††institutetext: CAFPE and Departamento de Física Teórica y del Cosmos,
Universidad de Granada,
E-18071 Granada, Spain

# Phenomenological implications of the new Littlest Higgs model with T-parity

José Ignacio Illana and José María Pérez-Poyatos<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the parameter space of the new Littlest Higgs model with
T-parity (NLHT) recently introduced to cure some pathologies of the original
LHT. The model requires extra fermion content and additional pseudo-Goldstone
bosons. While the heavy top quark sector is similar, there are both T-odd and
T-even heavy quarks and leptons with masses proportional to just two sets of
Yukawa matrices in flavor space, one more than in the LHT. The new scalars are
a singlet and real triplet, T-odd, with masses controlled by gauge and Yukawa
couplings, independent of the spontaneous symmetry breaking scale $f$, and
hence potentially light. Imposing that no mass exceeds the cutoff scale,
applying current lower bounds on vector-like quarks and assuming a simplified
model with mass degenerate heavy fermions compatible with the heavy photon as
dark matter constituent, we find that $f$ gets constrained within the interval
between 2 and 3 TeV, the common Yukawa coupling of heavy leptons gets fixed
and the Yukawa coupling of heavy quarks becomes greatly correlated to the top
quark Yukawa couplings. The particle spectrum is then bounded from below and
above, with the (lightest) heavy photon at about 0.5 TeV, not far from the
heavy leptons, the new scalars below 1 TeV, the usual complex scalar triplet
close to the heavy weak bosons at about 1.5 to 2.5 TeV, and the heavy quarks
and top quark partners between 2 and 5 TeV. The new scalars decay
predominantly to a standard and a T-odd lepton and have a width comparable to
that of the Higgs.

##  1 Introduction

Composite Higgs models panicoCompositeNambuGoldstoneHiggs2015  are well
motivated frameworks to study physics beyond the Standard Model (SM). In this
family of models, the Higgs boson and, typically, also additional scalar
degrees of freedom arise as pseudo-Nambu-Goldstone bosons (pNGBs) of a
spontaneously broken approximate global symmetry
colemanStructurePhenomenologicalLagrangians1969 ;
callanStructurePhenomenologicalLagrangians1969 . This global symmetry gets
explicitly broken by gauge and Yukawa interactions, so the pNGBs acquire a
mass proportional to the symmetry breaking couplings at one loop through the
Coleman-Weinberg potential colemanRadiativeCorrectionsOrigin1973 .

Within this class of models, the Littlest Higgs model with T-parity (LHT)
arkani-hamedConstructingDimensions2001 ; arkani-
hamedElectroweakSymmetryBreaking2001 ; ArkaniHamed:2002qy ;
chengTeVSymmetryLittle2003 ; chengLittleHierarchyLittle2004 ;
lowParityLittlestHiggs2004 ; chengTopPartnersLittle2006  is one of the most
elegant proposals. It is based on the global group SU(5) broken spontaneously
to SO(5) by the vacuum expectation value (vev) of a symmetric tensor at the
scale $f\sim 1-10$ TeV. The spontaneous breaking of the global group generates
14 pNGBs. The discrete T-parity symmetry is implemented to relax constraints
from electroweak precision data (EWPD) hubiszPhenomenologyLittlestHiggs2005 ;
hubiszElectroweakPrecisionConstraints2006 . To that end, all the SM particles
are T-even and most of the new particles are T-odd and hence pair-produced. A
subgroup
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{2}$
of the full SU(5) is gauged. This gauge group gets spontaneously broken to the
diagonal $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]$ giving rise to a set
of massive T-odd gauge bosons $W_{H}^{\pm}$, $Z_{H}$ and $A_{H}$. In
particular, the $A_{H}$ gauge boson is the lightest T-odd particle of the LHT
spectrum and is electrically neutral, constituting a dark matter candidate
hubiszPhenomenologyLittlestHiggs2005 ; 2006 ; Wang:2013yba ; Wu:2016rwz . On
the other hand, the Goldstone sector includes the T-even physical Higgs and
the would-be Goldstone bosons absorbed by the SM gauge bosons, together with
extra T-odd scalars: the would-be Goldstone bosons eaten by the heavy gauge
bosons and a physical complex triplet. The Little Higgs (with and without
T-parity) phenomenology has been extensively explored
hubiszPhenomenologyLittlestHiggs2005 ;
hubiszElectroweakPrecisionConstraints2006 ; 2006 ; Dey:2008dk ; Yue:2008zp ;
Wang:2013yba ; Reuter:2013iya ; Yang:2014gca ; Aranda:2015rya ; Kirca:2015iaa
; Cao:2015cdb ; Choudhury:2016wwt ; Aranda:2017raf ; Aranda:2017bgq ;
Wang:2017urv ; Yang:2018oek ; Yang:2018obf ; Dercks:2018hgz ; Long:2019oev ;
Yang:2019uea , also providing new sources of quark and lepton flavor violating
processes Belyaev:2006jh ; Blanke:2006sb ; Tarantino:2006afb ; Blanke:2006eb ;
Duling:2007sf ; Tarantino:2007ur ; delAguila:2008zu ; delAguila:2010nv ;
Blanke:2015wba ; Buras:2015hna ; delaguilaLeptonFlavorChanging2017 ;
delaguilaFullLeptonFlavor2019 ; Pacheco:2021djh ; Pacheco:2022ebc  and
neutrino mass generation delAguila:2005yi ; Han:2005nk ; Lee:2005kd ;
Blanke:2007db ; Hektor:2007uu ; deAlmeida:2007khx ;
delaguilaInverseSeesawNeutrino2019 .

However, the LHT has been shown to suffer from pathologies in the fermionic
sector. The fermion content breaks explicitly the gauge invariance of the LHT
pappadopuloTparityItsProblems2011 ; illana2022new . In this model, the left-
handed components of the SM and the mirror fermions come in incomplete SU(5)
multiplets whereas their right-handed counterparts come in a multiplet of
SO(5) Belyaev:2006jh ; Blanke:2006sb ; Blanke:2006eb ; delAguila:2008zu ;
delAguila:2010nv ; gotoTauMuonLepton2011 ; zhouFlavorChangingTop2013 ;
yangLeptonFlavorViolating2017 . The latter contains an SU(2) doublet of mirror
fermions, together with an SU(2) doublet of mirror partners and a gauge
singlet chengLittleHierarchyLittle2004 ; delaguilaLeptonFlavorChanging2017 .
The mirror fermions acquire a vector-like mass of order $f$ by a Yukawa
coupling to the non-linear scalar field $\xi$. The rest of fermion fields are
usually given large vector-like masses introducing their corresponding left-
handed counterparts in extra, incomplete SO(5) multiplets. However, this setup
breaks gauge invariance because the SO(5) multiplets transform non linearly.
In fact, it has been shown illana2022new  that a gauge transformation involves
all the SO(5) generators, not only those associated to the gauge group, which
implies that any SO(5) multiplet should be complete because in general a gauge
transformation will mix all its components. As a consequence, all the right-
handed fermions transforming in a SO(5) representation must have the same
mass. On the other hand, the singlet fermion field can be given either
T-parity assignment: some authors consider the T-even option
delaguilaLeptonFlavorChanging2017 ; delaguilaInverseSeesawNeutrino2019  while
others choose it T-odd chengLittleHierarchyLittle2004 ;
lowParityLittlestHiggs2004 ; Reuter:2013iya . However, only the T-even
realization is compatible with gauge invariance illana2022new .

The new Littlest Higgs model with T-parity (NLHT) is a minimal extension of
the LHT that fulfills the gauge invariance requirements illana2022new . To
that end, the LHT global symmetry group is minimally enlarged with an extra
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
spontaneously broken to
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$ at the same
scale $f$ due to the vev of another symmetric tensor. The gauge group is still
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{2}$
preserving the number of gauge bosons. As a consequence, the model includes
new particles: a T-odd real triplet and a T-odd singlet in the scalar sector
and a doublet of T-even mirror partners and a T-odd singlet in the fermionic
sector. All the new fermions and the usual T-odd mirror partners and T-even
singlet receive masses proportional to the scale $f$ through a Yukawa
Lagrangian.

In this work we explore the available parameter space compatible with EWPD and
cosmological constraints in a simplified version of the model in which the
Yukawa couplings of heavy leptons and heavy quarks are mass degenerate. In
particular all the new quarks must be heavier than 2 TeV to evade vector-like
quark constraints Vatsyayan:2020jan  while leptons could be lighter. On the
other hand, the new scalars are naturally light independent of $f$. Their
masses parametrically depend on the Yukawa couplings of the top quark and the
heavy fermions. However, they could take any value by tuning these couplings.
Using a naturalness argument and choosing the usual heavy photon $A_{H}$ as a
viable candidate for dark matter, we establish an upper and a lower bounds for
their masses. The paper is organized as follows. Section 2 presents a review
of the NLHT to define the framework and fix the notation. In section 3 we
study the parameter space and the particle spectrum, paying special attention
to the decay channels and lifetime of the new T-odd scalars. The last section
is devoted to our conclusions.

##  2 The new Littlest Higgs model with T-parity

###  2.1 Global symmetries

Let us briefly review the new Littlest Higgs model with T-parity (NLHT),
following closely the notation introduced in ref. illana2022new . The model is
based on the symmetric coset

$\displaystyle
G=\textrm{SU}(5)\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}\xrightarrow{\Sigma_{0},\widehat{\Sigma}_{0}}H=\textrm{SO}(5)\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$
(1)

parametrized by the vacuum expectation value of two symmetric tensors,

$\displaystyle\Sigma_{0}=\left(\begin{array}[]{ccc}0_{2\times
2}&0&{\bf{1}}_{2\times 2}\\\ 0&1&0\\\ {\bf{1}}_{2\times 2}&0&0_{2\times
2}\end{array}\right),\quad\widehat{\Sigma}_{0}=\Sigma_{0},$ (5)

where $\Sigma_{0}$ breaks spontaneously $\textrm{SU}(5)$ into $\textrm{SO}(5)$
leaving $14$ Goldstone bosons, while $\widehat{\Sigma}_{0}$ breaks
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
to $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$ leaving $4$
extra Goldstone bosons at the same energy scale $f$, the scale of new physics
(NP). On the other hand, one needs to extend the global symmetry with external
$\textrm{U}(1)^{\prime\prime\prime}_{1}\times\textrm{U}(1)^{\prime\prime\prime}_{2}$
in order to accommodate the proper hypercharges for the SM right-handed
charged leptons and all quarks (see for instance ref. gotoTauMuonLepton2011 ).
The unbroken generators preserve the vacuum satisfying the relations

$\displaystyle T^{a}\Sigma_{0}+\Sigma_{0}T^{aT}$ $\displaystyle=0,$ (6)
$\displaystyle\widehat{T}^{a}\Sigma_{0}+\Sigma_{0}\widehat{T}^{aT}$
$\displaystyle=0,$ (7)

where the first set of generators corresponds to $\textrm{SU}(5)$ and the
second to
$\left(\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}\right)^{2}$.
The set of broken generators are orthogonal to the unbroken ones satisfying

$\displaystyle X^{a}\Sigma_{0}-\Sigma_{0}X^{aT}$ $\displaystyle=0,$ (8)
$\displaystyle\widehat{X}^{a}\Sigma_{0}-\Sigma_{0}\widehat{X}^{aT}$
$\displaystyle=0.$ (9)

The broken generators span two Goldstone matrices $\Pi=\pi^{a}X^{a}$ and
$\widehat{\Pi}=\widehat{\pi}^{a}\widehat{X}^{a}$. This allows the introduction
of the non-linear fields $\xi$ and $\widehat{\xi}$ that transform under the
global symmetry group,

$\displaystyle\xi$ $\displaystyle=e^{i\Pi/f},\qquad\xi$
$\displaystyle\xrightarrow{G}V\xi U^{\dagger}=U\xi\Sigma_{0}V^{T}\Sigma_{0},$
(10) $\displaystyle\widehat{\xi}$
$\displaystyle=e^{i\widehat{\Pi}/f},\qquad\widehat{\xi}$
$\displaystyle\xrightarrow{G}\widehat{V}\widehat{\xi}\widehat{U}^{\dagger}=\widehat{U}\widehat{\xi}\Sigma_{0}\widehat{V}^{T}\Sigma_{0},$
(11)

where $V$ is a transformation of $\textrm{SU}(5)$, $U=U\left(V,\Pi\right)$ is
the compensating $\textrm{SO}(5)$ non-linear transformation and $\widehat{V}$
is a transformation of
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
with $\widehat{U}=\widehat{U}(\widehat{V},\widehat{\Pi})$ being the
corresponding $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$
non linear transformation.

Note that the second identities in eqs. (10) and (11) are derived from the
invariance of the symmetric tensor $\Sigma_{0}$ under the unbroken subgroup
SO(5), that implies $U\Sigma_{0}=\Sigma_{0}U^{*}$, together with the
characterization of the broken generators, $X^{a}\Sigma_{0}=\Sigma_{0}X^{aT}$,
that implies $\xi^{T}=\Sigma_{0}\xi\Sigma_{0}$. Substituting both relations
into the result of the transformation under the global symmetry group, $\xi\to
V\xi U^{\dagger}$ (CCWZ formalism
callanStructurePhenomenologicalLagrangians1969 ), we have
$\xi=\Sigma_{0}\xi^{T}\Sigma_{0}\to\Sigma_{0}(V\xi
U^{\dagger})^{T}\Sigma_{0}=\Sigma_{0}U^{*}\xi^{T}V^{T}\Sigma_{0}=U\Sigma_{0}\Sigma_{0}\xi\Sigma_{0}V^{T}\Sigma_{0}=U\xi\Sigma_{0}V^{T}\Sigma_{0}$
(and similarly for the hatted fields), as we wanted to prove.

We also define two tensor fields $\Sigma$, $\widehat{\Sigma}$ that transform
linearly under the global symmetry group,

$\displaystyle\Sigma$
$\displaystyle=\xi\Sigma_{0}\xi^{T}=\xi^{2}\Sigma_{0},\qquad\Sigma$
$\displaystyle\xrightarrow{G}V\Sigma V^{T},$ (12)
$\displaystyle\widehat{\Sigma}$
$\displaystyle=\widehat{\xi}\Sigma_{0}\widehat{\xi}^{T}=\widehat{\xi}^{2}\Sigma_{0},\qquad\widehat{\Sigma}$
$\displaystyle\xrightarrow{G}\widehat{V}\Sigma\widehat{V}^{T}.$ (13)

###  2.2 Gauge group

A subgroup
$G_{g}=\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{2}$
of the full global group $G$ is gauged. It is spanned by the Hermitian and
traceless generators

$\displaystyle Q_{1}^{a}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}\sigma^{a}&0&0\\\ 0&0&0\\\
0&0&0_{2\times 2}\end{array}\right)$ $\displaystyle,\quad
Y_{1}=\frac{1}{10}\textrm{diag}\left(3,3,-2,-2,-2\right),$ (17) $\displaystyle
Q_{2}^{a}$ $\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}0_{2\times
2}&0&0\\\ 0&0&0\\\ 0&0&-\sigma^{a*}\end{array}\right)$ $\displaystyle,\quad
Y_{2}=\frac{1}{10}\textrm{diag}\left(2,2,2,-3,-3\right),$ (21)

with $\sigma^{a}$ the three Pauli matrices.111This is possible because we have
taken the form of the generators in
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
the same as those which span the subgroup
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime}_{2}\subset\textrm{SU}(5)$.
A useful property that the gauge generators satisfy is

$\displaystyle Q^{a}_{1}=-\Sigma_{0}Q^{aT}_{2}\Sigma_{0},\quad
Y_{1}=-\Sigma_{0}Y_{2}^{T}\Sigma_{0},$ (22)

which relates the generators of both gauge subgroups. The vev along the
direction of $\Sigma_{0}$ and $\widehat{\Sigma}_{0}=\Sigma_{0}$ also breaks
spontaneously the gauge group down to the diagonal subgroup
$\textrm{SU}(2)_{L}\times\textrm{U}(1)_{Y}$ identified as the SM gauge group,
generated by the combinations
$\left\\{Q_{1}^{a}+Q_{2}^{a},Y_{1}+Y_{2}\right\\}\subset\left\\{T^{a},\widehat{T}^{a}\right\\}$.
The orthogonal combination are a subset of the broken generators,
$\\{Q_{1}^{a}-Q_{2}^{a},Y_{1}-Y_{2}\\}\subset\\{X^{a},\widehat{X}^{a}\\}$.

The set of broken generators of $\textrm{SU}(5)$, $\\{X^{a}\\}$, spans the
first Goldstone matrix,

$\displaystyle\Pi=\left(\begin{array}[]{ccccc}-\displaystyle\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&-\displaystyle\frac{\omega^{+}}{\sqrt{2}}&-i\displaystyle\frac{\pi^{+}}{\sqrt{2}}&-i\Phi^{++}&-i\displaystyle\frac{\Phi^{+}}{\sqrt{2}}\\\
-\displaystyle\frac{\omega^{-}}{\sqrt{2}}&\displaystyle\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&\displaystyle\frac{v+h+i\pi^{0}}{2}&-i\displaystyle\frac{\Phi^{+}}{\sqrt{2}}&\displaystyle\frac{-i\Phi^{0}+\Phi^{P}}{\sqrt{2}}\\\
i\displaystyle\frac{\pi^{-}}{\sqrt{2}}&\displaystyle\frac{v+h-i\pi^{0}}{2}&\sqrt{\displaystyle\frac{4}{5}}\eta&-i\displaystyle\frac{\pi^{+}}{\sqrt{2}}&\displaystyle\frac{v+h+i\pi^{0}}{2}\\\
i\Phi^{--}&i\displaystyle\frac{\Phi^{-}}{\sqrt{2}}&i\displaystyle\frac{\pi^{-}}{\sqrt{2}}&-\displaystyle\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}&-\displaystyle\frac{\omega^{-}}{\sqrt{2}}\\\
i\displaystyle\frac{\Phi^{-}}{\sqrt{2}}&\displaystyle\frac{i\Phi^{0}+\Phi^{P}}{\sqrt{2}}&\displaystyle\frac{v+h-i\pi^{0}}{2}&-\displaystyle\frac{\omega^{+}}{\sqrt{2}}&\displaystyle\frac{\omega^{0}}{2}-\frac{\eta}{\sqrt{20}}\end{array}\right),$
(28)

where $v$ is the Higgs vev. These Goldstone fields are only charged under
$\textrm{SU}(5)$. Under the SM gauge group this Goldstone matrix includes a
complex symmetric $\textrm{SU}(2)$ triplet with hypercharge $Y=1$ and its
hermitian conjugate,

$\displaystyle\Phi=\left(\begin{array}[]{cc}-i\Phi^{++}&-i\displaystyle\frac{\Phi^{+}}{\sqrt{2}}\\\
-i\displaystyle\frac{\Phi^{+}}{\sqrt{2}}&\displaystyle\frac{-i\Phi^{0}+\Phi^{P}}{\sqrt{2}}\end{array}\right),\quad\Phi^{\dagger}=\left(\begin{array}[]{cc}i\Phi^{--}&i\displaystyle\frac{\Phi^{-}}{\sqrt{2}}\\\
i\displaystyle\frac{\Phi^{-}}{\sqrt{2}}&\displaystyle\frac{i\Phi^{0}+\Phi^{P}}{\sqrt{2}}\end{array}\right),$
(33)

the SM Higgs doublet plus an $\textrm{SU}(2)$ triplet with zero hypercharge,

$\displaystyle H=\left(\begin{array}[]{c}i\pi^{+}\\\
\displaystyle\frac{v+h+i\pi^{0}}{\sqrt{2}}\end{array}\right),\quad\omega=\left(\begin{array}[]{cc}-\displaystyle\frac{\omega^{0}}{2}&-\displaystyle\frac{\omega^{+}}{\sqrt{2}}\\\
-\displaystyle\frac{\omega^{-}}{\sqrt{2}}&\displaystyle\frac{\omega^{0}}{2}\end{array}\right)$
(38)

and a singlet, $\eta$. The set of broken generators of
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$,
denoted $\\{\widehat{X}^{a}\\}$, spans the second Goldstone matrix,

$\displaystyle\widehat{\Pi}=\left(\begin{array}[]{ccccc}-\displaystyle\frac{\widehat{\omega}^{0}}{2}-\frac{\widehat{\eta}}{\sqrt{20}}&-\displaystyle\frac{\widehat{\omega}^{+}}{\sqrt{2}}&0&0&0\\\
-\displaystyle\frac{\widehat{\omega}^{-}}{\sqrt{2}}&\displaystyle\frac{\widehat{\omega}^{0}}{2}-\frac{\widehat{\eta}}{\sqrt{20}}&0&0&0\\\
0&0&\sqrt{\displaystyle\frac{4}{5}}\widehat{\eta}&0&0\\\
0&0&0&-\displaystyle\frac{\widehat{\omega}^{0}}{2}-\frac{\widehat{\eta}}{\sqrt{20}}&-\displaystyle\frac{\widehat{\omega}^{-}}{\sqrt{2}}\\\
0&0&0&-\displaystyle\frac{\widehat{\omega}^{+}}{\sqrt{2}}&\displaystyle\frac{\widehat{\omega}^{0}}{2}-\frac{\widehat{\eta}}{\sqrt{20}}\end{array}\right).$
(44)

These Goldstone fields are charged only under
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$.
$\widehat{\Pi}$ includes a new SU(2) triplet with zero hypercharge,

$\displaystyle\widehat{\omega}=\left(\begin{array}[]{cc}-\displaystyle\frac{\widehat{\omega}^{0}}{2}&-\displaystyle\frac{\widehat{\omega}^{+}}{\sqrt{2}}\\\
-\displaystyle\frac{\widehat{\omega}^{-}}{\sqrt{2}}&\displaystyle\frac{\widehat{\omega}^{0}}{2}\end{array}\right)$
(47)

and a singlet, $\widehat{\eta}$. A combination of the scalar fields in $\Pi$
and $\widehat{\Pi}$ with the same T-parity and quantum numbers under the
gauged subgroup will become the longitudinal modes of the heavy T-odd gauge
bosons. The other combination will remain in the physical spectrum.

###  2.3 Lagrangian

####  2.3.1 Gauge sector

In the construction of the Lagrangian we take into account the action of the
discrete T-parity symmetry introduced to keep the SM gauge bosons T-even and
light while the new ones are T-odd and heavy. The action of T-parity consists
in an interchange of the two gauge groups,

$\displaystyle G_{1}\overset{\rm T}{\longleftrightarrow}G_{2},$ (48)

where $G_{1}=\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{1}$ and
$G_{2}=\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{2}$. This requires that
the coupling constants of both copies must be the same
$g_{1}=g_{2}=\sqrt{2}g$, $g^{\prime}_{1}=g^{\prime}_{2}=\sqrt{2}g^{\prime}$,
with the first set of couplings referring to SU(2) and the second to U(1). The
gauge Lagrangian takes the usual form,

$\displaystyle{\cal
L}_{G}=\sum_{j=1}^{2}\left[-\frac{1}{2}\textrm{tr}\left(\widetilde{W}_{j\mu\nu}\widetilde{W}_{j}^{\mu\nu}\right)-\frac{1}{4}B_{j\mu\nu}B_{j}^{\mu\nu}\right],$
(49)

in terms of fields and field strength tensors,

$\displaystyle\widetilde{W}_{j\mu}=W^{a}_{j\mu}Q^{a}_{j},\quad\widetilde{W}_{j\mu\nu}=\partial_{\mu}\widetilde{W}_{j\nu}-\partial_{\nu}\widetilde{W}_{j\mu}-i\sqrt{2}g\left[\widetilde{W}_{j\mu},\widetilde{W}_{j\nu}\right],\quad
B_{j\mu\nu}=\partial_{\mu}B_{j\nu}-\partial_{\nu}B_{j\mu},$ (50)

where in the first expression the index $j$ is fixed. Before the electroweak
SSB, the SM gauge bosons come from the T-even combinations

$\displaystyle W^{\pm}=\frac{1}{2}\left[\left(W^{1}_{1}+W^{1}_{2}\right)\mp
i\left(W^{2}_{1}+W^{2}_{2}\right)\right],\quad
W^{3}=\frac{W^{3}_{1}+W^{3}_{2}}{\sqrt{2}},\quad
B=\frac{B_{1}+B_{2}}{\sqrt{2}},$ (51)

while the remaining T-odd combinations will define the heavy fields

$\displaystyle
W_{H}^{\pm}=\frac{1}{2}\left[\left(W^{1}_{1}-W^{1}_{2}\right)\mp
i\left(W^{2}_{1}-W^{2}_{2}\right)\right],\quad
W_{H}^{3}=\frac{W^{3}_{1}-W^{3}_{2}}{\sqrt{2}},\quad
B_{H}=\frac{B_{1}-B_{2}}{\sqrt{2}}.$ (52)

####  2.3.2 Scalar sector

In order to assign a T-even parity to the SM Higgs field and T-odd parities to
the rest of the scalar fields, one defines

$\displaystyle\Pi$
$\displaystyle\xrightarrow{\textrm{T}}-\Omega\Pi\Omega,\quad\Omega=\textrm{diag}\left(-1,-1,1,-1,-1\right),$
(53) $\displaystyle\widehat{\Pi}$
$\displaystyle\xrightarrow{\textrm{T}}-\widehat{\Pi}.$ (54)

It is important to remark that $\Omega$ is an element of the center of the
gauge subgroup and consequently commutes with the gauge generators. This fact
is directly related to the gauge invariance of the model, as pointed out in
ref. illana2022new . The T-parity transformation of the Goldstone matrices
defined above reads

$\displaystyle\xi$
$\displaystyle\xrightarrow{\textrm{T}}\Omega\xi^{\dagger}\Omega,\quad\Sigma\xrightarrow{\textrm{T}}\widetilde{\Sigma}\equiv\Omega\Sigma_{0}\Sigma^{\dagger}\Sigma_{0}\Omega,$
(55) $\displaystyle\widehat{\xi}$
$\displaystyle\xrightarrow{\textrm{T}}\widehat{\xi}^{\dagger}.$ (56)

With these ingredients one builds the scalar Lagrangian which is gauge and
T-parity invariant,222A term that mixes both scalar sectors is allowed by
gauge invariance. However it would lead to a quadratically divergent
contribution to the Higgs mass illana2022new .

$\displaystyle{\cal
L}_{S}=\frac{f^{2}}{8}\textrm{tr}\left[\left(D^{\mu}\Sigma\right)^{\dagger}D_{\mu}\Sigma\right]+\frac{f^{2}}{8}\textrm{tr}\left[\left(D^{\mu}\widehat{\Sigma}\right)^{\dagger}D_{\mu}\widehat{\Sigma}\right],$
(57)

where the covariant derivative is defined as

$\displaystyle
D_{\mu}\Sigma=\partial_{\mu}\Sigma-\sqrt{2}i\sum_{j=1}^{2}\left[gW^{a}_{j\mu}\left(Q^{a}_{j}\Sigma+\Sigma
Q^{aT}_{j}\right)-g^{\prime}B_{j\mu}\left(Y_{j}\Sigma+\Sigma
Y_{j}^{T}\right)\right]$ (58)

and the same for $\widehat{\Sigma}$.

####  2.3.3 Fermion sector

Here we will focus on the quark sector of the theory. Leptons were already
described in ref. illana2022new . First of all, one introduces two left-handed
SU(5) quintuplets in the anti-fundamental and fundamental representations,
respectively,

$\displaystyle\Psi^{q}_{1}=\left(\begin{array}[]{c}-i\sigma^{2}q_{1L}\\\
i\chi^{q}_{1L}\\\
-i\sigma^{2}\widetilde{q}^{c}_{1L}\end{array}\right),\quad\Psi^{q}_{2}=\left(\begin{array}[]{c}-i\sigma^{2}\widetilde{q}^{c}_{2L}\\\
i\chi^{q}_{2L}\\\ -i\sigma^{2}q_{2L}\end{array}\right).$ (65)

These SU(5) multiplets must be complete to ensure gauge invariant mass terms.
Under a gauge transformation,

$\displaystyle\Psi_{1}^{q}\xrightarrow{G_{g}}V_{g}^{*}\Psi^{q}_{1},\quad\Psi^{q}_{2}\xrightarrow{G_{g}}V_{g}\Psi^{q}_{2},$
(66)

while under T-parity,

$\displaystyle\quad\Psi^{q}_{1}\xrightarrow{\textrm{T}}\Omega\Sigma_{0}\Psi^{q}_{2}.$
(67)

Consequently, one can define T-even and T-odd combinations given respectively
by

$\displaystyle\Psi^{q}_{+}=\frac{\Psi^{q}_{1}+\Omega\Sigma_{0}\Psi^{q}_{2}}{\sqrt{2}},\quad\Psi^{q}_{-}=\frac{\Psi^{q}_{1}-\Omega\Sigma_{0}\Psi^{q}_{2}}{\sqrt{2}}.$
(68)

Except for the T-even combination of SM quark doublets,

$\displaystyle q_{L}=\frac{q_{1L}-q_{2L}}{\sqrt{2}},$ (69)

the rest needs to be paired with a right-handed fermion to get a vector-like
mass. To that end, an SO(5) and an
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$ right-handed
quintuplet are introduced,

$\displaystyle\Psi^{q}_{R}=\left(\begin{array}[]{c}-i\sigma^{2}(\widetilde{q}^{c}_{-})_{R}\\\
i\left(\chi^{q}_{+}\right)_{R}\\\
-i\sigma^{2}q_{HR}\end{array}\right),\quad\widehat{\Psi}^{q}_{R}=\left(\begin{array}[]{c}-i\sigma^{2}(\widetilde{q}^{c}_{+})_{R}\\\
i\left(\chi^{q}_{-}\right)_{R}\\\ 0_{2}\end{array}\right).$ (76)

We denote with a subscript $\pm$ the T-parity assignment of the fermion fields
except for the SM fermions, which are T-even, and the ‘mirror’ quarks $q_{H}$,
which are T-odd. The doublet $(\widetilde{q}^{c}_{\pm})_{R}$ describes the
‘mirror partner’ quarks and $\left(\chi^{q}_{\pm}\right)_{R}$ are SU(2)
singlets. The transformation under the gauged subgroup reads

$\displaystyle\Psi^{q}_{R}\xrightarrow{G_{g}}U_{g}\Psi^{q}_{R},\quad\widehat{\Psi}^{q}_{R}\xrightarrow{G_{g}}\widehat{U}_{g}\widehat{\Psi}^{q}_{R},$
(77)

where $U_{g}(V_{g},\Pi)$ and $\widehat{U}_{g}(V_{g},\widehat{\Pi})$ are in
general non linear transformations of SO(5) and
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$, respectively,
restricted to the gauged subgroup for a given $V_{g}$. In fact, according to
the CCWZ formalism, for any transformation $V\in G$ (the global symmetry
group) there is an $U\in H$ (the unbroken subgroup of $G$) such that $\xi\to
V\xi U^{-1}$. If in particular $V\in H$ then $U=V$, independent of the
Goldstone fields, so in this case $\xi$ transforms linearly. However if
$V\notin H$ then $U=U(V,\Pi)$ and the transformation is non linear. Therefore
if we restrict ourselves to gauge transformations $V_{g}\in G_{g}\subset G$
the corresponding $U_{g}$ is non linear unless $V_{g}$ is a Standard Model
gauge transformation, because then $V_{g}\in H$.

The T-parity transformation of these right-handed multiplets is given by

$\displaystyle\Psi^{q}_{R}\xrightarrow{\textrm{T}}\Omega\Psi^{q}_{R},\quad\widehat{\Psi}^{q}_{R}\xrightarrow{\textrm{T}}-\Omega\widehat{\Psi}^{q}_{R},$
(78)

in order to assign the proper T-parity to the quarks. With this in mind, one
can construct the two needed Yukawa Lagrangians to give masses to the heavy
quarks,

$\displaystyle{\cal L}^{q}_{Y_{H}}$
$\displaystyle=-\kappa_{q}f\left(\overline{\Psi}^{q}_{2}\xi+\overline{\Psi}^{q}_{1}\Sigma_{0}\xi^{\dagger}\right)\Psi^{q}_{R}+{\rm
h.c.},$ (79) $\displaystyle{\cal L}^{q}_{\widehat{Y}_{H}}$
$\displaystyle=-\widehat{\kappa}_{q}f\left(\overline{\Psi}^{q}_{2}\widehat{\xi}-\overline{\Psi}^{q}_{1}\Sigma_{0}\widehat{\xi}^{\dagger}\right)\widehat{\Psi}^{q}_{R}+{\rm
h.c.},$ (80)

tailored to provide the mirror quarks, the T-even singlet and the T-odd mirror
partner quarks with masses of order $\kappa_{q}f$, and the T-odd singlet and
the T-even mirror partner quarks with masses $\widehat{\kappa}_{q}f$. In this
way, only the first set couples to the Higgs field. As one can notice, this
construction is similar to that for leptons. This cancels quadratic
divergences of the Higgs mass and prevents unwanted infinite contributions to
lepton flavor changing Higgs decays as was discussed in refs.
delaguilaInverseSeesawNeutrino2019  and illana2022new .

The top quark is the heaviest SM fermion and the only contributing
significantly to the quadratic divergences of the Higgs mass at the loop
level. To cancel these corrections, in the LHT as well as in the NLHT, one
introduces a set of ‘cancelon’ fields: top partners $T_{1L}$, $T_{2L}$ and
their corresponding right-handed counterparts $T_{1R}$ and $T_{2R}$. In order
to provide masses to the top quark and its partners while avoiding quadratic
divergences to the Higgs mass, the following Lagrangian inspired in Han:2005ru
; hubiszPhenomenologyLittlestHiggs2005  can be introduced,

$\displaystyle{\cal L}^{t}_{Y}=$
$\displaystyle-i\frac{\lambda_{1}f}{4}\epsilon_{ijk}\epsilon_{xy}\left[\left(\overline{Q}^{t}_{1}\right)_{i}\Sigma_{jx}\Sigma_{ky}+\left(\overline{Q}^{t}_{2}\Sigma_{0}\Omega\right)_{i}\widetilde{\Sigma}_{jx}\widetilde{\Sigma}_{ky}\right]t_{R}$
$\displaystyle-\frac{\lambda_{2}f}{\sqrt{2}}\left(\overline{T}_{1L}\widehat{X}T_{1R}+\overline{T}_{2L}\widehat{X}^{*}T_{2R}\right)+{\rm
h.c.},$ (81)

where $\left\\{i,j,k\right\\}=1,2,3$ and $\left\\{x,y\right\\}=4,5$ and the
SU(2) singlet $t_{R}$ is the T-even right-handed top quark. The left-handed
quarks are embedded in the incomplete SU(5) multiplets

$\displaystyle
Q_{1}^{t}=\left(\begin{array}[]{c}-i\sigma^{2}\mathcal{T}_{1L}\\\ i\hskip
2.27626ptT_{1L}\\\ 0_{2}\end{array}\right),\quad
Q_{2}^{t}=\left(\begin{array}[]{c}0_{2}\\\ i\hskip 2.27626ptT_{2L}\\\
-i\sigma^{2}\mathcal{T}_{2L}\end{array}\right)$ (88)

where the SU(2) doublets

$\displaystyle\mathcal{T}_{rL}=\left(\begin{array}[]{c}t_{rL}\\\
b_{rL}\end{array}\right)$ (91)

have the same transformation properties under the gauge group and T-parity as
those in $\Psi^{q}_{1,2}$ of eq. (65) and
$(T_{1})_{L,R}\xrightarrow{\textrm{T}}(T_{2})_{L,R}$. The main difference with
respect to ref. Han:2005ru ; hubiszPhenomenologyLittlestHiggs2005 , is the
presence of the scalar field
$\widehat{X}=\widehat{\Sigma}^{-1/2}_{33}=e^{-i\sqrt{\frac{4}{5}}\widehat{\eta}}$
with hypercharges
$\left(Y_{1},Y_{2}\right)=\left(\frac{1}{5},-\frac{1}{5}\right)$ under the
gauge group, and its complex conjugate $\widehat{X}^{*}$. This field is
introduced in order to change the hypercharges of the right-handed top quark
partners $(T_{1,2})_{R}$ to those of the right-handed $t_{R}$. These new
hypercharges are more natural since all fermions in a linear representation of
the global group requiring extra hypercharge receive half of it in each of the
external $\textrm{U}(1)^{\prime\prime\prime}_{j}$, $j=1,2$ (see Table 1). The
same applies to fermions that transform in a non linear representation (see
Table. 2), which receive the extra hypercharge needed from the diagonal
$\textrm{U}(1)^{\prime\prime\prime}$. This is because the hypercharge under
the gauged part of
$\textrm{SU}(5)\times\left(\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}\right)^{2}$
is fixed by the form of the generators in eqs. (17) and (21).333Something
similar happens to the SM right-handed charged leptons $\ell_{R}$ and the down
type quarks $d_{R}$. They are singlets under
$\textrm{SU}(5)\times\left(\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}\right)^{2}$,
receiving their hypercharges
$\left(Y_{1},Y_{2}\right)=\left(-\frac{1}{2},-\frac{1}{2}\right)$ and
$\left(-\frac{1}{6},-\frac{1}{6}\right)$, respectively, from the external
$\textrm{U}(1)_{1}^{\prime\prime\prime}\times\textrm{U}(1)_{2}^{\prime\prime\prime}$.

| $\left[\textrm{SU}(2)^{\prime}\times\textrm{U}(1)^{\prime}\right]^{2}\subset\textrm{SU(5)}$ | $\left[\textrm{U}(1)^{\prime\prime\prime}\right]^{2}$ |  $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{2}$
---|---|---|---
$q_{2L}$ | $\left(1,2\right)_{\left(-\frac{1}{5},-\frac{3}{10}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,2\right)_{\left(\frac{2}{15},\frac{1}{30}\right)}$
$q_{1L}$ | $\left(2,1\right)_{\left(-\frac{3}{10},-\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(2,1\right)_{\left(\frac{1}{30},\frac{2}{15}\right)}$
$\chi^{q}_{2L}$ | $\left(1,1\right)_{\left(-\frac{1}{5},\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,1\right)_{\left(\frac{2}{15},\frac{8}{15}\right)}$
$\chi^{q}_{1L}$ | $\left(1,1\right)_{\left(\frac{1}{5},-\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,1\right)_{\left(\frac{8}{15},\frac{2}{15}\right)}$
$\widetilde{q}^{c}_{2L}$ | $\left(2,1\right)_{\left(\frac{3}{10},\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(2,1\right)_{\left(\frac{19}{30},\frac{8}{15}\right)}$
$\widetilde{q}^{c}_{1L}$ | $\left(1,2\right)_{\left(\frac{1}{5},\frac{3}{10}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,2\right)_{\left(\frac{8}{15},\frac{19}{30}\right)}$
$d_{R}$ | $\left(1,1\right)_{\left(0,0\right)}$ | $\left(-\frac{1}{6},-\frac{1}{6}\right)$ |  $\left(1,1\right)_{\left(-\frac{1}{6},-\frac{1}{6}\right)}$
$u_{R},T_{2R},T_{1R}$ | $\left(1,1\right)_{\left(0,0\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,1\right)_{\left(\frac{1}{3},\frac{1}{3}\right)}$
$T_{2L}$ | $\left(1,1\right)_{\left(-\frac{1}{5},\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,1\right)_{\left(\frac{2}{15},\frac{8}{15}\right)}$
$T_{1L}$ | $\left(1,1\right)_{\left(\frac{1}{5},-\frac{1}{5}\right)}$ | $\left(\frac{1}{3},\frac{1}{3}\right)$ |  $\left(1,1\right)_{\left(\frac{8}{15},\frac{2}{15}\right)}$
Table 1: Charge assignments under the different $\left[\textrm{SU}(2)\right]$’s and $\left[\textrm{U}(1)\right]$’s of quarks transforming in a linear representation. These fields are singlets under $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$. Note that $\mathcal{T}_{iL}$ and $t_{R}$ are the third family of $q_{iL}$ and $u_{R}$, respectively | $\textrm{SU}(2)^{\prime}\times\textrm{U}(1)^{\prime}\subset\textrm{SO(5)}$ | $\textrm{SU}(2)^{\prime\prime}\times\textrm{U}(1)^{\prime\prime}$ | $\left[\textrm{U}(1)^{\prime\prime\prime}\right]$ |  $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]$
---|---|---|---|---
$\left(\begin{array}[]{c}-i\sigma^{2}(\widetilde{q}^{c}_{-})_{R}\\\ i(\chi^{q}_{+})_{R}\\\ -i\sigma^{2}q_{HR}\end{array}\right)$ | $\left(\begin{array}[]{c}2_{\frac{1}{2}}\\\ 1_{0}\\\ 2_{-\frac{1}{2}}\end{array}\right)$ | $1_{0}$ | $\frac{2}{3}$ |  $\left(\begin{array}[]{c}2_{\frac{7}{6}}\\\ 1_{\frac{2}{3}}\\\ 2_{\frac{1}{6}}\end{array}\right)$
$(\widetilde{q}^{c}_{+})_{R}$ | $1_{0}$ | $2_{\frac{1}{2}}$ | $\frac{2}{3}$ |  $2_{\frac{7}{6}}$
$(\chi^{q}_{-})_{R}$ | $1_{0}$ | $1_{0}$ | $\frac{2}{3}$ |  $1_{\frac{2}{3}}$
Table 2: Charge assignments under the different
$\left[\textrm{SU}(2)\right]$’s and $\left[\textrm{U}(1)\right]$’s of right-
handed quarks transforming in a non linear representation.

There exist other possible ways to introduce fermions transforming under the
same symmetry group that are also free from quadratically divergent
contributions to the Higgs mass. For instance, in ref.
pappadopuloTparityItsProblems2011  they choose to embed the left-handed
doublets $q_{1L}$ and $q_{2L}$ in representations of the external
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
while the doublet of right-handed mirror fermions transforms under the
diagonal $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}$.
These are coupled through the analog to $\widehat{\xi}$ generating mirror
fermion masses of order $\kappa f$. This construction does not require the
introduction of mirror partner fermions nor the singlet $\chi$ and is gauge
invariant. Since the Higgs belongs to $\xi$, its mass is protected against
radiative corrections from this sector. Besides, the global invariance under
the external
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{\prime\prime}_{2}$
ensures that the new T-odd scalars do not develop a contribution to their mass
terms coming from mirror fermion interactions at one loop. However, we prefer
to keep the structure of the original Yukawa Lagrangian to give masses of
order $\kappa f$ to mirror fermions, compatible with gauge invariance
delaguilaLeptonFlavorChanging2017 ; delaguilaInverseSeesawNeutrino2019 ;
illana2022new . Regarding the SM fermions, which are the T-even combination of
$q_{1L}$ and $q_{2L}$, they are embedded in incomplete SU(5) representations
in order to provide them with masses by the same type of Yukawa Lagrangian as
in our model.

The CCWZ formalism provides us with the kinetic terms and the gauge
interactions for all quarks,

$\displaystyle{\cal L}^{q}_{F}={\cal L}^{q}_{F_{L}}+{\cal L}^{q}_{F_{R}}+{\cal
L}^{q}_{\widehat{F}_{R}}$ (92)

where

$\displaystyle{\cal L}^{q}_{F_{L}}$
$\displaystyle=i\overline{\Psi}^{q}_{1}\gamma^{\mu}D^{q*}_{\mu}\Psi^{q}_{1}+i\overline{\Psi}^{q}_{2}\gamma^{\mu}D^{q}_{\mu}\Psi^{q}_{2}$
(93) $\displaystyle{\cal L}^{q}_{F_{R}}$
$\displaystyle=i\overline{\Psi}^{q}_{R}\gamma^{\mu}\left[\partial_{\mu}+\frac{1}{2}\xi^{{\dagger}}\left(D^{q}_{\mu}\xi\right)+\frac{1}{2}\xi\Sigma_{0}D^{q*}_{\mu}\left(\Sigma_{0}\xi^{{\dagger}}\right)\right]\Psi^{q}_{R},$
(94) $\displaystyle{\cal L}_{\widehat{F}_{R}}^{q}$
$\displaystyle=i\overline{\widehat{\Psi}}^{q}_{R}\gamma^{\mu}\left[\partial_{\mu}+\frac{1}{2}\widehat{\xi}^{{\dagger}}\left(D^{q}_{\mu}\widehat{\xi}\right)+\frac{1}{2}\widehat{\xi}\Sigma_{0}D^{q*}_{\mu}\left(\Sigma_{0}\widehat{\xi}^{{\dagger}}\right)\right]\widehat{\Psi}^{q}_{R},$
(95)

with the covariant derivatives

$\displaystyle D^{q}_{\mu}=\partial_{\mu}$
$\displaystyle-\sqrt{2}ig\left(W^{a}_{1\mu}Q^{a}_{1}+W^{a}_{2\mu}Q^{a}_{2}\right)$
$\displaystyle+\sqrt{2}ig^{\prime}\left[B_{1\mu}\left(Y_{1}+\frac{1}{3}\mathbb{1}_{5\times
5}\right)+B_{2\mu}\left(Y_{2}+\frac{1}{3}\mathbb{1}_{5\times
5}\right)\right],$ (96) $\displaystyle D^{q*}_{\mu}=\partial_{\mu}$
$\displaystyle+\sqrt{2}ig\left(W^{a}_{1\mu}Q^{aT}_{1}+W^{a}_{2\mu}Q^{aT}_{2}\right)$
$\displaystyle-\sqrt{2}ig^{\prime}\left[B_{1\mu}\left(Y_{1}-\frac{1}{3}\mathbb{1}_{5\times
5}\right)+B_{2\mu}\left(Y_{2}-\frac{1}{3}\mathbb{1}_{5\times
5}\right)\right],$ (97)

where we have used the properties of gauge generators in eq. (22) and we have
taken into account the extra hypercharges of quarks showed in Tables 1 and 2.

Finally, the kinetic term and gauge interactions for the SM right-handed top
quark and its corresponding top partners are given by

$\displaystyle{\cal L}^{t}_{F^{\prime}}$
$\displaystyle=i\overline{t}_{R}\left(\partial_{\mu}-i\frac{2}{3}g^{\prime}B_{\mu}\right)t_{R},$
(98) $\displaystyle{\cal L}^{T_{1},T_{2}}_{F^{\prime}}$
$\displaystyle=i\overline{T}_{1L}\left[\partial_{\mu}-\sqrt{2}ig^{\prime}\left(\frac{8}{15}B_{1\mu}+\frac{2}{15}B_{2\mu}\right)\right]T_{1L}+i\overline{T}_{1R}\left(\partial_{\mu}-\frac{2}{3}ig^{\prime}B_{\mu}\right)T_{1R}$
$\displaystyle+i\overline{T}_{2L}\left[\partial_{\mu}-\sqrt{2}ig^{\prime}\left(\frac{2}{15}B_{1\mu}+\frac{8}{15}B_{2\mu}\right)\right]T_{2L}+i\overline{T}_{2R}\left(\partial_{\mu}-\frac{2}{3}ig^{\prime}B_{\mu}\right)T_{2R}.$
(99)

###  2.4 Physical particles

####  2.4.1 Gauge bosons

After the electroweak SSB, the SM gauge bosons are obtained from the T-even
fields of eq. (51) by diagonalizing the Lagrangian ${\cal L}_{S}$ of eq. (57),

$\displaystyle W^{\pm}=\frac{1}{\sqrt{2}}\left(W^{1}\mp
iW^{2}\right),\quad\left(\begin{array}[]{c}Z\\\
A\end{array}\right)=\left(\begin{array}[]{cc}c_{W}&s_{W}\\\
-s_{W}&c_{W}\end{array}\right)\left(\begin{array}[]{c}W^{3}\\\
B\end{array}\right)$ (106)

with

$\displaystyle W^{a}=\frac{W^{a}_{1}+W^{a}_{2}}{\sqrt{2}},\quad\quad
B=\frac{B_{1}+B_{2}}{\sqrt{2}}.$ (107)

To get the physical T-odd gauge bosons, one needs to expand ${\cal L}_{S}$ up
to order $v^{2}/f^{2}$ to derive the heavy physical fields from those in eq.
(52),

$\displaystyle W^{\pm}_{H}=\frac{1}{\sqrt{2}}\left(W^{1}_{H}\mp
iW^{2}_{H}\right),\quad\left(\begin{array}[]{c}Z_{H}\\\
A_{H}\end{array}\right)=\left(\begin{array}[]{cc}1&-x_{H}\frac{v^{2}}{f^{2}}\\\
x_{H}\frac{v^{2}}{f^{2}}&1\end{array}\right)\left(\begin{array}[]{c}W_{H}^{3}\\\
B_{H}\end{array}\right)$ (114)

with

$\displaystyle W^{a}_{H}=\frac{W^{a}_{1}-W^{a}_{2}}{\sqrt{2}},\quad
B_{H}=\frac{B_{1}-B_{2}}{\sqrt{2}},\quad
x_{H}=\frac{5gg^{\prime}}{8\left(5g^{2}-g^{\prime 2}\right)}.$ (115)

Their corresponding masses to order $v^{2}/f^{2}$ are

$\displaystyle M_{W}=\frac{gv}{2}\left(1-\frac{v^{2}}{12f^{2}}\right),\quad
M_{Z}=M_{W}/c_{W},\quad e=gs_{W}=g^{\prime}c_{W},$ $\displaystyle
M_{W_{H}}=M_{Z_{H}}=\sqrt{2}gf\left(1-\frac{v^{2}}{16f^{2}}\right),\quad
M_{A_{H}}=\sqrt{\frac{2}{5}}g^{\prime}f\left(1-\frac{5v^{2}}{16f^{2}}\right).$
(116)

Notice that even though the gauge bosons are the same as in the original
model, the masses of the T-odd combinations are at leading order a factor of
$\sqrt{2}$ larger (see for instance delaguilaLeptonFlavorChanging2017 ). This
is because the new extra scalar sector parametrized by $\widehat{\Sigma}$ also
takes the vev $\Sigma_{0}$ hence giving an additional contribution to the
heavy gauge boson masses. However the T-even gauge bosons couple only to the
Higgs field, belonging to $\Sigma$, so their masses do not change. Let us
point out that $A_{H}$ is always the lightest of the T-odd gauge bosons.

####  2.4.2 Scalars after gauge fixing

The kinetic terms of the scalar fields we have introduced are neither diagonal
nor canonically normalized after the SSB. Besides, the new T-odd scalars from
the extra group mix with old ones with the same quantum numbers. In order to
define the physical scalars and identify the actual would-be-Goldstone fields
one has to perform some redefinitions illana2022new . The redefined scalars
$\eta$, $\omega^{0}$ and $\omega^{\pm}$ are the would-be-Goldstone bosons of
the SSB of the gauge group down to the SM gauge group, eaten by $A_{H}$,
$Z_{H}$ and $W^{\pm}_{H}$. Similarly, $\pi^{0}$ and $\pi^{\pm}$ become the
would-be-Goldstone bosons of the SSB of the SM gauge group down to
$\textrm{U}(1)_{\textrm{em}}$, eaten by $Z$ and $W^{\pm}$. The remaining
scalar fields are all physical. They are the Higgs boson, the usual T-odd
triplet with hypercharge $Y=1$ composed of $\Phi^{\pm\pm}$, $\Phi^{\pm}$,
$\Phi^{0}$ and $\Phi^{P}$, and the four new T-odd scalars of hypercharge
$Y=0$, a singlet $\widehat{\eta}$ and a triplet composed of
$\widehat{\omega}^{0}$ and $\widehat{\omega}^{\pm}$. All of them get a mass by
gauge and Yukawa interactions from the Coleman-Weinberg potential.

The Higgs mass exhibits a dependence on the cut-off scale, that is just
logarithmic as a result of the collective symmetry breaking mechanism,

$\displaystyle M^{2}_{h}$
$\displaystyle=\frac{f^{2}}{8\pi^{2}}\log\Lambda^{2}\left(3\lambda_{1}^{2}\lambda_{2}^{2}-6g^{4}-\frac{2}{5}g^{\prime
4}-12T_{\kappa}\right),$ (117)

where
$T_{\kappa}=\textrm{tr}\left(\kappa_{l}\kappa_{l}^{\dagger}\widehat{\kappa}_{l}\widehat{\kappa}_{l}^{\dagger}\right)+3\textrm{tr}\left(\kappa_{q}\kappa_{q}^{\dagger}\widehat{\kappa}_{q}\widehat{\kappa}_{q}^{\dagger}\right)$
includes both heavy quark and heavy lepton Yukawa couplings illana2022new .
The influence of $T_{\kappa}$ on the Higgs mass corrections is unavoidable in
the NLHT. There is no dependence on $\kappa$’s in the Littlest Higgs model
without T-parity because mirror fermions are not needed in that case. The
implementation of T-parity requires mirror fermions with masses proportional
to $\kappa$. In the LHT the cancellation of quadratic corrections to the Higgs
mass is guaranteed if the right-handed quintuplet contains the singlet $\chi$
illana2022new , and logarithmic corrections cancel as long as $\chi$ is T-even
and has a mass $M_{\chi}=\sqrt{2}\kappa f$. However, in the NLHT with a
complete multiplet $\Psi_{R}$ the Yukawa Lagrangian $\mathcal{L}_{Y_{H}}$ is
SU(5) invariant because the left-handed fermion multiplets $\Psi_{1,2}$ are
also complete, so it cannot generate any contribution to scalar masses through
loops. On the other hand, $\mathcal{L}_{\widehat{Y}_{H}}$ shares the left-
handed multiplets with $\mathcal{L}_{Y_{H}}$ but it is not invariant under
global transformations of any of the symmetry group factors. As a consequence,
the breaking of the global symmetry due to $\mathcal{L}_{\widehat{Y}_{H}}$,
which does not contain the Higgs, can only propagate to the Higgs through
$\Psi_{1,2}$, resulting just in a logarithmically divergent contribution to
its mass proportional to $\kappa^{2}\widehat{\kappa}^{2}$ at one loop, a
manifestation of the collective symmetry breaking mechanism.

The rest of the scalar masses are then given in terms of the Higgs mass and
vev by

$\displaystyle M^{2}_{\Phi}$ $\displaystyle=2\frac{f^{2}}{v^{2}}M_{h}^{2},$
(118) $\displaystyle M^{2}_{\widehat{\omega}}$
$\displaystyle=8M^{2}_{h}\,\frac{g^{4}+T_{\kappa}}{3\lambda_{1}^{2}\lambda_{2}^{2}-6g^{4}-\frac{2}{5}g^{\prime
4}-12T_{\kappa}},$ (119) $\displaystyle M^{2}_{\widehat{\eta}}$
$\displaystyle=\frac{72}{5}M^{2}_{h}\,\frac{T_{\kappa}}{3\lambda_{1}^{2}\lambda_{2}^{2}-6g^{4}-\frac{2}{5}g^{\prime
4}-12T_{\kappa}},$ (120)

The components of both triplets are mass-degenerate at this order. Notice that
the masses of the new T-odd scalars are proportional to the Higgs mass and
independent of the scale $f$, and thus naturally light. This is because they
inherit part of the symmetry from the would-be Goldstone bosons to be eaten
after the SSB at the scale $f$. Similarly to the Higgs mass, they receive
contributions proportional to $T_{\kappa}$. On the other hand, the triplet
$\widehat{\omega}$ has zero hypercharge and the gauge couplings can only
generate a contribution to its mass proportional to $g^{4}$, while the singlet
$\widehat{\eta}$ does not receive any contribution from gauge interactions.
Numerically, their masses could take any value since they are strictly
increasing functions of $T_{\kappa}$ for fixed $\lambda_{1,2}$, keeping in
mind that a successful electroweak SSB requires that theHiggs mass in eq.
(117) remains positive. From these equations we find

$\displaystyle
M^{2}_{\widehat{\eta}}=\frac{9}{5}\frac{T_{\kappa}}{T_{\kappa}+g^{4}}M^{2}_{\widehat{\omega}}.$
(121)

One may argue that the new coupling between the top quark partners and the
scalar singlet $\widehat{\eta}$ in eq. (81) would add a correction to
$M_{\widehat{\eta}}$ not accounted in ref. illana2022new . However, a simple
one loop calculation reveals that this is not the case. Due to the
identification $\widehat{X}\sim e^{i\widehat{\eta}}$, the scalar
$\widehat{\eta}$ can be understood as the Goldstone boson of the spontaneous
breaking of the gauged $\left[\textrm{U}(1)\right]^{2}$ to $\textrm{U}(1)$.
This prevents the generation of a new contribution to its mass at one loop.

Finally, the Higgs quartic coupling is given by

$\displaystyle\lambda=\frac{1}{16\pi^{2}}\frac{\Lambda^{2}}{f^{2}}\left(g^{2}+g^{\prime
2}+3\lambda^{2}_{1}\right).$ (122)

This expression relates the cutoff scale $\Lambda$ to the top Yukawa coupling
$\lambda_{1}$. The value of this coupling is fixed to
$\lambda=\frac{1}{2}(M_{h}/v)^{2}$.

####  2.4.3 Quark masses

Once the fermion content of the model is extended beyond one family, the
Yukawa couplings must be understood as $3\times 3$ matrices in flavor space.
Lepton masses and mixings were tackled in ref. illana2022new . Here we will
restrict ourselves to the case of quarks. Omitting family indices, for each of
the three SM (T-even) left-handed quark doublets $q_{L}$ there is a vector-
like doublet of heavy T-odd mirror quarks $q_{H}$,

$\displaystyle q_{L}=\left(\begin{array}[]{c}u_{L}\\\
d_{L}\end{array}\right)=\frac{q_{1L}-q_{2L}}{\sqrt{2}},\quad
q_{HL}=\left(\begin{array}[]{c}u_{HL}\\\
d_{HL}\end{array}\right)=\frac{q_{1L}+q_{2L}}{\sqrt{2}},\quad
q_{HR}=\left(\begin{array}[]{c}u_{HR}\\\ d_{HR}\end{array}\right),$ (129)

where

$\displaystyle q_{rL}=\left(\begin{array}[]{c}u_{rL}\\\
d_{rL}\end{array}\right),\quad r=1,2$ (132)

are part of the SU(5) multiplets $\Psi^{q}_{r}$ in eq. (65) and $q_{HR}$ is
part of the SO(5) multiplets $\Psi^{q}_{R}$ in eq. (76). The SM right-handed
quarks $d_{R}$, $u_{R}$ are singlets under the full SU(5) but have
hypercharges under the external $\textrm{U}(1)^{\prime\prime\prime}$ groups
(see Table 1).

In addition there are two heavy right-handed mirror-partner doublets,

$\displaystyle(\widetilde{q}^{c}_{-})_{R}=\left(\begin{array}[]{c}(\widetilde{u}^{c}_{-})_{R}\\\
(\widetilde{d}^{c}_{-})_{R}\end{array}\right),\quad(\widetilde{q}^{c}_{+})_{R}=\left(\begin{array}[]{c}(\widetilde{u}^{c}_{+})_{R}\\\
(\widetilde{d}^{c}_{+})_{R}\end{array}\right).$ (137)

The first one is T-odd and it is necessary to complete the multiplets
$\Psi^{q}_{R}$ together with $(\chi^{q}_{+})_{R}$, while the second is T-even
and lays in the incomplete multiplet $\widehat{\Psi}^{q}_{R}$ in eq. (76)
along with $(\chi^{q}_{-})_{R}$. Their corresponding left-handed counterparts
come from the SU(5) multiplets,

$\displaystyle(\widetilde{q}^{c}_{-})_{L}=\left(\begin{array}[]{c}(\widetilde{u}^{c}_{-})_{L},\\\
(\widetilde{d}^{c}_{-})_{L}\end{array}\right)=\frac{(\widetilde{q}^{c}_{1})_{L}+(\widetilde{q}^{c}_{2})_{L}}{\sqrt{2}},\quad(\widetilde{q}^{c}_{+})_{L}=\left(\begin{array}[]{c}(\widetilde{u}^{c}_{+})_{L},\\\
(\widetilde{d}^{c}_{+})_{L}\end{array}\right)=\frac{(\widetilde{q}^{c}_{1})_{L}-(\widetilde{q}^{c}_{2})_{L}}{\sqrt{2}},$
(142)

with

$\displaystyle(\widetilde{q}^{c}_{r})_{L}=\left(\begin{array}[]{c}(\widetilde{u}^{c}_{r})_{L}\\\
(\widetilde{d}^{c}_{r})_{L}\end{array}\right),\quad r=1,2.$ (145)

The SU(2) singlets $(\chi^{q}_{+})_{R}$ and $(\chi^{q}_{-})_{R}$ complete the
set of quarks common to all families. Their left-handed counterparts are the
combinations with proper T-parities of the fields $\chi^{q}_{1L}$ and
$\chi^{q}_{2L}$ completing the SU(5) multiplets,

$\displaystyle(\chi^{q}_{+})_{L}=\frac{\chi^{q}_{1L}+\chi^{q}_{2L}}{\sqrt{2}},\quad(\chi^{q}_{-})_{L}=\frac{\chi^{q}_{1L}-\chi^{q}_{2L}}{\sqrt{2}}.$
(146)

Finally, the top sector includes a set of top partners $T_{1L}$ and $T_{2L}$,
belonging to $Q_{1}^{t}$ and $Q_{2}^{t}$ respectively, and their right-handed
counterparts $T_{1R}$, $T_{2R}$. Their corresponding T-parity eigenstates are

$\displaystyle\left(T_{+}\right)_{L,R}=\frac{\left(T_{1}\right)_{L,R}+\left(T_{2}\right)_{L,R}}{\sqrt{2}},\quad\left(T_{-}\right)_{L,R}=\frac{\left(T_{1}\right)_{L,R}-\left(T_{2}\right)_{L,R}}{\sqrt{2}}.$
(147)

In the following we present the masses for all quarks at leading order. For
simplicity we will assume flavor diagonal and degenerate Yukawa couplings in
the rest of this work. The mirror quarks $q_{H}$, the T-even SU(2) singlet
$\chi_{+}^{q}$ and the T-odd mirror partner quarks $\widetilde{q}^{c}_{-}$
receive degenerate masses from the Lagrangian in eq. (79) of size

$\displaystyle m_{d_{H}}=\sqrt{2}\kappa_{q}f$ (148)

while the T-odd SU(2) singlet $\chi_{-}^{q}$ and the T-even mirror partner
quarks $\widetilde{q}^{c}_{+}$ receive masses from the Lagrangian in eq. (80),

$\displaystyle m_{\widetilde{q}^{c}_{+}}=\sqrt{2}\widehat{\kappa}_{q}f.$ (149)

Finally, the top quark and its corresponding T-even partner mix in eq. (81).
After rotating the fields at leading order we have

$\displaystyle\left(\begin{array}[]{c}t_{R}\\\
\left(T_{+}\right)_{R}\end{array}\right)\rightarrow\left(\begin{array}[]{cc}c_{R}&s_{R}\\\
-s_{R}&c_{R}\end{array}\right)\left(\begin{array}[]{c}t_{R}\\\
\left(T_{+}\right)_{R}\end{array}\right),\quad
c_{R}=\frac{\lambda_{2}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}},\quad
s_{R}=\frac{\lambda_{1}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}},$ (156)

the physical mass of the top quark in terms of $\lambda_{1,2}$ reads

$\displaystyle
m_{t}=\frac{1}{\sqrt{2}}\frac{\lambda_{1}\lambda_{2}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}v,$
(157)

and its T-even and T-odd partners have masses

$\displaystyle
M_{T_{+}}=\sqrt{\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{2}}f,\quad
M_{T_{-}}=\frac{\lambda_{2}}{\sqrt{2}}f.$ (158)

Notice that the T-even top partner is always heavier than the T-odd one.

##  3 NLHT phenomenology

###  3.1 Parameter space

Before constraining the parameters of the model, some relations must be taken
into account. Let us focus first on the Yukawa couplings $\lambda_{1}$ and
$\lambda_{2}$ in eq. (81). They are not independent but connected to the top
quark mass $m_{t}\approx 173$ GeV Zyla:2020zbs  according to eq. (157). At
leading order in $v$ one finds

$\displaystyle\frac{1}{\lambda_{1}^{2}}+\frac{1}{\lambda_{2}^{2}}=\left(\frac{v}{\sqrt{2}m_{t}}\right)^{2}.$
(159)

The masses of the T-even and T-odd top partners ($T_{+}$ and $T_{-}$,
respectively) are given in eq. (158) in terms of these Yukawa couplings.
$T_{+}$ is heavier and we have to impose that its mass remains below the
cutoff scale $\Lambda$ or otherwise it should be integrated out and would not
be part of the spectrum of the theory. The cutoff scale is related to the
Higgs quartic coupling $\lambda=\frac{1}{2}(M_{h}/v)^{2}\approx 0.13$, the
Yukawa coupling $\lambda_{1}$ and the gauge couplings $g\approx 0.641$ and
$g^{\prime}\approx 0.344$ through eq. (122), where $g=e/s_{W}$ and
$g^{\prime}=e/c_{W}$ with $s_{W}^{2}=1-M_{W}^{2}/M_{Z}^{2}$ and
$e^{2}=4\pi\alpha$. This constrains the value of $\lambda_{1}$ to be in the
interval $\lambda_{1}\in\left[1.05,1.71\right]$, equivalently
$\lambda_{2}\in\left[1.22,3.10\right]$, as shown in fig. 1.

Figure 1: The interval of $\lambda_{1}$ yielding a top quark partner mass
$M_{T_{+}}$ below the scale $\Lambda$ on the plot of the left-hand side
determines the range of possible values of $\lambda_{2}$ from eq. (157) on the
plot on the right-hand side. These intervals are independent of $f$ but the
mass scales are proportional to $f$.

Since the cutoff scale and the top quark partner masses are proportional to
$f$, fig. 1 also shows that $\Lambda/f\in\left[1.49,2.32\right]$. These values
are compatible with the perturbative limit $\Lambda\lesssim 4\pi f$. In
addition, current EWPD impose that vector-like quarks must be heavier than
about 2 TeV Vatsyayan:2020jan . Therefore, we will force the lighter top
partner $T_{-}$ to have a mass $M_{T_{-}}>2$ TeV, implying $f\gtrsim 900$ GeV.

On the other hand, T-parity is exact and thus the lightest T-odd particle
(LTP) is stable. To be a viable dark matter candidate, the LTP must be
electrically neutral. In our model, the potentially light and neutral light
T-odd particles are the $A_{H}$ gauge boson, the neutral leptons, with masses
proportional to the free parameters $\kappa_{l}$ and $\widehat{\kappa}_{l}$,
the T-odd singlet $\widehat{\eta}$ and the neutral component of the T-odd real
triplet $\widehat{\omega}^{0}$. The neutral components of the original complex
triplet $\Phi$ do not play a role here since they are always heavier than
$A_{H}$ because comparing eqs. (2.4.1) and (118) we have that
$M_{\phi}/f=\sqrt{2}M_{h}/v\approx 0.718$ and
$M_{A_{H}}/f=g^{\prime}\sqrt{2/5}\approx 0.217$. At leading order the neutral
leptons share mass with the charged ones and the same happens with the neutral
and the charged components of the real triplet. This rules them out.
Therefore, the possible dark matter candidates are the singlet
$\widehat{\eta}$ and the $A_{H}$ gauge boson.

Figure 2: Left: Masses of the Higgs and the new T-odd scalars as a function of
$T_{\kappa}$ for a chosen value of $\lambda_{1}=1.5$. They are naturally light
except for $T_{\kappa}$ close to its maximum
($T^{\textrm{max}}_{\kappa}(\lambda_{1}=1.5)\approx 0.90$). Right: mass of the
T-odd singlet $\widehat{\eta}$ as a function of the allowed values of
$\lambda_{1}$ when $M_{\widehat{\eta}}=M_{\widehat{\omega}}$.

From eq. (121) we know that the singlet is lighter than the triplet if
$T_{\kappa}<\frac{5}{4}g^{4}\approx 0.211$. Then according to fig. 2 it has a
mass $M_{\widehat{\eta}}\lesssim 80$ GeV, and since $M_{A_{H}}\gtrsim 200$ GeV
for $f\gtrsim 900$ GeV, the singlet would be the LTP if T-odd leptons are
heavy enough. This opens a possible window to light scalar dark matter, as
already suggested in ref. Feng:2014vea . However, to compare with previous
works hubiszPhenomenologyLittlestHiggs2005 ; 2006 ; Wang:2013yba , we will
consider the scenario where the heavy photon $A_{H}$ is the dark matter
candidate. Then the new scalars must be heavier than $A_{H}$, which also
implies that the singlet is heavier than the triplet.

###  3.2 Simplified model

It is well known that, in order to reproduce within the LHT the current dark
matter relic density $\Omega h^{2}=0.120\pm 0.001$ Planck:2018vyg , it is
necessary the presence of co-annihilators if the LTP has $M_{A_{H}}\gtrsim
200$ GeV. In ref. 2006 , for instance, the T-odd leptons and quarks share
masses and are taken nearly mass-degenerate with the $A_{H}$ gauge boson.
However, according to the current constraints on the vector-like quark masses
already mentioned, this assumption would lead to a very heavy LTP. Here we
will analyze a simplified NLHT model in which all heavy lepton and quark
Yukawa couplings are diagonal and degenerate in flavor space, that is,
$\kappa_{l,i}=\widehat{\kappa}_{l,i}\equiv\kappa_{l}$ and
$\kappa_{q,i}=\widehat{\kappa}_{q,i}\equiv\kappa_{q}$, but they have different
masses $m_{\ell_{H}}=\sqrt{2}\kappa_{l}f\gtrsim M_{A_{H}}$ and
$m_{q_{H}}=\sqrt{2}\kappa_{q}f$.444Another difference is that the NLHT
includes more fermionic degrees of freedom to restore the gauge invariance of
the model. The chosen degeneracy implies that the traces appearing in the
expression for the masses of the new T-odd scalars in eqs. (119) and (120)
reduce to $T_{\kappa}=3\kappa_{l}^{4}+9\kappa_{q}^{4}$. We will consider that
only the T-odd heavy leptons can act as co-annihilators.

One should keep in mind that the contribution of a co-annihilator to the dark
matter relic density is exponentially suppressed as $e^{-\Delta/T_{\rm fo}}$,
where $\Delta$ is the mass splitting between the dark matter candidate and the
co-annihilator and $T_{\rm fo}\sim 20-30$ GeV is the freeze-out temperature
Griest:1990kh . To prevent that the scalars influence the relic density, we
safely impose a lower bound to the mass of the lighter T-odd real triplet
$M_{\widehat{\omega}}>M_{A_{H}}+M_{h}$ using the fact that the new scalar
masses can take any value according to fig. 2. This defines a minimum for the
traces,
$T^{\textrm{min}}_{\kappa,\widehat{\omega}}\left(\lambda_{1},f\right)$.

Figure 3: Region in the $M_{A_{H}}-m_{\ell_{H}}$ plane compatible with $A_{H}$
as dark matter candidate (white). The $A_{H}$ constitutes 100% (50%) of the
dark matter abundance for masses within the orange (green) bands. The grey
region is excluded because it would yield dark matter overabundance and the
blue region is excluded because $A_{H}$ would not be the LTP.

Implementing our model in FeynRules Alloul:2013bka , using micrOMEGAs
Belanger:2013oya  to evaluate the relic density and running T3PS
Maurer:2015gva  to scan the parameter space we obtain the plot in fig. 3. The
main contributions to the dark matter relic density come from the co-
annihilation of T-odd leptons to pair-produce SM gauge bosons $W^{\pm},Z$. The
narrow orange band covers the region where the relic abundance of $A_{H}$ is
compatible with 100% of the observed dark matter density, which can be
approximated by

$\displaystyle m_{\ell_{H}}\approx 1.16\;M_{A_{H}}$ (160)

for $M_{A_{H}}\in\left[200,800\right]$ GeV. From the expressions of
$M_{A_{H}}$ in eq. (2.4.1) and $m_{\ell_{H}}=\sqrt{2}\kappa_{l}f$ the heavy
lepton Yukawa coupling is then constrained to $\kappa_{l}\approx 0.185$ for
$f\gtrsim 900$ GeV.

We have already set a lower bound to the lighter T-odd scalar mass,
$M_{\widehat{\omega}}>M_{A_{H}}+M_{h}$. Using naturalness arguments one can
also expect upper bounds for the scalars. As previously emphasized, the new
scalar masses are proportional to the Higgs mass and are independent of the
scale $f$, so they should remain light. To be definite we impose
$M_{\widehat{\eta}}<1$ TeV, which provides the maximum trace values
$T_{\kappa,\widehat{\eta}}^{\rm max}(\lambda_{1})$ for which
$M_{\widehat{\eta}}=1$ TeV given $\lambda_{1}$. Thus the condition
$T^{\textrm{max}}_{\kappa,\widehat{\eta}}\left(\lambda_{1}\right)>T^{\textrm{min}}_{\kappa,\widehat{\omega}}\left(\lambda_{1},f\right)$
defines an upper bound on the allowed values of $f$ as a function of
$\lambda_{1}$, which is shown in fig. 4. The lower bound that follows from
$M_{T_{-}}>2$ TeV is also depicted.

Figure 4: Values of $f$ compatible with $M_{T_{-}}>2$ TeV as a function of
$\lambda_{1}$ (light blue). The region is further constrained (darker blue) by
the conditions
$T^{\textrm{max}}_{\kappa,\widehat{\eta}}\left(\lambda_{1}\right)>T^{\textrm{min}}_{\kappa,\widehat{\omega}}\left(\lambda_{1},f\right)$
(upper bound) and $T_{\kappa,\widehat{\eta}}^{\rm
max}(\lambda_{1})>T_{\kappa,q_{H}}^{\rm min}(f)$ (lower bound) as explained in
the text.

The condition that $A_{H}$ is the LTP implies $m_{q_{H}}/f>M_{A_{H}}/f$, that
is $\kappa_{q}>g^{\prime}/\sqrt{5}\approx 0.16$. However the current bound on
the heavy quark masses sets a limit $\kappa_{q}>\sqrt{2}\mbox{ TeV}/f$, which
is stronger for the values of $f\lesssim 3$ TeV allowed by fig. 4. From this
and the already constrained value of $\kappa_{l}\approx 0.185$ we derive the
restriction $T_{\kappa}>T_{\kappa,q_{H}}^{\rm min}(f)\equiv
0.0035+36\;\mbox{TeV}^{4}/f^{4}$. This additional condition for the traces
finally sets the allowed region in the $\lambda_{1}-f$ plane of fig. 4, which
features a possible window for $f$ between 2.0 and 3.1 TeV. The minimum for
the traces is actually given by
$T^{\textrm{min}}_{\kappa,\widehat{\omega}}\left(\lambda_{1},f\right)$.

###  3.3 Particle spectrum and scalar decays

As a consequence of all previous constraints, the Yukawa coupling of the heavy
quarks $\kappa_{q}$ and $\lambda_{1}$ get strongly correlated (fig. 5). This
correlation is nearly independent of $f$. This is because the maximum and
minimum values of $\kappa_{q}$ are extremely close due to the asymptotic
behavior of the new scalar masses with $T_{\kappa}$ and
$T_{\kappa,\widehat{\eta}}^{\max}$ does not depend on $f$. However, not all
values of $\lambda_{1}$ are available for every possible value of $f$, except
for $f\in[2.5,3.0]$, as can be seen in fig. 4 and is reflected on fig. 5.
Comparing figs. 1 and 5 one can explicitly check that heavy quarks are safely
below the cut-off scale because $m_{q_{H}}/f=\sqrt{2}\kappa_{q}<\Lambda/f$.
Their common masses are in practice a function of $\lambda_{1}$, ranging
between $0.8f$ and $f$.

Figure 5: Correlation between the heavy quark Yukawa coupling $\kappa_{q}$ and
the top quark Yukawa coupling $\lambda_{1}$ for different values of the scale
$f$.

Figure 6: Ranges of allowed masses of the new T-odd scalars for different
values of the scale $f$ as a function of $\lambda_{1}$ Figure 7: Typical NLHT
spectrum of our simplified model with $f\approx 2.7$ TeV and
$\lambda_{1}=1.2$.

Regarding the scalar masses, fig. 6 shows that they are constrained to the
intervals $M_{\widehat{\omega}}\in\left[600,800\right]$ GeV and
$M_{\widehat{\eta}}\in\left[800,1000\right]$ GeV. The maximum values are fixed
by $M_{\widehat{\eta}}<1$ TeV. The minimum values come from the condition
$M_{\widehat{\omega}}>M_{A_{H}}+M_{h}$ depending on $f$. The allowed values
for $\lambda_{1}$ are determined by $T_{\kappa,\widehat{\eta}}^{\rm
max}(\lambda_{1})>T_{\kappa,q_{H}}^{\rm min}(f)$ and
$T_{\kappa,\widehat{\eta}}^{\rm
max}(\lambda_{1})>T_{\kappa,\widehat{\omega}}^{\rm min}(\lambda_{1},f)$. For
completeness we show below the mass ranges of the T-odd gauge bosons, the
usual T-odd complex scalar triplet and the heavy leptons for the allowed
values of the scale $f$,

$\displaystyle M_{A_{H}}$ $\displaystyle\in\left[450,680\right]\textrm{ GeV},$
(161) $\displaystyle M_{W_{H}}=M_{Z_{H}}$
$\displaystyle\in\left[1850,2750\right]\textrm{ GeV},$ (162) $\displaystyle
M_{\Phi}$ $\displaystyle\in\left[1450,2150\right]\textrm{ GeV},$ (163)
$\displaystyle m_{\ell_{H}}$ $\displaystyle\in\left[530,800\right]\textrm{
GeV}.$ (164)

Our vector-like leptons decay to one standard lepton and a heavy photon, an
exotic decay that is not excluded in principle by current LHC searches within
these mass ranges Guedes:2021oqx . For $f\approx 2.7$ TeV, the typical
spectrum is shown in fig. 7. The top quark partners are always the heaviest
particles. The rest of heavy quarks (T-even and T-odd quarks share masses) are
lighter but above 2 TeV to comply with EWPD constraints. Apart from the top
quark partners, the T-odd $W_{H}$ and $Z_{H}$ are the heaviest gauge bosons
with a large gap with the $A_{H}$ gauge boson, the LTP. The heavy leptons are
only a few tens of GeV heavier than $A_{H}$. This is because, as in the quark
case, these T-even and T-odd leptons are mass degenerate in our simplified
model and the latter act as co-annihilators to reproduce the dark matter relic
density. Finally, in the scalar sector, the usual T-odd triplet $\Phi$ is
always the heaviest one, followed by the singlet $\widehat{\eta}$ and the
triplet $\widehat{\omega}$.

Now that we have studied the full model spectrum, let us discuss the decay
channels of the new T-odd scalars $\widehat{\omega}$ and $\widehat{\eta}$. At
leading order they decay mostly into a left-handed SM lepton and a right-
handed T-odd mirror lepton preserving both T-parity and electric charge. The
couplings involved are independent of the Higgs vev because the SM left-handed
leptons are SU(2) doublets and so are their mirror versions with same
hypercharge as the SM leptons. They are proportional to $\kappa_{\ell}$ and
scale as

$\displaystyle\widehat{\omega}^{\pm}:\widehat{\omega}^{0}:\widehat{\eta}\sim
1/\sqrt{2}:1/2:1/\sqrt{20}.$ (165)

Thus, the quantum numbers can combine to give those of a triplet for
$\widehat{\omega}$ and a singlet for $\widehat{\eta}$. The decays to other
leptons with opposite T-parities involve couplings suppressed by powers of
$v/f<0.12$ or are kinematically suppresed by the higher masses, of at least
$m_{\ell_{H}}\approx 530$ GeV in our simplified model. In addition, from eq.
(57), the charged components of the triplet $\widehat{\omega}^{\pm}$ can decay
into the $A_{H}$ gauge boson and a SM $W^{\pm}$ while the neutral component
$\widehat{\omega}^{0}$ can decay into a $A_{H}$ and a Higgs boson. The singlet
$\widehat{\eta}$ can decay in the same way as the neutral component of the
triplet but, being heavier, it can also decay to the neutral component of the
triplet and a Higgs boson. However those couplings are also suppressed by
powers of the Higgs vev. We have explicitly checked that these other channels
contribute less than a 3% to the total decay widths.

Figure 8: Decay widths for the triplet and singlet scalar fields normalized to
the Higgs width for $\lambda_{1}=1.1$ and $\lambda_{1}=1.5$ (lighter color).
The upper and lower bounds correspond to the maximum and minimum values of the
scalar masses.

In fig. 8 we show the range of values of the triplet and singlet decay widths
normalized to the Higgs boson width for a couple of values of $\lambda_{1}$.
The charged and neutral components of the triplet have the same width at this
order because the couplings to the neutral component are a factor $1/\sqrt{2}$
smaller than those for the charged components but for the neutral component
one can exchange particles for antiparticles in the final state adding a
contribution that compensates the factor $(1/\sqrt{2})^{2}$ in the decay
width. Note that the upper bounds for the width of triplet and singlet are of
the same order. This is because the singlet is heavier than the triplet but
the couplings of the singlet to leptons are a factor $\sqrt{5}$ smaller than
those for the neutral component of the triplet. On the other hand, the lower
bound of the triplet decay width is much smaller. This suppression comes from
the kinematic factor since the lower bound for the triplet mass is only a few
tens of GeV larger than the mass of the T-odd mirror leptons. In any case
their lifetimes are small, since in the worst case the triplet could live
twice more than the Higgs in the available parameter space.

Nevertheless, these new scalar particles would not be generated in a
significant amount at the LHC. They are produced by an electroweak interaction
together with another T-odd particle and the energy threshold for this process
is very high considering the usual spectrum in fig. 7.

##  4 Conclusions

In this work we have reviewed the new Littlest Higgs model with T-parity
(NLHT) illana2022new  and explored its main phenomenological consequences.
This model was developed to cure a non gauge invariance issue in the fermion
sector of the original LHT. For this purpose the global symmetry group
$\textrm{SU}(5)$ was extended with an
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{2}$ factor. The global group
gets spontaneously broken to
$\textrm{SO}(5)\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]$ at a high
energy scale $f$. The spontaneous breaking of the extra piece of the global
group gives rise to four new T-odd scalars. Only a subgroup
$\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{1}\times\left[\textrm{SU}(2)\times\textrm{U}(1)\right]_{2}$
of the full global group is gauged, that gets spontaneously broken to the SM
gauge group by the same vev as the global group.

Under the gauge group, the four new T-odd scalars share quantum numbers with
some of the would-be Goldstone bosons of the original LHT: a triplet with zero
hypercharge and a singlet. As a consequence they mix. After field
redefinitions, some combinations become the would-be Goldstone bosons of the
NLHT, absorbed by the heavy gauge bosons, and the rest are the usual triplet
$\Phi$ and a new triplet $\widehat{\omega}$ and a singlet $\widehat{\eta}$,
which are physical. Their corresponding masses are obtained evaluating the
gauge boson and the heavy fermion contributions to the Coleman-Weinberg
potential. The new scalars have masses proportional to the Higgs mass,
independent of the scale $f$. Parametrically, apart from gauge couplings, the
new scalar masses depend on the Yukawa couplings $\lambda_{1}$ and
$\lambda_{2}$, that provide masses to the top quark and its corresponding
T-even and T-odd partners, and the Yukawa couplings $\kappa$ and
$\widehat{\kappa}$, giving gauge invariant mass terms to the rest of non
standard heavy fermions (in principle different for quarks and leptons). These
are more than in the original model and have either of the T-parities.

The top quark Yukawa coupling $\lambda_{2}$ is a function of $\lambda_{1}$
given the top quark mass. The condition that the heavier top quark partner is
below the cutoff scale $\Lambda$ constrains the value of $\lambda_{1}$ to the
interval $\left[1.05,1.71\right]$. And the experimental lower bound on vector-
like quark masses above 2 TeV Vatsyayan:2020jan  pushes the allowed value of
$f\gtrsim 900$ GeV.

Since T-parity is exact, the LTP is stable. To have a viable dark matter
candidate, the LTP must be electrically neutral. The NLHT contains two
potential candidates, the singlet $\widehat{\eta}$ and the usual heavy photon
$A_{H}$. Although there is enough space to explore the singlet as a dark
matter component (with $M_{\widehat{\eta}}\lesssim 80$ GeV), we have chosen it
to be the heavy photon in order to compare with previous works
hubiszPhenomenologyLittlestHiggs2005 ; 2006 ; Wang:2013yba . The aforementiond
lower bound to $f$ implies that the LTP has $M_{A_{H}}\gtrsim 200$ GeV and
$M_{\widehat{\omega}}<M_{\widehat{\eta}}$. On the other hand, the scalar
masses do not depend on the high energy scale $f$ and thus they must remain
light by naturalness arguments. Therefore, to be conservative, we impose an
upper bound of 1 TeV to the mass of the singlet.

In order to reproduce the current relic density with heavy photons, the
presence of co-annihilators is necessary unless the LTP is lighter than the
lower bound set above 2006 . To put vector-like quarks beyond current bounds
and at the same time provide a not so heavy dark matter candidate yielding the
right relic abundance, we have adopted a simplified NLHT model with degenerate
and flavor diagonal Yukawa couplings $\kappa=\widehat{\kappa}$, different for
leptons ($\kappa_{\ell}$) and quarks ($\kappa_{q}$), such that the
corresponding heavy masses are $m_{\ell_{H}}=\sqrt{2}\kappa_{\ell}f\gtrsim
M_{A_{H}}$ and $m_{q_{H}}=\sqrt{2}\kappa_{q}f\gtrsim 2$ TeV, respectively. We
also require that only T-odd leptons act as co-annihilators by taking
$M_{\widehat{\omega}}>M_{A_{H}}+M_{h}$.

As a consequence of all these constraints, the allowed high energy scale $f$
lies approximately in the interval $f\in\left[2,3\right]$ TeV, the common
Yukawa coupling of the heavy quarks gets strongly correlated to the top Yukawa
coupling $\lambda_{1}$ and, demanding that all dark matter of the universe is
made of the NLHT heavy photons, the Yukawa coupling for all heavy leptons is
fixed to $\kappa_{l}\approx 0.185$. A typical spectrum of this scenario is
displayed in fig. 7.

Finally we studied the dominant decay channels of the new scalar particles. We
found that $\widehat{\omega}$ and $\widehat{\eta}$ decay mostly into a T-odd
mirror lepton and a SM lepton with the proper quantum numbers. Other channels
are negligible because they involve couplings that suffer from suppressions by
powers of $v/f<0.12$. They decay very fast; the (lighter) triplet
$\widehat{\omega}$ could live at most two times longer than the Higgs boson.
In any case, these new scalars are heavier than about 600 GeV and would be
generated by and electroweak interaction together with another heavy T-odd
particle at the LHC, so not very sizeable production rates are expected.

As a future research it would be interesting to further develop the idea of
the scalar singlet $\widehat{\eta}$ acting as dark matter candidate.
Phenomenologically there is still a window for light scalar dark matter
Feng:2014vea  and the NLHT model has enough room to accommodate this scenario
as well.

## Acknowledgments

We would like to thank M. Chala, A. Djouadi, T. Hahn, M. Masip and J. Santiago
for very helpful discussions. This work was supported in part by the Spanish
Ministry of Science, Innovation and Universities
(PID2019–107844GB-C21/AEI/10.13039/501100011033), and by Junta de Andalucía
(FQM 101, SOMM17/6104/UGR, P18-FR-5057).

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