Table Of ContentBaryogenesis 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.
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[email protected]
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[email protected]
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[email protected]
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[email protected]
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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,
