compat=1.1.0

# Higgs Inflation and the Electroweak Gauge Sector

Stephon Alexander<EMAIL_ADDRESS>Brown Center for Theoretical
Physics
Department of Physics, Brown University, Providence, Rhode Island 02912, USA
Cyril Creque-Sarbinowski<EMAIL_ADDRESS>Center for
Computational Astrophysics
Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA  Humberto
Gilmer<EMAIL_ADDRESS>Brown Center for Theoretical Physics
Department of Physics, Brown University, Providence, Rhode Island 02912, USA
Katherine Freese<EMAIL_ADDRESS>Texas Center for Cosmology and
Astroparticle Physics, Department of Physics, The University of Texas at
Austin, Austin, Texas 78712, USA

###### Abstract

We introduce a new method that allows for the Higgs to be the inflaton. That
is, we let the Higgs be a pseudo-Nambu-Goldstone (pNG) boson of a global coset
symmetry $G/H$ that spontaneously breaks at an energy scale $\sim 4\pi f$ and
give it a suitable $SU(2)\subset G$ Chern-Simons interaction, with $\beta$ the
dimensionless Chern-Simons coupling strength and $f$ an $SU(2)$ decay
constant. As a result, slow-roll inflation occurs via $SU(2)$-induced friction
down a steep sinusoidal potential. In order to obey electroweak $SU(2)_{\rm
L}\times U(1)_{Y}$ symmetry, the lowest-order Chern-Simons interaction is
required to be quadratic in the Higgs with coupling strength
$\propto\beta^{2}/f^{2}$. Higher-order interaction terms keep the full
Lagrangian nearly invariant under the approximate pNG shift symmetry.
Employing the simplest symmetry coset $SU(5)/SO(5)$, $N$ $e$-folds of
inflation occur when $N\approx
60\left(g/0.64\right)^{2}\left[\beta/\left(3\times
10^{6}\right)\right]^{8/3}\left[f/\left(5\times 10^{11}\ {\rm
GeV}\right)\right]^{2/3}$, with $g$ the weak isospin gauge coupling constant.
Small values of the decay constant, $f\lesssim 5\times 10^{11}{\rm GeV}$,
which are needed to address the Higgs hierarchy problem, are ruled out by
electric dipole measurements and so successfully explaining inflation requires
large $\beta$. We discuss possible methods to achieve such large couplings and
other alternative Higgs inflation scenarios outside the standard modified-
gravity framework.

††preprint: APS/123-QED

##  I Introduction

Inflation posits the existence of a slowly-rolling scalar field, the inflaton,
to drive a period of accelerated expansion in the early Universe as a means to
solve several problems in cosmology Guth (1981). Given that we already have a
scalar field in the Standard Model (SM), the Higgs Englert and Brout (1964);
Higgs (1964); Guralnik et al. (1964), a possible minimal realization of
inflation identifies the inflaton as the SM Higgs.111Other possible SM minimal
realizations include those that identify the inflaton as a neutrino condensate
Barenboim (2009) The SM Higgs potential, however, is not flat enough to both
reproduce the observed matter power spectrum and sustain a sufficiently long
inflationary phase, as its quartic self-interaction is too large Adams et al.
(1991); Bezrukov (2013); Rubio (2019); Linde (1982, 1983). As a result,
scenarios beyond the SM must be considered to achieve successful Higgs
inflation.

The original method to induce a slowly-rolling Higgs invoked a non-minimal
gravitational coupling that exponentially flattened the SM Higgs potential
upon a field redefinition Salopek et al. (1989); Bezrukov and Shaposhnikov
(2008a). Since its original formulation, a variety of extensions and
generalizations, such as those involving derivative-Higgs couplings Germani
and Kehagias (2010a, b); Nakayama and Takahashi (2011); Kamada et al. (2011,
2012), alternative formulations of General Relativity Bauer and Demir (2008,
2011); Rasanen and Wahlman (2017); Rasanen (2019); Raatikainen and Rasanen
(2019); Tenkanen (2019); Jinno et al. (2020); He et al. (2022), warm initial
conditions Samart et al. (2022); Eadkhong et al. (2023), additional particles
Shaposhnikov and Zenhausern (2009); Einhorn and Jones (2010); Garcia-Bellido
et al. (2011); Trashorras et al. (2016); He and Xianyu (2014); Ellis et al.
(2016); Aoki and Lee (2022), and other approaches involving SUSY or other non-
minimal couplings Yin (2022); Choudhury et al. (2014); Choudhury (2017) have
been explored, with all such varieties flattening the Higgs potential in a
similar manner. The original non-minimal proposal also violates perturbative
unitarity due to the small EFT scale of the coupling compared to the high-
energy scales of inflation Burgess et al. (2009); Barbon and Espinosa (2009);
Burgess et al. (2010); Hertzberg (2010). As a result, numerous perturbative
and non-perturbative unitarity resolutions inside Lerner and McDonald (2010a,
b); Bezrukov et al. (2011); Giudice and Lee (2011); Atkins and Calmet (2011);
Lerner and McDonald (2012); Lee (2013); Allison (2014); Barbon et al. (2015)
and outside Fumagalli et al. (2018); Shaposhnikov et al. (2020); McDonald
(2021); Mikura and Tada (2022); Ito et al. (2022) the original framework have
been proposed, with the predictiveness of non-minimal Higgs inflation
depending on the UV completion considered Burgess et al. (2014). Given this
predictive uncertainty, it is thus desirable to further investigate a variety
of alternative Higgs inflation models. The scope of all alternatives, however,
is not infinite, as one must be able to also solve the eta problem, whereby
higher-order corrections to the inflaton potential spoil its desired flatness
Freese and Kinney (2004a).

One paradigm meant to address the eta problem is natural inflation, where the
inflaton is endowed with an approximate shift symmetry that protects the shape
of its potential Freese et al. (1990). In natural inflation, the inflaton is a
pseudo-Nambu-Goldstone (pNG) boson with a sinusoidal potential derived from
the UV physics of a strongly-interacting vacuum. Cosmic microwave background
observations have pushed the original variant of this model to have large
super-Planckian excursions Ade et al. (2021), but its general form remains
attractive Arkani-Hamed et al. (2003a, b); Freese and Kinney (2004b); Ben-
Dayan et al. (2015); Grimm (2014); Choi et al. (2014); Czerny and Takahashi
(2014); Kappl et al. (2016); Linde et al. (2018); Freese et al. (2022). In
particular, if the inflaton is able to efficiently dissipate its kinetic
energy through friction, then sub-Planckian excursions, characterized by a
steep potential, do not pose an issue. Such a solution was first proposed
through the use of Abelian gauge fields Anber and Sorbo (2010); Alexander et
al. (2013) and then followed up with its non-Abelian variety Maleknejad and
Sheikh-Jabbari (2013, 2011), known as chromo-natural inflation Adshead and
Wyman (2012); Adshead et al. (2016); Bagherian et al. (2023), with a
combination of both ideas put forth recently Tishue and Caldwell (2023). It is
also possible to use thermally-induced Bastero-Gil et al. (2016, 2018) or
scalar-induced Creminelli et al. (2023) friction, rather than gauge-induced,
to remove unwanted fast roll.

Here we construct a model of Higgs inflation where the Higgs is a pNG boson,
based on Ref. Alexander (2022).222An inflationary setup where the Higgs is a
pNG was considered in Ref. Alanne et al. (2017), although there the Higgs does
not play the role as the inflaton. We use an effective field theory (EFT)
framework, in the spirit of chromo-natural inflation, to keep the Higgs slowly
rolling down the steep pNG potential. This friction mechanism is the new
ingredient which obviates the need for the flattening method of all previous
Higgs inflation models. With the aim of demonstrating the key inflationary
dynamics and putting such dynamics in the context of a realistic pNG Higgs
model, we consider two Higgs inflation variants of increasing complexity.
Specifically, we first consider a toy minimal pNG Higgs Inflation model where
the Higgs boson interacts with the weak isospin gauge fields. Second, we
investigate a more complete setup, based on the “littlest Higgs” model Arkani-
Hamed et al. (2002), and thus examine effects of additional gauge fields
(which are present in all pNG Higgs models). For specific parameters, such
realistic models, and its generalizations, are known solutions to the Higgs
hierarchy problem Arkani-Hamed et al. (2001a); Schmaltz and Tucker-Smith
(2005a); Perelstein (2007); Cheng and Low (2004). This model then also allows
us to engage with the possibility that both the Higgs and inflaton hierarchy
problems are resolved in the same manner. We note that simpler pNG Higgs
models have been ruled out experimentally Chen (2006); Hewett et al. (2003).
In all cases, we find that a sufficient number of $e$-folds can be achieved,
albeit with large EFT couplings.

This work is organized as follows. We present a minimal pNG Higgs and review
the littlest Higgs model in Sec. II. In doing so, we demonstrate how our
inflationary model can be implemented in an already-existing solution to the
hierarchy problem. We also show how to obtain EFT couplings consistent with
electroweak $SU(2)_{\rm L}\times U(1)_{Y}$ symmetry. Then, in Sec. III, we
present both the minimal pNG Higgs inflation model as well as its littlest
Higgs variant and delineate the necessary model parameters to achieve
successful inflation. We discuss potential methods to achieve large EFT
couplings, lay out future directions of study, and conclude in Sec IV.

Conventions and Notation: We let $\hbar=c=1$ and use $M_{\rm Pl}^{2}=1/(8\pi
G)$ as the reduced Planck mass. We let an overdot denote a derivative with
respect to cosmic time, $\dot{f}=\tfrac{\mathrm{d}f}{\mathrm{d}t}$, and a
prime denote a derivative with respect to the number of $e$-folds
$N=\int\mathrm{d}\log a$, $f^{\prime}=\tfrac{\mathrm{d}f}{\mathrm{d}N}$, with
$a$ the scale factor of the Friedmann-Lemâitre-Robertson-Walker (FLRW) metric
$g_{\mu\nu}=a^{2}{\rm diag}(-1,1,1,1)$.

##  II PNG Higgs

We review the standard structure of the littlest-Higgs model and show how to
express dimension-six Chern-Simons EFT couplings that obey the weak isospin
$SU(2)_{\rm L}$ symmetry. Given the overall complexity of this model, we begin
in Sec. II.1 by displaying the minimal particle setup that will be required
for inflation. We then address the entire particle-physics model in Sec. II.2,
delineating all new particles along with their interactions, calculating the
form of the Higgs doublet potential, and concluding with the Chern-Simons EFT
coupling. Readers interested in only the inflationary dynamics should go to
Sec. III.

###  II.1 Minimal pNG Higgs

We display a toy minimal pseudo-Nambu-Goldstone (pNG) Higgs model relevant for
inflation that encompasses a wide variety of pNG Higgs models. The components
common to this toy model consist of the usual Higgs doublet $H$, but now with
a periodic potential $V(H)$ arising from higher-order dynamics. In addition,
they contain a Higgs coupling to weak isospin gauge bosons through both
covariant derivatives and a weak-isospin Chern-Simons current. Altogether,
these terms take form in the Lagrangian

$\begin{split}\mathcal{L}&=D_{\mu}H^{\dagger}D^{\mu}H+V(H)+\frac{1}{4}W^{\mu\nu}_{a}W_{\mu\nu}^{a}+\frac{1}{4}B^{\mu\nu}B_{\mu\nu}\\\
&+\frac{g^{2}\beta^{2}}{2f^{2}}\left(H^{\dagger}H\right)\text{Tr}\left[W^{\mu\nu}_{a}\tilde{W}_{\mu\nu}^{a}\right]\end{split}$
(1)

where $D_{\mu}=\partial_{\mu}-igW_{\mu}^{a}\tau^{a}-(i/2)g^{\prime}B_{\mu}$,
$g$ and $g^{\prime}$ are the weak-isospin and hypercharge gauge couplings
associated with bosons $W_{\mu}^{a}$ and $B_{\mu}$, respectively, and
$\tau^{a}=\sigma^{a}/2$, with $\sigma^{a}$ the Pauli matrices. In addition,
$W_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}+f_{ij}^{a}W_{\mu}^{i}W_{\nu}^{j}$
is the field-strength tensor associated with the weak-isospin bosons. Finally,
$\tilde{W}_{\mu\nu}^{a}=(1/2)\epsilon^{\mu\nu\alpha\beta}W_{\alpha\beta}^{a}$
is the weak-isospin field-strength tensor Hodge dual with
$\epsilon^{\mu\nu\alpha\beta}$ the Levi-Civita tensor
$[\tilde{\epsilon}^{\mu\nu\alpha\beta}=\sqrt{-{\rm
det}\left(g_{\mu\nu}\right)}\epsilon^{\mu\nu\alpha\beta}$ is the flat-space
Levi-Civita symbol$]$, and $f_{ajk}=\tilde{\epsilon}_{ajk}$ the $SU(2)$
structure constants. Generically-speaking, the Higgs potential $V(H)$ may be
any periodic function; in the simplest case it is a cosine potential, but
other calculable potentials are possible and depend on the specific model
being considered, as will be seen in later sections of this work. We work in
the unitary gauge, $H=(0,h/\sqrt{2})^{\mathsf{T}}$, with $h$ the background
Higgs field. In this minimal setup, it will be the electroweak sector, by
itself, that drives inflation.

###  II.2 Littlest Higgs

We focus on a model that can supply all of the components described in the
previous section, known as the littlest Higgs Arkani-Hamed et al. (2002);
Perelstein et al. (2004); Han et al. (2003). This model is based on a global
$SU(5)/SO(5)$ coset and is the minimal enhancement to the Standard Model
particle content to realize a viable pNG Higgs. More specifically, the full
theory we obtain is given by the Lagrangian

$\begin{split}\mathcal{L}&=\frac{f^{2}}{8}\mathrm{Tr}\left[\left(D_{\mu}\Sigma\right)^{\dagger}D^{\mu}\Sigma\right]-\frac{1}{4}\sum_{j=1}^{2}W^{\mu\nu}_{aj}W^{aj}_{\mu\nu}\\\
&-i\psi^{\dagger}\not{D}\psi-i\bar{u}^{\dagger}_{i}\not{D}\bar{u}^{i}+\text{H.c.}\\\
&-\frac{\lambda_{1}}{4}f\psi^{\dagger}_{i}\epsilon_{ijk}\epsilon_{xy}\Sigma_{jx}\Sigma_{ky}\bar{t}^{\dagger}+\lambda_{2}fTi\sigma_{2}T+\text{H.c}\\\
&+\sum_{j=1}^{2}\frac{g_{j}^{2}\beta^{2}}{16}\text{Tr}\left[Y\Sigma\left(Y\Sigma\right)^{*}\right]\text{Tr}\left[W^{\mu\nu}_{ja}\tilde{W}_{\mu\nu}^{ja}\right].\end{split}$
(2)

To begin, recall that symmetries and symmetry breaking are key in many
theories of particle physics. Little Higgs theories are generically built upon
an interplay between spontaneously and explicitly broken global and gauged
symmetries. This interplay mirrors the physics that describes pions as pseudo-
Nambu-Goldstone bosons of a chiral flavor symmetry. A similar argument is
invoked in little Higgs models, whereby the smallness of the Higgs’s mass is
due to some underlying strong dynamics, but the particular details of the
underlying UV theory are negligible, instead encoding the physics in a semi-
UV-complete theory. We first review its standard formulation in Sec. II.2.1
before detailing our additional components in Sec. II.2.2.

####  II.2.1 Standard Littlest Higgs

The littlest Higgs theory is characterized by a partial UV completion of the
Standard Model using an approximate global $SU(5)$ symmetry along with a
gauged $SU(2)\times[SU(2)\times U(1)]$ subgroup. We note that the original
formulation of the littlest Higgs instead gauged a $[SU(2)\times U(1)]^{2}$
subgroup Arkani-Hamed et al. (2002). This subgroup differs from our choice by
one factor of $U(1)$, i.e. it has an additional $U(1)$ gauge field. The
removal of this additional $U(1)$ field does not affect the particular
conclusions presented here Perelstein et al. (2004); Csaki et al. (2003a). In
fact, the removal of the $U(1)$ gauge field relieves some of the
phenomenological tensions encountered by the original littlest Higgs; we thus
omit it. In terms of new field content, before global $SU(5)$ symmetry
breaking, the littlest Higgs contains a collection of $11$ massless scalar
fields, three massless gauge bosons, and a heavy vector-like fermion. We now
describe how this new field content is realized.

The 11 scalar fields, along with the 4 degrees of freedom in the Higgs
doublet, are embedded within a symmetric $5\times 5$ matrix scalar $\Sigma$
under the 15 representation of $SU(5)$. Upon spontaneous symmetry breaking at
an energy scale $\Lambda\sim 4\pi f$, this scalar obtains the vacuum
expectation value (VEV)

$\Sigma_{0}=\begin{pmatrix}&&\mathbb{1}_{2}\\\ &1&\\\
\mathbb{1}_{2}&&\end{pmatrix},$ (3)

with $\mathbb{1}_{N}$ the $N\times N$ identity matrix and blank entries zeros.
This VEV spontaneously breaks $SU(5)$ down to an $SO(5)$ subgroup.

$SU(5)$ has 24 generators, while $SO(5)$ has 10 generators. It follows that
$24-10=14$ massless scalars are produced in the particle spectrum, each
corresponding to a broken generator of $SU(5)$. Moreover, the VEV in Eq. (3)
is chosen so that these 14 fields live in the fundamental representation of
$SO(5)$ upon breaking. In addition to the 14 massless scalars, there is a
single massive scalar that represents deviations from the VEV along the
direction of symmetry breaking. We integrate this scalar out and make no
further mention of it henceforth. The Lagrangian for the matrix scalar alone
is that of a non-linear sigma model (NLSM):

$\mathcal{L}_{\rm
NLSM}=\frac{f^{2}}{8}\mathrm{Tr}\left[\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma\right],$
(4)

where

$\Sigma=\exp\left[\frac{2i}{f}\Pi\right]\Sigma_{0}=\exp\left[\frac{2i}{f}T_{a}\pi_{a}\right]\Sigma_{0},$
(5)

with $\Pi$ the ‘pion’ field matrix and $T_{a}$ the unbroken $SU(5)$ generators
($f$ is also called ‘pion’ decay constant). The pion matrix, fully written
out, is

$\Pi=\begin{pmatrix}\omega-\frac{\eta}{\sqrt{20}}\mathbb{1}_{2}&\frac{{H}}{\sqrt{2}}&\phi^{\dagger}\\\
\frac{{H}^{\dagger}}{\sqrt{2}}&\sqrt{\frac{4}{5}}\eta&\frac{{H}^{\mathsf{T}}}{\sqrt{2}}\\\
\phi&\frac{{H}^{*}}{\sqrt{2}}&\omega^{\mathsf{T}}-\frac{\eta}{\sqrt{20}}\mathbb{1}_{2}\end{pmatrix}.$
(6)

As promised, we have a theory of 14 scalar fields: $\mathbf{\phi}$, a complex
triplet, as a symmetric $2\times 2$ matrix, $H$ a complex doublet (our Higgs
candidate), $\omega$, a real triplet, as a Hermitian $2\times 2$ traceless
matrix, and the real scalar $\eta$. The above fields transform with a shift
symmetry under the broken generators of $SU(5)$, the specific field which does
so depending on which broken generator is used for the transformation. Under
the regime described thus far, the particles in $\Pi$ would be exactly
massless at all scales, per the Goldstone theorem, with Lagrangian as in Eq.
(4).

The symmetry is also explicitly broken, by gauging a $SU(2)\times[SU(2)\times
U(1)]$ subgroup. This breaking is done by promoting the derivative in Eq. (4)
to a covariant derivative. In so doing, the Goldstone bosons become pseudo-
Goldstone bosons that develop a mass proportional to the order parameter of
the explicit breaking (which in this case are the gauge couplings of the
gauged subgroup). This promotion also endows the Higgs with the requisite
gauge couplings. Thus, Eq. (4) becomes

$\mathcal{L}^{\rm gauged}_{\rm
NLSM}=\frac{f^{2}}{8}\mathrm{Tr}\left[\left(D_{\mu}\Sigma\right)^{\dagger}D^{\mu}\Sigma\right].$
(7)

Here the covariant derivatives are of the form

$D_{\mu}\Sigma=\partial_{\mu}\Sigma+ig_{j}W^{aj}_{\mu}\left(Q^{j}_{a}\Sigma+\Sigma
Q_{a}^{j\mathsf{T}}\right)+ig^{\prime}B_{\mu}\left(Y\Sigma+\Sigma Y\right)$
(8)

where $Q^{j}_{a}$ and $Y$ are the generators of the
$SU(2)\times\left[SU(2)\times U(1)\right]$ gauged subgroup (i.e.
$j\in\\{1,2\\}$ and $a\in\\{1,2,3\\}$); these generators are written
explicitly Appendix B. With this promoted setup, the vacuum in Eq. (3)
spontaneously breaks the $SU(2)\times[SU(2)\times U(1)]$ symmetry down to the
$SU(2)_{\rm L}\times U(1)_{Y}$ of the Standard Model prior to EWSB. When this
breaking occurs, the SM couplings are recovered by the relation
$g=\tfrac{g_{1}g_{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}}=e/\sin\left(\theta_{\rm
W}\right)=0.64$ and $g^{\prime}=e/\cos\left(\theta_{\rm W}\right)=0.34$.
Moreover, this breaking means that three of the generators are explicitly
broken and thus become the longitudinal modes of the corresponding $W$ bosons
post-symmetry-breaking. In other words, the $W_{1}$, $W_{2}$ bosons eat a
combination of the three $\omega$ and $\eta$ NGBs to spit out three massive
bosons (heavy counterparts of the SM $W$ bosons) and three massless bosons,
which are the weak isospin bosons of the SM prior to EWSB. As a result, there
is also a leftover massless Goldstone mode corresponding to the ungauged
$U(1)$ subgroup; other works that have considered this model Csaki et al.
(2003a) have ignored the effects of this mode and we do so as well. The heavy
gauge bosons have gauge couplings
$g_{H}=|g_{1}^{2}-g_{2}^{2}|/\left(2\sqrt{g_{1}^{2}+g_{2}^{2}}\right)$. The
$U(1)$ gauge boson corresponds to our hyperweak $U(1)_{Y}$. The kinetic terms
for the $SU(2)\times[SU(2)\times U(1)]$ gauge bosons are added in the
canonical manner, which we label by $\mathcal{L}_{\rm kin}^{\rm gauge}$.

Coset models, such as this one, are also distinguished by their implementation
of the coupling between the Higgs and top quark. The top is special because
its large Yukawa coupling could potentially disrupt the naturalness of the
Higgs potential. Other quarks, and leptons, are typically included according
to their SM description. The littlest Higgs model seeks a minimal
implementation (meaning the least number of new fermions). Hence, the $\Sigma$
field can be coupled to the top quark through a term of the form

$\mathcal{L}_{\text{top}}=-\frac{\lambda_{1}}{4}f\psi^{\dagger}_{i}\epsilon_{ijk}\epsilon_{xy}\Sigma_{jx}\Sigma_{ky}\bar{t}^{\dagger}+\lambda_{2}fTi\sigma_{2}T+\text{H.c},$
(9)

with $i,j,k\in\\{1,2,3\\}$ and $x,y\in\\{4,5\\}$. Here, $\psi$ is an $SU(3)$
triplet obtained by enhancing the usual left-handed third-generation quark
doublet $q_{3}$ with a new heavy top, labelled $T$,

$\psi_{i}=\begin{pmatrix}-i\sigma_{2}q_{3}\\\
T\end{pmatrix}=\begin{pmatrix}b\\\ t\\\ T\end{pmatrix}.$

Note the presence of two couplings, $\lambda_{1}$ and $\lambda_{2}$, as well
as the Dirac mass term for the new heavy top $T$. The top quark mass
eigenstate will be a mixture of the new heavy top and the usual left-handed
quark doublet top. As a result, the Higgs couples neither to $\lambda_{1}$ nor
$\lambda_{2}$, but to a product $\lambda_{1}\lambda_{2}$.

$\mathcal{L}\supset\lambda_{t}q_{3}h\bar{t}\qquad\text{where}\qquad\lambda_{t}=\frac{\lambda_{1}\lambda_{2}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}$
(10)

Thus if either coupling is turned off, the Higgs receives no quadratic
divergence from the top quark at all. This lack of divergence is a
manifestation of the generic collective symmetry breaking pattern of little
Higgs theories. When either coupling is turned off, the theory has an enhanced
global $SU(3)$ symmetry, which renders the coupling technically natural.

The other fermions of the Standard Model may be incorporated in the usual
fashion; although this does violate the collective symmetry breaking structure
that prevents quadratic divergences in the scalar sector, the relative
lightness of all other fermions compared to the Higgs that such divergences
are acceptable and to some extent negligible.

Now we must find the potential that the Higgs field experiences. Note that
most treatments of the little Higgs only consider the effective potential to
leading or next-to-leading order in the Higgs field $h$. Here we instead
consider a full treatment, in order to illustrate the periodic potential
experienced by the Higgs field, which plays an important role in the
inflationary dynamics.

A one-loop analysis yields additional operators that must be added to the
Lagrangian in Eq. (7) in order to account for quadratic divergences. These
terms are of the form

$\begin{split}\mathcal{L}&\supset cf^{2}\sum_{j=1}^{2}g_{j}^{2}{\rm
Tr}\left[\left(Q^{j}_{a}\Sigma\right)\left(Q^{j}_{a}\Sigma\right)^{*}\right]\\\
&+cf^{2}g^{\prime}{}^{2}\text{Tr}\left[\left(Y\Sigma\right)\left(Y\Sigma\right)^{*}\right]\end{split}$
(11)

from gauge boson loops and

$\displaystyle\mathcal{L}\supset
c^{\prime}\lambda_{1}\epsilon_{wx}\epsilon_{yz}\epsilon_{ijk}\epsilon_{\ell
mn}\Sigma_{iw}\Sigma_{jx}\Sigma^{*}_{my}\Sigma^{*}_{nz}+{\rm H.c.},$ (12)

from fermion loops. Here, $c$ and $c^{\prime}$ are undetermined coefficients,
consistent with Wilson renormalization analysis, whose exact value depends on
the UV completion. In the unitary gauge, the VEV of the $\Sigma$ field
$\Braket{\Sigma}$ may be written as

$\Braket{\Pi}=\begin{pmatrix}&\Braket{H}&\\\
\Braket{H}^{\dagger}&&\Braket{H}^{\mathsf{T}}\\\
&\Braket{H}^{*}&\end{pmatrix}.$ (13)

where the Higgs doublet $H=\left(0,h/\sqrt{2}\right)^{\mathsf{T}}$ (n.b.
defined as a column vector). Using Eq. (5), we can write the VEV of $\Sigma$
as a function of the Higgs boson $h$. Using the expression for $\Braket{\Pi}$
from Eq. (13) allows us to write

$\begin{split}\Braket{\Sigma}=\left[\mathbb{1}_{5}+\frac{i}{h}\sin\left(\frac{h}{f}\right)\Braket{\Pi}-\frac{2}{h^{2}}\sin^{2}\left(\frac{h}{f}\right)\Braket{\Pi}^{2}\right]\Sigma_{0}\end{split}$
(14)

Plugging $\Braket{\Sigma}$ into the terms shown in Eq.(11-12) yields a tree-
level potential for $h$ Perelstein et al. (2004); Espinosa et al. (2005):

$V_{0}(h)=D+f^{4}\left[cg^{\prime
2}\sin^{2}\left(\frac{2h}{f}\right)+\frac{1}{2}\lambda_{+}\sin^{4}\left(\frac{h}{f}\right)\right]$
(15)

where $D/f^{4}$ is an $\mathcal{O}\left(10^{-2}\right)$ constant and
$\lambda_{\pm}=c\left(g_{1}^{2}\pm g_{2}^{2}\right)\pm
16c^{\prime}\lambda_{1}^{2}$. This potential is periodic in $h$, and
electroweak symmetry breaking (EWSB) does not take place, as can be seen by
expanding the potential about $h\rightarrow 0$; this feature is in keeping
with the expectation of the Higgs as a Goldstone boson. A small non-zero Higgs
mass proportional to the $U(1)_{Y}$ coupling $g^{\prime}$ does arise at this
level, due to the explicit breaking of $SU(5)$ by the gauging of the
$U(1)_{Y}$ subgroup. It can be shown that this mass must be positive, meaning
EWSB does not take place at tree level Espinosa et al. (2005).

EWSB instead arises through the Coleman-Weinberg mechanism, resulting in a
negative loop-level Higgs mass and self-coupling terms. The full VEV of Eq.
(14) can be used to compute the mass matrices for the gauge bosons $M_{W}$,
fermions $M_{t}$ and remaining scalar fields $M_{s}$, in particular the
triplet field $\phi$ Perelstein et al. (2004). Note that these expressions
will be periodic in $h$, in accordance with its approximate shift symmetry.
These matrices can be used to compute the CW contributions which remain once
the heavy fields (the top partner, the Higgs triplet, etc) have been
integrated out, which have the form Kilic and Swaminathan (2016):

$\displaystyle V_{\rm
CW}=\frac{1}{64\pi^{2}}\text{STr}\left[\left(M^{\dagger}M\right)^{2}\left(\log\left(\frac{M^{\dagger}M}{\Lambda^{2}}\right)-\frac{3}{2}\right)\right]$
(16)

The supertrace takes into account both the statistics [$(+1)$ for bosons,
$(-1)$ for fermions] and the multiplicity of the fields (e.g., the weak boson
contributions are multiplied by a factor of three for the three
polarizations). The inclusion of these contributions leads to a negative Higgs
mass term and consequently to EWSB.

(a)

(b) Figure 1:  The periodic potential of the pNGB Higgs in the $SU(5)/SO(5)$
coset scenario as a function of $\theta=\frac{h}{f}$. The blue lines
correspond to the periodic potential generated by the Higgs VEV; the orange
line includes the contributions from the Coleman-Weinberg terms. The full
potential can be seen in the top figure. The potential near the origin,
exhibiting EWSB, is shown in the bottom figure. Both figures are made using
fiducial choices of parameters $g^{\prime}=0.34$,
$g_{1}=g_{2}=\sqrt{2}g=0.91$, $c=0.05$, $c^{\prime}=-0.05$ and $\lambda_{1}=1$
(so that $\lambda_{+}=-0.312$).

The contributions are from the heavy top quark, the heavy $W$ boson, and the
integrated-out Higgs triplet, respectively. These two potentials are shown in
Fig. 1.

####  II.2.2 Beyond Standard Littlest Higgs

The Higgs has an observed mass of $m_{H}=125\ {\rm GeV}$ Workman et al.
(2022). Hence, for $f\gg 10\ {\rm TeV}$, the tree-level potential in Eq. (15)
does not predict the correct Higgs mass. In order to explore a larger
phenomenological range of decay constants, while adhering to observations, we
make a small change, $f^{4}\rightarrow\mu^{4}$, leading to the following
potential

$\displaystyle V_{0}(h)=\mu^{4}\left[cg^{\prime
2}\sin^{2}\left(\frac{2h}{f}\right)+\frac{1}{2}\lambda_{+}\sin^{4}\left(\frac{h}{f}\right)\right],$
(17)

with $\mu$ a new parameter that satisfies

$m_{H}^{2}=8cg^{\prime 2}\left(\mu^{2}/f\right)^{2}$ (18)

so that $\mu/f\ll 1$ for the regime of interest (e.g. in Sec. III.1 we will
show that we require $f\gtrsim 10^{9}$ GeV implying $\mu/f\lesssim 10^{-3}$).
Such a scale difference between the width and height of a potential, required
by the amplitude of density perturbations arising in inflation, is common in
axionic theories, where $\mu$ would roughly be the strong coupling scale and
$f$ the axion decay constant Marsh (2016); Hook et al. (2020). Hence, for
convenience, we refer to $\mu$ as the strong coupling scale. In our case, we
posit that such a difference could occur once a full UV completion of the
Higgs is considered (i.e. in some composite Higgs model that embeds the little
Higgs structure).

In addition to the change in potential, we also employ an EFT analysis to
include two dimension-six Chern-Simons terms coupled to the scalar fields.
This term has the form

$\mathcal{L}_{\rm
CS}=\sum_{j=1}^{2}\frac{g_{j}^{2}\beta^{2}}{16}\text{Tr}\left[Y\Sigma\left(Y\Sigma\right)^{*}\right]\text{Tr}\left[W^{\mu\nu}_{ja}\tilde{W}_{\mu\nu}^{ja}\right],$
(19)

where we take both gauge fields to have the same dimensionless Chern-Simons
coupling $\beta$ and we explicitly point out that $*$ denotes complex (not
Hermitian) conjugation. The invariance of this expression under gauge
transformations is explicitly shown in Appendix B. Upon inserting the
expansion of the $\Sigma$ field, the lowest-order term gives the usual Chern-
Simons factor, which is a total derivative. The higher-order terms yield
couplings between the Chern-Simons current and the scalar sector of the
theory.

##  III pNG Higgs Inflation

Given the particle Lagrangians in Sec. II, we now turn to their inflationary
dynamics, i.e. we both solve for the dynamics and quantify the necessary
Lagrangian parameters to obtain successful pNG Higgs inflation. By successful
pNG Higgs inflation, we mean that at least $N\sim 60$ $e$-folds can be
achieved. To do so, we again first consider the minimal inflationary setup
displayed in Sec. III.1 and then its littlest-Higgs variant in Sec. III.2.

###  III.1 Minimal pNG Higgs Inflation

We begin with the Lagrangian in Eq. (1) and, for simplicity, take a Higgs
potential of the form

$V(\theta)=\mu^{4}\left[1+\cos\left(\theta\right)\right],$ (20)

where $\mu$ is the amplitude of the potential and $\theta=\frac{h}{f}$ is the
normalized Higgs boson. In this case, the Higgs mass is
$m_{H}=(\mu^{2}/f)^{2}$. If we treat $f$ as the scale of spontaneous symmetry
breaking, then we take $f\ll M_{\rm Pl}$ in order to safely neglect quantum
gravity corrections to our Lagrangian.333If instead we take $f/\beta$ as the
scale of spontaneous symmetry breaking, then it is possible to have $f\gtrsim
M_{\rm Pl}$. In this case, $f\ll M_{\rm Pl}$ is an assumption in the following
treatment.  In this limit, the Higgs-covariant derivative $D_{\mu}$ can be
treated as a flat-space derivative $\partial_{\mu}$ since the Higgs
interactions with weak isospin gauge fields are suppressed by a factor
$\left(f/M_{\rm Pl}\right)^{2}\ll 1$.

In order to maintain isotropy of the Universe, the inflaton is only a function
of time, $h(t,{\bf x})=h(t)$. For the same reason, a classical and
rotationally-invariant attractor gauge-field configuration is chosen as in
chromo-natural inflation Adshead and Wyman (2012):

$\displaystyle W_{0}^{a}(t)=0$ $\displaystyle,\quad
W_{i}^{a}(t)=a(t)\psi(t)\delta_{i}^{a},$ (21) $\displaystyle
W_{0i}^{a}(t)=\partial_{t}[a(t)\psi(t)]\delta_{i}^{a}$ $\displaystyle,\quad
W_{ij}^{a}(t)=-gf_{ij}^{a}[a(t)\psi(t)]^{2}.$ (22)

A general gauge-field configuration redshifts away its anistropic parts during
inflation, as the Chern-Simons term is only sensitive to the isotropic piece,
so that the rotationally-invariant configuration can be dynamically achieved
Maleknejad and Erfani (2014); Wolfson et al. (2020, 2021). The hypercharge
gauge boson is assumed to be identically zero, $B_{\mu}=0$.

Successful inflation occurs when the Hubble parameter $H$ slowly evolves,
$\varepsilon=-\dot{H}/H^{2}\ll 1$. Under this slow-roll condition, the
Friedmann equations are

$\displaystyle H^{2}$ $\displaystyle=\frac{\mu^{4}}{3M_{\rm
Pl}^{2}}\left[1+\cos(\theta)\right],$ (23) $\displaystyle\varepsilon$
$\displaystyle=\varepsilon_{h}+\varepsilon_{\psi}\left(1+2\eta_{\psi}+\eta_{\psi}^{2}+m_{\psi}^{2}\right),$
(24)

where we have introduced the dimensionless mass parameter
$m_{\psi}=\frac{g\psi}{H}$ and where

$\displaystyle\varepsilon_{h}$ $\displaystyle=\frac{\dot{h}^{2}}{2M_{\rm
Pl}^{2}H^{2}},\quad\varepsilon_{\psi}=\left(\frac{\psi}{M_{\rm
Pl}}\right)^{2},\quad\eta_{\psi}=\frac{\dot{\psi}}{H\psi},$ (25)

are slow-roll parameters Adshead et al. (2016). Since inflation requires
$\varepsilon<1$, each term in $\varepsilon$ must also be small. In addition,
for the slow-roll solution to persist, the acceleration of the fields must
also be small,

$\displaystyle-\frac{\ddot{h}}{H\dot{h}}=\varepsilon+\hat{\delta}_{\psi}\ll 1$
$\displaystyle,\quad-\frac{\ddot{\psi}}{H\dot{\psi}}=\varepsilon+\hat{\delta}_{\psi}\ll
1,$ (26)
$\displaystyle\implies\hat{\delta}_{h}=-\frac{h^{\prime\prime}}{h^{\prime}}\ll
1$
$\displaystyle,\quad\hat{\delta}_{\psi}=-\frac{\psi^{\prime\prime}}{\psi^{\prime}}\ll
1,$ (27)

where overdot indicates a cosmic time derivative and prime a conformal time
derivative. With the above conditions, the slow-roll equations of motion for
both the Higgs and the gauge-fields are then

$\displaystyle
3\theta^{\prime}-A\sin(\theta)=-3g^{2}\frac{\beta^{2}}{\left(f/M_{\rm
Pl}\right)^{2}}\varepsilon_{\psi}m_{\psi}\theta,$ (28) $\displaystyle
3\varepsilon_{\psi}^{\prime}+4(1+m_{\psi}^{2})\varepsilon_{\psi}=2g^{2}\beta^{2}m_{\psi}\theta\theta^{\prime}\varepsilon_{\psi},$
(29)

with $A=\mu^{4}/(H^{2}f^{2})$ and we have traded the gauge field $\psi$ for
its slow-roll parameter $\varepsilon_{\psi}$. The left-hand sides of these
equations are equivalent to those in Ref. Adshead and Wyman (2012), rewritten
in terms of $\varepsilon_{\psi}$. The right-hand sides are similar, but
different, due to a dimension six (rather than a dimension five) Chern-Simons
operator in Eq. (19) with dimensionless Chern-Simons coupling $\beta$. We seek
static gauge-field solutions, $\varepsilon_{\psi}^{\prime}=0$, so that the
second of the above equations becomes

$\displaystyle\theta^{\prime}$
$\displaystyle=\frac{2}{g^{2}\beta^{2}}\frac{1+m_{\psi}^{2}}{\theta
m_{\psi}}.$ (30)

Combining this equation with Eq. (28) yields

$\displaystyle\varepsilon_{\psi}m_{\psi}\theta$
$\displaystyle=\frac{\left(f/M_{\rm
Pl}\right)^{2}}{\beta^{2}}\left[\frac{1}{3}A\sin(\theta)-\frac{2}{g^{2}\beta^{2}}\frac{1+m_{\psi}^{2}}{m_{\psi}\theta}\right].$
(31)

This equation has a simple solution by further assuming
$g^{2}\beta^{2}m_{\psi}A\theta\sin(\theta)\gg 1$. That is, noting that
$H^{2}m_{\psi}^{2}=g^{2}M_{\rm Pl}^{2}\varepsilon_{\psi}$, we get

$\displaystyle\varepsilon_{\psi}$
$\displaystyle=\frac{\rho^{1/3}}{g^{2}\beta^{2}}\left\\{\frac{{\rm
sinc}(\theta)}{[1+\cos(\theta)]^{1/2}}\right\\}^{2/3},$ (32)

with $\rho=\left(\beta^{2}/3\right)\left(\mu/M_{\rm Pl}\right)^{4}$ and ${\rm
sinc}(\theta)=\sin(\theta)/\theta$. If inflation begins at the top of the
potential and ends at the bottom, so that $\theta\in[0,\pi]$, then it lasts
for

$\displaystyle N$
$\displaystyle=\int_{0}^{\pi}\frac{d\theta}{\theta^{\prime}}=\frac{g^{2}\beta^{2}}{2}\int_{0}^{\pi}d\theta\
\theta\left[\frac{m_{\psi}(\theta)}{1+m_{\psi}^{2}(\theta)}\right]$
$\displaystyle=\frac{g^{2}\beta^{2}}{2}\int_{0}^{\pi}d\theta\
\theta\frac{\rho^{1/3}[1+\cos(\theta)]^{2/3}{\rm
sinc}^{1/3}(\theta)}{\rho^{2/3}[1+\cos(\theta)]^{4/3}+{\rm
sinc}^{2/3}(\theta)},$ (33)

$e$-folds. The maximum number of $e$-folds occurs around $\rho\approx 1$ and,
in this case, $N\approx g^{2}\beta^{2}$. If instead $\rho\ll 1$, which is the
regime of interest, then

$\displaystyle N$ $\displaystyle\approx 60\
\left(\frac{g}{0.64}\right)^{2}\left(\frac{\beta}{3\times
10^{6}}\right)^{8/3}\left(\frac{\mu}{8\times 10^{6}\ {\rm GeV}}\right)^{4/3}.$
(34)

In addition to the above $e$-fold constraint, there is an additional
constraint from the electric dipole moment (EDM) of the electron $d_{e}$ Lue
et al. (1997); Zhang and Young (1994). More precisely, the dimension-six
Chern-Simons operator induces this EDM through the triangle diagram in Fig. 2.

Figure 2: Triangle diagram for the modified electric dipole moment (EDM) of
the electron in our model. The solid arrow lines represent electrons, the wavy
lines photons, and the dashed line the Higgs boson (post-EWSB). The black dot
is the Yukawa electron-Higgs interaction, the empty dot the covariant-
derivative electron-photon interaction, and the crossed dot the induced
electromagnetic Higgs-Chern-Simons interaction described in Appendix C.

Current bounds on this EDM come from spin precession measurements of polar
thorium monoxide (ThO) molecules by the Advanced Cold Molecule Electron EDM
(ACME) collaboration, which provide the limit $d_{e}\lesssim 1.1\times
10^{-29}e\ {\rm cm}$ Andreev et al. (2018).

We show both of these constraints in Fig. 3. The decay constant shown in the
lower horizontal axis must satisfy $f\gtrsim 5\times 10^{11}\ {\rm GeV}$; the
relation $m_{H}^{2}=\left(\mu^{2}/f\right)^{2}$ then leads to the values of
$\mu$ as shown in the upper horizontal axis. The dimensionless Chern-Simons
coupling shown in the vertical axis must satisfy $10^{5}\lesssim\beta\lesssim
10^{13}$.

Figure 3: The viable parameter space for minimal pNG Higgs Inflation, assuming
sub-Planckian decay constants, is shown in white. The lower horizontal axis is
the $SU(5)$ decay constant $f$, the corresponding upper horizontal axis is the
potential height $\mu$ [given by the relation
$m_{H}^{2}=\left(\mu^{2}/f\right)^{2}$], and the vertical axis is the
dimensionless Chern-Simons (CS) coupling $\beta$. The black line is the
maximum couplings allowed by electric-dipole moment (EDM) constraints. The
orange line is the minimum required couplings to achieve $N=60$ $e$-folds of
inflation. These two constraints imply that only decay constants satisfying
$f\gtrsim 5\times 10^{11}\ {\rm GeV}$ are not ruled out. The red dot indicates
the predicted regime given by the standard littlest Higgs potential with
$V\propto f^{4}$ instead of $V\propto\mu^{4}$.

###  III.2 Littlest Higgs Inflation

There are two main differences between the minimal pNG Higgs and littlest
Higgs. First, the Higgs potential is different. Second, there are additional
fields.

If the Higgs potential takes the form
$V(\theta)=c_{0}+c_{2}\sin^{2}\left(2\theta\right)+c_{4}\sin^{4}\left(\theta\right)$,
as in the littlest Higgs model, we numerically find that the number of
$e$-folds is roughly given by Eq. (34), up to small order unity corrections.
We also point out that $8cg^{\prime 2}\approx 1$, with $c=1$, so that the
minimal PNG and littlest-Higgs models have the same parameterization of the
Higgs mass, $m_{H}\approx(\mu^{2}/f)^{2}$.

The main additional fields during the inflating phase (i.e. after global
$SU(5)$ symmetry breaking) are the heavy $SU(2)$ bosons. We now address
whether the heavy gauge boson can be dynamical. A massive gauge boson with
mass $M_{H}$ changes all the equations of motions from
$m_{\psi}^{2}\rightarrow m_{\psi}^{2}+M_{H}^{2}/H^{2}$. For slowly-rolling
gauge fields, the fast oscillations of the mass term dominate the energy
density and lead to a suppressed number of $e$-folds by a factor of the
squared heavy-boson mass $M_{H}^{2}/H^{2}\sim\left(f/M_{\rm Pl}\right)^{2}$.
As a result, $\beta$ must be even larger to compensate for the smallness of
the boson mass. To avoid such a situation, we therefore conclude that the
heavy gauge-boson must have zero dynamics, so that littlest-Higgs inflation
reduces to the minimal set up in Sec. III.1.

##  IV Discussion and Conclusion

Why should the Higgs be the inflaton? One line of reasoning is that its
presence as the only scalar in the SM yields a minimal explanation for the
identity of the inflaton. Another is that the inflaton and Higgs boson both
suffer from a hierarchy problem: loop corrections to their respective
potentials spoil the viability of their theories. Previous work on Higgs
inflation has led to a rich series of models, rooted in modifications of
gravity, whose predictiveness is highly dependent on the precise resolution to
their violations of perturbative unitarity, which is largely unknown. Here, we
thus engaged with an alternative realization for Higgs inflation without
modifying gravity, whereby the Higgs has an approximate shift-symmetry as a
pNG boson and slow-roll is sustained through friction induced by a suitable
weak-isospin Chern-Simons operator. In our scenario, inflation thus happens
only within the electroweak sector.

In order to maintain a general framework, we consider two variants of pNG
Higgs inflation. One, where we use only the minimal ingredients as dictated by
our friction-assisted pNG description and another that embeds this minimal
model by explicitly placing the pNG Higgs into a non-linear sigma model, i.e.
we consider a little Higgs embedding. Specifically, we choose the simplest of
such models, the littlest Higgs. Moreover, in order to consider the widest
possible range of inflationary scales, we make a phenomenological replacement
and consider the height and width of the Higgs potential to be independent
parameters. However, we note that the standard description for the littlest
Higgs only works around $f\sim$TeV scales; larger values of $f$ wreck the
cancellations of quadratic divergences and restore the hierarchy problem.
Nevertheless, we imagine that our amplitude replacement is reasonable, as
similar scalings are seen in other pNG potentials (e.g. axions), and surmise
that it can be obtained in some true UV completion of the littlest Higgs which
would fully address the hierarchy problem (e.g. some composite Higgs theory).
We leave investigation into such UV completions for future work.

We find that, in both cases, successful Higgs inflation can occur at high
energies, with decay constants $f\gtrsim 5\times 10^{11}\ {\rm GeV}$, and with
a large dimensionless Chern-Simons couplings $10^{5}\lesssim\beta 10^{13}$
(see Fig. 3). It may be possible to achieve this large coupling through a
variety of means, such as by including other couplings between the Higgs and
the gauge fields, as in models with deconstructed dimensions Arkani-Hamed et
al. (2001b), or including multiple Higgs fields, such as in assisted Liddle et
al. (1998), $N$-flation Dimopoulos et al. (2008), or clockwork Kaplan and
Rattazzi (2016); Craig et al. (2017); Agrawal et al. (2018) models. The
additional fields (e.g. the massless $\eta$) may serve as one such assisted
field when properly considered. In the case of multiple Higgs, we would be
considering multiple copies of the model at hand. It may also be possible to
lessen the large dimensionless coupling with models that have additional
sources of friction, such as in warm inflation models Berera and Fang (1995);
Berera (1995); Visinelli (2011); Berghaus et al. (2020); Montefalcone et al.
(2023).

Instead of using SM field content, a dark Higgs and/or dark $SU(2)$ gauge-
fields can instead be used to achieve inflation. In the case of a dark Higgs,
the calculations we have presented remain the same, although the restriction
of recovering the SM Higgs mass and TeV electroweak physics is lifted. We note
that, in either case, it also is possible to use the little Higgs framework in
concert with the original non-minimal coupling of Higgs inflation, rather than
a new source of friction.

We have primarily focused here on the background evolution of inflation and
largely ignored predictions of the perturbations. Given the similarities
between this model and that of chromo-natural models, one immediate concern is
that those models are not in agreement with observations of the cosmic
microwave background; the original chromo-natural construction fails to
produce the correct spectral index for the primordial scalar power spectrum
and does not satisfy experimental constraints on the tensor-to-scalar ratio.
However, the original authors found that introducing a mass term for the gauge
field (by Higgsing) screens the gauge field fluctuations thus rendering
chromo-natural inflation observationally viable Adshead et al. (2016). In our
model, since the Higgs obtains a VEV during inflation, it naturally induces a
mass for the gauge fields and is likewise viable, by extension. We will
consider a full cosmological perturbation analysis for future work.

Moreover, while the additional fields of littlest Higgs inflation do not
change the dynamics of the background evolution, they can have an impact on
the perturbations through non-Gaussian signatures Lee et al. (2016). That is,
the additional fields in question have masses near the scale of inflation,
leading to oscillatory features in cosmological correlators generic to quasi-
single field inflation Chen and Wang (2010). In addition, interactions between
the Higgs and these other fields will yield cosmological collider signatures
Arkani-Hamed and Maldacena (2015); Lu et al. (2020).

This work lays the foundation for further studies with this model and its
extensions. For example, in one future work we will examine this model through
the lens of the analysis done in Ref. DeRocco et al. (2021); there exists the
possibility that this model, like many other models that employ a natural
inflation-like mechanism, must inherently take backreaction effects into
account, thus making them models of warm inflation. As mentioned previously,
this warming of inflation may be an avenue to lower the dimensionless Chern-
Simons coupling, making it a logical extension.

A simplifying assumption that underpins this work is that the Higgs rolls into
the same EW-breaking minimum throughout the entire universe; a priori this
need not be the case. Given that Higgs-dependent masses are $2\pi$-periodic in
$H$, these various vacua would be indistinguishable experimentally, but would
have lasting signals such as domain walls or tunnelling phenomena.

Finally, our approach has the virtue of providing a natural baryogenesis Lue
et al. (1997) and reheating mechanism, since the Higgs inflaton has gauge
invariant couplings to leptons and baryons and will decay into baryons,
leptons and gauge bosons at the end of inflation. A detailed analysis of such
a baryogenesis and reheating mechanism inherent to our model will also be
pursued in the near future.

### Acknowledgments

We thank Aaron Pierce and Robert McGehee for useful discussions. KF thanks Can
Kilic and Barmak Shams for extremely helpful conversations. CCS thanks Ken Van
Tilburg for reading a previous draft and providing useful comments. SA was
supported by the Simons Foundation award number 896696. HBG is supported by
the National Science Foundation, MPS Ascending Fellowship, Award 2213126. KF
is grateful for support from the Jeff and Gail Kodosky Endowed Chair in
Physics at the Univ. of Texas, Austin. KF acknowledges funding from the U.S.
Department of Energy, Office of Science, Office of High Energy Physics program
under Award Number DE-SC0022021. KF acknowledges support by the
Vetenskapsradet (Swedish Research Council) through contract No. 638- 2013-8993
and the Oskar Klein Centre for Cosmoparticle Physics at Stockholm University.

##  Appendix A Pion Physics

In this appendix we briefly review the physics of pions, as a reminder of the
structures employed in the construction of little Higgs models. At low
energies, the fundamental degrees of freedom of QCD are confined and
reorganized into composite degrees of freedom, namely the baryons and mesons.

At low energies ($\sim 100$ MeV), of particular relevance are the up, the
down, and, to a lesser extent, the strange quarks. Compared to their charm,
bottom, and top siblings, these quarks are effectively massless ($m_{u}\sim 2$
MeV, $m_{d}\sim 4$ MeV, $m_{s}\sim 95$ MeV). This (relative) degeneracy means
that the effective theory has a flavor symmetry, $SU(2)_{V}$ [$SU(3)_{V}$ if
the strange quarks are included in the discussion; we’ll proceed with just the
up/down case for this section].

Suppose that the quarks are taken to be massless. In this case, the symmetry
is exact and is furthermore enlarged to a chiral symmetry
$SU(2)_{\text{L}}\times SU(2)_{\text{R}}$. This symmetry has
$2\times(2^{2}-1)=6$ generators. Of course, quarks are not massless;
dimensional transmutation in QCD introduces a mass scale ($\Lambda_{\rm
QCD}$), so the massless limit chiral symmetry is spontaneously broken into the
diagonal flavor symmetry as above, $SU(2)_{V}$, which has $3$ generators.
Thus, by the Goldstone theorem, we would expect there to be 3 massless scalar
modes in the spectrum of the theory. Furthermore, these Goldstone modes remain
massless to all orders, which limits them to derivative couplings that may be
written down in the Lagrangian. In this case, these three modes are known as
the pions, $\pi^{\pm},\pi^{0}$. The dynamics of the pions are encoded as a
non-linear sigma model (NLSM), in a Lagrangian of the form

$\mathcal{L}=\frac{f^{2}}{8}\text{Tr}\left[\partial^{\mu}U^{\dagger}\partial_{\mu}U\right]$
(35)

where $U$ is a unitary operator $U=\exp\left[\frac{2\pi
i}{f}\vec{\pi}\cdot\vec{T}\right]$. Here $\vec{T}$ is the vector of broken
$SU(2)$ generators. Note that here it is very easy to see that under a
transformation $\vec{\pi}\rightarrow\vec{\pi}+f\vec{\alpha}$ the Lagrangian is
manifestly invariant.

Of course, the flavor symmetry is not exact; in addition to the spontaneous
symmetry breaking, the symmetry is also broken explicitly by the Yukawa
couplings which introduce mass differences between the quarks. The Goldstone
theorem may be extended to such a case, wherein the Goldstone modes are no
longer massless, but instead slightly massive, with the mass scale
proportional to the degree to which the symmetry is explicitly broken. In this
case, this means that the mass of the pions is controlled by the mass
difference between the up and down quark.

In little Higgs models, this structure is adapted; the overarching flavor
symmetry is replaced by a larger symmetry group of some UV-complete theory. In
the littlest Higgs this is the $SU(5)$ group. The breaking of this symmetry to
some ‘intermediate’ group, which happens spontaneously, corresponds in the
pion framework to the $SU(2)_{V}$ diagonal symmetry and to $SO(5)$ in the
littlest Higgs framework. Lastly, the explicit breaking of $SU(5)$ to the
$SU(2)\times[SU(2)\times U(1)]$ subgroup is analogous to the breaking of
$SU(2)_{\mathrm{L}}\times SU(2)_{\mathrm{R}}$ by the quark mass differences.

##  Appendix B Chern-Simons EFT Couplings

The centerpiece of the littlest Higgs model is the $SU(5)/SO(5)$ coset field,
whose dynamics are encoded as a NLSM model with Lagrangian

$\mathcal{L}=\frac{f^{2}}{8}\mathrm{Tr}\left[\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma\right]$
(36)

where $\Sigma$ is defined as

$\Sigma=\exp\left[\frac{2i}{f}\Pi\right]\Braket{\Sigma}=\exp\left[\frac{2i}{f}T_{a}\pi_{a}\right]\Braket{\Sigma}$
(37)

where $\Pi$ is the ‘pion’ field matrix made by contracting and $f$ is the
‘pion’ decay constant. This matrix has the following transformation property:

$\Sigma\rightarrow U\Sigma U^{\mathsf{T}}$ (38)

It can be shown that the generators of $SU(5)$ can be classed as either broken
or unbroken, $X_{a}$ and $T_{a}$, respectively, satisfying these relations

$\displaystyle
T_{a}\Braket{\Sigma}+\Braket{\Sigma}T_{a}^{\mathsf{T}}=0\qquad\text{and}\qquad
X_{a}\Braket{\Sigma}-\Braket{\Sigma}X_{a}^{\mathsf{T}}=0$

The pion matrix fully written out is

$\begin{split}&\Pi=\begin{pmatrix}-\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&-\frac{\omega^{+}}{\sqrt{2}}&\frac{H^{+}}{\sqrt{2}}&-i\phi^{++}&-\frac{i\phi^{+}}{\sqrt{2}}\\\
-\frac{\omega^{-}}{\sqrt{2}}&\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&\frac{H^{0}}{\sqrt{2}}&-i\frac{\phi^{+}}{\sqrt{2}}&\frac{-i\phi^{0}+\phi^{0}_{P}}{\sqrt{2}}\\\
\frac{H^{-}}{\sqrt{2}}&\frac{H^{0*}}{\sqrt{2}}&\sqrt{\frac{4}{5}}\eta&\frac{H^{+}}{\sqrt{2}}&\frac{H^{0}}{\sqrt{2}}\\\
i\phi^{--}&\frac{i\phi^{-}}{\sqrt{2}}&\frac{H^{-}}{\sqrt{2}}&-\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&-\frac{\omega^{-}}{\sqrt{2}}\\\
\frac{i\phi^{-}}{\sqrt{2}}&\frac{i\phi^{0}+\phi^{0}_{P}}{\sqrt{2}}&\frac{H^{0*}}{\sqrt{2}}&-\frac{\omega^{+}}{\sqrt{2}}&\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}\end{pmatrix}\\\
&=\begin{pmatrix}\omega-\frac{\eta}{\sqrt{20}}\mathbb{1}_{2}&\frac{H}{\sqrt{2}}&\phi^{\dagger}\\\
\frac{H^{\dagger}}{\sqrt{2}}&\sqrt{\frac{4}{5}}\eta&\frac{H^{\mathsf{T}}}{\sqrt{2}}\\\
\phi&\frac{H^{*}}{\sqrt{2}}&\omega^{\mathsf{T}}-\frac{\eta}{\sqrt{20}}\mathbb{1}_{2}\end{pmatrix}.\end{split}$
(39)

In addition to the spontaneous breaking induced by $\Braket{\Sigma}$, the
$SU(5)$ symmetry is also explicitly broken by gauging an
$SU(2)\times[SU(2)\times U(1)]$ subgroup. In the following, $Q^{a}_{1,2}$ and
$Y$ are the generators of the $SU(2)$ and $U(1)$ subgroups of $SU(5)$ that are
being gauged and $W^{\mu}_{i,a}$ and $B^{\mu}$ are the gauge fields. With four
gauge fields, there are four corresponding couplings: two ‘weak’ couplings
$g_{1},g_{2}$ and a hypercharge coupling $g^{\prime}$. These are generated by

$\displaystyle Q_{1}^{a}=\begin{pmatrix}\frac{\sigma_{a}}{2}&&\\\ &&\\\
&&\end{pmatrix},$ (40) $\displaystyle Q_{2}^{a}=\begin{pmatrix}&&\\\ &&\\\
&&-\frac{\sigma_{a}^{*}}{2}\end{pmatrix},$ (41) $\displaystyle
Y=\frac{1}{2}\begin{pmatrix}\mathbb{1}_{2}&&\\\ &&\\\ &&\end{pmatrix}.$ (42)

Thus the full covariant derivative is

$\begin{split}D_{\mu}\Sigma&=\partial_{\mu}\Sigma\\\
&+ig_{j}W^{a}_{j,\mu}\left(Q^{j}_{a}\Sigma+\Sigma
Q_{a}^{j\mathsf{T}}\right)\\\ &+ig^{\prime}B_{\mu}\left(Y\Sigma+\Sigma
Y\right)\end{split}$ (43)

so that the NLSM Lagrangian is

$\mathcal{L}=\frac{f^{2}}{8}\mathrm{Tr}\left[\left(D_{\mu}\Sigma\right)^{\dagger}D^{\mu}\Sigma\right].$
(44)

The explicit expressions showing the three- and four-point scalar-boson
couplings can be found by expanding the $\Sigma$ field

$\Sigma=\exp\left[\frac{2i}{f}\Pi\right]\Braket{\Sigma}\approx\Braket{\Sigma}+\frac{2i}{f}\Pi\Braket{\Sigma},$
(45)

where the $\Pi\Braket{\Sigma}$ term is given by

$\displaystyle\Pi\Braket{\Sigma}$
$\displaystyle=\begin{pmatrix}&\frac{H}{\sqrt{2}}&\phi^{\dagger}\\\
\frac{H^{\dagger}}{\sqrt{2}}&&\frac{H^{\mathsf{T}}}{\sqrt{2}}\\\
\phi&\frac{H^{*}}{\sqrt{2}}&\end{pmatrix}\begin{pmatrix}&&\mathbb{1}_{2}\\\
&1&\\\
\mathbb{1}_{2}&&\end{pmatrix}=\begin{pmatrix}\phi^{\dagger}&\frac{H}{\sqrt{2}}&\\\
\frac{H^{\mathsf{T}}}{\sqrt{2}}&&\frac{H^{\dagger}}{\sqrt{2}}\\\
&\frac{H^{*}}{\sqrt{2}}&\phi\end{pmatrix}.$ (46)

We can leverage what we’ve learned thus far to engender a coupling between the
scalar sector and the Chern-Simons term. The unique gauge-invariant term that
is second order in $\Sigma$ and yields a tree-level quadratic interaction is
of the form

$\text{Tr}\left[\left(Y\Sigma\right)\left(Y\Sigma\right)^{*}\right].$ (47)

This term is previously used when examining the gauge boson mass matrix; in
turn may be multiplied by $\text{Tr}[W\tilde{W}]$, which is itself already
gauge invariant:

$\sum_{j=1}^{2}\frac{g_{j}^{2}\beta^{2}}{16}\text{Tr}\left[\left(Y\Sigma\right)\left(Y\Sigma\right)^{*}\right]\text{Tr}\left[\tilde{W}_{j}^{a}W^{j}_{a}\right].$
(48)

Here $\beta$ is a constant determined by the UV complete theory, to which we
are agnostic. Expanding the trace of $\Sigma$ fields to lowest order in $\Pi$
yields a mass term for the Higgs doublet proportional to $g^{\prime}$, the
$U(1)_{Y}$ subgroup of $SU(5)$. This is to be expected, as the gauged subgroup
explicitly breaks $SU(5)$ and the shift symmetry prohibiting mass terms with
it. As shown in deriving Eq. (17), a quadratic Higgs interaction of the form

$\displaystyle\sum_{j=1}^{2}\frac{g^{2}_{j}\beta^{2}}{16}\text{Tr}\left[\left(Y\Sigma\right)\left(Y\Sigma\right)^{*}\right]\text{Tr}\left[\tilde{W}_{j}W_{j}\right]$
(49)
$\displaystyle\rightarrow\frac{\beta^{2}}{2f^{2}}\left(H^{\dagger}H\right)\sum_{j=1}^{2}g_{j}^{2}\text{Tr}\left[\tilde{W}W\right].$
(50)

arises, coupling to the CS current, as desired.

##  Appendix C Electron Electric Dipole Moment

We calculate the induced electric dipole moment (EDM) of the electron from the
new Chern-Simons operator. Below EWSB, the Higgs doublet is parameterized as
$H=[0,(v+h)/\sqrt{2}]^{\mathsf{T}}$, with $v=246\ {\rm GeV}$ the Higgs
electroweak VEV and $h$ now representing fluctuations from this value (not to
be confused with the background $h$ elsewhere in the text). Moreover, the
hypercharge gauge field $B_{\mu}$ and the third weak-isospin gauge boson
$W_{\mu}^{3}$ mix to form the massive $Z$-boson $Z_{\mu}$ and massless photon
$A_{\mu}$. In particular, the third weak-isospin gauge boson is given by the
linear combination $W_{\mu}^{3}=\sin\left(\theta_{\rm
W}\right)A_{\mu}+\cos\left(\theta_{\rm W}\right)Z_{\mu}$, giving rise to the
electromagnetic Chern-Simons operator

$\displaystyle\frac{g^{2}\beta^{2}}{2f^{2}}\left(H^{\dagger}H\right)\left[W_{a}^{\mu\nu}\tilde{W}_{\mu\nu}^{a}\right]\supset\frac{e^{2}\beta^{2}}{4f^{2}}\left(hv\right)F^{\mu\nu}\tilde{F}_{\mu\nu},$
(51)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field strength tensor and we used the gauge-coupling relation
$g=e/\sin\left(\theta_{\rm W}\right)$. Together with the SM Yukawa electron-
Higgs interaction,

$\displaystyle
y_{1}L_{1}He_{R}\supset\frac{y_{1}}{\sqrt{2}}h\bar{\Psi}_{e}\Psi_{e},$ (52)

as well as the covariant-derivative electron-photon interaction

$\displaystyle i\bar{\Psi}_{e}\gamma^{\mu}D_{\mu}\Psi_{e}\supset
eA_{\mu}\bar{\Psi}_{e}\gamma^{\mu}\Psi_{e},$ (53)

the electromagnetic Chern-Simons operator induces an electron EDM through the
triangle diagram in Fig. 2. In the above expressions,
$L_{1}=\left(\nu_{eL},e_{L}\right)$ is the first generation lepton doublet,
$\Psi_{e}=\left(e_{L},e_{R}\right)$ is the electron Dirac spinor, and $e_{i}$,
$i\in\\{L,R\\}$, are the corresponding electron chirality states.

The induced electron electric dipole moment is then Lue et al. (1997)

$\displaystyle\frac{d_{e}}{e}$
$\displaystyle=\frac{m_{e}\sin^{2}\left(\theta_{\rm
W}\right)}{8\pi^{2}}\frac{g^{2}\beta^{2}}{2f^{2}}\log\left(\frac{f^{2}+m_{H}^{2}}{m_{H}^{2}}\right).$
(54)

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