Stability of Global Monopoles Revisited

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  • VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000Stability of Global Monopoles Revisited

    Ana Achcarro1,2 and Jon Urrestilla11Department of Theoretical Physics, UPV-EHU, Bilbao, Spain

    2Institute for Theoretical Physics, University of Groningen, The Netherlands(Received 14 March 2000; revised manuscript received 24 July 2000)

    We analyze the stability of global O3 monopoles in the infinite cutoff (or scalar mass) limit. We ob-tain the perturbation equations and prove that the spherically symmetric solution is classically stable (orneutrally stable) to axially symmetric, square integrable, or power-law decay perturbations. Moreover,we show that, in spite of the existence of a conserved topological charge, the energy barrier between themonopole and the vacuum is finite even in the limit where the cutoff is taken to infinity. This feature isspecific of global monopoles and independent of the details of the scalar potential.

    PACS numbers: 11.27.+d, 11.10.Lm, 14.80.HvIntroduction. Global monopoles have been investi-gated for years as possible seeds for structure formation inthe Universe [1,2]. Although they appear to be ruled out bythe latest cosmological data [3], their appearance in con-densed matter and other systems and their peculiarproperties make them worthy of investigation. These ob-jects have divergent energy, due to the slow falloff of angu-lar gradients in the fields, which has to be cut off at acertain distance R (in practice, the distance to the nearestmonopole or antimonopole) and has two important conse-quences, in particular for cosmology. First, the evolution ofa network of global monopoles is very different from thatof gauged monopoles, as long-range interactions enhanceannihilation to the extent of eliminating the overabun-dance problem altogether [1]. Second, their gravitationalproperties include a deficit solid angle [4], which makesthem rather exotic.

    The stability of global O3 monopoles has been the sub-ject of some debate in the literature [57]. In this paperwe try to settle the issue by (a) analyzing the axial per-turbation equations in the limit where the cutoff is takento infinity, and (b) proving that the energy barrier betweenthe monopole and the vacuum (meaning the extra energyrequired by the monopole to reach an unstable configura-tion that decays to the vacuum) is finite. It is somewhatsurprising for different topological sectors to be separatedby finite energy barriers, but in this case it is a consequenceof the scale invariance of gradient energy on two dimen-sional surfaces (r const), and therefore independent ofthe details of the scalar potential.

    The model.We consider the simplest model that givesrise to global monopoles, the O3 model with Lagrangian

    L 12mF

    amFa 214ljFj2 2 h22, a 1, 2, 3 .


    Fa is a scalar triplet, jFj pFaFa, and m 0, 1, 2, 3.The O3 symmetry is spontaneously broken to O2,leading to two Goldstone bosons and one scalar excita-tion with mass ms

    p2lh. The set of ground states is0031-90070085(15)3091(4)$15.00the two-sphere jFj h and, since p2S2 Z, thereare field configurations with nontrivial topological charge.One such configuration with unit winding is the sphericallysymmetric monopole,

    Fa hfrxa

    r, (2)

    where f0 0 and fr ! ` 1. Its asymptotic be-havior is fr ! 0 ar, a 0.5, and fr ! ` 1 2 1r2 2 32r4, as can be seen from the equation ofmotion of fr,

    f 00 12rf 0 2


    f 2 f f2 2 1 0 . (3)

    The two parameters (h,l) appearing in the Lagrangiancan be absorbed by the rescaling Fa ! Fa Fah,xm ! xm plh2 xm, which amounts to choosing h asthe unit of energy and the inverse scalar mass as the unitof length (up to a numerical factor). Note, however, thatthe energy of a configuration with nontrivial winding suchas (2) is (linearly) divergent with radius, due to the slowfalloff of angular gradients, and has to be cut off at r R,say. Unlike h and l, the (rescaled) cutoff is an importantparameter which could affect the dynamics of solutionswith nontrivial topology. Dropping tildes,

    E Z R0


    amFa 114

    jFj2 2 12. (4)

    Since the energy diverges, Derricks theorem does not ap-ply in this case, and in [7] it was shown that the globalmonopole is stable towards radial rescalings. On the otherhand, the question of stability with respect to angular per-turbations has led to some discussion in the literature afterGoldhaber [5] pointed out that the ansatz

    F1 Fr, u sinur , u cosw ,

    F2 Fr, u sinur , u sinw ,

    F3 Fr, u cosur , u ,


    which describes axially symmetric deformations of thespherical monopole (2), leads to the following expression 2000 The American Physical Society 3091

  • VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000for the energy after a change of variables y ln tanu2:

    E Z 2p0

    dwZ `2`

    dyZ R0


    r1 1 r2 sech2 yr2 ,

    r1 F2y 1 F

    2sin2u 1 uy2 ,

    r2 F2r 1 F

    2ur2 112

    F2 2 12,


    with Fr rF, etc. The term in brackets in r1 is identicalto the energy of the sine-Gordon soliton, so translationalinvariance in y implies that configurations with

    Fr , y fr, tanur , y2 ey1j , j const(7)

    have the same energy as (2) (which corresponds to j 0).On a given r const shell, the effect of taking j ! ` isto concentrate the angular gradients in an arbitrarily smallregion around the North Pole. When the gradient energyis inside a region of size comparable to the inverse scalarmass, it is energetically favorable to undo the knot byreducing the modulus of the scalar field to zero and climb-ing over the top of the Mexican hat potential. Unwind-ing is estimated to occur at a critical value of j (say j0)whose dependence with r far from the core is logarith-mic, j0 lnr 1 const. Numerical simulation on individ-ual shells closer to the core gives j0 a0 1 b0 lnr withslowly varying b0 1 and a0 21.3. Unwinding is ex-pected if jr . j0r.

    Estimation of the energy barrier.As explained in[6,7], the shift (7) creates a tension pulling the monopolecore, and the apparent unwinding (which starts in the innershells) is only a manifestation of the cores translation.In order to stop the motion of the core, we consider ahybrid configuration such that the monopole core remainsunperturbed and the unwinding occurs in the outer shells.This is achieved by taking j 0 for, say, r , r1, astringlike configuration for r . r2, and some continuous3092interpolation in between. One such configuration [seeFig. 1a] would be

    Fr , y fr, jr