1 June 2000

Ž .Physics Letters B 482 2000 249–254

Lumps and p branes in open string field theory

R. De Mello Koch a, A. Jevicki b, M. Mihailescu b, R. Tatar b

a Physics Department, UniÕersity of Witwatersrand, Johannesburg, South Africab Department of Physics, Brown UniÕersity, ProÕidence, RI 02912, USA

Received 14 March 2000; accepted 17 April 2000Editor: M. Cvetic

Abstract

We describe numerical methods for constructing lump solutions in open string field theory. According to Sen, theselumps represent lower dimensional Dp branes and numerical evaluation of their energy can be compared with the expectedvalue for the tension. We take particular care of all higher derivative terms inherent in Witten’s version of open string fieldtheory. The importance of these terms for off shell phenomena is argued in the text. Detailed numerical calculations done forthe case of general p brane show very good agreement with Sen’s conjectured value. This gives credence to the conjectureitself and establishes further the usefulness of Witten’s version of SFT. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

Since its introduction, string field theory held outthe promise for nonperturbative studies of string

w xtheory. Recently Sen 1,2 has argued that openŽ .bosonic string field theory describes the dynamics

of DD system with the tachyon providing the insta-bility inherent in such pair. Tachyon condensationalso describes the decay of a single unstable D brane.

w xIn addition, according to 2 the kink and lumpsolutions of such field theory lead to lower dimen-

w xsional branes. In recent work Sen and Zweibach 3studied in detail tachyon condensation in 26D openstring theory. This follows the earlier, pioneeringwork of Samuel and Kostelecky who were the firstto consider the vacuum structure of string field the-

w xory 4,5 . They used a level truncation scheme togenerate an approximation for the tachyon effectivepotential. Following this scheme, results were foundw x3 that show great agreement in numerical values

with the expected exact results. Similar results werew xrecently obtained for superstring field theory 6,7 .

w xImpressive high level studies appeared in 8,9 .It is equally important to give a construction of

non-constant kink and lump-like solutions. One canexpect that it is for these that the stringy effectspresent in string field theory might play the mostimportant role. One of the characteristic featurespresent in the construction of Witten’s version of the

w xtheory 10–13 is the appearance in the interaction ofterms exponential in derivatives. These terms can bemoved from the interaction to give a nontrivial ki-netic term. They have frustrated early attempts for

Ž .construction of nonperturbative soliton or instantonsolutions of the theory.

In the present work, we consider this problem innumerical terms. Concentrating on the lump of openstring theory, we develop methods for its numericalsolution. In this we keep the nontrivial exponentialterms characteristic of the string vertex interaction

0370-2693r00r$ - see front matterq 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00521-9

( )R. De Mello Koch et al.rPhysics Letters B 482 2000 249–254250

and simultaneously perform a level truncation. This,we argue is to provide a very good approximation tothe exact result.

The content of the paper is as follows. After ashort description of open SFT, we discuss somefeatures relevant to the present work. We explainŽ w x.based on earlier observations 14 how and why theapproach of keeping higher derivatives and simulta-neous level truncation holds the promise for a goodapproximation. We then proceed to the numericalwork.

While this work was in progress, there appearedw xthe work of Ref. 15 which considers the problem in

its field theory limit.

2. Open string field theory

We begin by describing some features of openstring field theory which are of relevance to theinvestigation that follows. One has the cubic action:

g² < < : ² < < : < : < :Ss A Q A q V A A A 1Ž .33

with Q being the first quantized BRST operator. Thegeometric, three string interaction is realized in theHilbert space by the vertex

3Xr r s s 2² < ² <V s 0 exp a N a qa lng E . 2Ž .Ý Ý3 n nm m r½ 5

rs1

Here the Neumann coefficient N r s are determined interms of appropriate conformal mapping, their ex-

w xplicit values are determined in 10,11 .One of the main properties of the three-string

vertex is an explicit appearance of higher derivativeterms. They come in exponential form acting on eachstring field

< : X m < :A ™exp a lngE E A 3Ž .Ž .m

with the constant

'3 3gs . 4Ž .

4

The exponential terms are not relevant in studies ofvacuum structure but they can have a nontrivialeffect in any other nonzero momentum process. Con-cerning a systematic approximation or expansionscheme one notes the following. The masses of

Ž .tachyon and higher mass fields are proportional to1ra

X. In general aX serves as a scale, it can be

scaled out in front of the SFT action. At a nonpertur-bative level, there is no small free parameter and no

Žsystematic expansion. Since one is not able at pre-.sent to solve the theory in exact terms, one relies on

seemingly ad hoc approximations. Such is the pro-cess of level truncation. In order to understand moreclearly the procedure involved and the relative rele-vance of particular terms, let us recall the originalargument given for the level truncation in an unpub-

w xlished work of Ref. 14 . Considering an approximatecalculation of a nonzero momentum amplitude, forexample for four tachyons, one starts from

1 X Xysr2y2 R² < < :A s dx x V 3 b x V 3 , 5Ž . Ž . Ž .Hs 34 0 120

representing the s-channel Feynman diagram.Using the fact that x Ra xyR sx na , as well asn n

similar results for bXs and cXs, we have

1 ys r2y2 ˙ ˙² < :A s dx x V 3 b V 3 . 6Ž .Ž . Ž .Hs 34 0 120

The dot on V indicates that a ,b ,c haveyn yn yn

been replaced by x na , x nb , x nc . Expanding inyn yn yn

levels corresponds to expanding in powers of xseyr. With the use the appropriate Neumann coeffi-cients after some straightforward algebra, one has

1121 ys r2y2 2 EŽ x .A s dx x 1y x q PPP e . 7Ž .Hs 6ž /30

The term in the exponent has the expansion

sE x sy lngŽ .

2

s 23 22 P192y q2 y xq x q PPP3 6ž / ž /2 3 3

i 24 26 P723y q2 y xy x q PPP3 10ž / ž /2 3 3

24 232y xq x q PPP3 6ž /3 3

26P522q x q PPP . 8Ž .6ž /2P3

( )R. De Mello Koch et al.rPhysics Letters B 482 2000 249–254 251

This result can easily be rearranged into the form

Ž .z 1 ytr2y2ysr2y2A s dz z 1yz , 9Ž . Ž .Hs0

where

1 23 222z x s x 1y xq x q . . . . 10Ž . Ž .2 3 3ž /g 3 3

For agreement with the exact result one wouldŽ . Žneed to have z 1 s0.5 the s- and t-channel dia-

Ž ..grams are to cover the full range 0,1 . To theŽ .present order in the level expansion we have z 1 f

0.50480 which is indeed very close to the exactvalue. A more significant observation is the fact thatthe main effect is contained in the gy2 factor present

Ž .in the above expression. That term itself gives z 1f0.6. If we look up the origin of this factor, we seethat it comes directly from the exponential higherderivative operator present in the vertex: since theintermediate states in the four point amplitude calcu-lation are not on shell the exponential terms con-tribute giving the corresponding s dependence.

In this nonzero momentum example, we concludethat it is advantageous to keep the higher derivativesexactly and that this followed by a level truncation islikely to give a good overall approximation. Natu-rally one still expects this expansion to be good onlyfor a certain range of momenta.

Consider then the string field theory with theŽ .tachyon level 0 , but with the higher derivative

terms kept exactly. The action evaluated originally inw x11 reads

1 1 g2 2 3 3˜LLs E T q T y g TŽ . Xm2 2a 3

˜ lngE 2with Tse T . 11Ž .

The first sign that the presence of the exponentialterm in the cubic interaction profoundly influencesthe nature of the problem is seen in attempting toevaluate the asymptotics of a possible static solution.In ordinary field theories, kink and lump solution canbe asymptotically characterized as

f x ;f qa eym x , 12Ž . Ž .0 1

where f is the constant vacuum solution and m is0

the physical mass of the scalar field. Based on this,one can write a systematic expansion for the lump

`

yn m xf x s a e 13Ž . Ž .Ý nms0

with the classical equations providing a recursionformula for coefficients a . The exponential decayn

eyn m x has a physical meaning, the lump form factorreceives a contribution from n mesons. In attemptingan analogue expansion in the case of string fieldtheory, one meets a surprise. After moving the expo-nential into the kinetic term or equivalently denotingTsf the string theory tachyon field equation reads

2 22 ycE xE q1 e y2 f x sgf x 14Ž . Ž . Ž .Ž .ž /x

with cs2lngs ln 33r42. The Ansatzym xf x ;f qa,e 15Ž . Ž .0

leads to an eigenvalue equation22 ycmm q1 e y2s0 16Ž . Ž .

for m. This equation turns out to have no realsolution. With the exponential present the nature ofthe lump solution has changed from that of ordinaryfield theory. In particular the decay at asymptoticinfinity in string field theory has to be stronger thana simple exponential. The non-exponential decay of

Ž .the full lump kink solution signals that the formfactor will have a more complex physical meaning.This feature also necessitates a purely numericalapproach to the problem which we attempt in thenext section.

3. Calculations and results

The presence of the higher derivative terms inŽ .14 frustrate a numerical analysis directly in xspace. Upon transforming to momentum space, onefinds the following nonlinear integral equation

1yk 2 eck 2y2 f kŽ . Ž .Ž .

25ypsg d qf q f kyq , 17Ž . Ž . Ž .Hwhere k is a 25yp dimensional vector. To solve anintegral equation of this type, one typically dis-

( )R. De Mello Koch et al.rPhysics Letters B 482 2000 249–254252

Fig. 1. The wave function of the D20 brane as a function of theradial coordinate r. The plot was obtained by taking the Fourier

Ž .transform of the numerical solution of equation 18 .

cretizes the problem and solves the resulting matrixequation. In the case on hand, the resulting matrixequation is nonlinear and one is forced to search for

Ž .the solution f k using a numerical minimization. Itis computationally difficult and expensive to mini-mize functions with a large number of variables, andfor this reason we have worked directly with aspherically symmetric ansatz for the tachyon field.Stable numerical solutions were obtained using alattice having 81 points. This implies a nonlinearminimization problem with 81 parameters. Thechoice of a good objective function to be used in theminimization as well as an accurate initial guess arecrucial to obtain a nontrivial solution. Indeed, inpractice, we find that the majority of initial guesseslead to the trivial fs0 solution. Our objectivefunction was constructed as follows: Start by making

ˆŽ . Ž .an initial guess, G k for the right hand side of 17 .n

This initial guess is used to compute a value for thetachyon wave function

ˆgG kŽ .nf k s . 18Ž . Ž .2n 2 ck n1yk e y2Ž .ž /n

ˆŽ .The wave function f k can now be used to com-nŽ .pute the value of the left hand side of 17 . However,

this involves a convolution which must be performedwith care. Evaluating the convolution directly inmomentum space is not optimal: by exploiting thespherical symmetry of the wave function, the convo-lution can reduced to the integration over a single

angle and a radius. The integrand contains the factorˆ ˆŽ . Ž .f p f d withn n

2 2< < (d s kyp s p qk y2k p cos u . 19Ž . Ž .n n m m n l

In general, d does not correspond to a lattice pointn

and one is forced to interpolate from the knownˆŽ .points to find f d . A much more efficient way ofn

performing the convolution is by transforming to xspace, squaring the function and then transformingback to k space. By using the spherical symmetry ofwave function, the Fourier transform can be reducedto a single integration. Thus, the previous doubleintegration has been replaced by two single integra-tions, which is more efficient. In addition, the inte-

ˆgrations only require a knowledge of f at the latticeˆpoints. After convolving f with itself to obtain

Ž .G k , the error, EE to be minimized is constructedn

as

ˆ< <EEs G k yG k . 20Ž . Ž . Ž .Ý n nn

The minimization was performed using the fmins.msubroutine of MATLAB, which employs a simplexsearch method. A suitable initial guess is obtained bytaking a function which initially falls off slightly

Fig. 2. The error in the wave function of the D20 brane as afunction of the radial coordinate in momentum space p. The plot

ˆŽ Ž .shows EEr Ý G k as a function of k .n n n

( )R. De Mello Koch et al.rPhysics Letters B 482 2000 249–254 253

Table 1T yTp pValues of the Dp brane tensions computed using the numerical tachyon lump solution. The parameter DT is defined as DT s =100.p p

Tp

is the value of the wavefunction at the origin in position space.

ˆ Ž .p D T T rT f 0p p p

24 y29.4 0.706 y0.6323 y27.5 0.725 y0.8622 y14.3 0.857 y1.1921 y8.3 0.917 y1.6320 6.4 1.064 y2.2619 25.3 1.253 y3.1518 64.0 1.640 y4.46

faster than a Gaussian, but reaches zero at somefinite momentum. As an example we have shown thewavefunction for the D20 brane in Fig. 1.

The shown form for the wave function is typical.The value of the wave function at the origin inposition space decreased from fy0.62 for the D24brane to fy4.46 for the D18 brane. The point atwhich the wave function reaches zero is very nearlyconstant for all the Dp branes considered here. InFig. 2 we have shown the error in the D24 branesolution. The tensions of these solutions was evalu-ated directly in momentum space. For example, inthe case of the D24 brane, we compute

dp 1 22 2 c pT s2p T f p p y1 e f ypŽ . Ž .Ž .H24 25 ž2p 2

qf p f ypŽ . Ž .g dk

q f k f p f ypykŽ . Ž . Ž .H /3 2p

2p 2T dp 225 2 c ps f p p y1 e f ypŽ . Ž .Ž .ŽH6 2p

q2f p f yp . 21Ž . Ž . Ž ..In our conventions, a Dp brane has tension

25ypT s 2p T . 22Ž . Ž .p 25

In Table 1 we compare our numerically evaluatedˆtensions, T with the above values.p

Clearly there is no obstacle for the existence oflower p brane solutions. The trend is that the staticlump gives systematically a growing tension. This isactually opposite to what one finds in the extremefield theory limit. Concerning further improvement

Žof present results one expects especially for lower.p the relevance of higher massive levels. It is next

important to study the direction of their contribu-tions. It is also relevant to perform a similar study in

w xsuperstring theory 1,6,7,16 .

Acknowledgements

This research was supported by DOE grant DE-Ž .FG02r19ER40688- Task A , NRF grant GUN-

2034479 and by a Wits University FriedelSchellschop award.

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