#### Running non-minimal inflation with stabilized inflaton potential

Eur. Phys. J. C
Running non-minimal inflation with stabilized inflaton potential
Nobuchika Okada 0
Digesh Raut 0
0 Department of Physics and Astronomy, University of Alabama , Tuscaloosa, Alabama 35487 , USA
In the context of the Higgs model involving gauge and Yukawa interactions with the spontaneous gauge symmetry breaking, we consider λφ4 inflation with nonminimal gravitational coupling, where the Higgs field is identified as the inflaton. Since the inflaton quartic coupling is very small, once quantum corrections through the gauge and Yukawa interactions are taken into account, the inflaton effective potential most likely becomes unstable. In order to avoid this problem, we need to impose stability conditions on the effective inflaton potential, which lead to not only non-trivial relations amongst the particle mass spectrum of the model, but also correlations between the inflationary predictions and the mass spectrum. For concrete discussion, we investigate the minimal B − L extension of the standard model with identification of the B − L Higgs field as the inflaton. The stability conditions for the inflaton effective potential fix the mass ratio amongst the B − L gauge boson, the right-handed neutrinos and the inflaton. This mass ratio also correlates with the inflationary predictions. In other words, if the B − L gauge boson and the right-handed neutrinos are discovered in the future, their observed mass ratio provides constraints on the inflationary predictions.
1 Introduction
Current understanding about the origin of our universe is that,
for a very brief moment at the beginning, our universe went
through a period of rapid accelerated expansion known as
inflation. Inflation scenario [1–4] was originally proposed to
solve serious problems in the Standard Big-Bang
Cosmology, namely, the horizon, flatness and monopole problems.
In addition and more importantly from the view point of
the current cosmological observations, inflation provides a
mechanism to create primordial density fluctuations of the
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early universe which seed the formation of large scale
structure of the universe that we see today. In a simple inflation
scenario, inflation is driven by a single scalar field (inflaton)
that slowly rolls down to its potential minimum (slow-roll
inflation). During the slow-roll era, the inflaton energy is
dominated by a slowly varying potential, which causes the
universe to undergo a phase of an accelerated expansion.
Quantum fluctuations of the inflaton field are stretched to
macroscopic scales by inflation to yield the primordial
density fluctuations. After inflation, the inflaton decays to the
Standard Model (SM) particles and the decay products heat
up the universe (reheating). The success of big bang
nucleosynthesis scenario requires the reheating temperature to be
TR 1 MeV.
Recently the Planck 2015 results [5] have set an upper
bound on the tensor-to-scalar ratio as r 0.11 while the
best fit value for the spectral index (ns) is 0.9655 ± 0.0062
at 68% CL. Hence, the simple chaotic inflationary scenario
with the inflaton potentials V ∝ φ4 and V ∝ φ2 are
disfavored because their predictions for r are too large. Among
many inflation models, λφ4 inflation with non-minimal
gravitational coupling (ξ φ2R, where φ is inflaton, R is the scalar
curvature, and ξ is a dimensionless coupling) is a very simple
model, which can satisfy the constraints by the Planck 2015
with ξ 0.001 [6–8].
Given that we need interactions between SM particles and
inflaton for a successful reheating of the universe, a more
compelling inflation scenario would be where the inflaton
field plays another important role in particle physics. As an
example of such a scenario, we may consider the (general)
Higgs model, where a scalar (Higgs) field plays the
crucial role to spontaneously break the gauge symmetry of the
model, and we identify the Higgs field as inflation. The SM
Higgs inflation [9–18] is nothing but this scenario, where the
SM Higgs boson plays the role of inflaton with non-minimal
gravitational coupling. For the observed Higgs boson mass
of around 125 GeV, the stability of SM Higgs potential is
very sensitive to the initial value of top quark pole mass.
Hence, we need more precise measurements for top quark
pole mass [19] to obtain a conclusion about the SM Higgs
potential stability.1 For a large top quark pole mass, for
example, Mt = 173.34 GeV [24], the SM Higgs potential turns
out to be unstable, and some extension of the SM is necessary
to realize the SM Higgs inflation [25,26].2 It is interesting
to apply the same idea as the SM Higgs inflation to the
general Higgs model and identify the Higgs field of the model
(not the SM Higgs field) as the inflaton in the presence of
non-minimal gravitational coupling. For a simple example,
see [37].
As in the SM, the Higgs field in the general Higgs model
has the gauge, Yukawa and quartic Higgs interactions. For
complete analysis of inflation scenario in the Higgs model,
we consider the effective inflaton/Higgs potential by
taking quantum corrections into account. In fact, we see that
quantum corrections most likely cause an instability of the
effective inflaton potential. Note that unless the non-minimal
coupling (ξ ) is very large, the quartic inflaton coupling is
very small [6–8]. Hence, quantum corrections to the effective
potential are dominated by the gauge and Yukawa
interactions. We consider the renormalization group (RG) improved
effective potential described as
where φ denotes inflaton, and λ(φ) is the running quartic
coupling satisfying the (one-loop) RG equation of the form,
Cg g4 − CY Y 4.
Here, g and Y are the gauge and Yukawa couplings,
respectively, and Cg and CY are positive coefficients whose actual
values are calculable once the particle contents of the model
are defined. Since the quartic coupling is very small, we have
neglected terms proportional to λ (λ2 term and the
anomalous dimension term). The solution to the RG equation is
controlled by g and Y , which are much larger than λ and
independent of λ. Therefore, we expect that unless the beta
function is extremely small, the running inflaton quartic
coupling λ is driven to be negative in the vicinity of the inflation
initial value, in other words, the effective potential has true
minimum (far) away from the vacuum set by the Higgs
potential at the tree level.
A simple way to avoid this problem is to require the beta
function to vanish at the initial inflaton value (the stationary
1 See also [20–23] for the effect of possible Planck scale physics to the
effective Higgs potential.
2 Supersymmetric version of the Higgs inflation [27–35] is free from
this instability problem because of supersymmetry. It has been shown
in [36] that a large non-Gaussianity can be generated in this class of
models.
condition of λ with respect to φ), namely, Cg g4−CY Y 4 = 0.3
This condition leads to a relation between g and Y ,
equivalently, a mass relation between the gauge boson and fermion
in the Higgs model. Since the Higgs quartic coupling at low
energy is evaluated by solving the RG equation, in which the
gauge and Yukawa couplings dominate, the resultant Higgs
mass also has a relation to the gauge and fermion masses.
The stability of the effective potential also requires the
positivity of the second derivative of the potential, which leads
to another constraint on the gauge and Yukawa couplings. In
the slow-roll inflation, the inflationary predictions are
determined by the slow-roll parameters defined with the
potential and its derivatives, and therefore, the inflationary
predictions have a correlation with the mass spectrum of the Higgs
model.
In order to explicitly show the mass relation and the
correlation between the particle mass spectrum and inflationary
predictions, we take the minimal B − L model as an
example. This model is a very simple, well-motivated extension
of the SM, where the global B − L (baryon number minus
lepton number) in the SM is gauged. Three right-handed
neutrinos and the B − L Higgs field (which is identified as the
inflaton) are introduced for the cancellation of the gauge and
gravitational anomaly and the B − L gauge symmetry
breaking, respectively. Associated with the B − L gauge symmetry
breaking, the B − L gauge boson and the right-handed
neutrinos acquire their masses. With the generation of the Majorana
right-handed neutrino masses, the seesaw mechanism [40–
45] for the light neutrino mass generation is automatically
implemented in this model. Analyzing the RG evolutions of
the B − L sector and the effective inflaton (B − L Higgs)
potential, we show the particle mass spectrum and its
correlation to the inflationary predictions. Through the correlation,
the Planck 2015 results provide us with constraints on the
particle mass spectrum.
This paper is organized as follows. In the next section, we
briefly review the λφ4 inflation with non-minimal
gravitational coupling at the tree level, and discuss the inflationary
predictions in the light of the Planck 2015 results. In Sect. 3,
we introduce the minimal B − L extension of the SM and
calculate the RG improved effective Higgs potential. We show
the particle mass spectrum derived from the stability
conditions of the effective potential and its correlation to the
inflationary predictions. We then compare our results for various
values of the non-minimal gravitational coupling ξ with the
Planck 2015 results. In Sect. 5, we discuss reheating
scenario in the B − L Higgs inflation for the completion of our
inflationary scenario. Section 6 is devoted to conclusions.
3 It is interesting to notice that a similar condition is realized in the SM
Higgs inflation scenario with a special choice of parameters (a critical
point) [38,39].
2 Non-minimal λφ4 inflation at tree level
In the Jordan frame, the action of our inflation model is
given by (hereafter we always work in the Planck unit,
MP = MPl/√8π = 1, where MPl = 1.22 × 1019 GeV
is the Planck mass)
SJ =
d4x √−g
where f (φ) = (1 + ξ φ2) with ξ being a positive,
dimensionless parameter, and the inflaton potential is
Using the conformal transformation, gEμν = f (φ)gμν , the
action in the Einstein frame is descried as
SE =
1 1 V [φ (σ )]
−gE − 2 RE + 2 (∇Eσ )2 − f 2[φ (σ )] .
In the Einstein frame with a canonical gravity sector, we
describe the theory with a new inflaton field (σ ) which has
a canonical kinetic term. The relation between σ and the
original inflaton field φ is given by
where a prime denotes the derivative with respect to φ.
The inflationary slow-roll parameters in terms of the
original scalar field (φ) are expressed as
where VE is the potential in the Einstein frame in terms of
original field φ given by
where φI is the inflaton value at horizon exit corresponding to
the scale k0, and φe is the inflaton value at the end of inflation,
which is defined by max[ (φe), |η(φe)|] = 1. The value of
N depends logarithmically on the energy scale during
inflation as well as on the reheating temperature, and is typically
around 50–60.4
The slow-roll approximation is valid as long as the
conditions 1, |η| 1 and ζ 1 hold. In this case,
the inflationary predictions, the scalar spectral index ns, the
tensor-to-scalar ratio r , and the running of the spectral index
α = ddlnnsk , are given by
Here the slow-roll parameters are evaluated at φ = φI.
Figure 1 shows inflationary predictions for the
nonminimal λφ4 inflation at the tree level for N = 50 (dashed–
dotted) and N = 60 (solid). Top panels show r vs. ξ (left)
and ns vs. ξ (right). Both r and ns show asymptotic
behavior for both small and large ξ values. In the minimal λφ4
inflation limit with ξ = 0, we obtain r 0.31 (0.26) and
ns 0.942 (0.951) for N = 50 (60). The plots also show
that, for a larger e-holding number, we obtain a larger ns
while a smaller r . The bottom-left panel shows the tree-level
quartic coupling λ as a function of ξ . Note that λ is very
small unless ξ 1. The inflationary predictions for ns and r
for various values of ξ are depicted in the bottom-right panel
along with the results from the measurements by Planck 2015
and Planck+BICEP2/Keck Array [5]. We see that the
inflationary predictions for ξ 0.001 are consistent with the
observations.
3 Running B − L Higgs inflation and stability
of inflaton potential
In order to investigate the Higgs inflation with the stabilized
inflaton potential, in this paper we take the minimal B − L
extension of the SM as an example, where the
anomalyfree U(1)B−L gauge symmetry is introduced along with a
scalar field ϕ and three right-handed neutrinos NRi . The
particle contents of our model are listed in Table 1. This model
requires three generations of right-handed neutrinos to
cancel all the gauge and gravitational anomalies. The B − L
gauge symmetry is broken by the vacuum expectation value
(VEV) of ϕ in its Higgs potential of
4 For a small ξ 1, the inflaton is very light and its potential is approx
imately given by λφ4. In this case the inflaton energy density behaves
like the energy density of the radiation, and the e-folding number N can
be determined unambiguously [46]. In this case we find N 60.
The amplitude of the curvature perturbation
which should satisfy 2R = 2.195 × 10−9 from the Planck
2015 results [5] with the pivot scale chosen at k0 = 0.002
Mpc−1. The number of e-folds is given by
−1/3
−1/2
−1
−1/2
Associated with the gauge symmetry breaking, the
righthanded neutrinos acquire their Majorana masses through the
Yukawa interaction,
Fig. 1 The inflationary predictions for N = 50 (dot-dashed) and
N = 60 (solid) in the non-minimal λφ4 inflation. The top panels show r
vs. ξ (left) and ns vs. ξ (right). The bottom-left panel shows λ vs. ξ . The
inflationary predictions ns and r for various values of ξ are depicted
Table 1 Particle contents of the minimal B − L model. In addition to
the SM particle contents, the right-handed neutrino NRi (i = 1, 2, 3
denotes the generation index) and a complex scalar ϕ are introduced
.002 0.08
0
r
U(1)B−L
−1
−1
−1
in the bottom-right panel, along with the contours at the confidence
levels of 68% and 95% given by the results of Planck 2015 (solid) and
Planck+BICEP2/Keck Array (dot-dashed) [5]
where we have taken the degenerate mass spectrum for the
right-handed neutrinos, for simplicity. After the B − L
symmetry breaking with the Higgs VEV ϕ = vB L /√2, the
particle masses are given by
m Z = 2 g vB L ,
Let us now consider the B − L Higgs inflation scenario.
The action in the Jordan frame is given by
−g
where Dμ = ∂μ − i 2g Zμ, and f (|ϕ|) ≡ 1 + 2ξ ϕ†ϕ, and
√
ϕ = (vB L + φ)/ 2 in the unitary gauge with the physical
Higgs field φ identified as the inflaton. In the Einstein frame,
the RG improved effective inflaton potential at the one-loop
level is given by [47–49]
where ≡ φ/ 1 + ξ φ2, and we have neglected vB L , which
is much smaller than the Planck mass.5 The RG equations of
the couplings at the one-loop level are given by [37]
16π 2μ ddYμ = −6g2Y + 25 Y 3,
+ 96g4 − 3Y 4,
where a factor s defined as
18s2 + 2 λ2 − (48g2 − 6Y 2)λ
is assigned to each term in the RG equations associated with
only the physical Higgs boson loop corrections [9–16].6 We
find that this s-factor has no effect in our numerical analysis
for a parameter region satisfying g2, Y 2 λ, and one may
fix s = 1 as a good approximation. In our RG analysis, we
have neglected the RG evolution of ξ , whose effect is found
to be negligible. It has been pointed out in the last paper in [9–
13] and emphasized recently in [39,50], some undetermined
parameters are involved in the presence of the non-minimal
gravitational coupling, when we connect couplings at high
energies to those at low energies in the RG equations.
However, as we will see in the following, all couplings in our
model are very small, and for such small couplings, we can
expect the effect of the undetermined parameters is
negligibly small.
Let us now investigate the stability of the effective inflaton
potential. In our analysis throughout this paper, we set the
initial values of λ to be the one obtained in the tree-level
analysis at the initial inflaton value φ = φI, equivalently,
I = φI/ 1 + ξ φI2. Then, we consider the RG improved
effective inflaton potential by taking into account the RG
evolution of the quartic coupling with the initial condition at
I.
Since RGE evolves logarithmically, to a good
approximation we can analytically solve the RGE of λ around I such
that βλ is taken to be constant. The solution is given by
5 In this RG improved effective potential, we have identified the renor
malization scale with according to the prescription proposed in [47–
49], where it has been shown that the effective potential is
frameindependent with this choice. In fact, we are especially interested in
the parameter region of ξ 1 in this paper, so that φ and the
identification reduces to the usual one.
6 There are a few different prescriptions for computing quantum correc
tions in the presence of the non-minimal gravitational coupling [17,18].
For recent, detailed computations of quantum corrections, see [47–49]
and their results of 1-loop beta functions with the s-factor.
where I is the initial inflation scale and 0 is the scale such
that λ( 0) = 0. So we require βλ( I) 16π 2 × λ( I)
to ensure that the potential is stable. As we have seen in the
previous section (see Fig. 1), the quartic coupling is very
small unless ξ 1. Consider ξ = 1, such that λ( I)
10−10, we thus require βλ( I) 10−8. Hence, imposing
βλ( I) = 0 is a good approximation for stability of the
potential.
Unless ξ 1, the quartic coupling is very small, hence
the beta function of the quartic coupling is approximately
given by
18s2 + 2 λ2 − (48g2 − 6Y 2)λ + 96g4 − 3Y 4
when g2, Y 2 λ.7 The RG evolution is controlled by g and
Y , which are independent of the initial value of the inflaton
quartic coupling. Figure 2 shows the RG evolution of the
inflaton quartic coupling in the vicinity of the initial
inflaton value for various values of g and Y with a fixed ξ = 1.
In the left panel, the solid, the dashed and the dot-dashed
lines denote the RG evolutions for g = 0.01, 0.011 and
0.009, respectively, with the fixed value of Y = 0.0237.
In the right panel, the solid, the dashed and the dot-dashed
lines denote the RG evolutions for Y = 0.0237, 0.0214 and
0.0261, respectively, with the fixed value of g = 0.01. Note
that βλ( I) = 0 is satisfied with the parameter choice for the
solid lines, g = 0.01 and Y = 0.0237. We can see from Fig. 2
that if the condition of βλ = 0 is violated even with ±10 %
deviations for the values of g or Y , the running quartic
coupling quickly becomes negative in the vicinity of I (see the
dashed and the dot-dashed lines). This fact indicates that the
B−L gauge symmetry breaking vacuum at φ = vB L is
unstable and the effective potential develops a true vacuum with
a negative cosmological constant. The quantum corrections
through the gauge and Yukawa coupling completely change
our inflationary scenario from the one at the tree level.
In order to avoid this instability, we impose not only the
condition of βλ = 0 but also dβλ/d > 0 at I.8 From the
7 In this paper, we are interested in this case, otherwise the beta function
is so small that the inflaton quartic coupling is almost RG invariant.
Although the tree-level analysis is valid in this case, the gauge and
Yukawa couplings are too small to yield any impacts in the experimental
point of view.
8 To be precise, βλ is not necessary to be exactly zero to stabilize the
i3nYfl4atonλp2o. tSeinntciael.λWe m1,aβyλc=ons0idiseraagomoodreapgpernoexriamlactoinodni.tIionnad9d6igti4o−n,
although g, Y λ can approximately satisfy this condition, we are not
interested in such very small couplings for which RG evolutions are
negligible.
Fig. 2 The RG evolution of the inflaton quartic coupling λ for various
values of g and Y , with the fixed value of ξ = 1. The initial value of
λ( I) is fixed by the tree-level analysis. In the left panel, the solid, the
dashed and the dot-dashed lines show the running quartic couplings
for g = 0.01, 0.011 and 0.009, respectively, with the fixed value of
Y = 0.0237. In the right panel, the solid, the dashed and the
dotdashed lines show the running couplings for Y = 0.0237, 0.0214 and
0.0261, respectively, with the fixed value of g = 0.01. The stability
condition of βλ( I) = 0 is satisfied for g = 0.01 and Y = 0.0237, and
the corresponding running couplings are depicted as the solid lines
Fig. 3 The RG improved effective potential for various values of g and
a fixed ξ = 0.1. In the left panel, the effective potentials for g = 0.041
(solid) and 0.046 (dot-dashed) are shown. For g = 0.046 > gmax =
0.0425, the local minimal is developed. The vertical dashed line
indicates = I (in the Planck unit). In the right panel, the effective
potentials for various values of gmin < g < gmax are shown
first condition, the Yukawa coupling Y is determined by the
gauge coupling, which we take as a free parameter in our
analysis, along with the others, ξ and vB L . The second
condition ensures that the effective potential is monotonically
increasing in the vicinity of I, and yields a lower bound on
g > gmin.9 When we analyze the global structure of the
effective potential, we can notice that there exists an upper bound
on g < gmax. For a large g > gmax, the effective potential
develops a local minimum at < I, so that the inflaton
field will be trapped in this minimum after inflation.10 A
second inflation then takes place until the vacuum transition
9 From Eq. (3.9) we find gm2in ∼ λ. The discussion here is applicable
only for g2 λ, and gmin is not the general lower bound on the gauge
coupling. For g2, Y 2 λ, the running effect on all the couplings is
negligible and hence the inflaton potential is stable.
10 There is an interesting possibility that such a local minimum can be
lifted up once thermal effect on the scalar potential is taken into account
[50]. In our case, as we will see in Sect. 5, reheating temperature is not
high enough to remove this local minimum.
from this local minimum to the true B − L symmetry
breaking vacuum. To avoid this problem, the parameter region is
restricted to be in the range of gmin < g < gmax. Figure 3
shows the effective potential for various values of the gauge
coupling (g) for fixed ξ = 0.1. In the left panel, the solid
line depicts the effective potential for g = 0.041, while the
dot-dashed line is for g = 0.046. In this example, we find the
upper bound as gmax = 0.0425. We can see that the effective
potential develops a local minimum for g = 0.046 > gmax.
For various values of g < gmax, the effective potentials are
shown in the right panel.
4 Inflationary predictions and low energy observables
Under the stability conditions, gmin < g < gmax with various
value of ξ , we now calculate the inflationary predictions with
the effective inflaton potential. Since we refer the results in
the tree-level analysis for λ( I) for a fixed ξ and impose
the stability condition βλ( I) = 0, our prediction for the
tensor-to-scalar ratio r is the same as the one obtained in the
tree-level analysis. However, the RG evolution of the inflaton
quartic coupling alters the other inflationary predictions, ns
and α, from those obtained in the tree-level analysis, because
they are calculated by the second and third derivatives of the
effective potential (see Eqs. (2.5) and (2.9)).
Let us first derive an analytic formula for the deviations
of ns from the tree-level prediction. In the effective inflaton
potential, the inflation quartic coupling is not a constant, but
a function of = φ/ 1 + ξ φ2. With the stability condition
βλ( I) = 0, we calculate the derivatives of the effective
potential as
1
φ=φI + 4
E tree
level analysis, and we have used λ (φI) ∝ βλ( I) = 0 under
the stability condition. Rewriting d/dφ in terms of d/d , we
have
are evaluated to be the same as those in the
tree(φI) = tree, η(φI) = ηtree +
where tree and ηtree are the slow-roll parameters evaluated
in the tree-level analysis, and
(1 − ξ I2)2 λ¨( I) (1 − ξ 2I)2
η = 1 + 6ξ 2 I2 λ( I) = 1 + 6ξ 2 2
I
Here, a dot is the derivative with respect to . Since β˙λ is
non-zero, η is deviated from its tree-level value ηtree. Using
Eqs. (3.6) and (3.9), we obtain
where in the last expression we have used the stability
condition Y 4 32g4 at I. Finally, we arrive at an expression
for the spectral index as (see Eq. (2.9))11
11 In the same way, we can express the running of the spectral index,
α, in terms of ξ , I, λ and g. However, as we will see in the following
numeral analysis, the predicted α values are found to be very small
and always consistent with the Planck 2015 results. Thus, we omit the
expression for α.
For a fixed ξ , I and λ( I) are determined by the tree-level
analysis, so that the inflationary prediction is controlled by
the gauge coupling g, which is the free parameter in our
analysis.
In our model, there are only three free parameters, ξ , vB L
and g, with a fixed e-folding number N = 50/60. Once we fix
ξ and N , I and λ( I) are fixed by the tree-level analysis for
inflation, and the inflationary predictions except for r are
controlled by the gauge coupling g with its relation to the Yukawa
coupling Y led by the stability condition βλ( I) = 0. In the
B − L model, the particle mass spectrum is determined by
the gauge, Yukawa and inflation quartic couplings at the scale
vB L (see Eq. (3.3)), which are obtained by solving the RG
equations in Eq. (3.6) from μ = I to μ = vB L . Since λ( I)
is very small, its RG evolution is determined by g and Y as
we can see from its RG equation. Considering all of these
facts, we expect that there exists a non-trivial mass relation
in the particle spectrum and a non-trivial correlation between
the inflationary predictions and the particle mass spectrum.
In the following, the results of our numerical analysis will
show such non-trivial relations. For simplicity, we fix vB L
so as to yield m Z = 2g(vB L )vB L = 3 TeV, to be
consistent with the current results from the search for Z boson
resonance at the Large Hadron Collider [51–53].
Figure 4 shows the resultant inflationary predictions for
a variety of ξ values with the input values of g in the range
of gmin < g < gmax at I (N = 60), along with the
contours given by the Planck 2015 results. As we have discussed
above, the prediction for the tensor-to-scalar ratio is the same
as the one in the tree-level analysis, while the predicted
specFig. 4 Inflationary predictions for various values of ξ with the input of
g varied in the range of gmin < g < gmax, along with the contours given
by the experiments (same as in Fig. 1). The horizontal solid lines from
top to bottom correspond to the results for ξ = 1.5 × 10−3, 2.1 × 10−3,
2.9 × 10−3, 4.1 × 10−3, 6 ×10−3, 0.01, 0.02, 0.1 and 1. The diagonal
dashed line denotes the inflationary predictions in the tree-level analysis
Fig. 5 Inflationary predictions and low energy observables for
various values of g < gmax and ξ 0.0029, 0.01 and 0.1 (corresponding
r 0.108, 0.045 and 0.008), respectively. We have fixed N = 60 and
m Z to be 3 TeV. gmax increases with increasing ξ . Top two panels show
the inflationary predictions for spectral index ns (left) and the running of
tral index is altered by quantum corrections. In Fig. 4, we can
see that, for ξ 1, the results show sizable deviations for
g ∼ gmax from those at the tree-level analysis depicted as the
diagonal dashed line. Interestingly, the Planck 2015 results
provide upper bounds on g, which are more severe than gmax
faoprpξroach0e.0s011/.√Inξ ofruormnuamsmeraicllaelravnaalluyesiass, wwee icnacnresaeseetξhat 1I,
and hence the deviation of the predicted ns value from the
one in the tree-level analysis becomes smaller as we can see
from Eq. (4.6) with the limit 1 − ξ 2 I → 0.
In order to see the inflationary predictions as a function
of g, we show our results in Fig. 5 for ξ = 0.0029, 0.01
and 0.1, with N = 60. The top-left and top-right panels
show the inflationary predictions of ns and α = dns/d ln k
as a function of g in the range of gmin < g < gmax. The
prediction for r is the same in the tree-level analysis, and
r 0.108, 0.045 and 0.008, respectively, for ξ = 0.0029,
0.01 and 0.1. For a larger value of ξ , gmax becomes larger.
As g is lowered, the predicted ns value approaches the
treelevel prediction. By numerically solving the RG equations
for the couplings in Eq. (3.6) for a fixed g, we obtain the
particle mass spectrum with m Z = 3 TeV. The mass ratio
spectral index dns/d ln k (right), for decreasing r (top to bottom). The
bottom-left panel shows the mass of the inflaton mφ (top to bottom) for
decreasing r . The bottom-right panel shows mass ratio mNR/m Z (top
to bottom) for increasing r values
m N /m Z is shown in the bottom-left panel, while the
bottomright panel shows the inflaton mass.12 In both panels, the
solid lines from left to right correspond to the results for
ξ = 0.0029, 0.01 and 0.1, respectively. The resultant mass
ratio is almost independent of ξ , but shows a splitting for
g 5 × 10−4. For such a very small g, its corresponding Y
determined by the stability condition is also very small, and
hence both g and Y are almost RG invariant and the mass ratio
is determined by Y ( I)/g( I). However, in this case, the
condition g2, Y 2 λ is no longer valid, and Y determined
by βλ( I) = 0 depends on the input λ values. This is the
reason why the bottom-left panel shows the splitting among
three solid lines for g 5 × 10−4. For g 0.005, the
RG evolution of λ is mainly determined by g and Y in its
beta function, since λ( I) is extremely small. For a very
small g value 0.005, the effect of g and Y on the RG
evolution of λ becomes negligible, and λ(vB L ) λ( I).
12 Note that although the resultant mass ratios are shown at a very high
precision level, our RG analysis at the one-loop level has an uncertainty
of O(g2)% from quantum corrections at the next order. At this level
of precision, different lines are virtually indistinguishable, so we have
shown Figs. 5 and 6 at such a very high precision level.
V
e
G100
mφ 50
5 10 4 0.001
g
0.005 0.01
g
Fig. 7 Mass spectrum for various value of ξ = 1, 10, 50, 150, 500 and 1000 from left to right. The left panel shows the mass ratio mNR/m Z ,
while the right panel shows the mass of the inflaton. Here we have fixed m Z = 3 TeV
Since we have fixed m Z = 2g(vB L )vB L = 3 TeV, vB L =
1.5 TeV/g(vB L ) 1.5 TeV/g( I), and the inflaton mass
becomes larger proportionally to 1/g( I) as shown in the
bottom-right panel.
Same as Fig. 5 but for N = 50 and 60 with ξ = 0.1 is
depicted in Fig. 6. The dashed lines denote the results for
N = 50, while the solid lines for N = 60. The inflationary
predictions show a sizable deference for the two different N
values, as shown in the tree-level analysis in Sect. 2. On the
We show the results for the mass spectrum for large ξ
values and N = 60 in Fig. 7. The solid lines from left to
right corresponds to the results for ξ = 1, 10, 50, 150, 500
and 1000. For g 0.05, the resultant mass ratio in the left
panel shows ξ -dependence. This is because g2, Y 2 λ is
no longer valid for such a small g value, and the Y value
determined by βλ( I) depends on λ( I). We also show the
2
.00 0.08
0
r
Fig. 8 Inflationary predictions for various fixed g(vBL ) values along
with the results shown in Fig. 4. Here, for a fixed g(vBL ) value, the
inflationary predictions are calculated for various values of ξ taken in
Fig. 4. We have used m Z = 3 TeV. The diagonal solid lines correspond
to g(vBL ) = 0.0184, 0.0216, 0.023, 0.026, and 0.0425
inflation mass spectrum in the right panel, which show a
similar behavior to the result in the bottom-right panel in
Fig. 5. As we have seen in Fig. 4, the inflationary predictions
for ξ 1 are close to those obtained in the tree-level analysis,
ns 0.968 and r 0.003.
Finally, we show in Fig. 8 the contour plots (diagonal
solid lines) for the inflationary predictions with various fixed
g(vB L ) values, along with the results shown in Fig. 4. Here,
for a fixed g(vB L ) value, we calculate the inflationary
predictions from the effective inflaton potential for various values
of ξ . We have used m Z = 3 TeV. The diagonal solid lines
correspond to g(vB L ) = 0.0184, 0.0216, 0.023, 0.026, and
0.0425 from left to right.
5 Reheating after inflation
Any successful inflation scenario requires the transition to the
Standard Big Bang Cosmology after inflation. This happens
via the decay of the inflaton into the SM particles; during the
era the inflaton is oscillating around it potential minimum,
and the decay products then reheat the universe.13 We
estimate the reheating temperature after inflation TR by using
= H = TR2 π902 g∗, where is the inflaton decay width,
and g∗ is the effective degrees of freedom for relativistic SM
particles when the reheating occurs, so that
0.55 (100/g∗)1/4
13 In general, reheating can occur through the parametric resonance
more effectively than the perturbative decay of the inflaton. For a
detailed discussion of “preheating” in the Higgs inflation, see [54,55]. In
our analysis we do not consider the preheating and estimate the
reheating temperature using Eq. (5.1). The true reheating temperature could
be much higher than our result.
From the success of big bang nucleosynthesis, we take a
model-independent lower bound on the reheating
temperature as TR 1 MeV.
As we discussed in the previous section, the inflaton is
much lighter than the Z boson and right-handed neutrinos.
Thus, its decay width to the SM particles through off-shell
process mediated by the heavy particles is too small to satisfy
the lower bound TR 1 MeV. An efficient reheating process
is possible when the inflaton in general has a coupling with
the SM Higgs doublet (H ) such as (see, for example, Ref. [56,
57] for phenomenology of inflaton through the coupling with
the SM Higgs boson)
Although this coupling is crucial for the reheating process,
we assume λ 1 not to change our analysis for the B − L
model in the previous sections. Since the inflaton quartic
coupling is very small and hence the inflaton is light, it is
most likely that the inflaton can decay to the SM particles
only through its mixing with the SM Higgs boson through
the λ coupling. After the B − L and electroweak symmetry
breakings, we diagonalize the scalar mass matrix of the form
where h is the SM Higgs boson, and φ1 and φ2 are the mass
eigenstates. The relations among the mass parameters and
the mixing angle are the following:
2vB L vSMλ = (m2h − m2φ ) tan 2θ ,
where vSM = 246 GeV is the Higgs doublet VEV, and mh
and mφ are the masses for h and φ, respectively.
The left panel in Fig. 9 shows the mass eigenvalue mφ2 as a
function of sin θ for various mφ values with mh = 125 GeV.
In the θ = 0 limit, mφ2 = mφ , while mφ = mh for θ = π/2.
Although in this plot we show the results by using Eq. (5.5),
we only consider the case with θ 1 as mentioned above,
otherwise our results obtained in the previous sections are
changed in the presence of a sizable λ . For θ 1, the mass
eigenstate φ2 (φ1) is almost identical to φ (h).
The inflaton can decay to the SM particles through the
mixing with the SM Higgs boson. We calculate the inflaton
decay width as
where h (mφ2 ) is the SM Higgs boson decay width if the
SM Higgs boson mass were mφ2 . The reheating temperature
is then evaluated by Eq. (5.1). For various inputs of mφ , we
show the resultant reheating temperature in the right panel of
Fig. 9 Left the mass eigenvalue mφ2 as a function of sin θ for
various mφ values with mh = 125 GeV. We have used Eq. (5.5). Right
the reheating temperature as a function of sin θ for various values of
Fig. 8. The solid lines from bottom to top denote the results
for mφ = 0.32, 3.0 and 130 GeV, respectively, along with the
dotted horizontal line for the lower bound of TR = 1 MeV.
The sharp drop for each solid line corresponds to the fact that
mφ2 becomes zero for a certain sin θ value as shown in the
left panel. We find that the universe is sufficiently heated up
for a mixing angle in the suitable range shown in Fig. 8.
Finally, we check the theoretical consistency of our
analysis. When we introduce the coupling in Eq. (5.2), the beta
function of the inflaton quartic coupling is modified to
In order not to change our results in the previous sections by
the introduction of λ , λ 2 should be negligibly small in the
beta function. We then impose a condition
m2h , we obtain from Eq. (5.5)
where we have used m Z = 3 TeV, mh = 125 GeV and
vSM = 246 GeV. Combining this equation with Eq. (5.7),
we obtain
From Figs. 5, 7 and 9, we can see that this condition is
satisfied for a large potion of the parameters space.
6 Conclusions The inflationary universe is the standard paradigm in modern cosmology, which not only solves the problems in the Stan(5.6)
mφ = 0.32, 3.0 and 130 GeV (solid lines from bottom to top), along
with the dotted horizontal line for the lower bound of TR = 1 MeV
dard Big Bang Cosmology, but also provide the primordial
density fluctuations necessary for generating the large scale
structure of the present universe. As a simple and
successful inflationary scenario, we have considered the λφ4
inflation with non-minimal gravitational coupling. With a suitable
strength of the non-minimal coupling, the inflationary
predictions of this scenario becomes perfectly consistent with
the Planck 2015 results.
It is more interesting if the inflaton can also play some
crucial role in particle physics. We have considered the general
Higgs model with the gauge and Yukawa interactions with the
spontaneous gauge symmetry breaking. In the presence of the
non-minimal gravitational coupling, the Higgs field can also
play the role of inflaton. The analysis with the Higgs
potential at the tree-level leads to the inflationary predictions
consistent with the cosmological observations. However, once
we take quantum corrections, the effective inflaton potential
most likely becomes unstable. This is because the inflaton
quartic coupling is extremely small in a large portion of the
parameters space and the effective potential is controlled by
the gauge and Yukawa couplings independently of the
quartic coupling. In the renormalization group improved effective
potential, we see that the running quartic coupling becomes
negative in the vicinity of the initial inflaton value, indicating
the instability of the effective potential. In order to avoid this
problem, we have imposed the stability condition of
vanishing the beta function of the inflation quartic coupling. This
condition leads to a non-trivial relation between the gauge
and fermion masses. Since the renormalization group
evolution of the inflaton quartic coupling is mainly controlled by
the gauge and Yukawa coupling, the inflation mass at low
energy is determined by the couplings. Therefore, the mass
spectrum of the gauge boson, fermion and inflation shows a
non-trivial relation.
Since the inflaton potential is modified from the
treelevel one, the inflationary predictions are altered from those
obtained by the tree-level analysis. Although the prediction
of the tensor-to-scalar ratio remains the same under the
condition of the vanishing beta function, the predictions for the
scalar spectral index and the running of the spectral index can
be significantly altered. The fact that the effective potential
is controlled by the gauge and Yukawa couplings implies a
correlation between the inflationary predictions and the
particle mass spectrum. Therefore, the observables at the gauge
symmetry breaking scale correlate with the inflationary
predictions which determined by physics at an extremely high
energy compared to the gauge symmetry breaking scale.
By taking the minimal B − L extension of the Standard
Model as a simple example, we have shown such a
nontrivial relation in the particle mass spectrum driven by the
stability condition of the effective inflaton potential. We also
have calculated the inflationary predictions from the effective
potential and found their dependence of the B − L gauge
coupling. Therefore, the new particle mass spectrum of the B − L
model, once observed, has an implication to the inflationary
predictions. On the other hand, more precise measurements
of the inflationary predictions yield a constraint on the B − L
particle mass spectrum.
For completeness, we have also investigated reheating
after inflation. Since the inflation is lighter than the Z boson
and the right-handed neutrinos, its reheating process through
the heavy particles are not efficient, and the resultant
reheating temperature is too low to be consistent with the bound
from big bang nucleosynthesis. We then introduce a coupling
between the inflaton and the Standard Model Higgs doublet.
Through the mixing with the Standard Model Higgs boson,
the inflaton can decay into the Standard Model particles and
the universe can be heated up with a sufficiently high
reheating temperature. We have found that this happens with a
sufficiently small coupling between the inflaton and the Higgs
doublet and such a small has essentially no effect on our
analysis for the particle mass spectrum and the inflationary
predictions.
Acknowledgements This work is supported in part by the United
States Department of Energy Grant, No. DE-SC 0013680.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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1. A.A. Starobinsky , A new type of isotropic cosmological models without singularity . Phys. Lett. B 91 , 99 ( 1980 )
2. A.H. Guth , The inflationary universe: a possible solution to the horizon and flatness problems . Phys. Rev. D 23 , 347 ( 1981 )
3. A.D. Linde , Chaotic inflation . Phys. Lett. B 129 , 177 ( 1983 )
4. A. Albrecht , P.J. Steinhardt , Cosmology for grand unified theories with radiatively induced symmetry breaking . Phys. Rev. Lett . 48 , 1220 ( 1982 )
5. P.A.R. Ade et al. [Planck Collaboration], Planck 2015 results. XIII. Cosmological parameters . arXiv:1502. 01589 [astro-ph .CO]
6. N. Okada , M.U. Rehman , Q. Shafi , Tensor to scalar ratio in non-minimal φ4 inflation . Phys. Rev. D 82 , 043502 ( 2010 ). arXiv: 1005 .5161 [hep-ph]
7. N. Okada , V.N. Senoguz , Q. Shafi , The observational status of simple inflationary models: an update . arXiv:1403 .6403 [hep-ph] ;
8. T. Inagaki , R. Nakanishi , S.D. Odintsov , Non-minimal two-loop inflation . Phys. Lett. B 745 , 105 ( 2015 ). arXiv: 1502 .06301 [hepph]
9. F.L. Bezrukov , M. Shaposhnikov , The standard model Higgs boson as the inflaton . Phys. Lett. B 659 , 703 ( 2008 ). arXiv:0710.3755 [hep-th]
10. F.L. Bezrukov , A. Magnin , M. Shaposhnikov , Standard model Higgs boson mass from inflation . Phys. Lett. B 675 , 88 ( 2009 ). arXiv: 0812 .4950 [hep-ph]
11. F. Bezrukov , D. Gorbunov , M. Shaposhnikov , On initial conditions for the Hot Big Bang . JCAP 0906 , 029 ( 2009 ). arXiv: 0812 .3622 [hep-ph]
12. F. Bezrukov , M. Shaposhnikov , Standard model Higgs boson mass from inflation: two loop analysis . JHEP 0907 , 089 ( 2009 ). arXiv: 0904 .1537 [hep-ph]
13. F. Bezrukov , A. Magnin , M. Shaposhnikov , S. Sibiryakov , Higgs inflation: consistency and generalisations . JHEP 1101 , 016 ( 2011 ). arXiv: 1008 .5157 [hep-ph]
14. A.O. Barvinsky , A.Y . Kamenshchik , A.A. Starobinsky , Inflation scenario via the Standard model Higgs boson and LHC . JCAP 0811 , 021 ( 2008 ). arXiv: 0809 .2104 [hep-ph]
15. A.O. Barvinsky , A.Y . Kamenshchik , C. Kiefer , A.A. Starobinsky , C. Steinwachs , Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field . JCAP 0912 , 003 ( 2009 ). arXiv: 0904 .1698 [hep-ph]
16. A.O. Barvinsky , A.Y . Kamenshchik , C. Kiefer , A.A. Starobinsky , C. Steinwachs , Higgs boson, renormalization group, and naturalness in cosmology . Eur. Phys. J. C 72 , 2219 ( 2012 ). arXiv: 0910 .1041 [hep-ph]
17. A. De Simone , M.P. Hertzberg , F. Wilczek , Running inflation in the standard model . Phys. Lett. B 678 , 1 ( 2009 ). arXiv: 0812 .4946 [hep-ph]
18. T.E. Clark , B. Liu , S.T. Love , T. ter Veldhuis , The standard model Higgs Boson-inflaton and dark matter . Phys. Rev. D 80 , 075019 ( 2009 ). arXiv: 0906 .5595 [hep-ph]
19. A.V. Bednyakov , B.A. Kniehl , A.F. Pikelner , O.L. Veretin , Stability of the electroweak vacuum: gauge independence and advanced precision . arXiv:1507 .08833 [hep-ph]
20. V. Branchina , E. Messina , Stability, Higgs boson mass and new physics. Phys. Rev. Lett . 111 , 241801 ( 2013 ). arXiv: 1307 .5193 [hep-ph]
21. V. Branchina , E. Messina , Stability and UV completion of the standard model . arXiv:1507 .08812 [hep-ph]
22. V. Branchina , E. Messina , A. Platania , Top mass determination, Higgs inflation, and vacuum stability . JHEP 1409 , 182 ( 2014 ). arXiv: 1407 .4112 [hep-ph]
23. V. Branchina , E. Messina , M. Sher , Lifetime of the electroweak vacuum and sensitivity to Planck scale physics. Phys. Rev. D 91 , 013003 ( 2015 ). arXiv: 1408 .5302 [hep-ph]
24. [ ATLAS and CDF and CMS and D0 Collaborations], First combination of Tevatron and LHC measurements of the top-quark mass . arXiv:1403 .4427 [hep-ex]
25. N. Okada , Q. Shafi , Higgs inflation, seesaw physics and fermion dark matter. Phys. Lett. B 747 , 223 ( 2015 ). arXiv: 1501 .05375 [hepph]
26. S. Di Vita , C. Germani , Electroweak vacuum stability and inflation via non-minimal derivative couplings to gravity . arXiv: 1508 .04777 [hep-ph]
27. M.B. Einhorn , D.R.T. Jones , Inflation with non-minimal gravitational couplings in supergravity . JHEP 1003 , 026 ( 2010 ). arXiv: 0912 .2718 [hep-ph]
28. M.B. Einhorn , D.R.T. Jones , GUT scalar potentials for Higgs inflation . JCAP 1211 , 049 ( 2012 ). arXiv: 1207 .1710 [hep-ph]
29. S. Ferrara , R. Kallosh , A. Linde , A. Marrani , A. Van Proeyen , Jordan frame supergravity and inflation in NMSSM . Phys. Rev. D 82 , 045003 ( 2010 ). arXiv:1004.0712 [hep-th]
30. S. Ferrara , R. Kallosh , A. Linde , A. Marrani , A. Van Proeyen , Superconformal symmetry, NMSSM, and inflation. Phys. Rev. D 83 , 025008 ( 2011 ). arXiv:1008.2942 [hep-th]
31. M. Arai , S. Kawai , N. Okada , Higgs inflation in minimal supersymmetric SU(5) GUT . Phys . Rev . D 84 , 123515 ( 2011 ). arXiv: 1107 .4767 [hep-ph]
32. M. Arai , S. Kawai , N. Okada , Supersymmetric standard model inflation in the Planck era . Phys. Rev. D 86 , 063507 ( 2012 ). arXiv: 1112 .2391 [hep-ph]
33. M. Arai , S. Kawai , N. Okada , Higgs-lepton inflation in the supersymmetric minimal seesaw model . Phys. Rev. D 87 ( 6 ), 065009 ( 2013 ). arXiv: 1212 .6828 [hep-ph]
34. S. Kawai , N. Okada , TeV scale seesaw from supersymmetric Higgs-lepton inflation and BICEP2 . Phys . Lett . B 735 , 186 ( 2014 ). arXiv: 1404 .1450 [hep-ph]
35. C. Pallis , N. Toumbas , Non-minimal Higgs inflation and nonthermal leptogenesis in a supersymmetric Pati-Salam model . JCAP 1112 , 002 ( 2011 ). arXiv: 1108 .1771 [hep-ph]
36. S. Kawai , J. Kim , Testing supersymmetric Higgs inflation with nonGaussianity . Phys. Rev. D 91 ( 4 ), 045021 ( 2015 ). arXiv: 1411 .5188 [hep-ph]
37. N. Okada , M.U. Rehman , Q. Shafi , Non-minimal B-L inflation with observable gravity waves . Phys. Lett. B 701 , 520 ( 2011 ). arXiv: 1102 .4747 [hep-ph]
38. Y. Hamada , H. Kawai , K.Y . Oda , S.C. Park , Higgs inflation is still alive after the results from BICEP2 . Phys. Rev. Lett . 112 (24), 241301 ( 2014 ). doi:10.1103/PhysRevLett.112.241301. arXiv: 1403 .5043 [hep-ph]
39. F. Bezrukov , M. Shaposhnikov , Higgs inflation at the critical point . Phys. Lett. B 734 , 249 ( 2014 ). arXiv: 1403 .6078 [hep-ph]
40. P. Minkowski , Phys. Lett . B 67 , 421 ( 1977 )
41. T. Yanagida, in Proceedings of the Workshop on the Unified Theory and the Baryon Number in the Universe eds . by O. Sawada, A. Sugamoto (KEK, Tsukuba , 1979 ), p. 95
42. M. Gell-Mann , P. Ramond , R. Slansky , Supergravity ed. by P. van Nieuwenhuizen et al. (North Holland, Amsterdam, 1979 ), p. 315
43. S.L. Glashow , The future of elementary particle physics , in Proceedings of the 1979 Cargèse Summer Institute on Quarks and Leptons ed . by M. Lévy et al. (Plenum Press, New York, 1980 ) p. 687
44. R.N. Mohapatra , G. Senjanovic ´, Neutrino mass and spontaneous parity violation . Phys. Rev. Lett . 44 , 912 ( 1980 )
45. J. Schechter , J.W.F. Valle , Neutrino masses in SU(2) x U(1) theories. Phys. Rev. D 22 , 2227 ( 1980 )
46. F. Bezrukov , D. Gorbunov , Light inflaton after LHC8 and WMAP9 results . JHEP 1307 , 140 ( 2013 ). arXiv: 1303 .4395 [hep-ph]
47. D.P. George , S. Mooij , M. Postma , Quantum corrections in Higgs inflation: the real scalar case . JCAP 1402 , 024 ( 2014 ). arXiv:1310.2157 [hep-th]
48. D.P. George , S. Mooij , M. Postma , Top-Goldstone coupling spoils renormalization of Higgs inflation . arXiv:1408 .7079 [hep-th]
49. D.P. George , S. Mooij , M. Postma , Quantum corrections in Higgs inflation: the standard model case . arXiv:1508 .04660 [hep-th]
50. F. Bezrukov , J. Rubio , M. Shaposhnikov , Living beyond the edge: Higgs inflation and vacuum metastability . Phys. Rev. D 92 ( 8 ), 083512 ( 2015 ). doi:10.1103/PhysRevD.92.083512. arXiv: 1412 .3811 [hep-ph]
51. G. Aad et al. [ATLAS Collaboration], Search for high-mass dilepton resonances in pp collisions at √s = 8 TeV with the ATLAS detector . Phys. Rev. D 90 ( 5 ), 052005 ( 2014 ). arXiv:1405.4123 [hep-ex]
52. CMS Collaboration [CMS Collaboration], Search for resonances in the dilepton mass distribution in pp collisions at sqrt(s) = 8 TeV. CMS-PAS-EXO-12-061
53. V. Khachatryan et al. [CMS Collaboration], Search for physics beyond the standard model in dilepton mass spectra in protonproton collisions at √s = 8 TeV . JHEP 1504 , 025 ( 2015 ). arXiv:1412.6302 [hep-ex]
54. F. Bezrukov , D. Gorbunov , M. Shaposhnikov , On initial conditions for the Hot Big Bang . JCAP 0906 , 029 ( 2009 ). doi:10.1088/ 1475 - 7516 /2009/06/029. arXiv: 0812 .3622 [hep-ph] ;
55. J. Garcia-Bellido , D.G. Figueroa , J. Rubio , Preheating in the standard model with the Higgs-inflaton coupled to gravity . Phys. Rev. D 79 , 063531 ( 2009 ). doi:10.1103/PhysRevD.79.063531. arXiv: 0812 .4624 [hep-ph]
56. A. Anisimov , Y. Bartocci , F.L. Bezrukov , Inflaton mass in the nuMSM inflation . Phys. Lett. B 671 , 211 ( 2009 ). arXiv: 0809 .1097 [hep-ph]
57. F. Bezrukov , D. Gorbunov , Light inflaton Hunter's guide . JHEP 1005 , 010 ( 2010 ). arXiv: 0912 .0390 [hep-ph]