Could electromagnetic corrections solve the vorton excess problem?

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PHYSICAL REVIEW D 15 FEBRUARY 1998VOLUME 57, NUMBER 4

Could electromagnetic corrections solve the vorton excess problem?

Alejandro Gangui,* Patrick Peter,† and Celine Boehm‡

Departement d’Astrophysique Relativiste et de Cosmologie, Observatoire de Paris–Meudon, UPR 176, CNRS, 92195 Meudon, Franc~Received 30 April 1997; published 23 January 1998!

The duality between the master function and Lagrangian formalisms for the study of the dynamics of generalcosmic strings is reviewed to show their respective uses in various situations. We then consider an explicitmodel for which the Lagrangian method is untractable whereas the master function provides analytic solutions.From it, the modifications of circular cosmic string loop dynamics due to the electromagnetic self-interactionare then calculated and shown to reduce the available phase space for reaching classical vorton states. As itturns out, this effect might significantly decrease their remnant abundance.@S0556-2821~97!00824-2#

PACS number~s!: 11.27.1d, 98.80.Cq

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I. INTRODUCTION

Most particle physics theories, extensions of the so-castandard model of interactions, suggest that topologicalfects@1# should have been formed during phase transitionthe early universe@2#. Among those, the most fashionablbecause of their ability to solve many cosmological puzzare cosmic strings, provided couplings with other particare such that they are not of the superconducting kind@3#. Inthe latter case, however, they would not be able to decaypure gravitational radiation, terminating their life in the forof frozen vorton states which might be so numerous that twould cause a cosmological catastrophe@4–6#: a roughevaluation of their abundance yields a very stringent cstraint of a symmetry-breaking scale which, to avoid ancess, should be less than 109 GeV, which is incompatiblewith the idea of them being responsible for galaxy formatand leaving imprints in the cosmic microwave backgroun

There are many ways out of this vorton excess problethe most widely accepted relying upon stability considations: since vortons are centrifugally supported string loconfigurations, the origin of the rotation being hidden in texistence of a current, it is legitimate to first ask whethercurrent itself is stable against decay by quantum tunnelThis question, however, has not yet been properly addresand presumably depends on the particular underlying fimodel one uses, so that, although it is clearly an imporpoint to be clarified, it will not be considered in this worAnother issue, at a lower level, concerns the classical staity: a rotating string configuration in equilibrium may exhibunstable perturbation states; if it were the case in generaany equation of state, then one would expect vortons tosipate somehow, and the problem would be cured@7,8#. Thishope is not, however, fulfilled by the Witten kind of stringwhose equation of state@9,10# falls into the possibly stablecategory@11#.

Finally, another point worth investigating is that of vortoformation. It should be clear that an arbitrary cosmic strloop, endowed with fixed ‘‘quantum’’ numbers, will not i

*Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]

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general end up in the form of a vorton. This has to be qutified somehow, and one way of doing so is achievedlooking at some specific initial configuration, circular saand then letting it evolve until it reaches an equilibrium staif any. This has been done@12,13# for various neutralcurrent-carrying cases which showed again that many locan indeed end up as vorton states, and using many diffeequations of state@14#, moreover providing analytic solutions to the elastic kind of string equations which mayuseful in future sophisticated numerical simulations takinto account the possible existence of currents.

Our purpose here is twofold. First, we review the covaant macroscopic approach for general strings developedCarter @18#, and give the first explicit derivation of theequivalence between the master function and the Lagranformalisms for the study of string dynamics. The usual wof working out the dynamics of a current-carrying string~orany world sheet of arbitrary dimension living in a highedimensional space! consists in varying an action which iessentially the integral over the world sheet of a Lagrangfunction, itself seen as a function of the squared gradientscalar function living on the world sheet and representingvariations of the actual phase of some physical field@3,17#.There is, however, an ambiguity in this procedure in tsense that the phase gradient used can be chosen in a dent way by means of a Legendre kind of transformatwhereby one then considers the relevant dynamical variato be instead the current itself. This newer alternative produre provides a completely equivalent dual formulatiwhich turns out to be the only one that can deal with soinstances such as that of inclusion of the electromagncorrections. Second, it is also our aim to calculate the efof including these electromagnetic self-coupling correctioin the dynamics of a rotating loop. The reason for doing itthat this self-interaction modifies the equation of sta@10,15#, so that the evolution is indeed supposed to be dferent. In addition, it is the very first correction that canincluded without the bother of introducing the much motroublesome complications of evaluating radiation, andfact the radiation can only be consistently calculated pvided this first order effect has been properly taken intocount.

In the following section, we recapitulate this duality btween both descriptions, whose equivalence we show exp

2580 © 1998 The American Physical Society

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57 2581COULD ELECTROMAGNETIC CORRECTIONS SOLVE THE . . .

itly. Then, after a brief description of how electromagneself-corrections are included, we discuss the particular cof a circular rotating loop for which we calculate an effectipotential in view of resolving the dynamics. We concludeshowing that in general this corrective effect tends to redthe number of vorton states attainable for arbitrary initconditions.

II. THE DUAL FORMALISM

The usual procedure for treating a specific cosmic strdynamical problem consists in writing and varying an actwhich is assumed to be the integral over the world sheetLagrangian function depending on the internal degreesfreedom of the world sheet. In particular, for the structurelstring, this is taken to be the Goto-Nambu action@19#, i.e.,the integral over the surface of the constant string tensionmore general cases, various functions have been suggthat supposedly apply to various microscopic field configrations @14#. They share the feature that the descriptionachieved by means of a scalar functionw, identified with thephase of a physical field trapped on the string, whose squgradient, called the state parameterw}]aw]aw (a denotinga string coordinate index!, has values which completely determine the dynamics through a Lagrangian functionL$w%.This description has the pleasant feature that it is easilyderstandable, given the clear physical meaning ofw. How-ever, as we shall see, there are instances for which it is noeasily implemented and for which an alternative, equavalid, formalism is better adapted@18#.

In this section we will derive in parallel expressions fthe currents and state parameters in these two represtions, which are dual to each other. This section is essenta rephrasing of the formalism developed in@13# and @18#adapted to our present purposes. This will not be specifisuperconducting vacuum vortex defects, but is genervalid to the wider category of elastic string models@18#. Inthis formalism one works with a two-dimensional worsheet supported master functionL$x% considered as the duaof L$w%, these functions depending respectively onsquared magnitude of the gauge-covariant derivative ofscalar potentialsc andw as given by

x5 k0gabc uac ub↔w5k0gabw uaw ub , ~1!

wherek0 and k0 are adjustable, respectively, positive anegative dimensionless normalization constants that, aswill see below, are related to each other. The arrow inprevious equation stands to mean an exact correspondbetween quantities appropriate to each dual representaWe use the notationgab for the inverse of the induced meric, gab on the world sheet. The latter will be given, in termof the background spacetime metricgmn with respect to thefour-dimensional background coordinatesxm of the worldsheet, by

gab5gmnx ,am x ,b

n , ~2!

using a comma to denote simple partial differentiation wrespect to the world sheet coordinatesja and using Latinindices for the world sheet coordinatesj15s ~spacelike!,

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j05t ~timelike!. The gauge covariant derivativew ua wouldbe expressible in the presence of a background electromnetic field with Maxwellian gauge covectorAm byw ua5w ,a2eAmx ,a

m .In Eq. ~1! the scalar potentialsc andw are such that their

gradients are orthogonal to each other, namely,

gabw uac ub50, ~3!

implying that if one of the gradients, sayw ua is timelike, thenthe other one, sayc ua , will be spacelike, which explains thdifferent signs of the dimensionless constantsk0 and k0.

Whether or not background electromagnetic and gravtional fields are present, the dynamics of the system candescribed in the two equivalent dual representations@18,17#which are governed by the master functionL and the La-grangian scalarL, that are functions only of the state parametersx and w, respectively. The corresponding conservcurrent vectors,na and za say, in the world sheet, will begiven according to the Noetherian prescription

na52]L

]c ua↔za52

]L]w ua

. ~4!

This implies

Kna5 k0c ua↔Kza5k0w ua, ~5!

where we use the induced metric for internal index raisiand whereK andK can be written as

K21522dL

dx↔K21522

dLdw

. ~6!

As it will turn out, the equivalence of the two mutually dudescriptions is ensured provided the relation

K52K21, ~7!

holds. This means one can defineK in two alternative ways,depending on whether it is seen it as a function ofL or of L.We shall therefore no longer use the functionK in whatfollows.

The currentsna and za in the world sheet can be represented by the corresponding tangential current vectorsnm

andzm on the world sheet, where the latter are defined wrespect to the background coordinatesxm by

nm5nax ,am ↔zm5zax ,a

m . ~8!

The purpose of introducing the dimensionless scale cstantsk0 andk0 is to simplify macroscopic dynamical caculations by arranging for the variable coefficientK to tendto unity whenx and w tend to zero, i.e., in the limit forwhich the current is null. To obtain the desired simplificatiit is convenient not to work directly with the fundamentcurrent vectorsnm andzm that ~in units such that the DiracPlanck constant\ is set to unity! will represent the quantizedfluxes, but to work instead with the corresponding rescacurrentsvm andcm that are obtained by setting

nm5A2 k0vm↔zm5Ak0cm. ~9!

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2582 57ALEJANDRO GANGUI, PATRICK PETER, AND CE´ LINE BOEHM

Based on Eq.~3! that expresses the orthogonality of thscalar potentials we can conveniently write the relationtweenc andw as follows:

w ua5KA2 k0

Ak0

eabcub, ~10!

wheree is the antisymmetric surface measure tensor~whosesquare is the induced metric,eabe

bc5gac). From this and

using Eq.~1! we easily get the relation between the stavariables

w5K2x. ~11!

In terms of the rescaled currents, and using Eqs.~5! and ~8!we get

cmcm5w/K25x↔vmvm52K2x52w. ~12!

Both the master functionL and the LagrangianL arerelated by a Legendre type transformation that gives

L5L1Kx. ~13!

The functionsL andL can be seen@18# to provide values forthe energy per unit lengthU and the tensionT of the stringdepending on the signs of the state parametersx and w.~Originally, analytic forms@14# for these functionsL andLwere derived as best fits to the eigenvalues of the strenergy tensor in microscopic field theories@9,10#.! The nec-essary identifications are summarized in Table I.

This way of identifying the energy per unit length antension with the Lagrangian and master functions also pvides the constraints on the validity of these descriptions:range of variation of eitherw or x follows from the require-ment of local stability, which is equivalent to the demathat the squared speedscE

25T/U andcL252dT/dU of ex-

trinsic and longitudinal~sound type! perturbations be positive. This is thus characterized by the unique relation

LL

.0.dLdL

, ~14!

which should be equally valid in both the electric and manetic ranges.

Having defined the internal quantities, we now turn to tactual dynamics of the world sheet and prove explicitlyequivalence between the two descriptions.

TABLE I. Values of the energy per unit lengthU and tensionTdepending on the timelike or spacelike character of the currexpressed as the negative values of eitherL or L.

Equations of state for both regimes

Regime U T x andw Current

Electric 2L 2L ,0 TimelikeMagnetic 2L 2L .0 Spacelike

-

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III. EQUIVALENCE BETWEEN L AND L

The claim is that the dynamical equations for the strimodel can be obtained either from the master functionL orfrom the LagrangianL in the usual way, by applying thevariation principle to the surface action integrals

SL5E ds dtA2gL$x%, ~15!

and

SL5E ds dtA2gL$w%, ~16!

~whereg[det$gab%) in which the independent variables aeither the scalar potentialc or the phase fieldw on the worldsheet and the position of the world sheet itself, as speciby the functionsxm$s,t%.

The simplest way to actually prove this claim is to calclate explicitly the dynamical equations and show that thyield the same physical motion. To do this, we shall see tEq. ~7! is crucial by establishing a relation between the dnamically conserved currents in both formalisms. This nderivation of the equivalence between the two formaliswill also allow us to clarify the physical meaning of thconserved integer numbers below.

Independently of the detailed form of the complete stem, one knows in advance, as a consequence of the locglobal U~1! phase invariance group, that the correspondNoether currents will be conserved: namely,

~A2gna! ,a50↔~A2gza! ,a50. ~17!

For a closed string loop, this implies~by Green’s theorem!the conservation of the corresponding flux integrals,

N5 R djaeabnb↔Z5 R djaeabz

b, ~18!

meaning that for any circuit round the loop one will obtathe same value for the integer numbersN and Z, respec-tively. Z is interpretable as the integral value of the numbof carrier particles in the loop, so that in the charge coupcase, the total electric charge of the loop will beQ5Ze.

The loop will also be characterized by a second indepdent integer numberN whose conservation is trivially obvious. Thus we have the topologically conserved numbersfined by

2pZ5 R dc5 R djac ua5 R djac ,a

↔2pN5 R dw5 R djaw ua5 R djaw ,a ,

~19!

where it is clear thatN, being related to the phase of a physcal microscopic field, has the meaning of what is usuareferred to as the winding number of the string loop. The lequalities in Eqs.~19! follow just from explicitly writing thecovariant derivativeua and noting that the circulation integramultiplying Am vanishes. Note, however, that althoughZ and

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N have a clearly defined meaning in terms of underlymicroscopic quantities, because of Eqs.~18! and ~19!, theroles of the dynamically and topologically conserved intenumbers are interchanged depending on whether we deour equations fromL or from its dualL. Moreover, thosetwo equations, together with Eq.~10! yield

4p2k0k0521, ~20!

which confirms our original assumption.As usual, the stress momentum energy density distr

tions TLmn and TL

mn on the background spacetime are derable from the action by varying the actions with respectthe background metric, according to the specifications

TLmn[

2

A2g

dSL

dgmn[

2

A2g

]~A2gL!

]gmn~21!

and

TLmn[

2

A2g

dSLdgmn

[2

A2g

]~A2gL!

]gmn. ~22!

This leads to expressions of the standard form

A2gTmn5E ds dtA2gd~4!@xr2xr$s,t%# Tmn ~23!

in which thesurfacestress energy momentum tensors onworld sheet~from which the surface energy densityU andthe string tensionT are obtainable as the negatives ofeigenvalues! can be seen to be given@18,17# by

TLmn5Lhmn1K21vmvn↔ TL

mn5Lhmn1Kcmcn, ~24!

where the~first! fundamental tensor of the world sheetgiven by

hmn5gabx,am x,b

n . ~25!

Plugging Eqs.~9! into Eqs.~24!, and using Eqs.~7!, ~11!,and~13!, we find that the two stress-energy tensors coinci

TLmn5 TL

mn[ Tmn. ~26!

This is indeed what we were looking for since the dynamiequations for the case at hand, namely,

hmr ¹r Tmn50, ~27!

which hold for the uncoupled case, are then strictly equilent whether we start with the actionSL or with SL .

IV. INCLUSION OF ELECTROMAGNETICCORRECTIONS

Implementing electromagnetic corrections@15#, even atthe first order, is not an easy task as can already be seethe much simpler case of a charged particle for which a mrenormalization is required even before going on calculatanything in effect related to electromagnetic field. The saapplies in the current-carrying string case, and the requ

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renormalization now concerns the master functionL. How-ever, provided this renormalization is adequately performinclusion of electromagnetic corrections, at first order in tcoupling between the current and the self-generated elecmagnetic field, then becomes a very simple matter of shiftthe equation of state, everything else being left unchangLet us see how this works explicitly.

SettingKmnr[hm

t hns¹ths

r the second fundamental tensof the world sheet@18#, the equations of motion of a chargcoupled string read

TmnKmnr5'rmFmn j n, ~28!

where 'rm is the tensor of orthogonal projectio('m

r 5gmr 2hm

r ), Fmn52¹ [mAn] the electromagnetic tensoand j m the electromagnetic current flowing along the strinnamely, in our case

j m5rezm[qcm, ~29!

with r the effective charge of the current carrier in unit of telectron chargee ~working here in units wheree2.1/137).The self-interaction electromagnetic field on the world shitself can be evaluated@3# and one finds

Amustring

5l j m5lqcm, ~30!

with

l52ln~msD!, ~31!

where D is an infrared cutoff scale to compensate for tasymptotically logarithmic behavior of the electromagnepotential@10# andms the ultraviolet cutoff corresponding tothe effectively finite thickness of the charge condensate,the Compton wavelength of the current-carrierms

21 @9,10#.In the practical situation of a closed loop,D should at mostbe taken as the total length of the loop.

The contribution of the self-field~30! in the equations ofmotion ~28! can be calculated using the relations@15#

'nrhm

s¹sAnustring

5Anustring

Kmnr, ~32!

and

FmnUstring

52hs[m¹sAn]Ustring

1'r[mAn]UhabKabr, ~33!

which transform Eqs.~28! into

KmnrF Tmn1lq2S cmcn2

1

2hmncacaD G50, ~34!

which is interpretable as a renormalization of the stressergy tensor. This equation is recovered if, in Eq.~21!, oneuses

L→L11

2lq2x ~35!

instead ofL. This formula@15# generalizes the action renomalization originally obtained@16# in the special case fowhich the unperturbed model is of simple Goto-Nambu ty

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That the correction enters through a simple modificatof L$x% and not ofL$w% is understandable if one remembers thatx is the amplitude of the current, so that a pertubation in the electromagnetic field acts on the currentearly, so that an expansion in the electromagnetic field

current yields, to first order inq, L→L1 12 j mAm, which

transforms easily into Eq.~35!.Using this modification entitles us to consider essentia

noncoupled string world sheet dynamics at this order,uttermost simplification since we thus do not have to csider radiation backreaction. Note, however, that the cortion we are now going to take into account is necessary pto any evaluation of the radiation. We therefore still havedefine the circular motion but before that, let us specifymodel, i.e., the equation of state before the correctionsincluded.

V. EQUATION OF STATE

The underlying field theoretical model we wish to cosider is that originally proposed by Witten to describe tcurrent-carrying abilities of cosmic strings@3#. Although it isthe simplest possible model fulfilling that purpose, it is blieved to share most of the features that would be expefrom more realistic current-carrying cosmic string mod@20#. In essence, the microscopic properties of the stringdescribed by means of two complex scalar fields, the strforming symmetry-breaking Higgs field, and the chargecoupled~or not @9#! current-carrier@10#, whose phase gradient serves to calculate the state parameterw. Once thesefields are defined, it suffices to consider a stationaryaxisymmetric configuration and integrate the correspondrelevant stress-energy tensor components over a crosstion of the vortex to deduce the energy per unit length atension of the string. Repeating this operation for variovalues ofw as well as of the free parameters of the modone finds the required equation of state, albeit only numcally @9,10#.

For this model, it was shown that, in the electric regimwhere the current is timelike, the current diverges logarmically when the state parameter approaches the negativthe current-carrier mass squared,2m

*2 , say. Using this

property, it was then possible to propose a best fit tootherwise numerical equation of state@14#, fit which is amaz-ingly good for almost all values of the state parameter.particular, including a divergence in the electric regime wshown to also imply a current saturation in the magneregime. In the Lagrangian formalism, it reads, settingstring’s characteristic mass scale tom,

L$w%52m22m

*2

2lnH 11

w

m*2 J , ~36!

which, upon using Eqs.~11! and~13!, providesK as a func-tion of x in the form

K52dL

dx511

w

m*2 5

12A124x/m*2

2x/m*2

, ~37!

where, in the last equality, use has been made of Eq.~11! andthe minus sign in front of the squared root ensures thatK→1whenx→0 ~the Goto-Nambu limit of no current!. Integrat-

n

--d

yn-c-r

oere

-ed

reg--

dgec-dsl,i-

-of

e

nsce

ing Eq. ~37!, and normalizing in such a way thaL$0%5L$0%52m2, yieldsL as

L52m21m

*2

2 F12A124x/m*2

11

2lnS x

m*2

11A124x/m*2

12A124x/m*2 D G , ~38!

in which it now suffices to incorporate the shift~35! to ac-count for the inclusion of electromagnetic correction at fiorder @and note that nowK is modified toK1lq2, as oneclearly sees from Eqs.~6! and~7!#. As should be clear on thisparticular example, getting back to the original Lagrangformalism would be a very awkward task, and in fact donot lead to any analytically known form forL$w%. This un-pleasant feature, however, is not much bother since theformalism is available.

An important point needs be noted at this stage: it ccerns the relevant dimensionless parameters. The model~36!in fact depends solely on one such parameter, namely,ratio

a5S m

m*D 2

, ~39!

which, in a reasonable cosmic string microscopic model@3#,would be at least of order unity, and in many applications@6#largely in excess of unity. As it was shown@13# in a previousstudy not including electromagnetic corrections that thesults for the vortons themselves were not in any essenway dependent ona as long asa*1, we shall consider thecasea51 on the figures that follow.

As an illustration, Eqs.~11! and ~13! have been used tocalculate the equation of stateU(n) and T(n) for variousvalues of the electromagnetic correction parameterlq2, andthey are exhibited on Fig. 1.1 On this figure,U and T areplotted as functions ofn, which is defined as the square roof the state parameterw: n5sgn(w)Auwu. Its meaning isvery simple: for a straight string lying along thez axis say,one can set the phase of the current carrier asw5kz2vt,and there exist a frame in whichn is eitherk or 2v, i.e., itrepresents the momentum of the current-carrier alongstring’s direction, or its energy. The electromagnetic corrtion in this case is seen to enlarge the picture: a small~orvanishing! correction yields the usual form of the equationstate where the tension~hencecE

2) goes to zero for largenegativew ~phase frequency threshold! @9#, andcL

2 vanisheson the magnetic side forw5ws ~saturation!. The thresholdbecomes more and more negative with increasinglq2, andthe saturation point is reached for larger values ofw; boththese remarks show that inclusion of electromagnetic cor

1These curves are not only for the sake of completeness, aswere not published in Refs.@13# or @14#, but also to show explicitlythe overall features of the equation of state. Similar figures canfound in Refs.@9,10#, with the same axisn ~scales are differentbecause not normalized in the same way! for the numerically com-puted equation of state in the Witten bosonic superconductingmic string field-theoretic model.

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s

me

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tions can be interpreted as a rescaling ofw, which in Fig. 1is equivalent to rescaling thex axis.

VI. CIRCULAR MOTION IN FLAT SPACE

We now restrict our attention to the motion of a circulavortex ring in flat space. The analysis in this case has alreabeen done@13#, so we only need to summarize the resultsand eventually rephrase them in terms ofL instead ofL.

A. Equations of motion

The background and the solution admit two Killing vectors: one timelikekm normalized through

km]

]xm 5]

]t, ~40!

t being a timelike coordinate, and one spacelikel m,

l m]

]xm 52p]

]f, ~41!

with fP@0,2p# an angular coordinate; botht and f areignorable.The lengthl of the string loop is then given by

l 25l ml m , ~42!

while its total massM and angular momentumJ are definedby

M[2 R dxmemn Tnrkr ~43!

FIG. 1. Variation of the equation of state with the electromagnetic self-correctionlq2. It relates the energy per unit lengthU~upper set of curves! and the tensionT ~lower set of curves!, both inunits of m

*2 , the current-carrier mass, and is plotted againstn,

which is the square root of the state parameterw. Values used forthis correction are in the set@0,0.1,0.5,1,2,5,7,8,9,10,20#, and thefigure is calculated fora51. Increasing the value oflq2 enlargesthe corresponding curve in such a way that for very large values~inthis particular example, it is forlq2>7), the tension on the mag-netic side becomes negative before saturation is reached.

dy,

and

J[~2p!21 R dxmemn Tnrl r. ~44!

So long as we do not consider radiation of any kind~a re-quirement equivalent with the demand thatkm and l m beKilling vectors also for the string configuration!, these areconserved. Note also that the relationJ5NZ holds.

Other quantities need be introduced, related with the inger numbersN andZ, in which it turns out to be conveniento include the scale parameterk0 ~or equivalentlyk0) in thefollowing clearly dual definitions

B[N/A2 k0, C[Z/Ak0. ~45!

Now specifying to the particular case of a flat spacetibackground in which the circular string is confined onplane so that we can use circular coordinates$r ,u,f,t% andsetu5p/2, t5Mt andf5s ~recallt ands are the respec-tively timelike and spacelike internal coordinates!, use ofEqs.~19! imply that the phases vary as

c5s~ t !1Zf, w5 f ~ t !1Nf, ~46!

with s and f functions only of time and expressible in termof the conserved numbersB andC through

s5B

K

A12 r 2

2prA2 k0

, f 5CKA12 r 2

2prAk0

, ~47!

an overdot meaning a derivative with respect to the ticoordinatet. The variation of the string’s radiusr 5l /2pfollows from the equation@13#

MA12 r 25Y~r !, ~48!

from which we conclude that the string evolves in a sepotential Y(r ). Thus, it suffices to know the form of thipotential to understand completely the protovorton dynaics. It is given, in terms of our variables, by@13#

Y5B2

Kl2Ll , ~49!

while the string’s circumference reads

l 251

xS B2

K22C2D . ~50!

In order to express the results, it is simpler to resceverything by means of x r5x/m

*2 , l r5m* l /uCu,

L r5L/m*2 , Lr5L/m

*2 , and Y r5Y/(m* uCu) so that all

quantities of interest are dimensionless and depend onlythree arbitrary, also dimensionless, parameters, namelya,which we discussed already,b, defined byb25B2/C2, andthrough which one expresses the timelike or spacelike cacter of the current@from Eq. ~50!#, and the most importanparameter here, namely,lq2. We are now ready to examinthe actual electromagnetic correction to a string loop dynaics at first order in the couplingj mAm.

-

nt

ht,


FIG. 2. Constraints yielding minimum andmaximum values for the normalized currex/m

*2 , according to Eqs.~51!, ~52!, and~53!, for

valueslq251022 to 10. In the figure on the left,for each particular value oflq2, the unstable re-gion ~wherecT

2,0) lies below the correspondingcurve. The same is true for the figure on the rigbut in additionx/m

*2 ,1/4 for otherwisecL

2,0.

-

l-

ous

ci

2e

e

t,

g-onueyeree

eticen

le

es ofthinm-

nts

fect

e ton-

ing

mess

.

ndIf

her

B. The self-potential

The potentialY r as a function ofl r is derivable by meansof first expressingY r andl r as functions of the state parameter x r through Eqs.~49! and ~50!. In order to do this, oneneeds to know the range in whichx r varies, range given bythe requirements~14!, which can be rephrased into the folowing constraints:

2a2lq2x r211A124x r

21

2lnH x r

11A124x r

12A124x rJ .0 @T.0 magnetic#,

~51!

2a1lq2x r21

2lnH x r

11A124x r

12A124x rJ .0 @T.0 electric#,

~52!

12A124x r1lq22x rA124x r

A124x r@12A124x r1lq22x r #.0 @cL

2.0#. ~53!

The first and second of these constraints specify the cdition that the extrinsic squared perturbation velocity mbe positive~and therefore also the tensionT.0) in both themagnetic and the electric regimes~i.e., for spacelike andtimelike currents, respectively!. The third constraint Eq.~53!demands that the longitudinal squared perturbation velobe positive and is always satisfied providedx r,1/4. We plotthis as the vertical line in the picture on the right of Fig.the allowed range ofx r values being to the left of it. Thesconstraints were used in the calculation of Fig. 1.

The first thing to determine in order to plotY r(l r) iswhether the current is timelike or spacelike. This is achievby looking at Eq.~50! which states that the sign ofx, andhence the timelike or spacelike character of the currenalso the sign ofb22K2. Now Eq. ~37!, modified to accountfor Eq. ~35!, shows that the range of variation ofK is

x>0 ~magnetic!⇒11lq2<K<21lq2, ~54!

and

n-t

ty

,

d

is

x<0 ~electric!⇒lq2<K<11lq2. ~55!

Therefore, it is only possible forx to be positive ifb>11lq2, and negative otherwise. So we deal with a manetic configuration or an electric configuration dependingwhetherb is respectively greater or less than its critical valbc511lq2. Since x cannot change sign dynamically, bfixing b one also fixes the character of the current and, whthe physical constraints~51! to ~53! cease to be satisfied, thcurves plotted in the various figures end. As the quantitybcwas unity in the decoupled case, we see that electromagncorrections can modify the nature of the current for a givset of integer numbersZ andN.

Once the nature of the current is fixed, it is a simpmatter to evaluate the potentialY r , and it is found, as in Ref.@13#, that three cases are possible, depending on the valuthe free parameters. They correspond to whether thestring description holds for all values of the allowed paraeters or not, as illustrated on Fig. 3.

The general form of the potentialY r(l r) exhibits a mini-mum and two divergences, one at the origin which prevea collapse of the loop, and one forl r→` which holds theloop together and is responsible for the confinement ef@13#. The latter divergence, going asY r;l r , occurs what-ever the underlying parameters may be and is merely duthe fact that it requires an infinite amount of energy to elarge the loop to infinite size, its energy per unit length bebound from below (U>m2). The divergence atl r50 is,however, not generic, as is shown on Fig. 3, since for sosets of parameters, the quantityl r does not take values in itentire range of potential variations@0, @ . This can be seen afollows.

On the magnetic side, the functionl r2(x r) ranges from

1` in the limit where x r→01, to 4@b2

2~lq212)2]/(lq212) for x r→1/4 as sketched on Fig. 4Depending therefore on whetherb is less or greater than(lq212), the l r50 limit will or will not exist. In theformer case, the potentialY r , which diverges forl r→0,will have the form indicated as the thick curve on Fig. 3, athe string loop solution is in a vorton-forming zone.b.lq212, then the minimum value forl r

2 is nonzero sothe potential terminates at some point, which can be eit

a

g

inifi-orethisre-op

s ofa-

or

npectnot

end

-

ry

e

e

s

sly

chtheg is


sufficiently close to the origin that the minimum forY r canbe reached~still a vorton-forming configuration! or not. Inthe former case, the resulting configuration may reachequilibrium state of massM v ~when radiation is taken intoaccount, such a configuration will eventually loose enouenergy to settle down into a vorton state! provided its mass

FIG. 3. Variations of the self-potentialY r with the ring’s cir-cumferencel r and the electromagnetic self-couplinglq2 fora5b51. The thick curve stands for various values oflq2,0.1 forwhich they are undistinguishable, and in the vorton-forming casthe minimum value ofY r is thenM v , the vorton mass.Y r for thesame parametersa andb, this time forlq251 is represented as thefull thin line, where it is clear that we now are in a zone where thpotential has a minimum~new value forM v) but now terminates atsome point where it equalsMs . Finally the dashed curve representthe potential forlq2510, an unrealistically large value, and thistime the curve terminates even before reaching a minimum: this isituation in which all loops with such parameters will eventualdecay.

FIG. 4. The characteristic behavior ofl r as a function ofx r forlq251 in the magnetic regime wherex r.0: the curve down, in-dicatedb,lq212 is for b52.1, and the upper curve forb55.They all diverge aroundx r50.

n

h

M is less than that obtained forx r51/4, Ms say, with thesame set of parameters, whereas it will enter a regimewhich the thin string description is no longer validM.Ms . Finally, there is also the possibility that no minmum ofY r(l r) is attainable on the entire available range fl r ; whatever the value ofM , the loop again ends up in thregion where no string description holds anymore. Whenhappens, the topological stability of the vortex can bemoved dynamically and the quantum effects make the lodecay into a burst of Higgs particles.

The electric regime presents roughly the same featurehaving different zones, although for slightly different resons: the magnetic case ends either whencL

2→0 or cT2→0

whereas the electric case does so only in the casecT2→0. On

Fig. 5 is sketched the functionl r(x r) for x r,0 and variousvalues of the parametersb, givenlq2. What happens in theelectric regime is that the limiting case this time is fb5lq2, l r

2 behaving as (b22l2q4)/(l2q4x r) asx r→2`;thus, if b,lq2, asx r is negative,l r

2 is always nonzero andthere must exist a value forx r such that the string’s tensiovanishes, and the loop itself becomes unstable with resto transverse perturbations. Such a loop would clearlyform a vorton. On the other hand, ifb.lq2, thenl r

2 can getto zero, and for some values of the parameters~unfortunatelythe range is not derivable analytically!, it will do so beforethe tension vanishes. The corresponding loop might thenas a vorton.

Finally the values ofb that might lead to vorton formation, for all regimes taken together, is limited to

lq2<b<lq212, ~56!

a condition which is increasingly restrictive aslq2 increases,and may even forbid vorton formation altogether for a velarge coupling.

;

a

FIG. 5. The characteristic behavior ofl r as a function ofx r forlq251 in the electric regime wherex r,0: The curve down is fora vorton-forming configuration withb51.9.lq2, whereas the up-per curve, withb50.1,lq2, represents a set of parameters whicannot describe a vorton. The two upper curves terminate atpoint where the string tension becomes negative and the strinunstable with respect to transverse perturbations.

-ps

ate-r-onaceee


FIG. 6. A possible way out of the vorton excess problem: a sketch of a distribution of loowith b, and vorton-forming intervals@cf. Eq.~56!# for different values oflq2. It is clear thatthe actual number density of ensuing vortons,most proportional to the shaded areas, will dpend quite strongly on the location of this inteval. Note also that this electromagnetic correctimay reduce drastically the available phase spfor vorton formation since the maximum of thdN/db distribution is usually assumed to bpeaked aroundb51.

gtengneantar-nthtoalu

edreagulf

atn

on

icale,

-

xi-ghtisof

ingheracttumntssm,s,un-ess

toeot

hat

ve

enyrted

par-h

VII. CONCLUSIONS

We have exhibited explicitly the influence of electromanetic self-corrections on the dynamics of a circular vorline endowed with a current at first order in the couplibetween the current and the self-generated electromagfield, i.e., neglecting radiation. This is necessary beforeradiation can be taken into account and evaluated, awhich is still to be done. Moreover, use of the duality fomalism developed by Carter@18# has been made and showto be especially useful in this particular case in the senseit enabled us to derive the dynamical properties of a provorton configuration analytically. It is to be expected thsuch a dual calculation will prove indispensable when evaation of the higher electromagnetic orders will be perform

We have shown that most of the conclusions of our pvious paper on that subject actually hold when electromnetism is accounted for, at least at this order, with the resperhaps not intuitively obvious from the outset, that this seinteraction tends to destabilize the string loops towards stfor which a classical string description does not hold, cofigurations which are expected to decay into the string cstituents~the Higgs field in particular! when quantum effectsare taken into account.

As is clear from Eq.~56!, increasing the electromagnetcorrection is equivalent to reducing the available phase spfor vorton formation, asb of order unity is the most naturavalue @6#, situation that we sketch in Fig. 6. On this figurwe have assumed a sharply peakeddN/db distribution cen-tered aroundb51; with lq250, the available range for vor

re,

n

-x

ticysk

at-

t-.--

lt,-es--

ce

ton formation lies precisely where the distribution is mamal, whereas for any other value, it is displaced to the riof the distribution. Assuming a Gaussian distribution, theffect could easily lead to a reduction of a few ordersmagnitude in the resulting vorton density, the latter beproportional to the area below the distribution curve in tallowed interval. This means that as the string loops contand loose energy in the process, they keep their ‘‘quannumbers’’Z andN constant, and some sets of such constawhich, had they been decoupled from electromagnetiwould have ended up to equilibrium vorton configurationinstead decay into many Higgs particles, themselvesstable. This may reduce the cosmological vorton excproblem@6# if those are electromagnetically charged.

The present analysis, because of its being restrictedexactly circular configurations, is not sufficient to providgeneral conclusions as to whether vortons will form or nfor whatever original loop shapes, but clearly indicates teven though the cosmological vorton problem@6# cannot besolved by means of this destabilizing effect, it may well habeen slightly overestimated.

ACKNOWLEDGMENTS

We would like to thank B. Carter for his insights on thmatters dealt with here, and M. Sakellariadou for mastimulating discussions. The research of A.G. was suppoby the Foundation Robert Schuman and by the Socie´te deSecours des Amis des Sciences. He also acknowledgestial financial support from the program Antorchas/BritisCouncil ~Project No. 13422/1–0004!.

n,
@1# T. W. B. Kibble, J. Phys. A9, 1387~1976!; Phys. Rep.67, 183~1980!.
@2# E. P. S. Shellard and A. Vilenkin,Cosmic Strings and OtheTopological Defects~Cambridge University Press, CambridgEngland, 1994!.

@3# E. Witten, Nucl. Phys.B249, 557 ~1985!.@4# R. L. Davis and E. P. S. Shellard, Nucl. Phys.B323, 209

~1989!.@5# B. Carter, Ann.~N.Y.! Acad. Sci.647, 758~1991!; in Proceed-

ings of the XXXth Rencontres de Moriond, Villard-sur-Ollon,Switzerland, 1995, edited by B. Guiderdoni and J. Tran ThaVan ~Editions Frontie`res, Gif-sur-Yvette, 1995!.

h

@6# R. Brandenberger, B. Carter, A.-C. Davis, and M. TroddePhys. Rev. D54, 6059~1996!.

@7# B. Carter and X. Martin, Ann. Phys.~N.Y.! 227, 151 ~1993!.@8# X. Martin, Phys. Rev. D50, 7479~1994!.@9# P. Peter, Phys. Rev. D45, 1091~1992!.

@10# P. Peter, Phys. Rev. D46, 3335~1992!.@11# X. Martin and P. Peter, Phys. Rev. D51, 4092~1995!.@12# A. L. Larsen and M. Axenides, Class. Quantum Grav.14, 443

~1997!.@13# B. Carter, P. Peter, and A. Gangui, Phys. Rev. D55, 4647

~1997!.@14# B. Carter and P. Peter, Phys. Rev. D52, R1744~1995!.

cl.

-O

on


@15# B. Carter, Phys. Lett. B404, 246 ~1997!.@16# E. Copeland, D. Haws, M. Hindmarsh, and N. Turok, Nu

Phys.B306, 908 ~1988!.@17# B. Carter, inFormation and Interactions of Topological De

fects, edited by R. Brandenberger and A.-C. Davis, NATAdvanced Study Institute, Series B: Physics, Vol. 349~Ple-num, New York, 1995!, pp. 303–348.

@18# B. Carter, Phys. Lett. B224, 61 ~1989!; 228, 466 ~1989!.@19# Y. Nambu, in Proceedings of the International Conference

Symmetries and Quark Models, Detroit, Michigan, 1969~un-published!; T. Goto, Prog. Theor. Phys.46, 1560~1971!.

@20# A. Babul, T. Piran, and D. N. Spergel, Phys. Lett. B202, 307~1988!.

Could electromagnetic corrections solve the vorton excess problem?

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