Volume 171, number 4 PHYSICS LETTERS B 1 May 1986

D I L A T O N T A D P O L E S , S T R I N G C O N D E N S A T E S AND S C A L E I N V A R I A N C E

W. F I S C H L E R 1

Theory Group, Department of Physics, University of Texas, Austin, TX 78712, USA

and

L. S U S S K I N D 2

Department of Physics, Stanford University, Stanford, CA 94305, USA

Received 23 January 1986

The divergent dilaton tadpole occurring in certain string theories can be eliminated by introducing a background dilaton condensate such that the strings propagate in a background geometry which to leading order is equivalent to de Sitter space.

Closed strings contain a massless scalar called the dilaton. In the classical or tree approximation the conformal invariance of the two-dimensional world sheet field theory ensures a vanishing vacuum tadpole for the dilaton. However one-loop quantum corrections lead to a non-vanishing tadpole for either the bosonic string .1 or the su- perstring with supersymmetry broken by boundary conditions [2]. The combination of massless propagator and vacuum tadpole lead to the divergence. It is not clear whether a vacuum with N = 1 supersymmetry broken and zero cosmological constant exists for superstrings. But even if the true ground state has no cosmological constant one will have to face the question of tadpoles when studying cosmological questions.

We will show to O(g 2) how string theories cope with the presence of dilaton tadpoles. In pointlike field theories when a massless field ~(x) is absorbed or emitted by the vacuum, a condensate ~0b(X )

develops. In some cases a constant condensate is sufficient to deal with the zero-momentum pole. In other cases a non-homogeneous background has to be chosen. For example, the lagrangian for a complex scalar field

= IO~pl2(1 -g21~12) +J*~p +J~o*. (1)

Consider the correction to O(g 2) to the free part of this lagrangian:

g21D~121Ji2(1/k2)22 =0 ' (2)

arising from fig. 1. Shifting ¢(x) by a constant does not resolve this singularity, instead ~Ob(X ) = Jx2/2 which is a solution of ID~b = J does. Indeed to O(g 0) shifting ~o(x) ~ ~(x) + ~b(X) shifts the quadratic part of (1)

1Work supported by Robert A. Welch Foundation and NSF Grant 8304629. 2 Work supported by NSF Grant PHY-78-26847. * l See for example the review articles in ref. [1 ].

g2 ( ~ ~o)* Fig. 1. O(g 2 ) correction to the free lagrangian la#~ol 2 .

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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* -I- * lau¢l 2 -+ au~b(X)0u~ (x) 0u~b(X)0u~(x )

This leads to additional contributions of O(g 2) to the free part of (1):

[] * _ 12)(1/k2)2~_ 0 --g21Ou'pl2(J*[]~b + J Cb 1[3¢b - "

Therefore choosing [ ] t p b = J cancels the singular terms. A similar situation occurs in string theory, the dilaton tadpole cannot be cancelled by a constant dilaton back-

ground since this only redefines the string tension. Differences with the previous example are that the tadpole ap- pears at one loop in the string theory (conformal invariance guarantees vanishing tadpoles at the tree level) and there also arises additional singularities as the string functional is shifted by a non-trivial background which will be explained later.

Since the complete string action is only known in the light-cone gauge [3,4], we will work in that gauge. The action is

s= f dX+dX - f dx(o)a+~[x(o),x-,x+la_~[x(o),x-,x +] _ f dX+it(X(o),a/~X(o),X-) (3)

where H, the hamiltonian, is

H = H 2 + H 3 ,

where the free part is

1-12 = ~1 f dX(o)dX-i(a/aX-)~[X(o),X-] do[2~r82/SX2(a)+(1/87r)X'2(o)l~[X(G),X-I , (4)

and the interacting part is

-- ~ fdxl( ) dX2(o) dX~(o) dXfdXfdX2e[X~(o), Xf] e[X2(o), Xf]~,[X3(o), X~]

x ~ [ x 1 (o) - x 3 O)] 8 IX2 (o) - x 3 (o)1 ~ [ x f - x~- ] 6 [x~- - x ~ ] , (5)

where the 6 functions are non-vanishing when string 1 and part of string 3, string 2 and the remaining piece of string 3, exactly overlap.

The problem with the dilaton tadpole first shows up at one loop. We therefore consider the correction to the string propagator at that order. The diagram from the light-cone perturbation theory is shown in fig. 2. Recall that in light-cone quantization, the longitudinal momentum a is uniformly distributed along the string. Thus the relative "size" of intermediate strings 1 and 2 indicates how the total longitudinal momentum P+ is shared. In the sum over all intermediate states, the terms which are responsible for the tadpole are those where one of the strings (say #1) is small (t~ 1 ~ 0). These intermediate states are short lived because of large energy denominators. The string diagram looks then like a tube with a very tiny handle on it! These configurations are responsible for the dilaton tadpole since the average transverse distance from the center of mass of the small string to the Center of mass of the string from which it split off: ( [ f (da/2n)Xi (o) - X1M] 2)1/2 is logarithmically divergent. The diver-

To Fig. 2. O(g 2) correction to the sizing propagator.

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gence takes the form f dales, this has been interpreted as the dilaton propagator at zero momentum since 1/k 2 = f l da o~k 2 -1.

This tadpole singularity can be cancelled by shifting the string functional • [X(o), X - , r] by a condensate ~c [X(o), X - , r] which as we will show has to this order only the dilaton mode excited.

,I, c IX(o), x - , ~] = ~(XcM, x - , OCaU~to. [x(o)] ,

where

Cailaton[Xl] = [a-1 ~-1 / ( d - 2) 1/2 ] Co [Xt] ,

with C0 the tachyon wave function and a_ n , ~ -n creation operators for right- and left-moving modes. Shifting the quadratic part of the action S equation (3) by the condensate leads to a term in S linear in the fluctuating part of ~, we will call this term f dX+H1, where H 1 = f d X - dX(o)[cbK~i' e + qbeK~ ] with K the kinetic operator. In this expression • has been redefined such that it now describes the fluctuating part of the string functional. We now consider a correction to the string propagator where H 3 emits a string which is subsequently absorbed by HI , see fig. 3. We may require that this tree process gives a divergence of the same form as the closed-loop dilaton tad- pole. This implies a functional equation for the string condensate which at this order reduces to a differential equation for the dilaton field, very much like the pointlike field theory example given in the early part of the pa- per.

- O2/~X2M] ~dtlaton(XcM, X - , r) = gJp/(d - 2) I/2 (6)

where gJ is the one-loop correction to the string tension [5] and p will be fixed by requiring aU the divergences to cancel.

So

~p dilaton(XCM, X - , r) = [gJp/(d - 2) 1/2 ] ( r X - - X 2 ) / 2 d ,

where d is the number of space-time dimensions. We can in fact choose ~Odilato n such that the one-loop tadpole and tree dilaton process cancel each other (p = -1 ) , but this would leave uncancelled additional divergences, the origin of which will be explained. It is clear that the string condensate changes the hamiltonian: the quadratic part H 2 gets an O(g 2) modification whereas the source term produced is of 0(g 3) and therefore has been ne- glected in H 1 since we only work to O(g2).

~t h -- I-I f d X r dXr(O) ~<Xl(o) - X3<o)) ~(X2(o) - X3(o)) ~(Xi- - X ; ) ~(X~ - X~-) g

x [gJ/(d 2 ) l / 2 2 d ] ( r X { 2 X { ] ~ [ X 2 ( o ) , X 2 ] ~ [ X a ( o ) , X ; ] -- -- X 1 cM)Cdilaton [X 1 (o),

where r = 1,2, 3. Going to longitudinal Fourier components

6H 2 = f d o t r dXr(o ) 5(X 1 (o) - X3(o)) 5(X2(o ) - Xs(o))(gJ/2d)[76 '(or 1 ) ~(Pl CM)

+ (a2/ae12 cM) ~(Pl cM)'~(al)] 8(% + ~2 + °tS)Cdilaton [X1 (°)' °tl] ~ [X2(°)' °t2]cblX3(°), °t3] •

Let us now evaluate this change and determine its effect in terms of the two-dimensional field theory. The three strings-vertex can be represented by a state vector IV> (see e.g, reL [3 ] )

I H I

H 3 Fig. 3. Dilaton absorbed by the condensate.

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I V ) = g e x p ( 1 ~ Nrs ( , r at s + ~ r m ~ t S n ) + ~ N r (ot r +~rm)" ~ ) e x p ( ~r [(ff2 - 2)/otr]ro ) [O) m,n ran- - m -n - - m m - m -

(7)

where r, s = 1,2, 3, m = 1,2 ..... oo. r0 = Y'r o~ ln lo~l, a r is the longitudinal momentum of strings, Pr is the tran- verse momentum (CM) of string r,

= °t2P1 - a l P 2 ' [aim ' Ct]nS] = m~rS~i]~ m+n,O'

N m = _[mnotl Ot2ot3/(mot s + nOtr )]Nrm Nsn ' Nrm = (1/°tr) fm [-(°tr + 1)/°tr] exp(rnro/Otr) '

fm (7) = (1/m !)r(m-f)/r(m7 + 1 - m) . (8)

The overlap of the state IV) with the state associated with the string condensate (which we will take to be string number 1)

(1 IV) = (01ot~ ~] [i~'8 '(or I )~(P1 ) + (O2/ap2)5(P1)5 (°tl)] IV)g2jP/2d(d - 2). (9)

This implies a correction to the quadratic part of the hamiltonian operator. For the first term on the LHS of (9) we get up to a constant factor g2jp/2d(d - 2),

i(01~ ~ [rS'(~l)8(P1 CM)] IV)absorption of condensate + emission of condensate term

= - 2 i r e x p ( m~ ( - )m+l (ot2_mat3_m + ~ 2 ~ 3 _ m ) ) ( m ~ (at2_m + ( - ) m + l a t 3 m ) + ~/Otl)

X ( m ~ (~2_m+ (_)m+l ~3 m ) + ~/ot 1 )~,(ot2 + ot3)t~ (pCM + pCM)I0) '

where one has to use expressions (8) in the limit

O t l~0 , P1 -+ 0 .

This is easily seen to be *2

g2j f dX- f dX(o)i 2 d ( d - 2) p a X -

This corresponds to a change in the world sheet action:

(1/81r)(aX]Oo) 2 -~ (1/8rr)(1 + [g2JrX-/d(d - 2)] p}(OX]Oo) 2 .

The second term on the LHS of (9) up to a constant factor g2jp/2d(d - 2),

(O[~] ~] (a2 /aP2cM)~(P 1 CM)'~(~tl)[V)

equals

(lO)

*2 For an example of such a derivation see tef. [6] .

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lim { 2 [or2 o~,--0 ~2-12 + ( m ~ -m + (_)m+l.3_m] + ~O/o~1)( m~ [~2rn+(_ .--.~rn+l~3_rn ] + '~/0~I)

X [ 2 ( d - 2 ) [ l n ( t ~ l ) - l ] +( = {(ot2_m+~t2_m)/m+ [(--)m+l/m](~t3_m+~3m ))

× exp (m2 m m3 m + 10). rn - - - m - m

This can be recognized, when emission and absorption contributions are taken into account as

l f d x _ f d X ( o ) i a +[x(o),x-I lim c~i ~ 0 aX-

X (fdo{(2/ot2)exp[ikX(o)] k=O + [ - X 2 ( u ) + 2 ( d - 2 ) ( l n l ~ l l - 1 ) ] [(1/4rr)X'2(o) l ) )¢[X(o) ,X-] .

This leads to an additional change to (10) in the world sheet action.

(1 /8ro(ax /ao) 2 ~ (1/8~r)(ax/ao) 2

+ [g2jp/2d(d - 2)] (2/~z 2 + [(d - 2)/4rt] [2 lnlOtll - 2 - X2(o)/(d - 2)] (ax/ao) 2) c~1--'0 (11)

The 1/al 2 quadratically divergent term in (11) is a renormalization of the world sheet cosmological constant. This can be subtracted out since as will be clear we are working at a conformaUy invariant point, it is therefore techni- cally natural to do so.

The logarithmic divergent term in (11) is a correction to the string tension. An additional divergent contribu- tion to the string tension arises from the non-linear o-modal term x2(ax/ao) 2 .

Indeed the term _pg2 [jX2/2d(d _ 2)] OX/ao)2/Ir leads to a correction of the form OX/ao) 2 from fig. 4. This correction equals

-(pg2J/d)( II1r) log IKI (ax/ao) 2 , (12)

where IKI is the world sheet cut-off. If we identify IKI = 1/cq, then this divergent term has the same value as the previous correction to the string

tension. This identification is natural since ,v 1 represents small distances in the parameter space o and is allowed to tend to zero. The one-loop correction to the string propagator whose divergence comes from the dilaton tad- pole contributes to the string tension a term

[g2J/21r(d - 2)] log I ~ i I . (13)

The background dilaton field contributes to the string tension

ao ao Fig. 4. O(g ~) renormalization to the string tension.

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[pg2J/2rr(d - 2)] log I¢~11 • (14)

Requiring the divergent terms in (I 1), (12), (13) and (14) to cancel fixes the value of the unknown p,

p =-d /2 (d- 1).

The change to O(g 2) in the coefficient of (0X/8o) 2 in the hamfltonian due to the presence of a non-trivial string background, corresponds to a change in the geometry of space-time. To understand this let us consider the case of a classical string propagating in an arbitrary curved space-time. The lagrangian density

~? = v~gC~OGuv(x)OaXuO#X v .

The conjugate momentum to XU, Pu'

Pu = 2 v 'g gOOXvGuv (x) + 2 v'-g g O 1 X " G u , (x) "

The hamiltonian density,

egg = p u ~ u _ .~ , c~ = G u v p u p v / 4 1 x / ~ a l g O 0 + GuvX,UX,V lX/~t/gO0 _ (gOX /gOO)[GuvPuX'V ] .

The equations of motion are then

~PuPU + G u v ( X ) X ' u X ' V = O, Guv(X)PUX'V = O .

One can choose light-cone like coordinates X + = Fz where F is a constant. One can also choose the longitudinal momentum P_ to be uniformally distributed. Indeed one can always take as part of the #dimensional gauge con- dition G_ i = 0, then

P : 2 Ivq-~olg 00 G + _ ( X ) F .

Therefore provided gat3 is chosen such that g00 x / ~ a a l G + _ = constant, the longitudinal momentum will be uni- formally distributed.

We are now ready to read off from the quantum corrected hamiltonian the geometry of space-time. Since the hamiltonian

U = P+ = - (GiJPiPj + Gi jX ' iX 'J) /2(G + - P _ + G i*p i ) .

We then recognize G i+ = 0

Gij = - 8 i j [ 1 - g 2 J X 2 / ( d - 1 ) ( d - 2)] , G+_ = {[1 - g 2 J X 2 / ( d - 1 ) ( d - 2 ) ] .

Note that the d-dimensional coordinates are conformally flat and that one cannot choose the two-dimensional conformal coordinates if one requires P_ to be distributed uniformally.

The scalar curvature R = 2 g 2 j d / ( d - 2), the background is then a de Sitter ( J > 0) or anti-de Sitter ( J < 0) space.

The cosmological constant can be determined using Einstein's equation:

1 R v v - $gvv R = g2 Agvv ,

where A is the cosmological constant, then A = J in agreement with Polchinski [5] and Rohm [2]. As we have shown explicitly in the case of the 26-dimensional bosonic string, the string to O(g 2) propagates in

a de Sitter space, without violating scale invariance. It is then fascinating to notice that any tiny bit of string has aU the information on the behaviour of space-time at large distances. We also learn from the case treated in this paper that a non-trivial string background which arose dynamically, determines the geometry of space-time.

Two questions immediately come to mind. Does this pattern persist to higher order? Does this phenomenon apply to supersymmetric theories when supersymmetry is broken? We do not know the answer to either question. We believe that in at least one case our method does apply to broken supersymmetric theories and that is the case studied by Rohm [2] in which supersymmetry is broken by boundary conditions.

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References

[1] S. Mandelstam, Phys. Rep. 13 (1974) 260; J.H. Schwartz, Phys. Rep. 89 (1982) 223, and references therein.

[2] R. Robin, Nucl. Phys. B237 (1984) 553. [3] S. Mandelstam, NucL Phys. B64 (1973) 205; B69 (1974) 77; B83 (1974) 181. [4] E. Cremmer and J.L. C, ervais, NucL Phys. B76 (1974) 209; Bg0 (1975) 410;

M. Kaku and K, Kikkawa, Phys. Rev. D10 (1975) I 110. [5 ] J. Polchinski, University of Texas preprint UTTG-13-85. [6] M.B. Green and J.H. Schwartz, NucL Phys. B218 (1983) 43.

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