Baryogenesis via Elementary Goldstone Higgs Relaxation Helene Gertov,1,∗ Lauren Pearce,2,3,† Francesco Sannino,1,‡ and Louis Yang4,§ 1CP3-Origins & the Danish Institute for Advanced Study Danish IAS, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark. 2Department of Physics and Astronomy, Valparaiso University, Valparaiso, IN 46383 USA 3William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 USA 4Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA 6 We extend the relaxation mechanism to the Elementary Goldstone Higgs frame- 1 0 work. Besides studying the allowed parameter space of the theory we add the 2 minimal ingredients needed for the framework to be phenomenologically viable. n u The very nature of the extended Higgs sector allows to consider very flat scalar J potential directions along which the relaxation mechanism can be implemented. 0 2 Thisfacttranslatesintowiderregionsofapplicabilityoftherelaxationmechanism ] whencomparedtotheStandardModelHiggscase. h p Ourresultsshowthat,iftheelectroweakscaleisnotfundamentalbutradiatively - p generated,itispossibletogeneratetheobservedmatter-antimatterasymmetryvia e therelaxationmechanism. h [ Preprint: CP3-Origins-2016-002DNRF90&DIAS-2016-2FTPI-MINN-16/02 4 v 3 5 7 7 0 . 1 0 6 1 : v i X r a ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] 2 I. INTRODUCTION ThediscoveryoftheHiggsbosoncrownstheStandardModel(SM)ofparticleinteractions asoneofthemostsuccessfuldescriptionofphysicalphenomenabeloworataroundtheelec- troweak(EW)scale. However,severalpuzzlesremainunexplainedsuchasthenatureofdark matter,neutrinomassesandmixingaswellasthecosmologicalmatter-antimatterasymme- try of the universe. Solutions to any of these puzzles generically requires introduction of newphysicsbeyondtheSM. Herewefocusourattentionontheimportantcosmologicalmysteryofhowtheobservable universe came to be dominated by an excess of matter over antimatter. The necessary conditionsforbaryogenesisarewell-known[1]andseveralmodelsofbaryogenesisexist(for areview,seee.g. [2]). Amongthese,anappealingscenarioinvolvestherelaxationofascalar orpseudoscalarfieldinthepost-inflationaryuniverse. Suchfieldscanacquirelargevacuum expectationvaluesduetoflatpotentials[3]orbybeingtrappedinaquasi-stableminimum. Afterinflation,suchfieldsrelaxtotheirequilibriumvaluesviaacoherentmotion,andhigher / dimensional operators can couple the time-dependent condensate to baryon and or lepton number [4]. This can be done with the Higgs field ([5–7]) or an axion field ([8]). Similar models have been constructed using quintessence fields ([9, 10]) and MSSM flat directions ([11–13]). A novel feature of the Higgs scenario is that the chemical potential depends ffi on the time-derivative of the VEV-squared, which as we discuss, resolves some di culties with producing an asymmetry of the correct sign throughout the observable universe. An additional advantage of this scenario is that it makes use of fields whose existence is either knownorwell-motivated. HowevertheSMHiggssectorisfarfromestablishedandcouldhidenewexcitingphysics. In fact, several alternative paradigms have been put forward that are not only as successful astheSMinreproducingtheexperimentalresults,butalsocansimultaneouslyaddresssome oftheremainingexperimentalpuzzles. The Elementary Goldstone Higgs (EGH) paradigm established in [14, 15] allows one to disentangle the vacuum expectation of the elementary Higgs sector from the EW scale [14]. Here the Higgs sector symmetry is larger than the minimally required symmetry needed to spontaneouslybreaktheEWgaugesymmetry. Furthermore,thephysicalHiggsemergesas a pseudo Nambu Goldstone Boson (pNGB). A welcome feature is that once the SM gauge and fermion sectors are embedded in the larger symmetry, one discovers that calculable radiative corrections automatically induce the proper breaking of the EW symmetry by naturally aligning the vacuum in the pNGB Higgs direction. In this way the EW scale is not fundamental but radiatively induced1. The template Higgs sector leading to the → SU(4) Sp(4)patternofchiralsymmetrybreakingwasfirstintroducedin[18]. In this work, we successfully marry the EGH paradigm and the relaxation leptogenesis scenario[5,7]. Thereareatleasttwomotivationsforthismarriage: first,weobservethatina 1The EGH setup is profoundly different from the composite (Goldstone) Higgs scenario [16]. The main differences being: i) the elementary case is amenable to perturbation theory; ii) it is straightforward to endowtheSMfermionswithmassterms; iii)itispossibletoimmediatelyconsiderGrandUnifiedTheory extensions[17]. 3 modelsuchastheEGHscanario,whichhasanextendedscalarsector,therearenaturallynew scalar-field directions along which one can implement the relaxation mechanism; secondly, the relative freedom in the overall potential flatness translates into a wider region of appli- ff cability of the approximations and e ective theory used to derive successful baryogenesis (ascomparedtotheSMHiggscase). The structure of this paper is as follows: we begin by reviewing the EGH model, as introduced in [14, 15], with particular emphasis on the scalar sector and the Yukawa sector. We then review the Higgs relaxation scenario, focusing on the modifications necessary due totheextendedHiggssector. Finally,wepresentananalysisoftheavailableparameterspace inwhichHiggsrelaxationleptogenesisoccurswhenmarryingittotheEGHparadigm. Although we use a specific template to perform our analysis, the general results and features are expected to hold for generic realisations of successful baryogenesis via EGH drivenleptogensisscenarios. II. ELEMENTARYGOLDSTONEHIGGS:ABRIEFREVIEW TheEGHscenario[14,15]necessarilyextendstheSMHiggssectorsymmetry. Aworking template uses a linear realisation with SU(4) symmetry breaking spontaneously to Sp(4). / The SM Higgs doublet is now part of the SU(4) Sp(4) coset, while the EW symmetry, × SU(2) U(1) ,isembeddedinSU(4). L Y The relaxation leptogenesis mechanism [5–7] uses the scalar sector of the theory at very highenergies. Wethereforestartbyreviewingthescalarsectorofthetheory. The SM Higgs boson is identified with one of the Goldstone bosons which acquires mass via a slight vacuum misalignment mechanism induced by quantum corrections. The misalignment is due mostly to top-induced quantum corrections [14, 15] and therefore we willneglectheretheEWgaugesectorcorrections. ≤ The most general vacuum structure of the theory can be parametrised by an angle 0 θ ≤ π/ 2[14]and Eθ = cosθEB +sinθEH = −ETθ, (1) wherethetwoindependentvacuumdirectionsE andE are B H (cid:32) (cid:33) (cid:32) (cid:33) σ i 0 0 1 = 2 , = . E E (2) B − σ H − 0 i 1 0 2 θ The alignment angle is determined by radiative corrections after having constrained the model to reproduce the experimental results. It was found in [15] that the model naturally θ preferssmallvaluesof ,privilegingapNGBnatureoftheHiggs. WenotethatbecauseSU(4)isbrokenradiativelythroughcorrectionsfromtopandgauge interactions, the strength of SU(4) breaking increases at higher scales. Therefore, there is no highscaleinwhichitisappropriatetoneglectSU(4)breaking. 4 A. ScalarandFermionicsector Intheminimalscenariodescribedabove,thescalarsectoriscodifiedby (cid:20) √ (cid:21) 1 M = (σ+iΘ)+ 2(Πi +iΠ˜i)Xθi Eθ, (3) 2 whereXi (i = 1,...,5)arethebrokengeneratorsassociatedtothebreakingofSU(4)toSp(4), θ Π reported in Appendix A of [15], and the five are the five Goldtone bosons of the theory, i where after symmetry breaking, the first three become the longitudinal components of the gaugebosons,thefourthistheobservedHiggs,andthelastisadarkmattercandidate. The fullSU(4)invariant(tree-level)scalarpotentialcanbefoundin[15]. Having introduced the scalar sector of the model, we turn our attention to the fermionic sector. Our focus here is two-fold: first, we have explained above how the top-sector is θ ff responsible for setting , and therefore indirectly a ects the vacuum and the scalar sector moregenerally,andsecondly,theleptogenesisscenarioproducestheasymmetrythroughthe excessproductionofneutrinos,whichinvolveselectroweakinteractionsbetweenfermions. WeconstructtheYukawasectorofthetheorybyintroducingEWgaugeinvariantoperators that explicitly break the SU(4) global symmetry and correctly reproduce the SM fermion massesandmixing. First,weformallyaccommodateeachoneoftheSMfermionfamiliesin thefundamentalrepresentationofSU(4),namely (cid:16) (cid:17)T (cid:16) (cid:17)T Lα = L, ν˜, (cid:96)˜ αL ∼ 4, Qi = Q, q˜u, q˜d iL ∼ 4, (4) α = ,µ,τ = , , where e and i 1 2 3 are generation indices and the tilde indicates the charge conjugate fields of the Right Handed (RH) fermions, that is, for instance, ν˜αL ≡ (ναR)c, (cid:96)˜αL ≡ ((cid:96)αR)c,LαL ≡ (ναL,(cid:96)αL)T andsimilarlyforthequarkfields. NoticethataRHneutrinoναR for each family must be introduced in order to define Lα which transforms according to the fundamentalirrepresentationofSU(4). Given the embedding of quarks and leptons in SU(4), we now construct a Yukawa mass termfortheSMfermions. ForthiswemakeuseofSU(4)spurionfields[19]P andP ,where a a a = 1,2isanSU(2) index. Theytransformas(P) → (u†)T(P) u†,withu ∈ SU(4). Wehave L a a (cid:32) (cid:33) (cid:32) (cid:33) τ τ− 1 0 1 0 P = √ 2 3 , P = √ 2 , (5) 1 −τ 2 −τ+ 2 3 02 2 02 (cid:32) (cid:33) (cid:32) (cid:33) τ+ τ 1 0 1 0 P = √ 2 , P = √ 2 3 , (6) 1 −τ− 2 −τ 2 02 2 3 02 with σ ± σ +σ −σ i 1 1 τ± = 1 2, τ = 2 3, τ = 2 3 . and (7) 3 3 2 2 2 Then, using P1,2 and P1,2, we may write Yukawa couplings for the SM fermions which 5 preservetheSU(2) gaugesymmetry: L Yu (cid:16) (cid:17)† Yd (cid:16) (cid:17)† (cid:104) (cid:105) −LYukawa = √ij QTP Q Tr[P M]+ √ij QTP Q Tr P M i a j a i a j a 2 2 ν (cid:96) Yαβ (cid:16) (cid:17)† Yαβ (cid:16) (cid:17)† (cid:104) (cid:105) + √ LTαPaLβ Tr[PaM]+ √ LTαPaLβ Tr PaM + h.c. (8) 2 2 with the Yukawa matrices of quarks and leptons chosen in agreement with experimental measurements. This Lagrangian explicitly breaks the SU(4) global symmetry mentioned θ above, and therefore, it also contributes fixing the parameter which interpolates between the otherwise equivalent vacuum structures. In fact, in terms of the SM quark and lepton fields,Eq.(8)canbewrittenas (cid:16) (cid:17)† (cid:16) (cid:17)† (cid:104) (cid:105) −LYukawa = Yu Q q˜u Tr[P M]+Yd Q q˜d Tr P M ij iL jL a a ij iL jL a a (cid:16) (cid:17)† (cid:16) (cid:17)† (cid:104) (cid:105) + Yανβ LαLν˜βL Tr[PaM]+Yα(cid:96)β LαL(cid:96)˜βL Tr PaM + h.c. (9) a a where −1 (cid:16) (cid:17) Tr[P M] = √ σsinθ+Π cosθ+iΘsinθ−iΠ˜ cosθ+iΠ +Π˜ , (10) 1 4 4 3 3 2 1 (cid:16) (cid:17) Tr[P M] = √ iΠ +Π +Π˜ −iΠ˜ . (11) 2 1 2 1 2 2 Therefore,afterEWsymmetrybreaking,theSMfermionsacquirethemasses θ f sin m = y √ , (12) F F 2 = (cid:104)σ(cid:105) with f atlowenergiesand y beingtheSMYukawacouplingofquarksandleptonsin F the fermion mass basis. Comparing this expression with the corresponding SM prediction mF,SM = √yF vew weseethat f andθmustsatisfythephenomenologicalconstraint 2 θ = (cid:39) . f sin v 246GeV (13) EW Notice that a Dirac mass for neutrinos is generated as well. Ref. [15] also investigated the parameter space at low energy and found that (when keeping the masses of the scalars below five TeV) the most frequent value for f is f = 13.9+2.9 TeV corresponding to θ = −2.1 0.018+0.004. Althoughthesearethemostcommonvaluesthatgivetheappropriateelectroweak −0.003 phenomenology,thepointsofparameterspacewhichsatisfytheelectroweakconstraintsvary θ significantlyinvaluesfor f andsin( ). Thisisbecausetherearequiteafewcouplingsinthe SU(4) potential. To generate an acceptable electroweak phenomenology at values of f and θ ff sin( )significantlydi erentthanthese,itislikelynecessarytofine-tuneatleastsomeofthe parametersintheSU(4)potential. ThismodeldoesnotnaturallygenerateaMajoranamasstermfortheRHneutrinofields; however, one can be explicitly added. This provides an explicit breaking of the SU(4) symmetry, but preserves the EW gauge group and gives the standard seesaw mechanism. 6 AlthoughthisisnotstrictlyspeakingnecessaryfortheEGHbosonmodel,weincludeitinour analysis here. This is because a successful leptogenesis model must involve lepton-number violatingterms,andfollowing[5–7],wewillmakeuseoftheneutrino-sectorMajoranamass term. Inthiscase,themostgeneralmassLagrangianfortheleptonsis θ θ f sin f sin 1 −Llep = Yα(cid:96)β √2 (cid:96)αL(cid:96)βR +Yανj √2 ναLνjR + 2 (MR)jkνjR(νkR)c + h.c. (14) where M is the Majorana mass term of the three RH neutrinos. The couplings in Eq. (14) R allow to generate at tree-level a Majorana mass term for the LH neutrinos, in a manner similartothestandardtypeIseesawextensionoftheSM[20]. Thisyields 1 Lνmass = −2 (mν)αβ ναL(νβL)c + h.c. (15) with θ f sin v mν = −mDM−R1mTD and mD = Yν √ = Yν √EW . (16) 2 2 One can hope that this Majorana mass term would be generated by embedding the EGH modelintoalargermodel,perhapsaGrandUnifiedTheory. B. Radiativecorrections δ Φ Next we return our attention to the scalar sector. The one-loop correction V( ) to the scalarpotentialtakesthegeneralexpression (cid:34) (cid:32) (cid:33)(cid:35) 1 M2(Φ) δV(Φ) = Str M4(Φ) log −C (17) 64π2 µ2 Φ ≡ σ, Π whereinthiscase ( )denotesthebackgroundscalarfieldsthatweexpecttoleadto 4 M Φ the correct vacuum alignment of the theory and ( ) is the corresponding tree-level mass matrix. Thesupertrace,Str,isdefinedas (cid:88) (cid:88) (cid:88) = − + . Str 2 3 (18) scalars fermions vectors = / = / µ We have C 3 2 for scalars and fermions and C 5 6 for the gauge bosons, and is 0 a reference renormalization scale. As explained above, the Yukawa sector terms explicitly breaktheglobalSU(4)symmetryandgaugeinteractionswillalsoprovideexplicitsymmetry breaking. ThisexplicitbreakingwillgenerateanonzeromasstermfortheGoldstonebosons Π σ atthequantumlevelandamassmixingtermbetweenthe andthe fields. 4 At very high ene√rgy scales the background-dependent masses of all the scalars are the ≈ λσ λ same, namely, m , where is a linear combination of several quartic couplings. The renormalisation scale is fixed as a constant at the energy scale of inflation. Taking only the 7 top and scalar corrections into account, the one-loop corrections to the potential take the simpleform σ4 (cid:32) (cid:32) λσ2 2(cid:33) (cid:32) y2sin2θσ2 3(cid:33)(cid:33) δV = 7λ2 log − −3y4sin4θ log t − . (19) 64π2 µ2 3 t 2µ2 2 σ ff inthedirectionof . Thusthee ectivepotentialtoone-loopordercanbewrittenas λ σ V(σ) = eff( )σ4 (20) 4 ff wherethee ectivequarticcouplingis 4 (cid:34) (cid:32) (cid:32)λσ2(cid:33) 3(cid:33) (cid:32) (cid:32)y2sin2θσ2(cid:33) 3(cid:33)(cid:35) λeff(σ) = λ+ 64π2 7λ2 log µ2 − 2 −3y4t sin4θ log t 2µ2 − 2 . (21) λ Herey isthetopquarkYukawaand isnotyetexperimentallyconstrained. Atlowerenergy, t the one-loop potential is more involved and a detailed analysis of the one-loop potential at θ . low energy can be found in [15]. In that paper the authors found that for around 0018 λ . and around0007thereisaregionofparameterspacewiththemostEW-favorablepoints, butthatitisbynomeansrequiredforgoodEWbehaviorforthetaandlambdatotakethese values. In this work we will be primarily interested in the high energy regime since the scalar field will acquire a comparatively large vacuum expectation value. Below, we will find that in order to produce a baryon asymmetry of the appropriate size, it is desirable to λ σ haveasmallcoupling eff. Inordertoensurethestabilityofthepotentialatlarge ,itmaybe λ θ necessarytotuneboth and tobesmall,bychoosingtheparametersintheSU(4)potential λ appropriately. This is not a problem since, unlike in the SM, eff is not set by the observed Higgsmassbecauseoftheenlargedscalarsector. C. ThePhysicalHiggs For maximum clarity, we here pause to identify the physical Higgs boson states. At low σ Π energy there is a mass mixing between the and the fields as mentioned earlier. The 4 masseigenstatesofthismixingarethetwoHiggsparticles,handH,givenby (cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33) σ α − α cos sin h = , (22) Π α α sin cos H 4 α ,π/ where isthescalarmixingangle,chosenintheinterval[0 2]. TheobservedHiggsboson willbethelightesteigenstatewithamass = . ± . m 1257 04GeV (23) h α π/ whichin[15]wasfoundthat ispreferredtobeverycloseto 2;thatis,theobservedHiggs ismostlyapNGB. Asnoted,though,athighenergiesthesestatesarenearlydegenerateinmass. 8 III. RELAXATION-LEPTOGENESISFRAMEWORK Having introduced the EGH model, with particular attention to the scalar and fermionic sectors, we now introduce the Higgs relaxation leptogenesis framework, which has been exploredintheSMcontextin[5–7]. We outline the important steps of relaxation leptogenesis as follows: First, we need a scalar (or pseudo-scalar) field with a large vacuum expectation value (VEV). This can occur throughquantumfluctuationsduringinflation,orthefieldmaybetrappedinaquasi-stable minimum. Afterwards,thefieldrelaxestoitsequilibriumvalue. Duringthisrelaxation,achemicalpotentialforleptonnumbermaybeinducedviahigher dimensional operators. This lowers the energy of leptons and raises the energy of antilep- tons. Lepton-number-violatingprocesses,suchasthosemediatedbytheneutrinoMajorana massesintroducedabove,produceanexcessofleptonsoverantileptons. Theseinteractions canoccurwithintheparticleplasmaproducedduringreheating[5,7],orduringthedecayof theHiggscondensateitself[6]. Herewefocusonthefirstscenarioasanillustrativeexample. While we specifically consider the scalar sector here (which is of the most interest due to the extended scalar sector in the EGH model), we acknowledge that similar considerations applytoaxion-likedegreesoffreedom,whichhavebeenexploredin[8]. InourpreviousrealisationsofHiggs-relaxationleptogenesis,therelaxingfieldwasiden- tified with the SM Higgs, although we allowed for a modified potential at high scales. The recentobservationoftheHiggsbosonattheLHCsetsthequarticcoupling,althoughitissig- nificantlymodifiedatlargescales(asdescribedbytherenormalizationgroupequations)[21]. However, the EGH model has additional freedom as can be seen in Eq. (21). We will show below that in order to generate the observed baryonic asymmetry, while remaining in the regimeinwhichcertainapproximationsarevalid,thequarticcouplingmustbesignificantly smaller than the value preferred in the SM. This is not phenomenologically problematic in theEGHscenariobecausetheextendedHiggssectorallowsforadditionalflatdirections. A. LargeInitialVacuumExpectationValue(VEV)ofσ Aswementionedabove,duringinflation,scalarfieldsmayacquirelargevacuumexpecta- tionvalues(VEVs)throughquantumfluctuations: becauserelaxationviaacoherentmotion isaclassicalprocess,itstime-scalemaybesignificantlylongerthanthosetypicalofquantum fluctuations (see [3]). Concretely, quantum fluctations occur on a scale such that V(σ ) ∼ H4 (cid:112) √ I I where σ = (cid:104)σ2(cid:105) is the scalar field vacuum expectation value and H ≡ 8π/3Λ2/M is the I I I P Hubbleparameterduringinflation. TheVEVrollsdownclassicallytoitsminimumwiththe characteristictimescale (cid:34)d2V(σ)(cid:35)−1/2 τ ∼ / ∼ . roll 1 mσ,eff dσ2 (24) However, if τ (cid:29) H−1, there is insufficient time between quantum fluctuations for the roll I generated field VEV to roll down. In this case, the scalar field would develop a large VEV ∼ σ duringinflation. I 9 An alternative scenario for starting with a large scalar vacuum expectation value is that the scalar field may be trapped in a quasi-stable minimum in the early universe; this is particularly well-motivated in scenarios in which the initial scalar VEVs are distributed stochastically (provided the scalar potential does, indeed, have a high-scale quasi-stable minimum). The SM potential provides motivation for both scenarios: recent measurements of the Higgs mass suggest a rather flat potential, before turning over (and potentially becoming negative) [21]. The flat potential makes it easier for quantum fluctuations to generate a large VEV in the early universe; on the other hand, if the potential does turn over, higher- dimensional operators can stablize the potential in such a way as to produce a quasi-stable minimum. ff Here,though,weareinterestedintheEGHmodel,whichhasadi erentpotentialshape. σ Π Asnoted,thephysicalHiggsbosonhandHaremixturesofthe and degreesoffreedom, 4 althoughthereisanapproximaterotationalsymmetryathighenergies. Wewillconsiderthe σ ff case in which the field acquires a large VEV during inflation within the e ective potential σ given in Eq. (20). In fact at high energies can be seen as simply the modulus of the scalar fieldandthereforeitwouldnotmatterwhichdirectiononeselects. Furthermore,asalready ff explained,thee ectivequarticpotentialinthiscaseisnotfixedatlowenergies,becausethe ffi mass of the pNGB Higgs emerges radiatively via top corrections and there are no su cient experimentalconstraintsyettofixthisoverallcoupling. ff Next, we address an issue which a ects all relaxation leptogenesis models, which is discussed in more detail in [5, 7]. Namely, in both ways of generating large scalar VEVs, ff ff σ di erentpatchesoftheUniversegenericallyhavedi erentvaluesof attheendofinflation. I σ IftheleptonasymmetryislinkedtotheinitialVEVof ,eachpatchoftheUniversecouldhave ff a di erent final asymmetry. This would result in unacceptably large baryonic isocurvature perturbations [22], which are constrained by the cosmic microwave background (CMB) observations[23]. One solution to this problem, which was proposed in [5], is to couple the Higgs sector to the inflaton in such a way as to suppress the growth of the VEV until the end of slow-roll inflation. The resulting isocurvature perturbations are then on scales smaller than those which have been experimentally probed. In the EGH model, we adapt this solution by σ couplingthe fieldtotheinflatonI viaoperatorsoftheform L = In (cid:2) + (cid:3)m/2. σI cMm+n−4Tr M M (25) P Such a non-renormalizable operator can be generated by integrating out heavy states in loops; we can envision that these states arise by heavy SU(4)-preserving multiplets which arise when the EGH model is embedded into larger (perhaps grand unified) models. In the (cid:104) (cid:105) early stages of inflation, the VEV of the inflaton I can be large (superplanckian) and gives ff (cid:104) (cid:105) σ σ a large e ective mass mσ,eff( I ) to ; this suppresses the quantum fluctuations of the field. (cid:104) (cid:105) (cid:104) (cid:105) (cid:28) In the later stages of inflation, I decreases to a value such that mσ,eff( I ) HI, allowing a σ largeVEVfor todevelop. IfthedevelopmentoftheVEVoccursduringthelastN e-folds last 10 ofinflation,theVEVreachestheaveragevalue (cid:18) (cid:19) H (cid:112) σ = σ , I . min N (26) 0 I π last 2 Theresultingisocurvatureperturbationsappearonlyatthesmallestangularscales,andare (cid:46) notyetconstrainedforN 8. last Whileothersolutionstothisisocurvatureproblemwerenotedin[5],weconsiderthisone asanillustrativeexamplewhichalsoallowsthemostfreedominparameterspace. B. RelaxationofTheσField When the inflation is over, the inflaton begins oscillating coherently as it decays; conse- σ quently,theuniversebehavesasifitismatterdominates. Duringthisepoch,the fieldalso σ σ = relaxes from its starting value of and oscillates around 0 (the minimum of Eq. (20)) 0 σ withdiminishingamplitude. Theequationofmotionfor (t)(wherebyanabuseofnotation σ σ weuse (t)fortheVEVofthe field)is σ dV( ) σ+ σ+ = . ¨ 3H(t) ˙ 0 (27) σ d ff TheHubbleparameterH(t)isdeterminedbythesystemofdi erentialequations (cid:115) π ≡ a˙ = 8 (cid:0)ρ +ρ (cid:1), H(t) (28) r I a 3M2 P ρ + ρ = Γ ρ , ˙ 4H(t) (29) r r I I where ΓI is the decay rate of inflaton, and ρI = Λ4Ie−ΓIt/a(t)3 and ρr = (cid:0)g∗π2/30(cid:1)T4 are the energy densities of the inflaton field and the produced radiation respectively. We ensure σ that the energy density of the condensate never dominates the universe, so as to preserve that standard cosmological picture. Note that the maximum temperature during reheating (cid:16) (cid:17)1/4 and the reheat temperature can be estimated as [24] Tmax ≈ 0.618 Λ2I ΓImpl/g∗ and [25] TR (cid:39) (cid:0)3/π3g∗(cid:1)1/4 (cid:112)ΓImpl,respectively. C. EffectiveChemicalPotential σ Weconsiderthefollowingcouplingsbetweentheleptoncurrentandthe field L6 = −M12Tr(cid:2)M+M(cid:3)∂µjµB = −M12σ2∂µjµB, (30) n n where M is a potentially new scale. This coupling does not break SU(4) symmetry, and is n therefore consistent with the EGH picture. As this is a higher-dimensional operator, it may begeneratedbyintegratingoutheavystates;oneobviousmethodistoexpandtheminimal EGH model introduced above with heavy RH states that couple to a gauge boson anomaly,