I report on a critical analysis of the scenario of electroweak baryogenesis
mediated by nonsuperconducting cosmic strings. This mechanism relies upon
electroweak symmetry restoration in a region around cosmic strings, where
sphalerons would be unsuppressed. I discuss the various problems this scenario
has to face, presenting a careful computation of the sphaleron rates inside the
strings, of the chemical potential for chiral number and of the efficiency of
baryogenesis in different regimes of string networks. The conclusion is that
the asymmetry in baryon number generated by this scenario is smaller than the
observed value by at least 10 orders of magnitude.Comment: 5 pages, uses sprocl.sty. Talk given at SEWM98, Dec.98, Copenhage

Espinosa, J. R.

English

arXiv

I report on a critical analysis of the scenario of electroweak baryogenesis mediated by nonsuperconducting cosmic strings. This mechanism relies upon electroweak symmetry restoration in a region around cosmic strings, where sphalerons would be unsuppressed. I discuss the various problems this scenario has to face, presenting a careful computation of the sphaleron rates inside the strings, of the chemical potential for chiral number and of the efficiency of baryogenesis in different regimes of string networks. The conclusion is that the asymmetry in baryon number generated by this scenario is smaller than the observed value by at least 10 orders of magnitude.I report on a critical analysis of the scenario of electroweak baryogenesis mediated by nonsuperconducting cosmic strings. This mechanism relies upon electroweak symmetry restoration in a region around cosmic strings, where sphalerons would be unsuppressed. I discuss the various problems this scenario has to face, presenting a careful computation of the sphaleron rates inside the strings, of the chemical potential for chiral number and of the efficiency of baryogenesis in different regimes of string networks. The conclusion is that the asymmetry in baryon number generated by this scenario is smaller than the observed value by at least 10 orders of magnitude

Espinosa, J.R.

CERN Document Server

ELECTROWEAK BARYOGENESIS WITH COSMIC STRINGS?J.R. ESPINOSACERN TH-DivisionCH-1211 Geneva 23, SwitzerlandAbstractI report on a critical analysis of the scenario of electroweak baryoge-nesis mediated by nonsuperconducting cosmic strings. This mechanismrelies upon electroweak symmetry restoration in a region around cosmicstrings, where sphalerons would be unsuppressed. I discuss the variousproblems this scenario has to face, presenting a careful computation ofthe sphaleron rates inside the strings, of the chemical potential for chiralnumber and of the efficiency of baryogenesis in different regimes of stringnetworks. The conclusion is that the asymmetry in baryon number gen-erated by this scenario is smaller than the observed value by at least 10orders of magnitude.CERN-TH/99-05January 1999CERN-TH/99-05hep-ph/9901310ELECTROWEAK BARYOGENESIS WITH COSMIC STRINGS?J.R. ESPINOSACERN TH-DivisionCH-1211 Geneva 23, SwitzerlandI report on a critical analysis of the scenario of electroweak baryogenesis mediatedby nonsuperconducting cosmic strings. This mechanism relies upon electroweaksymmetry restoration in a region around cosmic strings, where sphalerons wouldbe unsuppressed. I discuss the various problems this scenario has to face, present-ing a careful computation of the sphaleron rates inside the strings, of the chemicalpotential for chiral number and of the efficiency of baryogenesis in different regimesof string networks. The conclusion is that the asymmetry in baryon number gen-erated by this scenario is smaller than the observed value by at least 10 orders ofmagnitude.1 IntroductionElectroweak baryogenesis 1 is a beautiful idea which fails (or is about to fail) inthe best motivated models we have for physics at the Fermi scale ( 100 GeV ).In the Standard Model, LEP II experiments set a lower bound on the massof the Higgs boson of about 97 GeV, implying that the electroweak phasetransition in that model is not rst order but rather a crossover 2. In theMinimal Supersymmetric Standard Model the electroweak phase transitioncan be rst order and suciently strong to allow for electroweak baryogenesis,but this occurs in a very small region of parameter space 3 which presumablywill be ruled out by LEP II in a couple of years.One may take the previous negative results as indication that the asym-metry in baryon number was not created at the electroweak epoch, but ratherrelated to the physics of B − L violation and neutrino masses. To stick toelectroweak baryogenesis one can consider extensions of the particle contentof the model to get a stronger electroweak phase transition (e.g. extensionswhich include singlets). In this talk I will consider another possibility: how theremnants of physics at energy scales higher than the electroweak scale (cosmicstrings in this case) can be useful to overcome the problems of having a weakelectroweak phase transition.Electroweak baryogenesis requires the co-existence of regions of large andsmall h’i=T , where T is the temperature and h’i the (T -dependent) Higgsvacuum expectation value. At small or zero h’i=T sphalerons are unsuppressedand mediate baryon number violation, while large h’i=T is needed to store the1created baryon number (for h’i=T  1 sphaleron transitions are ineectiveand baryon number is conserved). Below the critical temperature T EWc of theelectroweak phase transition and irrespective of whether it is rst or secondorder, h’i=T grows until sphaleron transitions are shut-o. For baryogenesisto be possible at those times, we need some region where h’i is forced toremain zero or small. The idea we examine in this talk is that this can bethe case along topological defects (like cosmic strings) left over from someother cosmological phase transition that took place before the electroweakepoch 4. If the electroweak symmetry is restored in some region around thestrings, sphalerons could be unsuppressed in the string cores while they wouldbe ineective in the bulk of space, away from the strings. The motion of thestring network, in a similar way as the motion of bubble walls in the usualrst-order phase-transition scenario, will leave a trail of net baryon numberbehind.Some problems with this scenario come immediately to mind. First, it isclear that the space swept by the defects is much smaller than the total volume,so there will be a geometrical suppression factor with respect to the usualbubble-mediated scenario 4. Another suppression factor arises from the factthat there is a partial cancellation between front and back walls of the string,which tend to produce asymmetries of opposite signs 4. Another problem comesfrom the condition that the symmetry restoration region (which naively wouldbe of size Rrest  1=ph’i, where  is the quartic Higgs coupling) should belarge enough to contain sphalerons (which in the symmetric phase have sizeRsph  1=g2T ), while outside the strings, sphalerons should be suppressed(h’i=T  1). Combining both conditions one obtains   g4, which meansthe scenario would require small values of the Higgs mass, in conflict withexperimental bounds. LEP II tells us that  is at least of order g2, so thatsphalerons won’t t in the restoration region. In other words, for realistic valuesof the Higgs mass sphalerons are not going to be fully unsuppressed. We willmeasure how eective they are by writing the rate of sphaleron transitionsper unit time and unit of string length as Γl = l2wT 2. For a string withRrest = Rsph, one has ΓlR2rest equal to the rate in the symmetric phase,corresponding to l  1. Values of l much smaller than 1 would mean thatsphalerons are not really unsuppressed inside the strings.In the rest of the talk I review the careful analysis of this mechanismcontained in ref. 5, to which I refer the interested reader for further details.22 Strings with electroweak symmetry restorationCosmic strings 6 are 1-dimensional solitons, stable by topological reasons, thatcan form in the spontaneous breaking of a symmetry G where I consider thesimplest case, G = U(1), in this talk. A model with a complex scalar S andlagrangianL = @S@S − S(SS − S20)2; (1)admits global strings: congurations with S = 0 along some line (say the z-axis) and S(r) = f(r)S0ei, with f(1) ! 1, where r is the distance to thez-axis and  the azimuthal angle. The radius of these strings (where most ofthe energy is trapped) is set by the scale 1=mS  1=pSS0.If the U(1) is made local, in addition to the S eld, a non-zero gauge eld isalso present, A = −a(r)@=qS , with a(1) = 1, where qS is the U(1) chargeof the S eld. This gauge eld is such that the covariant derivative DS goesto zero for large r resulting in a nite energy per unit length of string.We assume that S-strings (global or local) form at some temperature T Sc >T EWc and are present at the time of the electroweak phase transition. To forceh’i ! 0 in the cores of the strings, the Higgs eld must interact either withthe S eld or the A eld (if the strings are local):2.1 S − ’ interactionSuppose the scalar potential has the formV (S; ’) = S(jSj2 − S20)2 − γ(jSj2 − S20)(j’j2 − ’20) + (j’j2 − ’20)2; (2)with γ > 0. The mass squared of the Higgs eld in the string backgroundis m2’(r)  γ(S20 − jS(r)j2) − 2’20, which is negative outside the string corebut can be positive inside, so that electroweak symmetry tends to be restoredalong the strings. Exploring the (S0; S ; γ; ) parameter space, the typicalcase, with SS20  ’20 leads to Rrest  1=m’(1). The best posible caseto get a large restoration region has S  γ   and S0  ’0 and givesRrest √γ=S=m’(1).2.2 S −A interactionIn this case we assume that the Higgs eld carries a charge q’ under the extraU(1) responsible for the strings, so that its covariant derivative has an extrapiece. As we saw, the A eld in the string goes like −1=qSr at large r tocancel the azimuthal derivative of S, give vanishing DS and minimize energy.In D’, the A contribution is now proportional to q’=qS and the azimuthal3derivative of ’ can cancel D’ only if q’=qS is an integer. If that is notthe case, a Z boson condensate is induced until the covariant derivative iscancelled 7. In any case, a non-zero winding of ’ forces ’ ! 0 in the stringcore (r = 0). The restoration region around r = 0 is larger in the presence ofa non-zero Z string (case of non-integer q’=qS).3 Sphaleron rates and CP asymmetry in the string coresIn general, with no tuning of potential parameters nor a Z condensate, h’iis zero only at the string core (r = 0) and rises inmediately away from thatline. As the symmetry is never really restored in a wide region, the energy ofthe sphaleron in such background (it can be computed in the lattice lookingfor a saddle point of the energy functional) is only about a factor 0.7 smallerthan the sphaleron energy in the broken phase (alternatively l  10−6: thatis, sphalerons are not really unsuppressed in this type of strings).The situation is better when a Z-eld is induced, in which case l 1=30 for h’i=T  1 (this number can be obtained in the lattice using a fullynon-perturbative approach and tracking Chern-Simons number in real timeevolution). However this number is very sensitive to T and drops signicantlywhen T decreases.Fully unsuppressed sphalerons can only be obtained in the global U(1)case for large enough γ=S . In fact, to obtain an asymmetry of the order ofthe observed one, one would need γ=S  1014. On the other hand, stabilityof the potential requires 4=γ > γ=S, so that =S  1028. Such an ad-hocand wild ne-tuning of the parameters prevents us from taking this particularcase seriously.Unsuppressed sphaleron transitions inside the string cores are not sucientto generate the baryon asymmetry: they must occur in a background with CPasymmetric particle distributions so that the sign of the B-violation is biased.This asymmetry comes about if the interactions between the particles in theplasma and the string walls violate CP. In that case the walls of a movingstring act as sources of chiral-number flux (which would be zero if the stringvelocity vS were zero). This asymmetry diuses away from the walls andonly that inside the string is useful to create baryons (for geometrical reasonsit is also clear that this diusion eect is less ecient for strings than forbubbles). In conclusion, we have to compute the chemical potential  forchiral number inside the strings. General arguments (conrmed by detailedanalysis of particular models) give the result  = Kv2ST for small vS , withK < 0:01 and  = K 0T for vS  1 with K 0 of order 1.44 Evolution of string networks and efficiency of baryogenesisTo get a nal number for the asymmetry generated by this mechanism, we needto know how many strings there are and how quickly they are moving (the bestcase being that of a dense network of fast moving strings). We can describe thestring network by a mean average separation between strings R(t) and a meanaverage velocity vS(t). The evolution of these quantities with time t is governedby Hubble expansion (H  1=2t); energy loss by loop formation; and frictionwith the plasma. The friction force goes like F  vST 3: it is important at earlytimes when it dominates the dynamics of the evolution. This is the frictiondominated or Kibble regime, with R(t)  t5=4 and vS(t)  t1=4  HR(t).Eventually, friction will no longer be important and a scaling regime is reachedwith R(t)  1=H and vS  1.In conclusion, to get the nal number for the baryon asymmetry we startwith the equation for the rate of change of baryon number NB per unit timeand unit length of string:dNBdLdt= 1:5[l2wT2]T: (3)If we use the results for l and  previously discussed, and integrate eq.(3) inone Hubble time (this is because l is shut-o quickly with decreasing T ) usingthe network evolution results just presented we end up with the result that[NBNγ]strings< 10−10[NBNγ]observed: (4)That is, the mechanism just studied is uncapable of generating a sucientlylarge matter-antimatter asymmetry.AcknowledgmentsI thank J.M. Cline, G.D. Moore and A. Riotto for an enjoyable collaborationon the topic presented.References1. A.G. Cohen, D.B. Kaplan and A.E. Nelson, Ann. Rev. Nucl. Part. Sci.43, 27 (1993); V.A. Rubakov and M.E. Shaposhnikov, Phys. Usp. 39,461 (1996).2. Z. Fodor, these proceedings.53. M. Quiros, these proceedings. J.M. Cline, these proceedings.4. M. Trodden, A. Davis and R. Brandenberger, Phys. Lett. B 335, 123(1994), Phys. Lett. B 349, 131 (1995); R. Brandenberger, A. Davis, T.Prokopec and M. Trodden, Phys. Rev. D 53, 4257 (1996).5. J.M. Cline, J.R. Espinosa, G.D. Moore and A. Riotto, [hep-ph/9810261].6. M. Hindmarsh and T. Kibble, Rept. Prog. Phys. 58, 477 (1995).7. W. Perkins and A. Davis, Nucl. Phys. B 406, 377 (1993).8. C. Martins and E. Shellard, Phys. Rev. D 54, 2535 (1996); C. Aulakh,M. Nagasawa and V. Soni, [hep-ph/9807435].6

Electroweak Baryogenesis with Cosmic Strings ?

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