Missouri University of Science and Technology Scholars' Mine

Physics Faculty Research & Creative Works Physics

01 Aug 1995

Instability of the R³xS¹ Vacuum in Low-Energy Effective String Theory

Mariano Cadoni

Marco Cavaglia Missouri University of Science and Technology, [email protected]

Follow this and additional works at: https://scholarsmine.mst.edu/phys_facwork

Part of the Physics Commons

Recommended Citation M. Cadoni and M. Cavaglia, "Instability of the R³xS¹ Vacuum in Low-Energy Effective String Theory," Physical Review D, vol. 52, no. 4, pp. 2583-2586, American Physical Society (APS), Aug 1995. The definitive version is available at https://doi.org/10.1103/PhysRevD.52.2583

This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Physics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. PHYSICAL REVIEW D VOLUME 52, NUMBER 4 15 AUGUST 1995

instabi]ity of the R3 x Sx vacuum in low-energy effective string theory

Mariano Cadoni Dipartimento di Scienze Fisiche, Universita di Cagliari, Cagliari, Italy and INFN, Sezione di Cagliari, Via Ada ¹gri 1'8, I-09187 Cagliari, Italy

Marco Cavagliat Sissa Inter-national School for Advanced Studies, Via Beirut 2-g, I 8401-8 Trieste, Italy (Received 14 February 1995) We present and discuss a Euclidean solution of the low-energy efFective string action that can be interpreted as a semiclassical decay process of the ground state of the theory.

PACS number(s): 11.25.Mj, 04.20.Jb

In Ref. [1] the authors found an instanton solution of Action (1) follows froxn the modulus-dependent low- a four-dimensional, modulus Geld-dependent, low-energy energy effective string theory considered in [5] once one effective string theory. That solution describes either a eliminates the modulus &om the action by choosing an wormhole connecting two asymptotically fIat regions or appropriate ansatz consistent with the Geld equations the nucleation of a baby universe starting Rom an origi- [6,1]. The action describes a Jordan-Brans-Dicke the- nal flat region. Our aim here is to show how this instan- ory coupled to the electromagnetic field and reduces to ton can also describe a different physical process taking well-known theories according to the value of k [6—9]. place in the theory. Indeed, using a different analyti- The meaning of the parameter e needs some further cal continuation to the hyperbolic space, the solution of explanation. As shown in [1], in order to write the con- Ref. [1] can be interpreted as a semiclassical decay pro- tribution of the EM field to the Lagrangian in a space cess of the ground state (vacuum) of the theory. The with a signature (+, +, +, +), we have to choose the sign existence of a process of semiclassical decay is important of the term E according to the electric or magnetic con- since it may lead to the instability of the vacuum of the Gguration of the Geld. Indeed, the EM Geld in Euclidean theory. Furthermore, a careful analysis of the geometric space is not analytically related to the EM Geld in hy- and topological features of the instanton will enable us to perbolic space by the simple transformation t ~ i7, but identify the wormhole solution of Ref. [1] as a Hawking- in general we have type wormhole [2] connecting two asymptotic regions of 2 = 2 2 = —~ 2 B x S topology. hyp EucI & hyp Eucl (2) In this paper we will follow an approach similar to the one used by Witten in Ref. [3] to prove the semiclassi- Since we wish to deal with real analytical continuations cal instability of the Kaluza-Klein vacuum in five dimen- of (real) hyperbolic fields in Euclidean space, we allow sions. Even though the theory considered here has little for a different sign in &ont of the E term in the action, according to the configuration of the EM Geld. We will to do with the Kaluza-Klein theory in five dimensions, — both instantons have common geometrical and topologi- choose e = 1 for a purely magnetic configuration and cal features and consequently most of the mathematical c = 1 for a purely electric one. Now us techniques used in [3] can also be implemented in our let consider a four-dimensional Riemannian case. manifold described by a line element of the form Our starting point is the Euclidean action [(16' G) ds = A EI r2d022, Mp2x/16~ = 1] (r)dt + (r)dy2+ where is the coordinate of the one-sphere, 0 & 8k k y y & 2', — ~ — V' 2 3+ — S~ — d x ge R+ — +e — E and dO& —d0 + sin Ody represents the line element 0 1 k 1 k of the two-sphere S . Choosing for the EM field the magnetic monopole S2 = —2 d xv he ~(K —Ko), configuration on (and thus e 2A —1), where B is the curvature scalar, P is the dilaton field, F = Q sin Od0h dp, I"~„is the usual electroxnagnetic (EM) field tensor, and k is a coupling constant, —1 & k & 1. The boundary term is required by unitarity (see, e.g. , [4]). e' = kl is a parameter whose meaning will be clear in a moment. Also duality invariance arguments support this prescription (see [1] for details). These arguments are similar to those used in Ref. [10] for the case of the axion. The key point is that 'Electronic address: CADONICA. INFN. IT F ~ *E and the continuation to Euclidean space do not t Electronic address: CAVAGLIATSMI19. SISSA.IT commute.

0556-2821/95/52(4)/2583(4)/$06. 00 52 2583 1995 The American Physical Society 2584 BRIEF REPORTS 52 the solution of the field equations derived from (1) is Equation (11) represents the maximal extension of (5). As before, when x, t oc the manifold is asymptoti- —1 ~ cally Bat with topology B x S . The critical surfaces d — d2 =/ / "') are two: x + t = Q and x + t = 0. Using the coor- dinate transformation, it is easy to verify that the first +Q' — 1 — one corresponds to r = Q. The second critical surface ~ d~'+ I l+ I I . "df)' (5) ") & "') corresponds to r = oo. Hence Eq. (11) describes two asymptotically Bat regions smoothly joined through the surface = This structure is related the (A: —i)/2 r Q. strange to — ( g e2(4 4o) (6) existence of a conformal equivalence between the region inside x2 + t2 = Q2 and the region outside. In fact, the Euclidean line element is invariant under the trans- has been redefined (ll) where the magzietic charge Q formation through = — y" m 0" y", 2, (i4) Q 2/l kQ. y2 p=1, 3, — with The crucial point for the identification of (4) (6) where y" are Cartesian coordinates of three-dimensional a vacuum decay process is the analytical continuation space (t, x, &p), y = t, y = x cosy, y3 = x sing, and 0& line element to Therefore, let of the hyperbolic space. is a 3 x 3 rotation matrix. Hence solution (11) repre- us discuss the geometric and topological properties of sents a Hawking-type wormhole [2] with a minimum ra- the Euclidean manifold described by Since the lat- (5). dius equal to Q connecting two asymptotically flat spaces ter has definition signature r can take by (+, +, +, +), with topology B3 x Si. Note that (14) is an invariance of values only in the range oo[. For r -+ oo the [Q, the entire solution (11)—(13), not only of the metric (11). with x . For space is asymptotically Bat topology B S Indeed, also the expression (12), (13) for the dilaton and = the metric tensor is singular. However, in r = r Q Q EM field do not change under the transformation (14). the manifold is smooth, as can be shown by putting r — How can we recover the vacuum decay interpretation? gQ + & (r C] oo, oo[) and defining y as a peri- In order to answer this question, we have to back to 2i " go odic variable with period 2vr x [1]. This conclusion (5) and continue analytically the Euclidean solution to a coordinate seems to indicate that the system (r, y, 8, p) hyperbolic spacetime. In Ref. [1] the analytical contin- does not cover the whole manifold. In order to obtain the uation was performed first by defining r = gr2 —Q2, maximal extension of the Euclidean metric we have (5), thereafter by the complexification of 7 7 —+ i7. The to perform an appropriate coordinate transformation: resulting hyperbolic manifold was interpreted as a baby (x2+ t3) + Q2 x universe of spatial topology S x S nucleated at 'T = 0. r tane= —. However, the is not continuation 2/x'+ t' (8) latter the only analytic =, we can perform. For instance, we can complexify the 6t = The inverse of (8) is coordinate of the two-sphere S . In this case, since 0 0 is a coordinate singularity of the metric, it is convenient x = f(r)sin8, t = f(r)cos8, to choose as a symmetry plane the surface 8 = n/2 and to put where jr 8-+ —+i(. (i5) f(r) = gx2+ t2 = Q exp[arccosh(r/Q)] . (io) 2 The coordinate transformation (9) is never singular. Us- After the replacement (15) we obtain the hyperbolic so- ing (8), the Euclidean solution (4)—(6) becomes lution

1 ( Q2) ds = — 1+ [dt +dx +x dp 4 f2) ] —r d( + r cosh (d&p, (16) 2Q' f' + Q' f' + Q' ("—i)/2 ) ( ) e2(4 —4o) )

— The EM two-form is now — g ) (k i)/2 ,2(~ So) (12) f'+ Q') I" = Q cosh(d( h dy .

The EM field is real, as a result of the choice r = —1 in 1 x & I' = —v 1 —kQ —[x dt A dp —t dx A dp] . (13) the action (1). For r Q this spacetime is nonsingular, 2 f3 the coordinate singularity at r = Q being as harmless 52 BRIEF REPORTS 2585 as it was for the Euclidean space (5). The solution (16) are analogous to the Euclidean ones, and their conformal for r ) Q represents the spacetime in which the Bs x S equivalence can be proved using a coordinate transfor- vacuum decays. The topology of the initial = 0 surface mation similar to (14). 2 1 ( is B x S . Note that the analytic continuation to the Region II is the starting point for the vacuum decay hyperbolic space of Ref. [1], even though it was obtained interpretation of the Euclidean instanton. As one can from the Euclidean instanton (5), has instead a spatial easily verify, the origin of the Euclidean plane (x, t), coin- topology S2 x Si. ciding with an asymptotically flat infinity, is not the only The topology of the analytic continuation to hyper- surface we can use to perform the analytic continuation bolic space depends thus on the coordinate chosen to in hyperbolic space. At t = 0 we can join the Euclidean complexify. A better understanding of the features of manifold described by (11) with a hyperbolic spacetime, this space can be achieved starting from a hyperbolic namely, the region x —t2 ) Q of the spacetime (20) line element that covers only the region r ) Q. Using (region II in Fig. 1). Indeed, at t = 0 the metric, dila- the coordinate transformation ton Geld, and EM Geld assume a minimal configuration, and so the extrinsic curvature vanishes and the joining is x = f(r)cosh(, t = f(r)sinh(, possible. The hyperbolic spacetime in which the vacuum — decays is region II in Fig. 1. I et us explore in detail its where f(r) = gx2 t2 is defined as a function of r as in properties. Because of the maximal analytic extension, Eq. (10), we obtain the regions on the left and right of the plane (x, t) are identical, and so we will focus our attention on one of I( them. Choosing for simplicity y = const, the line ele- ds2= — 1+ —dt ~dx +x dy ~') [ ] ment (20) becomes conformally equivalent to a R Hat Minkowskian spacetime. Of course, the manifold is not geodesically complete, since there exist geodesics crossing — f2+Q2 f2 Q2 the boundary x t = Q . The meaning of the boundary ) can be understood following its time evolution. Starting (20) at t = 0, as t becomes larger and larger, the coordinate x of the boundary grows according to x = QQ2 + t2. Since the coordinate x corresponds to a radius in the cylindri- ~&, + f2 + Q2) (21) cal system of coordinates (t, x, p), the boundary can be interpreted as a hole in space starting with radius Q at t = 0 and growing up for t ) 0. At t = 0 the EM field is 1 x a purely electric field in the direction, = /x; as E = —gl —kQ —[x dt A de —t dx A d . (22) p E~ Q p) the time t flows and E~ changes in intensity, the latter — generates a magnetic Geld in the perpendicular y direc- Since 1 & t/x ( 1, the new coordinates (x, t) do not tion. Finally, when x, t —+ oo, the EM field vanishes, cover the whole plane. They cover only the region out- as expected because the spacetime is asymptotically flat. side the light cone, x = +t, corresponding to the physical The Euclidean line element (11) represents thus the de- region. As for the Euclidean case, the critical surfaces cay process of the flat spacetime of topology B x S in x22 — 2 — 2 are two: t = Q, corresponding to r = Q, and a spacetime with a growing hole. 2 —2— x t = 0, representing infinity (see Fig. 1). Of course, In conclusion, the Euclidean instanton we are dealing the manifold described by (20) is geodesically complete with represents either a wormhole or a vacuum decay pro- and its topology is B x S . Regions I and II in Fig. 1 cess according to the null-extrinsic curvature surface used for the analytic continuation to hyperbolic spacetime. The previous results can be straightforwardly extended to the purely electric EM Geld configuration. Choosing s = 1 in the action (1) and using an electric field along the y direction, we obtain a line element which divers from the previous one for the purely magnetic case only through the conformal factor e ~'(1 —Q/r) " and so all conclusions remain unchanged. At this stage we can ask ourselves if the semiclassical vacuum decay process is consistent with energy conser- vation. Since the R x S vacuum has zero energy, the space (16) in which it decays must also have zero en- ergy. Using the Arnowitt-Reser-Misner (ADM) formula generalized to dilaton-gravity theories, the total energy of (16)—(18) can be calculated as usual by means of a FIG. 1. Two-dimensional section of hyperbolic space de- surface integral depending on the asymptotic behavior of scribed by the metric (20). The physical region corresponds the gravitational and dilaton fields. The line element (20) to the shaded part (region II) of the picture enclosed by the is not static with respect to t, and so the integral must hyperbola y —t = Q . be evaluated at the initial t = 0 surface, corresponding 2586 BRIEF REPORTS 52 in to = 0. The result of the integration is zero. (16) ( ~ k —1 Indeed, the terms of the gravitational and dilaton fields d3~ g(3) g R( ) + 4 /0 0 2 which contribute to the total energy of the solution are those of order 1/r. However, in our case these terms give a null contribution to the energy, owing to the R2 x S (24) topology of the ( = 0 surface. The space described by (16) has therefore zero energy. This feature makes the where rr = P + g/2, rI = @ + 2$(k + 1)/(k —1), and we Rs x Si vacuum not stable for the theory defined by (1), have dropped the boundary terms. since there exists a solution with zero and the energy A solution of the ensuing equations of motion is same asymptotic behavior as the R x S vacuum. An important consequence of this result is that the positive 1/ q&1' energy theorem [11] does not hold for the theory (1) if ds = — 1+ [dt +dx +x d(p], one considers vacua with topology R x S . The positive energy theorem states that every non8at, asymptotically (25) Minkowskian solution of the Einstein equations has zero f' /' & energy. However, its validity for spaces with arbitrary ( .) + Q', ) f + Q i („„, topology and for theories such as (1) is difficult to prove. f' —Qz (,f —Qy In the case under consideration, the failure of the positive theorem seems related to the presence of the EM where f = gxz + t2 and we have chosen the EM ten- energy I' Geld: In the R x S vacuum there exists excitations of sor as in (13). The solution of the three-dimensional the EM Geld for which the positive energy theorem does theory is thus a Hawking-type wormhole connecting two not hold. asymptotic regions of topology R . The interpretation of the Euclidean solution (5) as Now let us calculate the decay rate of the vacuum. an instability of the vacuum has been established us- Evaluating the action (1) on the Euclidean solution (4)— ing the analytical continuation (15). Considering a sec- (6), we have ond analytical continuation to a hyperbolic spacetime, S5 = 4vr e ~'Q (k+ 1) . (26) we have also seen that the instanton can be interpreted as a Hawking-type wormhole. The latter has an intrin- This result has been obtained by integrating r and 0 in sically three-d. imensional nature because its topology is the range Q ( r & oo, 0 ( 0 ( vr/2, the appropriate one Bs x Si and the radius of S is equal to in the two for the vacuum decay process. The rate of decay of the Q S» asymptotic regioiis f = oo, f = 0 and shrinks to zero for R x vacuum is r = Q. Hence the most natural interpretation of this so- I'vD = exp 47r e —'Q (k+ 1) . (27) lution can be found in the context of a 3+1 Kaluza-Klein The vacuum is long lived for values of much greater theory. Q than the Planck length and becomes unstable when Starting from the action with s = —1, setting to l~ (1) is of the same order of magnitude of Finally, it is zero the components of the EM field along the direc- Q l~. y interesting to compare the vacuum decay rate I'~D with tion, and splitting the four-dimensional line element as the probability for the nucleation of a baby universe I'BU (see Ref. [1]): ds ) =ds()+q e @dy (23) fBU = (lvD) (28) after some manipulations we obtain the three- Hence the probability of nucleation of a baby universe is dimensional action smaller than the probability of the vacuum decay.

[1] M. Cadoni and M. Cavaglia, Phys. Rev. D 50, 6435 (1994). (1994). [7] D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. [2] S. W. Hawking, Phys. Rev. D 37, 904 (1988). Rev. D 43, 3140 (1991). [3] E. Witten, Nucl. Phys. B195, 481 (1982). [8] G. W. Gibbons and K. Maeda, Nucl. Phys. B298, 741 [4] S. W. Hawking, in General Relativity, an Einstein Gen (1988). Senary Survey, edited by S. W. Haw'king and W. Is- [9] M. Cadoni and S. Mignemi, Phys. Rev. D 51, 4319 rael (Cambridge University Press, Cambridge, England, (1995). 1979). [10] S. B. Giddings and A. Strominger, Nucl. Phys. B308, [5] M. Cadoni and S. Mignemi, Phys. Rev. D 48, 5536 890 (1988). (1993). [11] P. Schoen and S. T. Yau, Commun. Math. Phys. 65, 45 [6] M. Cadoni and S. Mignemi, Nucl. Phys. B427, 669 (1979).