# Low-energy probes of small CMB amplitude in models of radiative Higgs
mechanism

Sunghoon Jung  Kiyoharu Kawana

###### Abstract

The small CMB amplitude $A_{s}\simeq 10^{-9}$ (or, small temperature
fluctuation $\delta T/T\simeq 10^{-5}$) typically requires an unnaturally
small effective coupling of an inflaton $\lambda_{\phi}\sim 10^{-14}$. In
successful models, there usually is extra suppression of the amplitude, e.g.
by large-field inflaton with non-minimal coupling $\xi$, so that
$\lambda_{\phi}$ can be much larger. But $\lambda_{\phi}$ and $\xi$ cannot be
$\sim{\cal O}(1)$ simultaneously; the naturalness burden is shared between
them. We show that the absence of new physics signals at TeV scale may prefer
a more natural size of $\xi\lesssim{\cal O}(1-100)$ with
$\lambda_{\phi}\lesssim{\cal O}(10^{-4}-10^{-8})$, constraining larger $\xi$
with larger $\lambda_{\phi}$ more strongly. This intriguing connection between
low- and high-energy physics is made in the scenarios with $U(1)_{X}$ where
inflaton’s renormalization running also induces Coleman-Weinberg mechanism for
the electroweak symmetry breaking. We particularly work out the prospects of
LHC 13 and 100 TeV $pp$ colliders for probing the parameter space of the small
CMB amplitude.

##  1 Introduction

Although slow-roll is almost an inevitable requirement of inflation, the
observed small temperature fluctuation in the universe (or, more precisely the
small CMB amplitude) is not. Slow-roll inflation could have explained
homogeneity, causality, and the quantum origin of density perturbations
without predicting the small CMB amplitude. The observed $\delta T/T\sim
10^{-5}$ (or $A_{s}\sim 10^{-9}$) [1, 2] is typically translated to
$\lambda_{\phi}\sim 10^{-14}\ll 1$ for an effective quartic potential
description $V=\lambda_{\phi}\phi^{4}/4$ of an inflaton $\phi$. Such a small
coupling is seemingly unnatural even though it is not a necessary prediction
of slow-roll. Why is our universe realized so? Could this small CMB amplitude
be related to other physics?

Anthropically, it was argued that if $\delta T/T$ were larger, there would
have been too much black hole formations; while if smaller, there would have
been too little structures within the age of the universe [3]. It was even
argued that smaller $\delta T/T$ is more likely realized since it usually
requires a flatter inflaton potential, inducing longer inflation and
exponentially more numbers of Hubble patches [4].

Higgs inflation has shed some light in this regard. The measured Higgs boson
mass (combined with the top quark mass) surprisingly implies an almost
vanishing Higgs potential near the Planck scale [5, 6, 7, 8, 9], via the
renormalization group (RG) evolution in the Standard Model (SM). This not only
made the electroweak vacuum possibly being metastable, but also made the case
of the Higgs boson as an inflaton [10, 11, 12, 13] more plausible. As the
first derivative of the Higgs potential is also expected to vanish near the
same scale, some of the slow-roll conditions as well as small potential height
are automatically satisfied. Although Higgs inflation in its minimal form in
the SM is not completely successful (so that generalized with an approximate
inflection point [14, 15, 16, 17, 18], or a non-minimal coupling $\xi$ [19,
20, 21, 22, 23, 24, 25, 26, 27, 28], or various other features [29, 30, 31,
32, 33]), it became noticed that small CMB amplitude may have some connections
with other physics at distant energy scales.

Further, this interesting coincidence between high- and low-energy potential
shapes was attempted to be explained by multiple point principle [34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44] (i.e., why we seem to have almost degenerate
minima at the electroweak and Planck scales) or by classical conformal
invariance at the Planck scale (i.e., why the dimensionful Higgs quadratic
term almost vanishes there). These hypotheses triggered a possibility of all
dimensionful parameters in the SM being induced by RG evolutions. In
particular, electroweak symmetry breaking (EWSB) was successfully induced
starting from even a vanishing Higgs potential (all potential terms vanish at
the Planck scale) just with an additional $U(1)$ gauge symmetry, a charged
fermion, and a symmetry breaking scalar [45, 46]. Also, it was shown that RG
evolution can be used to realize (saddle-point) inflation together with the
spontaneous symmetry breaking of $U(1)_{B-L}$ at a low scale [47, 16, 17, 18].

In this paper, along this light, we study whether small CMB amplitude can have
meaningful connections or correlations with low-energy physics in the models
where $\phi$ inflaton’s renormalization running induces both slow-roll
inflation and radiative breaking of the electroweak symmetry (Coleman-Weinberg
mechanism). These well measured high- and low-energy physics may induce non-
trivial constraints on $\phi$ through perturbative quantum corrections (RG
evolutions). In Sec. 2, we start by introducing our model and describing CW
mechanism, then in Sec. 3 we match this model to simple inflation models, and
in Sec. 4 we interpret low-energy searches to the constraints on the parameter
space of inflation sector that can explain the small CMB amplitude.

##  2 Coleman-Weinberg mechanism with $U(1)_{X}$

Our model consists of Coleman-Weinberg (CW) mechanism for the electroweak
symmetry breaking (EWSB) with vanishing dimensionful parameters at $M_{\rm
Pl}$. This may represent one class of resolutions to the Planck-weak hierarchy
problem.

CW mechanism is realized with an additional $U(1)_{X}$, a linear combination
of $U(1)_{B-L}$ and $U(1)_{Y}$ hypercharge, parameterized by $x$ as

$X\,=\,(B-L)-xY.$ (2.1)

This model in our framework has been worked out in [45] (including RG
equations), with two small but necessary changes discussed below. To
summarize, one generation (for simplicity) of right-handed neutrino $\nu_{R}$
and the SM-singlet scalar $\Phi$ with $X(\Phi)=2$ are introduced, having
Yukawa interactions

${\cal
L}\,\ni\,-y_{\nu}\bar{\ell}\nu_{R}H\,-\,y_{N}\overline{(\nu_{R})^{c}}\nu_{R}\Phi$
(2.2)

and scalar potentials and the portal mixing

$V\,=\,m_{H}^{2}|H|^{2}\,+\,m_{\Phi}^{2}|\Phi|^{2}\,+\,\lambda_{h}|H|^{4}\,+\,\lambda_{\phi}|\Phi|^{4}\,+\,\lambda_{h\phi}|H|^{2}|\Phi|^{2}.$
(2.3)

In this work, quadratic terms vanish at $M_{\rm Pl}$: $m_{H}^{2}(M_{\rm
Pl})=0$ and $m_{\Phi}^{2}(M_{\rm Pl})=0$, and they are not even RG induced at
lower scales (since we work with a mass-independent scheme of dimensional
regularization) but only by spontaneous symmetry breaking as will be
discussed. We use the unitary gauge with $H=h/\sqrt{2}$ and
$\Phi=\phi/\sqrt{2}$.

The one-loop CW effective potential of $\phi$ is

$\displaystyle V(\phi)$ $\displaystyle\,=\,$
$\displaystyle\frac{1}{4}\lambda_{\phi}(M_{\rm
Pl})\phi^{4}+\frac{\phi^{4}}{64\pi^{2}}\left(10\lambda_{\phi}^{2}+48g_{X}^{4}-8y_{N}^{4}\right)\ln\frac{\phi^{2}}{M_{\rm
Pl}^{2}}\,+\,\Delta V.$ (2.4)

This can also be approximated in terms of the running coupling
$\lambda_{\phi}(\phi)$ as

$V(\phi)\,\simeq\,\frac{1}{4}(\lambda_{\phi}(\phi)+C)\phi^{4}+\Delta V,$ (2.5)

with the running

$\lambda_{\phi}(\phi)\,\simeq\,\lambda_{\phi}(M_{\rm
Pl})+\beta_{\lambda_{\phi}}\ln\frac{\phi}{M_{\rm Pl}}.$ (2.6)

We use $\overline{\rm MS}$ scheme $C=0$ so that $\lambda_{\phi}(M_{\rm Pl})$
directly measures the CMB amplitude $A_{s}\propto\lambda_{\phi}(M_{\rm
Pl})$111In [45] focusing on the CW mechanism for the EWSB,
$C=-\frac{25}{12}\beta_{\lambda_{\phi}}(v_{\phi})$ was chosen by the
renormalization condition at low-energy
$\left.\frac{\partial^{4}V}{\partial\phi^{4}}\right|_{\phi=v_{\phi}}=6\lambda_{\phi}(v_{\phi})$.
$C$ effectively only shifts the value of $\lambda_{\phi}$., and $\Delta V$
will be determined soon. $M_{\rm Pl}$ is assumed to be the high-energy
inflation scale, for simplicity. The form in Eq. (2.5) is convenient in
matching the potential at the inflation scale. It is a good approximation to
the correct Eq. (2.4) as field-strength renormalizations (not explicitly
appearing in Eq. (2.4)), denoted by $\cdots$ in

$(4\pi)^{2}\beta_{\lambda_{\phi}}\,=\,20\lambda_{\phi}^{2}+96g_{X}^{4}-16y_{N}^{4}\,+\,\cdots,$
(2.7)

are small ${\cal O}(\lambda_{\phi}g_{X}^{2},\lambda_{\phi}y_{N}^{2})$.

The potential is minimized at $v_{\phi}$ ($U(1)_{X}$ spontaneously broken)
when

$\lambda_{\phi}(v_{\phi})\,=\,-\left(\frac{\beta_{\lambda_{\phi}}}{4}+C\right)\,=\,-\frac{1}{64\pi^{2}}\left(20\lambda_{\phi}^{2}+96g_{X}^{4}-16y_{N}^{4}\right)(v_{\phi}),$
(2.8)

where $C=0$. This requires $\lambda_{\phi}(v_{\phi})\sim
g_{X}^{4},y_{N}^{4}\ll g_{X}^{2},y_{N}^{2}$, and such a hierarchical structure
can be RG induced. Consequently, $v_{\phi}$ is exponentially suppressed
compared to $M_{\rm Pl}$, by dimensional transmutation

$v_{\phi}\,\simeq\,M_{\rm
Pl}\,e^{-\frac{1}{4}}\exp\left(-\frac{4\pi^{2}\lambda_{\phi}(M_{\rm
Pl})}{24g_{X}^{4}(M_{\rm Pl})-4y_{N}^{4}(M_{\rm Pl})}\right),$ (2.9)

resolving the hierarchy problem – it is the CW mechanism. The symmetry
breaking is subsequently transmitted to the SM Higgs potential through the
portal mixing as

$m_{H}^{2}(v_{\phi})\,=\,\frac{1}{2}\lambda_{h\phi}(v_{\phi})v_{\phi}^{2}.$
(2.10)

Thus, the negative $\lambda_{h\phi}(v_{\phi})$ is needed to break the
electroweak symmetry too; such a negative value will also be RG induced (Eq.
(4.2)). The electroweak vev $v_{EW}\equiv\langle h\rangle$ is finally defined
as

$v_{\rm EW}\,=\,\sqrt{-\frac{m_{H}^{2}(v_{\rm EW})}{\lambda_{h}(v_{\rm
EW})}},$ (2.11)

which is required to be 246 GeV.

The potential at the minimum $\phi=v_{\phi}$ is

$V(\phi=v_{\phi})\,=\,\frac{v_{\phi}^{4}}{4}(\lambda_{\phi}(v_{\phi})+C)+\Delta
V\,=\,-\frac{v_{\phi}^{4}}{16}\beta_{\lambda_{\phi}}(v_{\phi})+\Delta V,$
(2.12)

where the second equality used the minimization condition Eq. (2.8) and holds
independently on $C$. The potential is non-zero and negative
($\beta_{\lambda_{\phi}}(v_{\phi})>0$) without $\Delta V$. Thus, we set
$\Delta V=+\frac{v_{\phi}^{4}}{16}\beta_{\lambda_{\phi}}(v_{\phi})$ to make it
zero and avoid unnecessary dark energy. This also necessarily affects the
potential height at the inflation scale, but as will be discussed in the next
section this is negligible at the inflation scale.

By this mechanism, even a strict flatland scenario starting with
$\lambda_{\phi}(M_{\rm Pl})=0$ and $\lambda_{h\phi}(M_{\rm Pl})=0$ can
successfully induce EWSB [45, 46]. $\lambda_{\phi}$ has to be positive before
it breaks $U(1)_{X}$ so that $\beta_{\lambda_{\phi}}<0$ at high scales, but
$\lambda_{\phi}$ must turn to be negative to break $U(1)_{X}$ requiring
$\beta_{\lambda_{\phi}}$ to flip its sign at some intermediate scale. This
flip is achieved by the fine interplay of bosonic $g_{X}$ and fermionic
$y_{N}$ contributions to the beta function

$(4\pi)^{2}\beta_{\lambda_{\phi}}\,\simeq\,96g_{X}^{4}-16y_{N}^{4}$ (2.13)

with their respective running

$(4\pi)^{2}\beta_{g_{X}}\,\simeq\,\left(12-\frac{32}{3}x+\frac{41}{6}x^{2}\right)g_{X}^{3},\qquad(4\pi)^{2}\beta_{y_{N}}\,\simeq\,6y_{N}(y_{N}^{2}-g_{X}^{2}).$
(2.14)

As a result, for almost any $g_{X}(M_{\rm Pl})$, there exists a unique
solution $y_{N}(M_{\rm Pl})$ for successful EWSB. The resulting collider
phenomena depend on the value of $g_{X}(M_{\rm Pl})$. As shown in [45], the
smaller $g_{X}(M_{\rm Pl})$ is, the smaller corresponding $y_{N}(M_{\rm Pl})$
is. And more relevantly, $M_{X}=2g_{X}(v_{\phi})v_{\phi}$ becomes larger and
the Higgs mixing angle smaller (among many observables). But there is no
definite range of such predictions since any value of $g_{X}(M_{\rm Pl})$ can
induce EWSB. This is one crucial difference of our work.

In this paper, above all, we relax the condition $\lambda_{\phi}(M_{\rm
Pl})=0$ in order to also explain inflation with the inflaton $\phi$. As will
be discussed, this brings definite ranges of low-energy predictions, leading
to intimate connection between high- and low-energy physics. [47] considered
such a scenario but with a different focus; [17, 18] without EWSB. Later, we
also relax $\lambda_{h\phi}(M_{\rm Pl})=0$, mainly to explore the dependence
of our conclusion on this model parameter. A wider range of $x$ is allowed now
(see [46] for the allowed narrow range in a flatland), but only $x\sim 1$ will
be considered to avoid too large $X$ charges; models with $x=4/5$ and $2$ can
be obtained from the $SO(10)$ grand unified gauge group.

##  3 Inflation models for interpretation

In this section, we introduce benchmark models of inflation, which are simple
enough to represent a large range of models and allow simple interpretations
of low-energy results. The inflaton potential must also match with the CW
potential in Eq. (2.4) at low energy.

We start with a quartic chaotic inflation (as a warm up and to show basic
features)

$\displaystyle V(\phi)\,=\,\frac{\lambda_{\phi}(\mu=\phi)}{4}\phi^{4}+\Delta
V~{},$ (3.1)

where $\lambda_{\phi}(\mu=\phi)$ is the running quartic coupling evaluated at
$\mu=\phi$. This potential obviously matches with the CW potential Eq. (2.5)
at low energy, again up to small field-strength renormalizations. We first
show that the effect from $\Delta V$ (introduced in Eq. (2.5) and determined
in Eq. (2.12)) is negligible. By rewriting the potential as $V+\Delta
V=\frac{\lambda_{\phi}}{4}(1+\delta)M_{\rm Pl}^{4}$, the fractional correction

$\delta\,\equiv\,\frac{\Delta V}{V(M_{\rm
Pl})}\,=\,\frac{\beta_{\lambda_{\phi}}(v_{\phi})}{4\lambda_{\phi}(\phi_{I})}\left(\frac{v_{\phi}}{M_{\rm
Pl}}\right)^{4}\,\ll\,1$ (3.2)

is negligible for only mildly suppressed $v_{\phi}\lesssim M_{\rm Pl}$ because
$\beta_{\lambda_{\phi}}\sim g_{X}^{4}\sim y_{N}^{4}$ and
$\lambda_{\phi}(M_{\rm Pl})=10^{-4,-8},\,g_{X}=10^{-3}\sim 10^{-1}$ are
relevant in this work (see next section).

This simple model shows that the CMB amplitude directly measures the potential
height at the inflation scale

$\quad A_{s}\,=\,\frac{1}{24\pi^{2}}\frac{V}{M_{\rm
Pl}^{4}}\frac{1}{\epsilon}\,\simeq\,5.76\times
10^{5}\,\frac{\lambda_{\phi}(M_{\rm Pl})}{\pi^{2}},$ (3.3)

so that very small $\lambda\simeq 4\times 10^{-14}$ is needed to explain the
observed Planck 2018 data [1, 2] as alluded:

$\displaystyle A_{s}$ $\displaystyle\,=\,2.101^{+0.031}_{-0.034}\times
10^{-9}$ $\displaystyle(68\%\ {\rm CL}),$ (3.4)

where CMB observables are evaluated at the pivot scale $k_{*}=0.05\,{\rm
Mpc}^{-1}$ with $N=60$. But this minimal model predicts too large tensor-to-
scalar ratio $r$

$\displaystyle r\,=\,16\varepsilon\,\simeq\,\frac{16}{N}~{},\quad
n_{s}\,=\,1-6\varepsilon+2\eta\,\simeq\,1-\frac{3}{N}~{},$ (3.5)

compared to the Planck data

$\displaystyle r$ $\displaystyle\,<\,0.056,$ $\displaystyle(95\%\ {\rm CL})$
$\displaystyle n_{s}$ $\displaystyle\,=\,0.9665\pm 0.0038,$
$\displaystyle\frac{dn_{s}}{d\ln k}$ $\displaystyle\,=\,0.013\pm 0.024$
$\displaystyle(68\%\ {\rm CL}).$ (3.6)

$\bullet$ Inflation with non-minimal coupling
Our main benchmark model is the quartic potential with a non-minimal coupling
$\xi\phi^{2}R/M_{pl}^{2}$ (with dimensionless $\xi>0$) between inflaton and
gravity [10, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. This model is known to
realize successful inflation, and the single new parameter $\xi$ allows easy
interpretation of our results.

This model works as $\xi$ effectively suppresses the quartic potential at the
inflation scale

$V_{E}\,=\,\frac{\lambda_{\phi}}{4}\frac{\phi^{4}}{\Omega^{4}},\qquad\Omega^{2}\,=\,1+\xi\left(\frac{\phi}{M_{\rm
Pl}}\right)^{2}>1,$ (3.7)

where the subscript $E$ refers to the Einstein frame; a canonical
normalization of $\phi$ brings additional modifications but this is a basic
structure. Thus, $\xi$ becomes effective for large-field inflation
$\phi\gtrsim M_{\rm Pl}/\sqrt{\xi}$. In this limit, (similarly to the
conventional Higgs inflation case) CMB observables are

$\displaystyle r\,\simeq\,\frac{12}{N^{2}}~{},\quad
n_{s}\,\simeq\,1-\frac{2}{N}~{},\quad
A_{s}\,\simeq\,\frac{\lambda_{\phi}(M_{\rm Pl})N^{2}}{72\pi^{2}\xi^{2}}~{}.$
(3.8)

Not only can $r$ and $n_{s}$ be consistent with Planck data, but we also
obtain the following relation between $\lambda_{\phi}(M_{pl})$ and $\xi$ by
the CMB amplitude $A_{s}\simeq 2.1\times 10^{-9}$ in Eq. (3.4)

$\displaystyle\frac{\lambda_{\phi}(M_{\rm Pl})}{\xi^{2}}\,\simeq\,4.1\times
10^{-10}\left(\frac{60}{N}\right)^{2}.$ (3.9)

The larger $\lambda_{\phi}(M_{\rm Pl})$, the larger field value at the
inflation scale, hence the larger suppression by $\xi$ is needed.

How large or small values of $\xi$ are natural, or most preferred? Since we do
not specify a fundamental theory that might be able to calculate the value of
$\xi$, it is naively reasonable to consider $\xi\sim{\cal O}(1)$ as a natural
value. If we restrict $\xi\lesssim 100$ for example, the required size of
$\lambda_{\phi}(M_{\rm Pl})\lesssim 10^{-6}$ can be significantly larger
(hence, more natural or likely) than the naive translation $10^{-14}$
mentioned in the introduction, albeit still too small to be perfectly natural.
On the other hand, there exists a lower bound on $\xi$. In the limit of
$\xi\to 0$, the theory asymptotes to a pure quartic potential, which is
inconsistent with observations as discussed in Eq. (3.5). We have numerically
checked that $\xi\gtrsim 0.01$ in order to be consistent with CMB
observations; the above large-$\phi$ approximation starts to break down for
$\xi\lesssim 0.1$ (or, $\lambda_{\phi}(M_{\rm Pl})\lesssim 10^{-11}$). It is
also possible to have $\lambda_{\phi}\sim{\cal O}(1)$ natural while with much
larger $\xi\sim 10^{5}$, but usually too strong interactions can produce
various unexpected corrections too. Thus, we will regard
$0.01\lesssim\xi\lesssim 100$ (or, $10^{-12}\lesssim\lambda_{\phi}(M_{\rm
Pl})\lesssim 10^{-6}$) as the most desired (natural and comfortable) parameter
space of inflation. Later, we will see that this is exactly the parameter
space that is preferred by low-energy constraints.

Lastly, we comment on the reheating after inflation. In usual $U(1)_{X}$
models, the transfer of energy from inflatons to radiation can occur through
perturbative decays $\phi\rightarrow Z^{\prime}Z^{\prime}~{}(\nu_{R}\nu_{R})$,
or non-perturbative particle productions caused by the oscillation of $\phi$.
For $\lambda_{\phi}(M_{\rm Pl})\gtrsim 10^{-3}$ (or, $\xi\gtrsim 10^{3})$, the
qualitative behaviors of the preheating are expected to resemble those of
Higgs inflation because $\phi$ couples to $Z^{\prime}~{}(\nu_{R})$ in the
similar way as $H$ couples to $W,Z~{}(\nu_{R})$. Thus, the reheating can occur
instantaneously and the reheat temperature $T_{R}$ can be high ${\cal
O}(\rho_{\rm inf}^{1/4})={\cal O}(\lambda_{\phi}^{1/2}M_{\rm Pl}/\xi^{1/2})$
[48, 49, 50, 51]. On the other hand, for $\lambda_{\phi}(M_{\rm Pl})\ll
10^{-3}$ (or, $\xi\ll 10^{3})$, the $U(1)_{X}$ gauge coupling $g_{X}$ can
become much smaller than SM gauge couplings. Such a small coupling prevents
rapid perturbative decays of $Z^{\prime}$ into SM fermions as well as reduces
the efficiency of parametric resonances. But some of the conventional studies
[52, 53] still predict sufficiently large reheat temperature222For example,
[52] shows that for $\xi\sim 1(\lambda_{\phi}\sim 10^{-8})$ and $g_{X}\sim
0.01$, the ratio of energy densities between $Z^{\prime}$ and $\phi$ is
roughly given by $\sim 10^{-4}(2.7)^{j}/\sqrt{j}$, where $j$ is the number of
zero crossings of $\phi$. This becomes ${\cal O}(1)$ for $j\sim 10$, which
corresponds to large radiation energy
$\rho_{R}\sim\rho_{\phi}\sim\lambda_{\phi}M_{\rm
Pl}^{4}/(6\pi^{2}\xi^{2}j^{2})\sim(10^{-3}M_{\rm Pl})^{4}$., and [54] shows
that the existence of higher dimensional operators such as
$(\partial\phi)^{4}/(M_{\rm Pl}/\sqrt{\xi})^{2}$ can significantly alter
preheating dynamics. Thus, (p)reheating with non-minimal couplings is model
dependent, and more dedicated estimations are beyond the scope of this paper.

$\bullet$ $\alpha$-attractor models
The $\alpha$-attractor model [30, 31, 32, 33] is another benchmark model. Its
inflation predictions are universal as long as the inflaton potential is
smooth around the pole $\phi=\sqrt{6}M_{\rm Pl}$ of the kinetic term. The
predictions are

$\displaystyle r\,\simeq\,\frac{12\alpha}{N^{2}}~{},\quad
n_{s}\,\simeq\,1-\frac{9\alpha}{2N^{2}}-\frac{2}{N}~{},\quad
A_{s}\,\simeq\,\frac{V_{0}N^{2}}{18\pi^{2}\alpha M_{\rm Pl}^{4}}~{},$ (3.10)

where $V_{0}$ is the height of the inflaton potential at $\phi=\sqrt{6}M_{\rm
Pl}$. In the case of quartic potential Eq. (3.1), we have

$\displaystyle A_{s}\,=\,\frac{\lambda_{\phi}(M_{\rm
Pl})N^{2}}{2\pi^{2}\alpha}\qquad\therefore\ \frac{\lambda_{\phi}(M_{\rm
Pl})}{\alpha}\,\simeq\,1.2\times
10^{-11}\times\left(\frac{60}{N}\right)^{2}~{},$ (3.11)

where the running of $\lambda_{\phi}(\phi)$ is neglected. Thus, as in the non-
minimal coupling case, $\alpha$ is determined as a function of
$\lambda_{\phi}(M_{\rm Pl})$. From $n_{s}$ data in Eq. (3.6), $\alpha\lesssim
10$ at 2$\sigma$ level, hence $\lambda_{\phi}(M_{\rm Pl})\lesssim 10^{-10}$.
Thus, the allowed values of $\lambda_{\phi}(M_{\rm Pl})$ are somewhat smaller
than those of the non-minimal coupling case.

As for the preheating, it was found that the effective mass of $\phi$ becomes
tachyonic after inflation [55], so that careful analyses are often necessary
to estimate particle productions and resulting reheat temperature; see for
example [56, 57, 58] and references therein.

##  4 Probing small CMB amplitude by low-energy physics

###  4.1 The connection between low and high energy physics

Figure 1:  $M_{X}$ (left) and Higgs mixing $\sin\theta$ (right) predicted by
solutions of correct EWSB, parameterized by $\lambda_{\phi}(M_{\rm Pl})$ and
$g_{X}(M_{\rm Pl})$ for two different choices of $x=4/5$ and 2. For each
$\lambda_{\phi}$ shown as different color (also marked with $\xi$ value for
correct CMB amplitude), the minimum $g_{X}$ and corresponding maximum $M_{X}$
(and minimum $\sin\theta$) are marked as square dots; large-$g_{X}$ regions of
different colors overlap. Recasted collider bounds from $Z^{\prime}$ searches
and precision Higgs coupling measurements are shown as dashed lines.
$\lambda_{h\phi}(M_{\rm Pl})=0$.

The most crucial reason for the existence of this intriguing connection is
that for given $\lambda_{\phi}(M_{\rm Pl})\neq 0$ there exists a _maximum_
$M_{X}$ (strongest low-energy constraints among many observables) consistent
with successful EWSB. Consequently, a definite range of $\lambda_{\phi}(M_{\rm
Pl})$, which is directly related to the CMB observables at the inflation
scale, can be probed with low-energy constraints on $M_{X}$.

The explanation begins with the existence of the minimum $g_{X}(M_{\rm Pl})$
for given $\lambda_{\phi}(M_{\rm Pl})\neq 0$. A smaller $g_{X}(M_{\rm Pl})$
generally induces smaller (negative) RG corrections to $\lambda_{h\phi}$
($\lambda_{h\phi}(M_{\rm Pl})=0$) so that a larger $v_{\phi}$ is needed to
produce a correct $v_{\rm EW}=246$ GeV from Eqs. (2.10) and (2.11)

$v_{EW}\,\sim\,\sqrt{-\frac{\lambda_{h\phi}(v_{\phi})}{\lambda_{h}(v_{EW})}}\,v_{\phi}\,\lesssim\,v_{\phi},$
(4.1)

where the last inequality is due to $|\lambda_{h\phi}|\ll\lambda_{h}\sim 0.1$.
A larger $v_{\phi}$ needs larger $24g_{X}^{4}(M_{\rm Pl})-4y_{N}^{4}(M_{\rm
Pl})$ in the dimensional transmutation of $v_{\phi}$ in Eq. (2.9). In any
case, too small $g_{X}(M_{\rm Pl})$ would require too large $v_{\phi}\gtrsim
M_{\rm Pl}$ to be realized in this model, hence the existence of the min
$g_{X}(M_{\rm Pl})$333Note that Eq. (2.9) depends directly on
$\lambda_{\phi}(M_{\rm Pl})$ so that this argument does not directly apply to
flatland scenarios with $\lambda_{\phi}(M_{\rm Pl})=0$.. The existence is
proved numerically in Fig. 1, where the min $g_{X}(M_{\rm Pl})$ is marked with
a square dot for each $\lambda_{\phi}(M_{\rm Pl})$ and for each $x=4/5$ and
$2$. The values of the min $g_{X}(M_{\rm Pl})$ depend weakly on $x$ as the
exponential dimensional transmutation of $v_{\phi}$ does not strongly depend
on $x$.

The min $g_{X}(M_{\rm Pl})$ leads to the max $M_{X}$ for given
$\lambda_{\phi}(M_{\rm Pl})$. Is it because the smaller $g_{X}(M_{\rm Pl})$
corresponds to the larger $v_{\phi}$, as discussed (see also [45]). Since
$v_{\phi}$ depends exponentially on $g_{X}$, the resulting
$M_{X}=2g_{X}(v_{\phi})v_{\phi}$ is larger (Higgs mixing smaller). Thus, the
min $g_{X}(M_{\rm Pl})$ is translated to the max $M_{X}$, as also proved
numerically in Fig. 1. Moreover, the larger $v_{\phi}\gg v_{\rm EW}$ leads to
smaller Higgs mixing corrections to Higgs couplings. The values of max $M_{X}$
depend on $\lambda_{\phi}(M_{\rm Pl})$ and $x$. First, a larger
$\lambda_{\phi}(M_{\rm Pl})$ typically needs a larger $g_{X}(M_{\rm Pl})$ to
RG-drive $\lambda_{\phi}$ negative (for symmetry breaking). This also
necessarily induces a larger portal mixing $|\lambda_{h\phi}(v_{\phi})|$, thus
a smaller $v_{\phi}$ yields a correct EW scale via Eq. (4.1). Thus, larger
$\lambda_{\phi}(M_{\rm Pl})$ predicts smaller max $M_{X}$ (and larger Higgs
mixing). Second, the $x$ dependence arises mainly from the running of
$\lambda_{h\phi}$

$(4\pi)^{2}\beta_{\lambda_{h\phi}}\,\simeq\,12g_{X}^{4}x^{2}.$ (4.2)

Roughly speaking, the larger $x$, the larger $|\lambda_{h\phi}(v_{\phi})|$ at
$v_{\phi}$, so that a smaller $v_{\phi}$ can induce a correct $v_{EW}$,
resulting in smaller max $M_{X}$. These behaviors are as shown in Fig. 1.

This is the main feature of the models where inflation and CW mechanism of
EWSB are induced by a common particle.

###  4.2 Main results

It turns out that the constraints on $M_{X}$ (from $Z^{\prime}$ collider
searches) provide the strongest probe. The current bound on the mass of
$Z^{\prime}$ having the same interactions as the SM $Z$ is 4.5 TeV from LHC13
35/fb [59, 60]. It is recasted to the bound on $M_{X}$ for the same LHC13
35/fb, high-luminosity upgrade of LHC13 with 10/ab, and future 100 TeV $pp$
collider with 1/ab; these are shown in Fig. 1 as dashed lines for each $x$.
The recasted mass bound is where the same number of signal events are produced
as the current bound; it is based on the assumption that signal and background
compositions and cut efficiencies remain similar, which is reasonable for
narrow resonance searches. The production ($q\bar{q}$-initiated) times BR
($e^{+}e^{-},\mu^{+}\mu^{-}$) is rescaled according to $x$-dependent charges,
and the parton luminosity is taken from MSTW2008NLO [61].

$Z^{\prime}$ searches provide a meaningful probe of the inflation sector. For
the model with $x=2$ shown in Fig. 1, a 100 TeV collider can probe
$\lambda_{\phi}(M_{\rm Pl})\gtrsim 10^{-7}$ ($\xi\gtrsim 10$) _definitely_.
This means first that larger values of $\lambda_{\phi}$ and $\xi$ cannot
induce correct EWSB while being consistent with CMB and $Z^{\prime}$ searches.
But this does not mean that a whole parameter space with a smaller
$\lambda_{\phi}(M_{\rm Pl})$ can explain all CMB, EWSB and $Z^{\prime}$;
rather, there exists some working parameter space, which usually yields small
$g_{X}$ and heavy $M_{X}$. In this sense, we will say that a 100 TeV collider
with 1/ab has a (definite) reach down to $\lambda_{\phi}(M_{\rm Pl})\simeq
10^{-7}$ and $\xi\sim 10$ for $x=2$; less natural size of $\xi\gtrsim 10$ will
be more strongly constrained by 100 TeV collider searches.

Figure 2:  Collider constraints on the parameter space of inflation with non-
minimal coupling $\xi$ and Coleman-Weinberg Higgs mechanism. On each point on
the plane of $\lambda_{\phi}(M_{\rm Pl})$ and $x$, minimum $g_{X}(M_{\rm Pl})$
for correct EWSB and corresponding maximum $M_{X}$ (blue dashed) are used to
obtain collider constraints from $Z^{\prime}$ searches. Red shaded regions are
definite bounds, within which no parameter space can induce correct EWSB while
being consistent with $Z^{\prime}$ searches. From darkest to lightest shaded
regions are the bounds from current LHC13 35/fb, high-luminosity LHC13 10/ab
projection, and future 100 TeV $pp$ collider 1/ab projection. Also shown as
horizontal dashed lines are the required value of $\xi$ for correct CMB
amplitude as in Eq. (3.9); the most natural range of $\xi\lesssim{\cal
O}(1-100)$ is preferred. $\lambda_{h\phi}(M_{\rm Pl})=0$.

Fig. 2 shows such _definite_ bounds, more generally in the plane of
$x-\lambda_{\phi}(M_{\rm Pl})$. Shaded regions are such excluded regions by
recasted bounds on $M_{X}$; for given $\lambda_{\phi}(M_{\rm Pl})$ and $x$
within these regions, the max $M_{X}$ is lighter than the recasted bounds.
Note that both recasted bounds and max $M_{X}$ (dotted contours) vary with $x$
and $\lambda_{\phi}(M_{\rm Pl})$ as well as with collider spec. Also
overlapped are the contours of $\xi$ (horizontal dashed) for the correct CMB
amplitude. In general, $\xi$ can be more strongly constrained for larger $x$,
as discussed; a pure $B-L$ model with $x=0$ is much harder to probe. Due to
this $x$-dependence, it is not appropriate to find a strict upper bound on
$\xi$. But we conclude that a large part of $\lambda_{\phi}\gtrsim{\cal
O}(10^{-4}-10^{-6})$ or $\xi\gtrsim{\cal O}(10-100)$ can be probed with
current LHC13 35/fb and LHC13 10/ab; and $\xi\gtrsim{\cal O}(1-10)$ with 100
TeV $pp$ collider 1/ab. As discussed in Sec. 3, the allowed range of $\xi$ may
be considered most natural. Likewise, in the $\alpha$ attractor model, using
Eqs. (3.9) and (3.11), we conclude that a large part of $\alpha\gtrsim
10^{6}-10^{5}$ and $10^{5}-10^{3}$ can be probed, respectively.

Lastly, Higgs-related physics gives weaker bounds but can still be relevant.
Higgs physics is modified by small portal mixing
$\lambda_{h\phi}(v_{\phi})=10^{-9}\sim 10^{-4}$, for $g_{X}=0.005-0.1$
respectively. The resulting Higgs mixing angle is

$\sin\theta\,\simeq\,\lambda_{h\phi}(v_{\phi})v_{\phi}v_{EW}/m_{h}^{2}\,\sim\,\sqrt{\lambda_{h\phi}(v_{\phi})/2\lambda_{h}}\,=\,10^{-4}\sim
10^{-1}$ (4.3)

as also shown in Fig. 1 right panel. But these are too small to be probed with
current LHC precision $\sin\theta\lesssim 0.27$ from ATLAS+CMS with 35/fb [62,
63] as well as with expected precision $\sin\theta\lesssim 0.045$ from ILC 250
S2 stage with 2/ab [64, 65]. In addition, $\phi$ is expected to be light
$M_{\phi}\simeq\sqrt{\frac{6}{11}\lambda_{\phi}(v_{\phi})}v_{\phi}={\cal
O}(0.1-10)$ GeV for most parameter space, but $h\to\phi\phi$ decay rate is
still too small
$\Gamma(h\to\phi\phi)\,\simeq\,\frac{\lambda_{h\phi}(v_{\phi})^{2}v_{EW}^{2}}{32\pi
m_{h}}\sqrt{1-\frac{4m_{\phi}^{2}}{m_{h}^{2}}}\,\lesssim\,10^{-3}\,{\rm MeV}$
(or, its branching ratio $\lesssim 10^{-3}$) to be probed even at ILC 250,
whose expected precision on invisible decay branching ratio is $\sim 0.003$
[64, 65]. LHCb, BaBar, and Belle are sensitive to GeV-scale dark photons with
its interaction strength $\epsilon\gtrsim 10^{-3}-10^{-4}$ [66], but the
$\phi$-lepton interaction yields too small
$\epsilon=\frac{y_{e}}{e}\sin\theta\simeq 9.2\times 10^{-6}\sin\theta\ll
10^{-3}$ in this scenario. Nevertheless, as discussed, Higgs-related physics
can be important for proper reheating.

###  4.3 Variation with non-zero portal mixing

We now assess the variation with $\lambda_{h\phi}(M_{\rm Pl})\neq 0$. It
cannot be arbitrarily large positive for given $g_{X}(M_{\rm Pl})$. Since
$\lambda_{h\phi}(v_{\phi})<0$ has to be negative to induce EWSB, its RG
running should be large enough

$\Delta\lambda_{h\phi}\,\simeq\,\beta_{\lambda_{h\phi}}\cdot\log\left(\frac{v_{\phi}}{M_{\rm
Pl}}\right)\,\simeq\,\frac{12g_{X}^{4}x^{2}}{(4\pi)^{2}}\cdot{\cal
O}(10)\,\gtrsim\,\lambda_{h\phi}(M_{\rm Pl}).$ (4.4)

Thus, for given $g_{X}(M_{\rm Pl})$ (and $\lambda_{\phi}(M_{\rm Pl})$), there
exists a maximum $\lambda_{h\phi}(M_{\rm Pl})$ inducing correct EWSB.
Importantly, this still yields a max $M_{X}$ for the given parameters because
larger positive $\lambda_{h\phi}(M_{\rm Pl})$ yields a smaller
$|\lambda_{h\phi}(v_{\phi})|$, hence requiring a larger $v_{\phi}$, yielding
heavier $M_{X}$. The max $M_{X}$ will be larger for $\lambda_{h\phi}(M_{\rm
Pl})>0$, but the maximum still exists. In the other limit of negative
$\lambda_{h\phi}(M_{\rm Pl})<0$, the max $M_{X}$ is smaller so that more
strongly constrained. These are numerically proved in Fig. 3; the modified max
$M_{X}$ is shown in the left panel as shaded regions, and the max $M_{X}$ as a
function of $\lambda_{h\phi}(M_{\rm Pl})$ is shown in the right panel where
the max positive $\lambda_{h\phi}(M_{\rm Pl})$ for EWSB is marked as a
vertical line. The new max $M_{X}$ is larger by a factor 1.5 – 2.0.

Figure 3:  (Left): Same as Fig. 1 but showing variations of low-energy
predictions with $\lambda_{h\phi}(M_{\rm Pl})>0$ (shaded regions). The shaded
regions still have definite ranges determined by maximum positive
$\lambda_{h\phi}(M_{\rm Pl})$ in Eq. (4.4) that can induce correct EWSB, which
yields somewhat heavier $M_{X}$ for given $g_{X}(M_{\rm Pl})$. (Right): An
example $M_{X}$ prediction as a function of $\lambda_{h\phi}(M_{\rm Pl})$ for
fixed $g_{X}(M_{\rm Pl})=0.02$ and $\lambda_{\phi}(M_{\rm Pl})=10^{-6}$. The
maximum $\lambda_{h\phi}(M_{\rm Pl})$ consistent with EWSB is marked as a
vertical dashed line. See text for more discussion.

Figure 4:  Same as Fig. 2 but now with $\lambda_{h\phi}(M_{\rm Pl})\neq 0$:
maximum positive $\lambda_{h\phi}(M_{\rm Pl})>0$ for EWSB as in Eq. (4.4) and
Fig. 3 yielding heavier max $M_{X}$ (left panel), and negative
$\lambda_{h\phi}(M_{\rm Pl})<0$ with the twice magnitude yielding lighter max
$M_{X}$ (right panel).

In Fig. 4, we show definite bounds with max positive $\lambda_{h\phi}(M_{\rm
Pl})$ (worst constraints) as well as with negative $\lambda_{h\phi}(M_{\rm
Pl})$ with the twice magnitude (stronger constraints). Again, shaded regions
are definite bounds, within which no parameter space can be consistent with
all CMB, EWSB and collider searches. For the former case, current LHC13 35/fb
can still probe a large part of $\xi\gtrsim{\cal O}(100)$, and future high-
luminosity LHC13 and 100 TeV 1/ab can probe a large part of $\xi\gtrsim
100-10$. But this is for the max $\lambda_{h\phi}(M_{\rm Pl})$ yielding the
worst constraints; but $\lambda_{h\phi}$ would more likely take a smaller or
negative value. In such a case (right panel), the constraints/prospects are
stronger, and the majority of $\xi\gtrsim 1-10$ can be probed with future 100
TeV 1/ab, and current LHC13 35/fb can already probe a large part of
$\xi\gtrsim{\cal O}(100)$. In any case, smaller $\xi$ will be preferred by
collider experiments.

##  5 Summary

We have shown that the parameter space for the small CMB amplitude can be
probed by collider experiments. In the inflation model with non-minimal
coupling $\xi$, the absence of collider signals of $Z^{\prime}$ at LHC 13 and
100 TeV $pp$ will prefer a smaller and more natural size of $\xi\lesssim
1-100$ with $\lambda_{\phi}(M_{\rm Pl})\lesssim 10^{-4}-10^{-8}$ even though
$\xi$ could equally well take a much larger and less natural value to explain
the CMB amplitude. Another possibility of $\lambda_{\phi}(M_{\rm Pl})\sim{\cal
O}(1)$ with $\xi\sim 10^{5}\gg 1$ will be strongly constrained. A similar
conclusion for an $\alpha$ attractor model is obtained, albeit more weakly.

This probe was possible in models where an inflaton $\phi$ also induces EWSB
via perturbative quantum corrections (CW mechanism). The restrictions on the
$\phi$ potential from well-measured high-energy inflation and low-energy
electroweak physics lead to intimate correlations, in particular between low-
energy $M_{X}$ (collider observables) and high-energy $\lambda_{\phi}(M_{\rm
Pl})$ (CMB amplitude $A_{s}\propto\lambda_{\phi}(M_{\rm Pl})$). More
crucially, there exists an upper range of $M_{X}$ prediction for each value of
$\lambda_{\phi}(M_{\rm Pl})$, so that definite collider constraints on
$\lambda_{\phi}(M_{\rm Pl})$ could be derived.

This work not only proves an interesting possibility of probing the inflation
sector with low-energy experiments, but may also relate the absence of new
physics signals to the physics at a disparate energy scale. Although this
connection does not explain why our universe is realized with such small CMB
amplitude, the fact that such particular realization at high-energy scale also
has consequences in low-energy physics is intriguing enough, worthwhile to be
pursued in a more varieties of models.

## Acknowledgments

We are supported by Grant Korea NRF-2019R1C1C1010050, 2019R1A6A1A10073437, and
SJ also by POSCO Science Fellowship.

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