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R. Holmana,b, T.W.B. Kibblea,c and Soo-Jong Reya,d,∗

arXiv:hep-ph/9203209v1 11 Mar 1992NSF-ITP-09-92YCTP-P06-92Imperial/TP/91-92/18CMU-HEP92-03February, 1992HOW EFFICIENT IS THELANGACKER-PI MECHANISMOF MONOPOLE ANNIHILATION?R. Holmana,b, T.W.B.

Kibblea,c and Soo-Jong Reya,d,∗aInstitute for Theoretical Physics, University of California,Santa Barbara CA 93106bPhysics Department, Carnegie-MellonUniversity, Pittsburgh PA 15213cBlackett Laboratory, Imperial College, London SW7 2BZ, UKdCenter for Theoretical Physics, YaleUniversity, New Haven CT 06511ABSTRACT: We investigate the dynamics of monopole annihilation by the Langacker-Pi mechanism.We find that considerations of causality, flux-tube energetics and thefriction from Aharonov-Bohm scattering suggest that the monopole annihilation is mostefficient if electromagnetism is spontaneously broken at the lowest temperature (Tem ≈106 GeV) consistent with not having the monopoles dominate the energy density of theuniverse.submitted to Physical Review Letters* Yale-Brookhaven SSC Fellow

As is well known, all grand unified theories (GUT’s) must of necessity give rise to ’tHooft-Polyakov magnetic monopole solitonsmonopoles. As a practical matter, these willarise whenever a U(1) subgroup appears after spontaneous symmetry breaking (a moregeneral criterion involves the second homotopy group of the vacuum manifoldvilenkinrev).From a cosmological viewpoint, these monopoles are disastrous.

They have a massmM ∼MGUT ∼1016 GeV and since they are created via the misalignment of the Higgsfields in different horizon volumeskibble, we expect to have at least one monopole per horizonat the time of the GUT phase transition giving rise to the monopoles. These two factsthen lead us to the conclusion that the universe would have become monopole dominatedlong ago and recollapsed shortly thereaftermonopoleproblem.Historically, the monopole problem was an important factor in arriving at the infla-tionary universe scenario.Indeed, with an appropriate amount of supercooling (as inthe case of a first order phase transition), the monopole number density could be dilutedaway.

However, there are other solutions to the monopole problem. In particular, Lan-gacker and Pilangackerpi proposed such a solution some time ago.

They argued that if theelectromagnetic gauge group U(1)em were broken for a period of time and then restored,then monopole-antimonopole pairs would become bound by flux tubes and then annihilateeach other. Recently, there has been a revival of interest in this work from a variety ofstandpointsvilenkin,sriva,weinberg,sher,kephart,turok.Our aim in this Letter is to elucidate some points concerning the efficiency of theLangacker-Pi mechanism and in particular, discuss the issue of when U(1)em should bebroken.

The results of our analysis are rather surprising (at least to us! ): the time tem atwhich U(1)em is broken should be postponed as long as possible, i.e., , until just beforethe monopoles begin to dominate the energy density of the universe!This is rather counter-intuitive; the natural expectation, given the energetics of themonopole-flux tube system is that the temperature Tem corresponding to the time tem2

should be as close to the GUT phase transition temperature TM as possible. The reasonfor this is that the tension in the flux tube is ∼T 2em.

Thus the force between monopolesis stronger for larger Tem. However, this cursory analysis neglects some important fac-tors, such as the role of Aharonov-Bohm scattering by the flux tube, in determining theannihilation efficiency.

It is to these issues we now turn.1. Causality Efficiency: Let us suppose that U(1)em is broken spontaneously at a tem-perature Tem well below the monopole production scale TM.

The magnetic monopoles wereproduced with an initial density nM(TM) ≈O(1)ξ−3(TM) , where ξ(T) is the correlationlength of the Higgs field at temperature T. While the actual value of ξ(T) depends sensi-tively on the nature of the GUT phase transition, we can use causality to bound it aboveby the horizon size 2t(TM), where t(T) ≈0.03MPl/T 2 during the radiation dominated era.This yields the following lower bound on the monopole number density at creation:nM(TM) ≥O(104) T 6MM3Pl. (1)If U(1)em were broken immediately right after the GUT phase transition, there would notbe enough monopoles available to be connected by the flux tubes within a Hubble timescale.

On the other hand, at later times when the Universe cools down to a temperatureT, the total monopole number inside the horizon grows asNM(T) ≈O(1)TMT3. (2)The ever increasing total monopole number inside the horizon at temperature T << TMimplies that the flux tube network is easily formed within a Hubble time scale.

For example,when the temperature T ≈106 GeV, at which the Universe starts to become monopole-dominated, the total monopole number inside a horizon is ≈1030!2. Energetic Efficiency: When U(1)em is spontaneously broken, the flux tube connect-ing a monopole–anti-monopole pair provides a linearly increasing confining potential.

The3

string tension µ isµ ≈T 2em. (3)If Tem is much less than TM, the motion of the monopole pair is described by Newton’sequation of motionmMd2l(t)dt2= Fconf ≈−T 2em.

(4)Here l(t) denotes the monopole–anti-monopole separation (which is the same as the fluxtube length). The initial separation l(tem) should be of the same order of magnitude asthe mean separation distance among the monopoles:⟨l(tem)⟩≈[nM(Tem)]−1/3≈( TMTem)ξ(TM)≈MPl20TemTM.

(5)The energy stored inside the flux tube isEflux ≡µ(tem)⟨l(tem)⟩≈MPl20TM· Tem. (6)We should mention that if the length in Eq.

(5) is long enough so that the energy containedin the flux tube is larger than 2mM, it becomes energetically favorable for the tube to breakvia monopole pair creation. We see from Eq.

(5) that this happens when Tem > 400T 2M/MPl ≈TM/25. In this case the flux tube may move relativistically and the mean separationafter monopole pair creation by the tube is⟨l(tem)⟩r ≈20TMT 2em≈(20TMTem) · ( 1Tem).

(7)We should emphasize that this only happens if Tem is rather close to TM.From Eq. (4), we find that the characteristic time scale τa for mono poles and anti-monopoles to annihilate (assuming an efficient energy dissipation mechanism; see below)4

isτa ≈mM⟨l(tem)⟩T 2em1/2≈MPlT 3em1/2. (8)Comparing this with the Hubble time scale τH ≈2tem, we findτaτH≈30 TemMPl1/2.

(9)Hence, the monopole annihilation rate becomes larger as Tem becomes lower!Intuitively, this can be understood as follows. The energetics argument based on theflux tube string tension effect favors having Tem as close to TM as possible.

On the otherhand, the formation of a network of monopoles connected by flux tubes favors lower valuesof Tem as can be seen from Eq. (2).

This is a direct consequence of the slowing expansionrate of the Universe. The two effects compete with each other, but the latter dominatesat lower temperatures.

Indeed, using Eq. (2), one can rewrite Eq.

(9) as τaτH3≈3 × 104 TMMPl3/21pN(tem). (10)This clearly shows that the monopole annihilation rate depends only upon the instanta-neous total monopole number within the horizon.3.

Thermal Fluctuations: So far, we have not taken into account the effects of the ther-mal bath on the monopoles. These are important since the thermal energy of monopolesprovides transverse velocity to the flux tubes, and thus nonzero angular momentum to themonopole pair connected by the flux tube.

First of all, monopoles at a temperature Temare expected to be in good thermal contact with the background photons and the ambientplasma. Indeed, the strength of monopole-photon interaction is of order unity, and thecross-section for charged plasma-monopole interactions is correspondingly O(α−1em) largerthan that among charged particles.5

Thus, the initial kinetic and potential energies of the magnetic monopoles at temper-ature Tem << 125TM areK ≈Tem,V ≈T 2em⟨l(Tem)⟩≈500Tem. (11)The typical transverse momentum of the monopoles due to thermal motion is P⊥(Tem) ≈(20TMTem)1/2.

Thus, the initial angular momentum of the flux tube-monopole pair readsL ≈⟨l(tem)⟩P⊥(tem)≈ M2Pl20TMTem!1/2. (12)In the absence of friction, energy and angular momentum conservation lead to a final meanseparation⟨⟨l(Tem)⟩⟩≈120MPlTM1/21Tem,(13)in which the double bracket denotes an average with thermal fluctuations taken into ac-count.

It is seen that the final mean separation of the monopole-pair is larger by a factorof 100 than the flux tube thickness 1/eTem. At the same time, the final transverse mo-mentum of monopoles at the above separation is of order110(MPlTem)1/2 << TM, show-ing that the monopoles are always nonrelativistic.

For relativistic monopoles (i.e., , if125TM ≤Tem ≤TM), the transverse momentum P⊥≈E ≈MPlTem/TM and v⊥≈1. Theflux tubes whose original length was given by Eq.

(7) shrink to a mean separation⟨⟨l(Tem)⟩⟩r ≈10TMTem1/21Tem. (14)They are longer than the flux tube thickness by a factor of ≥3.In both the relativistic and the nonrelativistic cases, it is seen that the final monopole-pair is separated by a centrifugal barrier due to the angular momentum.

Thus the wave-function overlap and the annihilation cross-section are exponentially suppressed.6

This leads us to a crucial point: in order for the monopole pair to be confined by theflux tube and annihilate efficiently, the initial angular momentum must be dissipated byfriction.4. Friction from Aharonov-Bohm Scattering: There are several mechanisms for dis-sipating the initial angular momentum: (1) radiation of long-range gluons and/or weakgauge bosons, (2) interactions between the magnetic monopole and the ambient plasma,and (3) the interaction between the flux tube and the plasma through Aharonov-Bohmscattering.The interaction between magnetic monopole and the plasma gives rise to a frictionforce FM(T) ≈ρ(T)σCRv ≈T 2emv where ρ is the background plasma energy density, σCRthe Callan-Rubakovcallanrubakov,adavis cross-section of the monopole and v the monopoleterminal velocity.

Thus, the monopole dissipation rate isΓMon ≈ TemMPl1/2Tem(nonrelativistic), TMMPlTem(relativistic). (15)The monopole dissipation rate from radiation of gluons and weak gauge bosons is foundto bevilenkinrevΓrad ≈1αTemTM21⟨⟨l⟩⟩≈102T 2emT 3MMPl1/2 Tem(nonrelativistic),(TemTM)5/2 Tem(relativistic).

(16)The Aharonov-Bohm (AB) scatteringabscattering arises because the magneti c field isconfined inside the flux tube while the color and the weak gauge field are not. Due to thefractional electric charges Qu = 2e/3 and Qd = −e/3 carried by the quarks, the flux tubeconnecting the monopoles experiences nontrivial AB scattering with a cross sectiondσABdθ=sin2 Qu,de π2πk sin2 θ2.

(17)7

This result does not contradict the Dirac quantization condition as the latter applies tothe total sum of color, weak isospin and electromagnetic quantum numbersours. The ABdissipation rate isΓAB ≈ρσAB⟨⟨l⟩⟩vE≈TemTM1/2Tem(nonrelativistic),T 3/2MMPlT 1/2em(relativistic).

(18)Thus, we find that radiation dissipation is negligible while monopole-plasma dissipation issuppressed by a geometrical factor TM/MPl or Tem/TM relative to dissipation due to ABscattering.From Eq. (18), we find thatτABτa≈ TMMPl1/2≈10−2(nonrelativistic),TemTM1/2(relativistic).

(19)AB dissipation is most efficient for nonrelativistic monopoles, i.e., , for Tem << TM. Simi-larly, comparing τAB with the Hubble expansion time, we findτABτH≈30(TMTem)1/2MPl(nonrelativistic),30TemTM3/2(relativistic).

(20)From Eqs. (19) and (20), we thus come to our main conclusion: the monopole annihi-lation by the Langacker-Pi mechanism is most efficient for the lowest possible Tem << TM.Recall that the Hubble time scale increases as t ∝T −2, which is faster than themonopole annihilation time.

This was responsible for the efficiency of the annihilation atthe lower temperature of EM breaking. We have now found that the friction due to the ABscattering not only dissipates the angular momentum efficiently but also helps monopole8

annihilation at lower temperature scales!The time scales involved in the annihilationdynamics satisfy the following hierarchy:τH >> τa >> τAB(19)for temperatures Tem << TM, thus explaining why the highest efficiency for monopoleannihilation occurs at the lowest possible temperature. Of course, the scale Tem cannot betoo low since the monopoles will eventually dominate the energy density of the Universe.With the initial monopole density given by Eq.

(1), we find that the temperature atwhich monopoles dominates energy density of the the Universe (i.e., ρM/ρtotal ≈1) ist−1c≈106 GeV. Therefore, we can safely set the lower bound of Tem as Tem ≥106 GeV.In this Letter, we have examined the detailed dynamics of the Langacker-Pi mech-anism.

Due to the unusual temperature dependence of the characteristic time scales assummarized in Eq. (19), we find the counter-intuitive result that the most efficient scenarioof monopole annihilation occurs when U(1)em is broken just before the monopoles domi-nate the energy density of the Universe.

The fact that the photon is massive and electriccharge is spontaneously broken leads us to expect that charge nonconserving processes mayprovide novel signatures of the phase, which should be left over until today. In addition,the Callan-Rubakov effectcallanrubakov may provide additional bayron-asymmetry genera-tion at a relatively low energy scalesher,kephart, and we expect sizable entropy generationfrom the monopole and anti-monopole annihilation.

We are currently investigating theseissues, and will report as a separate publicationours. After this work was completed wewere informed that E. Gates, L.M.

Krauss and J. Terningkrauss have recently studied themonopole annihilation efficiency using W-condensate flux tubes.We are grateful for the hospitality of the Institute for Theoretical Physics at SantaBarbara, where this work was initiated.S.J.R. thanks M. Alford and S. Coleman foruseful discussions.T.W.B.K.

thanks A.C. Davis for helpful comments.This research9

was supported in part by the National Science Foundation under Grant No.PHY89-04035. R.H. was supported in part by DOE grant DE-AC02-76ER3066, while S.J.R.

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