Physics Letters B 534 (2002) 163–166

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A note on nonperturbative instability in string theory

Oleg Andreev1

Humboldt-Universität zu Berlin, Institut für Physik, Invalidenstraße 110, D-10115 Berlin, Germany

Received 19 December 2001; accepted 8 March 2002

Editor: L. Alvarez-Gaumé

Abstract

We investigate the possibility that stringy nonperturbative instabilities are described by worldsheet methods. We focus on thecase of open bosonic string theory, where the D-instanton plays a role of the bounce, i.e., it describes barrier penetration. In theprocess, we compute the exponential factor in a decay probability. 2002 Elsevier Science B.V. All rights reserved.

PACS: 11.25.Sq

Keywords: Nonperturbative instability; D-instanton

It is well-known since the seventies that the per-turbative stringS-matrix defined as expectation valuesof worldsheet vertex operators reproduces field theoryscattering amplitudes [1]. If one thinks of string the-ory as field theory with infinite number of fields livingin higher dimensions then this statement is equivalentto saying that scattering amplitudes of string theorynear its trivial perturbative vacuum are reproduced byworldsheet methods. It is a big problem to understandthe vacuum structure of string theory. It is clear thatsome vacua may be unstable in perturbative or even innonperturbative sense. In the last case, it seems nat-ural to ask whether one can compute decay probabil-ities as, for example, it was done by instanton meth-ods in ordinary field theory [2]. It is clear that thisis in principle a solvable problem. However, in prac-tice it turns out to be very hard to deal even with

E-mail address: andreev@physik.hu-berlin.de (O. Andreev).1 Permanent address: Landau Institute, Moscow, Russia.

several fields not saying an infinite number of them.A hope maybe that worldsheet methods are appro-priate again. A good motivation for this comes fromthe fact that there are representations of string the-ory effective actionsS via worldsheet objects [3,4].These representations are based on partition functionsof strings propagating in background fields. Moreover,the actions evaluated at solutions of the correspondingequations of motion coincide with the partition func-tions, namely,

(1)S(λi∗

) = cZ(λi∗

),

whereλi are string fields andZ is the partition func-tion. Here, we also include a normalization constantc.The extrema of the actions have the known meaningwithin the worldsheet theory: they are conformal back-grounds. So, the right-hand side of Eq. (1) representsa partition function of two-dimensional theory at itsfixed point. This simplifies explicit computations ofpartition functions since there are no UV divergenciesanymore. Let us now assume that equations of motion

0370-2693/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0370-2693(02)01564-2

164 O. Andreev / Physics Letters B 534 (2002) 163–166

associated with the Euclidean actionSE obtained fromS by analytic continuation admit a solutionλi

B whichfrom the field theory point of view can be recognizedas the bounce, i.e., it describes a decay of some falsevacuumλi

0 [2]. Then, assuming that the relation (1)is also valid in the Euclidean case, we can immedi-ately get the following representation for the exponen-tial factor in a decay probability per unit time per unitvolume

(2)w ∼ exp[−cZE

(λi

B

)].

Notice that we do not includeSE(λi0) into the expo-

nent.The purpose of this note is to give an example of

explicit computations. To do so, we consider openbosonic string theory where a big progress has beenrecently achieved in understanding of D-brane decayas open string tachyon condensation [5]. In particular,this was achieved by using a toy field theory model [6]and a background independent open string field theory[4]. Indeed, they turned out to be useful tools to gainintuition on the physics and carry out some explicitcalculations. It turns out that they are useful for ourpurpose as well.

To gain some intuition, we begin with a scalar fieldtheory in Euclidean(p + 1)-dimensional space withaction

(3)Sft = τp

∫dp+1x e−T

(1+ T + 1

2α′∂iT ∂iT

).

This theory was used as a toy model for tachyoncondensation on unstable branes in [6]. In this context,it describes the open string tachyon living on anunstablep-brane whose tension isτp. Note that thetension includes a factor of the dilaton (string couplingconstantg).

One remarkable observation is that the theorybelongs to a set of field theory models whose equationsof motions admit exact solutions [9]. In particular, a setof exact spherically symmetric solutions associatedwith the action (3) is given by

(4)Tn(x) = 1

2α′(x2

0 + · · · + x2n−1

) − n,

wheren ranges from 1 top + 1. In the context oftachyon condensation, these solutions are interpretedas the lower-dimensional branes. Indeed, they almost

Fig. 1. The nonderivative part of the Lagrangian,V , for a theorywith the Euclidean action (3). It gives the tachyon potential.

reproduce the famous descent relations for the D-branetensions [6].

It is easy to find the bounce among the set ofthe solutions. It corresponds ton = p + 1. Indeed,only in this case the Euclidean action (3) evaluated atthe solution is finite. Moreover, one can immediatelycheck thatTp+1 obeys the boundary conditions for thebounce in the sense of Coleman2

(5)limx0→±∞Tp+1(x) = +∞, ∂0Tp+1|x0=0 = 0,

where∂0 = ∂/∂x0 andx0 is treated as Euclidean time.At this point we should mention that in the context

of tachyon condensationTp+1(x) is called as theD-instanton. Thus, what we have learned from this toyfield theory model is a hint on the physical meaningof the D-instanton: it might describe a decay of anunstable vacuum through barrier penetration. In ourcase, the unstable vacuum corresponds toT = +∞(see Fig. 1).

To complete our discussion of the field theorymodel, let us compute the exponential factor in a decayprobability of the unstable vacuum. Evaluating theaction atTp+1(x), we obtain

(6)Sft(Tp+1) = τp ep+1(2πα′)(p+1)/2

that results in the following expression for the expo-nential factor

(7)w ∼ exp[−τp ep+1(2πα′)(p+1)/2

].

2 To complete the picture, one can compute the spectrumfluctuations nearTp+1(x) and show that it includes only onenegative mode as it should be for the bounce. In doing so, there isa great simplification as the corresponding equation reduces to theequation for a harmonic oscillator (see, e.g., [6]).

O. Andreev / Physics Letters B 534 (2002) 163–166 165

SinceSft evaluated atT = +∞ vanishes, there is nothe corresponding contribution in (7).

So far we have just noticed that the D-instantonmight be interpreted as a bounce in the sense that itdescribes a decay of some unstable vacuum. However,our discussion given within the field theory model hastwo disadvantages: it cannot in principle provide uswith the desired representation of the decay factor (2).The relation of the solutionTp+1 with the D-instantonmight seem not sufficiently convincing. Fortunately,both of these disadvantages disappear in the back-ground independent open string field theory [4]. Tothe leading order in derivatives, its Euclidean actionis simply [7,8]

(8)SE = τp

∫dp+1x e−T (1+ T + α′∂iT ∂iT + · · ·),

where the dots stand for an infinite number of higherderivative terms. These terms can in principle beconsidered as a result of integration over the otheropen string modes that modifies the action (3). It turnsout that this modification of the action does not havea strong influence on the existence of a set of exactspherically symmetric solutions like (4). We now have

(9)Tn(x) = t

2α′(x2

0 + · · · + x2n−1

) + a,

where a and t are some parameters which will bedetermined later.

We will again specialize toTp+1 because it resultsin a finite action and obeys the boundary conditions(5). Due to these reasons, we will call it the bounce.It turns out that the action evaluated at the bouncecan be rewritten as a function of the parameters in thefollowing form [4]

(10)SE(a, t) = τp

[1+ βa

∂

∂a+ βt

∂

∂t

]ZE(a, t),

whereβa = −a − (p + 1)t , βt = −t and

ZE(a, t) =(

2πα′

t

)(p+1)/2

e−a[eγ tΓ (1+ t)

]p+1.

γ denotes the Euler’s constant.The parameters in (10) are determined by demand-

ing thatSE(a, T ) is stationary under their variations. Itis a simple task to do so fora [8]. Indeed, in this casea simple algebra leads to

a(t) = (p + 1)

[−t + t

∂

∂t

(γ t + lnΓ (1+ t)

)].

As a consequence, the action for the bounce reducesto

(11)SE(a(t), t

) = τpZE(a(t), t

).

Before going on, it is time to remind the meaningof the entries on the right-hand side of Eq. (10)as objects of the underlying worldsheet theory [4].Consider open bosonic string in Euclidean target spacein the presence of the tachyon background whoseprofile is similar toTp+1.3 This is a simple choiceof the background for which the partition functioncan be computed exactly. In a special scheme, it isgiven byZE(a, t) [4]. As to βa andβt , they are therenormalization group (RG) beta functions. A simpleRG analysis shows that the parameters flow from zeroin the UV to infinity in the IR. The last means thatall Xi in the path integral are subject to the Neumannboundary conditions at the UV and the Dirichletboundary conditions at the IR.

Having reminded the worldsheet theory, we haveall at our disposal to achieve the purpose. First, let usnote that the formula (11) is the desired representationfor the effective action.4 At this point, we have only tocheck that it results in a finite action for the bounce.t is easily found from the correspondence betweenthe fixed points of RG on the worldsheet and extremaof the effective action. It is unique and given by itsvalue in the IR fixed point. Next, pluggingt = ∞ intothe partition function, we getZE(a(∞),∞) ≡ ZD =(4π2α′)(p+1)/2. From the worldsheet point of view,it is of course the expected result as the string pathintegral is finite for the Dirichlet boundary conditions.Second, the desired relation between the bounce andthe D-instanton follows from a canonical constructionof the D-instanton within the worldsheet theory (see,e.g., [10]) as att = ∞ all Xi are subject to theDirichlet boundary conditions. Finally, we finds for theexponential factor in the decay probability

(12)w ∼ exp[−τpZD].Let us conclude by several short remarks.

3 Note that at the beginning this assumes the use of parameters(bare couplings) which differ from the parameters inTp+1. Theparametersa andt are the renormalized couplings.

4 This relation is obvious for the UV as it directly follows fromEq. (10) but it is far from obvious for the IR.

166 O. Andreev / Physics Letters B 534 (2002) 163–166

(i) So far, there is not known any partition functionrepresentation for string theory effective action thatincludes all string modes. It is the reason why wedealt with the background independent open stringfield theory.

(ii) Our analysis of the actions for the bounce is infact similar to the computation of the descent relationsbetween brane tensions in [6,8].

(iii) The idea that the D-instantons lead to nonper-turbative effects like exp(−O(1/g)) is an old one (see,e.g., [11,12]). In particular, from the point of view[12], the right-hand side of Eq. (12) withp = 25 is in-terpreted as the open string partition function with theDirichlet boundary conditions for allXi . This meansthat what we found can be called as the partition func-tion representation for the exponential factor in a de-cay probability.

(iv) We have made no attempt to study quantumcorrections. It is clear that it would include highergenera of two-dimensional surfaces and, as a con-sequence, appearance of closed string modes. Thismakes the problem more involved than even includ-ing gravity within the field theory analysis [13].

(v) A perturbative instability of the standard boson-ic open string vacuum (T = 0 of Fig. 1) has been dis-cussed in [14], where its decay rate has been evaluatedvia one-loop computations.

Acknowledgements

We have been benefited from discussions withL. Alvarez-Gaumé, I. Bars, and J. Polchinski. We alsowould like to thank H. Dorn and A.A. Tseytlin for

comments and reading the manuscript. This work issupported in part by DFG under grant No. DO 447/3-1and the European Commission RTN ProgrammeHPRN-CT-2000-00131.

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