The Higgs Portal to Cosmology

Oleg Lebedev

Department of Physics and Helsinki Institute of Physics,
Gustaf Hällströmin katu 2a, FI-00014 Helsinki, Finland

Abstract

The discovery of the Higgs boson has opened a new avenue for exploring physics
beyond the Standard Model. In this review, we discuss cosmological aspects of
the simplest Higgs couplings to the hidden sector, known as the Higgs portal.
We focus on implications of such couplings for inflation, vacuum stability and
dark matter, with the latter including both the traditional weakly interacting
massive particles as well as feebly interacting scalars. The cosmological
impact of the Higgs portal can be important even for tiny values of the
couplings.

###### Contents

  1. 1 Introduction
  2. 2 Generalities
    1. 2.1 Renormalization group evolution
    2. 2.2 Radiative generation of the Higgs–inflaton couplings
    3. 2.3 $Z_{2}$ symmetric scalar potential
      1. 2.3.1 Some phenomenological implications
  3. 3 Higgs portal inflation
    1. 3.1 Large non–minimal couplings to gravity
      1. 3.1.1 Inflationary predictions
    2. 3.2 General non–minimal couplings at $\tau=0,\infty$
    3. 3.3 Unitarity issues
  4. 4 Vacuum stability and inflation
    1. 4.1 Higgs potential in the Standard Model and quantum fluctuations
    2. 4.2 Cosmological challenges
    3. 4.3 Stabilizing the Higgs potential during inflation
  5. 5 Vacuum stability after inflation
    1. 5.1 Parametric resonance
      1. 5.1.1 Particle creation in a parabolic potential
    2. 5.2 Parametric resonance and vacuum stability
      1. 5.2.1 Basics
      2. 5.2.2 Mathieu equation
      3. 5.2.3 Vacuum destabilization
    3. 5.3 Tachyonic resonance
    4. 5.4 Effect of the non–minimal Higgs–gravity coupling
    5. 5.5 Discussion
  6. 6 Stabilizing the Higgs potential via scalar mixing
    1. 6.1 $Z_{2}$ symmetric potential with a large singlet VEV
    2. 6.2 General potential and Higgs–inflaton mixing
      1. 6.2.1 Higgs–inflaton mixing vs cosmology
  7. 7 Higgs and singlet condensates
  8. 8 Postinflationary Higgs and dark matter production
    1. 8.1 Particle production by an oscillating background
      1. 8.1.1 Quartic Higgs–inflaton interaction
      2. 8.1.2 Trilinear Higgs–inflaton interaction
    2. 8.2 Effect of the non–minimal inflaton–gravity coupling
    3. 8.3 Resonant vs non–resonant dark matter production and reheating
      1. 8.3.1 $\phi^{2}$ inflaton potential
      2. 8.3.2 $\phi^{4}$ inflaton potential
    4. 8.4 Matter production via standard inflaton decay
  9. 9 Higgs portal dark matter
    1. 9.1 Scalar dark matter
      1. 9.1.1 Minimal real scalar dark matter
      2. 9.1.2 Pseudo–Goldstone dark matter
    2. 9.2 Vector dark matter
      1. 9.2.1 U(1) gauge symmetry
      2. 9.2.2 SU(2) gauge symmetry
      3. 9.2.3 SU(3) and larger groups
      4. 9.2.4 Phenomenology of vector dark matter
      5. 9.2.5 “Secluded” dark matter: evading direct detection
    3. 9.3 Fermion dark matter
      1. 9.3.1 Dirac fermion dark matter
      2. 9.3.2 Majorana fermion dark matter
  10. 10  Feebly interacting dark scalars
    1. 10.1 Relativistic treatment
      1. 10.1.1 Generalizing the Gelmini–Gondolo formula
    2. 10.2 Decoupled scalar dark matter
      1. 10.2.1 Thermalization constraints
      2. 10.2.2 Freeze–out and relic abundance
    3. 10.3 Higgs portal freeze–in
      1. 10.3.1 Dark scalar production and non–thermalization constraints
      2. 10.3.2 Dark matter abundance
    4. 10.4 Constraints on a decaying Higgs portal scalar: an example
  11. 11 Other cosmological implications of the Higgs portal
  12. 12 Conclusion

##  1 Introduction

The Standard Model (SM) of particle physics provides us with a successful and
precise framework to describe microscopic processes at particle accelerators.
It was made complete with the discovery of the Higgs boson in 2012 [1],[2]
predicted by Brout and Englert [3], Higgs [4], and Guralnik, Hagen and Kibble
[5]. Even though the Standard Model is exceptionally successful, it leaves a
number of important questions unanswered. Among them, some are of cosmological
nature. These include the origin of dark matter (DM), inflation as well as
matter–antimatter asymmetry. Addressing these fundamental questions requires
physics beyond the Standard Model in some form.

The discovery of the Higgs boson has opened an important avenue for exploring
physics beyond the Standard Model. Its current mass stands at [6]

$m_{h_{0}}=125.10\pm 0.14\;{\rm GeV}$ (1)

and its properties appear to match the corresponding SM predictions within the
experimental uncertainties. Nevertheless, this does not preclude the Higgs
boson from having additional interactions with fields beyond the SM. These can
belong to the “hidden sector” in the sense that they have no SM quantum
numbers and thus escape detection. Such fields can, however, have an important
cosmological role: they can drive inflation or constitute dark matter.

In fact, on general grounds, one expects couplings between the Higgs field $H$
and the hidden sector scalars in quantum field theory (QFT). One of the
special features of the Higgs field in the Standard Model is that

${\cal O}_{2}=H^{\dagger}H$ (2)

is the only Lorentz and gauge invariant operator of dimension 2. Therefore,
given a scalar $\phi$, the coupling $H^{\dagger}H\phi^{2}$ or
$H^{\dagger}H\phi^{\dagger}\phi$, if the scalar transforms under some
symmetry, is renormalizable and consistent with all the symmetries. As such,
it must be included in the Lagrangian, although not much can be said of the
coupling strength. In much of the present review, we will consider the
simplest possibility that the scalar $\phi$ is real and has no quantum
numbers. In this case, the allowed renormalizable terms are

$V_{\phi h}=\lambda_{\phi h}H^{\dagger}H\,\phi^{2}+\sigma_{\phi
h}H^{\dagger}H\,\phi\;.$ (3)

$\phi$ can play the role of an inflaton, in which case this interaction has
important implications for inflation, reheating and vacuum stability. If the
system is endowed with a $Z_{2}$ symmetry, $\phi\rightarrow-\phi$, the
trilinear term is forbidden and the scalar becomes a viable dark matter
candidate. Depending on the coupling, it can be a traditional weakly
interacting massive particle (WIMP) or constitute feebly interacting dark
matter. We will also consider the possibility that $\phi$ is a multiplet
transforming under some hidden sector symmetry and plays the role of a
messenger between the observable and dark sectors.

The Higgs coupling to the hidden sector scalar was first considered by
Silveira and Zee in the context of dark matter in Ref. [7], and as an
auxiliary tool in Ref. [8]. Such a coupling can also induce a Higgs–singlet
mixing leading to specific signatures at the Large Hadron Collider (LHC) [9].
Renewed interest in this framework was triggered by Patt and Wilczek’s paper
[10], where the phrase “Higgs portal” was coined. The common definition of the
Higgs portal, also adopted in this review, is the coupling of the Higgs
bilinear ${\cal O}_{2}$ to fields neutral under the SM symmetries, a prime
example of which is given by Eq. 3.

In what follows, we review, at times complicated, physics of simple Higgs
portal couplings, focussing on their cosmological implications.

##  2 Generalities

In this review, we focus on renormalizable interactions between the Higgs
field and the hidden sector. On general grounds, these are expected to be
present in QFT. In addition, we take into account non–minimal scalar couplings
to gravity [11]. Although these are effectively higher dimensional operators,
the couplings are dimensionless and their presence can be motivated by scale
invariance of the theory at large field values [12]. Furthermore, their impact
is important in the Early Universe, when the space–time curvature is
significant [13],[14],[15].

Consider the system of the Higgs field and a real scalar $\phi$. The scalar
may, for example, be an inflaton. The action that includes the most general
renormalizable potential and lowest order non–minimal couplings to gravity has
the form [16]

${\cal L}_{J}=\sqrt{-\hat{g}}\left(-{1\over 2}\Omega\hat{R}\,+{1\over
2}\,\partial_{\mu}\phi\partial^{\mu}\phi+\,(D_{\mu}H)^{\dagger}D^{\mu}H-{V(\phi,H)}\right)\;.$
(4)

For many purposes, it is convenient to use the unitary gauge

$H(x)={1\over\sqrt{2}}\left(\begin{matrix}0\\\ h(x)\end{matrix}\right)\;,$ (5)

in which the potential takes the form

$V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over 2}\sigma_{\phi h}\phi
h^{2}+{1\over 3}b_{3}\phi^{3}+{1\over 2}m_{h}^{2}h^{2}+{1\over
2}m_{\phi}^{2}\phi^{2}+b_{1}\phi\;.$ (6)

We assume that all the dimensionful parameters are far below the Planck scale.
The function $\Omega$ has the general form
$\Lambda^{2}+\Lambda^{\prime}\phi+\xi_{\phi}\phi^{2}+\xi_{h}h^{2}$. The linear
term can be eliminated by field redefinition. Then, the constant term can be
identified with the Planck mass squared, as long as the vacuum average of
$\xi_{\phi}\phi^{2}+\xi_{h}h^{2}$ can be neglected. Therefore, we may take

$\Omega=M_{\rm Pl}^{2}+\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\;.$ (7)

In the presence of odd in $\phi$ terms in the potential, the $\phi\hat{R}$
coupling is still generated at loop level. However, if the above form is
enforced at the high scale, say the Planck scale, the resulting coupling at
the inflationary scale is suppressed by a loop factor as well as by the
dimensionful couplings $\sigma_{\phi h}$, $b_{i}$ which we assume to be far
below the Planck scale. On the other hand, at late stages of the Universe
evolution, the impact of the non–minimal couplings to gravity is negligible so
they can be dropped altogether. Thus, it is sufficient for our purposes to
adhere to $\Omega$ of the form (7).111Some effects of the linear in $\phi$
coupling were explored in [17].

It is important to remember that the unitary gauge is singular at $\langle
h\rangle=0$, and all 4 Higgs degrees of freedom have to be considered in this
case. Equivalently, at energies far above $\langle h\rangle$, effects of the
longitudinal components of $W$ and $Z$ must be taken into account. In loop
computations, it is necessary to include the Goldstone contributions, which
are made obscure by the unitary gauge.

One rather general consequence of the potential (6) is that a Higgs–inflaton
mixing is expected [16]. This is due to the presence of the trilinear $\phi
h^{2}$ term coupled with the fact that the Higgs develops a non–zero VEV. The
$\phi h^{2}$ interaction itself is also quite generic: if the inflaton decays
into SM fields, as assumed in most models of reheating, this coupling is
generated radiatively [18]. The effects of the Higgs–inflaton mixing may be
too small to be observed depending on the size of the model–dependent mixing
angle. Having said that, even a small mixing can have a major impact on vacuum
stability by increasing the Higgs self–coupling.

In what follows, we will discuss the most important properties of the system.

###  2.1 Renormalization group evolution

The couplings that appear in the above Lagrangian enter observables which are
associated with vastly different energy scales. It is therefore necessary to
take the scale dependence of the couplings into account. For our purposes, it
suffices to use the leading log corrections encoded in the 1–loop
renormalization group (RG) equations.

The 1–loop renormalization group running of the relevant couplings in the
potential and the SM couplings is given by [16]222Note a different convention
for $\lambda_{\phi h}$ and $\sigma_{\phi h}$ in [16].

$\displaystyle 16\pi^{2}{\dfrac{d\lambda_{h}}{dt}}$
$\displaystyle=24\lambda_{h}^{2}-6y_{t}^{4}+\dfrac{3}{8}\left(2g^{4}+(g^{2}+g^{\prime
2})^{2}\right)+(12y_{t}^{2}-9g^{2}-3g^{\prime 2})\lambda_{h}+{1\over
2}\lambda_{\phi h}^{2}\;,$ $\displaystyle 16\pi^{2}\dfrac{d\lambda_{\phi
h}}{dt}$ $\displaystyle=4\lambda_{\phi h}^{2}+12\lambda_{h}\lambda_{\phi
h}-\dfrac{3}{2}(3g^{2}+g^{\prime 2})\lambda_{\phi h}+6y_{t}^{2}\lambda_{\phi
h}+6\lambda_{\phi}\lambda_{\phi h}\;,$ $\displaystyle
16\pi^{2}\dfrac{d\lambda_{\phi}}{dt}$ $\displaystyle=2\lambda_{\phi
h}^{2}+18\lambda_{\phi}^{2}\;,$ $\displaystyle 16\pi^{2}\dfrac{d\sigma_{\phi
h}}{dt}$ $\displaystyle=\sigma_{\phi h}\left({12\lambda_{h}+4\lambda_{\phi
h}-\frac{3g^{\prime 2}}{2}-\frac{9g^{2}}{2}+6y_{t}^{2}}\right)+2\lambda_{\phi
h}b_{3}\;,$ (8) $\displaystyle 16\pi^{2}\dfrac{db_{3}}{dt}$
$\displaystyle=6\sigma_{\phi h}\lambda_{\phi h}+18\lambda_{\phi}b_{3}\;,$
$\displaystyle 16\pi^{2}\dfrac{dy_{t}}{dt}$
$\displaystyle=y_{t}\left(\dfrac{9}{2}y_{t}^{2}-\dfrac{17}{12}g^{\prime
2}-\dfrac{9}{4}g^{2}-8g_{3}^{2}\right)\;,$ $\displaystyle
16\pi^{2}\dfrac{dg_{i}}{dt}$
$\displaystyle=c_{i}\,g_{i}^{3}\quad\textrm{with}\quad(c_{1},c_{2},c_{3})=(41/6,-19/6,-7)\;,$

where $t=\ln\mu$ with $\mu$ being the RG energy scale,
$g_{i}=(g^{\prime},g,g_{3})$ denote the gauge couplings and $y_{t}$ is the top
quark Yukawa coupling. The standard input values of the couplings at the top
quark mass scale $M_{t}$ are
$g(M_{t})=0.64,\;g^{\prime}(M_{t})=0.35,\;g_{3}(M_{t})=1.16$ and
$y_{t}(M_{t})=0.93$, although $y_{t}$ is subject to tangible uncertainties. As
emphasised above, the RG running includes the Goldstone contributions as well.

The non–minimal couplings to gravity also receive significant leading log
corrections. Such couplings are generated even if their tree level values are
zero, while in the conformal limit $\xi_{i}=-1/6$ they remain scale–invariant.
The 1–loop RG running is given by [19]

$\displaystyle 16\pi^{2}{d\xi_{h}\over dt}=\left(\xi_{h}+{1\over
6}\right)\Bigl{(}12\lambda_{h}+6y_{t}^{2}-{3\over 2}(3g^{2}+g^{\prime
2})\Bigr{)}+\left(\xi_{\phi}+{1\over 6}\right)\lambda_{\phi h}\;,$
$\displaystyle 16\pi^{2}{d\xi_{\phi}\over dt}=\left(\xi_{\phi}+{1\over
6}\right)6\lambda_{\phi}+\left(\xi_{h}+{1\over 6}\right)4\lambda_{\phi h}\;.$
(9)

These corrections do not include gravitons in the loop, so their computation
parallels that for the scalar couplings. For example, one notices the same
combination of the SM couplings that contributes to the beta functions of
$\lambda_{\phi h}$ and $\xi_{h}$.

The above results are valid at low scalar background values,
$\xi_{\phi}\phi^{2}+\xi_{h}h^{2}\ll M_{\rm Pl}^{2}$. If this is not the case,
the scalar propagators get modified by the curvature term and the RG equations
receive corrections. However, connecting the small and large field regimes can
be challenging due to the unitarity issues to be discussed in Section 3.3,
hence we focus in this section on the former.

The effect of the running can be crucial. A widely appreciated example is
provided by the Higgs self–interaction: the RG running can turn the coupling
negative indicating vacuum metastability. Analogous corrections to other
couplings can also play an important role. For example, the Higgs portal
coupling $\lambda_{\phi h}$ generates the inflaton self–coupling
$\lambda_{\phi}$ at one loop, so the latter cannot be too small, at least in a
substantial interval of energy scales.

On the other hand, for many applications in this review, what is relevant is
the coupling value at a particular scale. For instance, the non–minimal
couplings to gravity are important during inflation or preheating, but not at
low energies. Unless specified otherwise, in what follows, the couplings are
to be understood as the running couplings evaluated at the scale of the
considered process.

###  2.2 Radiative generation of the Higgs–inflaton couplings

The Higgs–inflaton interactions are expected on general grounds in QFT.
Specifically, Lorentz and gauge invariant interactions of dimension 4 or lower
must be included in a renormalizable theory, and the Higgs–portal couplings
belong to this class (unless additional symmetries are imposed on the inflaton
sector). In cosmology, there is a further argument in favor of these couplings
[18]. The inflaton energy must be transferred to the SM sector at the end of
inflation due to the process known as reheating. In most models of reheating,
the transfer is accomplished at least partly via inflaton decay into SM
matter. This implies a linear inflaton coupling to some SM fields or other
fields that couple to the SM. Loop effects then induce a coupling between the
Higgs and the inflaton. Typically, such loop corrections are divergent and
require the corresponding counterterms. This means that the Higgs–inflaton
couplings are a priori arbitrary parameters and cannot be ignored.

Figure 1:  Higgs–inflaton couplings and inflaton self–interaction generated
via the right–handed neutrinos [18]. (Other diagrams with the same topology
are not shown).

To illustrate this point, let us set the tree level Higgs–inflaton couplings
to zero and consider a set–up in which reheating occurs due to the inflaton
coupling to heavy right–handed Majorana neutrinos $\nu_{R}$. The neutrinos
produced in inflaton decay subsequently decay into leptons and Higgses,
leading to thermalization of the SM fields. The relevant neutrino interactions
read

$-\Delta{\cal L}_{\nu}={\lambda_{\nu}\over
2}~{}\phi\;\nu_{R}\nu_{R}+{y_{\nu}}~{}\bar{l}_{L}\\!\cdot\\!H^{*}\,\nu_{R}+{M\over
2}~{}\nu_{R}\nu_{R}+{\rm h.c.}~{},$ (10)

where $l_{L}$ is the lepton doublet and $M$ is taken to be real for
simplicity. At 1 loop, this Lagrangian generates various scalar couplings,
including Higgs–inflaton interactions (Fig. 1). Such corrections are divergent
and require a renormalization condition. Setting the couplings to zero at the
Planck scale, at the inflationary scale $H$ we have [18]:

$\displaystyle\lambda_{\phi h}$ $\displaystyle\simeq$
$\displaystyle{|\lambda_{\nu}y_{\nu}|^{2}\over 2\pi^{2}}\ln{M_{\rm Pl}\over
H}\;,$ $\displaystyle\sigma_{\phi h}$ $\displaystyle\simeq$
$\displaystyle-{M|y_{\nu}|^{2}{\rm Re}\lambda_{\nu}\over 2\pi^{2}}\ln{M_{\rm
Pl}\over H}\;,$ (11)

where we keep only the leading log terms. An analogous correction to the
inflaton self–coupling is also generated,

$\Delta\lambda_{\phi}\simeq{|\lambda_{\nu}|^{4}\over 4\pi^{2}}\ln{M_{\rm
Pl}\over H}\;.$ (12)

The latter is important since it sets an upper bound on $\lambda_{\nu}$,
depending on the model of inflation. The radiative contribution to the
inflaton potential should not spoil its flatness. For example, for chaotic
$\phi^{2}$–inflation, one requires
$m_{\phi}^{2}\phi^{2}/2\gg\Delta\lambda_{\phi}\phi^{4}/4$ during the last 60
e–folds. This results in $\lambda_{\nu}<10^{-3}$. Assuming $M<10^{13}$ GeV and
$y_{\nu}<0.6$ as required by the bound on the neutrino masses in seesaw
models, one finds $\lambda_{\phi h}\lesssim 10^{-7}$ and $\sigma_{\phi
h}\lesssim 10^{-4}M$. These results are model–dependent and in other models
such as Higgs–like inflation the couplings are allowed to be much larger. The
radiative corrections control the $natural$ size of the couplings so that
their smaller values would require cancellations and thus finetuning.

Similar conclusions apply to other popular reheating mechanisms. For instance,
inflaton decay may proceed through non–renormalizable interactions

$O_{1}={1\over\Lambda_{1}}\phi\;\bar{q}_{L}\\!\cdot\\!H^{*}\,t_{R}~{}~{},~{}~{}O_{2}={1\over\Lambda_{2}}\phi\;G_{\mu\nu}G^{\mu\nu}~{},$
(13)

where $\Lambda_{1,2}$ are some scales, $G_{\mu\nu}$ is the gluon field
strength and $q_{L},t_{R}$ are the third generation quarks. Closing the quarks
or gluons in the loops, one finds Higgs–inflaton couplings of the size similar
to that obtained above and analogous conclusions apply [18].

Even small Higgs–inflaton couplings can have a significant impact on the Early
Universe physics. In particular, these terms affect stability of the Higgs
potential when the inflaton field takes on large values. Furthermore,
non–perturbative Higgs production during preheating is efficient even for tiny
couplings, e.g. $\lambda_{\phi h}<10^{-8}$. It is therefore necessary to
account for the Higgs–inflaton interactions in a realistic setting.

###  2.3 $Z_{2}$ symmetric scalar potential

The vacuum structure in the most general Higgs–inflaton system is rather
complicated [20]. A useful simpler limit to consider is the $Z_{2}$ symmetric
scalar potential, that is, the potential invariant under
$\phi\rightarrow-\phi$. Although this eliminates some of the couplings which
can be important for reheating, it provides a helpful perspective on the
Higgs–inflaton mixing and applies more generally to a singlet extended
Standard Model. A special case of this potential with no dimensionful
parameters is relevant to the conformal extension of the Standard Model
[21],[22].

Consider the $Z_{2}$ symmetric potential

$V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over
2}m_{h}^{2}h^{2}+{1\over 2}m_{\phi}^{2}\phi^{2}\;.$ (14)

Denoting the vacuum expectation values

$\langle h\rangle\equiv v~{}~{},~{}~{}\langle\phi\rangle\equiv w,$ (15)

one finds the following classification of the minima [39]:

$\displaystyle v\not=0,w\not=0~{}:~{}\lambda_{\phi
h}m_{\phi}^{2}-2\lambda_{\phi}m_{h}^{2}>0~{},~{}\lambda_{\phi
h}m_{h}^{2}-2\lambda_{h}m_{\phi}^{2}>0~{},$ $\displaystyle
v\not=0,w=0~{}:~{}\lambda_{\phi
h}m_{\phi}^{2}-2\lambda_{\phi}m_{h}^{2}<0~{},~{}m_{h}^{2}<0~{},$
$\displaystyle v=0,w\not=0~{}:~{}\lambda_{\phi
h}m_{\phi}^{2}-2\lambda_{\phi}m_{h}^{2}<0~{},~{}m_{\phi}^{2}<0~{},$
$\displaystyle v=0,w=0~{}:~{}m_{h}^{2}>0~{},~{}m_{\phi}^{2}>0~{}.$ (16)

These conditions are mutually exclusive such that there is only one local
minimum at tree level (barring the reflected minimum
$\phi\rightarrow-\phi$)333The couplings are assumed to satisfy the
“positivity” constraints (19).. If $w=0$, there is no Higgs–inflaton mixing
and the system is straightforward to analyze. Let us consider the case
$v,w\not=0$ in detail. The stationary point condition requires

$\displaystyle v^{2}=\frac{2\lambda_{\phi
h}m_{\phi}^{2}-4\lambda_{\phi}m_{h}^{2}}{4\lambda_{h}\lambda_{\phi}-\lambda_{\phi
h}^{2}}\,,\qquad w^{2}=\frac{2\lambda_{\phi
h}m_{h}^{2}-4\lambda_{h}m_{\phi}^{2}}{4\lambda_{h}\lambda_{\phi}-\lambda_{\phi
h}^{2}}\,.$ (17)

The corresponding mass matrix is

${\cal M}^{2}=\left(\begin{array}[]{cc}2\lambda_{h}v^{2}&\lambda_{\phi h}vw\\\
\lambda_{\phi h}vw&2\lambda_{\phi}w^{2}\end{array}\right)\,.$ (18)

Its eigenvalues are positive if

$\lambda_{h},\lambda_{\phi}>0~{}~{},~{}~{}4\lambda_{h}\lambda_{\phi}-\lambda_{\phi
h}^{2}>0\;.$ (19)

The mass matrix can be diagonalised by the orthogonal transformation
$O^{T}{\cal M}^{2}O=\textrm{diag}(m_{1}^{2},m_{2}^{2}),$ where

$O=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{array}\right)\;$ (20)

and the angle $\theta$ is given by

$\tan 2\theta=\frac{\lambda_{\phi
h}vw}{\lambda_{\phi}w^{2}-\lambda_{h}v^{2}}\,.$ (21)

The mass squared eigenvalues are

$m_{{1,2}}^{2}=\lambda_{h}v^{2}+\lambda_{\phi}w^{2}\mp\frac{\lambda_{\phi}w^{2}-\lambda_{h}v^{2}}{\cos
2\theta}\,.$ (22)

In our convention, $\textrm{sign}(m_{2}^{2}-m_{1}^{2})=\textrm{sign}(\cos
2\theta)\,\textrm{sign}(\lambda_{\phi}w^{2}-\lambda_{h}v^{2})$. The mass
eigenstates $h_{1},h_{2}$ are given by

$\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right)=\left(\begin{array}[]{c}\left(h-v\right)\cos\theta-\left(\phi-w\right)\sin\theta\\\
\left(\phi-w\right)\cos\theta+\left(h-v\right)\sin\theta\end{array}\right)\,.$
(23)

Finally, it is important to note that the leading log corrections can be
included by using the running couplings evaluated at the relevant scale. These
are particularly significant for the conditions imposed at large field values,
when the logarithms are large. Such corrections are relevant to the asymptotic
behaviour of the potential, especially in regard to vacuum stability. Clearly,
the quartic couplings must be positive (semidefinite) for the potential to be
bounded from below, while the constraint on the Higgs portal coupling depends
on the sign of $\lambda_{\phi h}$. Neglecting the quadratic terms, one finds

$V\simeq{1\over
4}\Bigl{[}\left(\sqrt{\lambda_{h}}h^{2}-\sqrt{\lambda_{\phi}}\phi^{2}\right)^{2}+\left(\lambda_{\phi
h}+2\sqrt{\lambda_{h}\lambda_{\phi}}\right)h^{2}\phi^{2}\Bigr{]}\;.$ (24)

Thus, for $\lambda_{\phi h}>0$ there is no extra constraint, whereas at
negative $\lambda_{\phi h}$ there exists a run–away direction unless
$|\lambda_{\phi h}|<2\sqrt{\lambda_{h}\lambda_{\phi}}$. As a result, one finds
the following constraints on the running couplings

$\lambda_{h}(\mu),\lambda_{\phi}(\mu)>0~{}~{},~{}~{}4\lambda_{h}(\mu)\lambda_{\phi}(\mu)+{\rm
sign}\left[\lambda_{\phi h}(\mu)\right]\;\lambda_{\phi h}(\mu)^{2}>0\;,$ (25)

where $\mu$ is some large energy scale such as the Planck scale. To avoid deep
minima at other scales, one imposes this condition for all $\mu\gg v,w$. Note,
however, that the above constraint may be too restrictive: our vacuum can be
metastable and one can still have a well defined theory at small field values.
In particular, $\lambda_{h}({\rm M_{Pl}})<0$ does not pose any fundamental
problems although certain cosmological issues do arise.

####  2.3.1 Some phenomenological implications

For phenomenological applications, it is often convenient to analyze the model
in terms of the set $(m_{1}^{2},m_{2}^{2},\sin\theta,v,\lambda_{\phi h})$
instead of the input parameters in the scalar potential [23]. The
self–couplings can then be found via

$\displaystyle\lambda_{h}$ $\displaystyle=$
$\displaystyle\frac{m_{1}^{2}}{2v^{2}}+\sin^{2}\theta\
\frac{m_{2}^{2}-m_{1}^{2}}{2v^{2}}\;,$ $\displaystyle\lambda_{\phi}$
$\displaystyle=$ $\displaystyle\frac{2\,\lambda_{\phi
h}^{2}}{\sin^{2}2\theta}\frac{v^{2}}{m_{2}^{2}-m_{1}^{2}}\left(\frac{m_{2}^{2}}{m_{2}^{2}-m_{1}^{2}}-\sin^{2}\theta\right)\;.$
(26)

These are useful for analyzing perturbativity and stability of the model. The
most important couplings of the mass eigenstates to the SM matter are

${\cal L}\supset{h_{1}\cos\theta+h_{2}\sin\theta\over
v}\left[2m_{W}^{2}W_{\mu}^{+}W^{\mu-}+m_{Z}^{2}Z_{\mu}Z^{\mu}-\sum_{f}m_{f}\bar{f}f\right]\,.$
(27)

Compared to the SM Higgs boson partial decay widths, these for the $h_{1}$ and
$h_{2}$ scalars are universally suppressed by $\cos^{2}\theta$ and
$\sin^{2}\theta$, respectively. The trilinear couplings between $h_{1}$ and
$h_{2}$ are also important since they can lead to the decays $h_{2}\to
h_{1}h_{1}$ and $h_{1}\to h_{2}h_{2}$, if kinematically allowed. We define
them as

${\cal L}\supset-{\kappa_{112}\over
2}v\sin\theta~{}h_{1}^{2}h_{2}-{\kappa_{221}\over
2}v\cos\theta~{}h_{2}^{2}h_{1}~{},$ (28)

with

$\displaystyle\kappa_{112}$ $\displaystyle={2m_{1}^{2}+m_{2}^{2}\over
v^{2}}\left(\cos^{2}\theta+{\lambda_{\phi h}v^{2}\over
m_{2}^{2}-m_{1}^{2}}\right)~{},$ (29a) $\displaystyle\kappa_{221}$
$\displaystyle={2m_{2}^{2}+m_{1}^{2}\over
v^{2}}\left(\sin^{2}\theta+{\lambda_{\phi h}v^{2}\over
m_{1}^{2}-m_{2}^{2}}\right)\,.$ (29b)

In the kinematically allowed regime, the decay widths are given by

$\displaystyle\Gamma(h_{2}\to h_{1}h_{1})$
$\displaystyle={\sin^{2}\theta~{}\kappa_{112}^{2}v^{2}\over
32\pi\,m_{2}}\sqrt{1-{4m_{1}^{2}\over m_{2}^{2}}}~{},$ (30a)
$\displaystyle\Gamma(h_{1}\to h_{2}h_{2})$
$\displaystyle={\cos^{2}\theta~{}\kappa_{122}^{2}v^{2}\over
32\pi\,m_{1}}\sqrt{1-{4m_{2}^{2}\over m_{1}^{2}}}\,.$ (30b)

The model is subject to a range of constraints from various particle
experiments. Both $h_{1}$ and $h_{2}$ behave as Higgs–like particles, and
$h_{1}$ is identified with the scalar observed at the LHC, given that
$\cos\theta$ is close to one. This, together with the correct electroweak
symmetry breaking requires

$m_{1}=125\;{\rm GeV}~{}~{},~{}~{}v=246\;{\rm GeV}\;,$ (31)

which leaves only three free parameters in the model: $m_{2},\sin\theta$ and
$\lambda_{\phi h}$. A comprehensive analysis of the relevant constraints is
presented in [23] and here we only outline the most important bounds.

Combined measurements of the Higgs couplings [24], encapsulated in the “signal
strength” $\mu>0.92$ (ATLAS) [25] and $\mu>0.90$ (CMS) [26], impose an upper
bound on the mixing angle:

$|\sin\theta|\lesssim 0.3$ (32)

at the 2$\sigma$ level. This bound is independent of $m_{2}$, while if $h_{2}$
is light, it is superseded by other constraints. The LEP Higgs searches
require $|\sin\theta|\lesssim 0.1-0.2$ for $m_{2}<90$ GeV, whereas B–physics
imposes a strong bound $|\sin\theta|\lesssim 10^{-2}-10^{-3}$ for $m_{2}<5$
GeV. If $h_{2}$ is very heavy, the constraints are also strong: the
electroweak precision measurements set an upper bound on $|\sin\theta|$ that
scales as

$|\sin\theta|\lesssim 0.3/\sqrt{1+\ln(m_{2}/{\rm TeV})}\;,$ (33)

with $m_{2}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}1$ TeV. This
behaviour may appear counterintuitive: the constraint gets stronger for larger
$m_{2}$. It is nevertheless easily understandable since the Higgs contribution
to the gauge boson propagators gets reduced by $\cos^{2}\theta$, so $h_{2}$
should be light enough to make up the deficit. The scaling then follows from
the corresponding bound on the $S,T$ Peskin–Takeuchi variables [27].444For
light and moderate $m_{2}$, the $S,T$ variable approximation is inadequate and
one should use the full corrections as in [23]. This reference performs a
global electroweak fit to 15 observables including the $Z$–pole and $W$–data.
We conclude that, unless $h_{2}$ is too heavy or too light, a significant
Higgs–inflaton mixing up to $|\sin\theta|\simeq 0.3$ is allowed by experiment
[23].

Some parameter space of the model can further be probed at the LHC [28]. The
general predictions are (i) a universal reduction of the SM couplings of
$h_{1}$ compared to those of the Higgs, (ii) the existence of the Higgs–like
resonance $h_{2}$, (iii) possible resonant di–Higgs production
$h_{2}\rightarrow h_{1}h_{1}$ or $h_{1}\rightarrow h_{2}h_{2}$, if
kinematically allowed. In particular, for $500\,{\rm GeV}>m_{2}>2m_{1}$, the
di–Higgs production rate can exceed the SM prediction by an order of magnitude
[29],[23]. The Higgs couplings are expected to be measured within about 5% at
HL-LHC [30], which would tighten the bound on $\sin\theta$ to about 0.2.
Searches for the “heavy Higgs” $h_{2}$ are likely to cover the mass range up
to about 1 TeV, unless the mixing angle is very small.

A recent analysis of the singlet scalar model without the $Z_{2}$ symmetry can
be found in [31]. Many of the above conclusions apply to this more general
set–up as well.

##  3 Higgs portal inflation

Inflation is one of the cornerstones of modern cosmology [32],[33],[34]. It
provides us with a compelling explanation why the Universe is so big and flat,
why causally disconnected regions happen to have the same temperature and also
seeds fluctuations for structure formation [35] and the Cosmic Microwave
Background (CMB) (see [36],[37] for reviews). To realize inflation one
normally needs a scalar field with a flat enough potential,

${V^{\prime}\over V}\,M_{\rm Pl}\ll 1~{}~{},~{}~{}{V^{\prime\prime}\over
V}\,M_{\rm Pl}^{2}\ll 1~{}~{},$ (34)

where the prime denotes a derivative with respect to the field. The Universe
dominated by the potential energy of this field expands exponentially with an
almost constant Hubble rate $H=\dot{a}/a$, where $a$ is the scale factor,
while the associated quantum fluctuations eventually lead to the observed CMB
spectrum. The current CMB data disfavor simple polynomial potentials and tend
to prefer concave ones, in the relevant field range [38].

The Higgs portal framework provides an excellent setting for inflationary
model building. Even though the polynomial terms in the scalar potential
cannot fit the data themselves, the presence of the non–minimal couplings to
gravity changes the situation [14]. As in the case of well known Higgs
inflation [12], the potential becomes exponentially close to a flat one at
large field values, which is favored by the inflationary data. In what
follows, we apply this idea to the Higgs–singlet system where the role of the
inflaton can be played by a combination of the Higgs and the singlet [39], in
addition to the Higgs or the singlet itself [19],[40]. For simplicity, we
consider the $Z_{2}$ symmetric version of the scalar potential, although the
main results apply more generally (see Section 6.2). A special case of this
system has been studied in the context of the $\nu$MSM [41],[42].

In the unitary gauge

$H(x)={1\over\sqrt{2}}\left(\begin{matrix}0\\\ h(x)\end{matrix}\right)\;,$
(35)

the Lagrangian in the Jordan frame reads

${\cal L}_{J}=\sqrt{-\hat{g}}\left(-{1\over 2}\hat{R}\,(M_{\rm
Pl}^{2}+\xi_{h}h^{2}+\xi_{\phi}\phi^{2})+{1\over
2}\,\partial_{\mu}\phi\partial^{\mu}\phi+{1\over
2}\,\partial_{\mu}h\partial^{\mu}h-{V(\phi,h)}\right)\;,$ (36)

where we assume for simplicity a $Z_{2}$–symmetric scalar potential

$V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over
2}m_{h}^{2}h^{2}+{1\over 2}m_{\phi}^{2}\phi^{2}\;.$ (37)

Here $\hat{R}$ is the scalar curvature based on the Jordan frame metric
$\hat{g}_{\mu\nu}$. To avoid a singularity at large field values, we take
$\xi_{h},\xi_{\phi}>0$. Although there are indications that $\lambda_{h}$ in
the pure SM is negative at high energies, here we assume that both
$\lambda_{h}$ and $\lambda_{\phi}$ are positive at the inflationary scale.
This may be due to the RG effects associated with the singlet, Higgs–singlet
mixing (Section 6) or simply a somewhat lower top quark mass. The precise
mechanism of vacuum stabilization is unimportant for our purposes.

It is often more convenient to work in the Einstein frame, where the “Planck
mass” is constant, that is, the only coupling to the scalar curvature is
$-1/2M_{\rm Pl}^{2}R$. To simplify formulas, let us choose the Planck units in
this section,

$M_{\rm Pl}=1\;.$ (38)

Then, the transition to the Einstein frame is accomplished by rescaling the
metric

$g_{\mu\nu}=\Omega\,\hat{g}_{\mu\nu}~{}~{}\;,~{}~{}\Omega=1+\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\;.$
(39)

This transformation affects the kinetic terms making them non–canonical and
also rescales the scalar potential. For a general $\Omega$, the kinetic
function multiplying
$1/2\,g^{\mu\nu}\partial_{\mu}\phi_{i}\,\partial_{\nu}\phi_{j}$ and the
potential become [15]:

$\displaystyle K^{ij}={3\over
2}\;{\partial\log\Omega\over\partial\phi_{i}}\,{\partial\log\Omega\over\partial\phi_{j}}+{\delta^{ij}\over\Omega}\;,$
$\displaystyle V_{E}={V\over\Omega^{2}}\;,$ (40)

where $i,j$ label scalar fields. Consider now large field values such that

$\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\gg 1\;.$ (41)

In this case, $\Omega\simeq\xi_{h}h^{2}+\xi_{\phi}\phi^{2}$ and the Lagrangian
in the Einstein frame is given by

${\cal L}={3\over
4}\Bigl{(}\partial_{\mu}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})\Bigr{)}^{2}+{1\over
2}{1\over\xi_{h}h^{2}+\xi_{\phi}\phi^{2}}\Bigl{(}(\partial_{\mu}h)^{2}+(\partial_{\mu}\phi)^{2}\Bigr{)}-{V\over(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})^{2}}\;.$
(42)

Introduce new variables [39]

$\displaystyle\chi=\sqrt{3\over 2}\ln(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})\;,$
$\displaystyle\tau={h\over\phi}\;.$ (43)

In terms of $\chi$ and $\tau$, the kinetic terms read

$\displaystyle{\cal L}_{\rm kin}$ $\displaystyle=$ $\displaystyle{1\over
2}\left(1+{1\over
6}{\tau^{2}+1\over\xi_{h}\tau^{2}+\xi_{\phi}}\right)\,(\partial_{\mu}\chi)^{2}+{1\over\sqrt{6}}{(\xi_{\phi}-\xi_{h})\tau\over(\xi_{h}\tau^{2}+\xi_{\phi})^{2}}\,(\partial_{\mu}\chi)(\partial^{\mu}\tau)$
(44) $\displaystyle+$ $\displaystyle{1\over
2}{\xi_{h}^{2}\tau^{2}+\xi_{\phi}^{2}\over(\xi_{h}\tau^{2}+\xi_{\phi})^{3}}\,(\partial_{\mu}\tau)^{2}\;.$

Since the kinetic functions depend on $\tau$ only, the mixing term can be
eliminated by the shift

$\chi\rightarrow\chi+f(\tau)\;.$ (45)

The explicit form of $f(\tau)$ will not be necessary for our applications. Let
us focus on a few special cases in which the mixing vanishes and the kinetic
functions simplify: $\xi_{\phi}+\xi_{h}\gg 1$, $\tau=0$ and $\tau=\infty$.
This also applies to the case $\xi_{h}=\xi_{\phi}$, however, this relation is
not radiatively stable.

###  3.1 Large non–minimal couplings to gravity

Consider the limit $\xi\equiv\xi_{\phi}+\xi_{h}\gg 1$, i.e. at least one of
the non–minimal couplings to gravity is large, and expand the kinetic terms in
“$1/\xi$”. The term $(\partial_{\mu}\tau)^{2}$ scales as $1/\xi$, so does the
mixing term $(\partial_{\mu}\chi)(\partial^{\mu}\tau)$. Therefore, in terms of
canonically normalized variables, the mixing is suppressed and can be
neglected. To leading order in $1/\xi$, we have

${\cal L}_{\rm kin}={1\over 2}(\partial_{\mu}\chi)^{2}+{1\over
2}{\xi_{h}^{2}\tau^{2}+\xi_{\phi}^{2}\over(\xi_{h}\tau^{2}+\xi_{\phi})^{3}}\,(\partial_{\mu}\tau)^{2}\;.$
(46)

We see that variables $\chi$ and $\tau$ fully separate, while $\chi$ is
already canonically normalized. One may introduce a canonically normalized
$\tau^{\prime}$ by integrating the kinetic function, however it cannot be
written in closed form nor is it necessary. In a few interesting limits, the
expression for $\tau^{\prime}$ simplifies:

$\displaystyle\xi_{\phi}\gg\xi_{h}~{}~{}{\rm or}~{}~{}\tau\rightarrow
0~{}:~{}~{}~{}\tau^{\prime}={\tau\over\sqrt{\xi_{\phi}}}~{},$
$\displaystyle\xi_{h}\gg\xi_{\phi}~{}~{}{\rm
or}~{}~{}\tau\rightarrow\infty~{}:~{}~{}\tau^{\prime}={1\over\sqrt{\xi_{h}}\tau}~{},$
$\displaystyle\xi_{\phi}=\xi_{h}~{}:~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau^{\prime}={1\over\sqrt{\xi_{h}}}\arctan\,\tau~{}.$
(47)

In order to study certain properties of the scalar potential, e.g. stability,
using the canonically normalized $\tau^{\prime}$ is not necessary and we may
work with $\tau$. Let us consider large $h,\phi$ field values and neglect the
mass terms. Then, the potential in the Einstein frame $V_{E}$ becomes

$V_{E}={\lambda_{h}\tau^{4}+\lambda_{\phi h}\tau^{2}+\lambda_{\phi}\over
4(\xi_{h}\tau^{2}+\xi_{\phi})^{2}}\;.$ (48)

Its minima are classified according to [39]555We are assuming no special
relations among the couplings, e.g. $2\lambda_{h}\xi_{\phi}=\lambda_{\phi
h}\xi_{h}$. Such relations are not radiatively stable.

$\displaystyle(1)~{}2\lambda_{h}\xi_{\phi}-\lambda_{\phi
h}\xi_{h}>0,~{}2\lambda_{\phi}\xi_{h}-\lambda_{\phi
h}\xi_{\phi}>0~{},~{}~{}~{}\tau_{\rm
min}=\sqrt{2\lambda_{\phi}\xi_{h}-\lambda_{\phi h}\xi_{\phi}\over
2\lambda_{h}\xi_{\phi}-\lambda_{\phi h}\xi_{h}}\;,$
$\displaystyle(2)~{}2\lambda_{h}\xi_{\phi}-\lambda_{\phi
h}\xi_{h}>0,~{}2\lambda_{\phi}\xi_{h}-\lambda_{\phi
h}\xi_{\phi}<0~{},~{}~{}~{}\tau_{\rm min}=0\;,$
$\displaystyle(3)~{}2\lambda_{h}\xi_{\phi}-\lambda_{\phi
h}\xi_{h}<0,~{}2\lambda_{\phi}\xi_{h}-\lambda_{\phi
h}\xi_{\phi}>0~{},~{}~{}~{}\tau_{\rm min}=\infty\;,$
$\displaystyle(4)~{}2\lambda_{h}\xi_{\phi}-\lambda_{\phi
h}\xi_{h}<0,~{}2\lambda_{\phi}\xi_{h}-\lambda_{\phi
h}\xi_{\phi}<0~{},~{}~{}~{}\tau_{\rm min}=0,\infty\;.$ (49)

Cases (2) and (3) correspond to the singlet and Higgs inflation, respectively
while both are possible in case (4) due to the existence of two local minima.

Let us focus on case (1), where inflation is driven by a combination of $\phi$
and $h$ with their ratio being fixed. The potential takes on the value

$V_{E}\Bigl{|}_{\rm min\;(1)}={1\over
16}{4\lambda_{\phi}\lambda_{h}-\lambda_{\phi
h}^{2}\over\lambda_{\phi}\xi_{h}^{2}+\lambda_{h}\xi_{\phi}^{2}-\lambda_{\phi
h}\xi_{\phi}\xi_{h}}\;.$ (50)

This energy density is positive: the numerator is positive as required by the
absence of run–away directions in the scalar potential, while the denominator
is positive according to conditions (1). For large non–minimal couplings to
gravity, the field $\tau$ is a spectator during inflation. Indeed, the Hubble
rate scales as $\sqrt{V_{E}}\sim 1/\xi$, while the mass for the canonically
normalized $\tau^{\prime}\sim\tau/\sqrt{\xi}$ scales as $1/\sqrt{\xi}$. Hence,

$m_{\tau^{\prime}}^{2}\gg H^{2}$ (51)

and it evolves quickly to the minimum. In other words, $\tau$ can be
“integrated out”.

In this framework, inflation is driven by the $\chi$ field. At leading order
in large $\xi$ and $\xi_{h}h^{2}+\xi_{\phi}\phi^{2}$, the potential is flat
with respect to $\chi$. At next-to-leading order, mild $\chi$–dependence
appears. Retaining the
$1/(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})=\exp(-{2}\chi/\sqrt{6})$ term in
$\Omega$, one finds that the potential is modified in two ways: first, the
rescaling $V_{E}=V/\Omega^{2}$ introduces $\chi$–dependence; second,
$\tau=h/s$ mixes kinetically with $\chi$ thereby bringing additional
$\chi$–dependence in. However, at the minimum of the potential,

${\partial V_{E}\over\partial\tau}\Bigl{|}_{\rm min}=0\;,$ (52)

and the shift in $\tau$ does not affect the value of the potential. Thus, the
$\chi$–dependence at this order comes from the first source and

$V_{E}={\lambda_{\rm eff}\over
4\xi_{h}^{2}}\left(1+\exp\left(-{2\chi\over\sqrt{6}}\right)\right)^{-2}\;,$
(53)

where

$\lambda_{\rm eff}={1\over 4}{4\lambda_{\phi}\lambda_{h}-\lambda_{\phi
h}^{2}\over\lambda_{\phi}+\lambda_{h}x^{2}-\lambda_{\phi h}x}$ (54)

and

$x={\xi_{\phi}\over\xi_{h}}\;.$ (55)

This notation is convenient to draw a parallel with Higgs inflation [12]. We
observe that the inflationary potential is exponentially close to the flat one
at large $\chi$. As we show below, potentials of this type are favored by
cosmological data.

The above considerations can trivially be extended to $\tau_{\rm min}=0$ and
$\infty$ by replacing the potential value at the minimum by
$\lambda_{\phi}/(4\xi^{2}_{\phi})$ and $\lambda_{h}/(4\xi^{2}_{h})$,
respectively.

####  3.1.1 Inflationary predictions

The potential (53) leads to specific inflationary predictions. The slow roll
parameters are given by

$\displaystyle\epsilon={1\over 2}\left({\partial V_{E}/\partial\chi\over
V_{E}}\right)^{2}\simeq{4\over 3}\exp(-4\chi/\sqrt{6})\;,$
$\displaystyle\eta={\partial^{2}V_{E}/\partial\chi^{2}\over
V_{E}}\simeq-{4\over 3}\exp(-2\chi/\sqrt{6})\;,$ (56)

where we have neglected $\exp(-2\chi/\sqrt{6})$ compared to 1. During
inflation $e^{\chi}\gg 1$ and $\epsilon,\eta\ll 1$. Inflation ends when
$\epsilon$ approaches 1. This corresponds to

$\chi_{\rm end}\simeq\sqrt{3\over 8}\ln{4\over 3}\;.$ (57)

The number of e–folds is found through

$N=\int_{\rm in}^{\rm end}H\,dt=-\int^{\rm end}_{\rm in}{V_{E}\over\partial
V_{E}/\partial\chi}\,d\chi\simeq{3\over 4}\exp(2\chi_{\rm in}/\sqrt{6})\;,$
(58)

which implies that the initial field value is

$\chi_{\rm in}\simeq{\sqrt{6}\over 2}\ln{4N\over 3}\;.$ (59)

The last 60 or so e–folds of inflation are constrained by the observed
inflationary perturbations. In particular, the COBE constraint requires
$V_{E}/\epsilon=0.027^{4}$ at $\chi_{\rm in}$ [37], which implies

${\lambda_{\rm eff}\over 4\xi_{h}^{2}}=4\times 10^{-7}\,{1\over N^{2}}\;.$
(60)

For $N=60$, this gives ${\lambda_{\rm eff}/(4\xi_{h}^{2})}\simeq 10^{-10}$.
Therefore, either the coupling has to be small or $\xi$ large enough. If
inflation is Higgs–like, $\tau_{\rm min}\gg 1$, the non–minimal coupling
$\xi_{h}$ is required to be very large, $\xi_{h}\sim 10^{4}$. On the other
hand, for mostly–singlet inflation, $\tau_{\rm min}\ll 1$, the COBE constraint
can be satisfied with a small $\lambda_{\phi}$ and moderate $\xi$’s. Models of
this type can be embedded into interesting particle physics frameworks which
address various problems of the SM [43],[44].

Figure 2:  PLANCK constraints on inflationary models [38]. The $R^{2}$
(Starobinsky) inflation is largely equivalent to inflation driven by a
non–minimal scalar coupling to gravity. Figure credit: Y. Akrami et al.
[Planck], Astron. Astrophys. 641, A10 (2020); reproduced with permission ©
ESO.

The predictions of the model are conveniently formulated in terms of the
number of e–folds $N$. Using

$\displaystyle\epsilon\simeq{3\over 4N^{2}}\;,$
$\displaystyle\eta\simeq-{1\over N}$ (61)

at $\chi_{\rm in}$, the spectral index $n$ and and the tensor to scalar
perturbation ratio are computed via

$\displaystyle n=1-6\epsilon+2\eta\simeq 1-2/N\simeq 0.97\;,$ $\displaystyle
r=16\epsilon\simeq 12/N^{2}\simeq 3\times 10^{-3}\;.$ (62)

Note that $\epsilon\ll|\eta|$ and the gravitational wave production is
suppressed. These values are consistent with and even preferred by the PLANCK
cosmological data [38]. Figure 2 shows constraints on $n,r$ for different
inflationary models. The predictions of models based on the non–minimal scalar
coupling to gravity are largely equivalent to those of Starobinsky $R^{2}$
inflation [32] given by the green line and are well within the preferred
region.

###  3.2 General non–minimal couplings at $\tau=0,\infty$

At specific points in field space such as $\tau=0$ and $\infty$, the analysis
can be performed for arbitrary non–minimal couplings. These correspond to
singlet and Higgs inflation, respectively. One needs to make sure, however,
that these points are stable and, in order to have single–field inflation, the
$\tau$ variable is sufficiently heavy.

Consider the large field limit $\xi_{h}h^{2}+\xi_{\phi}\phi^{2}\gg 1$. If the
scalar potential has a minimum at $\tau=0$ or $\tau=\infty$, the kinetic terms
simplify. At $\tau=0$, the mixing term disappears and the canonically
normalized variables are given by

$\chi^{\prime}={\chi\;\sqrt{1+{1\over
6\xi_{\phi}}}}~{}~{},~{}~{}\tau^{\prime}={\tau\over\sqrt{\xi_{\phi}}}\;.$ (63)

Let us now determine under what circumstances $\tau=0$ is a local minimum and
$\tau$ can be integrated out. The scalar potential at large $\chi$ reads

$V_{E}={\lambda_{h}\tau^{4}+\lambda_{\phi h}\tau^{2}+\lambda_{\phi}\over
4(\xi_{h}\tau^{2}+\xi_{\phi})^{2}}\;.$ (64)

Expanding it at small $\tau$, one finds that for

$\lambda_{\phi h}\xi_{\phi}-2\lambda_{\phi}\xi_{h}>0\;,$ (65)

$\tau=0$ is a local minimum. The mass squared of the canonically normalized
$\tau^{\prime}$ is then given by

$m_{\tau^{\prime}}^{2}={\lambda_{\phi h}\xi_{\phi}-2\lambda_{\phi}\xi_{h}\over
2\xi_{\phi}^{2}}\;.$ (66)

On the other hand, the Hubble rate is found from

$3H^{2}=V_{E}(\tau=0)={\lambda_{\phi}\over 4\xi_{\phi}^{2}}.$ (67)

The $\tau^{\prime}$ variable can be integrated out if
$m_{\tau^{\prime}}^{2}\gg H^{2}$ or

$\lambda_{h\phi}\gg\lambda_{\phi}\;{12\xi_{h}+1\over 6\xi_{\phi}}\;.$ (68)

In this case, inflation is driven by $\chi$ and $\tau^{\prime}$ is a static
heavy spectator. Note that small as well as order one non–minimal couplings
are consistent with this inequality.

The inflationary potential is derived along lines of the previous subsection.
The $\chi$–dependent correction to the potential comes from the Weyl rescaling
$V_{E}=V/\Omega^{2}$, where the subleading
$1/(\xi_{h}h^{2}+\xi_{\phi}\phi^{2})$ term at large
$\xi_{h}h^{2}+\xi_{\phi}\phi^{2}$ is retained. The kinetic mixing does not
contribute at this order and, in terms of the canonically normalized
$\chi^{\prime}$, we get

$V_{E}={\lambda_{\phi}\over
4\xi_{\phi}^{2}}\left(1+\exp\left(-{2\gamma\chi^{\prime}\over\sqrt{6}}\right)\right)^{-2}\;,$
(69)

where

$\gamma=\sqrt{6\xi_{\phi}\over 6\xi_{\phi}+1}\;.$ (70)

Since $\gamma<1$, the potential is less steep than $V_{E}$ at large
non–minimal couplings.

The potential of this form was considered in Section 3.1.1 except for the
$\gamma$ factor. Repeating the steps described in that section, one finds that
the COBE constraint becomes

${\lambda_{\phi}\over 4\xi_{\phi}^{2}}=4\times
10^{-7}\,{1\over\gamma^{2}N^{2}}\;.$ (71)

The slow roll parameters are now given by hand

$\displaystyle\epsilon\simeq{3\over 4\gamma^{2}N^{2}}\;,$
$\displaystyle\eta\simeq-{1\over N}\;,$ (72)

such that

$\displaystyle n\simeq 1-{2\over N}-{9\over 2\gamma^{2}N^{2}}\;,$
$\displaystyle r\simeq{12\over\gamma^{2}N^{2}}\;.$ (73)

The resulting modifications of the inflationary predictions can be significant
while still well within the PLANCK bounds. They tend to increase the tensor to
scalar ratio and decrease the spectral index. Note, however, that $\gamma\ll
1$ is not consistent with our approximations: in this case, inflation proceeds
at least partly in the region $e^{-2\gamma\chi^{\prime}/\sqrt{6}}\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}1$.

Inspection of the $\tau=\infty$ point proceeds analogously with the help of
the inversion $\tau\rightarrow 1/\tau$. The main difference is that the Higgs
coupling that appears in the inflationary potential cannot be too small,
barring accidental cancellations.666For special values of $m_{t}$ and $m_{h}$,
the SM Higgs potential develops a plateaux at large field values
[45],[46],[47], in which case a separate analysis is required.  Indeed, in the
SM one typically has $\lambda_{h}\sim{\cal O}(10^{-2})$ at relevant energy
scales, although this number is sensitive to the top quark mass. The coupling
to the singlet tends to increase $\lambda_{h}$ via renormalization group
running. Therefore, $\xi_{h}$ has to be large to fit the COBE normalization
and the corresponding $\gamma$ is close to 1.

We see that the Higgs portal framework endowed with non–minimal couplings to
gravity provides an excellent setting for inflationary model building. The
resulting inflaton potential is concave and exponentially close to the flat
one, which fits very well with the PLANCK constraints.

###  3.3 Unitarity issues

The non–minimal coupling to gravity generates non–renormalizable operators
leading to scattering amplitudes growing with energy. This signifies the
effective field theory nature of our description which breaks down above a
certain energy scale. Since we are interested in the inflationary dynamics,
the corresponding cut–off must lie above the inflationary energy scale. As we
show below, this issue becomes critical for large $\xi_{i}$ characteristic of
Higgs inflation [48],[49].

Let us determine the cut–off of our theory in the presence of a large
non–minimal Higgs coupling to gravity, $\xi\gg 1$. To make the discussion more
transparent, restore the Planck scale in our formulae and consider linearized
gravity,

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}/{M_{\rm Pl}}\;,$ (74)

where $\eta_{\mu\nu}$ is the Minkowski metric and $h_{\mu\nu}$ is a small
perturbation. In the Jordan frame, the coupling $\xi h^{2}R$ leads to the
dimension–5 operator

${\xi\over M_{\rm Pl}}h^{2}\eta^{\mu\nu}\partial^{2}h_{\mu\nu}\;.$ (75)

The scattering processes mediated by this operator, e.g.
$h+h_{\mu\nu}\rightarrow h+h_{\mu\nu}$, exhibit rapid growth of the rate with
energy. This leads to unitarity violation [48],[49] at energies above

$\Lambda_{U}\sim{M_{\rm Pl}\over\xi}\;.$ (76)

The same conclusion is reached in the Einstein frame. The conformal
transformation of the metric $g_{\mu\nu}\rightarrow\Omega\,g_{\mu\nu}$, with
$\Omega=1+\xi h^{2}/M_{\rm Pl}^{2}$, generates the term
$(\partial_{\mu}\,\ln\Omega)^{2}$, i.e. the dim–6 operator

${\xi^{2}\over M_{\rm Pl}^{2}}\,{h^{2}\over(1+\xi h^{2}/M_{\rm
Pl}^{2})^{2}}\,(\partial_{\mu}h)^{2}\;.$ (77)

The cutoff is again $M_{\rm Pl}/\xi$.

In these considerations, we are expanding the fields around the flat
background with a zero Higgs expectation value. The presence of a non–trivial
background affects these results [50]. In this case, one expands the fields in
terms of the average values and fluctuations,

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}/{M_{\rm
Pl}}~{}~{},~{}~{}h=\bar{h}+\delta h\;.$ (78)

At large values of the background, the unitarity cutoff changes. This is
clearly seen in the Einstein frame for $\xi h^{2}\gg M_{\rm Pl}$: in this
case, the results of Section 3.1 apply and, for a canonically normalized
$\chi$ and $\tau=\infty$, the higher dimensional operators are encoded in the
potential $V_{E}\propto 1-2\exp\Bigl{(}-2\chi/(\sqrt{6}M_{\rm Pl})\Bigr{)}$.
Expanding it around $\bar{\chi}\gg M_{\rm Pl}$, one finds a Taylor series that
behaves like

$\exp\Bigl{(}-2\bar{\chi}/(\sqrt{6}M_{\rm Pl})\Bigr{)}\;\left({\delta\chi\over
M_{\rm Pl}}\right)^{n}\;.$ (79)

At sufficiently large $n$, the overall prefactor is unimportant and the
unitarity cutoff is

$\Lambda_{U}^{\rm infl}\sim{M_{\rm Pl}}\;.$ (80)

The corresponding bound in the Jordan frame, $\sqrt{\xi}h$, can be obtained by
noting that the cutoffs are related via the metric rescaling $\Omega$:
$\Lambda_{U}\rightarrow\Omega^{1/2}\Lambda_{U}$. Since in the Jordan frame the
scalar curvature is multiplied by $M_{\rm Pl}^{2}+\xi h^{2}$, the cutoff of
the theory in both cases coincides with the cutoff of the gravitational sector
[50]. This can be compared to the scale of inflation,

$V_{E}^{1/4}\sim\lambda_{h}^{1/4}\,{M_{\rm Pl}\over\sqrt{\xi}}\;.$ (81)

We thus conclude that inflation proceeds in a controllable manner below the
unitarity cutoff.

Although during inflation the unitarity cutoff is high, it relaxes to (76) as
the inflaton background value drops at the end of inflation. In the inflaton
oscillation epoch, violent particle production takes place. At large $\xi$,
the particle momentum can be as high as $\sqrt{\lambda_{h}}M_{\rm Pl}$
[51],[52]. This exceeds the cutoff casting doubt on the validity of our
approach. While the theory is well–behaved at large and small field values,
the intermediate field range is problematic calling for a UV completion. These
problems do not arise in the singlet–driven inflation at smaller $\xi_{\phi}$
and $\lambda_{\phi}$ subject to the COBE relation (60).

Quantum corrections [53],[54],[55],[56] are also controllable at small and
large field values. For $\xi h^{2}\ll M_{\rm Pl}^{2}$, we recover the Standard
Model, while at $\xi h^{2}\gg M_{\rm Pl}^{2}$, one can use the approximate
shift symmetry in the Einstein frame,

$\chi\rightarrow\chi+{\rm const}~{},$ (82)

to organize the perturbation series in the effective field theory (EFT) form
[50]. The effective Lagrangian is expanded in powers of the inverse cutoff for
fluctuations, $1/M_{\rm Pl}$, and the symmetry violating corrections
$e^{-n\chi/M_{\rm Pl}}$. The theory is then renormalizable in the EFT sense.
However, in the intermediate field range, there is no organizing principle to
control the quantum corrections. Again, one concludes that a UV completion is
needed for a consistent description of the entire field range [50],[57],[58].
A discussion of related issues and their possible solutions can be found in
[59],[60],[61],[62].

In conclusion, the Higgs portal allows us to build viable inflationary models,
where the role of the inflaton is played by a combination of the Higgs and
singlet fields. Such models fit the PLANCK data and satisfy the tree level
unitarity constraint as long as the effective quartic coupling is sufficiently
small.

##  4 Vacuum stability and inflation

The issue of vacuum stability has become one of the central questions in Higgs
physics in recent years. The current data favor vacuum metastability, which
entails a number of cosmological puzzles. Even if our vacuum is very long
lived, one should explain how the Universe ended up in this energetically
disfavored state in the first place. In what follows, we formulate the
problems and discuss their possible solutions within the Higgs portal
framework.

Figure 3:  Vacuum stability in the Standard Model in terms of the top quark
and Higgs masses. The measured values are marked by a dot. The figure is from
Ref. [68] © CC–BY.

###  4.1 Higgs potential in the Standard Model and quantum fluctuations

The value of the Higgs mass is intimately related to stability of the
electroweak (EW) vacuum in the Standard Model. If the Higgs is light, the
corresponding quartic coupling is small and driven negative at high energy by
the top quark loop [63],[64],[65]. This implies that the potential turns
negative at large field values and the EW vacuum is not absolutely stable
[66]. The current Higgs and top quark masses $m_{h}\simeq 125$ GeV,
$m_{t}\simeq 173$ GeV 777The current LHC top quark mass measurements tend to
give a slightly lower value than the world average, i.e. close to 172.5 GeV,
as reviewed in [67]. favor its metastability (Fig. 3), meaning that the vacuum
decays on a long time scale by tunnelling to the energetically favored state.
Its lifetime is controlled by the Lee-Weinberg bounce [69] with the Euclidean
action $S_{E}=8\pi^{2}/(3|\lambda_{h}(\mu)|)$, where $\mu$ is such that
$\beta_{\lambda_{h}}(\mu)=0$ [66]. The result is that the decay takes much
longer than the age of the Universe, $\tau\gg 10^{100}$ years, so it can be
considered stable for most practical purposes.888Vacuum decay can be catalyzed
by primordial black holes [70],[71].

The absolute vacuum stability bound on the Higgs mass according to [72] reads

$m_{h}>129.6~{}{\rm GeV}+2.0(m_{t}-173.34~{}{\rm GeV})-0.5~{}{\rm
GeV}~{}{\alpha_{3}(M_{Z})-0.1184\over 0.0007}\pm 0.3~{}{\rm GeV}\;,$ (83)

where $m_{t}$ is the top quark mass and $\alpha_{3}=g_{3}^{2}/4\pi$ is the
color structure constant. The uncertainty of this lower bound quoted in [72]
is $\pm 1.5$ GeV meaning that absolute stability of the EW vacuum is excluded
at around 3$\sigma$. Other evaluations of the stability bound confirm that the
currently preferred top quark and Higgs masses lead to a metastable
electroweak vacuum, although the error bars and the statistical preference for
metastability vary [73],[74],[75] with the former reaching 6 GeV in [76],[77].
A notable complication in these calculations concerns gauge dependence
[78],[79].

It should be noted that the decay rate of the EW vacuum is sensitive to Planck
suppressed operators $h^{n}/{M_{\rm Pl}^{n-4}}$ as well as the Higgs
non–minimal coupling to gravity [80],[81]. Such UV sensitivity is due to the
tunnelling rate being an exponential function of the potential. Since
non–renormalizable operators can be generated by quantum gravity, this creates
an additional source of uncertainty in our considerations. The state-of-the-
art calculation of the decay rate in the pure Standard Model is presented in
[82],[83].

The Standard Model Higgs potential at large field values can be approximated
by $V(h)\simeq\lambda_{h}(h)h^{4}/4$, where $\lambda_{h}(h)$ is the running
coupling evaluated at the scale $\mu\sim h$ [84]. A more careful analysis
shows that the effective potential around the value $h_{\rm max}$, where the
potential is maximized, reads [85]

$V(h)\simeq-b\ln\left(h^{2}\over h^{2}_{\rm max}\sqrt{e}\right)\;{h^{4}\over
4}\;,$ (84)

with $b\simeq 0.16/(4\pi^{2})$. $h_{\rm max}$ defines the position of the
barrier separating the two vacua, with its typical value being $h_{\rm
max}\sim 5\times 10^{10}$ GeV. This form of the potential is important for
analyzing stability of the EW vacuum under fluctuations.

Even though vacuum metastability does not pose any immediate danger in the
current epoch, the situation was different in the Early Universe when the
field fluctuations were large [86]. During inflation, light scalar fields are
subject to quantum fluctuations of order the Hubble rate. It is then natural
to expect that the EW vacuum gets destabilized if the Hubble rate exceeds the
size of the barrier $h_{\rm max}$. This can be viewed as tunnelling to the
true vacuum in de Sitter space [87],[88],[89],[90], where the fluctuations are
described by the effective temperature $T=H/(2\pi)$ [91]. The problem can also
be seen from the viewpoint of the Higgs effective potential in de Sitter
space: the barrier disappears at large $H$ [92],[93].

Let us consider this issue more carefully by analyzing the Higgs fluctuations
[85]. If the effective Higgs mass is below the Hubble rate, the evolution of
its long wavelength modes is described by the Langevin equation [94],

${dh\over dt}+{1\over 3H}\,{dV(h)\over dh}=\eta(t)\;,$ (85)

where $h=\sqrt{2H^{\dagger}H}$ is the Higgs radial mode treated as a classical
field and $\eta(t)$ represents random Gaussian noise,

$\langle\eta(t)\eta(t^{\prime})\rangle={H^{3}\over
4\pi^{2}}\,\delta(t-t^{\prime})\;.$ (86)

As a result, the Higgs field experiences a random walk in a classical
potential. Indeed, if the potential is neglected, one can “square” Eq. 85 and
compute its statistical average. One then finds that $h$ changes on the
average by ${\cal O}(H)$ every Hubble time $H^{-1}$. The average Higgs value
then grows as the square root of time. It is convenient to formulate the
problem in terms of the probability density of finding value $h$ after $N=Ht$
e–folds of inflation, $P(h,N)$, which satisfies the Fokker-Planck equation

${\partial P\over\partial N}={\partial^{2}\over\partial
h^{2}}\left({H^{2}\over 8\pi^{2}}P\right)+{\partial\over\partial
h}\left({V^{\prime}\over 3H^{2}}P\right)\;.$ (87)

This equation can be solved numerically, while the salient features of the
solution can be understood analytically. For the Hubble rate similar to
$h_{\rm max}$, the potential contribution is subleading and the scalar is
effectively free. In this case, $P(h,N)$ satisfies the heat equation. Its
solution is well known,

$P(h,N)={1\over\sqrt{2\pi\langle h^{2}\rangle}}\,\exp\left(-{h^{2}\over
2\langle h^{2}\rangle}\right)~{},~{}~{}~{}\sqrt{\langle h^{2}\rangle}={H\over
2\pi}\sqrt{N}\;,$ (88)

where the initial value $h=0$ has been assumed. Numerical analysis shows that
this Gaussian shape is maintained even for $h$ significantly above $h_{\rm
max}$. Then, the probability of finding $h$ beyond the barrier after $N$
e–folds of inflation is999Here, the probability distribution is correctly
normalized for $h$ ranging from $-\infty$ to $\infty$, although $h$ is
initially taken as positive semidefinite or, in other words, $h$ and $-h$ are
gauge–equivalent. Thus, for consistency, the bound has to be imposed on the
absolute value, $|h|$.

$p\bigl{(}|h|>h_{\rm max}\bigr{)}=1-{\rm erf}\left({\sqrt{2}\pi h_{\rm
max}\over H\sqrt{N}}\right)\;,$ (89)

where the error function is defined by

${\rm erf}(z)={2\over{\sqrt{\pi}}}\int_{0}^{z}e^{-t^{2}}dt\;.$ (90)

The observable Universe is composed of $e^{3N}$ causally independent patches
formed during inflation. To make sure that the Higgs field does not fall into
the true vacuum in any of them during $N$ e–folds, we require

$p\bigl{(}|h|>h_{\rm max}\bigr{)}\;e^{3N}<1\;.$ (91)

Using the large $z$ asymptotics $1-{\rm erf}(z)\simeq
e^{-z^{2}}/(x\sqrt{\pi})$, this implies [85]

${H\over h_{\rm max}}<\sqrt{2\over 3}{\pi\over N}\simeq 0.04\;.$ (92)

Since the bound requires $h_{\rm max}\gg H$, the large $z$ expansion is indeed
justified. This condition gives us the criterion for vacuum stability during
inflation.101010In the context of eternal inflation, implications of vacuum
metastability have been studied in [95].

A few clarifications are in order. First, the Gaussian approximation for
$P(h,N)$ breaks down at large field values, where the classical evolution
takes over quantum fluctuations. Only if $h$ is above the corresponding
critical value instead of $h_{\rm max}$, does the system evolve irreversibly
to the true vacuum [96],[97],[98]. This refinement however makes an
insignificant correction of about +10% to the bound (92) as shown in [85]. It
also implies that there is a range of $h$ at the end of inflation, in which
the fate of the field is not entirely certain: if $h$ is not far above $h_{\rm
max}$, the potential favors its evolution to the true minimum, whereas thermal
effects during reheating may overcome this tendency and bring it back to the
stable region [85],[99]. To evaluate viability of this option would require a
detailed understanding of the preheating and thermalization processes with
heavy fields coupled to the Higgs.

Another point concerns the fate of the patches evolving to the true vacuum.
These do not pose any danger during inflation because the distance between the
bubbles of true vacuum increases. However, the situation changes after
inflation: depending on the initial conditions for the bubble formation, many
such regions expand in flat space with the speed of light eventually
swallowing the entire Universe [85]. This justifies our condition that these
bubbles should not be allowed to form in the first place.

Finally, the EW vacuum could in principle be destabilized by thermal effects
after inflation. Indeed, while the thermal Higgs mass favors $h\sim 0$, the
thermal fluctuations grow exponentially at high temperatures leading to the
formation of the true vacuum bubbles. For realistic values of the Higgs and
top quark mass, however, the resulting upper bound on the reheating
temperature is uninformative, $T\lesssim M_{\rm Pl}$ [86].

We conclude that stability of the EW vacuum during inflation requires the
inflation scale to be quite low: according to Eq. 92, the Hubble rate should
typically be below $10^{9}$ GeV or so. Traditional inflationary models, on the
other hand, prefer higher values of $H$, often by orders of magnitude as long
as $H\lesssim 10^{14}$ GeV [38], in which case the fluctuations are
catastrophically large. This conclusion, however, is sensitive to the presence
of the Higgs–inflaton and Higgs–gravity couplings which are expected on
general grounds. Their effect is the subject of the next sections.

###  4.2 Cosmological challenges

Figure 4: Schematic view of the Higgs potential. The barrier separating the
two minima is located at $h\sim\Lambda\ll M_{\rm Pl}$. The figure is from Ref.
[100].

Let us formulate the cosmological challenges one faces if the EW vacuum is
metastable. The existence of the deep minimum in the Higgs potential at large
field values (Fig. 4) poses two problems [100]:

– why has the energetically disfavored vacuum at $v=246$ GeV been chosen?

– why has the system remained in this shallow vacuum despite fluctuations?

The first problem concerns the initial conditions for the Higgs field at the
beginning of inflation. The second minimum is much deeper than the electroweak
one and, for most Higgs initial values, the system would evolve to the wrong
vacuum. In order to end up at $v=246$ GeV, the Higgs field must start its
evolution at very small field values, $h_{0}<10^{-8}M_{\rm Pl}$, while its
“natural” range is expected to be around the Planck scale. Thus, one may
estimate the degree of finetuning required to be roughly $1/10^{8}$.

The second problem concerns stability of the electroweak vacuum with respect
to field fluctuations in the Early Universe [86]. The two vacua are separated
by a tiny barrier which can be overcome by quantum fluctuations during
inflation or preheating. Unless there exists a stabilizing mechanism, such
fluctuations have to be small enough. The scale of these fluctuations is tied
to the scale of inflation, so vacuum stability requires low scale inflation.
While this is certainly possible, it disfavors most of the existing
inflationary models.

Although the above problems are related, they are not the same: solving the
second problem with low scale inflation does not address the first problem.
Conversely, setting up the right initial conditions does not guarantee
stability against quantum fluctuations.

###  4.3 Stabilizing the Higgs potential during inflation

The problems formulated above can be addressed within the Higgs portal
framework [100]. Given that the inflaton field takes on large values during
inflation, the Higgs portal couplings induce an effective mass term for the
Higgs field which can drastically change its behaviour. At large $\phi$, the
most important inflaton–induced contribution to the Higgs potential is
provided by the quartic coupling,

$\Delta V={1\over 4}\lambda_{\phi h}\phi^{2}h^{2}\;.$ (93)

For $\lambda_{\phi h}>0$, it has a stabilizing effect and dominates the Higgs
potential when

$\phi>\sqrt{|\lambda_{h}|\over\lambda_{\phi h}}\,h\;.$ (94)

If the initial values of the inflaton and the Higgs satisfiy this inequality,
the Higgs field will quickly roll down to $h\sim 0$. We may neglect quantum
gravity effects as long as the energy density is far below ${\rm M_{Pl}}^{4}$
implying that the Higgs field range is bounded by about ${\cal O}(10^{-1}){\rm
M_{Pl}}$.111111We assume that higher dimensional in $h$ operators may be
neglected in this field range. Above this bound, our field–theoretic
description is meaningless and the problem we are addressing cannot even be
formulated. Notably, the inflaton field is allowed to take on values above the
Planck scale due to flatness of its potential. To get a feeling how large
$\phi$ should be to stabilize the Higgs potential, let us take $h=0.1{\rm
M_{Pl}}$, $|\lambda_{h}|=10^{-2}$ and $\lambda_{\phi h}=10^{-6}$. The required
initial value of the inflaton is then 10 MPl or larger. This range is quite
typical for chaotic inflation and much larger values are still consistent with
a classical description of gravity .

The above coupling should not affect the inflaton potential and spoil the
inflationary predictions. In particular, radiative corrections due to
$\lambda_{\phi h}$ must be small enough. The consequent constraint on the
coupling is strongly model–dependent. For illustration, let us consider the
quadratic inflaton potential, $V_{\phi}={1\over 2}m^{2}_{\phi}\phi^{2}$ with
$m\sim 10^{-5}$MPl. Closing the Higgs in the loop, one finds the resulting
Coleman–Weinberg potential

$\Delta V_{\rm 1-loop}\simeq{\lambda_{\phi h}^{2}\over
64\pi^{2}}\phi^{4}\ln{\lambda_{\phi h}\phi^{2}\over m_{\phi}^{2}}\;,$ (95)

where we have included 4 Higgs degrees of freedom. Requiring the correction
not to exceed the tree potential in the last 60 e–folds of inflation, one
finds

$\lambda_{\phi h}\lesssim 10^{-6}\;.$ (96)

This bound, however, relaxes significantly in other models of inflation, e.g.
those based on the non–minimal scalar coupling to gravity [100]. In these
models, the inflaton quartic coupling ${1\over 4}\lambda_{\phi}\phi^{4}$ is
already present at tree level and the leading log correction $\lambda_{\phi
h}^{2}/(16\pi^{2})\,\ln(\mu/\mu_{0})^{2}$ should not exceed $\lambda_{\phi}$.
The latter can be significant, depending on the non–minimal coupling, and the
consequent bound on $\lambda_{\phi h}$ relaxes compared to (96).

Let us consider the evolution of the Higgs–inflaton system with a quadratic
inflaton potential in more detail. If the initial Higgs value is large enough,
the Higgs portal term may dominate the energy density of the Universe and thus
affect the inflationary dynamics. The equations of motion together with the
Friedmann equation read

$\displaystyle\ddot{h}+3H\dot{h}+{\partial V\over\partial h}=0\;,$
$\displaystyle\ddot{\phi}+3H\dot{\phi}+{\partial V\over\partial\phi}=0\;,$
(97)

with the Hubble rate $H$ in Planck units  (M${}_{\rm Pl}=1$) given by

$3H^{2}={1\over 2}\dot{h}^{2}+{1\over 2}\dot{\phi}^{2}+V$ (98)

and

$V\simeq{1\over 4}\lambda_{\phi h}h^{2}\phi^{2}+{1\over
2}m_{\phi}^{2}\phi^{2}\;.$ (99)

Suppose that initially both $\dot{h}$ and $\dot{\phi}$ are small. For
$\lambda_{\phi h}$ not far from the upper bound (96) and the initial Higgs
value $h_{0}\sim 0.1$, the Hubble rate is dominated by the cross term,
$H_{0}\simeq\sqrt{\lambda_{\phi h}/12}\;\phi_{0}h_{0}$. Then, taking
$\phi_{0}$ which satisfies (94), one finds the hierarchy

$m_{\phi}\ll H_{0}\ll m_{h}^{\rm eff}\;,$ (100)

where the effective Higgs mass is $m_{h}^{\rm eff}=\sqrt{\lambda_{\phi
h}/2}\;\phi_{0}$. Although the inflaton receives an effective mass
contribution from the $\lambda_{\phi h}$ coupling, it is small in the regime
we are considering. In general, fields with masses above the Hubble rate
evolve quickly while those with masses below the Hubble rate are effectively
“frozen”. Thus, the above hierarchy implies that $h$ decreases while $\phi$
undergoes a “slow roll”.

The Higgs evolution at the initial stage is determined by

$\ddot{h}+\sqrt{3\over 2}\;\sqrt{\dot{h}^{2}+(m_{h}^{\rm
eff\,})^{2}\;h^{2}}\;\dot{h}+(m_{h}^{\rm eff\,})^{2}\;h=0\;,$ (101)

where $m_{h}^{\rm eff}$ varies slowly. This equation is well known as it
describes the inflaton evolution in a quadratic potential during the inflaton
oscillation epoch (see, e.g. [101]). It has a simple solution at $m_{h}^{\rm
eff}t\gg 1$,

$h\simeq C\;{\cos(m_{h}^{\rm eff}t)\over m_{h}^{\rm eff}t}\;,$ (102)

with order one $C$. Since $m_{h}^{\rm eff}\gg H$, the asymptotic behaviour
sets in after a few Hubble times. In about 10 Hubble times, $h$ reduces by an
order of magnitude and at that point ${1\over 2}m^{2}_{\phi}\phi^{2}$ takes
over the energy density. After that, the usual slow roll inflation takes
place. The Hubble rate is almost constant and $h$ evolves according to

$\ddot{h}+3H\dot{h}+(m_{h}^{\rm eff})^{2}h=0\;,$ (103)

with $H\simeq m_{\phi}\phi_{0}/\sqrt{6}$. The solutions are linear
combinations of

$h=C_{\pm}\,\exp\Big{(}-3/2\;H\pm\sqrt{9/4\;H^{2}-(m_{h}^{\rm
eff})^{2}}~{}\Big{)}t\;.$ (104)

Since the Higgs field is heavy, its amplitude of oscillations decays
exponentially,

$|h(t)|\sim e^{-{3\over 2}Ht}|h(0)|\;.$ (105)

The field becomes of electroweak size after about 20 $e$–folds. The inflaton,
on the other hand, evolves slowly until the end of inflation. The quantum
fluctuations of the Higgs field are unimportant since $m_{h}^{\rm eff}\gg H$
and the barrier separating the two Higgs potential minima is located at

$h_{\rm crit}\sim\sqrt{\lambda_{\phi h}\over|\lambda_{h}|}\;\phi\gg H\;.$
(106)

The field is stuck at the origin and, therefore, stability of the Higgs
potential has been achieved even for sub–Planckian initial Higgs values. A
numerical analysis of the Higgs–inflaton system supports this conclusion
[100].

To summarize, we find that the Higgs evolves quickly to small field values and
stays there till the end of inflation. The details of this mechanism depend on
the initial field values: for instance, if $h_{0}$ is below $10^{-2}$, the
first stage in the Higgs evolution, when $h\propto 1/t$, is absent. The main
ingredients are a positive Higgs portal coupling and a large initial value of
the inflaton. The coupling cannot be too small: the Higgs effective mass must
be above the Hubble rate during the slow roll. Thus, for a quadratic inflaton
potential, the coupling must lie in the range

$10^{-6}>\lambda_{\phi h}>10^{-10}\;.$ (107)

Similar conclusions hold for other large field inflation models, although the
allowed range of $\lambda_{\phi h}$ depends strongly on the specifics of the
model. Note that small field models are disfavored by Eq. 94. Even though the
Higgs quantum fluctuations can be suppressed, such models do not address the
problem of initial conditions: the deep minimum at large Higgs field values
still persists during (and after) inflation and the system is overwhelmingly
likely to evolve there.

An analogous stabilizing effect can be provided by the non–minimal
Higgs–gravity coupling ${1\over 2}\xi_{h}h^{2}R$ [86]. Since the scalar
curvature $R=12H^{2}$ is large in the Early Universe, this effective mass term
can dominate the Higgs potential and lead to the field evolution described
above.121212The full analysis of the initial stage would require specifying
the inflaton sector in order to assess the backreaction of the Higgs on the
scalar curvature. Requiring the effective Higgs mass to exceed the Hubble
rate, one finds the lower bound on the coupling, $\xi_{h}\gtrsim 10^{-1}$
[86]. Similar results have been obtained using a more sophisticated effective
potential analysis in de Sitter space [92],[93] as well as bubble nucleation
during inflation [102].

The discussion above focuses on the leading effects. Understanding the
complete dynamics of the system requires the knowledge of the UV completion as
well as inclusion of more subtle effects. Some of them have been considered in
[103],[104],[105].

##  5 Vacuum stability after inflation

The issue of vacuum stability remains relevant even after inflation,
especially during the inflaton oscillation epoch. In this period, the Higgs
quanta production can have an explosive character leading to large field
fluctuations and eventual vacuum destabilization. The reason is that the Higgs
portal couplings create an effective Higgs mass,

$\bigl{(}m_{h}^{\rm eff}\bigr{)}^{2}={1\over 2}\lambda_{\phi
h}\phi^{2}+\sigma_{\phi h}\phi\;,$ (108)

where $\phi$ is an oscillating background. This leads to resonant production
of the Higgs field: as the inflaton goes through the origin, the effective
Higgs mass turns zero and the system becomes highly non–adiabatic. If the
$\lambda_{\phi h}$ term dominates, the resonance is parametric
[106],[107],[108], while for a significant $\sigma_{\phi h}$ term, it is
tachyonic [109],[110],[111]. In the latter case, the effective Higgs mass
squared goes negative at $\phi<0$ leading to exponential growth of the Higgs
amplitude. Analogously, the non–minimal Higgs coupling to gravity also results
in tachyonic Higgs production after inflation [112].

Let us note that the physics of vacuum destabilization is semiclassical as it
deals with collective phenomena. In the postinflationary Universe, the main
quantity that controls it is the statistical variance, $\langle h^{2}\rangle$,
such that if it exceeds some critical value, fluctuations destabilize the
system. The semiclassical approach requires the occupation numbers to be
sufficiently large, which is indeed the case during the resonance. On the
other hand, vacuum destabilization cannot be caused by a few quanta, even if
they are very energetic (such as cosmic rays) [66].

In what follows, we mostly specialize to the case of a quadratic inflaton
potential in the postinflationary epoch, while many conclusions apply more
generally. In this case [101],

$\phi(t)\simeq\Phi(t)\,\cos(mt)\;,$ (109)

where the amplitude decreases as $\Phi(t)\simeq\Phi_{0}/(mt)$ and $m$ is the
inflaton mass.

Let us start by reviewing the theory of parametric resonance.

###  5.1 Parametric resonance

Here we follow the discussions in [101] and [113] while providing somewhat
more details to make the exposition pedagogical.

In order to focus on important features of the resonance, let us neglect the
Universe expansion as well as the extra Higgs couplings,
$\lambda_{h},\sigma_{\phi h}\rightarrow 0$. Then, expanding the semiclassical
Higgs field in spacial Fourier modes (see Section 5.2.1 for details), one
finds that the evolution of the momentum mode $X_{k}$ is given by the Mathieu
equation

$\ddot{X}_{k}+\omega^{2}(t)X_{k}=0\;,$ (110)

where

$\omega^{2}(t)={k^{2}\over a^{2}}+{1\over 2}\lambda_{\phi
h}\Phi^{2}\cos^{2}mt\;,$ (111)

and $a$ is the scale factor. Due to the periodic time dependence of
$\omega(t)$, $X_{k}$ can undergo resonant growth. This phenomenon is known as
parametric resonance.

The physics of the resonance is largely captured by a simpler problem of wave
scattering in a parabolic potential. In a semiclassical system, the solution
to the equations of motion is given by the WKB formula

$X_{k}\propto{1\over\omega}\exp\left(\pm i\int\omega\,dt\right)\;,$ (112)

where the adiabatic condition

$\left|{\dot{\omega}\over\omega^{2}}\right|\ll 1$ (113)

is assumed. For large momenta or large enough couplings, $\sqrt{\lambda_{\phi
h}}\Phi\gg m$, this condition is satisfied away from the inflaton zero
crossing, $mt=\pi/2+n\pi$ . The solution then describes a constant number of
particles with a slowly varying $\omega(t)$. Close to (but not at) a zero
crossing, adiabaticity is violated, which can be interpreted as particle
production. Let us expand the cosine close to $mt=\pi/2$. It is convenient to
introduce

$\displaystyle k_{*}=\left(\sqrt{\lambda_{\phi h}\over
2}m\Phi\right)^{1/2}\;,$ $\displaystyle\tau=\left(mt-{\pi\over
2}\right)\,{k_{*}\over m}\;,$ $\displaystyle\kappa={k/a\over k_{*}}\;.$ (114)

In terms of these variables, the equation of motion takes the form
${d^{2}X_{k}\over d\tau^{2}}+(\kappa^{2}+\tau^{2})X_{k}=0\;,$ where higher
order terms in $\tau$ have been neglected. Particle creation takes place for

${\tau\over(\kappa^{2}+\tau^{2})^{3/2}}\geq 1\;,$ (115)

which can happen only for $\kappa\leq 1$ and $\tau\leq 1$. This implies, in
particular, that only particles with comoving momenta below $k_{*}$ can be
created. Parametric resonance is efficient for $k_{*}\gg m$. Let us now
consider this process in more detail neglecting expansion of the Universe.

####  5.1.1 Particle creation in a parabolic potential

Consider a semiclassical field $\chi$ with the equation of motion

${d^{2}\chi\over d\tau^{2}}+(\kappa^{2}+\tau^{2})\chi=0\;.$ (116)

The problem is equivalent to that of wave scattering in a $negative$ parabolic
potential [113]. The WKB solution to the above equation reads

$\chi_{\pm}={1\over(\kappa^{2}+\tau^{2})^{1/4}}\exp\left(\pm
i\int^{\tau}\sqrt{\kappa^{2}+\tau^{2}}\,d\tau\right)\;,$ (117)

which can be analytically continued to complex $\tau$. At sufficiently large
$|\tau|$, we may expand these functions in $\kappa/\tau$,

$\chi_{\pm}\simeq\tau^{-{1\over 2}\pm{1\over
2}i\kappa^{2}}\exp\left(\pm{i\tau^{2}\over 2}\right)\;.$ (118)

Note that our approximate description of the resonance makes sense when $\tau$
is large enough for the WKB method to be applicable, $\tau>1$, but not too
large such that higher order terms in $\omega(\tau)$ are subdominant, $\tau\ll
k_{*}/m$. In terms of $\Delta t=t-\pi/(2m)$, this requires

$m^{-1}\gg\Delta t\gg k_{*}^{-1}\;,$ (119)

which is a large range if the resonance is efficient. For the purposes of this
section, we may use asymptotic expressions for $\tau\gg 1$.

Figure 5: Contours for analytic continuation of the WKB solutions. The circle
radius is much greater than 1.

Suppose we start with $\chi_{+}$ at $\tau\rightarrow-\infty$. At late times,
$\tau\rightarrow+\infty$, the solution is a linear combination of $\chi_{+}$
and $\chi_{-}$ such that

$A_{+}\chi_{+}\rightarrow B_{+}\chi_{+}+C_{+}\chi_{-}\;,$ (120)

where $A_{\pm},B_{\pm},C_{\pm}$ are some constants. Analogously,

$A_{-}\chi_{-}\rightarrow B_{-}\chi_{-}+C_{-}\chi_{+}\;.$ (121)

Since the WKB approximation works at sufficiently large $|\tau|$ in the
complex plane, one can use a trick from quantum mechanics [113]. Starting at
$\tau\ll-1$, we can analytically continue our solution along an appropriate
contour. For complex $\tau$, the exponential in (118) develops a real part.
Since the WKB approximation corresponds to an expansion in $\hbar$ which
retains the two leading terms, the exponentially suppressed contribution
compared to the leading one must be dropped in the process of analytic
continuation. This procedure is sensitive to the contour choice, in
particular, how the real axis is approached. Let us use the polar coordinates

$\tau=\rho e^{i\phi}\;,$ (122)

where $\phi$ varies from $0$ to $\pi$ from right to left in the upper half
plane. In order to relate $A_{+}$ and $B_{+}$, we can use the “+” contour in
Fig. 5. In this case, the phase varies from 0 to $-\pi$ from right to left,

$\tau\Big{|}_{\rm right}\rightarrow\tau\Big{|}_{\rm left}\;e^{-i\pi}.$ (123)

The $\chi_{+}$ solution is exponentially larger than $\chi_{-}$ as
$\phi\rightarrow 0_{-}$, thus the $C_{+}$ term must be dropped. The $B_{+}$
and $A_{+}$ terms match if

$B_{+}=-iA_{+}e^{-\pi\kappa^{2}/2}\;.$ (124)

Similarly, the “-” contour is used to determine $B_{-}$, in which case
$\tau\Big{|}_{\rm right}\rightarrow\tau\Big{|}_{\rm left}\;e^{i\pi}$ and

$B_{-}=iA_{-}e^{-\pi\kappa^{2}/2}\;.$ (125)

The $C_{\pm}$ terms are lost in this process, but they can be found via the
Wronskian

$W\equiv\dot{\chi}\chi^{*}-\chi\dot{\chi}^{*}\;.$ (126)

It is easy to see that $\dot{W}=0$ for real $\tau$ and therefore $W$ is
conserved. For the linear combination $B_{+}\chi_{+}+C_{+}\chi_{-}$, the
leading term in $W$ at large $|\tau|$ is
$2i\tau\bigl{(}|B_{+}|^{2}-|C_{+}|^{2}\bigr{)}|\chi_{\pm}|^{2}$. This implies

$|A_{+}|^{2}=|C_{+}|^{2}-|B_{+}|^{2}\;.$ (127)

In quantum mechanics, this can be interpreted as “flux” conservation since the
$A_{+}$ and $C_{+}$ terms correspond to incident waves while the $B_{+}$ term
corresponds to the reflected wave [113].131313In our convention, the direction
of propagation is determined by the phase decrease along the real axis.
Similarly,

$|A_{-}|^{2}=|C_{-}|^{2}-|B_{-}|^{2}\;.$ (128)

Thus,

$C_{\pm}=\sqrt{1+e^{-\pi\kappa^{2}}}\,e^{i\alpha_{\pm}}\,A_{\pm}\;,$ (129)

where $\alpha_{\pm}$ are some phases. For a general initial configuration, by
linearity we have

$A_{+}\chi_{+}+A_{-}\chi_{-}\rightarrow(B_{+}+C_{-})\chi_{+}+(C_{+}+B_{-})\chi_{-}\;.$
(130)

The relations among the coefficients remain the same as above. Indeed, Eq. 116
has a unique solution when $\chi$ and its derivative are fixed at some
$\tau_{0}\ll-1$. This is equivalent to fixing $A_{+}$ and $A_{-}$. When one of
them vanishes, we can use the above considerations to find $B_{\pm},C_{\pm}$.
Linearity of the differential equation and the initial conditions allows us to
add the solutions, which makes sure that the right hand side of (130)
satisfies Eq. 116 with the right initial conditions.

Having determined how the amplitude changes upon passing the non–adiabatic
region, we can interpret the result as particle creation. Our solution
corresponds (approximately) to a collection of harmonic oscillators with
frequency $\omega=\sqrt{\kappa^{2}+\tau^{2}}$. Since $\omega$ changes slowly
far from the origin, the corresponding occupation numbers $n$ are constant in
the asymptotic regions. They are found through the harmonic oscillator
relation

${\cal E}=(n+1/2)\,\omega\;,$ (131)

where ${\cal E}$ is the energy stored in $\chi$,

${\cal E}={1\over 2}|\dot{\chi}|^{2}+{1\over 2}\omega^{2}|\chi|^{2}$ (132)

averaged over a sufficiently large number of oscillations. (Here the dot
denotes differentiation with respect to $\tau$.) We thus have

$n+{1\over 2}={{\cal E}\over\omega}\simeq\omega|\chi|^{2}$ (133)

at large $|\tau|$. We can now compute the change in the occupation number upon
passing through the non–adiabatic region,

${{n}_{\rm out}+1/2\over{n}_{\rm
in}+1/2}\simeq{|B_{+}+C_{-}|^{2}+|B_{-}+C_{+}|^{2}\over|A_{+}|^{2}+|A_{-}|^{2}}\;.$
(134)

Constancy of the Wronskian requires Re $B_{+}C_{-}^{*}=$Re $B_{-}C_{+}^{*}$,
which means

${{n}_{\rm out}+1/2\over{n}_{\rm in}+1/2}\simeq
1+2e^{-\pi\kappa^{2}}+{4|A_{+}||A_{-}|\over|A_{+}|^{2}+|A_{-}|^{2}}\,e^{-\pi\kappa^{2}\over
2}\sqrt{1+e^{-\pi\kappa^{2}}}\,\cos\delta\;,$ (135)

where $\delta$ is some phase. The term $1/2$ in $n+1/2$ represents “vacuum
fluctuations”. It shows that even if one starts initially with no particles,
$A_{-}=0$, non–zero $|\chi|$ results in particle creation according to the
above formula. Every time the system passes through $\tau=0$, particles get
created and “flux conservation” requires

$\bigl{|}A_{+}^{(0)}\bigr{|}^{2}=\bigl{|}A_{+}^{(1)}\bigr{|}^{2}-\bigl{|}A_{-}^{(1)}\bigr{|}^{2}=\bigl{|}A_{+}^{(2)}\bigr{|}^{2}-\bigl{|}A_{-}^{(2)}\bigr{|}^{2}=...,$
(136)

where the superscript denotes the number of zero crossings. Both $A_{+}$ and
$A_{-}$ grow with time reaching $|A_{+}|\simeq|A_{-}|\gg|A_{+}^{(0)}|$. The
particle number grows exponentially,

$n={1\over 2}\,e^{2\pi\mu N}~{}~{},~{}~{}\mu\simeq{1\over
2\pi}\ln\Bigl{(}1+2e^{-\pi\kappa^{2}}+2\cos\delta\,e^{-\pi\kappa^{2}\over
2}\;\sqrt{1+e^{-\pi\kappa^{2}}}\Bigr{)}\,,$ (137)

where $N\gg 1$ is the number of zero crossings. Here we take $n=1/2$ as the
initial value. The Floquet exponent $\mu$ has the maximal value of about 0.28,
at $\kappa=0$ and $\delta=0$. Due to the presence of the phase $\delta$, the
particle number may also decrease, $\mu<0$. However, if $\delta$ takes on
random values, the particle number grows on the average. For instance, at
$\kappa=0$, $\mu>0$ if $|\delta|<3\pi/4$, so the particle number increase is
significantly more likely. Averaging over $\delta$ gives

$\bar{\mu}\simeq{1\over 2\pi}\ln\Bigl{(}1+2e^{-\pi\kappa^{2}}\Bigr{)}\,.$
(138)

The particle number grows exponentially, which can have interesting
implications for vacuum stability.

###  5.2 Parametric resonance and vacuum stability

Let us apply the above quantum mechanical approach to the Higgs field in QFT
and study its implications for vacuum stability.

####  5.2.1 Basics

The Heisenberg representation for the Higgs field is given by
[114],[115],[116],[117]

$\hat{h}={1\over(2\pi)^{3/2}}\int d^{3}k\left(\hat{a}_{k}h_{k}(t)e^{-i{\bf
k\cdot x}}+\hat{a}_{k}^{\dagger}h_{k}(t)^{*}e^{i{\bf k\cdot x}}\right)\;,$
(139)

where $\hat{a}_{k}$ and $\hat{a}_{k}^{\dagger}$ are the (time–independent)
annihilation and creation operators, respectively, with the commutation
relation
$\left[\hat{a}_{k},\hat{a}_{k^{\prime}}^{\dagger}\right]=\delta^{(3)}({\bf
k}-{\bf k^{\prime}})$ and other commutators vanishing. The comoving frame
momentum ${\bf k}$ is related to the physical momentum ${\bf p}$ by ${\bf
p}={\bf k}/a$. The rescaled momentum modes $X_{k}\equiv a^{3/2}h_{k}$ satisfy

$\ddot{X}_{k}+\omega_{k}^{2}X_{k}=0\;,$ (140) $\omega_{k}^{2}(t)={k^{2}\over
a^{2}}+{1\over 2}\lambda_{\phi
h}\Phi^{2}(t)\cos^{2}mt+3\lambda_{h}a^{-3}\langle X^{2}\rangle+\left({3\over
2}\right)^{2}wH^{2}\;,$ (141)

where $w$ is the equation of state parameter of the Universe,
$w=p/\rho=-\left(1+{2\dot{H}\over 3H^{2}}\right)$. We have used the Hartree
approximation, that is, we have split $h(x)$ into the average field and the
fluctuations, and averaged over the latter: $h^{4}\rightarrow 6h^{2}\langle
h^{2}\rangle$. The last term in (141) is small during the resonance, $\ll
m^{2}$, and can be omitted. Note also that $w=0$ as long as the energy density
is dominated by the non–relativistic inflaton.

The WKB solution for the Higgs momentum modes is given by

$X_{k}={\alpha_{k}\over\sqrt{2\omega_{k}}}\,e^{-i\int\omega_{k}dt}+{\beta_{k}\over\sqrt{2\omega_{k}}}\,e^{i\int\omega_{k}dt}\;,$
(142)

where $\alpha_{k},\beta_{k}$ are coefficients normalized as
$|\alpha_{k}|^{2}-|\beta_{k}|^{2}=1$, which are constant in the adiabatic
regime and change only close to the inflaton zero crossing. They can be
identified with the coefficients of the Bogolyubov transformation [118]

$\hat{A}_{k}(t)=\alpha_{k}\hat{a}_{k}+\beta_{k}^{*}\hat{a}_{-k}^{\dagger}\;,$
(143)

which describes particle creation by a time dependent background. In terms of
the time dependent annihilation and creation operators
$\hat{A}_{k}(t),\hat{A}_{k}^{\dagger}(t)$,

$\hat{h}={1\over(2\pi)^{3/2}}\int d^{3}k\left(\hat{A}_{k}(t)g_{k}(t)e^{-i{\bf
k\cdot x}}+\hat{A}_{k}^{\dagger}(t)g_{k}(t)^{*}e^{i{\bf k\cdot x}}\right)\;,$
(144)

with $g_{k}(t)=1/\sqrt{2\omega_{k}a^{3}}\exp(-i\int\omega_{k}dt)$. These
operators satisfy the standard commutation relations
$\left[\hat{A}_{k},\hat{A}_{k^{\prime}}^{\dagger}\right]=\delta^{(3)}({\bf
k}-{\bf k^{\prime}})$, etc. and allow one to define the particle number
operator [114],[115]

$\hat{n}_{k}(t)=\hat{A}_{k}^{\dagger}(t)\hat{A}_{k}(t)\;,$ (145)

whose average over the time–independent vacuum $|0\rangle$ gives the particles
density

$n(t)=\int{d^{3}k\over(2\pi a)^{3}}\,\langle
0|\hat{A}_{k}^{\dagger}(t)\hat{A}_{k}(t)|0\rangle=\int{d^{3}k\over(2\pi
a)^{3}}\,|\beta_{k}|^{2}\;.$ (146)

Here the vacuum satisfies $a_{k}|0\rangle$ for all $k$ since there are no
particles initially. Due to spacial translational invariance, the total
3–momentum is conserved and particles are created in pairs with momenta ${\bf
k}$ and ${\bf-k}$.

The occupation numbers can also be found via the harmonic oscillator analogy:
one can simply divide the energy of the ${\bf k}$–mode in the comoving frame
by the energy of a single quantum, $\omega_{k}$, and subtract the vacuum
contribution,

$n_{k}={\omega_{k}\over
2}\,\left({|\dot{X}_{k}|^{2}\over\omega_{k}^{2}}+|X_{k}|^{2}\right)-{1\over
2}\simeq|\beta_{k}|^{2}\;,$ (147)

where the terms proportional to $\dot{\omega}_{k}$ have been dropped in the
adiabatic approximation. Initially there are no Higgs particles in our system,
so $\alpha_{k}=1$ and $\beta_{k}=0$.

The vacuum expectation value of $h^{2}$, that is, the variance, is given by

$\langle h^{2}\rangle={1\over(2\pi a)^{3}}\int
d^{3}k\;|X_{k}|^{2}\simeq{1\over(2\pi a)^{3}}\int
d^{3}k\;{n_{k}\over\omega_{k}}\,.$ (148)

Clearly, it grows when particle production is efficient indicating large
fluctuations of the Higgs field. These fluctuations may lead to vacuum
destabilization [119].

The variance and the energy density are finite in this adiabatic approach: the
vacuum contribution corresponding to $n_{k}=1/2$ is subtracted. The resonance
excites particles with momenta up to $k_{*}$ and the corresponding occupation
numbers are large, so the quanta with momenta beyond $k_{*}$ do not play any
role and can be “subtracted”, while the vacuum contribution to $n_{k}$ for
$k<k_{*}$ is negligible.

####  5.2.2 Mathieu equation

For our purposes, the Higgs field can be treated semiclassically. Due to the
resonance, the occupation numbers $n_{k}$ become large quickly and the Higgs
dynamics can be extracted from classical equations of motion.

Introducing

$z\equiv{1\over 2}mt\;,$ (149)

we can rewrite the equation of motion in the form

${d^{2}\over dz^{2}}X_{k}+\left[A(k,z)+2q(z)\cos 4z\right]X_{k}=0\;.$ (150)

Here

$\displaystyle q(z)={\lambda_{\phi h}\Phi^{2}(z)\over 2m^{2}}\;,$
$\displaystyle A(k,z)=\left(2{k\over
am}\right)^{2}+2q(z)+{12\lambda_{h}a^{-3}\langle X^{2}\rangle\over m^{2}}\;.$
(151)

Note that $\omega^{2}$ is positive semidefinite as long as $\langle
X^{2}\rangle$ is negligible. In the limit of slow Universe expansion and
$\langle X^{2}\rangle\rightarrow 0$, coefficients $q$ and $A$ are constant. We
thus obtain the Mathieu equation which describes an oscillator with a
periodically changing frequency. It is intuitively clear that such a system
can exhibit resonant behavior. In particular, $q\gg 1$ corresponds to the
broad resonance regime as opposed to the narrow resonance with $q\ll 1$. In
the former case, particles within a large range of momenta get created.

Since the Mathieu equation is homogeneous, the initial value of $X_{k}$ must
be non–zero for particle production to occur. Such initial conditions are
provided by the quantum fluctuations. In practice, one can simulate these
classically with a Rayleigh probability distribution
$P(X_{k})\propto\exp(-2\omega_{k}|X_{k}|^{2})$ such that the average value of
$|X_{k}|^{2}$ is exactly $1/(2\omega_{k})$ as in the vacuum [120],[121]. To
regularize the vacuum contribution to various physical quantities, one may set
$X_{k}$ to zero for $k>k_{*}$ as these modes do not get amplified and play no
role in our discussion.

Figure 6: Stability chart of the Mathieu equation. White/dark regions
correspond to unstable/stable solutions.

The solutions to the Mathieu equation demonstrate drastically different
behavior depending on $q$ and $A$. For some of their values, the solutions
grow exponentially, while for others the solutions oscillate in time,

$X_{k}(z)=a_{+}(k,z)e^{\mu_{k}z}+a_{-}(k,z)e^{-\mu_{k}z}\;,$ (152)

where $a_{\pm}$ are periodic in $z$ and $\mu_{k}$ is the Floquet exponent,
which can be purely imaginary or have a real part depending on $q,A$. This
behavior is conveniently represented by the stability chart shown in Fig. 6:
the dark regions correspond to “stable” oscillatory solutions, while in white
regions the solutions grow exponentially. In the latter, $\mu_{k}$ develops a
real part, Re $\mu_{k}\not=0$.

In practise, the resonance is relatively short and intense so the Universe
expansion can be accounted for adiabatically. For $\lambda_{\phi h}\Phi^{2}\gg
m^{2}$ and moderate momenta, both $q$ and $A$ are large initially with
$A\simeq 2q$. Since the inflaton amplitude decreases, $\Phi\propto 1/t$, the
system follows a trajectory in the $(q,A)$ plane that ends at the origin.
Along the way, it goes through periods with exponential growth of $X_{k}$ and
those where $X_{k}$ oscillates. When it reaches the last stability band at
$q\sim 1$, the growth stops. The stability chart also shows that particles
with large momenta corresponding $A\gg q$ are never excited since the
instability bands become very narrow. In fact, the resonance is efficient if
$A-2q\lesssim\sqrt{q}$ such that only the modes satisfying

${k\over a}<k_{*}\sim mq^{1/4}~{}~{},~{}~{}q\gg 1$ (153)

get excited. This agrees with our discussion of particle creation in a
parabolic potential. By the same token, backreaction effects due to $\langle
X^{2}\rangle$ can suppress the resonance.

The Universe expansion has the effect that the resonance becomes stochastic.
Within one inflaton oscillation, the field jumps over many instability bands
and its phase becomes effectively random. The particle number can sometimes
decrease while increasing on the average. The resonant behaviour can be viewed
as a collective effect due to large Bose enhancement of the reaction rates.
Even though the momenta of the created particles redshift, they remain in the
resonant bands long enough for the Bose enhancement to take effect. This is in
contrast to the narrow resonance case.

####  5.2.3 Vacuum destabilization

The broad resonance is active until

$q\simeq 1\;,$ (154)

at which point the growth of $X_{k}$ stops and the occupation numbers $n_{k}$
remain constant. The resulting field variance can be approximated by

$\langle h^{2}\rangle\simeq{1\over(2\pi a)^{3}}\int
d^{3}k\;{n_{k}\over\omega_{k}}\sim{n(t)\over\sqrt{\lambda_{\phi
h}/2}\,|\Phi|}\,,$ (155)

where we use the typical inflaton–dominated frequency
$\omega\sim\sqrt{\lambda_{\phi h}/2}\,|\Phi|$. As discussed earlier, the
occupation numbers grow exponentially during the resonance. Using the steepest
descent method, (146) can be integrated and one finds $n(t)\propto
e^{2\bar{\mu}mt}\,{k_{*}^{3}\over a^{3}\sqrt{mt}}$. Therefore, the Higgs
fluctuations grow exponentially fast during the resonance and can lead to
vacuum destabilization. The most important terms in the Higgs potential are

$V_{h}\simeq{1\over 4}\lambda_{\phi h}h^{2}\phi^{2}+{1\over
4}\lambda_{h}(h)h^{4}\;,$ (156)

where $\lambda_{h}(h)$ turns negative above some critical value which we can
take in the ballpark of $10^{10}$ GeV. Therefore, the barrier separating the
two Higgs vacua is located at

$h_{c}\simeq\sqrt{\lambda_{\phi h}\over|\lambda_{h}|}\,|\phi|\;.$ (157)

It is natural to expect that for Higgs fluctuations greater than $h_{c}$ the
system becomes unstable. The situation is more subtle however: the position of
the barrier is modulated by $\cos mt$ and a more careful analysis is needed.

Figure 7: Time dependence of the Higgs variance during and after parametric
resonance simulated with LATTICEEASY. For $\lambda_{\phi h}>3\times 10^{-8}$,
vacuum destabilization occurs.

The stability condition can be derived as follows. As $\langle h^{2}\rangle$
builds up, the Higgs mass term gets dominated by $3\lambda_{h}\langle
h^{2}\rangle$ close to each inflaton zero crossing. This is the case during a
small time interval

$\Delta t\simeq\sqrt{6|\lambda_{h}|\langle
h^{2}\rangle\over\lambda_{h\phi}\Phi^{2}m^{2}}$ (158)

and the corresponding $m^{2}_{\rm eff}\simeq 3\lambda_{h}\langle
h^{2}\rangle$. The Higgs amplitude grows as $\exp(|m_{\rm eff}|\Delta t)$
during this period. As long as

$|m_{\rm eff}|\Delta t<1,$ (159)

the growth is slow, however if this product exceeds one, the Higgs amplitude
explodes quickly. We are interested in vacuum stability during the resonance,
thus we impose this condition as long as $q\gtrsim 1$. Then $\lambda_{\phi
h}\Phi^{2}\gtrsim 2m^{2}$ and we may rewrite (159) as

${3|\lambda_{h}|\langle h^{2}\rangle\over m^{2}}<1\;.$ (160)

which is not far from the requirement $\sqrt{\langle h^{2}\rangle}<h_{c}$. In
fact, this condition can be obtained directly from $V_{h}$ in Hartree
approximation by requiring ${1\over 4}\lambda_{\phi h}h^{2}\phi^{2}>{3\over
2}\lambda_{h}h^{2}\langle h^{2}\rangle$ at the end of the resonance. The above
inequality leads to the upper bound [119],[123]

$\lambda_{\phi h}\lesssim 3\times 10^{-8}\;$ (161)

for a typical $\lambda_{h}=-10^{-2}$ above the critical scale. Since $\langle
h^{2}\rangle$ is an exponential function of time and, thus, $\lambda_{\phi
h}$, the $\lambda_{h}$–dependence is only logarithmic making this result quite
robust.

This estimate is confirmed by lattice simulations with LATTICEEASY [121]. Fig.
7 shows the time evolution of the Higgs variance in units of $h_{c}$ for a few
Higgs portal couplings. The input parameters are $m=6\times 10^{-5}$ in Planck
units ($M_{\rm Pl}=2.4\times 10^{18}$ GeV) and $\Phi(t=0)=M_{\rm Pl}$. We see
that $\lambda_{\phi h}=3.5\times 10^{-8}$ and $4.5\times 10^{-8}$ lead to
vacuum destabilization, that is, $\langle h^{2}\rangle$ blows up before the
resonance ends at $mt\sim 40-50$. The corresponding critical $\sqrt{\langle
h^{2}\rangle}$ is not far from $h_{c}$, as expected. We also observe that the
strength of the resonance depends non–linearly on $\lambda_{\phi h}$: for
$\lambda_{\phi h}=3.5\times 10^{-8}$, the variance grows faster than it does
for $\lambda_{\phi h}=4.5\times 10^{-8}$.

It is important to note that the lattice simulations do not resort to the
Hartree approximation, which can make a significant impact on the outcome. In
the presence of quartic interactions such as $h^{4}$, the equations of motion
for the different momentum modes do not decouple which makes the problem
impossible to treat analytically (beyond simple approximations). Due to the
coupling between the different modes, the resonance usually proceeds more
violently compared to the Hartree limit. A detailed discussion of lattice
simulations of parametric resonance can be found in [122].

###  5.3 Tachyonic resonance

The Higgs–inflaton interaction contains, in general, a trilinear term $\phi
h^{2}$. During preheating, this induces a Higgs mass term whose sign
alternates in time leading to efficient particle creation [123]. This is in
contrast to the parametric resonance case, where the sign is fixed. The
particle creation process can be analyzed semiclassically, as before. With the
trilinear term, the equation of motion for the Higgs momentum modes becomes

${d^{2}\over dz^{2}}X_{k}+\left[A(k,z)+2p(z)\cos 2z+2q(z)\cos
4z\right]X_{k}=0\;,$ (162)

where

$p(z)={2\sigma_{\phi h}\Phi(z)\over m^{2}}$ (163)

and the other parameters are defined in (151). In the limit of slow Universe
expansion, this is known as the Whittaker–Hill equation.

Figure 8: Examples of 2d stability charts of the Whittaker–Hill equation for
$q=0$ and $q=1$. The dark/white regions correspond to stable/unstable
solutions. The contours show the magnitude of the real part of the Floquet
exponent (see Eq. 152). The figure is from Ref. [123] © CC–BY.

The behaviour of the solutions of the Whittaker–Hill equation depends on $p,q$
and $A$. Since the Higgs mass is a periodic function of time (albeit with two
periods), the solution can be written in terms of the Floquet exponent as in
Eq. 152. The corresponding stability charts for fixed $q$ are shown in Fig. 8.
They exhibit a rather complicated pattern compared to that of the Mathieu
equation.

For most excited Higgs modes, one may take $A\simeq 2q$ as before. Then, one
finds that the end of the resonance corresponds to $|p|<1,q<0.5$ with the
boundary of the last stability region described by

$q\simeq 0.5\,(1-|p|)\;.$ (164)

Note that $p$ decreases slower with time than $q$ does, therefore at late
times the tachyonic resonance dominates.

For a sufficiently large initial $p$, the system stays long enough in the
unstable bands for the vacuum to be destabilized. Hence, stability imposes an
upper bound $\sigma_{\phi h}$. An estimate of this bound can be obtained as
follows. As we saw in the previous subsection, one expects that as long as the
$\sigma_{\phi h}$–term is greater than $|\lambda_{h}|h^{4}/4$, the system
remains stable although the variance grows rapidly. In Hartree approximation,
this requires

$\langle h^{2}\rangle\lesssim{|\sigma_{\phi h}|\Phi\over 3|\lambda_{h}|}\;.$
(165)

If the fluctuations do not reach the critical value by the end of the
resonance, no destabilization occurs. Since it ends at $|p|\sim 1$, we can
replace the above condition with

${6|\lambda_{h}|\langle h^{2}\rangle\over m^{2}}\lesssim 1\;.$ (166)

The Higgs variance at the end of the resonance can be estimated as

$\langle h^{2}\rangle\simeq{k_{*}^{2}\Delta k_{*}\over
a^{3}}\,{n_{k_{*}}\over\omega_{k_{*}}}\;,$ (167)

where $k_{*}$ is the characteristic momentum and $\Delta k_{*}\sim k_{*}/2$ is
the width of the resonant band. Towards the end of the resonance $k_{*}\sim
m$, while $\omega_{k_{*}}$ can be approximated by the inflaton–induced term
$\omega_{k_{*}}\sim\sqrt{|\sigma_{\phi h}\Phi|}$. Further, the occupation
number $n_{k_{*}}$ is an exponential function of time $\sim\exp(\mu_{*}mt)$
and the duration of the resonance is dictated by $|p|\gtrsim 1$. Putting all
the ingredients together, one finds [123]

$|\sigma_{\phi h}|\lesssim 10^{9}\;{\rm GeV}\;,$ (168)

with the $\lambda_{h}$–dependence being only logarithmic.

Figure 9:  Upper bound on $\sigma_{\phi h}$ from vacuum destabilization as a
function of the Higgs–inflaton quartic coupling. The red dots correspond to
the simulation results obtained with LATTICEEASY. The figure is from Ref.
[123] © CC–BY.

This estimate gives the right ballpark of the bound on $\sigma_{\phi h}$, as
lattice simulations show [123]. However, the true bound exhibits a significant
non–linear dependence on $\lambda_{\phi h}$ due to a complicated stability
band structure of the Whittaker–Hill equation. This is shown in Fig. 9. The
initial conditions for the simulation are described in the previous
subsection. Note that the range of $\lambda_{\phi h}$ in the figure is such
that this coupling does not lead to destabilization by itself, so the
trilinear term is the main driver of the $\langle h^{2}\rangle$ growth.

###  5.4 Effect of the non–minimal Higgs–gravity coupling

The non–minimal coupling to gravity also has an important effect on vacuum
stability [112]. Suppose we add to the Lagrangian the term

$\Delta{\cal L}={1\over 2}\sqrt{-\hat{g}}\,\xi h^{2}\hat{R}\;,$ (169)

where $\hat{R}$ is the scalar curvature in the Jordan frame. This coupling is
generated radiatively even if absent at tree level. To go over to the Einstein
frame, one performs a conformal metric transformation

$g^{\mu\nu}=\Omega^{-1}\hat{g}^{\mu\nu}~{}~{},~{}~{}\Omega=1-\xi{h^{2}\over
M_{\rm Pl}^{2}}\;,$ (170)

where $\hat{g}^{\mu\nu}$ is the Jordan frame metric. This transformation
modifies the kinetic terms and the potential such that in the Einstein frame
one finds

${\cal L}_{\rm sc}=\sqrt{-g}\left({1\over
2\Omega}\,\partial_{\mu}\phi\partial^{\mu}\phi+{1\over 2}\,{6(\xi
h)^{2}+\Omega\over\Omega^{2}}\,\partial_{\mu}h\partial^{\mu}h-{V(\phi,h)\over\Omega^{2}}\right)\;.$
(171)

For the Higgs field values far below the Planck scale,

$|\xi|{h^{2}\over M_{\rm Pl}^{2}}~{},~{}\xi^{2}{h^{2}\over M_{\rm Pl}^{2}}\ll
1\;,$ (172)

we may expand the Lagrangian in powers of $h/M_{\rm Pl}$. One then finds that
the canonically normalized Higgs field $h_{c}$ is given by

$h_{c}\simeq h\left[1+\left(\xi+{1\over 6}\right)\,\xi\,{h^{2}\over M_{\rm
Pl}^{2}}\right]\;,$ (173)

and the potential becomes [124]

$V\simeq{1\over 2}m^{2}\phi^{2}+\left({1\over 4}\lambda_{\phi
h}+\xi{m^{2}\over M_{\rm Pl}^{2}}\right)\phi^{2}h_{c}^{2}+{1\over
2}\sigma_{\phi h}\phi h_{c}^{2}+{1\over 4}h_{c}^{2}\;,$ (174)

where we are keeping terms up to fourth order in $h_{c}$. We see that the
Higgs portal coupling $\phi^{2}h_{c}^{2}$ is generated via the non–minimal
coupling to gravity when $\lambda_{\phi h}$ is set to zero. Even though $\xi
m^{2}/M_{\rm Pl}^{2}$ is Planck–suppressed, $m$ is large enough to make it
comparable to $\lambda_{\phi h}$ considered in the previous subsections.

The equation of motion for $h_{c}$ reads

$\ddot{h}_{c}+3H\dot{h}_{c}-\partial_{i}\partial^{i}h_{c}+\left[\left({1\over
2}\lambda_{\phi h}+2\xi{m^{2}\over M_{\rm
Pl}^{2}}\right)\phi^{2}-\xi{\dot{\phi}^{2}\over M_{\rm Pl}^{2}}+\sigma_{\phi
h}\phi\right]h_{c}+\lambda_{h}h_{c}^{3}=0\;.$ (175)

The impact of the non–minimal coupling to gravity amounts to the presence of
the two Planck–suppressed terms proportional to $\xi\phi^{2}$ and
$\xi\dot{\phi}^{2}$. Their effect on the Higgs dynamics can be very
significant since, for $\phi\propto\cos mt$, their size is determined by $\xi
m^{2}/M_{\rm Pl}^{2}\sim\xi\times 10^{-10}$. In Hartree approximation, the
parameters of the Mathieu equation in Eq. 151 get shifted according to

$\displaystyle q(z)=\left(\lambda_{\phi h}+6\xi\,{m^{2}\over M_{\rm
Pl}^{2}}\right)\,{\Phi^{2}(z)\over 2m^{2}}\;,$ $\displaystyle
A(k,z)=\left(2{k\over am}\right)^{2}+2q(z)-4\xi\,{m^{2}\over M_{\rm
Pl}^{2}}\,{\Phi^{2}(z)\over m^{2}}+{12\lambda_{h}a^{-3}\langle
X^{2}\rangle\over m^{2}}\;.$ (176)

For $k\simeq 0$ and $\langle h_{c}^{2}\rangle\simeq 0$, the relation $A=2q$ no
longer holds. Due to the extra contribution to $A$, the effective mass squared
can turn negative depending on the sign of $\xi$ and therefore the resonance
can be tachyonic. For a negative $\xi$, the effect is instead stabilizing:
increasing $A$ at fixed $q$ brings one into a stable region. The $p$–parameter
of the Whittaker–Hill equation is not affected by the non–minimal coupling to
gravity, so we may set $p=0$ for the present discussion.

Figure 10:  Vacuum stability region in the $(q,A)$ plane. The initial
conditions corresponding to points in the shaded region ensure a stable vacuum
throughout preheating with $\lambda_{h}(h_{c})=-10^{-2}$. The red dots are
obtained with LATTICEEASY. The figure is from Ref. [124] © IOP Publishing Ltd
and SISSA Medialab Srl. Reproduced by permission of IOP Publishing. All rights
reserved.

Vacuum stability is controlled mainly by the behaviour of $X_{k}$ for moderate
momenta, which can be neglected in the expression for $A_{k}$. Fig. 10 shows
the region in the $(q,A)$ plane in which the EW vacuum remains stable
throughout preheating. That is, the boundary of this region determines the
maximal $|q|$ allowed for a given $A$ at the initial time. Since it is above
the $A=2q$ line, the required $\xi$ is negative. For a fixed $\lambda_{\phi
h}$, it cannot be too large in magnitude such that $A$ remains positive. As a
result, some cancellation between the $\lambda_{\phi h}$– and
$\xi$–contributions is required, unless both of them are small. Allowing for
such cancellations, one finds that $|\xi|$ values up to $10^{4}$ are in
principle consistent with vacuum stability [124]. In the absence the Higgs
portal coupling, the stability constraint is $\xi\lesssim 10$ [119],[127].

###  5.5 Discussion

The obtained bounds on the Higgs–inflaton couplings are quite robust since
they depend on $\lambda_{h}$ only logarithmically. The SM instability scale
$\Lambda_{\rm SM}$, which is close to $h_{\rm max}$ of Section 4.1, appears in
these calculations implicitly: we are assuming that the Higgs quartic coupling
at the scale $h_{c}\simeq\sqrt{\lambda_{\phi h}/|\lambda_{h}|}\,\Phi$ (or
$h_{c}\simeq\sqrt{\sigma_{\phi h}\Phi/|\lambda_{h}|}$ in the trilinear case)
during the resonance is negative and the destabilization occurs when

$\displaystyle\sqrt{\langle h^{2}\rangle}$ $\displaystyle\gtrsim$
$\displaystyle h_{c}\;,$ $\displaystyle h_{c}$ $\displaystyle\gg$
$\displaystyle\Lambda_{\rm SM}\;.$ (177)

Clearly, this is only possible if $\Lambda_{\rm SM}$ is below the inflationary
scale. As $|\lambda_{h}|$ decreases and approaches zero, $h_{c}$ becomes so
large that no fluctuation can reach it and vacuum stability gets restored.

In our analysis, we have taken the benchmark value $\lambda_{h}=-10^{-2}$.
Then, according to Fig. 7, $h_{c}\sim 10^{14}$ GeV and our considerations
apply as long as

$\Lambda_{\rm SM}\ll 10^{14}\;{\rm GeV}\;.$ (178)

In the present setting, one finds that $\sqrt{\langle h^{2}\rangle}<10^{15}$
GeV for any $\lambda_{\phi h}$. Therefore, for $\Lambda_{\rm SM}$ of this
order or above, the system remains stable.

There are a number of effects that we have neglected. In particular, the Higgs
decay into the top quarks dilutes the Higgs population and weakens the
resonance. This decay is kinematically allowed since the Higgs mass is
dominated by the large inflaton–induced term, while the Higgs VEV is small
making the fermions effectively massless. Although this makes an impact on the
dynamics of the resonance, the stability bounds remain essentially unaffected
[123]. Further, we have included a single degree of freedom for the Higgs
field à la unitary gauge. At $\langle h\rangle=0$, there are 4 Higgs degrees
of freedom. Again, this complication does not affect the above results in any
significant way.

There is a degree of model dependence in our analysis. To illustrate the
effect of the Higgs portal couplings on vacuum stability, we have chosen the
quadratic inflaton potential during preheating. This is not directly connected
to the shape of the inflaton potential during inflation and various
inflationary models can give the same small–field limit. Therefore, our
results apply more generally. On the other hand, the specifics of
destabilization are sensitive to the shape of the potential during preheating.
In some cases, the potential is dominated by the quartic term during most of
the preheating period. Then, the resonance is described by the Lamé equation
for which the instability band structure and backreaction effects are rather
subtle [108],[125]. This applies, in particular, to preheating in Higgs
inflation [126]. Other related work can be found in [128],[129],[130].

While an exhaustive survey of all the possibilities is still lacking, the
vacuum stability analysis for inflation driven by a non–minimal coupling to
gravity has been carried out in [131]. The conclusion is qualitatively similar
to what we find in the quadratic case: there is a coupling range, where
stability is achieved both during and after inflation. The main relevant
parameter is $\lambda_{\phi h}/\xi_{\phi}$ such that for $\lambda_{\phi
h}/\xi_{\phi}$ between $10^{-10}$ and $10^{-8}$ no destabilization occurs,
although there is also some dependence on $\xi_{\phi}$.

##  6 Stabilizing the Higgs potential via scalar mixing

The couplings that stabilize the Higgs potential during inflation can have an
opposite effect during preheating. Although there is a range of couplings
where stability is achieved in both epochs, this tendency motivates one to
look for further solutions.

The simplest possibility is to modify the Higgs potential in a
time–independent manner. As argued in Section 2, one expects a Higgs–inflaton
mixing on general grounds. This, among other things, modifies the Higgs
self–coupling which, within the Standard Model, is inferred from the Higgs
mass of 125 GeV. Even though such a correction is small, it can completely
change the asymptotic behaviour of the Higgs potential [132],[133].

Let us first consider the simple case of a $Z_{2}$ symmetric scalar potential
where an SM singlet develops a large VEV.

###  6.1 $Z_{2}$ symmetric potential with a large singlet VEV

In this Section, we illustrate the sensitivity of the Higgs potential
stability to a tiny Higgs–singlet mixing. Consider the scalar potential (14),
where $\phi$ is a general SM singlet and not necessarily the inflaton. Suppose
$w\gg v$, then to leading order in $v^{2}/w^{2}$ we have

$\displaystyle m_{1}^{2}\simeq 2\left(\lambda_{h}-{\lambda_{\phi h}^{2}\over
4\lambda_{\phi}}\right)v^{2}\,$ (179) $\displaystyle m_{2}^{2}\simeq
2\lambda_{\phi}w^{2}\;,$

and the mixing angle is given by

$\theta\simeq{\lambda_{\phi h}v\over 2\lambda_{\phi}w}\;.$ (180)

The main feature that we observe is that, while the mixing angle is
suppressed, the lighter mass eigenvalue receives a significant correction.
Since $m_{1}=125$ GeV,

$\lambda_{h}-{\lambda_{\phi h}^{2}\over 4\lambda_{\phi}}\simeq 0.13\;.$ (181)

The resulting $\lambda_{h}$ is $larger$ than that in the Standard Model. This
is a tree–level effect which affects the boundary condition for the subsequent
RG evolution of the Higgs self–coupling. One finds that about a 10% correction
is sufficient to make $\lambda_{h}$ positive all the way to the Planck scale
[132]. It is remarkable that this can be achieved with tiny couplings
$\lambda_{\phi h},\lambda_{\phi}$ as long as $\lambda_{\phi
h}^{2}/\lambda_{\phi}$ is substantial. As a result, the deep minimum in the
Higgs field direction is eliminated and complete scalar potential stability
can be achieved for a wide range of couplings.

It is instructive to look at this system from an effective field theory
viewpoint [133]. If $\phi$ is very heavy, it can be integrated out leaving an
effective Higgs potential. The form of the latter is constrained by gauge
invariance, so it takes on the usual form albeit with a modified
self–coupling. Specifically, let us first trade the mass parameters
$m_{h},m_{\phi}$ for the VEVs using the stationary point condition:

$\displaystyle m_{h}^{2}=-{1\over 2}\lambda_{\phi h}w^{2}-\lambda_{h}v^{2}\,,$
$\displaystyle m_{\phi}^{2}=-{1\over 2}\lambda_{\phi
h}v^{2}-\lambda_{\phi}w^{2}\;.$ (182)

Now, restore all four Higgs components and use the equation of motion for
$\phi$ neglecting the momentum,

$\phi^{2}=-{1\over\lambda_{\phi}}\left(\lambda_{\phi
h}H^{\dagger}H+m_{\phi}^{2}\right)\;.$ (183)

Plugging this back into the scalar potential, one finds

$V=\left(\lambda_{h}-{\lambda_{\phi h}^{2}\over
4\lambda_{\phi}}\right)\,\left(H^{\dagger}H-{v^{2}\over
2}\right)^{2}~{}+~{}{\rm const}\;.$ (184)

We thus recover the usual Higgs potential with a redefined coupling at low
energies. The relation between the Higgs mass and the Higgs self–coupling
remains unaffected. The impact of the singlet amounts to a threshold
correction: it shifts the Higgs self–coupling in the infra–red (IR) theory
with respect to that in the ultra–violet (UV) theory, where both $h$ and
$\phi$ are active degrees of freedom. The singlet may be very heavy as long as
the threshold occurs below the SM instability scale, $m_{2}\ll h_{\rm crit}$.

On the other hand, if the singlet is not too heavy and remains an active
degree of freedom at the TeV scale, the relation (179) implies that the
mass–coupling relation for the Higgs gets modified by the presence of the
singlet [132]. The correction could in principle be as large as 100% which may
be probed by the Higgs self–coupling measurement at the HL-LHC.

###  6.2 General potential and Higgs–inflaton mixing

In general, the Higgs–inflaton potential also contains terms odd under
$\phi\rightarrow-\phi$ as in Eq. 6. This naturally leads to the Higgs–inflaton
mixing even in the absence of the inflaton VEV and thus affects the potential
stability [16].

The scalar potential is generally minimized at a non–zero inflaton VEV
$\langle\phi\rangle=w$. To simplify the analysis, it is convenient to
eliminate it by redefining the inflaton field [20],

$\phi^{\prime}=\phi-w\;,$ (185)

and the dimensionful parameters,141414Note the differences between our
convention and those of Refs. [20],[16].

$\displaystyle
b_{1}^{\prime}=b_{1}+\lambda_{\phi}w^{3}+b_{3}w^{2}+m_{\phi}^{2}w\,,$
$\displaystyle b_{3}^{\prime}=b_{3}+3\lambda_{\phi}w\,,$
$\displaystyle\sigma_{\phi h}^{\prime}=\sigma_{\phi h}+\lambda_{\phi h}w\,,$
$\displaystyle
m_{\phi}^{\prime\,2}=m_{\phi}^{2}+3\lambda_{\phi}w^{2}+2b_{3}w\,,$
$\displaystyle m_{h}^{\prime\,2}=m_{h}^{2}+{1\over 2}\lambda_{\phi
h}w^{2}+\sigma_{\phi h}w\,.$ (186)

The new field has a vanishing VEV, $\langle\phi^{\prime}\rangle=0$. One then
proceeds to minimizing the potential along the lines of Section 2.3. The mass
eigenstates are given by

$\left(\,\begin{matrix}h_{1}\\\
h_{2}\end{matrix}\,\right)=\left(\,\begin{matrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{matrix}\,\right)\left(\,\begin{matrix}h-v\\\
\phi^{\prime}\end{matrix}\,\right)\,.$ (187)

One finds the following relations among the mass eigenvalues $m_{1},m_{2}$ and
$\theta$,

$\displaystyle
2\lambda_{h}v^{2}=m_{1}^{2}\cos^{2}\theta+m_{2}^{2}\sin^{2}\theta\,,$ (188)
$\displaystyle{1\over 2}\lambda_{\phi
h}v^{2}+m_{\phi}^{\prime\,2}=m_{1}^{2}\sin^{2}\theta+m_{2}^{2}\cos^{2}\theta\,,$
(189) $\displaystyle\sigma_{\phi h}^{\prime}v=\dfrac{\sin
2\theta}{2}\left({m_{2}^{2}-m_{1}^{2}}\right)\,.$ (190)

The first relation implies that for $m_{2}>m_{1}$, the Higgs self–coupling is
larger than that in the SM. This leads to a stable Higgs potential for a
significant range of $m_{2}$ and $\theta$. Fig. 11 illustrates the impact of
these parameters on the RG evolution of $\lambda_{h}$. The stabilizing action
is twofold: first, the boundary condition at the top quark mass scale $M_{t}$
changes; second, the Higgs portal coupling contributes positively to the
$\lambda_{h}$–running (see Section 2.1). Unlike in the previous Section, here
we consider substantial mixing angles: the upper bound from experiment is
about 0.3 unless the inflaton is too light or too heavy (see Section 2.3.1).
We see that a rather small mixing of order $10^{-1}$ is sufficient to
stabilize the Higgs potential for a sub–TeV inflaton. Eq. 188 shows that even
a tiny mixing can lead to the desired result for a large enough $m_{2}$. This
is consistent with electroweak precision measurements which impose the bound
on $\theta$ (33) growing stronger only logarithmically with $m_{2}$.

Figure 11:  Higgs quartic coupling scale dependence for different
Higgs–inflaton mixing angles. The figure is from Ref. [16].

Our conclusion is that the Higgs–inflaton mixing, which is expected on general
grounds, can fully stabilize the scalar potential. It is, however, not obvious
that such a mixing is allowed by cosmology, i.e. inflation itself. In the next
subsection, we analyze this issue in detail.

####  6.2.1 Higgs–inflaton mixing vs cosmology

The main ingredient in the following discussion is the energy scale separation
between the Early Universe phenomena and the mixing effects which occur after
the electroweak phase transition. Indeed, in traditional models, inflation and
reheating take place at large field values, while the field mixing discussed
above is relevant when the fields are close to their vacuum values. It is
therefore clear that these phenomena are not $directly$ related.

To illustrate this point let us take the general Higgs portal Lagrangian
including the non–minimal scalar–gravity couplings (4) [16]. At large $h$ and
$\phi$ field values, we may neglect all the dimensionful parameters leaving us
with 5 variables $\lambda_{i}$ and $\xi_{i}$. This is the Higgs portal
inflation set–up discussed in Section 3. Depending on the relations among
these parameters, $h$ and/or $\phi$ can drive inflation. For example,
inflation takes place along the $\phi$ direction for

$\xi_{\phi}\gg\xi_{h}$ (191)

as long as $2\lambda_{\phi}\xi_{h}-\lambda_{\phi h}\xi_{\phi}<0$. The scalar
potential is then stabilized at $\tau\equiv h/\phi=0$. Taking $\xi_{\phi}\gg
1$, the inflationary potential in Planck units (${\rm M_{Pl}}=1$) is

$V_{E}\simeq{\lambda_{\phi}\over 4\xi_{\phi}^{2}}\left(1-e^{-\sqrt{2\over
3}\chi}\right)^{2}\;,$ (192)

with $\chi\simeq\sqrt{3\over 2}\ln\xi_{\phi}\phi^{2}$. The couplings are
subject to a number of constraints. First, the CMB normalization requires
$\lambda_{\phi}/(4\xi_{\phi}^{2})\simeq 10^{-10}$. Second, the scale of
inflation $(\lambda_{\phi}/4\xi_{\phi}^{2})^{1/4}$ should not exceed the
unitarity cutoff $1/\xi_{\phi}$. This implies the following bounds on the
running couplings:

$\xi_{\phi}(H)\lesssim 300~{}~{},~{}~{}\lambda_{\phi}(H)\lesssim 4\times
10^{-5}\;,$ (193)

where we (loosely) identify the running scale with $H$. To avoid large
radiative corrections to the inflaton potential, we also require
$\lambda_{\phi h}^{2}/16\pi^{2}<\lambda_{\phi}$ at the inflationary scale.
This results in $\lambda_{\phi h}(H)\lesssim 10^{-2}$. On the other hand, the
Coleman–Weinberg potential induced by $\sigma_{\phi h}$ is suppressed by
$(\sigma_{\phi h}/\phi)^{2}$ and can be neglected.

As discussed in Section 3, this model predicts the spectral index and the
tensor to scalar ratio that fit the PLANCK data very well. The
postinflationary evolution is also “healthy”: a significant $\lambda_{\phi h}$
allows for efficient Higgs production and subsequent thermalization of the
Higgs–inflaton system [16]. Eventually, due to the $\phi h^{2}$ coupling, the
inflaton decays into the Higgs pairs or other SM states, if the main channel
is kinematically closed. The Higgs–inflaton mixing appears only after the
electroweak phase transition making no significant impact on the Early
Universe cosmology. We therefore conclude that the mixing can have a
stabilizing effect on the Higgs potential while being consistent with standard
cosmology.

##  7 Higgs and singlet condensates

In the Early Universe, light scalars develop a background value comparable to
the Hubble rate [94]. This may apply to the Higgs [134],[135] as well as
scalar dark matter [136],[137]. Let us consider under what circumstances this
is possible.

Given the uncertainty in the top quark mass, one cannot exclude the
possibility that the SM Higgs potential is stable at large field values. If,
in addition, the Higgs–inflaton interaction and the non–minimal Higgs coupling
to gravity are negligible, the Higgs field is light and experiences large
fluctuations during inflation. As discussed in Section 4.1, the evolution of
its long wavelength (“background”) modes is governed by the Langevin equation
[94]

${dh\over dt}+{1\over 3H}\,{dV(h)\over dh}=\zeta(t)\;,$ (194)

where $h=\sqrt{2H^{\dagger}H}$ and $\zeta(t)$ represents random noise,
$\langle\zeta(t)\zeta(t^{\prime})\rangle={H^{3}\over
4\pi^{2}}\,\delta(t-t^{\prime})\;.$ However, unlike in Section 4.1, the value
of the potential plays an important role. The above picture applies if $h$ is
“light” enough in this regime, $V^{\prime\prime}(h)<9H^{2}/4$, such that
super–Hubble fluctuations are generated. The field experiences a random walk
and its various values can be found according to the (late–time) probability
distribution [94]

$\rho(h)=C\;\exp\left(-{2\pi^{2}\lambda_{h}h^{4}\over 3H^{4}}\right)$ (195)

with the variance

$\langle h^{2}\rangle\simeq 0.13\,{H^{2}\over\sqrt{\lambda_{h}}}\;.$ (196)

The corresponding induced Higgs mass squared is given by $3\lambda_{h}\langle
h^{2}\rangle$, which is far below $9H^{2}/4$ as required by self–consistency
of our approximation. Also, the condensate (196) contributes very little to
the energy balance of the Universe since its fraction is suppressed by
$H^{2}/M_{\rm Pl}^{2}$.

If the Higgs initial value is large so that $V^{\prime\prime}(h)>9H^{2}/4$, it
evolves classically to smaller values as long as its energy density does not
affect inflation.151515This is not strictly speaking necessary before the last
60 e–folds of inflation. At the early stage, the energy density can be
dominated by the Higgs field, see the discussion in Section 4.3. Then, its
effective mass becomes small and the above considerations apply.

During inflation, the field is “frozen” in the sense that $\langle
h^{2}\rangle$ remains constant. After inflation, the Hubble rate decreases and
the Higgs evolution gets soon dominated by the classical term. The field
oscillates around $h=0$ in a quartic potential inducing an oscillating mass
term for the SU(2) gauge bosons. This results in resonant production of the
latter and the condensate decays quickly. Lattice simulations show that this
occurs after ${\cal O}(10)$ oscillations [135] (see [138] for the Abelian
case). Since the Higgs energy density is very small, in standard cosmology the
Higgs condensate evaporates without leaving any significant trace.161616In
some cases, the Higgs VEV modulation may lead to kinematic blocking of some of
the reheating channels thereby inducing observable density perturbations
[139],[140].

Similar considerations apply to other light scalars, except these can be
stable and constitute secluded dark matter [141],[142],[143]. Although such
scenarios are subject to strong isocurvature perturbation constraints, they
can be viable for certain parameter choices. Suppose we have a light scalar
$s$ with mass $m_{s}$ and self–interaction $\lambda_{s}s^{4}/4$. At weak
coupling, the field goes through the following main stages in its evolution: