Higgs Scalaron Mixed Inflation
Yohei Ema









Department of Physics, Faculty of Science, The University of Tokyo



We discuss the inflationary dynamics of a system with a non-minimal coupling between the Higgs and the Ricci scalar
as well as a Ricci scalar squared term. There are two scalar modes in this system, 
i.e. the Higgs and the spin-zero mode of the graviton, or the scalaron.
We study the two-field dynamics of the Higgs and the scalaron during inflation, 
and clarify the condition where inflation is dominated by the Higgs/scalaron.
We also find that the cut-off scale at around the vacuum 
is as large as the Planck scale,
and hence there is no unitarity issue,
although there is a constraint on the couplings from the perturbativity of the theory
at around the vacuum.




Higgs Scalaron Mixed Inflation

    Yohei Ema
    December 30, 2023
================================

UT 17-04









§ INTRODUCTION




After the observation of the cosmic microwave background (CMB) anisotropy, 
inflation plays a central role in the modern cosmology. 
It is usually assumed that inflation is caused by potential energy of a scalar field, or the inflaton,
but there is no candidate within the standard model (SM). 
Hence we need to go beyond the SM to cause inflation.
Among a variety of such inflation models, 
the Higgs-inflation <cit.> 
and the R^2-inflation <cit.> models are intriguing
because of their minimality as well as consistency with the CMB observation.
In the Higgs-inflation model, the SM Higgs boson plays the role of the inflaton
thanks to a large non-minimal coupling to the Ricci scalar.
In the R^2-inflation model, a spin-zero component of the metric (or the scaralon)
obtains a kinetic term and plays the role of the inflaton once we introduce a Ricci scalar squared term
in the action. It is known that both models predict a similar value of the spectral index
which is in good agreement with the Planck observation <cit.>.
It is also attractive that both models predict the tensor-to-scalar ratio that may be detectable
in the future CMB experiment (CMB-S4) <cit.>.

In the actual analysis of these models, 
it is sometimes assumed that the Higgs or the scalaron is 
the only scalar degree of freedom during inflation.
In reality, however, the Higgs must always be there 
even if we consider the R^2-inflation model.
In addition, if we consider the Higgs-inflation model, 
the large non-minimal coupling of the Higgs to the Ricci scalar 
may radiatively induce a large Ricci scalar squared term <cit.>
that makes the scalaron dynamical as well. Hence it is more realistic 
to consider the dynamics of both the Higgs and the scalaron simultaneously.[

	A similar study with an additional scalar field instead of the scalaron
	with a non-minimal coupling to the Ricci scalar has been 
	performed in literature.
	See, e.g. Refs. <cit.> 
	and references therein.

] In this paper we will thus study the Higgs-scalaron two-field inflationary dynamics,[

	An analysis in this direction is also performed in Ref. <cit.>
	although some aspects we discuss in this paper such as
	the unitarity/perturbativity and the implication of 
	the electroweak vacuum metastability are not addressed there.
	See also Refs. <cit.> 
	as other treatments.

] and derive the parameter dependence of the inflationary predictions in our system.
We will clarify the quantitative condition where inflation is dominated by the Higgs or the scalaron.
In addition, we will address the unitarity structure of our system, 
which is much different from that of the Higgs-inflation.

The organization of this paper is as follows.
In Sec. <ref>, we discuss the inflationary dynamics of the Higgs-scalaron two-field system.
We first study the dynamics analytically, 
and later confirm it by numerical calculation.
In Sec. <ref>, we study the unitarity structure of this system.
We find that the cut-off scale of our system is as large as the Planck scale,
which is similar to the case of the R^2-inflation rather than the Higgs-inflation.
In Sec. <ref>, we concentrate on the dynamics of the Higgs
when the electroweak (EW) vacuum is metastable.
The last section <ref> is devoted to the summary and discussions.







§ INFLATIONARY DYNAMICS



In this section, we study the two-field dynamics of the Higgs and the scalaron during inflation.




 §.§ Action in Jordan/Einstein frame


We start from the following action in the Jordan frame:


    S = ∫ d^4x √(-g_J)   [
    	M_P^2/2(1+ξ_h h^2/M_P^2)R_J . 
       .
    	+ ξ_s/4R_J^2
    	-1/2g^μν_J∂_μ h∂_ν h - λ_h/4h^4
    	],


where g_Jμν is the metric (with the “almost-plus” convention), g_J is the determinant of the metric,
R_J is the Ricci scalar, M_P is the reduced Planck mass
and h is the Higgs in the unitary gauge.
We add the subscript J for the quantities in the Jordan frame.
We consider only the case


    ξ_s, |ξ_h |≫ 1.


In particular, we concentrate on the case ξ_s > 0 since otherwise there is a tachyonic mode.
On the other hand, we do not specify the signs of ξ_h and λ_h.
Concerning the sign of λ_h, 
the current measurement of the top and Higgs masses indicates that 
it becomes negative at a high energy region, resulting in
the metastable EW vacuum <cit.>,
although the stable EW vacuum is also still allowed.
In view of this, we consider both λ_h > 0 and λ_h < 0 in this paper.


By introducing an auxiliary field s, the action (<ref>) 
is rewritten as <cit.>[

This choice of the dual description is unique up to the shift and the rescaling of the auxiliary field s. 
For more details, see App. <ref>.

]

    S = ∫ d^4x √(-g_J)   [
    	M_P^2/2(1+ξ_h h^2 + ξ_s s/M_P^2)R_J . 
       .
    	- ξ_s/4s^2
    	-1/2g^μν_J∂_μ h∂_ν h - λ_h/4h^4
    	].


Note that the variation with respect to s gives


    s = R_J,


and we restore the original action (<ref>) after substituting it to Eq. (<ref>).
The field s corresponds to a spin-zero mode of the graviton that is dynamical due to 
the presence of the Ricci scalar squared term.
We call it a “scalaron” in this paper.

First we perform the Weyl transformation to
obtain the action in the Einstein frame. We define the metric in the Einstein frame as


    g_μν = Ω^2 g_Jμν,   Ω^2 = 1+ξ_h h^2 + ξ_s s/M_P^2.


The Ricci scalar is transformed as


    R_J = Ω^2[R + 3lnΩ^2 
    	- 3/2g^μν∂_μlnΩ^2 ∂_νlnΩ^2],


where R and  are the Ricci scalar and the d'Alembert operator constructed from g_μν, respectively.
The action now reads


    S = ∫ d^4x √(-g)[
    	M_P^2/2R - 3M_P^2/4g^μν∂_μlnΩ^2 ∂_νlnΩ^2 
    	. 
    .
    	-g^μν/2Ω^2∂_μ h∂_ν h - U(h, s)
    	],


where the potential in the Einstein frame is given by


    U(h, s) ≡λ_h h^4 + ξ_s s^2/4Ω^4.


We define a new field ϕ as


    ϕ/M_P   ≡√(3/2)lnΩ^2.


It corresponds to the inflaton degree of freedom in our system.
By eliminating s in terms of ϕ, we finally obtain


    S = ∫ d^4x √(-g)   [
    	M_P^2/2R - 1/2g^μν∂_μϕ∂_νϕ. 
       .
    	-1/2e^-χg^μν∂_μ h∂_ν h
    	- U(ϕ, h)
    	],


where the potential now reads


    U(ϕ, h) = 1/4e^-2χ[
    	λ_h h^4 
    	+ M_P^4/ξ_s(e^χ-1-ξ_h h^2/M_P^2)^2
    	],


and we have defined


    χ≡√(2/3)ϕ/M_P.


This is the master action in our system.
Note that so far we have not used any approximation.
In the following, we study the inflationary dynamics of this action in the Einstein frame.





 §.§ Two-field dynamics


Now we study the inflationary dynamics of the action (<ref>).
An analysis for a similar system is performed in Ref. <cit.>,
and we follow that procedure here.
The action (<ref>) contains the kinetic mixing term 
between ϕ and h, and hence we define the following field τ to solve the mixing:


    τ≡s/h^2.


Note that τ = 0 corresponds to the pure Higgs-inflation,
while τ = ∞ corresponds to the pure R^2-inflation.
The kinetic terms now read


    ℒ_kin 
    	=   
    	-1/2(1+1/6(ξ_h + ξ_s τ)e^χ/e^χ - 1)(∂ϕ)^2 
    	
       - M_P^2/8ξ_s^2(1-e^-χ)/(ξ_h+ ξ_s τ)^3(∂τ)^2
    	+ M_P/2√(6)ξ_s/(ξ_h + ξ_s τ)^2(∂ϕ) (∂τ).


Since we are interested in the inflationary dynamics, we concentrate on the case


    ξ_h h^2 + ξ_s s ≫ M_P^2,
    	  or  
    	e^χ≫ 1,


in this section. 
Then, the kinetic terms are approximated as


    ℒ_kin 
    	=   
    	-1/2(1+1/6(ξ_h + ξ_s τ))(∂ϕ)^2 
    	
       - M_P^2/8ξ_s^2/(ξ_h+ ξ_s τ)^3(∂τ)^2
    	+ M_P/2√(6)ξ_s/(ξ_h + ξ_s τ)^2(∂ϕ) (∂τ).


Note that τ satisfies


    ξ_h + ξ_s τ≫M_P^2/h^2 > 0,


when the condition (<ref>) is satisfied,
and hence the kinetic term of τ has a correct sign.
In the following we assume


    ξ_h + ξ_s τ≫ 1,


which is true for h ≲ M_P.[

	It might not be true for, e.g. the critical case <cit.> 
	where ξ_h ∼𝒪(10) since the Higgs field value is 
	of order M_P during inflation in that case.

] Then after defining the canonically normalized field τ_c as


    dτ_c ≡ξ_sM_P/2(ξ_h + ξ_sτ)^3/2dτ,


the kinetic mixing term between ϕ and τ_c is suppressed by 1/√(ξ_h + ξ_s τ). 
Therefore, to the leading order in it, 
we can approximate the kinetic term as


    ℒ_kin
    	=
    	-1/2(∂ϕ)^2 
    	- M_P^2/8ξ_s^2/(ξ_h τ^2 + ξ_s)^3(∂τ)^2.


There is no kinetic mixing between ϕ and τ to the zero-th order in 
1/√(ξ_h + ξ_s τ).

Armed with these diagonalized fields, we now study the structure of the potential,
which is expressed as


    U(ϕ, τ)    = M_P^4/4λ_h + ξ_sτ^2/(ξ_h + ξ_sτ)^2[1-exp(-√(2/3)ϕ/M_P)]^2.


To the leading order in Eq. (<ref>), 
its derivative gives


    ∂ U/∂τ_c   = -λ_h + ξ_h τ/(ξ_h + ξ_sτ)^3/2M_P^3.


Note that we take the derivative with respect to τ_c, not τ.
Hence the extrema are


    τ = ∞, τ_min,


where we have defined


    τ_min≡λ_h/ξ_h.


In particular, τ = 0 is not an extremum of the potential. 
It means that the pure Higgs-inflation is never realized in our system,
although the pure R^2-inflation is possible.
Nevertheless, there is some parameter region where inflation is caused mostly by the Higgs as we will see below.

In order to be the inflationary trajectory,
these extrema must be minima of the potential.
The second derivative at each extremum is given by


    .∂^2 U/∂τ_c^2|_τ = ∞   = -ξ_h/ξ_sM_P^2, 
    	
    .∂^2 U/∂τ_c^2|_τ = τ_min   = 2ξ_h/ξ_sM_P^2.


Note again that we take the derivative with respect to τ_c, not τ.
Thus, τ = ∞ is the minimum for ξ_hξ_s < 0, 
while τ = τ_min is the minimum for ξ_hξ_s>0.
Also, the potential at the minimum must be positive to cause inflation.
The potential at each extremum is given by


    .U/M_P^4|_τ = ∞   = 1/4ξ_s, 
    .U/M_P^4|_τ = τ_min   = 1/4(ξ_h^2/λ_h + ξ_s).


The former is always positive in the case of our interest,
while the latter together with Eq. (<ref>) gives a non-trivial constraint on the parameters.
In fact, Eq. (<ref>) in particular means


    ξ_h + ξ_s τ_min > 0,


and hence we obtain the condition


    λ_h ξ_h > 0,


by combining the requirement that Eq. (<ref>) is positive.
Thus, the trajectory with τ = τ_min causes inflation only if
ξ_h > 0 and λ_h > 0. Note that we consider only the case ξ_s > 0.[

	It is sufficient to require only these conditions since Eq. (<ref>)
	is trivially satisfied under them.

]
 
In summary, there are three cases that are relevant for inflation: (a) ξ_h > 0, ξ_s > 0 and λ_h > 0,
(b) ξ_h < 0, ξ_s>0 and λ_h>0, and (c) ξ_h<0, ξ_s>0 and λ_h<0.
From now we mainly concentrate on the case (a) since τ = τ_min is the minimum only in this case.
We briefly comment on the case (b) in the end of this subsection. 
The case (c) is also interesting since it corresponds to the metastable EW vacuum. 
Hence we discuss the case (c) in detail in Sec. <ref>.





  §.§.§ Case (a): ξ_h>0,  ξ_s>0  and  λ_h>0







In this case, the potential minimum for τ is given by


    τ = τ_min≠ 0,


and hence the inflaton ϕ is a mixture of the Higgs and the scalaron.
We first assume that τ sits at this minimum during inflation.
Later we will see that this assumption is actually valid.
Recalling that τ = s/h^2, for ξ_h ≫ξ_s τ_min, the inflaton is dominated by the Higgs,
while in the other limit, it is mostly composed of the scalaron. Thus, the situation is as follows:[

	A similar condition is obtained for a scale invariant model with an additional dilaton field
	in Ref. <cit.>. 
	It may be reasonable because the theory is almost scale invariant in our case
	as long as we consider the inflationary dynamics.
	It is also consistent with the rough estimation in Ref. <cit.>.

]


    λ_h ξ_s ≪ξ_h^2:    Higgs-inflation like, 
    
    λ_h ξ_s ≫ξ_h^2:     R^2-inflation like.


It is also seen from the potential. The potential for τ = τ_min is given by


    .U|_τ = τ_min 
    	= M_P^4/41/ξ_h^2/λ_h + ξ_s[1-exp(-√(2/3)ϕ/M_P)]^2,


and hence it is the same as the Higgs-inflation for λ_hξ_s ≪ξ_h^2, 
while the same as the R^2-inflation for λ_hξ_s ≫ξ_h^2.
In the intermediate case, the inflaton is a mixture of the Higgs and the scalaron,
which we call a “Higgs scalaron mixed inflation.”
In order to reproduce the normalization of the CMB anisotropy, 
the parameters should satisfy <cit.>


    ξ_h^2/λ_h + ξ_s ≃ 2× 10^9.


It is well-known that this type of model is in good agreement with the spectral index
observed by the CMB experiments.
In Fig. <ref>, we show the schematic picture of the parameter region in the case (a).

Now we investigate the assumption that τ sits at the minimum of its potential during inflation.
It is valid if the mass squared of τ_c at around the minimum 
∼ξ_h M_P^2/ξ_s (see Eq. (<ref>))
is much larger than the Hubble parameter squared during inflation H_inf^2.
The ratio is estimated as


    ξ_h M_P^2/ξ_s/H_inf^2   ∼ξ_h/ξ_s(ξ_h^2/λ_h + ξ_s) > ξ_h ≫ 1.


Thus, τ sits at the minimum of its potential during inflation as long as ξ_h ≫ 1 is satisfied,
which is the case of our interest, and hence we have verified our assumption.
It means that the inflationary dynamics effectively reduces to a single field case, 
whose potential is given by Eq. (<ref>).





  §.§.§ Case (b): ξ_h<0,  ξ_s>0  and  λ_h>0


In this case, the potential minimum for τ is


    τ = ∞,


and hence inflation is caused solely by the scalaron. 
Actually, the potential is given by


    .U|_τ = ∞ 
    	= M_P^4/4ξ_s[1-exp(-√(2/3)ϕ/M_P)]^2,


which is nothing but the potential of the R^2-inflation model.
It is consistent with the CMB for ξ_s ≃ 2× 10^9 <cit.>.




 §.§ Numerical confirmation


In this subsection, we perform numerical calculation to confirm the analysis in the previous subsection.
In particular, we numerically study the case (a), or the Higgs scalaron mixed inflation case.
While some approximations are used in the previous subsection, 
we emphasize that we use no approximation in this subsection.
More explicitly, we directly solve the background equations of motion 
derived from the action (<ref>), which read


    0    = ϕ̈ + 3Hϕ̇ + e^-χ/√(6)M_Pḣ^2 + ∂ U/∂ϕ, 
    
    	0    = ḧ + (3H - √(2/3)ϕ̇/M_P)ḣ + e^χ∂ U/∂ h, 
    Ḣ   = -1/2M_P^2(ϕ̇^2 + e^-χḣ^2), 
    
    	H^2    = 1/6M_P^2(ϕ̇^2 + e^-χḣ^2 + 2U),


where the potential is


    U(ϕ, h) = 1/4e^-2χ[
    	λ_h h^4 
    	+ M_P^4/ξ_s(e^χ-1-ξ_h h^2/M_P^2)^2
    	].


Here we take the background metric in the Einstein frame to be
the Friedmann-Lemaître-Robertson-Walker (FLRW) one without spatial curvature, 
H is the Hubble parameter and the dots denote the derivatives with respect to the time.
For the convenience of readers, we write down the explicit forms of the derivatives of the potential:


    ∂ U/∂ϕ   = 
    	e^-2χM_P^3/√(6) ξ_s[
    	(1+ξ_h h^2/M_P^2)(e^χ - 1 - ξ_h h^2/M_P^2)
    	- λ_h ξ_s h^4/M_P^4], 
    ∂ U/∂ h   =
    	e^-2χM_P^2 h/ξ_s[
    	-ξ_h(e^χ -1 -ξ_h h^2/M_P^2)
    	+ λ_h ξ_s h^2/M_P^2].


Actually, these equations of motion are redundant, and hence we have used the last one (<ref>)
to check the consistency of our numerical calculation. We have numerically solved the first three equations of motion, 
and checked that Eq. (<ref>) is satisfied at least better than 10^-5 level.









In Fig. <ref>, we show the time evolution of τ. 
The blue line corresponds to our numerical calculation, 
while the red dashed line is τ_min given in Eq. (<ref>).
The top panel corresponds to the Higgs-inflation like case, 
while the bottom one does to the R^2-inflation like case. 
See the caption for more details on the parameters and the initial conditions.
As we can see from the figures, after hundreds or thousands of oscillations 
τ eventually settles down to its potential minimum. 
It happens within a few e-foldings, 
and hence τ has almost no effect on the inflationary dynamics at all.
This result confirms our analysis in the previous subsection.
In Fig. <ref>, we also show the inflationary dynamics in the Higgs-scalaron field space.
The blue line again corresponds to our numerical calculation, 
while the red dashed line is the ϕ direction that is orthogonal to the τ direction.
We can see that the fields oscillate in the τ direction at first, but after τ settles down to its potential minimum,
the system follows the trajectory of ϕ. 
Thus, it is again consistent with our analytical treatment in the previous subsection.
In summary, we have verified that our analysis in the previous subsection well describes
the actual inflationary dynamics of the present system.




§ CUT-OFF SCALE AND PERTURBATIVITY



In this section, we discuss the cut-off scale and the perturbativity 
of our model at around the vacuum.
Note that the discussion in this section holds for all the cases (a), (b) and (c).
We start our discussion with the exact action in the Einstein frame (<ref>).
The minimum of the potential (<ref>) is at (ϕ, h) = (0, 0),
so we expand the action around that point:


    S 
    	   ≃∫ d^4x √(-g)[ M_P^2/2R - 1/2(∂ϕ)^2 
    	- 1/2(1-χ)(∂ h)^2
    	. 
    	
       .
    	-λ_UV/4(1-2χ)h^4
    	-m_ϕ^2/2ϕ^2 +m_ϕ/3√(2ξ_s)ϕ^3
    	.
    	
       .
    	-7/108ξ_s(1-3χ/7)ϕ^4
    	+ξ_h m_ϕ/√(2ξ_s)ϕ h^2
    	-ξ_h/2ξ_s(1-7χ/9)ϕ^2h^2
    	],


where we have defined the Higgs quartic coupling in the original ultraviolet (UV) theory
and the inflaton mass squared as[

	Although λ_UV is shifted due to the mixing between ϕ and h, 
	it does not help to stabilize the EW vacuum for λ_h < 0.
	This is because the CMB observation fixes m_ϕ∼ 10^13 GeV,
	which is much higher than the instability scale of the Higgs potential <cit.>.
	Note that inflation is dominated by the scalaron for λ_h < 0,
	and hence m_ϕ is fixed.

]


    λ_UV   ≡λ_h + ξ_h^2/ξ_s, 
    
    	m_ϕ^2    ≡M_P^2/3ξ_s.


Neglected terms are suppressed by higher powers of M_P.
Thus, recalling that χ = √(2/3) ϕ/M_P, we can see that 
the cut-off scale of our model is as high as the Planck scale.
It is in contrast to the case of the Higgs-inflation,
where a power counting argument[

	Analysis beyond the power counting might change the situation <cit.>.

] suggests that the cut-off scale Λ_cut;h
at around the vacuum is of 
𝒪(M_P/ξ_h) <cit.>.[

	It does not necessarily mean an inconsistency of the Higgs-inflation during inflation <cit.>
	although the unitarity can be broken during the inflaton oscillation epoch <cit.>.
	See, e.g. Refs. <cit.> 
	for construction of a UV completed model and
	also
	Refs. <cit.> for the discussion of UV effects on the Higgs-inflation.

] Rather, our model is similar to the case of the R^2-inflation.
In the R^2-model, the cut-off scale is the Planck scale because the scalaron is just an auxiliary field 
in the Jordan frame, and hence it can absorb the large non-minimal coupling 
with the curvature <cit.>.
Similarly in our case, the scalaron absorbs the large non-minimal couplings ξ_s and ξ_h
so that it can avoid the unitarity issue.
An interesting point of our model is that, Higgs can be the dominant component of the inflaton while keeping
the cut-off to be the Planck scale.

Still, there is a constraint on the parameters if we require the theory to be perturbative.
In order to see this, let us keep only the renormalizable terms in the action at around the vacuum:


    S 
    	≃∫ d^4x √(-g)   [ M_P^2/2R - 1/2(∂ϕ)^2 
    	- 1/2(∂ h)^2
    	. 
    	
       .
    	-λ_UV/4h^4
    	-m_ϕ^2/2ϕ^2 +m_ϕ/3√(2ξ_s)ϕ^3
    	.
    	
       .
    	-7/108ξ_sϕ^4
    	+ξ_h m_ϕ/√(2ξ_s)ϕ h^2
    	-ξ_h/2ξ_sϕ^2h^2
    	].


The Higgs quartic coupling in our system is λ_UV, 
and hence it should be smaller than ∼ 4π for the theory to be perturbative.
Thus, we may require the following condition:[

	We implicitly assume |λ_h |≲𝒪(0.1).

]


    ξ_h^2/ξ_s≲ 4π,


otherwise it is unreasonable to use the tree-level potential such as Eq. (<ref>).
It means that the theory with non-zero ξ_s is qualitatively different from that with ξ_s = 0 even
if we take the limit ξ_s → 0 because of the presence of the scalaron.
Once we have the scalaron degree of freedom, we need Eq. (<ref>)
to keep the theory perturbative at around the vacuum,
while there is no such a constraint if ξ_s = 0 from the beginning.
It might be interesting to see that Eq. (<ref>) requires the inflaton mass as


    m_ϕ^2 ≲M_P^2/ξ_h^2∼Λ_cut;h^2,


where the right-hand-side is the cut-off scale of the Higgs inflation model at around the vacuum.
Since we consider the case |ξ_h|≫ 1, the inflaton-Higgs quartic coupling is always
in the perturbative region once Eq. (<ref>) is satisfied.


We should note that the perturbativity of our system is important 
to safely obtain the SM at the energy scale below m_ϕ.
In order to see this point, we now derive the infrared (IR) theory from our system
(or the UV theory) below the energy scale of m_ϕ.
We may define the IR theory as


    S_IR =
    	∫ d^4x √(-g)   [ M_P^2/2R - 1/2(∂ h)^2
    	-λ_IR/4h^4
    	].


Then, λ_IR may be determined by matching, e.g. the scattering process
hh→ hh in the IR and UV theories.[

	We consider only the degrees of freedom of h and ϕ for simplicity.

] If the UV and IR theories are perturbative, 
we may obtain λ_IR = λ_h
by comparing the tree level processes in the IR and UV theories.
For more details on this point, see App. <ref>.
Thus, the IR theory is nothing but the SM
with the Higgs quartic coupling given by λ_h.
If the UV theory is strongly coupled, however, 
the tree-level matching does not make sense.
We need to sum an infinite numbers of diagrams to calculate the scattering,
which is a complicated task.
Moreover, it is even non-trivial whether the Higgs remains as an asymptotic state,
and the IR theory might be totally different from the one described 
by Eq. (<ref>). 
Therefore it is at least secure to consider only the parameter region ξ_h^2/ξ_s ≲ 4π.















§ METASTABLE ELECTROWEAK VACUUM



In this section, we discuss the two-field dynamics of our system in the case (c): 
ξ_h<0, ξ_s>0 and λ_h<0.
It corresponds to the case of the metastable EW vacuum.
Note that 
the current measurement of the top and Higgs masses actually indicates that the Higgs quartic
coupling becomes negative at a high energy region, 
resulting in the metastable EW vacuum <cit.>.

We now study the dynamics of the Higgs during and after inflation.
During inflation, the potential minimum for τ is


    τ = ∞,


and hence it corresponds to the usual R^2-inflation.
In this case, as long as |ξ_h |≫ 1, 
the Higgs stays at h = 0 during inflation
since τ is heavy enough (see Eq. (<ref>)).
It is nothing but the stabilization mechanism discussed, e.g. in 
Refs. <cit.>.
It is well-known that the EW vacuum metastability has some tension 
with high-scale inflation models including the R^2-inflation if there
is no coupling between the inflaton and the Higgs 
sectors <cit.>.
However, once we introduce couplings between the inflaton and/or the Ricci scalar and the Higgs,
the EW vacuum is stabilized during inflation since they induce an effective mass of the Higgs.
Here we have explicitly shown that such a stabilization mechanism does work
for the R^2-inflation model.

After inflation, however, resonant Higgs production
typically occurs due to the inflaton oscillation, 
and hence the EW vacuum may be destabilized
during the preheating epoch <cit.>.
Here we consider the dynamics of the Higgs after inflation in our system.
Soon after inflation ends, the ϕ^3 and ϕ^4 terms as well as the Planck suppressed terms
become negligible due to the cosmic expansion.
In the same way, the Higgs-inflaton quartic coupling becomes less important
than the Higgs-inflaton trilinear coupling.
Therefore we may approximate the action as


    S 
    	≃∫ d^4x √(-g)   [ M_P^2/2R - 1/2(∂ϕ)^2 
    	- 1/2(∂ h)^2
    	. 
    	
       .
    	-λ_UV/4h^4
    	-m_ϕ^2/2ϕ^2
    	+ξ_h m_ϕ/√(2ξ_s)ϕ h^2
    	].


There are two cases depending on the sign of λ_UV.
If λ_UV is negative, or ξ_h^2 < |λ_h |ξ_s,
this system is studied in Ref. <cit.>.[

	Actually in that study they also include the positive Higgs-inflaton quartic coupling,
	but the trilinear coupling eventually becomes more important than the quartic coupling,
	so we can safely apply their result to our system.

] In this case, the so-called tachyonic preheating occurs since the effective mass squared 
of the Higgs oscillates between positive and negative values <cit.>.
As a result, the EW vacuum is destabilized
during the preheating epoch if the trilinear coupling satisfies <cit.>


    |ξ_h m_ϕ/√(ξ_s)|≳𝒪(10)×m_ϕ^2/M_P.


In order to prevent such a catastrophe, ξ_h must satisfy


    |ξ_h |≲𝒪(10).


Note that the Higgs-inflaton quartic coupling makes things even worse in our case
since it contributes negatively to the effective mass squared of the Higgs for ξ_h < 0.




If λ_UV is positive, or ξ_h^2 > |λ_h |ξ_s,
the situation is more complicated.
The direction ϕ = 0 considered 
in Refs. <cit.>
is absolutely stable in this case. 
Nevertheless, the potential has an unstable direction ϕ = ξ_h h^2/M_P
(or more precisely e^χ - 1 = ξ_h h^2/M_P)
since λ_h is negative. Fluctuations in this direction might be enhanced
since the inflaton inevitably couples to this direction.
Hence it might be possible that the EW vacuum is destabilized even in this case for large enough |ξ_h|,
although we leave a detailed study as a future work.







§ SUMMARY AND DISCUSSIONS




In this paper, we have considered the inflationary dynamics of a system with the non-minimal coupling
between the Higgs and the Ricci scalar ξ_h h^2 R as well as the Ricci scalar squared term ξ_sR^2.
In such a system, there are two scalar degrees of freedom, 
i.e. the Higgs and the scalar part of the metric, or the scalaron.
We have shown that inflation successfully occurs 
in the following three cases: (a) ξ_h > 0, ξ_s > 0 and λ_h > 0,
(b) ξ_h < 0, ξ_s>0 and λ_h>0, and (c) ξ_h<0, ξ_s>0 and λ_h<0,
where λ_h is the Higgs quartic coupling in the Jordan frame.
We have seen that in every case the inflationary dynamics effectively reduces to a single field one
since the direction orthogonal to the inflaton is heavy enough for |ξ_h |≫ 1.
In particular, in the case (a), the inflaton is a mixture of the Higgs and the scalaron, 
which we call a Higgs scalaron mixed inflation. 
The inflaton potential in this case is given by


    U	
    	= M_P^4/41/ξ_h^2/λ_h + ξ_s[1-exp(-√(2/3)ϕ/M_P)]^2,


where ϕ is the inflaton and M_P is the reduced Planck mass,
and hence it is consistent well with the CMB observation as long as 
ξ_h^2/λ_h + ξ_s ≃ 2× 10^9.

We have also addressed the unitarity structure of our system at around the vacuum,
and found that the cut-off scale is as large as the Planck scale.
This is in contrast to the Higgs-inflation where the cut-off scale at around the vacuum
is M_P/ξ_h ≪ M_P.
Rather, it is similar to the R^2-inflation.
Still, the parameters must satisfy ξ_h^2/ξ_s ≲ 4π if we require the perturbativity of our system,
and hence the inflaton mass should be less than ∼ M_P/ξ_h.


Finally we have briefly discussed the implications of the metastable EW vacuum
to the R^2-inflation.
We have explicitly shown that if |ξ_h |≫ 1, 
the EW vacuum is not destabilized during inflation albeit it is metastable.
Still, however, it is possible that the EW vacuum is destabilized during the inflaton oscillation
epoch due to a resonant enhancement of the Higgs quanta.
In order to avoid such a catastrophe, we might require |ξ_h |≲𝒪(10),
although the situation is unclear 
when |ξ_h| is large enough so that ξ_h^2 ≳|λ_h|ξ_s.
It may be interesting to study further on this respect.

We have several remarks.
First of all, although we have mainly concentrated on the inflationary dynamics in this paper,
the (p)reheating dynamics after inflation is also important.
The inflationary predictions of our system depend
on the reheating temperature thorough the number of e-foldings.
Actually, it is the dynamics after inflation that makes 
the differences of the inflationary predictions 
between the Higgs- and R^2-inflation <cit.>.
We leave a detailed study of the reheating dynamics as a future work.


Related to the reheating dynamics, we point out the difference of our system
to the Higgs inflation during the inflaton oscillation epoch.
In general, if there is a large non-minimal coupling between 
the inflaton and the Ricci scalar (without a scalaron) as in the Higgs-inflation,
the inflaton dynamics shows a peculiar behavior called a “spike”-like feature
during the inflaton oscillation 
epoch <cit.>,
which was not taken into account in the previous studies <cit.>.[

	Actually there is a comment on the energy scale ∼√(λ_h) M_P
	related to this spike-like feature in Sec. 3.4 in Ref. <cit.>.
	Nevertheless, possible effects on the preheating dynamics are underestimated there.

] In the Jordan frame, this is because the kinetic term of the inflaton 
(or the Higgs) suddenly changes when the inflaton passes 
the points |ϕ|∼ M_P/ξ_h,
which has some influence even in the Einstein frame.
In particular, if the inflaton is gauge-charged such as the Higgs,
longitudinal gauge bosons with extremely high-momentum ∼√(λ_h) M_P 
are efficiently produced at the first oscillation so that the unitarity may be violated <cit.>.
Here	it is essential to note that the longitudinal gauge boson mass is different from
the transverse gauge boson one if the symmetry breaking field (the Higgs in our case)
is time-dependent <cit.>.
In our case with the scalaron, however, 
such a violent phenomena does not occur thanks to the presence of the scalaron.
Thus, we can trust our system from inflation until the present universe.


It is also valuable to note that some Higgs condensation is unavoidably  produced
in the Higgs scalaron mixed inflation case even if the Higgs is sub-dominant. 
The amplitude of the Higgs condensation at the beginning of the inflaton oscillation
is estimated for the Higgs sub-dominant case as follows.
During inflation, the ratio s/h^2 is fixed to be λ_h/ξ_h, and hence


    exp(√(2/3)ϕ/M_P) - 1 = ξ_h h^2 + ξ_s s/M_P^2≃λ_h ξ_s/ξ_hh^2/M_P^2.


The left-hand-side is of order unity at the beginning of the inflaton oscillation,
and thus the amplitude of the Higgs condensation at that time is estimated as


    h_osc∼√(ξ_h/λ_h ξ_s)M_P ∼ 10^-5√(ξ_h/λ_h)M_P,


where we substitute ξ_s ≃ 2× 10^9 from the CMB observation in the last similarity.
Although the dynamics of the Higgs condensation after inflation is non-trivial
due to the couplings to the inflaton,
it might have some phenomenological consequences such as 
the spontaneous leptogenesis <cit.> and
the gravitational wave <cit.>.
Note that in our case the Higgs has no isocurvature perturbation 
as opposed to the case considered 
in Refs <cit.>
since τ is massive during inflation.





Note Added: While finalizing this paper, another paper <cit.> 
was submitted to the arXiv that calculated the curvature perturbation in the same system.





§ ACKNOWLEDGMENTS


YE thanks Kazunori Nakayama for useful discussions.
YE also acknowledges Oleg Lebedev and the members of the theoretical high energy physics group 
in the University of Helsinki for their hospitality, where some part of this work was done.
This work was supported by the JSPS Research Fellowships for Young Scientists
and the Program for Leading Graduate Schools, MEXT, Japan.







§ REDUNDANCY OF DUAL DESCRIPTION



In this appendix, we comment on redundancy of dual description of Eq. (<ref>). 
Instead of Eq. (<ref>), we may consider the following action:


    S = ∫ d^4x √(-g_J)   [
    	M_P^2/2(1+ξ_h' h^2 + ξ_s' s/M_P^2)R_J . 
       .
    	- ξ_s”/4s^2-λ_sh/2s h^2
    	-1/2g^μν_J∂_μ h∂_ν h - λ_h'/4h^4
    	],


where the parameters ξ_h', ξ_s', ξ_s” and λ_h' are in general different
from ξ_h, ξ_s and λ_h.
As long as they satisfy


    ξ_h' - ξ_s'/ξ_s”λ_sh   = ξ_h,   ξ_s'^2/ξ_s” = ξ_s,   λ_h' - λ_sh^2/ξ_s” = λ_h,


the action (<ref>) reproduces Eq. (<ref>)
after integrating out the scalaron s, and hence they are actually redundant.
This redundancy corresponds to the shift and the rescaling of the scalaron s,
and the physics does not depend on this ambiguity of the dual description.
We can also see this in the Einstein frame.
By following the same procedure we obtained Eq. (<ref>),
we can show that the Einstein frame action obtained from Eq. (<ref>)
is exactly Eq. (<ref>) thanks to Eq. (<ref>).
In the main text, we have chosen the parameters as


    ξ_h' = ξ_h,   ξ_s' = ξ_s” = ξ_s,   λ_sh = 0,   λ_h' = λ_h,


which of course satisfy Eq. (<ref>).




§ MATCHING








In this appendix, we explain how to obtain the relation λ_IR = λ_h.
In order to express the coupling in the IR theory in terms of those in the UV theory,
we may compare the tree-level scattering amplitude of the process hh→ hh.
The corresponding diagrams in the IR and UV theories are shown in Fig. <ref>.
In the IR theory, the tree-level amplitude is given by


    iℳ_IR = -6iλ_IR,


where the numerical factor comes from the permutation of the external Higgs particles.
In the UV theory, it is given by


    iℳ_UV   = -6iλ_UV
    	-2ξ_h^2 m_ϕ^2/ξ_s(i/s-m_ϕ^2+i/t-m_ϕ^2+i/u-m_ϕ^2) 
       ≃ -6i(λ_UV - ξ_h^2/ξ_s) 
    	+ 𝒪(k^2/m_ϕ^2),


where the second term comes from the inflaton-Higgs trilinear coupling with 
s, t and u being the Mandelstam variables.
In the last line, we have taken only
leading order terms in 𝒪(k^2/m_ϕ^2)
with k being the typical momentum of the scattering.
Thus, by matching the scattering amplitude, we obtain


    λ_IR = λ_UV - ξ_h^2/ξ_s = λ_h.


Note that this procedure is reasonable only when the IR and UV theories are perturbative.




apsrev4-1




