Volume 133B, number 3,4 PHYSICS LETTERS 15 December 1983

ZERO COSMOLOGICAL CONSTANT FROM MINIMUM ACTION

Eric BAUM Department o f Mathematics, University o f California, Berkeley, CA 94720, USA

Received 2 August 1983

We impose minimum euclidean action on a scalar field coupled to classical gravity and find zero effective cosmological constant without fine tuning, as well as a new mechanism for symmetry breaking.

We would like to suggest a mechanism by which gravity coupled classically to quantum fields would enforce zero cosmological constant and break internal symmetries. We consider a scalar field with euclidean action:

I =f [ ( - m2/16rr) (R - 2Ao) + ~bV] ~b + V(~b)] x/g-d4x.

(1)

We assume that we wish to minimize this action with the metric treated as a function of the quantum field q~ found by solution o f the Einstein equations. This situation might arise from the path integral:

f (d~b) (dgab) exp [ - I (¢ , gab)], (2) Z

where we first perform the integral over (dgab) and here keep only the classical term. This truncation is justified if we want to consider physics below the Planck mass mp where quantum gravitational effects should be unimportant. Alternatively, if we coupled N scalar fields, graviton loops are second order in the 1IN expansion and gravity may be treated classically [1 ].

We will first search for a vacuum with ~b = 4~ inde- pendent o fx . If we define

Aef f = 10 + (16rr/m 2) V(~), (3)

we must compute the minimum of

f(-m2/l@r) (R - 2 Aeff) V~- d 4x. (4)

Solving the Einstein equations

0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

1 Rab -- "£ gab R + Aeffgab = 0 (5)

by contraction with gab we find

- R + 2Aef f = - 2 A e f f. (6)

Thus

I = -2Ae f f (m2 /16rOfx / rg d4x. (7)

Now for Aef f = const. > 0 the constant curvature solution is de Sitter space S 4 with volume (1/Aeff)2 and action - 2 / A e f f. For Aef f = 0 we have flat R4 but I = 0. For Aef f < 0 we have in general an infinite vol- ume (anti-de Sitter space) and I = +oo. This situation is displayed in fig. 1. Provided that the minimum of Aeff(q5 ) over ~ is less than 0, or that

Aeff

Fig. 1. Action I as a function of Aef f.

185

Volume 133B, number 3,4 PHYSICS LETTERS 15 December 1983

A 0 < - rn~m (161r/m 2) V(~), (8)

the minimum of the action will occur for ~ such that

Aef f = 0..

This means that we need not fine tune. For any pa- rameter value A 0 satisfying eq. (8), the vacuum will be such that the effective cosmological constant is zero. Although this configuration is not a saddle point but rather an essential singularity in exp ( - I ) , one would expect the path integral, eq. (2), to be heavily weight- ed around this point so long as the measure d~b is an analytic function of qS. That is Greens functions or ex- pectation values, say the expectation value o f Aeff, will be computed almost entirely on spaces with Aef f = 0.

This minimum will in general break symmetries of the lagrangian. Even if V has a minimum at ~b = 0, the minimum o f / w i l l occur for ~ 4 :0 [provided A 0 satis- fies eq. (8)].

Actually, if we search for general ¢(x) rather than constant vacuum solutions, the argument is unchanged The Einstein equations become:

(m2 /16rr)(Rab - -1Rgab + A0 g a b ) - - Oa(Pab dp

1 V(dp)gab = O. (9) + ½ gab ~ e ~) ~ e dp + -ff

Contracting we find

(m2p/161r) ( - R + 2A 0) = - ~a ¢ oa~_ 2 V(~) - 2A 0 m2/167r.

(10)

The action, after substituting for R by eq. (10)be- comes

/ [ - V(~b) -- 2A 0 m2/167r] X/ff d4x. (1 I)

If we assume V contains no derivative terms, V will be maximized or minimized for constant q~, so that eq. (1 1) reduces again to eq. (8):

--(m2/167r) Aef f fx ' / f f d4x. (12)

But for compact solutions of the Einstein equations the euclidean action is minimized by the standard metric on S 4 (ref. [2] ) and we find the action is in- deed minimized by the solution for constant q~.

It is not clear to us that a minimum euclidean ac- tion principle such as eq. (2) is the correct way to de- scribe physics in our lorentzian world, particularly when gravitation is considered. Our original motivation for this research was consideration of GUT inflation- ary models [3]. Here one varies the effective potential to find a vacuum without including any energy for the gravitational field, since we know of no useable defini- tion for such. Yet one interprets the vacuum potential as a cosmological constant. This curves space and thus intuitively should increase the energy in the gravita- tional field. This is the old problem of how one deter- mines the quantum vacuum in a gravitational field. The approach of taking a euclidean path integral has led to some successes, notably in reproducing black hole thermodynamics, and also to problems [4]. Our re- marks have demonstrated some new, suggestive results when a euclidean path integral is taken over a quantum scalar field with gravity coupled classically.

I would like to thank Bruce Allen for very useful dis- cussions, Andy Strominger for discussions concerning ref. [1], and Stephen Hawking for pointing out the relevance of ref. [2] and suggesting that these results were independent of the measure d~b. Professor Hawking informed me that he has had and will publish similar ideas.

References

[1] E. Tomboulis, Phys. Lett. 70B (1977) 361; 97B (1980) 77. [2] S.W. Hawking, Nucl. Phys. B144 (1978) 349. [3] A.H. Guth,lPhys. Rev. D22 (1980) 347. [4] S.W. Hawking, Euclidean quantum gravity, in: Recent

developments in gravitation (Carg~se, 1978) eds. M. L6vy and S. Deser (Plenum, New York).

186