Postinflationary Higgs relaxation and the origin of matter-antimatter asymmetry
Alexander Kusenko,1, 2 Lauren Pearce,3 and Louis Yang1
1

Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547, USA
2
Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa, Chiba 277-8568, Japan
3
William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455 USA
The recent measurement of the Higgs boson mass implies a relatively slow rise of the Standard
Model Higgs potential at large scales, and a possible second minimum at even larger scales. Consequently, the Higgs field may develop a large vacuum expectation value during inflation. The
relaxation of the Higgs field from its large postinflationary value to the minimum of the effective
potential represents an important stage in the evolution of the universe. During this epoch, thetime-dependent Higgs condensate can create an effective chemical potential for the lepton number,leading to a generation of the lepton asymmetry in the presence of some large right-handed Majorana neutrino masses. The electroweak sphalerons redistribute this asymmetry between leptons andbaryons. This Higgs relaxation leptogenesis can explain the observed matter-antimatter asymmetryof the universe even if the Standard Model is valid up to the scale of inflation, and any new physics
is suppressed by that high scale.
PACS numbers: 98.80.Cq,98.80.-k,11.30.Fs,14.80.Bn

The recent discovery of a Higgs boson with mass
125 GeV [1, 2] implies that the Higgs potential is very
shallow and may even develop a second minimum, assuming that the Standard Model is valid at high energy
scales [3]. During cosmological inflation, the Higgs field
may be trapped in a quasi-stable second minimum, or alternatively, may develop a stochastic distribution of vacuum expectation values due to the flatness of the potential [4–6]. In both scenarios, the Higgs field relaxes to its
vacuum state after inflation via a coherent motion. In
this work we explore this epoch of Higgs relaxation.
We show that the observed matter-antimatter asymmetry could arise during this epoch. The Sakharov conditions [7], necessary for baryogenesis, are generically satisfied in the presence of the out-of-equilibrium Higgs condensate evolving with time [8, 9] and the neutrino Majorana masses that violate the lepton number.
The Standard Model Higgs boson has a tree-level potential V (Φ) = m2 Φ† Φ + λ(Φ† Φ)2 , where Φ is an SU(2)doublet. Using a gauge transformation,
one can write√
the classical field as Φ = 1/ 2){eiθ φ, 0 , where φ(x) isreal. Loop corrections substantially modify this potential at large values. We will include one-loop and finite
temperature corrections to the Higgs potential, although
two-loop effects may also be important near the metastability boundary [3]. For the experimentally
preferred toppand Higgs mass values, the hφ2 i = vEM = 246 GeV
minimum appears to be metastable [3], which entails anumber of important ramifications [10]. However, a stable vacuum is still possible within the experimental uncertainties [3]. Furthermore, higher-dimensional operators can modify the potential at large vacuum expectation value (VEV) [29] and make the vacuum stable. During inflation, a scalar field may develop a nonzero VEV
φ2 for more than one reason. We will consider two

cosmological scenarios that lead to two types of initialconditions.
Initial Condition 1 (IC-1) occurs for the central values of measured Higgs and top quark masses. A false
vacuum can appear at large φ due to the negative effective coupling λeff (φ) in the potential, stabilized by somehigher-dimensional operator(s). If the VEV prior to inflation is large (similar to the initial VEV of the inflaton
in chaotic inflation), then the field can evolve toward thefalse vacuum from above and may become trapped in thefalse vacuum. When reheating begins, finite-temperatureeffects eliminate this minimum, and the field rolls down
to the global minimum at φ2 = 0.
Initial Condition 2 (IC-2). During inflation, scalar
fields with slowly rising potentials develop large VEVs [4–6, 11, 12]. The qualitative reason is that, in a de Sitter
space, a scalar field can perform a quantum jump via a
Hawking-Moss instanton [4, 13]. The subsequent relaxation by means
p of a classical motion requires time of the2order τφ ∼ d2 V /dφ
p . If 2the Hubble parameter during inflation, HI = 8π/3ΛI /MP , is much greater than
τφ , then relaxation is too slow and quantum jumps occur
frequently enough to maintain a large VEV. Averaged
over superhorizon scales, the mean Higgs VEV is suchthat V (φI ) ∼ HI4 [4, 13, 14]. This scenario occurs if
the standard Higgs vacuum is stable (which is consistent
with the Higgs and top mass measurements, although notwith the central values), or if the φ2 = 0 minimum isquasi-stable and the scale of inflation is not sufficiently
high to probe the false vacuum.
We will see that the Higgs relaxation in both cases(IC-1 or IC-2) can explain the baryon asymmetry of theuniverse, and the asymmetry
pdepends on the initial valueof the VEV, denoted by
hφ2 i ≡ φ0 . As quantum

2
fluctuations of the Higgs field were ongoing during inflation, different patches of the universe began with slightly
different φ0 values and consequently developed different
baryon asymmetries. This could result in unacceptably
large baryonic isocurvature perturbations [15], which areconstrained by CMB observations [16]. These constraints
can be satisfied as follows.
In the case of IC-1, we make use of the additional termthat stabilizes the second minimum; we choose a termthat also ensures meff > HI in the false vacuum. As aconcrete example, one such term for the experimentally
preferred values mh = 126 GeV and mt = 173.07 GeV is
Llift =(φ† φ)5.
(6.52 × 1015 GeV)6(1)

In this scenario, the parameters must be chosen such thatthe barrier is large enough that the Hawking-Mass instanton does not destabilize the false vacuum during inflation,
but the reheat temperature does destabilize the vacuum.
In the IC-2 case, we consider a coupling between the
Higgs field and the inflaton via one or several operatorsof the form(φ† φ)m/2 (I † I)n/2,
LφI = cMPm+n−2(2)

which increases the effective mass of the Higgs field during inflation. While hIi is large (superplanckian, in thecase of chaotic inflation), this term limits the Higgs VEV
and can be chosen to give meff (φI )  HI during mostof the inflationary epoch. During the last Nlast e-folds ofinflation, the inflaton VEV is small enough√for the Higgs
VEV to grow to some value φ0 = min[φI , Nlast HI /2π].
If Nlast is sufficiently small, the baryonic isocurvature
perturbations develop only on the smallest angular scales,on which they have not yet been constrained. Since thechange in hIi during the slow-roll phase of inflation ismodel-dependent, the choice of parameters c, m, n differsfrom model to model and may require some fine-tuning.
We note that both operators (1) and (2) may be viewed
as effective operators arising from loops after integratingout heavy states. We also note that, although Higgs relaxation commences at the early stages of reheating, theenergy density is never dominated by the Higgs field. Inflaton oscillations dominate until the transition to theradiation dominated era, which occurs when the inflaton
decay width ΓI ∼ HRH , at a much later√ time [17]. The
resulting reheat temperature TRH ∼ ΓI MP  HI isallowed over a broad range of values [18].
The large VEV of the Higgs field makes its dynamics sensitive to higher-dimensional operators which arenormally suppressed by some power of a high scale. A
number of different operators can play an important rolein Higgs relaxation. We will consider the following operator, familiar from spontaneous baryogenesis models [9]:O6 = −

1(∂µ |φ|2 ) j µ .
Mn2

(3)

j µ is the fermion current whose zeroth component is thedensity of (B + L). This operator is equivalent to


3
2
2
02 1
O6 = −|φ | g W W̃ − g AÃ ,
(4)
16π 2
2
(where W and A are the the SUL (2) and UY (1) gauge
fields, respectively) through the mixed SU(2)×U(1)anomaly [9]. A term of this form may be generated
through a loop with fermions which couple to SUL (2)vectorially and have soft mass terms with CP-violatingphases, as higgsinos and gauginos in supersymmetricmodels. We also note that thermal loops can produce
a similar term with T in place of Mn [19–22]. Provided
that the temperature evolution is slow with respect to thetime evolution of the Higgs VEV, one may approximate∂t (|φ(t)|2 /T (t)2 ) ≈ (∂t |φ|2 )/T 2 .
While the Higgs VEV φ(t) is in motion, the Lagrangian
0
contains the term (−µeff jB+L), whereµeff = ∂t |φ|2 /Mn2 .

(5)This term spontaneously breaks CPT [8] and acts as achemical potential, shifting the energy levels of leptons ascompared to antileptons. Lepton number violating processes allow the system to lower its free energy at somevalue of (B + L) 6= 0. Since (∂t |φ|2 ) changes sign during the oscillations of the Higgs VEV φ, there is partial
cancellation during the oscillation of the Higgs VEV, but
this cancellation is not exact due to the decrease in theamplitude of the VEV. The sign of the final asymmetry
is determined by the first, large swing of the field. Thissign is the same everywhere in the universe because |φ|2
decreases with time and ∂t |φ|2 is negative.
We assume the standard seesaw [23] mass matrix forneutrinos, and require that the Majorana mass MR 
T, MR  meff (φ0 ) to forbid production of right-handed
neutrino in thermal plasma and in the condensate decay.
The Majorana mass allows for processes violating the lepton number L (and, therefore, (B + L)). Such processes
involve exchange of virtual heavy Majorana neutrinos;
some of them are shown in Fig. 1. In the presence of Majorana mass terms, the effective lepton number L is thesum of the lepton numbers of the charged leptons andthe helicities of the light neutrinos. This is conserved in
the limit MR → ∞, but not conserved for a finite MR .
These lepton number violating processes change thedensity of (B − L) charge, which is conserved in all otherprocesses, including the sphaleron transitions. At thesame time, the U(1) symmetry corresponding to (B + L)
is anomalous, and the sphaleron transitions can change
this quantity at a characteristic rate per unit volume∼ (αW T )4 exp{−gW φ(t)/T }. During the relaxation of
the Higgs VEV this rate can change dramatically froma slow rate, when B and L are conserved separately(for a large VEV), to a rapid rate, at which B and
L densities approach the equilibrium values such that

3

h0

νL

h0
NR

NRc

νL

h0

νℓc

νL

h0

νLc

NR

NRc



2
d
2
nL + 3HnL ∼= − 2 T 3 σR nL − 2 µeff T 2 .
dt
π
π

νℓc

h0
NRc

NR

NRc

NR
νℓ

mate equation

h0

0

h

νℓc

h0

FIG. 1: Some diagrams that contribute to lepton number
violation via exchange of a heavy Majorana neutrino.

The effective temperature of the plasma during reheating is defined through its radiation density, whichcomes from decays of the inflaton hI and evolves
with
2
 i
MP
ΓI 1
g∗ π 2 4
ti 5/3time as [24] ρR = 30 T ' 10π t 1 − t, wherep2
ti = (2/3) 3/8πMP /ΛI is the effective end of inflation.
For t  ti , the temperature evolves asT =nB = (28/79)nB−L . However, once the asymmetry in L
develops, the sphaleron transitions do not change L by
more than a factor of order 1. Therefore, it is appropriate
to concentrate on either the lepton number or on (B − L)for order-of-magnitude estimates.
We note that, depending on the value of φ(t), weak interactions, mediated by heavy weak bosons with masses
MW ∝ φ(t), may be in or out of equilibrium in the plasma
created by inflaton decay. When φ(t) is close to zero,
weak interactions equilibrate the distributions of charged
and neutral leptons. Neutrinos and Higgs bosons may
also be produced by the decay of the inflaton directly.
For definiteness, we assume a thermal number density
for each of these species. Consequently, the rate of lepton number violating (and (B − L) violating) processes
per unit volume due to the diagrams shown in Fig. 1
can be estimated as (ΓL/ /Vol) ∼ T 6 σR , where σR is thethermally-averaged cross section of the interactions involving the heavy neutrino exchange. The exact ratedepends on the temperature of the plasma and the flavor structure in the Yukawa matrix Yij that enters in theneutrino mass matrix. However, one can estimate
P 2|(Y † Y )jj |2
j mν,jσR ''∼ 10−31 GeV−2 ,2
4
4 × 16πMR,j
16πvEM(6)

where MR,j are the right-handed neutrino masses, andthe reheat temperature is assumed to be too low to produce on-shell right-handed neutrinos. Since the righthanded neutrinos are not present in plasma and not produced via Higgs condensate decay, the contribution fromstandard leptogenesis [26] is strongly suppressed. (Wenote that Eq. (6) neglects several O(1) factors, most notably from thermal averaging and the resonance in thes-channel diagram.)
Based on detailed balance, one can describe the evolution of the particle number densities by a system of
Boltzmann equations. One expects the lepton number
density nL = nν − nν of each species to relax to its equilibrium value ∼ µeff T 2 linearly, which gives the approxi-(7)

3 ΓI MP2g∗ π 3 t

1/4
,

(8)

√
until it reaches the reheat temperature TR ∼ ΓI MP , atwhich point the radiation dominates the energy density.
After this the temperature evolves as

T =

45
16π 3 g∗

1/4

p

MP /t.

(9)

The approximate equation (7) can be analyzed in
two regimes: during the relaxation of the Higgs VEV(µeff (t) 6= 0) and during the subsequent cooling of theuniverse (µeff = 0). During the Higgs relaxation, which
occurs on the time scale of the order of Hφ−1 ,√ 3
λφ0
Hφ φ20∂t |φ2 |∼∼.
µeff =
Mn2
Mn2
Mn2

(10)

As the Higgs VEV oscillates, the equilibrium value is
√ 3 2
√ 3
λφ0 Tmax
λφ0 TR ΛI
nL,eq ∼ µeff T 2 ∼
∼
(11)
Mn2
Mn2
However, the relevant reactions may not be fast enoughto equilibrate to this value before the Higgs VEV approaches zero at trlx . In this case, the maximum asymmetry reached by the end of Higgs relaxation at time trlx
3is nL,eq σR Trlx
trlx , where Trlx is temperature at trlx . In
either case,3nrlx ∼ nL,eq × min 1, (σR Trlx
trlx ) ,(12)

After the Higgs relaxation is completed at trlx , the generated lepton asymmetry can be partially washed out by
heavy neutrino exchanges, until these go out of equilibrium. During washout, Eq. (7) can be re-written as
2dNL (t)' − 2 T 3 σR NL (t),
dt
π

(13)

where NL ≡ nL a3 is the lepton number per comoving
volume. Using the functions T (t) from Eq. (8) or (9),one can rewrite and solve Eq. (13) for the evolution of

4

g∗ π

(14)
provided that trlx  ti . At the end of reheating,#"rg∗ π 324
σR MP TR ,NL (TR ) = NL (Trlx ) exp −g∗ π 53(15)
where we set T = TR , assumedpthat Trlx  TR , and used
the relation ΓI /TR = TR /MP g∗ π 3 /3. The asymptotic
value at low temperature is"#√ !
24 + 3 15p
NL (T → 0) = NL (Trlx ) exp −σR MP TR .
3g∗ π 7(16)
To calculate the asymmetry we include the dilution byentropy production from the time trlx to the time of reheating. The comoving entropy is conserved for T < TR
until the Standard Model degrees of freedom go out of
equilibrium. Therefore, taking into account the dilution
by factor (arlx /aR )3 ≈ (trlx /tR )2 = t2rlx Γ2I , the number
density can be evaluated as nL (0)/nL (Trlx ) = (NL (0)/NL (Trlx ))(a3rlx /a3R ) = (NL (0)/NL (Trlx ))(t2rlx Γ2I ). This
leads to an approximate expression for the asymmetry:η≡nL
2(2π /45)g∗ T 3√

45 λφ30 ΛI 2 23trlx ΓI × min 1, Trlx
trlx σR= 222
2π Mn TR"#√ !
24 + 3 15p
× exp −σR MP TR ,3g∗ π 7

(17)

This analytical estimate agrees within one order of magnitude with the numerical results presented below.
We have analyzed the evolution of the asymmetry numerically. The equation of motion for φ(t) isφ̈ + 3H(t)φ̇ + Vφ0 (φ, T (t)) = 0.

(18)

Here the Hubble parameter is determined by the systemof equations
s
ȧ
8π
H≡ =
(ρr + ρI ),
(19)
a
3MP2
ρ̇r + 4H(t)ρr = ΓI ρI ,(20)

where ρI = Λ4I e−ΓI t /a(t)3 is the energy density of theinflaton field. We have included one-loop corrections [3]

-2
-4
-6

,
logHΗLNL as a function of temperature T :ih224(σR MP
)ΓI
−1
−1

(T
−
T
)
,
N
(T
)
exp
−
 L rlxrlxg∗ π 5
NL (T ) = √
 T ≥ TR
NL (TR ) exp − 3 √5σR M7 P (TR − T ) , T < TR

-8
- 10
- 12
- 14
- 16

- 14

- 12

logHt @GeV-1 DL

- 10

-8

-6

FIG. 2: Numerical solutions for the lepton number asymmetry. The IC-1 scenario is shown by the blue solid line forΛI = 1015 GeV, ΓI = 109 GeV, and MR = 9 × 1015 GeV,
which results in Tmax = 6 × 1013 GeV, sufficient to destabilize the second minimum. The initial VEV is φ0 = 1 ×1015 GeV. The IC-2 case is shown by the red dashed line forΛI = 1017 GeV, ΓI = 108 GeV, Mn = 5 × 1012 GeV, and
MR = 1016 GeV. The maximum temperature during reheating is Tmax = 3 × 1014 GeV, and φ0 = 1 × 1015 GeV. The
vertical blue dot-dashed (IC-1) and red dotted (IC-2) lines
denote (from left to right) the effective end of inflation ti ,the first crossing of zero (φ(t) = 0), and the beginning ofradiation-dominated era.

and finite temperature corrections [25] in the Higgs potential. The solution of (18) determines the effective
chemical potential µeff (t) via Eq. (5), which we then usedin solving Eq. (7) numerically. The results are shown in
Fig. 2.
For the IC-1 case shown in Fig. 2 by red dashed line, we
set Mn = T in the denominator of Eq. (5), as expected if
the operator O6 is generated by thermal loops. For thevalues of ΛI and ΓI listed in the caption, the effective
mass of the Higgs field in the false vacuum is indeed larger
than HI , suppressing the baryonic isocurvature perturbations. For IC-2 case, shown by the blue solid line, weused a constant value of Mn in Eq. (5). The lepton asymmetries at the end of reheating are ∼ O(10−8 ) in bothcases. As the universe cools down, the Standard Modeldegrees of freedom go out of equilibrium, and the entropy
production reduces the value of the baryon asymmetry
by a factor ξ ≈ 30. This brings the final asymmetry to
the observed value of O(10−10 ). As discussed above, thesphaleron processes redistribute this asymmetry between
lepton and baryon numbers, as in the case of thermal leptogenesis [26], leading to a successful baryogenesis.
We note that the reheat temperature controls thedilution of η in Eq. (17) via an exponential factor
exp[−0.036 σR MP TR ] = exp[−TR /3 × 1013 GeV]. Thisimplies an upper bound on TR . 3 × 1014 GeV, to avoid
excessive dilution.
Finally, we note that the epoch of Higgs relaxation

5
can have other observable consequence; in particular, it
has the necessary conditions for primordial magnetogenesis [27], although it may be challenging to obtain the large
correlation length necessary to explain the seed magnetic
fields and the fields observed in the voids [28].
In summary, we have shown that the matterantimatter asymmetry could be generated during Higgs
relaxation, assuming that the Standard Model is valid up
to some very high scales.
Acknowledgements. The authors thank J. Cornwall,
K. Harigaya, M. Ibe, M. Kawasaki, R. Peccei, K. Schmitz,
H. Sugawara, and T.T. Yanagida for helpful comments
and discussions. This work was supported by the U.S.
Department of Energy Grant DE-SC0009937 and by theWorld Premier International Research Center Initiative(WPI Initiative), MEXT, Japan, as well as the National
Science Foundation under Grant No. PHYS-1066293 andthe hospitality of the Aspen Center for Physics.

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