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Metastable Cosmic Strings in Realistic Models

arXiv:hep-ph/9208245v1 25 Aug 1992Fermilab-Pub-92/228-AHUTP-92/A032hep-ph/9208245Metastable Cosmic Strings in Realistic ModelsRichard HolmanDepartment of Physics, Carnegie MellonUniversity, Pittsburgh PA 15213.Stephen Hsu*Lyman Laboratory of Physics, HarvardUniversity, Cambridge, MA 02138Tanmay VachaspatiTufts Institute of Cosmology, Department of Physics and Astronomy,Tufts University, Medford, MA 02155.Richard Watkins**Department of Astronomy and Astrophysics, Enrico Fermi Institute,The University of Chicago, Chicago, IL 60637andNASA/Fermilab Astrophysics Center,Fermi National Accelerator Laboratory,Batavia, IL 60510ABSTRACT: We investigate the stability of the electroweak Z-string at high tempera-tures. Our results show that while finite temperature corrections can improve the stabilityof the Z-string, their effect is not strong enough to stabilize the Z-string in the standardelectroweak model.

Consequently, the Z-string will be unstable even under the conditionspresent during the electroweak phase transition. We then consider phenomenologically* Junior Fellow, Harvard Society of Fellows** AddressafterSept.1:PhysicsDepartment,University of Michigan, Ann Arbor MI 48109

viable models based on the gauge group SU(2)L × SU(2)R × U(1)B−L and show thatmetastable strings exist and are stable to small perturbations for a large region of theparameter space for these models. We also show that these strings are superconductingwith bosonic charge carriers.

The string superconductivity may be able to stabilize seg-ments and loops against dynamical contraction. Possible implications of these strings forcosmology are discussed.2

Over the last two decades, cosmic strings have evoked a great deal of interest bothas possible remnants of a Grand Unified era in the early universe as well as a possiblemechanism for structure formation in the universe1.However, no compelling particlephysics models exist that give rise to such defects. Recently, a defect that is closely relatedto the “ordinary” cosmic string has been found2,3 in what may be the most compelling ofall particle physics models - the standard electroweak model.

The defect is identical in itsstructure to the cosmic string solution found by Nielsen and Olesen4 and may be thoughtof as a cosmic string embedded in the standard electroweak model5. The difference now isthat the defect does not owe its existence to topology and consequently may not be stable.The stability of the defect depends on the parameters of the electroweak model6.In Ref.

7 the stability of the string in the standard electroweak model was analyzed.This resulted in a map of parameter space demarcating the regions in which the string isstable to small perturbations from the regions in which it is unstable. Given the knownvalue of the Weinberg angle, sin2 θW ≈0.23, and the constraints on the Higgs mass, mH >57 GeV, it is clear that the electroweak string is unstable.

However, the analysis in Ref.7 was limited to the bare electroweak model. The issue of stability must be reconsideredwhen one takes quantum and thermal corrections to the potential into account.

In essence,the question is whether the strings can be stable at temperatures close to the electroweakphase transition temperature. If this is true, the strings may be relevant to cosmology.We answer this question in Sec.

2 where we map the parameter space as in Ref. 7 for thecase when thermal and quantum corrections are taken into account.

The results show thatthese corrections increase the region of stability, but not to the extent of allowing for stableelectroweak strings in the standard model even near the electroweak phase transition.We then consider the question of whether there is any realistic particle physics modelin which one might expect stable embedded strings. We show that left-right models8 aregood candidates.

In Sec. 3 we consider an SU(2)L × SU(2)R × U(1)B−L model.

The3

parameters may be chosen such that this model gives acceptable particle physics and alsocontains stable strings. This gives us a concrete example of a realistic particle physicsmodel with stable embedded strings.The cosmology of embedded strings will be very different from that of topologicalstrings.The basic reason for this has to do with the metastability of the embeddedstrings versus the complete stability of topological strings.

In Sec. 4 we speculate on thecosmology of embedded strings.

The results of this section should not be thought of as firmconclusions but only as first guesses intended to inspire future work. Section 5 containsour conclusions.2.

Stability of the Z-String at Finite TemperatureThe addition of quantum corrections to the Higgs potential can have a drastic effect onthe vacuum structure of the standard model9. The most important correction is in theform of a φ4 log(φ/M) term, which can destabilize the potential at large φ.

However, ourinterests lie at relatively small φ where this term is quite small. We have done a stabilityanalysis including this term and found that there is very little effect.

Therefore for theremainder of this discussion we shall ignore quantum corrections and concentrate on thoseinduced by finite temperature effects.The one loop finite temperature effective potential for the Higgs field can be writtenas10VT (φ) = λφ†φ −η222+ DT 2φ†φ −ET(φ†φ)3/2(2.1)where D and E are functions of the particle masses, and can be approximated byD =18η22m2W + m2Z + 2m2t,(2.2)E =14πη32m3W + m3Z(2.3).4

Here η = 246 GeV is the expectation value of φ at the minimum of the zero-temperaturepotential. Here we have chosen to ignore temperature corrections to λ, which are logarith-mic and should not effect our results significantly.As in the zero temperature case, Z-string solutions will take the formW µ1 = W µ2 = Aµ = 0,Zµ = −v(r)rˆeθφ = f(r)eiθΦ,Φ =01,(2.4)where we have assumed the string to be aligned along the z axis, and r and θ are polarcoordinates in the xy-plane.

The functions f and v are determined by the equations ofmotion:f′′ + f′r −1 −α2 v2 fr2 −2λf2 −η22f + DT 2f −32ETf2 = 0(2.5)v′′ −v′r + α1 −α2 vf2 = 0(2.6)where primes denote differentiation by r. Here α is given by g = α cos(θW).The functions also satisfy the boundary conditions:f(0) = v(0) = 0,f(∞) = f∞,v(∞) = 2α(2.7)where f∞is the magnitude of the global minimum of VT .In order to study the solutions to these equations numerically, it is convenient tointroduce the dimensionless quantities:P ≡f/f∞,V ≡α2 v,R ≡αη2√2r(2.8)so that the equations take the simple formP ′′ + P ′R −(1 −V )2 PR2 −β(P −1)(P −Pe)P = 0. (2.9)5

V ′′ −V ′R + (1 −V ) P 2 = 0(2.10)where β = 8λα2 andPe = −1 +2ETET +rT 2E2 −4βα29D+ β2α418 η2. (2.11)The parameter Pe carries all of the information about finite temperature effects in therescaled potential.

It takes on values between −1 at T = 0 up to 0.5, above which P = 0becomes the true vacuum. For 0 < Pe < 0.5, P = 0 is a local minimum, separated fromthe global minimum at P = 1 by a potential barrier.

This assumes the phase transition tobe first order; in models with a second order transition Pe ≤0.Eqns. (2.9-2.10) can be solved using standard methods.

The string configurations thatresult, even in the extreme Pe = 0.5 case, are not qualitatively different than T = 0 strings.In particular, both P(r) and V (r) remain monotonically increasing functions of r.The stability of electroweak strings at finite temperature can be determined in a similarmanner to that for zero temperature strings as described in Ref. 7.

Here we will give ashort review of the procedure, referring the reader to Ref. 7 for details.The energy functional for two-dimensional static solution in the electroweak modelmay be written in the standard notation of Ref.

11:E =Zd2x14GaijGaij + 14FBijFBij + (Djφ)†(Djφ) + VT (φ)(2.12)where, i, j = 1, 2 and a = 1, 2, 3. The string solution that extremizes this energy functionalis given in (2.4) and we now perturb around that solution.It can be shown that the only relevant perturbations are those in which the uppercomponent, φ1, of the Higgs doublet and the W fields are perturbed.

These fields can beexpanded in modes:φ1 = χm(r)eimθ(2.13)6

for the mth mode where m is any integer. For the nth mode of the gauge fields we have,⃗W 1 = ¯fn1 (r) cos(nθ) + fn1 sin(nθ)ˆer + 1r−¯hn1 sin(nθ) + hn1 cos(nθ)ˆeθ(2.14)⃗W 2 =−¯fn2 (r) sin(nθ) + fn2 cos(nθ)ˆer + 1r¯hn2 cos(nθ) + hn2 sin(nθ)ˆeθ(2.15)Of these, the m = 0, n = 1 mode is the most dangerous and it is sufficient to look at thismode alone.

Further, it turns out that the barred variables separate from the unbarredones and it is sufficient to look at the problem in the unbarred variables alone.Define the quantitiesF± = f2 ± f12(2.16)ξ± = h2 ± h12(2.17)ζ = (1 −γv)χ + 12gfξ+(2.18)Then, after substituting the mode expansions in the energy functional and a lot of algebra,we find that the energy variation around the unperturbed solution is,δE = 2πZdrrhnζ′2P++U(r)ζ2o+sum of whole squaresi(2.19)where primes denote differentiation with respect to r,P+ = (1 −γv)2 + 12g2r2f2 ,(2.20)U(r) =f′2P+f2 +2S+g2r2f2 + 1rddr rf′P+f,(2.21)and,S+ = g2f22−γ2v′2P++ γr ddrv′r(1 −γv)P+. (2.22)The energy variation in (2.19) is minimized if the sum of squares is chosen to vanish.

Thisfixes the modes F± and also requires ξ−= 0. Then we are simply left with a problem in7

ζ. (Note that although the explicit form of the Higgs potential does not appear anywherein the ζ dependent terms, it does appear implicitly in the unperturbed solution given byP and V .

)By performing an integration by parts, we can now rewrite the ζ part of δE as,δE[ζ] = 2πZdr rζOζ(2.23)where, O is the differential operator:O = −1rddr rP+ddr+ U(r) . (2.24)Now we have to determine if the differential equationOζ = ωζ(2.25)has any negative eigenvalues (ω).

The boundary conditions on ζ are: ζ(r = 0) = 1 andζ(r = ∞) = 0.In this way, the whole stability analysis has been reduced to the single differentialequation (2.25). This equation can be solved numerically using the shooting method.For a given Pe and β, one can use (2.25) to find the value of θW for which there existsa zero eigenmode.

This determines the critical value of θW which marks the boundarybetween stability and instability for the strings. In Fig.

1 we plot a number of curvesshowing the regions of stability for different values of Pe. In Fig.

2, we plot regions ofstability for different temperatures, assuming fixed values of D/α2 and E/α2.These results show a number of significant features. First, we see that the region ofstability grows significantly as the temperature increases, including regions where β > 1.Thus finite temperature effects have a stabilizing effect on the strings.

This allows oneto consider scenarios where strings only exist during a specific epoch in the evolution ofthe universe. Stable strings can form in a phase transition, but then become unstable and8

disappear as the temperature drops below some critical value. It is important to note,however, that even at finite temperature the standard model value of sin2(θW ) = 0.23 andmH > 57GeV is still deep within the region of instability.Another feature of the plot is that there appears to be a lower bound to the sin2(θW) atwhich there are stable strings.

The stability region for small β is unknown, due to the factthat in this limit the strings become very thick and are difficult to treat numerically. For theT = 0 case it was unclear whether the region of stability extended down to small sin2(θW)at small β.

For the finite temperature case, all of the stability curves corresponding todifferent Pe should converge at β = 0. This is because the potential becomes unimportantin the lagrangian in this limit.

From the figure it seems likely that the critical value ofsin2(θW ) at β = 0 is around 0.92. The absolute lower bound for stable strings is thusgiven by the Pe = 0.5 curve, and is ≃0.91.3.

Metastable Strings in Left-Right ModelsAs we have seen in Sec. 2, the Z-strings of the electroweak theory are unstable to smallperturbations for physically reasonable values of the Higgs mass and the Weinberg angle,even when quantum and finite temperature corrections are taken into account.

This leadsus to wonder if there are any well motivated particle physics models that admit (meta)stable, embedded strings.A hint as to how to go about finding such a model comes from the analysis in Sec.2.There we were essentially trying to stabilize the string through modification of the Higgspotential. Our failure to obtain stable strings was due in some part to the small value ofsin2(θW ) required by the standard model.

Thus, we are motivated to look for extensionsof the standard model where the gauge sector of the theory is enlarged, allowing for thepresence of other Weinberg-type angles which can take somewhat larger values.9

A well-known extension of the gauge sector of the standard model is the left-rightmodel based on the gauge group SU(2)L × SU(2)R × U(1)B−L8. One of the interestingfeatures of this class of theories is that they can be compatible with known experimentalresults even if the scale of SU(2)R ×U(1)B−L breaking to U(1)Y is quite low i.e.

500 GeV- 1 TeV.12The field content of the model we consider is as follows (the quantum numbers are theSU(2)L × SU(2)R × U(1)B−L representation assignments). The left handed quarks andleptons transform as (2, 1; 1/3), (2, 1; −1) respectively while their right-handed companionstransform as (1, 2; 1/3), (1, 2; −1) (note that we have added a right handed neutrino state).The minimal Higgs content required for the phenomenological viability of the model is:φ ∼(1, 2; −1), χ ∼(1, 3; 2), ∆∼(2, 2; 0).

The right handed triplet χ is required togive the right handed neutrino a large Majorana mass so as to implement the see-sawmechanism13, while φ is needed to yield the correct pattern of symmetry breaking and∆induces the Dirac masses of all other fermions. The phenomenology of this model wasconsidered in Ref.12.

We assume that the vacuum expectation values (VEV’s) fφ, fχ, vof φ, χ and ∆respectively satisfy the following hierarchy: v << fχ << fφ.From the viewpoint of constructing models with embedded metastable strings, theadvantages of the left-right model described above are clear. We can just take the Z-stringfound in the electroweak model and embed it in the right handed sector.

To the extent thatv, fχ are much smaller that fφ, we can neglect the backreaction of ∆and χ on the stringconfiguration described by φ and the right-handed neutral gauge boson ZR (these effectsare proportional to fχ/fφ).Thus the stability analysis of Sec.2 goes through withoutany changes except for the replacement (g, g′) →(gR, gB−L). This implies that there isa non-trivial region of the parameter space for which our model will admit metastablestrings.While the strings of the left-right model are stable to small perturbations, as it10

stands, it would appear that they are unstable to perturbations along the string. In otherwords, they are unstable to contraction (which leads to the annihilation of the monopole-antimonopole pair at the ends of the string).

Since we expect that the contraction timescale will be at most a Hubble time, if these strings are to be of any cosmological signifi-cance, we must find a way to make them more stable against this mode of instability. Onepossibility is that if the strings are superconducting, a standing wave of charge carrierscan be set up along the string, which reflects offthe monopoles at either end.

While thereflection coefficient is not unity, we can imagine that it is large enough so that it willtake some time before the string can rid itself of enough current so as to allow for it tocontract away. This mechanism for preventing the dynamical collapse of string loops hasbeen studied in some detail in earlier work and it has been shown that there is a regionof parameter space where current carrying loops can form static rings, or, vortons.

As wewill see below, it is these loops that will be most relevant from a cosmological perspective.To show that the string is superconducting we start by displaying the Higgs potentialfor the coupled φ −χ system (note that this system is remarkably similar to the tripletmajoron model14,15):V (φ, χ) = λ1(φ†φ −f2φ/2)2 + λ2(tr(χ†χ) −f2χ/2)2 + λ3(φ†φ −f2φ/2)(tr(χ†χ) −f2χ/2)+λ4(φ†φtr(χ†χ) −φ†χχ†φ) + λ5((tr(χ†χ))2 −tr(χ†χχ†χ)). (3.1)We parametrize φ and χ in the following fashion:φ =φ0φ−,(3.2)χ =χ+/√2χ++χ0−χ+/√2,(3.3)where the factors of√2 in χ have been chosen so that if Dµχ is the χ covariant derivative,then tr((Dµχ)†Dµχ) is the correctly normalized χ kinetic term.

Note that χ can alsobe written as χ =√2(T3χ+ + T+χ++ + T−χ0), where {T+, T−, T3} are the generators of11

SU(2)R, satisfying; [T3, T±] = ±T±, [T+, T−] = T3. The action of SU(2)R on χ is viacommutator: T aW aµ · χ ≡[T a, χ]W aµIf λ1,2,4,5 > 0 and |λ3| < 2√λ1λ2, then V (φ, χ) is positive definite and φ, χ acquirethe following VEV’s: ⟨φ0⟩= fφ/√2, ⟨χ0⟩= fχ/√2.We now claim that given this potential, there are large regions of the λ1,2,3,4,5 pa-rameter space for which χ+,++ act as bosonic charge carriers on the φ string.

To showthis, we use the following argument, first given by Witten16. First we show that it canbe energetically favorable for the components of χ to be nonzero in the core of the stringwhere φ = 0.

If φ = 0, the potential for χ reads:V (φ = 0, χ) = −λ1f2φ/2 + λ2(tr(χ†χ) −f2χ/2)2−λ3(f2φ/2)(tr(χ†χ) −f2χ/2) + λ5((tr(χ†χ))2 −tr(χ†χχ†χ))(3.4).This is extremized if (i) χ = 0, or (ii) either |χ0|2 = |χ++|2 or 2λ2(tr(χ†χ) −f2χ/2) =λ2f2φ/2. It is easy to see that if λ2f2χ + λ3f2φ/2 > 0, then χ = 0 is a maximum of thepotential.

Thus, in this case, non-zero values of χ are energetically preferred in the stringcore.The above analysis is not sufficient to show the existence of bosonic charge carriers.We must check to see that the kinetic term for χ also allows for a nonzero value of χ in thestring. We do this by showing that the equations of motion for χ, linearized around χ = 0,admit growing solutions.

This will then show that in the background of the φ string, χ isunstable to the formation of a nonzero condensate on the string. Let us first consider theχ++ equation of motion:−∂µ(∂µχ++ −iαR cos 2θRZµRχ++) = 2λ2(tr(χ†χ) −f2χ/2)χ+++λ3(fNO(ρ)2 −f2φ/2)χ++ + λ5(χ+2 + 2χ0χ++)χ0∗.

(3.5)Here αR ≡qg2R + g2B−L, θR is the right-handed version of the Weinberg angle and fNO(ρ)is the φ part of the string configuration.In the string, ZR takes the form ZR(ρ) =12

−(v(ρ)/ρ) ˆeθ, where v(ρ) is the Nielsen-Olesen configuration for the vector field. We nowlinearize eqn.

(3.5) around χ = 0 and take the following form for the perturbation of χ++:δχ++ = exp(−iω++t)g++(ρ). The linearized equation of motion for g++(ρ) reads:−∇2g++(ρ) + V(ρ)g++(ρ) = ω2++g++(ρ)(3.6),where V(ρ) is given by:V(ρ) = −λ2f2χ + λ3(fNO(ρ)2 −f2φ/2)(3.7),and ∇2 is the two dimensional Laplacian.

We see that at ρ = 0, V = −(λ2f2χ + λ3f2φ/2)and that V increases monotonically with ρ until it reaches the asymptotic value of −λ2f2χ.Thus, as long as λ2f2χ + λ3f2φ/2 > 0, V is negative definite and, as in Witten’s originalanalysis16, the two dimensional Schroedinger equation above for g++ will admit at leastone bound state with ω2++ < 0. Thus, χ++ is unstable to forming a condensate on thestring.

A similar analysis can be repeated for the other components of χ, with the resultthat under certain conditions, they too can condense onto the string (except for χ0, sinceit has a nonzero expectation value away from the string).13

4. Cosmological SpeculationsHere we speculate as to the possible cosmological implications of embedded strings.There are many uncertainties in outlining any cosmological scenario involving these stringsbecause of the model dependence of many of their characteristics.

Here we will contentourselves with outlining one of several possible scenarios in which embedded strings mightbe cosmologically relevant. We will also try to compare and contrast the formation andevolution of embedded strings with that of standard topological strings.Consider the production of strings in a phase transition in the early universe.Attemperatures above the phase transition temperature, the thermal fluctuations in the fieldswill spontaneously produce string-like configurations which will, however, decay just as fast.As the temperature decreases and the universe goes through the phase transition, some ofthe string-like configurations that were undergoing thermal fluctuations will freeze out andthus not decay.

It is these string configurations that may survive the phase transition andbe important for cosmology. This process of thermal production is the same for topologicalas well as embedded strings.There is, however, an important difference between topological and embedded strings.This is that topological strings cannot end whereas embedded strings may end on monopoles.Hence, after the phase transition, topological strings can only occur as closed loops or in-finite strings, while embedded strings can also occur as finite segments of strings withmonopoles attached at their ends.

This is the crucial difference between the two kinds ofstrings.We now discuss the formation of embedded strings. The first question is: what isthe size distribution of the embedded strings after the phase transition?

This questioncannot be answered with any certainty but some reasonable guesses can be made. As wediscussed above, the production of the strings is thermal and is similar in some ways to the14

production of topological strings; hence, it is prudent to first look at topological strings.In this case17, the string network upon formation consists of a network of infinite stringsthat contain about 80% of the entire string length. The remaining 20% goes into a scaleinvariant distribution of closed loops.

These results were obtained by using an argumentfirst given by Kibble18. In this argument, if the boundary conditions on a spatial contourare fixed, they determine whether there is a string passing through the contour almostunambiguously.19 So to detect the presence of a string, all one needs to check are theboundary conditions.This “Kibble” mechanism does not apply in the case of embedded strings because theboundary conditions are not sufficient to determine the presence of a string.

However,one might assume that there is a certain probability of a string passing through any givencontour. On this basis, one could attempt to use the results of Ref.

20. In the case thatthe probability of string formation is sufficiently low, there is a population of loops whoselength distribution is given by,dn(l) = ae−bl/ξξ2l2 dl .

(4.1)The dimensionless parameters a and b will depend on the probability of string formationwhile ξ is the correlation length at the phase transition which we will assume is givenroughly by T −1cwhere Tc is the temperature at which the phase transition occurs. If thephase transition is second order, the correlation length can actually exceed T −1cby ordersof magnitude, leading to a small value of b in the above equation.

Note that the exponentof l in the denominator is 2 and not 5/2 as might be expected from a scale invariantdistribution. When the string formation probability is low, the number density of openstrings will be similar to that in (4.1) but the overall amplitude will be suppressed by afactor exp(−(m−µm−1)/Tc) where, m is the mass of the monopole necessary to terminatea string and µ is the mass density of the string.

(The exponent is derived by the followingconsiderations: the energy cost in terminating a string is the mass of the monopole m but15

were the string not to terminate, the energy cost would have been the string density µmultiplied by the size of the monopole ≈m−1. ) If the mass of the monopole is large - thatis, if the strings are stabilized by a large potential barrier - the open segments of string arenegligible in number as compared to the closed loops.

It should also be remarked that theexponential suppression of long loops (and open segments) may be viewed as a Boltzmannfactor in the thermal production of embedded strings.If the probability of string formation is large, the loop distribution will be given by ascale-invariant distributiondn(l) = α e−βl/ξξ3/2l5/2dl . (4.2)If strings cannot terminate, a network of infinite strings would also be present.

However,since embedded strings can terminate, the length that would have been in infinite stringswould now be in finite segments of string. The length distribution of the finite segmentswould also be exponential since, at every step, there is a certain probability for the stringto end.

But, for large string formation probabilities, the total length in open segmentswould exceed the total length in loops. In the limit that the monopole becomes infinitelyheavy, the open segments would be infinitely long and the fraction of length in open stringswould approach 80%.There is another important feature of the string network that we have ignored so far:the strings are superconducting.

Then, during the phase transition, random currents willbe induced on the strings. The net current on a loop of size l is expected to be proportionalto T 2c (l/ξi)1/2 where ξi ∼T −1cis the correlation length of the random currents.

There willbe currents on the open strings also. However, it is not clear what happens to the currentwhen it encounters the monopole at the end of the string.

We expect that the current couldbe reflected offthe monopole and, in this way, a standing wave would be set up on the openstring. (In addition to the reflection, there might be a small transmission amplitude andthe current would slowly leak out from the string.) Another way of saying this is that the16

zero modes are a solution to the Dirac equation (or the Klein-Gordon equation for bosonicsuperconductivity) in the presence of the string. The string provides a potential well forthe zero mode carriers.

In the case of an open string, one might envisage the presence ofstanding wave solutions while in the case of a loop, one can imagine traveling waves goingaround the loop in addition to the standing waves.In the following we will assume that, after the phase transition, there is a loop distri-bution given by (4.1) and a strongly suppressed open string distribution also given by theform in (4.1). In addition, all the strings carry currents in proportion to the square rootof their length.What is the evolution of this system?Let us first consider the loops.

The dynamics of the loop is governed by the tensionof the string, frictional forces and the Hubble expansion.We are justified in ignoringthe Hubble expansion since it is unimportant on scales much smaller than the horizon. (The embedded string distribution is exponentially suppressed at long lengths and so itis unlikely to find strings that span the horizon.

Therefore, the Hubble expansion has nosignificant effect on the dynamics of embedded strings). Initially the frictional forces arevery large and so the string motion is highly damped.

This would lead to a collapse ofthe loops under their own tension. However, the effective tension of the string is a sum ofthe bare tension and the square of the current on the string21,22.

As the loop collapses,the current builds up and the effective tension becomes smaller. Now two possibilities canoccur21,22,23,24: (i) the current is so large that the charge carriers can leave the string -that is, the current can saturate, and, (ii) the effective tension goes to zero and the loopdoes not collapse any further.

If possibility (i) is realized, the loops continue to collapseand eventually disappear. Depending on the lifetime of the loops and their decay products,their cosmology may be of some interest.

If possibility (ii) is realized, the loops form staticring configurations that can survive until some quantum tunneling event causes the charge17

to leak. In this case, the rings would have a magnetic dipole moment and perhaps somenet electric charge and could survive for a very long time.

Depending on the net chargethat a ring carries, the rather severe constraints on charged dark matter (CHAMPS) wouldapply25.The evolution of the open segments of strings is even less certain than that of the loopsbut we shall indicate some possible scenarios. The initial dynamics of the monopoles andopen segments will be heavily damped due to the friction from the ambient plasma.

Thelong range magnetic field of the monopoles will be frozen into the cosmological plasma.The tension in the open strings will shrink the segments, bringing the monopoles andantimonopoles at the ends together. One possibility is that the current and the chargein the segment would prevent the segment from shrinking any further as happens in thecase of the loop.

Then the segment would form a dumbbell and survive for a very longtime. On the other hand, if the current leaks out through the monopoles, the segmentwould collapse rather quickly since the frictional forces cannot slow down the longitudinalmotion of the string but only the transverse motion.

In this scenario, the open segmentsdecay soon after forming and disappear. The disappearance of open segments (and loops)would also be hastened by the breaking up of long strings by the spontaneous nucleationof monopole and antimonopole pairs.

However, we might assume that this process will beslow (compared to the direct collapse of a segment) since it requires a monopole pair tonucleate by a quantum process.There is yet another alternative to this entire scenario which follows from the stabilityanalysis in Sec. 2.

From Fig. 2, we see that it is possible for the strings to be stable at hightemperatures and unstable at low temperatures.

Then the string network - the rings anddumbbells - would behave like unstable particles with a life-time given by the time it takesfor the universe to cool down to the temperature of instability. Unstable particles havebeen considered on numerous occasions in cosmology, particularly as a means for generating18

additional entropy. It is amusing that embedded strings would be natural candidates forsuch unstable particles.What may be the consequences of long-lived rings and dumbbells?

The most obviousconsequence is that these objects may be the dark matter of the universe and may stillbe around today.They may be lurking in stars and in galactic halos.On the otherhand, since these objects are formed in the early universe, there is a chance that theywill come to dominate the universe rather early (since they redshift as matter). In thiscase, their cosmology might be useful to constrain particle physics models - though, giventhe uncertainties, this promises to be a difficult task.

Finally, the decay of the rings anddumbbells would produce energetic exotic particles. These decay products might lead tointeresting effects.

Finally, the presence of strings for some period of time could lead tobaryogenesis26.5. ConclusionsTopological strings can have dramatic consequences in the early universe but can occuronly in certain specially constructed particle physics models.

On the other hand, embeddedstrings are almost universal in their occurrence but their consequences depend on theirstability. For the embedded string to have some affect on cosmology, it should survive forone Hubble time at the very least.

This criterion makes it necessary to study the stabilityof the electroweak Z-string at high temperatures.We have analyzed the stability of the Z-string at high temperatures and also takenquantum corrections to the scalar potential into account. The analysis shows that thermalcorrections tend to enhance stability but the effect is too small to stabilize the Z-stringin the standard electroweak model with sin2 θW ≈0.23 and Higgs mass larger than 57GeV.

Hence, we come to the conclusion that the Z-string is unstable at all temperatures.Then, even if a Z-string configuration is formed during the electroweak phase transition,19

it will quickly decay into particles and the string will not survive for more than a Hubbletime. This means that Z-strings are probably irrelevant for cosmology after the electroweakphase transition; their role during the electroweak phase transition is still unclear.We found that it is possible to construct phenomenologically acceptable left-rightmodels that also admit stable embedded strings (ZR−strings).

Due to its stability theZR−string may survive for a large number of Hubble times and may be cosmologicallysignificant.The cosmology of embedded strings was discussed in Sec. 4.

Here, we pointed outthat embedded strings would be produced thermally during the phase transition. TheBoltzmann suppression of long string segments and large loops means that there is apossibility that all the loops and segments will collapse dynamically and decay into ordinaryparticles.

On the other hand, the pressure from bosonic and fermionic zero modes on thestrings might prevent this collapse and serve to stabilize the “rings” and “dumb-bells”. Weconsidered the more interesting possibility that some of these remnants may have survivedfor a few Hubble times and perhaps even until the present epoch.Acknowledgements:We would like to thank Ed Copeland, Rick Davis, and Mark Hindmarsh for discussionand the ITP, Santa Barbara where this work was partly done.

This work was supportedin part by the National Science Foundation under Grant No.PHY89-04035. RH waspartially supported by DOE contract DE-FG02-91ER40682.

SDH acknowledges supportfrom the National Science Foundation under grant NSF-PHY-87-14654, the state of Texasunder grant TNRLC-RGFY106 and the Harvard Society of Fellows. The work of RW wassupported in part by the NASA (NAGW-1340 at Fermilab) and by the DOE (at Chicagoand Fermilab).20

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Figure Captions1. Regions of stability for various values of Pe.Plotted are the boundaries betweenstability and instability for (from left to right) Pe = 0.5, 0.4, 0.25, 0, −0.5, −1.0.

Stringsare (meta)stable for parameters to the right of the curves.2. Regions of stability for various temperatures for fixed D/α2 = 0.224 and E/α2 = 0.019.Strings are stable in regions below the solid lines and to the right of the dashed line.The dashed line corresponds to the boundary of stability at the phase transition.24

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