The static tachyon potential in the open bosonic string theory

Volume 207, number 2 PHYSICS LETTERS B 16 June 1988

THE STATIC TACHYON POTENTIAL IN THE OPEN BOSONIC STRING THEORY

V. Alan KOSTELECKq Physics Department, Indiana University, Bloomington, IN 47405, USA

and

Stuart SAMUEL Physics Department, City College ofNew York, New York, NY 10031, USA

Received 25 January 1988

In an effort to understand whether the open bosonic string theory has a stable vacuum, the four-point contribution to the static

tachyon potential is computed. This off-shell calculation is carried out using covariant string field theory.

1. Introduction

String theory [ 1,2] is a proposal for a consistent theory of gravity [ 3 1, that can contain anomaly-free gauge groups large enough to include SU( 3) x

SU(2) xU( 1) [4,5]. It is at present the unique theory having the possibility of combining the four fun- damental interactions in a unified quantum theory.

However, the physics of the simplest string models hardly resembles the standard model: the gauge groups generally have high rank, the spectrum is not realistic, the symmetry breaking mechanism to a GUT group and/or the SU(3)xSU(2)xU(l) group is not evident, and even the spacetime dimension ( 10 or 26) is different from four. These differences may be due to our lack of understanding of the string vacuum structure [ 6 1. Stated differently, one hopes that the perturbative string vacuum is unstable and spontaneously undergoes compactification, gauge group symmetry breaking, etc. One should thus endeavor to understand as fully as possible string vacuum physics.

In this paper, we perform a calculation with this goal in mind, albeit for an unrealistic model. We analyze the 26-dimensional open bosonic string which cannot describe the world since it contains no fer- mions. Nevertheless, our efforts are among the first

in this direction and may be helpful in analyzing more realistic string theories.

The bosonic string theory is an ideal arena for such investigations. Since it contains a tachyon, the perturbative vacuum is unstable. Our goal is to try to determine, at tree level, whether a stable vacuum exists. Unfortunately, we are unable to incorporate all the tree-level effects. Instead, we include tree-level diagrams up to four external legs.

Since the tachyon is the lowest state and the source of vacuum instability, its behavior is important. If it acquires a finite vacuum expectation value then the bosonic string theory is stable. If this expectation value is non-zero, which is the case at tree-level, then the gauge group associated with Paton-Chan [ 7 ] factors is spontaneously broken.

If interactions are neglected, the theory is incon- sistent due to the negative mass-squared state. It is energetically favorable to pair-produce tachyons. That the tachyon expectation value, (p), is infinite is a manifestation of this inconsistency. The interesting question is whether interactions stabilize this singu- lar behavior.

Since the value of (v) is determined by the tachyon effective potential at zero momentum, an off-shell formalism must be used. Off-shell computations can be performed [ 8,9] via covariant string field theory

0370-2693/M/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

169


[ lo] in the Siegel-Feynman gauge [ 111. The qua- dratic and trilinear terms in the tachyon potential are found from the lagrangian in the Fock-space representation [ 12- 191. The trilinear term stabilizes the vacuum in one direction but further destabilizes it in the other; depending on the sign of the trilinear coupling, (p) = *co. However, these are not the only tree-level contributions. Others arise from the interactions of excited states with the tachyon.

One way of thinking about these interactions is to adopt a functional-integral viewpoint. Integrate over all fields except the tachyon field. The procedure gen- erates a complicated non-local tachyon potential. When Taylor expanded, the potential contains arbitrary powers of the tachyon field, i.e., the higher-mass states generate tachyon n-point functions for all IZ. There are also arbitrary numbers of derivatives act- ing on the tachyon fields. The static potential differs from the effective potential in that the derivative terms are set to zero.

Diagrammatically, these n-point functions have n external tachyon lines and one or more internal lines. The internal lines are necessarily excited states. Tachyons are not allowed in internal lines because such diagrams are generated from the tachyon effective potential. Including them would be over counting.

We restrict ourselves to the classical or tree-level contribution, i.e., those graphs without loops. Fur- thermore, we determine only the four-point contribution. Although our techniques probably can obtain the five- and six-point contributions, more sophisti- cated methods are required to get the full tree-level tachyon static potential. Our truncation is not sys-

tematic; it is shown in section 3 that the n-point function contribution to (p) is of order l/a’g, independent of n.

Let us summarize the results before presenting details. We find that the four-point contribution fails to stabilize the vacuum; see eqs. ( 10) and ( 11). With our truncation, no stable vacuum exists. It is conceiv- able that higher-point contributions rectify this.

Our computation is aided by ref. [ 8 1, in which the off-shell four-tachyon scattering amplitude was determined. Although it might seem straightforward to extract from this the four-point term in the static potential, there is one subtle point: naively, it is infinite.

The four-point contribution is obtained from the

170

amplitude by setting to zero the external momentum and by explicitly subtracting out the term with the tachyon on the internal line. The amplitude is a sum of the six Feynman graphs, shown in figs. la-If of ref. [ 8 1. The value of diagram 1 a is [ 8 ]

I

g2a’ -- - 7 F ~W,P',P2,P3,P4,4) 9 (1) L

1;2

where

I(4 P’> P2,P3, P4, n)

EX ol'(p'-p~+p~.p~)(~_X)a'(pl+p4)2-2

x [$(X)1 cY(ZL, P’.P’)--n

2 (2)

and where the pi are the external momenta. The known function, K(X), specified in ref. [ 8 1, goes to a constant at X= 1. As a result, the integral diverges at p'= 0 due to the behavior of the integrand.

One way to understand this is as follows. The contribution from 1 a at zero momentum is

zla:=Ala(pi=O)= ;$, S

where m, is the mass of the intermediate state s and g, is its coupling to two tachyons at zero momentum. For all states except the tachyon and the massless vector, the terms in the sum in eq. (3) are of the same sign. In fact, the massless vector does not contribute because its coupling to two tachyons vanishes in the absence of Paton-Chan factors. Since the string contains an infinite number of states, the sum in eq. (3) minus the tachyon contribution has an infinite number of terms of the same sign and may diverge. Ap- parently, the couplings g, do not decrease sufficiently rapidly for convergence.

Actually, eq. (1) diverges fors:= (p'+~~)~< l/a’. The same is true for the Veneziano formula [ 201. The full amplitude has poles at s = n/a' for n = 1, - 1, - 3, - 5 etc. #l. In the integral representation, the Vene- ziano amplitude is defined for s < 1 /(.u’ by analytic continuation, The static tachyon potential must be defined similarly. By analytic continuation around the

R’ Individual channel diagrams have poles at even integers as well, and the same is true of the full amplitude if Paton-Chan fac-

tors are present.


tachyon pole at s < 1 /cu’ , the value of the tachyon PO- tential at s=O is obtained. The sign of result is then not known a priori, even though eq. (3) with the tachyon term removed is positive.

We now present the details of the calculation.

2. The calculation

The analytic continuation is performed with standard methods: integrating by parts or performing a Taylor series subtraction on the integrand at x= 1. Using these two methods, we arrive at two formulae:

I

z,a=- ;fo(l)+ j d.x ti(x>-fo(l)l(l-x)-2~ L/2

(4)

and

I,,=-2f,(t)+~f,(l)- j dxfb(x)(l-x)-l, l/2

(5)

where

fo(x)=jcr’g2 [2/K(X)14. (6)

The prime onfo indicates differentiation with respect to x. It can be shown that eqs. (4) and ( 5 ) are equal.

To evaluate these equations, rc must be known as a function of x. It is given as an integral in eq. (3.13) in ref. [ 81 in terms of the variables y and (Y used in ref. [ 2 11. The variable cx is related to x via eq. ( 14) of ref. [ 2 11. The variable y is not independent of (Y but is determined implicitly by eqs. ( 10) and ( 11) of ref. [ 2 11. We have written computer programs to evaluate numerically y, u, and the contributions from eqs. (4) and (5). Figs. 1 and 2 display y and K as functions of x. The variable K is a slowly varying function of x that monotonically goes from ic(;)=,/&oic(1)=8/3fi~l.54 [8].

We find

ZLa= ;a’g’( -2.94kO.02) , (7)

where the kO.02 comes from numerical uncertain- ties in the computer calculations.

We must remove from eq. (7 ) the contribution from the internal tachyon line. From eq. (4.33) of

0.6-

0.4

0.2-

0.04 I . I I

0.5 0.6 0.7 0.8 0.9 ' X

Fig. 1. The variable y versus the Koba-Nielsen variable x.

0

ref. [8], it is found to be -(39/2’2)+cx’g2% -4.81 (dcx’g’). The coefficient 39/2’2 differs from unity, which is the on-shell result obtained from fac- torization because the trilinear tachyon couplings are evaluated at p2=0 instead ofp2= 1 /a’.

In the tachyon static potential, after integrating out the heavy fields, there is a term

(8)

in the functional integral in euclidean space. The static tachyon potential up to the four-point contribution is

Fig. 2. The function K(X).

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If A> 0 the vacuum instability is stabilized, at least within our truncation.

Each graph in fig. 1 of ref. [ 8 ] contributes equally, thereby introducing a factor of six. The final result is

A= -6(4 oCg2 39/2’2+1,,) , (10)

which numerically is

II= fcX’g2;Z= @‘g2(-11.2+0.1). (11)

Since il is negative, the four-point coupling fails to stabilize the vacuum.

3. Discussion

In this paper, we have endeavored to understand the vacuum structure of the open bosonic string theory by calculating the four-point coupling in the static tachyon potential. We did this by analyzing the four- tachyon scattering amplitude for zero external momenta. Since the on-shell condition for tachyons is p2= 1 /a’, an off-shell representation [ 81 of the amplitude was required. Our calculation was straightforward except for the necessity of an analytic continuation in s=(p’+~~)~ and the use of com- puters to evaluate numerically various quantities. Our results are summarized in eqs. (9) and ( 10). The four-point coupling has the wrong sign to stabilize the tachyon field, p.

This result does not mean that a finite value of the expectation value (v) is excluded since there is no systematic approximation scheme by which higher- point couplings can be neglected. This can be seen as

1 ,

follows. Let us analyze the powers of (Y’ and g that enter a tree-level diagram at zero momentum. For each three-point vertex, there is a factor of g. For each propagator, there is a factor of 1 /rn 2 N (Y ’ . A tree-level diagram with IZ external legs has n-2 vertices and n - 3 propagators; hence, it has a factor of gnP2 CY’“-~. The static tachyon potential is necessarily of the form

V,tatic(P) = f cng”-2 Cl!’ n-3 VI” > (12) n=2

where c, are constants independent of g and (Y’ . If we rescale

q=X/@‘g 3 (13)

then a factor of l/g2at3 appears in front of the potential with no further dependence on g or CY’ . For example, under ( 13)) eq. (9) becomes

1 vs/static= - cYt3g2 (

-4x2+ f$$+-$x4+... . >

(14)

It is impossible to choose g or a!’ such that a truncation which keeps terms up to order x4 is a good approximation; all tree-level n-point contributions are equally important. If v, does achieve a finite expectation value, then by eq. ( 13)) it is proportional to 1 /go!’ and hence is non-perturbative in the coupling, g. A complete determination of ( p) would be one of the first non-perturbative calculations in string theory. It is unlikely that such a calculation can be done from a first-quantized formalism.

Although the four-point contribution does not sta-

b

Fig. 3. (a) The tachyon static potential from the tree-level lagrangian to order x3. (b) The tachyon static potential to orderx4.

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bilize the vacuum, it has a dramatic effect on the PO- tential. In fig. 3a, we plot the first two terms in eq. ( 14). As one can see, there is a locally stable minimum at ~=2~/3~/~~0.91, about which one might imagine doing physics. When the four-point contribution is included, fig. 3a becomes fig. 3b and the local minimum disappears. Within our truncation, the criterion that the minimum remains is

1x1 <3”/215= 1.80, (15)

which is not satisfied for the value of 2 in eq. ( 11). Again, we emphasize that, since we do not know the higher-point couplings, we do not know whether local or global minima exist.

Although our investigations have been exploratory in nature, we hope that our techniques will be useful in future calculations. We have demonstrated, in a modest way, that string field theory can yield results not obtainable from a first-quantized approach by finding the tree-level contribution to the four-point term in the tachyon static potential.

Acknowledgement

This research was supported in part by the United States Department of Energy under contracts DE- AC02-84ER40125 and DE-AC02-83ER40107.

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The static tachyon potential in the open bosonic string theory