Quantum Field Theory and Gravitation

On Tachyon Condensation in String Theory:Worldsheet Computations

Oleg Andreev1Þ

Humboldt-Universitat zu Berlin, Institut fur Physik, Invalidenstra�e 110,D-10115 Berlin, Germanyandreev@physik.hu-berlin.de

Abstract

I briefly review some recent results of computations of the tachyon potentials by worldsheet methods.

1. Introduction

It is a big problem to better understand the vacuum structure of string/M theory. Even inthe case of bosonic string theory which contains the tachyon near its perturbative vacuum,it turned out to be not a simple deal to find a stable vacuum. Note that hopes for theexistence of such a vacuum have their origins already in the seventies [1]. However, foropen strings a real breakthrough has been achieved in recent years when the open stringtachyons were related with annihilation or decay of D-branes via the process of tachyoncondensation [3] (see [3] for a review and a list of references).In studying the phenomenon of tachyon condensation, string field theory methods are the

most appropriate ones (see, e.g., [4] for recent developments). However, in practice it turnsout to be very hard to deal with them. This was a motivation for finding simpler methods,for example toy models [5], that can allow one to gain some intuition on the physics of thephenomenon. Recently, it has been realized that the background independent open stringfield theory (BIOSFT) of Witten [6, 7] is a powerful tool to attack the problem. In particu-lar, it allows one to compute the tree level effective tachyon potentials [8––10] which amus-ingly coincide with the potentials found from the toy models based on the exactly solvableSchrodinger problem [5]. In fact, it is also possible to take a more intuitive approach (see,e.g., [11]) which generalizes the ordinary sigma model approach to string theory (see [12]for a review).

2. Tree Level

The BIOSFT is based on the Batalin-Vilkovisky formalism whose master equation providesthe effective action of the theory. In bosonic string theory, the tree level action of thetachyon field living on an unstable p-brane is given by [6, 8, 9]

S ¼ tpÐd pþ1x e�Tð1þ T þ a0 @iT @iT þ . . .Þ ; ð1Þ

Fortschr. Phys. 50 (2002) 5––7, 670 –– 674

# WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0015-8208/02/5-705-0670 $ 17.50þ.50/0

1) On leave from Landau Institute for Theoretical Physics, Moscow.

where the dots stand for an infinite number of higher derivative terms. These terms can inprinciple be considered as a result of integration over the other open string modes. tpstands for the p-brane tension.

The tachyon potential is

VðTÞ ¼ ð1þ TÞ e�T : ð2Þ

It has two extrema. The standard perturbative vacuum corresponds to the first one, T ¼ 0.A new vacuum to which the tachyon condenses is related to the second extremum,T ¼ þ1.

In superstring theory, the tree level action of unstable D-brane is [10]

SS ¼ ttpÐd pþ1x e�T2ð1þ 4 ln 2a0 @iT @iT þ . . .Þ ; ð3Þ

where the dots again stand for an infinite number of higher derivative terms. Note that theodd and even p’s are referred to type IIA and IIB, respectively. In other words, these un-stable D-branes are simply the D-branes with the ‘‘wrong” p’s.

The tachyon potential is now symmetric under reflection T ! �T and given by

VVðTÞ ¼ e�T2

: ð4Þ

It has three extrema. As in the bosonic case, the standard perturbative vacuum correspondsto the first one, T ¼ 0. A new vacuum to which the tachyon condenses is related to therest, T ¼ �1.

I finish the discussion of the tree level results with a few comments:

(i) It is easy to incorporate the abelian gauge field. To the leading order in a0, the actionsare given by [11, 13, 14]

S ¼ tpÐd pþ1x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet ð1þ FÞ

pe�T ð1þ T þ a0Gij @iT @jT þ . . .Þ ð5Þ

and

SS ¼ ttpÐd pþ1x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidet ð1þ FÞ

pe�T2 ð1þ 4 ln 2a0Gij @iT @jT þ . . .Þ ; ð6Þ

where Gij ¼ ½ð1� F2Þ�1�ij.(ii) It is also easy to incorporate a constant B field that slightly modifies the closed string

background. As a result, unstable branes are now described by noncommutative field the-ories. Their actions have the same form as the actions (1) and (3) but the product of thetachyon field is taken to be the star product [14, 15]. To be more precise, the star product

is defined as TðxÞ � TðxÞ ¼ ei2 q

ij @@yi

@@zi TðyÞ TðzÞjy¼z¼x, where q is the antisymmetric matrix

defined in terms of the B field.(iii) In superstring theory, there is another class of unstable systems than the D-branes

with the ‘‘wrong” p’s. These are the brane-antibrane (D �DD) systems. It turns out to be diffi-cult even at the leading order in a0 to compute their couplings to NS-NS backgrounds inthe form that includes all powers of the fields like (6). However, what one can do now is tocompute the couplings to a constant R-R background [16]. The result is a simple general-ization of the well-known couplings of the D-branes to the R-R potentials. The only no-velty is that the field strength of the gauge field is replaced to a curvature associated withthe superconnection A which is defined as a 2� 2 matrix whose elements expressed interms of the tachyon and gauge fields.

Fortschr. Phys. 50 (2002) 5––7 671

3. One-Loop

A deeper understanding of the process of tachyon condensation requires information aboutquantum corrections to the tree level potentials (2) and (4). Unfortunately, the quantummaster equation which is a proper machinery for doing so in the framework of the BIOSFTis still missing. Thus one is forced to use more intuitive approaches [17––19].For example, Tassilo Ott and I [19] have recently used the consistency with the field

theory results as a basic principle in studying one-loop corrections to the potentials. In thebosonic case, we have found the one-loop correction to the potential (2) in the followingform

V1�loopðTÞ ¼ � 1

2

ð1

0

dl

lð8p2a0lÞ�

pþ12 e�plT hðilÞ�24 ; ð7Þ

where h is the Dedekind eta function.A detail analysis of (7) shows that the one-loop correction develops an imaginary part.

There are in fact two contributions to it. The first one is due to the open string tachyon,while the second one comes from massive open string modes or, equivalently, from theclosed string tachyon. The imaginary part of V1�loopðTÞ signals that the theory is perturba-tively unstable. A physically meaningful quantity to compute in this case is the decay rateper unit volume of an initial unstable state. Explicitly, it is defined as G ¼ 2 ImV1�loop.In the supersymmetric case, the one-loop correction to the potential (4) takes the form

VV1�loopðTÞ ¼ � 1

2

ð1

0

dl

lð8p2a0lÞ�

pþ12 e�plT2

hðilÞ�8½ f 8NSðlÞ � f 8R ðlÞ� ; ð8Þ

where fNSðlÞ ¼ epl24Q1

n¼1 ð1þ e�plð2n�1ÞÞ and fRðlÞ ¼ffiffiffi2

pe�

pl12Q1

n¼1 ð1þ e�2plnÞ.As in the bosonic case, VV1�loop develops an imaginary part. This time the open string

tachyon only contributes. Indeed, there is no the closed string tachyon because of the corre-sponding cancellation between the NS and R sectors. The decay rate associated with thisperturbative instability is found to be

G ¼ pð8pa0Þ�ðpþ1Þ=2 ð1� T2Þpþ12

Gð1þ ðpþ 1Þ=2Þ Hð1� T2Þ ; ð9Þ

where H stands for the Heaviside step function.I conclude the discussion of the one-loop corrections by making a couple of remarks:

(i) We have found the one-loop corrections by computing the open string partition func-tions in the presence of a constant tachyon background. Doing the standard modular trans-formation, it is easy to rewrite the partition functions via closed string modes. The resultmight seem curious. Indeed, it reveals the transverse directions to the brane and looks likeclosed string modes appearing from one brane, propagating a distance R2Þ, and then disap-pearing on the other brane. In other words, the description in terms of closed strings as-sumes that the original brane splits into two parallel branes separated with a distance de-pending on the vev of the tachyon field.It is of some interest to notice that there is an analogy with field theory models that

undergo spontaneous symmetry breaking [20] that helps to understand the effect of brane

O. Andreev, On Tachyon Condensation in String Theory672

2Þ R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p2a0T

pfor bosonic string and R ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2p2a0

pjTj for superstring.

splitting. Indeed, given an effective potential, for example Veff ��T2 � T2

0

�2, one can study

the dynamics of spontaneous symmetry breaking by setting an initial narrow distribution ofthe scalar field at the maximum of the potential (T ¼ 0). Then, by quantum fluctuations itspreads out and shows two maxima. To reveal such a picture for the branes, let me notetwo key facts: The first is that in terms of closed strings T transforms into R (a distanceinto a transverse direction to a brane). The second is that a brane itself can be consideredas a lump solution of the higher dimensional worldvolume theory that allows one to saythat this solution plays a role of the initial distribution in field theory. Finally, let me notethat such a description of the decay shows that these lumps interact by the closed stringexchanges.

(ii) The instabilities I have discussed so far are perturbative ones. The form of the poten-tial in the bosonic case (see Eq. (2)) assumes that the new vacuum T ¼ þ1 is also un-stable but now in the nonperturbative sense because of the tunneling through the potentialbarrier. It is well-known that in field theory one can compute decay probabilities of suchunstable vacua by the instanton methods (see, e.g., [21]). Recently, I have computed theexponential factor in the decay probability of the vacuum T ¼ þ1 [22]. It is simply givenby

w � e�tpZD ; ð10Þ

where ZD is the standard partition function of open bosonic string on the unit disk with theDirichlet boundary conditions.

Acknowledgment

I thank the Alexander von Humboldt Foundation for inviting me to the Heisenberg Symposium. A partof the work reported here was done in collaboration with Tassilo Ott. This work was supported in partby DFG under grant No. DO 447/3-1 and the European Commission RTN Programme HPRN-CT-2000-00131.

References

[1] K. Bardakci, Nucl. Phys. B68 (1974) 331;K. Bardakci and M. B. Halpern, Phys. Rev. D10 (1974) 4230; Nucl. Phys. B96 (1975) 285;K. Bardakci, Nucl. Phys. B133 (1978) 297;V. A. Kostelecky and S. Samuel, Nucl. Phys. B336 (1990) 263.

[2] A. Sen, Int. Jour. Mod. Phys. A14 (1999) 4061 [hep-th/9902105]; JHEP 9912 (1999) 027[hep-th/9911116].

[3] K. Ohmori, A Review on Tachyon Condensation in Open String Field Theories‚ [hep-th/0102085].[4] http://online.itp.ucsb.edu/online/mtheory. This is a link to talks of D. Gross, A. Sen, W. Taylor,

and B. Zwiebach delivered at Avatars of M-Theory, Santa Barbara, California.[5] J. A. Minahan and B. Zwiebach, JHEP 0009 (2000) 029 [hep-th/0008231]; JHEP 0103 (2001)

038 [hep-th/0009246].[6] E. Witten, Phys. Rev. D46 (1992) 5467 [hep-th/9208027]; Phys. Rev. D47 (1993) 3405 [hep-th/

9210065];S. L. Shatashvili, Phys. Lett. B311 (1993) 83 [hep-th/9303143]; On the Problems with Back-ground Independence in String Theory, [hep-th/9311177].

[7] M. Marino, JHEP 0106 (2001) 059 [hep-th/0103089];V. Niarchos and N. Prezas, Nucl. Phys. B619 (2001) 51 [hep-th/0103102].

[8] A. A. Gerasimov and S. Shatashvili, JHEP 0010 (2000) 034 [hep-th/0009103].[9] D. Kutasov, M. Marino, and G. Moore, JHEP 0010 (2000) 045 [hep-th/0009148].

[10] D. Kutasov, M. Marino, and G. Moore, Remarks on Tachyon Condensation in SuperstringTheory, [hep-th/0010108].

Fortschr. Phys. 50 (2002) 5––7 673

[11] A. A. Tseytlin, J. Math. Phys. 42 (2001) 2854 [hep-th/0011033].[12] A. A. Tseytlin, Int. J. Mod. Phys. A4 (1989) 1257.[13] K. Li and E. Witten, Phys. Rev. D48 (1993) 853 [hep-th/9303067].[14] O. Andreev, Nucl. Phys. B598 (2001) 151 [hep-th/0010218].[15] L. Cornalba, Phys. Lett. B504 (2001) 55 [hep-th/0010021].

K. Okuyama, Phys. Lett. B499 (2001) 167 [hep-th/0010028].[16] P. Kraus and F. Larsen, Phys. Rev. D63 (2001) 106004 [hep-th/0012198];

T. Takayanagi, S. Terashima, and T. Uesugi, JHEP 0103 (2001) 019 [hep-th/0012210];M. Alishahiha, H. Ita, and Y. Oz, Phys. Lett. B503 (2001) 181 [hep-th/0012222].

[17] T. Suyama, Tachyon Condensation and Spectrum of Strings on D-branes, [hep-th/0102192];R. Rashkov, K. S. Viswanathan, and Y. Yang, Background Independent Open String FieldTheory with Constant B Field On the Annulus, [hep-th/0101207];K. Bardakci and A. Konechny, Tachyon condensation in boundary string field theory at oneloop, [hep-th/0105098];B. Craps, P. Kraus, and F. Larsen, JHEP 0106 (2001) 062 [hep-th/0105227].

[18] K. S. Viswanathan and Y. Yang, Phys. Rev. D64 (2001) 106007 [hep-th/0104099];M. Alishahiha, Phys. Lett. B510 (2001) 285 [hep-th/0104164];G. Arutyunov, A. Pankiewicz, and B. Stefanski, jr, JHEP 0106 (2001) 49 [hep-th/0105238];T. Lee, K. S. Viswanathan, and Y. Yang, Boundary String Field Theory at One-Loop[hep-th/0109032].

[19] O. Andreev and T. Ott, On One-Loop Approximation to Tachyon Potentials, [hep-th/0109187].[20] G. Felder, J. Garcia-Bellido, P. Greene, L. Kofman, A. D. Linde, and I. Tkachev, Phys. Rev.

Lett. 87 (2001) 011601 [hep-th/ hep-th/0106179].[21] M. B. Voloshin, I. Yu. Kobzarev, and L. B. Okun’, Yad. Fiz. 20 (1974) 1229;

S. Coleman, Phys. Rev. D15 (1977) 2929;C. G. Callan and S. Coleman, Phys. Rev. D16 (1977) 1762.

[22] O. Andreev, A Note on Nonperturbative Instability in String Theory [hep-th/0112088].

O. Andreev, On Tachyon Condensation in String Theory674