The tachyon potential in string theory

Nuclear Physics B361 (1991) 166-172 North-Holland

THE TACHYON POTENTIAL IN STRING THEORY T. BANKS*

Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA Received 4 April 1991

We argue that the tachyon potential in string theory is exactly given by the unstable quadratic mass term calculated perturbatively around the critical string. The argument is given in terms of the sigma model formulation. The same result follows from the exact Wilson renormalization group equations. The discrepancy with previous calculations of the tachyon potential is explained by the fact that other authors worked near the tachyon mass shell where it is impossible to distinguish a potential from derivative terms in the effective action.

1. Introduction

Most known classical solutions of bosonic string theory are unstable. This is known as the tachyon problem. In terms of the space-time effective action, the statement means that for small tachyon field the tachyon potential behaves like - T 2. Numerous attempts [1-7] have been made to compute higher-order corrections to the potential and find a stable minimum. In recent years it has become apparent that this is a difficult if not impossible task. Martinec [8] argued that any classical solution of string theory with a flat Minkowski subspace would have a tachyon instability. The solution has the form of a free-field theory tensor product with another conformal field theory, and the unit operator of this other theory can be dressed to make a gauge-invariant tachyonic perturbation of the tensor product theory. More recently, Seiberg [9] has used modular invariance to prove a generalization of this result to certain time-dependent solutions of string theory in which the time component of the string is described by an interacting conformal field theory. He showed that the existence of tachyons followed whenever the theory had an infinite number of physical states*. In the present paper we approach the same problem from a somewhat different point of view. We argue that the nonderivative term in the space-time action for *Supported in part by the Department of Energy under grant No. DE-FG05-90ER40559. * To be more precise, whenever it has "more" states than a quantum mechanical system with a finite number of degrees of freedom. Elsevier Science Publishers B.V. (North-Holland)

T. Banks / Tachyonpotential

167

the tachyon is exactly given by - T 2. This shows that there can be no homogeneous stable solutions of bosonic string theory. The essence of the argument is very simple. The equations of motion of string theory come from the requirement that the lagrangian for motion of the string be a generally covariant quantum field theory. The difficulty of constructing such field theories stems from the fact that quantum field theories require regulators for their definition, which in turn implies an a priori notion of length. If the metric is one of the quantum fields*, then we must introduce a fictitious "fiducial" metric to regulate the theory in a covariant way. Quantum mechanical covariance is achieved when the theory is independent of the fiducial metric, a requirement that is encoded in the vanishing of the fl-functions of the theory. The dilaton field is defined by the term SaD = D ( x ) ~ - / ~

(1.1)

in the two-dimensional lagrangian. Here o~ is the background metric and /~ its scalar curvature. As a consequence, D ( x ) satisfies a well-known ,low-energy theorem" in classical string theory. A constant shift in the dilaton produces a term in the lagrangian which is independent of the quantum fields and factors out of the functional integral. It is also independent of the fiducial metric on a space of fixed topology, its only function being to rescale the value of terms in the string loop expansion. This means that the space-time lagrangian has the form

S a= e - ° . ~ ( V D ) .

(1.2)

We claim that a similar result holds for constant shifts of the tachyon field T(x). T ( x ) is defined by a term in the two-dimensional lagrangian of the form

SaT = T( X)x/-~"

(1.3)

A constant shift in T again produces a term independent of the quantum fields, but now dependent on the fiducial metric. Quantum mechanics of the two-dimensional field theory produces another term of the form c/gr~ in the partition function. Its coefficient, c, is cutoff dependent and nonuniversal. The full dependence on the fiducial metric is then (t is the constant shift in the tachyon) ( t - c)V~- + terms depending on fiducial curvature.

(1.4)

There is no way for the curvature dependent terms to cancel the variation of this fiducial cosmological constant for general fiducial metrics [10]. Thus the fl-function * We take the point of view that the conformal factor of the physical metric can be viewed as one of the coordinates of target space.

168

T. Banks /

Tachyonpotential

equation for the constant mode of the tachyon must read t = c.

(1.5)

This follows from a space-time lagrangian* .gP=e - n f G [ ( T -

c) 2 +.~'(VT,

ITD)].

(1.6)

We can shift away the constant c, since all other terms in the lagrangian depend only on derivatives of the tachyon field. This is consistent with its. cutoff dependence. Nonuniversality always shows up in the space-time equations of motion as a field redefinition ambiguity. This simple argument, if correct, implies that it is fruitless to search for stable, homogeneous solutions of the tachyon equations of motion. If stable solutions exist, they must be time dependent. In the next section we will show how this result can be derived in another framework for writing string equations of motion.

2. E x a c t r e n o r m a l i z a t i o n

group equations

In ref. [11] it was proposed that Wilson's exact renormalization group equations might form the basis for off-shell string field theory. Not much work has been done to implement this proposal, but the authors of ref. [12] showed that it could be used to compute the tree-level Shapiro-Virasoro amplitudes**. A version of the Wilson equations appropriate for describing local scale transformations can be obtained as follows: Consider a two-dimensional field theory on a world-sheet of spherical topology with a fiducial metric ~. Write the lagrangian of the theory as .~=½vl-gg~O,,XO~X+.~1, and regulate it by replacing the first term by vfgXK(A)X. Here A -- (I/~fg)O~(~/'gg ~° 08) is the covariant laplacian and K(D) is 1 a function behaving as - 7 D for small values of its argument and growing exponentially for large values. Assume further that the local measure for the X variables is flat***. We can then ask what the conditions are on the interaction that guarantee that the partition function be independent of the metric ~. Going to a coordinate system where the fiducial metric is conformally fiat (g~0 = 8~0 e2~) and integrating out the Faddeev-Popov ghosts to give an explicit local term in the * Actually we have not proven that the lagrangian has this form, only that it must lead to eq. (1.5). **The primary reason that I have given up on this approach is the difficultyof incorporating chiral fermions into it. One must also make a commitment to the number of degrees of freedom in the cutoff version of string theory, a difficultythat is avoided in lattice approaches to the problem [13]. *** It may be that any nontrivial measure can be incorporated into the interaction lagrangian in an appropriately regulated fashion. Howeverwe have not examined this problem carefully.

T.Banks / Tach),onpotential

169

action, we can write this condition as

f[dX

]eS(½f

=

dz dwX(z)X(w)

6K( z, w) 8~(~)

~SI 1 + 8~(~) J

f[dX]e~fdz dwF[X; z,slK(z,w)X(w) 6S,

+f dz F[X;z,S]6x(z----~+

6F[X;z,s] ) 6X(z)

(2.1)

In this equation S = - ½f d z dwX(z)X(w)K(z, w) + S n and F[X; z, s] is an arbitrary function(al) of the indicated arguments. The equation says that a change in the fiducial metric can be compensated by a change of variables in the functional integral and thus does not affect the partition function. It can be satisfied if we choose

F[ X; z,s] =

k[f

dwdtX(w)

6K(w,t) K_t(t,z) + f dw 8S,

6K-t(w,z)

8x(~)

8~(s)

8~(s)

' (2.2)

8S, =½fdzdw[I

~(s)

6Sn 6K-I(z'w) 8x(z)

8¢(s)

6Sn ~X(w)

6K(w,z) 62Sl 6K-l(w,z) +2K-t(z'w) 6tr(s) + 6(X)z6X(w) 6tr(s) ]"

(2.3)

The latter is the exact equation of motion of string fields in this formulation of the dynamics of string theory. S n will contain terms that depend on X as well as or-dependent terms which are independent of the fields. The term containing the constant mode of the tachyon is of the latter type. Let us write S l -- S O+ S~, where S O is independent of X. The equation for S O can then be written schematically as

= K-' --+sK [ 8sI 2 + 82s ] 8~

8~

~-X

8X2 ]x-o'

(2.4)

while the equation for S~ does not depend on S °. Imagine now solving the field equations for all field modes contained in S~. This should give us the equation of motion for the modes in S O that follows from the effective action for these

170

T. Banks / Tachyonpotential

variables. For any such solution the right-hand side of eq. (2.4) is just a o'-dependent number, independent of S o. Thus, the exact effective equation of motion for these variables is linear, in agreement with the argument given in sect. 1.

3. Relation to previous calculations There have been many previous calculations of the tachyon potential in a variety of formulations of string theory. Many of them find a nontrivial potential for the tachyon field. The purpose of the present section is to try to reconcile our calculation with these previous results. In this endeavour the most important thing to remember is the ambiguity in any lagrangian due to field redefinition. The results we proved above used a particular definition of the tachyon field. We will see that at least some previous results differ from our own through field redefinitions*. The simplest derivation of nonlinear terms in the renormalization group equation for the tachyon field comes from conformal perturbation theory. One considers the flat space solution of the bosonic string or the deformation of it by a linear dilaton field, and perturbs by a general tachyon potential. To leading order in perturbation theory the renormalization group equation requires the tachyon to be on shell, (A + 2 ) T = 0.

(3.1)

The next to leading term in the 0-function comes from the operator product expansion of the on-shell tachyon operator with another of different three momenta. There is a pole in the OPE corresponding to an on-shell tachyon operator with three momentum equal to the sum of the momenta of the two multiplicands. This gives rise to a logarithmically divergent contribution to coupling constant renormalization, and thus a contribution to the Callan-Szymanzik O-function. The calculation can be continued to all orders in perturbation theory in the tachyon field. Like all calculations of nonlinear renormalization effects in continuum field theory, this calculation only makes sense when the tachyon operator is marginal, that is when the tachyon is on shell. More precisely, the tachyon is allowed to be off shell only by amounts of order T", n > 0. If this is the case, a term T 2 in the 0-function differs from ½O2T2 only by terms of order T 3 and to each order in the expansion one can make a redefinition of T which converts the equation into one depending only on derivatives of T. The transformation is of the form T ( x ) Zo(d2)T(x) + Z~(02)T2 + ... with Z i ( - 2 ) = 1. Since it leaves the on-shell tachyon field unchanged, it will not affect the space-time S-matrix. * We have not examined all previous calculations of the tachyon potential.

T. Banks / Tachyonpotential

171

The above paragraph essentially describes the calculation of ref. [14]. Klebanov and Susskind [15] start with a more ambitious point of view. They introduce a cutoff and attempt to calculate the tachyon/3-function off shell. Nonetheless, since they only calculate divergent terms, their calculation is only valid near the tachyon mass shell. Thus they cannot distinguish T from Z(O2)T either. Similar remarks apply to the calculation of the tachyon potential from string field theory. There the tachyon field is defined by the linearized equations of motion Q ~ = 0. The quadratic term in the action is constructed by inventing a bilinear form for string fields ( ~ l , gz2) with the properties (1, Q $ ) = 0 and (qJ~,Q$2) = (Q$1,~2). Other forms obeying these rules can be constructed by defining (01,02)M = ([1-I-M]l/tl,~/2) where M is any operator obeying M Q = Q M = 0 and (Ol, M O 2 ) = (MOI, ~2)*. In particular, any field redefinition which coincides with the usual fields on shell will induce an M of this sort. The correspondence between fields in the string theory action and the fields in the o-model has never been worked out explicitly. Thus, calculations of a nontrivial tachyon potential in string field theory do not contradict the claims made in this paper. Although general arguments thus show that there is no contradiction between our results and those of previous authors, it would be comforting to work out the explicit connection between the various formalisms to check that they are indeed consistent.

4. Conclusions

We have given an argument based on the general structure of renormalization group equations that the tachyon potential in string theory is (for a particular definition of the tachyon field) purely quadratic. An interesting application of this argument allows us to understand an otherwise puzzling feature of the exact solutions of string theories in two-dimensional target spaces [17-20]. In these theories the tachyon is related by bosonization to a fermion field with bilinear action. The fermion number current is realized as the gradient of the scalar tachyon field. To all orders in the string loop expansion, the effective action is local in the fermion number current, and therefore depends only on derivatives of the tachyon field. This tachyon is related to the usual tachyon of the o--model formalism by multiplication by the exponential of the dilaton field**. The resulting equation of motion for the o-model tachyon contains precisely one linear nonderivative term, in accordance with our general argument. It is intriguing that our tachyon low-energy theorem is related to a hidden symmetry of string theory in this * It is conceivablethat such a redefinition of the bilinear part of string field action would remedy the problem of the one-loop vacuum energy in string field theory. See ref. [16] for a discussion. ** In two dimensions general coordinate invariance allows us to map a general dilaton background without critical points into a field which depends linearlyon the coordinates. Thus multiplicationof the tachyon by an exponential is a universal operation in two dimensions.

172

T. Banks / Tachyon potential

simple example, a symmetry that acts nontrivially only on the fermionic degrees of freedom. These new degrees of freedom are invisible in the usual description of strings propagating in a target space. Perhaps the general constraint on the tachyon potential is a hint that this symmetry and the associated hidden degrees of freedom exist in any string theory. I would like to thank A. Cooper, L. Susskind, and L. Thorlacius for valuable discussions. This research was supported in part by a grant from the Department of Energy. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

K. Bardacki and M.B. Halpern, Phys. Rev. D10 (1974) 4230 K. Bardacki, Nucl. Phys. B68 (1974) 331; B70 (1974) 397 E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117 J.H. Schwarz, Nucl. Phys. B48 (1972) 525 C. Lovelace, Nucl. Phys. B273 (1986) 413 S. Das and B. Sathiapalan, Phys. Rev. Lett. 56 (1986) 2664 A. Cooper, L. Susskind and L. Thorlacius, SLAC-PUB 5413, Jan. 1991 E. Martinet, unpublished, 1985 N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Rutgers preprint RU-90-29; Lectures given at the 1990 Yukawa International Seminar, Common Trends in Mathematics and Quantum Field Theories, and at the Carg~se Workshop on Random surfaces, quantum gravity and strings C. Callan and L. Thorlacius, Nucl. Phys. B319 (1989) 133 T. Banks and E. Martinec, Nucl. Phys. B294 (1987) 733 J. Hughes, J. Liu and J. Polchinski, Nucl. Phys. B316 (1989) 15 T. Banks, Lecture presented at the Carg~se Workshop on Random surfaces, quantum gravity and strings, May 28-June 1, 1990 S. Das and B. Sathiapalan, Phys. Rev. Lett. 57 (1986) 1511; 56 (1986) 2664 I. Klebanov and L. Susskind, Phys. Lett. B200 (1988) 446 R. Woodard, Phys. Lett. B213 (1988) 144 E. Brezin, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59 (1978) 35 D.J. Gross and I. Klebanov, Princeton preprint PUPT-1198 A.M. Sengupta and S.R. Wadia, Tata Inst. preprint 90-33, July 1990, Int. Journal Mod. Phys., to be published G. Moore, Rutgers/Yale preprint, RU-91-12, YCTP-P8-91