Solitons in extensions of the Standard Model can serve as localized sources
of CP violation. Depending on their stability properties, they may serve either
to create or to deplete the baryon asymmetry. The conditions for existence of a
particular soliton candidate, the membrane solution of the two-Higgs model, are
presented. In the generic case, investigated by Bachas and Tomaras, membranes
exist and are metastable for a wide range of parameters. For the more viable
supersymmetric case, it is shown that the present-day existence of CP-violating
membranes is experimentally excluded, but preliminary studies suggest that they
may have existed in the early universe soon after the electroweak phase
transition, with important consequences for the baryon asymmetry of the
universe.Comment: Talk given by Ola Tornkvist, to appear in the proceedings of
  Fundamental Physics at the Birth of the Universe, II, in Rome, Italy, 19-24
  May 1997. Revtex, 7 pages, 1 postscript figure, uses epsf.tex, aps.sty,
  prl.sty, preprint.sty. Preprint number correcte

Riotto, Antonio

Tornkvist, Ola

English

arXiv

Solitons in extensions of the Standard Model can serve as localized sources of CP violation. Depending on their stability properties, they may serve either to create or to deplete the baryon asymmetry. The conditions for existence of a particular soliton candidate, the membrane solution of the two-Higgs model, are presented. In the generic case, investigated by Bachas and Tomaras, membranes exist and are metastable for a wide range of parameters. For the more viable supersymmetric case, it is shown that the present-day existence of CP-violating membranes is experimentally excluded, but preliminary studies suggest that they may have existed in the early universe soon after the electroweak phase transition, with important consequences for the baryon asymmetry of the universe

Tornkvist, O.

Riotto, A.

UNT Digital Library

F Fermi National Accelerator LaboratoryFERMILAB-Conf-97/179-ACP-Violating Solitons in the Early UniverseOla Törnkvist and Antonio RiottoFermi National Accelerator LaboratoryP.O. Box 500, Batavia, Illinois 60510July 1997Presented at the Fundamental Physics at the Birth of the Universe, II,Rome, Italy, May 19-24, 1997Operated by Universities Research Association Inc. under Contract No. DE-AC02-76CH03000 with the United States Department of EnergyDisclaimerThis report was prepared as an account of work sponsored by an agency of the United StatesGovernment. Neither the United States Government nor any agency thereof, nor any oftheir employees, makes any warranty, expressed or implied, or assumes any legal liability orresponsibility for the accuracy, completeness, or usefulness of any information, apparatus,product, or process disclosed, or represents that its use would not infringe privately ownedrights. Reference herein to any specic commercial product, process, or service by tradename, trademark, manufacturer, or otherwise, does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the United States Government or any agencythereof. The views and opinions of authors expressed herein do not necessarily state or reectthose of the United States Government or any agency thereof.DistributionApproved for public release; further dissemination unlimited.FERMILAB{Conf-97/179-Aastro-ph/9706271CP-Violating Solitons in the Early UniverseOla Tornkvistzand Antonio Riottox ,NASA/Fermilab Astrophysics Center,Fermi National Accelerator Laboratory,Batavia, Illinois 60510-0500, USAJune 4, 1997AbstractSolitons in extensions of the Standard Model can serve as localized sourcesof CP violation. Depending on their stability properties, they may serveeither to create or to deplete the baryon asymmetry. The conditions forexistence of a particular soliton candidate, the membrane solution of the two-Higgs model, are presented. In the generic case, investigated by Bachas andTomaras, membranes exist and are metastable for a wide range of parameters.For the more viable supersymmetric case, it is shown that the present-dayexistence of CP-violating membranes is experimentally excluded, but prelim-inary studies suggest that they may have existed in the early universe soonafter the electroweak phase transition, with important consequences for thebaryon asymmetry of the universe.Talk given by O. T. at the conference \Fundamental Physics at the Birth of the Universe, II"in Rome, Italy, 19-24 May 1997; contribution to the proceedings.zAddress after 24 Sept. 1997: DAMTP, Univ. of Cambridge, Silver Street, CambridgeCB3 9EW, England. Email: olat@fnal.govxPPARC Advanced Fellow, Oxford Univ. from Sept. 1997. From 1 Dec. 1997 on leave of ab-sence at CERN, Theory Division, CH{1211 Geneva 23, Switzerland. Email: riotto@fnal.govIn this talk I report on work done in collaboration with Antonio Riotto[1] on CP-violating solitons and their possible consequences for the baryon asymmetry of the Uni-verse. The talk is organized as follows: After reviewing some well-known sources ofCP violation in models of particle physics, I propose that solitons in extensions of theStandard Model can serve as localized sources of CP violation. From then on I limitthe discussion to the type of soliton considered in Ref. [1] { membranes. The conditionsfor existence of CP-violating membranes are presented in two important cases, corre-sponding to the smallest possible extensions of the Standard Model: (1) a generic modelwith two Higgs doublets, and (2) the supersymmetric two-Higgs-doublet model, betterknown as the Minimal Supersymmetric Standard Model (MSSM). Finally, I discuss theimplications of the existence of CP-violating solitons in the early Universe and argue thatthey may, depending on their stability properties, serve either to create or to deplete thebaryon asymmetry.Sources of CP ViolationIn the Standard Model (SM) of particle physics, CP non-conserving processes are theresult of an irreducible complex phase in the Cabibbo-Kobayashi-Maskawa matrix, whichdescribes the mixing of quark avors of dierent generations. Because this phase entersonly in higher-loop processes involving massive charged W bosons, CP violation in theSM is strongly suppressed. Except in highly speculative scenarios, it cannot account forthe observed baryon asymmetry of the Universe [2].Extensions of the SM with two Higgs doubletsH1 andH2 provide an additional sourceof CP violation in the relative phase  of their expectation values in the ground state,hH1i =0B@ 0v11CA ; hH2i  ei0B@ 0v21CA : (1)A non-zero value of  arises from loop corrections to the Higgs potential that determinesthe minimum. At nite temperature, the corrections can be large and lead to \sponta-neous" CP violation [3] as H1, H2 acquire expectation values in the electroweak phasetransition. It is generally agreed that extended models of particle physics provide sourcesof CP-violation that are suciently strong for the purpose of baryogenesis.Localized Sources of CP ViolationFor the sake of simplicity take  = 0. Then the ground state (1) contains no spatiallyuniform source of CP violation. However, solitonic excitations exist that include a phase1 which is non-zero only in a localized region of space. In such a case, CP violation isconned to the interior of the soliton.As a rst example, consider a texture [Copeland]1 of the formH1 =0B@ 0f1(x)1CA ; H2 = U(x)0B@ 0f2(x)1CA ; (2)where U(x) 2 SU(2) and 3[SU(2)] = Z. A spherical solution of this form has beeninvestigated by Bachas et al. [4], who found it to be unstable for all parameters of themodel.For the rest of the talk I will concentrate on membranes, which are wall-like solutionssimilar to domain walls. The important dierence is that a membrane, much like asoap lm, separates two regions that are in the identical state, whereas a domain wallseparates dierent states. Viewed in a local rest frame, the at membrane is a staticsolution depending only on the coordinate x orthogonal to the wall:H1 =0B@ 0f1(x)1CA ; H2 = ei(x)0B@ 0f2(x)1CA ; (3)where ( 1) = 0 and (1) = 2. The phase  changes from 0 to 2 within a distancesynonymous with the thickness of the membrane M 1A , where MA is the mass of thelightest CP-odd neutral Higgs boson. Therefore, ei 6= 1 inside the membrane whileei = 1 outside.The presence of such a kink solution can be easily understood by noticing that theLagrangian of the model contains terms similar to those of the sine-Gordon model La-grangian. In fact, using the form (3) and the approximation f1  v1, f2  v2 the termsjDH1j2+ jDH2j2+m 23 [Hy1H2+Hy2H1] reduce to2 v1v2[cos sin (0)2+2m23 cos ]. Heretan = v2=v1 is the usual constant parameter dened by the ratio of the two expectationvalues. The corresponding potential V /   cos  is known to give rise to kink solutionsthat interpolate from one minimum at x =  1 to the adjacent minimum at x = +1.The analytic solution is the sine-Gordon kink, given by (x) = 4 tan 1[exp (MAx)], whereM2A = m23=(cos sin) determines the characteristic thickness. This solution is identicalto the time evolution (t) of the angle of a pendulum released at rest from its highestvertical position and given a small kick so that it swings through a full circle within atime M 1A and comes to rest again asymptotically in its highest position as t! +1.1A single name in brackets refers to that person's contribution to these proceedings.2Care must be taken to solve for the gauge potential in terms of 0 [1].2Conditions for the Existence of MembranesLet me rst discuss the generic model with two Higgs doublets. In this case the Higgssector contains 9 parameters, and the membrane solutions of this model were investigatedby Bachas and Tomaras [5]. They restricted themselves to a hyperplane in parameterspace by xing 6 of the parameters, including tan(= 1). For the remaining 3 parametersthey picked Mh0=MA, MH0=MA and MH=MA, where h0, H0, H+ and H  are the CP-even Higgs bosons of the model, and were able to show that there exist classically stablemembranes when Mh0=MA > 2, MH=MA> 2:2 and MH0=MA> g(Mh0=MA), g being amonotonically decreasing function. I want to emphasize that membranes thus exist in alarge region of parameter space, especially since 6 of the parameters were xed in thisstudy.Because these solitons are non-topological, classical stability means that they consti-tute a local minimum of the energy, that they are metastable, and that they will decayafter a nite life-time via the process of tunneling.I now proceed to the case of the supersymmetric two-Higgs model, or the MSSM.This model is, of course, much more attractive from the particle physics point of view,since supersymmetry is a major motivation for having two Higgs doublets in the rstplace. Supersymmetry provides ways to solve many of the puzzles of the SM such asthe electroweak symmetry breaking, the problem of quadratic divergences in the Higgsmass, and stability of the weak scale under radiative corrections without ne-tuning. Inaddition, supersymmetric particles serve as candidates for dark matter [Sadoulet].Together with Antonio Riotto, I have shown that there exist no membrane solutionsin the MSSM for any parameters unless one includes one-loop quantum corrections tothe potential [1]. These corrections are highly signicant because supersymmetry leadsto accidental conspiracies in the classical tree-level potential. In order to address theexistence of membranes, one therefore must consider the one-loop corrected potential.The low-energy limit of this model can be described by three parameters,MA, tan, andMSUSY, while the rest of the parameters are constrained by supersymmetry. Here MSUSYis the scale of supersymmetry breaking, typically in the range MW  MSUSY < O(few)TeV.The issue of existence of membrane solutions hinges crucially on whether the magni-tudes f1 and f2 of eq. (3) are able to satisfy their boundary conditions fi = vi at x = 1.This turns out to require a low value of the massMA, corresponding to thick membranewalls. Fig. 1 shows our results [1] for the region of parameter space (MA; tan) wheresolutions exist, for one \low" and one \high" value of MSUSY. We cannot choose MSUSYmuch higher without reintroducing the need for ne-tuning. Also in the gure are twobounds. The vertical bound is from LEP measurements of e+e  ! h0A0; h0Z, requiring3MA > 62:5 GeV for tan > 1. The horizontal bound results from assuming that physicsis described by a Grand Unied Theory (GUT) at a scale of about 1016 GeV. In sucha case the value of the top Yukawa coupling ht at low energies is predicted by the the-ory, and combining this with the measured value of the top-quark mass Mt = (175 6)GeV implies that tan > 1:1. This minimal value corresponds to a gauge-mediatedsupersymmetry-breaking mechanism.From Fig. 1 it is apparent that no CP-violating membranes exist in the MSSM atzero temperature (for which our analysis was done), provided that one abides with theGUT hypothesis.One may now ask oneself whether the non-existence of membranes at zero temper-ature is in fact an attractive feature, and whether membranes could have existed attemperatures near that of the electroweak phase transition, Tc  102 GeV, and vanishedat low temperatures. Our preliminary investigations show that such a scenario is possibleand would result from large nite-temperature quantum corrections to M2A, the parame-ter that most directly inuences the existence of the membranes. TakingM2A;T=0 slightlylarger than the experimental limit of (62:5)2GeV2, the question is whether one can sup-press the high-temperature value M2A;T>0 =M2A;T=0+ M2A;T>0 by means of a sucientlylarge negative correction M2A;T>0 so as to bring it down to values< 202 GeV2 requiredfor membrane solutions.The dominant contribution to M2A;T>0 comes from stop loops and is given by [6]M2A;T>0 = Ath2t f(MQT; MUT)=(sin cos), wheref(x; y)   32"J 0+(x2)  J 0+(y2)x2   y2+4 1x+ y 1~x+ ~y!#(4)and J+(x2) R10 dt t2 ln[1   exp( pt2 + x2)]. Here , At, MQ and MU are parametersin the stop mass matrix, and a tilde (~) implies the use of 1-loop self-energy correctedmasses ~MQ and ~MU [6]. For a concrete example, take   = At =MQ =MU = T = 145GeV and tan = 1. We then nd f(1; 1) = :0433 . . ., M2A;T>0   3644 GeV2, andM2A;T>0  162 GeV2. This indicates that membranes may exist in the MSSM at T < Tc.Riotto and I are currently investigating this issue more carefully.Implications for the Baryon AsymmetryIn electroweak models, the chiral anomaly permits processes at high temperaturethat do not conserve the baryon (B) and lepton (L) numbers, but conserve B   L [2].These B-violating sphaleron transitions involve passing over an energy barrier of heightEsph / (jH1j2 + jH2j2)1=2, where Esph  10 TeV at zero temperature. The B-violatingtransition probability is proportional to exp( Esph=T ).4As we can see from Fig. 2, the magnitudes fi = jHij, i = 1; 2, decrease inside amembrane, leading to a lower barrier and higher transition rate. Moreover, our resultsindicate thatMA  MW in order for membranes to exist. This means that the membraneis thick enough to accommodate the eld conguration at the top of the barrier, thesphaleron, which has a characteristic size of M 1W .From now on let us assume that membrane solutions do exist at high temperatures.The picture I want to present features membranes forming inside expanding criticalbubbles in a rst-order phase transition [Gleiser]1 at temperatures just below the criticaltemperature. The membranes are bounded by string loops, in analogy with axionic walls[Sakharov], or (more rarely) form closed surfaces. Because the size of a critical bubble ismuch larger than M 1A , many membranes form inside a bubble.The implications of these membranes are dierent depending on their stability proper-ties. If they are metastable, they can move across the plasma and create non-equilibriumconditions as they strike particles in their way. Since CP is violated in the interior ofthe membrane, and the B-violation rate is enhanced there, all three Sakharov conditions[Riotto] may be satised, and a net B(= L) can be generated that will survive if thephase transition is strongly enough rst-order.If, on the other hand, the membranes are unstable, they will still be created as thermaluctuations inside a critical bubble. The nucleation rate for a size-R membrane is of theorder of T exp[ F (R)=T ]. In order for membranes to play a role, this rate must be largerthan the expansion rate of the universe, H = T 2=MPl, which is true if F (R)=T < 40. Thismeans that small membranes are ubiquitous as uctuations and provide the conditionsfor B and CP violation in the critical bubble. Because this scenario occurs in thermalequilibrium, no net B + L can be generated. The membranes can, however, serve todeplete pre-existing baryon asymmetry.3ConclusionsIn summary, I have demonstrated that both the generic and the supersymmetric two-Higgs Standard Model contain CP-violating soliton solutions. If Nature is described bya generic two-Higgs model, then there exist metastable membranes that provide a newmechanism for creating the baryon asymmetry of the universe. If Nature is supersym-metric, such membranes are excluded at zero temperature, but may exist (unstable ormetastable) near the electroweak phase transition temperature.3This does not aect the asymmetry in models of GUT baryogenesis where a net B   L iscreated at the GUT scale.5FIGURESI emphasize that CP-violating solitons, whatever their particular realization, can playan important role in critical bubbles immediately after a rst-order phase transition,serving either to create or deplete the baryon asymmetry.ACKNOWLEDGMENTSAntonio and I thank F. Occhionero, L. Amendola and C. Baccigalupi for organizing asuccessful conference. The work of O.T. and A.R. is supported in part by the DOE andNASA under Grant NAG5{2788. O.T. is also supported by the Swedish Natural ScienceResearch Council (NFR).REFERENCES[1] A. Riotto and O. Tornkvist, CP-Violating Solitons in the Minimal SupersymmetricStandard Model , hep-ph/9704371, to be published in Phys. Rev. D.[2] For a recent review, see for instance, V. A. Rubakov and M. E. Shaposhnikov, Usp.Fiz. Nauk. 166, 493 (1996).[3] D. Comelli, M. Pietroni, Phys. Lett. B306 (1993) 67; D. Comelli, M. Pietroni,A. Riotto, Nucl. Phys. B412 (1994) 441; J.R. Espinosa, J.M. Moreno, M. Quiros,Phys. Lett. B319 (1993) 505.[4] C. Bachas, P. Tinyakov, T.N. Tomaras, Phys. Lett. B385 (1996) 237.[5] C. Bachas and T.N. Tomaras, Phys. Rev. Lett. 76 (1996) 356.[6] A. Brignole, J.R. Espinosa, M. Quiros, F. Zwirner, Phys. Lett. B324 (1994)181.6

CP Violating Solitons in the Early Universe

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