Vortex loops: Are they always doomed to die?

PHYSICAL REVIEW D VOLUME 51, NUMBER 6 15 MARCH 1995

Vortex loops: Are they always doomed to die?

Uri Ben-Ya'acov * International Solvay Institutes for Physics and Chemistry, Campus Plaine-CP 231, Universitt. Libre de Bruxelles,

Boulevard du Triomphe, B-1050 Brussels, Belgium (Received 20 September 1994)

The effective equations of motion of relativistic strings in material media are derived and applied to moving rings with a time-dependent radius. The equations contain the Magnus force, due to the motion of the ring relative to the medium, whose eventual effect is the possible stabilization of the ring against shrinking. A constant solution is identified, and small fluctuations around it are bound, demonstrating the stability of the solution. If the string loops created in the cosmological cosmic string scenario interact via this mechanism with a formed-up Higgs particle condensate, then the stabilizing velocities are - 6~,op/R~o,p, and the overall effect of this phenomenon is to stabilize large loops and reduce the general disappearance rate of the string loops.

PACS number(s): 11.27.+d, 47.32.Cc, 67.40.Vs, 98.80.Cq

I. INTRODUCTION

Cosmic string loops, formed as part of the cosmic string scenario in the early Universe, are well known to always shrink and disappear, inevitably [I]. On the other hand, it is well known that vortex rings in classical hydrodynamics [2] and in superfluids [3] can maintain their form by moving with respect to the medium, as the Magnus force, which is due to the relative motion, counterbal- ances the shrinking.

A similar situation can happen for vortices in relativistic superfluids [4,5], and it is extendable to the cosmic string scenario in the early Universe. There, at first, as the Higgs field evolves in a vacuum, no such effect exists. But as time evolves and Higgs particles are created, they may form a condensate which becomes a medium with which the vortices, which appear in the Higgs field, interact via a Magnus force, in exactly the same man- ner as in a superfluid. Then, the interaction with the condensate will affect the time evolution of the strings.

In the following the effective equation of motion of a relativistic string in a material medium is derived from the exact equations of motion for vortices in a Lorentz- invariant U(1)-symmetric field theory [6,7]. These equations are applied to a moving ring with a time-dependent radius. A constant solution, and the Magnus force with its stabilizing effect are identified. Small fluctuations around the constant solutions are bound, demonstrating the stability of the solution. The possible application to the cosmic string scenario is discussed in the conclusions.

11. VORTICES IN A MATERIAL MEDIUM

We consider vortices that appear in Lorentz-invariant U(1)-symmetric field models, with complex field d(x)

'Fax: 32-2-6505767 Electronic address: [email protected]

in a Minkowski space-time with the metric g,, = diag(- l , l , 1, 1), /L = 0,1,2 ,3 . The asymptotic (away from the vortices) field, which also corresponds to the vacuum or ground state of the model, is $(x) = doeiv'", with constant do and V, [6,8]. In the case of a material medium V, is the energy-momentum vector of the medium's particles.

A vortex solution, vanishing on the vortex's world sheet x, = (p(ra), a = 0, 1, and written as +(x) = ld(x) 1 ei+'(") = ~Oei+'(x)-.JI(x), leads, due to field equations, to the equation of motion of any single vortex [6,7]:

where f is the winding direction of the vortex, yab = (,, . [ J , is the induced metric on the vortex's world sheet, and

This equation is exact, since no approximations were made in its derivation, as it makes use of the fields p (x) and $(x) and the gauge field Ap which are determined by field equations from the distributions of other vortices, boundary conditions, and the medium [6,7]. It has the re- markable feature that, although the fields p ( ~ ) and $(x) are, separately, singular on the vortex, their singularities cancel each other, leaving an automatically regularized self-interaction.

For a single vortex whose curvature is not too high, the effect of the medium that enters through p(z), namely, pmedium(%) = V . X, can be separated from the self- interaction, in which the only effect of the $ field is to regularize the vortex's self-interaction which turns out to be proportional to the generalized acceleration 151. The right-hand side (RHS) of Eq. (2.1) (except for the Vp- dependent term) is then shown in Ref. [5] to be approx- imately equal to (~j ln(R, /G) . ["I:, where K is the (in- teger) vortex's winding number (the topological charge),

2918 @ 1995 The American Physical Society

5 1 - VORTEX LOOPS: ARE THEY ALWAYS DOOMED TO DIE? 2919

R, is of the order of the coherence length for the local strings and of the order of the local radius of curvature for global strings, and 6 is of the order of the core radius. Altogether, the effective vortex equation becomes

For local strings the effective string tension p(c) is roughly a constant, while for global strings it depends on the structure of the world sheet.

It is important to emphasize that Eq. (2.2) applies equally well for vortices in locally, or globally, symmetric field models. If the field model is only globally symmetric, the interactions between the vortices are long ranged, and the assumption that the vortex is single is essential. If the field model is locally symmetric the interaction between the vortices is only short ranged, and if the vortices are not too close, the presence of other vortices can be ignored. In any case, the interaction that resides in the RHS of Eq. (2.2) is the interaction of the vortex with the medium, the latter represented by the vector V,, and it is the relativistic version [6,9,10] of the well-known Magnus force in superfluids and classical hydrodynamics.

An immediate result of Eq. (2.2) is obtained by contracting it with V,, which implies, due to the vanishing of the RHS, that V .E is always a harmonic function on the world sheet:

In a trivial medium (i.e., the vacuum for cosmic strings), in which V" = 0, Eq. (2.2) reduces to the well- known equation of a Nambu string. In the present paper we are interested in the motion of vortices which aDDear

A

in a material medium, say, a superfluid or a supercon- ductor. Then V, is a timelike four-vector. which in the medium's rest kame may be written as Vp = (pol O ) , where po is the relativistic chemical potential of the medium [I 11.

The freedom in the parametric description of the man- ifold xJ' = E,(c) implies that it can be represented, in any Lorentz frame, in a parametrization x, = E, = (t, X( t , a)) satisfying X,t . X,, = 0. In this parametrization the metric on the world sheet is

Computing Eq. (2.3) in this parametrization in the medium's rest frame yields

Since B / A is independent of t , there is still a parametrization freedom which allows a to be redefined such that A = B. Then all the parametrization freedom is ex- hausted, and from (2.4) follows the relation

The spatial components of Eq. (2.2) yield the vortex's equation

Contracting with X, t yields

With a similar result for a, it follows that (2.6) is a first integral of Eq. (2.7). It is interesting to note that, as- suming p % const, Eq. (2.7) can be obtained by variation from

The energy-momentum tensor on the ring that corresponds to this action is

its conservation being assured by Eq. (2.6)

111. STABILITY OF MOVING RINGS

The vortex's equation of motion (2.7) is now applied to the study of vortex rings with time-varying radii. In the (t , a) parametrization, the general description of such a ring may be given by

Fixing for convenience sign ( a ) = i and substituting (3.1) in Eq. (2.7) yields [12]

with a first integral from (2.6):

Equations (3.2) admit a constant solution, with constant R = Ro and i = ,B, given by

URI BEN-YA'ACOV - 5 1

This solution has been already obtained by Davis [4]. The rest of our discussion, however, differs from that of Davis, primarily in the dynamics developed in the preceding sec- tion and in their application to the time-dependent ring represented by Eq. (3.1).

If the initial condition differs from (3.4) then Eqs. (3.2) have a stabilizing nature. Assume that a ring is initially formed a t rest with radius Ro. Initially R = -a2R < 0 and the ring begins to shrink, but then, because of R < 0, 2 > 0 and that introduces a positive contribution to R. The eventual result, depending on the initial conditions and the competition between the two terms in Eq. (3.2a), can then be either the ultimate shrinking of the ring, or its halt and then fluctuation of the ring's radius and velocity around some central values. From Eq. (3.3): the halting condition is i2 = 1 - a 2 R 2 . The smaller the ring the harder it is to satisfy the halting conditions. Thus it is easy to stabilize large rings, difficult to halt the shrinking of small ones.

To demonstrate the stabilization of the ring, we use the fact that the string tension p depends (at the most, for global vortices) only logarithmically on the radius, and if the radius does not change drastically and is not too small, we may assume, as a first approximation, that still p M const around some central value. Equation (3.2b) is then easily integrated,

and substitution in (3.3) yields

with

The important feature of the solution is that the radius (3.4) R(t) and the velocity i ( t ) both fluctuate around some

central value, thus proving the stability of the constant solution (3.4).

Equation (3.6) can be solved in terms of elliptic func- tions, with R2 ( t ) confined to the interval

IV. CONCLUSIONS

When vortex loops are formed in various processes (phase transitions, intercommuting of tangled vortices), they come out in different shapes and velocities. The ultimate fate of a ring depends on its initial conditions at formation, and the consequent result of the competition between the shrinking on the one hand and the stabilizing Magnus force on the other hand. Although not all the loops thus formed are prevented from fatal shrinking, the stabilization effect of their interaction with the medium has a slowing effect on the total rate of shrinking and disappearing of loops. Since the Magnus force grows with the radius of the ring, large rings (which are also harder to create) will remain for longer times than the small ones.

In the cosmic string scenario in the early Universe [1,13-151, the Higgs field starts in a vacuum, in which V'" 0 a ~ l d no Magnus force exists. But as time evolves, Higgs particles are created and they may form a condensate, for which V W f 0 and a Magnus force becomes possible. Vortices which appear in this Higgs field nec- essarily interact with the condensate. The stabilization velocity for a loop of size R can be estimated from Eq. (3.4), giving a value v -- l / m s R -- SIR, where m s is the mass of the scalar Higgs particles and S is a measure of the width or core radius. Since loops with such velocities are very likely to be created in the cosmic string scenario, this phenomenon may have a non-ignorable effect. In a locally symmetric theory the creation of a Higgs condensate involves the creation of net electric charge, which is unlikely, unless there is some mechanism that neutralizes the created charges and yet leaves the condensate. In a globally symmetric theory such difficulties do not exist, and the formation of a condensate is more likely. A more detailed answer requires the study of the possible formation of the massive Higgs condensate and its evolution with the expansion of the Universe.

ACKNOWLEDGMENTS

The author thanks T. Vachaspati, T . Kibble, and M. Hindmarsh for useful discussions, and the Isaac Newton Institute for Mathematical Sciences for the hospitality

and i ( t ) confined by during the NATO AS1 on "Formation and interactions of topological defects,', where this work was completed. This research was supported by the Belgian Government

(3.8) under the "Poles D'Attraction Interuniversitaires" pro- gram.

5 1 - VORTEX LOOPS: ARE THEY ALWAYS DOOMED TO DIE? 292 1

[I] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 1994).

[2] H. Lamb, Hydrodynamics (Dover, New York, 1945). [3] W. I. Glaberson and R. J. Donnelly, in Progress in Low

Temperature Physics, edited by D. F. Brewer (North- Holland, Amsterdam, 1985), Vol. 9, p. 1.

[4] R. L. Davis, Phys. Rev. D 40, 4033 (1989). [5] U. Ben-Ya'acov, Nucl. Phys. B382, 616 (1992). [6] U. Ben-Ya'acov, Nucl. Phys. B382 , 597 (1992). [7] U. Ben-Ya'acov, Phys. Lett. B 274, 352 (1992). [8] U. Ben-Ya'acov, Phys. Rev. D 4 4 , 2452 (1991). [9] R. L. Davis and E. P. S. Shellard, Phys. Rev. Lett. 63 ,

2021 (1989). 1101 U. Ben-Ya'acov, J. Phys. A 27, 7165 (1994). [Ill E. I. Guendelman, Mod. Phys. Lett. A 4 , 2225 (1989). [12] Davis, in Ref. [4], used the same type of parametriza-

tion as Eq. (3.1). However, in applying it directly in the effective equation (2.2), his expression for is incor- rect, leading consequently to results differing from those of the present paper. The correct expression is given in Eq. (3.2).

[I31 T. W. B. Kibble, J. Phys. A 9, 1387 (1976). [14] T . W. B. Kibble, Phys. Rep. 67, 183 (1980). [15] A. Vilenkin, Phys. Rep. 121, 263 (1985).

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Vortex loops: Are they always doomed to die?