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MONOPOLE ANNIHILATION AT THE

arXiv:hep-ph/9203208v1 10 Mar 1992MONOPOLE ANNIHILATION AT THEELECTROWEAK SCALE—NOT!Evalyn Gates, Lawrence M. Krauss∗, and John Terning†Center for Theoretical PhysicsSloane Physics LaboratoryYale UniversityNew Haven, CT 06511February 1992YCTP-P3-92AbstractWe examine the issue of monopole annihilation at the electroweakscale induced by flux tube confinement, concentrating first on the sim-plest possibility—one which requires no new physics beyond the standardmodel. Monopoles existing at the time of the electroweak phase transi-tion may trigger W condensation which can confine magnetic flux intoflux tubes.

However we show on very general grounds, using several inde-pendent estimates, that such a mechanism is impotent. We then presentseveral general dynamical arguments constraining the possibility of mo-nopole annihilation through any confining phase near the electroweakscale.∗Also Department of Astronomy.

Research supported in part by the NSF, DOE, and the TexasNational Research Laboratory Commission. Bitnet: Krauss@Yalehep.†Research supported by NSERC.1

The “monopole problem” has been with us since the advent of Grand UnifiedTheories (GUTs), which allow the formation of these non-singular stable topologicaldefects when a semi-simple gauge group is broken to a lower symmetry group thatincludes an explicit U(1) factor. These objects typically have a mass mM ≃mX/α,where mX is the mass of the gauge bosons in the spontaneously broken GUT theory,and α is the fine structure constant associated with the gauge coupling of the theory.Shortly after it was recognized that monopoles could result as stable particlesin spontaneously broken GUT models [1], and also that they would be produced inprofusion during the phase transition associated with the GUT symmetry breakingin the early universe [2], it was also recognized that they posed a potential problemfor cosmology.

Comparing annihilation rates with the expansion rate of the universeafter a GUT transition, it was shown [3, 4] that the monopole to photon ratiowould “freeze out” at a level of roughly 10−10. Not only would such an initial levelresult in a cosmic mass density today which is orders of magnitude larger than thepresent upper limit, but direct observational limits on the monopole abundance inour neighborhood are even more stringent [5].This cosmological problem was one of the main motivations for the originalinflationary scenario[6].

However one of the chief challenges to the original infla-tionary solution of the monopole problem was the necessity of having a reheatingtemperature which is high enough to allow baryogenesis, but low enough to suppressmonopole production. In addition, recent work on large scale structure, includingobserved galaxy clustering at large scales, large scale velocity flows, and the ab-sence of any observable anisotropy in the microwave background, has put strongconstraints on such models.With the recent recognition that even something as exotic as baryogenesismay be possible within the context of the standard electroweak theory (supple-mented by minor additions), it is worth examining the issue of whether the mo-nopole problem may be resolved purely through low energy physics.

A canonicalmethod by which one might hope to achieve complete annihilation is by confiningmonopole-antimonopole pairs in flux tubes, such as might occur if U(1)em were bro-ken during some period. Proposals along this line, based on introducing new physics2

have been made in the past, eg. [7, 8].

Most recently, the possibility that such aphase might briefly occur near the electroweak breaking scale, for multi-Higgs mod-els, has also been raised [10]. By far the simplest possibility, however, is that fluxtube confinement of monopoles might occur in the standard model unsupplementedby any new physics.

We explore this issue in detail here, and then go on to examinethe general dynamical obstacles facing any model involving monopole confinementat the electroweak scale.1. Monopole Confinement in the Standard Model:It has been known for some time that the electroweak vacuum in the bro-ken phase is unstable in the presence of large (≥m2W/e) magnetic fields[11].

Theinstability is due to the coupling between the magnetic field H and the magneticmoment of the massive W gauge bosons. Due to this coupling the effective mass ofthe W at tree level ism2Weff = m2W −eH(1)where e = g sin θW (all expressions are given in Heaviside-Lorentz units for electro-magnetism).

This effective mass squared becomes negative for H(1)c≥m2W/e. Thegeneral resolution [11] of the instability is the formation of a condensate of W and Zbosons, which sets up currents that antiscreen the magnetic field.

The vacuum thenacts as an anti-type II superconductor, and the energy is minimized by the formationof a periodic network of magnetic flux tubes. As we shall describe in some lengthlater, Ambjørn and Olesen have also shown, at least for the special case mH = mZ,that if the magnetic field increases above H(2)c=m2We cos2 θW , the full SU(2)L ⊗U(1)symmetry is restored [12].

Thus for an external magnetic field H(1)c< H < H(2)c ,the electroweak vacuum passes through a transition region where a W condensateexists and the magnetic field is confined in a periodic network of flux tubes.It is possible to imagine how such a phase might arise naturally in a waywhich might lead to monopole-antimonopole annihilation at the electroweak scalein the early universe (This idea has also been suggested elsewhere in the literature[9]). First of all, the magnetic field necessary to produce such a phase could comefrom the monopoles themselves, provided the electroweak transition is second order.3

In this case, the mass of the W boson generically has a temperature dependence ofthe formm2W(T) ≈m2W(0)[1 −T 2/T 2c ](2)where Tc (≈300 GeV) is the critical temperature associated with electroweak break-ing. Thus, just below the transition temperature Tc, relatively small magnetic fieldscould trigger W condensation.

A remnant density of GUT-scale monopoles couldprovide such a magnetic field. Once the condensate forms, monopoles would becomeconfined to the network of flux tubes, whose width is related to the W mass, as weshall describe.

Once the width of the flux tubes is of the same order as the distancebetween monopoles, the monopoles would experience a linear potential and beginto move towards each other. If the flux tubes exist for a sufficiently long time, themonopoles could annihilate, and their density would correspondingly decrease.This picture is very attractive in principle.

However, we now demonstrate,using a series of arguments which probe this scenario in successively greater detail,that the parameters associated with such a transition at the electroweak scale gener-ally preclude it from being operational. Moreover, we present dynamical argumentsrelevant for any scenario involving monopole annihilation via flux tube confinementat the electroweak scale.2.

Kinemetic Arguments: Non-annihilation via Magnetic Instabilities:(a) A Global Argument: In figure 1, we display a phase diagram describingthe W condensation picture discussed above, as a function of both temperature Tand background magnetic field H. At T = 0, for the case examined by Ambjørnand Oleson[12, 13, 14], in the region 1 < He/m2W < 1/ cos2 θW a magnetic fluxtube network extremizes the energy and both the φ (Higgs) and |W|2 fields developnon-zero expectation values. For finite temperature the phase boundaries evolve asshown, in response to the reduction in the W mass with temperature, up to T = Tc,where they meet.

Thus, the phase in which flux tubes and a W condensate areenergetically preferred falls in between these two curves.While the actual magnetic field due to the presence of a density of monopolesand anti-monopoles will be complicated and inhomogenous, we first approximate4

it by a homogenous mean field Hm, whose precise value is not important for thisdiscussion. (As we will later show, given the remnant density of monopoles predictedto result from a GUT transition, the value of this field will be well below the zerotemperature critical field m2W(0)/e at the time of the electroweak phase transition.

)As the universe cools from above Tc, this background magnetic field will eventuallycross the upper critical curve for the existence of a flux tube phase.We now imagine that immediately after this happens, flux tubes form,and monopole annihilation instantaneously begins. We shall later show that thisis far from the actual case.Nevertheless, this assumption allows us to exam-ine constraints on monopole annihilation even in the most optimistic case.Asmonopole-antimonopole annihilation proceeds, the mean background magnetic fieldfalls quickly.

At a certain point this mean field will fall below the lower critical curve,and if it is this background field which governs the energetics of W condensation, theW condensate will then become unstable, the magnetic field lines will once againspread out, and monopole-antimonopole annihilation will cease. As can be seenin the figure, the net reduction in the magnetic field expected from this period ofannihilation will be minimal.

Quantitatively the final field (neglecting dilution dueto expansion during this period) will be a factor of cos2 θW smaller than the initialfield. This is hardly sufficient to reduce the initial abundance of monopoles by themany orders of magnitude required to be consistent with current observations.

(b) A Local Argument: The above argument points out the central problemfor a monopole annihilation scenario based on magnetic field instabilities at theelectroweak scale. In order to arrange for flux tubes to form, and confinement ofmonopoles to occur, the field must be tuned to lie in a relatively narrow region ofparameter space.

Nevertheless, a potential problem with the above argument, evenif it were less sketchy, is that flux tube formation, and monopole annihilation, maymore likely be related to local and not global field strengths. For example, evenif the globally averaged magnetic field is reduced by annihilation, the local fieldbetween a monopole-antimonopole pair connected by a flux tube may remain abovethe critical field, so that the flux tube will presumably persist, and annihilation canproceed.

We now demonstrate that even under the most optimistic assumptions5

about the magnitude of local fields, for almost all of electroweak parameter space,local flux tube formation at a level capable of producing a confining potential be-tween monopole- antimonopole pairs will not occur. We first consider the case forwhich solutions (involving a periodic flux tube network) were explicitly obtained byAmbjørn and Oleson[13].The area A of flux tubes forming due to the W condensate can be obtained byminimizing the classical field energy averaged over each cell in the periodic networkin the presence of a background H field [13]:EminA = m2WeZceℓℓf12d2x −m4W2e2 A + λ −g28cos2θW!

Z φ2 −φ202 d2x,(3)where f12 is the magnetic field, and λ is the φ4-coupling in the Lagrangian, and φ0is the Higgs VEV. Utilizing the topological restrictions on the flux contained in theflux tubes (containing minimal flux 2π/e),Zceℓℓf12d2x =I⃗A · ⃗dℓ= 2π/e,(4)this yields an expression for A, determined by the energy density Emin, which is inturn a function of the external magnetic field:A =2πm2We2hEmin + m4W/2e2 −λ −g28cos2θW R (φ2 −φ20)2 d2xi.

(5)Taking the Bogomol’nyi limit[15] λ =g28cos2θW , corresponding to mH = mZ, the clas-sical field equations simplify, and the properties of the flux tubes can be derived.In particular, one can show [12] that the area of the flux tubes is restricted to lie inthe range2π cos2 θW < Am2W < 2π. (6)From our point of view, it is important to realize that this result is equivalent tothe statement that a W condensate can only exist between the two critical valuesof the magnetic fieldm2We cos2 θW> H > m2We .

(7)Moreover, it gives a one to one correspondence between the area of the flux tube andthe background magnetic field value in this range. We shall use this correspondence,6

both in the Bogomol’nyi limit and beyond, to examine the confinement propertiesof such a flux tube network connecting monopole-antimonopole pairs.Magnetic monopoles are formed at the GUT transition with a density ofabout one monopole per horizon volume.This corresponds to a value ofnMs=10.4g∗1/2(TGUT/MP l)3 ∼102(TGUT/MP l)3, where nM is the number density of mo-nopoles, g∗is approximately the number of helicity states in the radiation at the timetGUT, MP l is the Planck mass, and s is the entropy of the universe at this time. SinceTGUT could easily exceed 1015 GeV for SUSY GUTs, it is quite possible that the ini-tial monopole abundance left over from a GUT transition is nMs > 10−10.

Preskill hasshown that in this case monopoles will annihilate shortly after the GUT transitionuntil nMs ∼10−10[3], and this value remains constant down to the electroweak scale.Since s = (2π2/45)g∗T 3, the monopole number density at the electroweak transition(Tc ∼300GeV ) of ≈0.13 GeV 3 (assuming g∗(Tc) ≈100) corresponds to a meanintermonopole spacing of L ≈2 GeV −1. From this, we can calculate the meanmagnetic field produced by the monopoles with Dirac charge h = 2π/e.

In general,because the monopole background is best described as a “plasma” involving bothmonopoles and antimonopoles, the mean magnetic field will be screened at distanceslarge compared to the intermonopole spacing. However, because we will demonstratethat even under the most optimistic assumptions, monopole-antimonopole annihi-lation will not in general occur, we ignore this mean field long-range screening, andconsider the local field in the region between a monopole-antimonopole pair to bepredominantly that of nearest neighbors, i.e.

a magnetic dipole. While the fieldis not uniform in the region between the monopole and antimonopole, we will beinterested in the minimum value of the field here.

We shall make the (optimistic)assumption that if this field everywhere exceeds the critical value m2W/e on the linejoining the two monopoles, that an instability of the type described above, involvinga condensate of W fields and an associated magnetic flux tube, can occur along thisline.For a monopole-antimonopole pair separated by a distance L, the minimumfield will be halfway between them, and will have a magnitude H = 2h/πL2 = 4/eL2.For this field to exceed the minimum Ambjørn-Oleson field m2W/e then implies7

the relation: L < 2/mW. For a value mW = 81GeV this relation is manifestlynot satisfied for the value of L determined above.However, assuming a secondorder transition, as we have described, the W mass increases continuously from zeroas the temperature decreases below the critical temperature, implying some finitetemperature range over which the (fixed) background field due to monopoles will liein the critical range for flux tube formation.

In this case, the magnetic field wouldenter this range from above. In order that the magnetic field lie in the range givenby inequality (7), we find2/mW < L < 2 cos θW/mW(8)Nevertheless, even if a flux tube forms connecting the monopole-antimonopolepair, this will not result in a confining linear potential until the width of the fluxtube 2r < L. A bound on this width can be obtained from the lower bound on thearea of the flux tube (equation (6)):2r > 2√2 cos θW/mW .

(9)when the magnetic field is at its upper critical value of m2W/e cos2 θW. This impliesthe constraintL > 2√2 cos θW/mW .

(10)As can be seen, inequalities (8) and (10) are mutually inconsistent. Hence,there appears to be no region in which both a Ambjørn-Oleson type superconduct-ing phase results, and at the same time monopole-antimonopole pairs experiencea confining potential.

We expect the situation will be similar to the quark-hadronphase transition when the transition is second order. In that case, it is impossibleto distinguish between a dense plasma of confined quarks and a gas of free quarks,because the mean interquark spacing is small compared to the confinement scale.Here there will be no physical impact of a short superconducting phase, because theconfinement scale is larger than the distance between monopoles required to triggerthe phase transition.

We expect no significant monopole annihilation during theshort time in which this phase is dynamically favored as the W mass increases.8

This result has been derived in the Bogomol’nyi limit, when mH = mZ. Whatabout going beyond this limit?

First, note that the energy density of the externalmagnetic field, E = H2/2, provides an upper bound on Emin. Then from equation(5) one can show that as long as λ > g2/8 cos2 θW (mH > mZ), the flux tube area,for a fixed value of the field, is larger than it is in the Bogomol’nyi limit.

While wehave no analytic estimate of the upper critical field, and hence no lower bound on theflux tube area, the scaling between area and magnetic field will still be such that fora given monopole-antimonopole spacing, and hence a given magnetic field strength,the area of the corresponding flux tube will be larger than in the Bogomol’nyi limit.Hence the inconsistency derived above will be exacerbated.Only in the narrowrange mZ/2 <∼mH < mZ (still allowed by experiment) is there a remote possibilitythat even in principle, flux tube areas may be reduced sufficently so that confiningpotentials may be experienced by monopoles triggering a W condensate. However,in this range, the energy (4) can be reduced by increasing φ, so we expect thatinstabilities arise in this range which are likely to make a W condensate unstable inany case.3.

Dynamical Arguments Against Annihilation:Even if a confining potential may be achieved through flux tube formation,there are dynamical reasons to expect monopole annihilation will not be complete.These arguments apply to any scenario involving a confining phase for monopoles,and suggest that estimates based on the efficacy of monopole annihilation may beoverly optimistic. In the first place, we can estimate the energy of a monopole-antimonopole pair separated by a string of length L. For a long flux tube of radiusr, considerations of the electromagnetic field energy trapped in the tube imply a netenergy stored in the flux tube ofE =L2αr2.

(11)Considering the case when L ≈2r, when confinement would first begin, we find theenergy associated with the string tension is E =2αL ≈130 GeV . This is significantlysmaller than the mean thermal energy associated with a transition temperature Tc ≈300 GeV .

Hence, if the string tension does not vary significantly over the period9

during which the magnetic field exceeds the critical field, the string tension exerts aminor perturbation on the mean thermal motion of monopoles, and hence will notdramatically affect their dynamics. The only way this would not be the case wouldbe if the monopole-antimonopole pair moved towards one another at a rate whichcould keep the magnetic field between them sufficiently large so as to track theincrease in the minimum critical field as mW(T) increased to its asymptotic value.However, this cannot in general occur, because thermal velocities are sufficientlylarge so as to swamp the motion of the monopole-antimonopole towards each other.Using the mean thermal relative velocity of monopoles at T = Tc, we can calculatehow much time, δt, it would take to traverse a distance equal to the initial meandistance between monopoles.

Since the thermal velocity is ≪1, non-relativisticarguments are sufficient. We find δt/t ≈4.6 × 10−6, for mM ≈1017GeV , and Tc ≈300 GeV .

During such a small time interval, mW(T) remains roughly constant, andhence so does the string tension. We find that during the time δt the flux tubeinduced velocity of the monopole-antimonopole pair remains a small fraction of themean thermal velocity, for mM > 1015GeV .

Thus, monopoles and antimonopoleswill not in general move towards one another as mW increases.Since r(T) willnot change significantly between H(1)cand H(2)cas mW increases, if the mean inter-monopole spacing remains roughly constant, monopole annihilation will, on average,not proceed before the field drops below its critical value.What about the more general case of a brief superconducting phase whichmight result if U(1)em is broken for a small temperature range around the elec-troweak scale [10]? In this case, the flux tube area is not driven by the strengthof the background magnetic field, and hence is not tied to monopole-antimonopolespacing.

Nevertheless, dynamical arguments suggest that annihilation, even in thiscase, may be problematic. We describe three obstacles here: (a) as above, the fieldenergy contained in the string may not be enough to significantly alter the dynam-ics of a thermal distribution of monopoles; (b) even in the event that this energy issufficiently large, the time required to dissipate this energy will in general exceedthe lifetime of the universe at the time of the U(1)em breaking transition; (c) thetime required for monopoles to annihilate even once they have dissipated most ofthe string energy and are confined within a “bag” may itself be comparable to the10

lifetime of the universe at the time of the transition. (a) Consider the energy (11) stored in the flux tube.The radius, r, willdepend upon the magnitude of the VEV of the field responsible for breaking U(1)em.If this symmetry breaking involves a second order transition, then until this fieldachieves a certain minimum value, flux tubes will not be sufficiently thin to producea confining potential for monopoles.

Moreover, even if this VEV quickly achievesits maximum value, one must investigate whether or not this field energy is largecompared to the thermal energy at that time, in order to determine whether themonopoles will be dynamically driven towards each other. As long as r−1 ≈eφ0 ≈eTc, where in this case φ0 represents the VEV if the field associated with U(1)embreaking and Tc represents the transition temperature, then E ≫T, so that thecondition of a confining potential is in general satisfied.

Nevertheless, one mustalso verify that this inequality is such that the Boltzmann tail of the monopoledistribution with velocities large enough to be comparable to this binding energy issufficiently small (i.e. that sufficiently few monopoles have thermal motion whichis not significantly affected by the confining potential).If we assume that suchmonopoles do not annihilate, then to avoid the stringent limits on the monopoledensity today probably requires E > O(30)T. Determining L by scaling from theinitial density, we find that if φ0 = ρTc, then the ratio of the binding energy to thetransition temperature, E/Tc ≈3800ρ2, independent of Tc.

This implies a rathermild constraint on the VEV of the field associated with U(1)em breaking: ρ > 0.09. (b) Monopole’s must dissipate the large energy associated with the stringfield energy if they are to annihilate.

There are two possible ways in which thisenergy can be dissipated: thermal scattering, and the emission of radiation [3, 16].Utilizing the estimates of energy loss by radiation given by Vilenkin [16] we findthat this process requires ≈1015 times longer to dissipate the string energy thanthe lifetime of the universe at the time of the transition.1 Hence, we concentrateon the possibility of dissipating the energy by thermal scattering. We shall assumehere that ρ ≈1, so that the initial average monopole- antimonopole pair energy1This calculation itself is probably an underestimate (unless the monopole couples to masslessor light particles other than the photon), since it assumes the photon is massless, which it is notin this phase.11

is ≈3800T. The energy loss by collisions with thermal particles in the bath is[16] dE/dt ≈−bT 2v2, where b = 3ζ(3)/(4π2) P (qi/2)2, and the sum is over allhelicity states of charged particles in the heat bath.

At T ≈100 GeV , b ≈0.7.Utilizing the relationship between temperature and time in a radiation dominatedFriedmann-Robertson-Walker universe, we then findlnEfEi= 0.03bMP l2mMln titf!. (12)We will take Ef to be the string energy (11) when L = 2r, i.e.

the energy whenthe string has become a “bag”. This implies that the time required to dissipate theinitial string energy is O(50) ti for mM ≈1017GeV .

Unless the phase of brokenU(1)em lasts for longer than this time (which does depend sensitively upon themonopole mass), not all the string energy will be dissipated. We have ignored herepossible transverse motion of the string.

This energy must also be dissipated byfriction, which may be dominated by Aharanov-Bohm type scattering[17].2 In anycase, this is a rather severe constraint on the temperature range over which theU(1)em breaking phase must last. (3) Once the string energy is dissipated so that the mean distance betweenmonopole-antimonopole pairs is of order of the string width, they will be confinedin a “bag”, and one must estimate the actual time it takes for the pair to annihilatein such a “bag” state.

(The monopole “crust”, of characteristic size mW −1, isassumed to play a negligible role here. In any case, inside this “bag” it is quitepossible that the electroweak symmetry may be restored, in which case such a crustwould not be present.) In a low lying s-wave state, the annihilation time is veryshort.

However, in an excited state, involving, for example, high orbital angularmomentum, this may not be the case, since the wave function at the origin will behighly suppressed. We provide here one approximate estimate for the annihilationtime based on the observation that the Coulomb capture distance ac ≈1/4αE is 8times smaller than the “bag” size, for a monopole whose “bag” energy is inferredfrom equation (11) with L = 2r.It is reasonable to suppose that annihilationmight proceed via collapse into a tightly bound Coulomb state.

Thus, for the sake2We have been informed that this issue is being treated in detail by R. Holman, T. Kibble, andS.-J. Rey[18].12

of argument one might roughly estimate a lower limit on the annihilation time byutilizing the Coulomb capture cross section[3] inside the “bag”. This capture timeis τ ≈(4e/3πT)(mM/T)11/10, and is slightly longer than the lifetime of the universeat temperature T ≈300GeV , for mM = 1017GeV .

Again, this suggests that thetime during which the U(1)em breaking phase endures must be long compared tothe lifetime of the universe when this phase begins3. If capture into a Coulomb statehas not occured by the time the U(1)em breaking phase is over, previously confinedmonopole pairs separated by more than the Coulomb capture distance will no longerbe bound.

The annihilation rate for these previously confined pairs compared tothe expansion rate will remain less than order unity, so that monopoles will againfreeze outThese considerations suggest that monopole-antimonopole annihilation byflux tube formation at the electroweak scale is far from guaranteed. In particular,monopole confinement triggered by monopole induced magnetic fields seems un-workable.

More generally, in any confining scenario, dissipation of the initially largeflux tube energies requires times which are generally long compared to the horizontime at the epoch of electroweak symmetry breaking. This places strong constraintson the minimum range of temperatures over which a confining phase for monopolesmust exist.3If one imagines that because of the monopole outer crust, emission of scalars is possible, thecapture cross section may be increased[3] to ≈(Tc)−2.

This would decrease the capture time by asignificant amount (≈106). However, once again, this requires that the scalars are light, otherwisephase space suppression might be important.13

Figure CaptionsFig. 1.

Phase diagram for W condensation as a function of external magneticfield and temperature assuming a second order electroweak phase transtion.References[1] G. ’t Hooft, Nucl. Phys.

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