D. Kabat and P.Pouliot, hepth/9603127. 11