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Instability of the Râ³xsâ¹ Vacuum in Low-Energy Effective String Theory

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Instability of the Râ³xsâ¹ Vacuum in Low-Energy Effective String Theory 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 .
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