aainstitutetext:  II. Institut für Theoretische Physik, Universit t Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany bbinstitutetext:  DESY,
Notkestraße 85, D-22607 Hamburg, Germany

# The fate of the Littlest Higgs Model with $T$-parity under 13 TeV LHC Data

Daniel Dercks a Gudrid Moortgat-Pick b J rgen Reuter b and So Young Shim
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We exploit all LHC available Run 2 data at center-of-mass energies of 8 and 13
TeV for searches for physics beyond the Standard Model. We scrutinize the
allowed parameter space of Little Higgs models with the concrete symmetry of
$T$-parity by providing comprehensive analyses of all relevant production
channels of heavy vectors, top partners, heavy quarks and heavy leptons and
all phenomenologically relevant decay channels. Constraints on the model will
be derived from the signatures of jets and missing energy or leptons and
missing energy. Besides the symmetric case, we also study the case of T-parity
violation. Furthermore, we give an extrapolation to the LHC high-luminosity
phase at 14 TeV as well.

###### Keywords:

Little Higgs models, LHC searches, New Physics Signatures

††arxiv: 1801.XXXXX

DESY 17–192

##  1 Introduction

The main legacy of the 7/8 and 13 TeV runs of the Large Hadron Collider (LHC)
is the discovery of a 125 GeV Higgs boson Aad:2012tfa ; Chatrchyan:2012xdj  as
well as the absence of any signals for other new particles. This is in
accordance with the measurements of electroweak precision observables (EWPO)
which agree very well with a Standard Model (SM) containing a light Higgs
boson and no further degrees of freedom in the range up to a TeV. Besides the
EWPO from the pre-LHC era and flavor physics observables, both direct searches
and the ever more precise measurements of the properties of the Higgs boson
(as well as the top quark and weak gauge bosons) are the tools to search for
physics beyond the Standard Model (BSM) at the LHC. These are used to
constrain any type of BSM model.

In this paper we study the Littlest Higgs Model with $T$-parity (LHT). This is
an attractive representative of Little Higgs models ArkaniHamed:2001nc ;
ArkaniHamed:2002pa  since fine tuning problems in the Higgs potential can be
avoided via a discrete global $Z_{2}$ symmetry. Little Higgs models in general
regard a naturally light Higgs boson as a pseudo–Nambu-Goldstone boson (pNGB)
arising from a (new) global symmetry at high scale, see e.g. Ref.
Georgi:1975tz ; Kaplan:1983fs . However, such a mechanism would require new
strong interactions to tie the constituents of the Higgs boson together, which
unavoidably would show up in electroweak precision observables. In order to
avoid such strong constraints from EWPO, the mechanism of so-called collective
symmetry breaking has been applied, i.e. interweaving several global
symmetries which all have to broken in order to give mass to the pNGBs charged
under them. This means that the Higgs mass achieves only a logarithmic
sensitivity to the cutoff scale at one-loop , while a quadratic sensitivity
only arises at the two-loop level, thereby shifting the strongly-interacting
UV completion scale from the multi-TeV to the multi-10 TeV-region. Note,
however, that Little Higgs model are effective field theories (with new
degrees of freedom beyond the SM like heavy vectors, scalars and quarks) that
not necessarily have a direct strongly coupled UV completion, but could also
have weakly-coupled sectors at the next scale Kaplan:2004cr .

In this paper we consider the LHT model just as sucg an effective (low-energy)
field theory consisting of the SM degrees of freedom augmented by ($T$-odd)
heavy vector bosons, heavy quarks (and leptons) as well as additional heavy
pNGBs (which turn out to be irrelevant for the phenomenology of that model).
All of these particles just have the SM gauge interactions as well as
generalizations of the SM Yukawa couplings, which reflect the implementation
of both the Little Higgs collective symmetries as well as $T$-parity. We
consider all phenomenologically relevant production mechanisms for the heavy
new particles, including all relevant decays in order to compare the
predictions within the LHT model with the LHC 13 TeV data from Run 2\. In
addition, we reproduce the constraints from the EWPO. For completeness, we
review the status from the 8 TeV Run 1 data. Because of the possibility of
$T$-parity breaking in a strongly coupled UV completion of the LHT, as well as
tensions from dark matter (DM) constraints, we also take signatures and limits
from a scenario with $T$-parity breaking into account which is different than
the Littlest Higgs model without $T$-parity. We also give prospects for the
upcoming high-luminosity runs at the LHC at 14 TeV.

The outline of the paper is as follows: in order to make the paper self-
contained, in Sec. 2 we briefly summarize the model-building setup of the
Littlest Higgs model (with $T$-parity) needed to understand the
phenomenological analyses later on. In Sec. 3 we review the existing limits
from EWPO on the LHT model. In the next section, Sec. 4, we discuss the tool
chain for generating events and recasting the LHC analyses. We then collect
the relevant collider topologies along with cross sections and branching
ratios for different regions of parameter space in Sec. 5. Our main collider
results are collected in Sec. 6, and compared to the sensitivity from
electroweak precision data in Sec. 7. Finally, we give a summary and outlook
in Sec. 8.

##  2 Little Higgs Models with $T$-Parity

The Littlest Higgs model ArkaniHamed:2002qy  is based on a non-linear sigma
model with a single field $\Sigma$ parameterizing a $SU(5)/SO(5)$ symmetry
breaking structure.111For different implementations of Little Higgs models in
terms of product group and simple group models and a way to distinguish them,
cf. e.g. Kilian:2004pp ; Perelstein:2005ka ; Kilian:2006eh . The vacuum
expectation value (vev) causing the breaking from $SU(5)$ to $SO(5)$,
$\Sigma_{0}$, can be cast into the form of the $5\times 5$ matrix

$\Sigma_{0}=~{}\left(\begin{array}[]{ccc}&&{\mathbf{1}}_{2\times 2}\\\ &1&\\\
{\mathbf{1}}_{2\times 2}&&\end{array}\right).$ (1)

The gauge group of the Littlest Higgs is $G_{1}\times
G_{2}~{}=~{}(SU(2)_{1}\times U(1)_{1})\times(SU(2)_{2}\times U(1)_{2})$
embedded in $SU(5)$ as a subgroup such that the vev in Eq. (1) above breaks it
down into the diagonal subgroup $SU(2)_{L}\times U(1)_{Y}$ which is identified
with the SM electroweak group. The kinetic term for the non-linear sigma model
field is

$\mathcal{L}_{kin}~{}=~{}\frac{f^{2}}{4}{\rm{Tr}}|D_{\mu}\Sigma|^{2}$ (2)

with

$\Sigma~{}=~{}e^{i\Pi/f}~{}\Sigma_{0}~{}e^{i\Pi^{T}/f}~{}=~{}e^{2i\Pi/f}~{}\Sigma_{0}\;$
(3)

where $f$ is the Nambu-Goldstone-Boson (NGB) decay constant of the model. At
this scale the symmetry breakings $SU(5)\to SO(5)$ and $G_{1}\times G_{2}\to
SU(2)_{L}\times U(1)_{Y}$ take place. The covariant derivative in Eq. (2) is
given by

$D_{\mu}\Sigma~{}=~{}\partial_{\mu}\Sigma-i\sum_{j}[g_{j}W_{j}^{a}(Q_{j}^{a}\Sigma+\Sigma
Q_{j}^{a^{T}})+g_{j}^{\prime}B_{j}(Y_{j}\Sigma+\Sigma Y_{j})]$ (4)

with the generators

$\displaystyle
Q_{1}^{a}=~{}\frac{1}{2}\left(\begin{array}[]{cc}\sigma^{a}&{\mathbf{0}}_{2\times
3}\\\ {\mathbf{0}}_{3\times 2}&{\mathbf{0}}_{3\times
3}\end{array}\right),\quad$ $\displaystyle\quad
Q_{2}^{a}=~{}\frac{1}{2}\left(\begin{array}[]{cc}{\mathbf{0}}_{3\times
3}&{\mathbf{0}}_{2\times 3}\\\ {\mathbf{0}}_{3\times
2}&-(\sigma^{a})^{*}\end{array}\right)$ (9) $\displaystyle
Y_{1}=~{}\frac{1}{10}\,\text{diag}\left(-3,-3,2,2,2\right),\quad$
$\displaystyle\quad
Y_{2}=~{}\frac{1}{10}\,\text{diag}\left(-2,-2,-2,3,3\right)\quad,$ (10)

where $\sigma^{a}$ are Pauli matrices. The $SU(5)\to SO(5)$ symmetry breaking
generates a total of 14 NGBs $\Pi^{a}$ which decompose under the unbroken EW
group $SU(2)_{L}\times U(1)_{Y}$ as $\mathbf{1_{0}\oplus 3_{0}\oplus
2_{\pm\frac{1}{2}}\oplus 3_{\pm 1}}$. Four of these NGBs are eaten by the
extra gauge bosons, $Z_{H}$, $W_{H}$ and $A_{H}$, which get masses of the
order $f$. The remaining ten physical (p)NGBs decompose into the complex Higgs
doublet and a hypercharge one complex triplet. The latter is
phenomenologically irrelevant as the production cross section for these
particles is negligibly small, cf. Ref. Reuter:2013iya .

Like many other BSM models, the Littlest Higgs model suffers from constraints
by electroweak precision observables, particularly as the heavy hypercharge
boson, $A_{H}$, has an accidentally small prefactor, cf. the right hand side
of Eq. (14). To alleviate these constraints, a discrete symmetry, TeV- or
short $T$-parity has been added Cheng:2003ju ; Cheng:2004yc , which
phenomenologically plays a similar role as $R$-parity in supersymmetry (SUSY).
$T$-parity is an involutary automorphism that exchanges the sets of the two
different gauge algebras $G_{1}$ and $G_{2}$, or alternatively, their gauge
bosons:

$W_{1\mu}^{a}~{}\xleftrightarrow{T}~{}W_{2\mu}^{a},\qquad\qquad
B_{1\mu}~{}\xleftrightarrow{T}~{}B_{2\mu}.$ (11)

This fixes the gauge coupling constants of the two different $SU_{1,2}(2)$ and
$U_{1,2}(1)$ to be equal:

$\displaystyle g_{1}$ $\displaystyle=~{}g_{2}~{}=~{}\sqrt{2}g,$ (12)
$\displaystyle g_{1}^{\prime}$
$\displaystyle=~{}g_{2}^{\prime}~{}=~{}\sqrt{2}g^{\prime}.$ (13)

The mass eigenstates are then just the (normalized) sum and difference of the
two gauge fields, respectively, with mixing angles of ${\pi}/{4}$. This
results in the mass terms of the heavy gauge bosons

$\displaystyle m_{W_{H}}=m_{Z_{H}}$ $\displaystyle=gf,$ (14a) $\displaystyle
m_{A_{H}}$ $\displaystyle=\frac{g^{\prime}f}{\sqrt{5}}.$ (14b)

In order to implement collective symmetry breaking in the fermion fields, a
partner state to the third generation quark doublet has to be introduced,
forming an incomplete $SU(5)$ multiplet $\Psi$ and its $T$-parity partner
$\Psi^{\prime}$

$\Psi=\left(\begin{array}[]{c}ib_{L}\\\ -it_{1L}\\\ t_{2L}\\\
\mathbf{0}_{2\times 1}\end{array}\right)=\left(\begin{array}[]{c}q_{L}\\\
t_{2L}\\\ \mathbf{0}_{2\times
1}\end{array}\right),\qquad\Psi^{\prime}=\left(\begin{array}[]{c}\mathbf{0}_{2\times
1}\\\ t^{\prime}_{2L}\\\ ib^{\prime}_{L}\\\
-it^{\prime}_{1L}\end{array}\right)=\left(\begin{array}[]{c}\mathbf{0}_{2\times
1}\\\ t^{\prime}_{2L}\\\ q^{\prime}_{L}\end{array}\right)\;,$ (15)

which are related via

$\displaystyle\Psi\xleftrightarrow{T}-\Sigma_{0}\Psi^{\prime}$ (16)

Here, $q_{L}$ denotes the quark doublet of the SM following the conventions in
ArkaniHamed:2002qy , while $q_{L}^{\prime}$ and $t_{2}^{\prime}$ are the
$T$-parity partner fermions needed to reconcile both $T$-parity and the
collective symmetry breaking mechanism. The $T$-parity invariant Lagrangian
then reads as

$\mathcal{L}_{Y}\,\supset\;\frac{\lambda_{1}f}{2\sqrt{2}}\epsilon_{ijk}\epsilon_{xy}(\overline{\Psi}_{i}\Sigma_{jx}\Sigma_{ky}-(\overline{\Psi^{\prime}})_{i}\tilde{\Sigma}_{jx}\tilde{\Sigma}_{ky})t_{1R}+\lambda_{2}f(\overline{t}_{2L}t_{2R}+\overline{t^{\prime}}_{2L}t_{2R})+h.c.,$
(17)

where $\lambda_{1,2}$ denote the top-quark Yukawa couplings, respectively. The
$T$-parity eigenstates are now the (normalized) differences (even states) and
sums (odd states) of the primed and unprimed fermion fields
$t_{+}=(t_{1L,+},t_{R})$, $t_{=}(t_{1L,-},t_{1R,-})$,
$T^{-}=(t_{2L,-},t_{2R,-})$ and $T^{\prime}_{+}=(t_{2L,+},t_{2R,+})$.
Diagonalizing the left-handed $T$-even fermions yields the (SM) top quark and
the heavy $T$-even top quark, $T^{+}$. The $t_{-}$ gets a mass with the help
of the so-called mirror fermions, cf. below for the first and second
generation fermions, while the masses for the SM top quark and the other top
partners are given by,

$\displaystyle m_{t_{SM}}$
$\displaystyle=m_{t+}=\;\frac{\lambda_{2}R}{\sqrt{1+R^{2}}}v,$ (18)
$\displaystyle m_{T^{-}}$
$\displaystyle=\lambda_{2}f=\;\frac{m_{t_{+}}}{v}\frac{f\sqrt{1+R^{2}}}{R}$
(19) $\displaystyle m_{T^{+}}$
$\displaystyle=\;\frac{m_{t_{+}}}{v}\frac{f(1+R^{2})}{R}=\;m_{T-}\sqrt{1+R^{2}}\quad.$
(20)

$R$ is defined as the ratio between the Yukawa coefficients of the two
different possible terms, $R=\lambda_{1}/\lambda_{2}$ and is one of the
parameters used for investigating the parameter space in this paper.

Up-type quarks for the first and second generations have a similar Lagrangian
than the top quark except for the vector-like quark, which is not present as
there is no need to cancel the contribution from light quarks to Higgs self
energies:

$\displaystyle\mathcal{L}_{Y}\supset\frac{i\lambda_{d}f}{2\sqrt{2}}\epsilon_{ij}\epsilon_{xyz}(\overline{\Psi^{\prime}}_{x}\Sigma_{jy}\Sigma_{jz}X-(\overline{\Psi}\Sigma_{0})_{x}\tilde{\Sigma}_{iy}\tilde{\Sigma_{jz}}\tilde{X})d_{R}$
(21)

The $SU(2)_{1,2}$ singlet $X$ with $U(1)_{1,2}$ charges
$(Y_{1},Y_{2})=(1/10,-1/10)$ renders the term gauge invariant. There are two
different $X$ embeddings as $(3,3)$ component into the NGB multiplet, namely
$X=(\Sigma_{33})^{-1/4}$ [_Case A_] and $X=(\Sigma_{33})^{1/4}$ [_Case B_].
These cases do not differ in the context of BSM collider phenomenology which
is why we choose Case A in this study. Differences only arise in the
discussion of constraints from the Higgs sector and electroweak precision
observables and more details can be found in Ref. Reuter:2013iya .

To give rise to mass terms for the $T$-odd fermions without introducing any
anomalies, another $SO(5)$ multiplet $\Psi_{c}$ is introduced as

$\displaystyle\Psi_{c}=\left(id_{c},\,-iu_{c},\,\chi_{c},\,i\tilde{d}_{c},\,-i\tilde{u}_{c}\right)^{T}=\left(q_{c},\,\chi_{c},\,\tilde{q}_{c}\right)^{T}\,,\qquad\Psi_{c}\xleftrightarrow{T}-\Psi_{c}\;.$
(22)

The $q_{c}$ fields are called mirror fermion.

The $T$-parity invariant Lagrangian for the light fermions is

$\displaystyle\mathcal{L}_{\kappa}=-\kappa
f(\overline{\Psi^{\prime}}\xi\Psi_{c}+\overline{\Psi}\Sigma_{0}\Omega\xi^{\dagger}\Omega\Psi_{c})+h.c..$
(23)

This Lagrangian not only adds the $T$-odd mass terms but also imposes new
interactions between Higgs boson and up-type partners.

$\displaystyle\mathcal{L}_{\kappa}\supset$ $\displaystyle-\sqrt{2}\kappa
f(\overline{d}_{L-}\tilde{d}_{c}+\frac{1+c_{\xi}}{2}\overline{u}_{L-}\tilde{u}_{c}-\frac{s_{\xi}}{\sqrt{2}}\overline{u}_{L-}\chi_{c}-\frac{1-c_{\xi}}{2}\overline{u}_{L-}u_{c})+h.c.+\cdots$
(24)

where $c_{\xi}=\cos((v+h)/\sqrt{2}f)$, $s_{\xi}=\sin((v+h)/\sqrt{2}f)$.

The parameter $\kappa$ characterizing the coupling between the Higgs and the
$T$-odd fermions is another degree of freedom in the model parameter space we
investigated. We will distinguish between $\kappa_{q}$ for the light quarks
and $\kappa_{l}$ for the leptons.

The mass spectrum for heavy $T$-odd fermions is given (at order
$\mathcal{O}(v^{2}/f^{2})$) by

$\displaystyle m_{u,-}$
$\displaystyle=\;\sqrt{2}\,\kappa_{q}f\Big{(}1-\frac{1}{8}\frac{v^{2}}{f^{2}}\Big{)},$
(25) $\displaystyle~{}~{}m_{d,-}$ $\displaystyle=\;\sqrt{2}\,\kappa_{q}f$ (26)
$\displaystyle m_{\ell,-}$ $\displaystyle=\;\sqrt{2}\,\kappa_{l}f$ (27)

###  2.1 $T$-parity Violation

For the phenomenology of the LHT model, we will also consider $T$-parity
violation. There are two reasons for that: first, in the context of strongly
interacting UV completions $T$-parity violation can naturally occur via an
anomalous Wess-Zumino-Witten term, Hill:2007zv ; Hill:2007nz , secondly, there
is a certain tension for the case that the lightest $T$-odd particle, the
heavy photon $A_{H}$ is absolutely stable from relic density calculations and
direct detection dark matter experiments Hubisz:2004ft ; Wang:2013yba . In
order to avoid any constraints from dark matter bounds, one can assume that
the $A_{H}$ only has a microscopic lifetime and that dark matter instead is
made up of an axion-like particle in the strongly interacting UV completion of
the Little Higgs model.

As has been studied in Hill:2007zv ; Freitas:2008mq , $T$-parity violation
generates decays the heavy photon partner $A_{H}$ into the electroweak gauge
bosons $WW$ and $ZZ$ similar to the decay of the pion into two photons. Above
the kinematic threshold for these $A_{H}$ decays, the partial width is given
by:

$\displaystyle\Gamma(A_{H}\rightarrow ZZ)$
$\displaystyle=\left(\frac{Ng^{\prime}}{80\sqrt{3}\pi^{3}}\right)^{2}\frac{M_{A_{H}}^{3}m_{Z}^{2}}{f^{4}}\left(1-\frac{4m_{Z}^{2}}{M_{A_{H}}^{2}}\right)^{\frac{5}{2}},$
(28) $\displaystyle\Gamma(A_{H}\rightarrow W^{+}W^{-})$
$\displaystyle=\left(\frac{Ng^{\prime}}{40\sqrt{3}\pi^{3}}\right)^{2}\frac{M_{A_{H}}^{3}m_{W}^{2}}{f^{4}}\left(1-\frac{4m_{W}^{2}}{M_{A_{H}}^{2}}\right)^{\frac{5}{2}}.$
(29)

Here, the integer $N$ depends on the UV completion of the theory. As we are
only interested in branching ratios, shown in a later section, the precise
choice of this number does not matter for our analysis.

If the mass of $A_{H}$ is below the $WW$ and $ZZ$ thresholds, it will decay
into the SM fermions via $WW$\- and $ZZ$-induced triangle loops leading to the
partial widths:

$\displaystyle\Gamma(A_{H}\rightarrow ff)$
$\displaystyle=\left(\frac{N_{C,f}M_{A_{H}}}{48\pi}\right)\left[c_{-}^{2}\left(1-\frac{4m_{f}^{2}}{M_{A_{H}}^{2}}\right)+c_{+}^{2}\left(1+\frac{2m_{f}^{2}}{M_{A_{H}}^{2}}\right)\right]\left(1-\frac{4m_{f}^{2}}{M_{A_{H}}^{2}}\right)^{\frac{1}{2}},$
(30)

with $c_{\pm}:=c_{R}\pm c_{L}$, and $c_{L}$ and $c_{R}$ being the left- and
right-handed fermion couplings, shown in Table 1. $N_{C,f}$ is the number of
colors of the final state fermions. For the range
$f~{}\sim~{}1-10\,\mathrm{TeV}$ and $N=\mathcal{O}(1)$, the total $A_{H}$
width $\Gamma_{A_{H}}$ ranges between 0.01-1 eV which corresponds to a
lifetime of order 10-17 s. This excludes $A_{H}$ from being a viable dark
matter candidate. On the other hand, it leads to a mean free path of
approximately 10 nm, resulting in nearly prompt decays which do not produce
observable displaced vertices in the LHC detectors.

Particles |  $c_{L}^{f}$ |  $c_{R}^{f}$
---|---|---
$A_{H}e^{+}e^{-}$ |  $\frac{9\hat{N}}{160\pi^{2}}\frac{v^{2}}{f^{2}}g^{4}g^{\prime}(4+(c_{w}^{-2}-2t_{w}^{2})^{2})$ |  $-\frac{9\hat{N}}{40\pi^{2}}\frac{v^{2}}{f^{2}}g^{\prime 5}$
$A_{H}\overline{\nu}\nu$ |  $\frac{9\hat{N}}{160\pi^{2}}\frac{v^{2}}{f^{2}}g^{4}g^{\prime}(4+c_{w}^{-4})$ |  0
$A_{H}\overline{u}_{a}u_{b}$ |  $-\frac{\hat{N}}{160\pi^{2}}\frac{v^{2}}{f^{2}}g^{4}g^{\prime}(36+(3c_{w}^{-2}-4t_{w}^{2})^{2})\delta_{ab}$ |  $-\frac{\hat{N}}{10\pi^{2}}\frac{v^{2}}{f^{2}}g^{\prime 5}\delta_{ab}$
$A_{H}\overline{d}_{a}d_{b}$ |  $-\frac{\hat{N}}{160\pi^{2}}\frac{v^{2}}{f^{2}}g^{4}g^{\prime}(36+(3c_{w}^{-2}-2t_{w}^{2})^{2})\delta_{ab}$ |  $-\frac{\hat{N}}{40\pi^{2}}\frac{v^{2}}{f^{2}}g^{\prime 5}\delta_{ab}$
Table 1: Coefficients for the $A_{H}$ TPV decays, cf. Eq. (30). The indices
$a,b$ refer to the color of the respective quarks and we use
$\hat{N}=N/48\pi^{2},c_{W}=\cos\theta_{W},t_{W}=\tan\theta_{W}$.

###  2.2 Naturalness and Fine Tuning

Together with the model setup, we discuss in this section the definition of
fine tuning, that is sometimes used as a guideline for the naturalness of a
model or of certain regions of parameter space. The naturalness is generally
tied to the radiative corrections to the scalar potential in quantum field
theories. In order for a model to be considered natural, those corrections
should be of the same order as the scalar mass term from the mechanism that
originally created that mass term (the explicit breaking of the global
symmetries in Little Higgs models). A fine-tuning measure usually compares the
size of the radiative corrections to this bare mass term. In the absence of a
special cancellation mechanism, this measure depends quadratically on the
typical scale of these corrections; cancellation by means of a symmetry turns
this into a logarithmic dependence, or even zero if the symmetry is mighty
enough like exact supersymmetry or conformal symmetries.

In Little Higgs models, the cancellation comes from SM partner particles of
like statistics by means of nonlinearly realized global symmetries. The most
severe SM radiative corrections from the top quark are cancelled by the
$T$-odd and even top partners, $T^{\pm}$, followed by the cancellations of the
EW gauge bosons due to the heavy new gauge bosons, $A_{H}$, $Z_{H}$, and
$W_{H}$. In this paper, we adopt the fine-tuning measure defined in
ArkaniHamed:2002qy , which only accounts for the top partners, and neglects
the contributions from the gauge boson partners as well as from the heavy
pNGBs and the light fermion partners. The fine tuning is then defined as the
ratio of the experimentally measured Higgs mass squared and the absolute value
of the radiative corrections from the top partners to the Higgs quadratic
operator:

$\displaystyle\Delta~{}=~{}\frac{\mu_{exp}^{2}}{|\delta\mu^{2}|},\qquad\delta\mu^{2}~{}=~{}-\frac{3\lambda_{t}M_{T}^{2}}{8\pi^{2}}\log\frac{\Lambda^{2}}{M_{T}^{2}}.$
(31)

Here $\Lambda=4\pi f$ is the cut-off scale of the LHT model, i.e. the
equivalent to $\Lambda_{\text{QCD}}$ in a strongly-interacting embedding of
the LHT, $\lambda_{t}$ is the SM top Yukawa coupling and $M_{T}$ is a generic
mass scale of the top partner sector. Note that this definition of the fine-
tuning measure leads to the fact that smaller values of that measure (provided
in per cent in general) constitute a higher amount of fine tuning, hence a
more finely tuned point of parameter space. While the LHC Run 1 datasets at 7
and 8 TeV together with electroweak precision observables still allowed
parameter space with $\mathcal{O}(1\%)$ Reuter:2013iya , we will see in this
paper that the fine tuning including LHC Run 2 data is now everywhere around
one per cent or even in the sub-per cent regime. This is still comparable with
or better than the amount of fine tuning in generic parameter regions of the
minimal supersymmetric SM (MSSM), and it is generically (much) better than the
fine tuning for Composite Higgs models.

##  3 Electroweak Precision Constraints

Even before the start of data taking at the LHC, Little Higgs models were
already grossly constrained by comparing their predictions to precise
measurements in the electroweak sector, the so-called electroweak precision
observables (EWPO) Csaki:2002qg ; Hewett:2002px ; Han:2003wu ; Kilian:2003xt .
Additional constraints come from flavor data (in the $K$, $D$ and $B$ sector),
as well as for the models with $T$ parity and stable massive particles from
dark matter searches. We will not discuss the first point here as this has
been studied elsewhere Blanke:2006eb ; Blanke:2015wba , and the second point
has been addressed in the last section.

EWPO mainly contain a list of measurements from $e^{+}e^{-}$ colliders like
LEP1, LEP2, SLC, and TRISTAN, and a few selected measurements from hadron
colliders where the precision has superseded that from lepton colliders, like
the $W$ mass, or was only possible there, like the Higgs mass and couplings.
In Refs. Reuter:2012sd ; Reuter:2013iya ; Tonini:2014dza , both the EWPO as
well as the latest Higgs data have been scrutinized in order to give the then
best constraints on the parameter space of the LHT model.

We will not repeat the complete table of the EWPO fit of the LHT model from
Reuter:2013iya  here, but just remind that the two main observables with the
highest pull in the fit giving the highest constraint are the total hadronic
cross section at the $Z$ pole as well as the left-right asymmetry on the $b$
quarks, $A^{(b)}_{LR}$. Higgs observables in general do not give any further
constraints beyond that as EWPO already drive the Little Higgs scale $f$ in a
region where the deviations of the Higgs couplings are well within the LHC
experimental uncertainties. The only exception to this statement comes from
the case when the decay $H\to A_{H}A_{H}$ is possible which is ruled out by
the LHC limits on Higgs invisible branching ratios and excludes
$m_{A_{H}}<62.5\,\mathrm{GeV}$, i.e. $f<480\,\mathrm{GeV}$ Reuter:2013iya .

The first EWPO constraints that have been applied to Little Higgs models came
from oblique corrections, the so-called Peskin-Takeuchi $\Delta S$, $\Delta T$
and $\Delta U$ parameters Peskin:1990zt ; Peskin:1991sw . These parameterize
corrections to the self energies of EW gauge bosons, that are measured in
two-(and four-) fermion processes at lepton colliders. $T$-parity was
specifically introduced to minimize the contributions from Little Higgs heavy
particles to the oblique parameters as far as possible, as no $T$-odd particle
can contribute to them at tree level. However, at loop-level there are
contributions from $T$-odd heavy quarks, the $T$-even top quark, the mirror
fermions and the heavy gauge bosons. These have been calculated in
Hubisz:2005tx ; Berger:2012ec .

One interesting feature derived in Reuter:2012sd ; Reuter:2013iya  from the
contribution of the heavy top partners to the $\Delta T$ parameter, is the
exclusion limit from EWPO as a function of the parameter $R$, the ratio of the
two different Yukawa couplings $\lambda_{1}$ and $\lambda_{2}$ in the top
sector. There is an accidental cancellation to the EWPO in terms of $R$ for
the value of $R$=1. This gives an only relatively weak exclusion limit for
$f\gtrsim 405$ GeV at 95% confidence level from EWPO only. For $R\ll 1$ this
bound goes up to roughly 750 GeV while for large $R\sim 3$ the bound from EWPO
goes up to 1.3 TeV.

For our discussion in this paper and the motivation into which regions of
parameter space to look at, even more relevant are the contributions from the
mirror fermions:

$\Delta
T_{q_{H},\ell_{H}}=-\sum_{q_{H},\ell_{H}}\frac{\kappa_{q,\ell}^{2}}{192\pi^{2}\alpha_{w}}\frac{v^{2}}{f^{2}}\quad.$
(32)

These expressions come from box diagrams contributing to four-fermion
operators with heavy quark and lepton mirror fermions running in the loop:

$\mathcal{O}_{4-\text{ferm.}}=-\frac{\kappa^{2}_{q,\ell}}{128\pi^{2}f^{2}}\Bigl{(}\overline{\psi}_{L}\gamma^{\mu}\psi_{L}\Bigr{)}\Bigl{(}\overline{\psi}^{\prime}_{L}\gamma_{\mu}\psi^{\prime}_{L}\Bigr{)}$
(33)

Here, $\psi$ and $\psi^{\prime}$ are any combinations of different SM
fermions. These four-fermion operators can be reinterpreted in terms of a
contribution to the oblique $\Delta T$ parameter. The peculiar feature about
them is that they increase with the mass of the mirror fermions for fixed
scale $f$. This is clear from the fact that in that case the Yukawa-type
coupling which enters the box diagrams has to be enlarged leading to a larger
contribution from the box diagrams. The $\kappa$ is usually assumed to be a
diagonal matrix in flavor space or even proportional to the unit matrix. In
this paper, we do not lift the degeneracy in generation space, however, we
investigate different values for the $\kappa$ couplings for mirror quarks and
mirror leptons.

As was shown in Reuter:2012sd ; Reuter:2013iya ; Tonini:2014dza , the end of
LHC Run 1 was sort of a turning point where limits from direct searches of
heavy particles in Little Higgs models started to become competitive with
EWPO, and now with Run 2 even superseded them. As the only relevant EWPO
result is Eq. (33) and the $R$ dependence from the top partner contributions
to the $\Delta T$ parameter, we do not discuss EWPO any further here, and take
Eq. (33) as a motivation to look into different scenarios of combinations of
all-light degenerate mirror fermions, heavy mirror quarks, as well as light
mirror leptons and decoupled quarks and vice versa.

##  4 Tool Framework and Scan Setup

The main goal of this paper is to derive limits on the LHT model from all
available LHC run II data. In this section we describe the framework that we
used in order to derive numerically the current LHC bounds on the LHT model.

###  4.1 Used Software

To be able to generate Monte-Carlo events for our model, we make use of the
FeynRules implementation of the LHT model as in Ref. Reuter:2012sd ;
Reuter:2013iya ; Tonini:2014dza . We slightly extended the model definition
such that the heavy fermion Yukawa couplings $\kappa$ are transformed into
independent coupling constants $\kappa_{\ell}$ and $\kappa_{q}$. We then
exported the LHT model to the event generators MG5_aMC@NLO Alwall:2011uj  and
WHIZARD Kilian:2007gr ; Moretti:2001zz ; Nejad:2014sqa ; Kilian:2011ka
222WHIZARD recently also has been extended towards next-to-leading order
functionality, cf. Kilian:2006cj ; Robens:2008sa ; Binoth:2009rv ;
Greiner:2011mp ; Nejad:2016bci ; Bach:2017ggt . via the UFO file format.333The
model file is available on demand from the authors.

The collider phenomenology of the LHT model studied in this paper depends on
the mass scale $f$, the two Yukawa coupling parameters $\kappa_{\ell}$ and
$\kappa_{q}$, as well as the ratio of top Yukawa couplings $R$. For these four
parameters we derive the corresponding masses according to Eqs. (14), (20),
(26) and store these in a spectrum file which follows the definitions of the
UFO model. The branching ratios and corresponding decay tables for all LHT
particles are calculated analytically using the formulae in the above linked
model file. These include all 2-body decays for all relevant particles. Note
that within the parameter space that we analyze, no 3-body decays need to to
be considered as there is always a dominating 2-body final state. The only
difference is the anomaly-mediated decay of $A_{H}$ in the case of $T$-parity
violation, see Sec. 2.1. For this, we use the branching ratios as functions of
$f$ taken from Ref. Freitas:2008mq  which will be shown later in this work.
For decays into gauge bosons, we assume that for $m(A_{H})>185\,\mathrm{GeV}$,
i.e. for $f\gtrsim 1080\,\mathrm{GeV}$, $A_{H}$ decays via 2-body decays into
$WW$ and $ZZ$. For smaller masses, we formulate 3-body decays for the decay
table as follows: we consider all possible decay modes of the $W$ or $Z$,
replace one of the final state gauge bosons with the corresponding decay
products and multiply the branching ratio accordingly.

For the main tasks of this numerical study, we make use of the collider
analysis tool CheckMATE Drees:2013wra ; Kim:2015wza ; Dercks:2016npn . This
program is useful to test a given BSM model in an automatized way. It makes
again use of the aforementioned generator MG5_aMC@NLO to simulate partonic
events. By making use of the UFO model description file format, MG5_aMC@NLO or
WHIZARD are able to simulate partonic events for a given BSM model which was
implemented in a model building framework like FeynRules Christensen:2008py ;
Alloul:2013bka  or SARAH Staub:2013tta , e.g. via the WHIZARD-FeynRules
interface Christensen:2010wz . The showering and hadronization of these events
is subsequently performed by Pythia8 Sjostrand:2007gs , followed by the fast
detector simulation Delphes deFavereau:2013fsa  which considers the effect of
measurement uncertainties, finite reconstruction efficiencies and the jet
clustering of the observed final state objects. These detector events are then
quantified by various analyses from both ATLAS and CMS at center-of-mass
energies of 8 and 13 TeV (more details below). Events are categorized in
different signal regions and CheckMATE determines which signal region provides
the strongest expected limit. If the input model predicts more signal events
than are allowed by the observed limit of that signal region, CheckMATE
concludes that the model is _excluded_ at the 95% confidence level, otherwise
the model is _allowed_. For more details on the inner functionality of
CheckMATE, we refer to the manual papers in Refs. Drees:2013wra ; Kim:2015wza
; Dercks:2016npn .

###  4.2 Details on Event Generation

For the event generation, we consider the production of all relevant two-body
final states. In the following, we use $q_{H}$ for all heavy fermion squarks
$\\{d_{H},u_{H},s_{H},c_{H},b_{H},t_{H}$}, $\ell_{H}$ for all other heavy
fermions $\\{e_{H},\mu_{H},\tau_{H},\nu_{eH},\nu_{\mu H},\nu_{\tau H}\\}$,
$V_{H}$ for all heavy gauge bosons $\\{W_{H},Z_{H}$ and $A_{H}\\}$ and
$T^{\pm}$ for the additional heavy $T$-even/odd top partner, respectively. We
analyzed the following processes for the LHC (cf. also Reuter:2012sd ;
Reuter:2013iya )

$\begin{array}[]{rl}1.&\quad pp\rightarrow
q_{H}q_{H},q_{H}\bar{q}_{H},\bar{q}_{H}\bar{q}_{H}\\\ 2.&\quad pp\rightarrow
q_{H}V_{H}\\\ 3.&\quad pp\rightarrow\ell_{H}\bar{\ell}_{H}\\\ 4.&\quad
pp\rightarrow V_{H}V_{H}\\\ 5.&\quad pp\rightarrow
T^{+}\bar{T}^{+},T^{-}\bar{T}^{-}\\\ 6.&\quad pp\rightarrow
T_{+}\bar{q},\bar{T}_{+}q,T_{+}W^{\pm},\bar{T}_{+}W^{\pm}\end{array}$ (34)

At this stage we give some remarks on the choice of these final states.

  * •

If $T$-parity is conserved, $T$-odd particles need to be produced in pairs.
Therefore, the $T$-even top partner $T^{+}$ is the only LHT particle which can
be produced in association with Standard Model particles. This rule also holds
in case of anomaly-triggered $T$-parity violation as the corresponding TPV
couplings $A_{H}-V-V$ are too small to result in another $T$-odd final state
with experimentally accessible cross section.

  * •

We focus our discussion on certain benchmark scenarios and within these
scenarios, some processes are expected to be negligible compared to others. We
give more details on this when we discuss the individual scan setups below.

  * •

While the production of color-charged objects is expected to be dominant at
the LHC in case $q_{H}$ and $V_{H}$ have similar masses, heavy gauge boson
production can become dominant in regions of parameter space where the heavy
gauge bosons are significantly lighter than the heavy quarks (i.e. for large
$\kappa$). We discuss the parameter dependence of the respective cross
sections below.

  * •

Processes with additional hard radiation in the final state, e.g. the process
$pp\rightarrow q_{H}q_{H}j$, are not considered here. They are expected to be
relevant in regions with strong mass degeneracy between the produced particle
and the stable particle it decays into as in such a case the process
$pp\rightarrow q_{H}q_{H}$ produces too soft jets to be observed. By requiring
an additional hard jet in the event, $pp\rightarrow q_{H}q_{H}j$, the
additional jet can boost the $q_{H}q_{H}$ system and create a new, potentially
observable multijet topology (see e.g. Ref. Dreiner:2012gx ). However, in our
case the gauge bosons $W_{H}$ and $Z_{H}$ are always predicted to be at least
100 GeV heavier than the $A_{H}$, cf. Eqs. (14a), (14b). Similarly, the
$q_{H}-A_{H}$ and $T^{-}-A_{H}$ mass splittings are always large enough in the
studied parameter regions. Therefore, we do not need to look at these peculiar
topologies which have a significantly smaller cross section than our discussed
two-body final states.

  * •

Another interesting final state is $pp\rightarrow A_{H}A_{H}j$ whose analysis
is motivated because of the distinct and typical monojet signature as
generally expected in models with a dark matter candidate (see e.g. Ref.
Aad:2015zva ). However, since our dark matter candidate has a mass of the
order of about 100 GeV and since it couples directly to quarks via the
$A_{H}-q-q_{H}$ vertex, we do not expect LHC searches for dark matter final
states to be more constraining than existing bounds from direct detection
searches. Furthermore, the consideration of this final state is technically
involved as double-counting with the decay topology $pp\rightarrow
q_{H}A_{H},q_{H}\rightarrow qA_{H}$ could occur in specific parts of the
parameter region and needs to be under precise control within the simulation.
The detailed discussion of such a decay topology is postponed to a forthcoming
study.

All simulations have been done automatically byCheckMATE using the event
generator MG5_aMC@NLO and have been cross-checked with WHIZARD.

###  4.3 Scan Benchmark Scenarios

The LHT model —as already described earlier— depends on the following four
parameters

  1. 1.

the symmetry breaking scale $f$ which affects the masses of all
$q_{H},\ell_{H}$, $V_{H}$ and $T^{\pm}$.

  2. 2.

the Yukawa parameter $\kappa_{q}$ which affects the masses of the heavy quarks
$q_{H}$,

  3. 3.

the Yukawa parameter $\kappa_{\ell}$ which affects the masses of the color-
neutral heavy fermions $\ell_{H}$ and

  4. 4.

the Yukawa parameter $R$ which affects the masses of the heavy top partners
$T^{+},T^{-}$.

Furthermore we distinguish models in which a) $T$-parity is exactly conserved
and b) models where gauge anomalies introduce the $T$-parity violating
couplings $A_{H}-W-W$ and $A_{H}-Z-Z$.

In order to reduce the number of free parameters we focus on particular
benchmark scenarios with different theoretical and/or phenomenological
motivation and with different assumptions on the fermion sector, the heavy top
sector and the validity of $T$-parity. These scenarios result in $3\times
2\times 2=12$ different benchmark cases, summarized in Tab. 2.

Sector | Model | Constraint |  Phenomenology | Considered Topology
---|---|---|---|---
$f_{H}$ | _Fermion Universality_ | $\kappa_{l}=\kappa_{q}$ |  $\bullet$ mass degeneracy of $q_{H},\ell_{H}$ | Exclude process 3
$\bullet$ $\ell_{H}$ production negligible
_Heavy $q_{H}$_ | $\kappa_{q}=3.0$ |  $\bullet$ $q_{H}$ decoupled | Exclude processes 1, 2
$\bullet$ $\ell_{H}$ production relevant
_Light $\ell_{H}$_ | $\kappa_{l}=0.2$ |  $\bullet$ $\ell_{H}$ very light | Exclude process 3
$\bullet$ $V_{H}$ branching ratios change
$T^{\pm}$ | _Light $T^{\pm}$_ | $R=1.0$ |  $\bullet$ $T^{\pm}$ are light/accessible | Include process 4, 5
_Heavy $T^{\pm}$_ | $R=0.2$ |  $\bullet$ $T^{\pm}$ are heavy/inaccessible | Exclude process 4, 5
$A_{H}$ | _TPC_ | No TPV |  $\bullet$ $A_{H}$ is stable and invisible |  $A_{H}$ stable
_TPV_ | With TPV |  $\bullet$ $A_{H}$ is unstable |  $A_{H}\rightarrow VV$ decays
Table 2: Definitions of the considered benchmark models of this study. In this
work we consider all $3\times 2\times 2$ combinations of the options given in
this table. The process numbers refer to the list in Eq. (34).

##### Heavy Fermion Sector:

We first discuss the different assumptions on the heavy fermion sector. In the
_Fermion Universality_ model we set the two coefficients
$\kappa_{q}=\kappa_{l}$ equal and hence get a mass degeneracy in the heavy
fermion sector. Due to their color charge, the production cross sections for
processes involving heavy quarks are significantly higher than the respective
cross sections for final states with color-neutral heavy fermions. Hence, we
do not consider process 3 of our list in 34.

The masses of the heavy fermions have two important consequences for the
phenomenology: they affect their production cross sections and they change the
branching ratios of the heavy gauge bosons
$V_{H}\rightarrow\ell_{H}^{(*)}\ell^{\prime}$. To get an understanding which
role this plays when setting bounds on the model we choose two further
benchmark cases, each taking into account one of these effects.

In the _Heavy $q_{H}$_ model we decouple the heavy quarks from the model by
fixing $\kappa_{q}=3.0$. This raises the heavy quark masses to the multi-TeV-
scale and hence makes them experimentally inaccessible. Therefore, we do not
consider production modes which involve $q_{H}$, i.e. processes 1 and 2 of 34,
but take into account $\ell_{H}$ pair production, process 3, instead. The
results of this benchmark scenario should give insight to which degree the LHC
sensitivity relies on the presence of the color-charged objects and which
limits can be determined from searches looking for color-neutral particles
only.

The _Light $\ell_{H}$_ benchmark is also designed to lift the degeneracy of
the color-charged and color-neutral objects. Here, by fixing $\kappa_{\ell}$
to a small value of 0.2, the latter are light enough for the heavy gauge
bosons to decay into them. We are interested to see how this change in the
expected decay patterns affects the bounds compared to the _Fermion
Universality_ model. Note that even though the $\ell_{H}$ are light we do not
take into account the bounds from $\ell_{H}$ production as we are interested
in how only a change in the decay pattern affects the resulting bounds. The
bounds resulting from direct $\ell_{H}$ production are determined in the
previously discussed _Heavy $q_{H}$_ benchmark.

The results of these three benchmark cases should be sufficient to
qualitatively determine the resulting bounds for other
$\kappa_{q}-\kappa_{\ell}$ combinations and to avoid a full 3D parameter scan
in the $f-\kappa_{q}-\kappa_{\ell}$ plane.

##### Heavy Top Partner Sector:

The main phenomenological difference between the heavy top partners $T^{\pm}$
and the other heavy fermions $q_{H}$ is that their mass depends on $R$ instead
of $\kappa$. We choose two benchmark values for this parameter in such a way
that one results in experimentally accessible top partners ($R=1.0$) while the
other ($R=0.2$) does not. The value $R=1.0$ also corresponds to a case where
minimal fine-tuning can be achieved, see Reuter:2012sd ; Reuter:2013iya , and
thus this benchmark case tests the natural regions of parameter space of the
LHT model. In the _Heavy $T^{\pm}$_ scenario we ignore any processes which
involve these particles as they are too heavy to result in an LHC exclusion.
The comparison of the two bounds at $R=1.0$ and $R=0.2$ gives insight to which
degree the masses of the particles in this sector are relevant for the overall
sensitivity.

#####  $T$-Parity Violation:

As discussed in Sec. 2.1, gauge anomalies in the heavy sector can result in
anomalous $T$-parity violating $A_{H}-W-W$ and $A_{H}-Z-Z$ couplings. The
presence of these operators may drastically change the expected collider
phenomenology as the final state not necessarily contains an invisible
particle any more. Supersymmetry motivated searches are however still expected
to be sensitive as the leptonic decays of the $W$ and the invisible decays of
the $Z$ boson can still produce a significant amount of missing energy. We are
interested to see by how much the bounds derived for the $T$-parity conserving
case are changed due to these anomaly-mediated decays. For that reason we
analyze each of the above discussed benchmark scenarios once with a stable
$A_{H}$ and once with enabling $A_{H}\rightarrow VV$ decays.

##  5 Collider Topologies

For the discussion of the LHC results, it is useful to understand both the
values of the production cross sections for all the processes we listed in the
last section and the dominant branching ratios of the relevant final state BSM
particles. Collider bounds are expected to be set by processes with a large
production cross section times a decay topology with only a small Standard
Model contamination. In this section we review the parameter dependence of
these observables in order to determine the theoretically expected collider
topologies of our LHT benchmark scenarios. Many of them are relevant for the
discussion of the exclusion bounds that we determine with CheckMATE in the
upcoming section.

###  5.1 Cross Sections

Figure 1: LHC production cross sections ($\sqrt{s}=13\,\mathrm{TeV}$) for
benchmark models _Fermion Universality/Light $\ell_{H}$_ \+ _Light $T^{\pm}$_.
Left: Dependence on $f$ for fixed $\kappa=1.0$ (solid), $\kappa=2.0$ (dashed).
Right: Dependence on $\kappa$ for fixed $f=1\,\mathrm{TeV}$ (solid),
$f=2\,\mathrm{TeV}$ (dashed). Labels in the legend appear in decreasing order
of the respective maximum value of the solid lines.

Figure 2: Same as Fig. 2 for benchmark model _Heavy $q_{H}$_ \+ _Light
$T^{\pm}$_.

We start with a discussion of the production cross sections for all the
process sets listed in Sec. 4.2. In Figs. 2,2 we show the cross sections for
$\sqrt{s}=13\,\mathrm{TeV}$ as a function of the symmetry breaking parameter
$f$ with fixed $\kappa$ and vice versa. As the benchmark case _Light
$\ell_{H}$_ does not affect any production mode, the cross sections are
identical to those in the _Fermion Universality_ benchmark. In all cases we
show the results in the _Light $T^{\pm}$_ subscenario for which the $T^{\pm}$
are kinematically accessible and the cross sections would nearly vanish in the
case of _Heavy $T^{\pm}$_. Note that $\kappa$ refers to
$\kappa_{q}=\kappa_{\ell}$ in the _Fermion Universality_ case and to
$\kappa_{q}(\neq\kappa_{\ell})$ in the _Light $\ell_{H}$_ scenario. $T$-parity
violation does not play a role in the discussion of LHT particle production
which is why we do not distinguish _TPC_ and _TPV_ here. Results for center-
of-mass energies of 8 and 14 TeV are provided in App. A.

Since the mass of all heavy sector particles increases with $f$, the cross
sections for all processes drop with increasing $f$.444Small fluctuations in
the $f$-dependent $q_{H}q_{H}$ production cross section are caused by
numerical noise. Similarly, since the mass of the heavy fermions depends
linearly on $\kappa$, the cross sections for producing these particles becomes
smaller for larger values of this parameter. As both mass and couplings of the
$T^{\pm}$ only depend on $f$ and the fixed parameter $R$, no dependence on
$\kappa$ can be seen.

Interestingly, even though the mass of the vector bosons $V_{H}$ also depends
on $f$ only, their production cross sections show a small $\kappa$-dependence
in the _Fermion Universality_ scenario. This is due to contributions of
$t$-channel $q_{H}$ which interfere destructively with the $s$-channel vector-
boson diagrams. Since all masses scale linearly with $f$, this effect appears
nearly independently of $f$ at the position $\kappa\approx 0.5$. As a result,
the cross section for $V_{H}$ pair production is roughly a factor 5 smaller
for small $\kappa\approx 0.5$ than for large values $\kappa\gtrsim 4$ when the
heavy fermions are decoupled. As the $q_{H}$ are by construction decoupled in
the _Heavy $q_{H}$_ benchmark scenario, the $\kappa$ dependence of the
$V_{H}V_{H}$ production cross section vanishes in the resulting distribution
shown in Fig. 2.

The production cross sections can reach values up to $10^{3}$ fb and we thus
expect the $\sqrt{s}=13\,\mathrm{TeV}$ LHC to be sensitive to large regions of
the parameter space we considered. Even for values of $f\approx
3\,\mathrm{TeV}$, cross sections of order $10^{-1}$ fb and thus detectable
event rates can be expected which improves results from LHC Run 1 which were
insensitive to values of the symmetry breaking scale above 2 TeV Reuter:2012sd
; Reuter:2013iya . Comparing the results of both the $f$-$\sigma$ and the
$\kappa$-$\sigma$ planes, it becomes clear that there is no dominant process
with a universally largest cross section. The cross sections have very
different dependencies on $\kappa$ and $f$ and thus different regions in
parameter space are expected to have different dominating final states.

Generally, regions with small values of $\kappa$ and thus with light
$q_{H},\ell_{H}$ predict a large rate of produced heavy fermions. As expected
for a hadron collider, the $q_{H}$ production is about two to three orders of
magnitude larger than the production of heavy leptons $\ell_{H}$ and the
latter appear only to be relevant for small values $f\lesssim
1\,\mathrm{TeV},\kappa\lesssim 0.5$. In regions with larger values of
$\kappa$, the production of heavy vector bosons becomes more important as
their mass is independent of $\kappa$. If heavy top partners $T^{\pm}$ are
accessible, they are produced with comparable abundance as the heavy vector
bosons.555Note that this statement in general depends on the specific value of
the additional parameter $R$ which we fixed to $1.0$ in our benchmark
scenario. Since the $T^{-}$ is always lighter than the $T^{+}$, the production
of the latter appears to be negligible in comparision.

Figure 3: Branching ratios of $d_{H}$ in the _Fermion Universality/Light
$\ell_{H}$_ model. Items in legend appear in decreasing order of the maximum
value of the respective curve. Left: Fixed $f=1\,\mathrm{TeV}$ (solid),
$f=2\,\mathrm{TeV}$ (dashed). Right: Fixed $\kappa=1$ (solid), $\kappa=2$
(dashed).

Figure 4: Branching ratios of $u_{H}$ in the _Fermion Universality/Light
$\ell_{H}$_ model. Parameters as in Fig. 3.

Figure 5: Branching ratios of $\ell_{H}$ in the _Fermion Universality_ model.

Figure 6: Branching ratios of $\nu_{e,H}$ in the _Fermion Universality_ model.

Figure 7: Branching ratios of $Z_{H}$ (left) and $W_{H}$ (right) in the
_Fermion Universality_ (The _Heavy $q_{H}$_ scenario is very similar, cf.
text). Parameters as in Fig. 3. In both plots, curves corresponding to decays
with $\nu,\ell$ or $b$ are nearly identical.

Figure 8: Branching ratios of $Z_{H}$ (left) and $W_{H}$ (right) in the _Light
$\ell_{H}$_ model. $f$ is chosen as in Fig. 3, left. In both plots, the curves
corresponding to decays with $\nu$ or $\ell$ are nearly identical. Figure 9:
Branching ratios of $T^{+}$ in the _Light $T^{\pm}$_ scenario. Figure 10:
Branching ratios of $A_{H}$ in the _TPV_ benchmark.

###  5.2 Branching Ratios

We now continue with a discussion of the branching ratios for the relevant
partner particles within the given benchmark cases. Note that we combine
phenomenologically similar branching ratios which involve $q:=u,d,c,s$, (so we
particularly do not distinguish heavy up- and down-type quarks here)
$\ell=e,\mu,\tau$, $\nu:=\nu_{e},\nu_{\mu},\nu_{\tau}$ and their respective
heavy partner fermions.666It is only in this section where we distinguish
between the charged heavy fermion $\ell_{H}$ and the neutral particle
$\nu_{H}$. In the rest of this work, $\ell_{H}$ refers to both heavy charged
and heavy neutral leptons. Also, we only discuss those decays with a branching
ratio of at least 1 % anywhere in the discussed parameter space. Though we do
not show it in the plots, we analytically calculated all decay widths and
considered all kinematically allowed $2$-body final states in the decay tables
used in our scans in order to get correct values for the branching ratios. We
mainly discuss results for the _Fermion Universality_ and the _Light
$\ell_{H}$_ scenarios as the _Heavy $q_{H}$_ scenario does not show any
differences in the observable decay pattern - except for one difference which
we mention along the way. Obviously, it is only the decay of the $A_{H}$ which
shows different behaviour in the benchmark cases _TPV_ and _TPC_. These two
benchmarks are hence not disinguished in the discussion regarding the decays
for the other particles.

Within the parameter ranges that we focus on, the particles $T^{-},\ell_{H}$
and $\nu_{H}$ each only have one decay mode in some scenarios:

Light $T^{-}$: $\displaystyle\qquad\text{BR}(T^{-}\rightarrow tA_{H})=1$ (35)
Light $\ell_{H}$:
$\displaystyle\qquad\text{BR}(\ell_{H}/\nu_{H}\rightarrow\ell/\nu A_{H})=1$
(36)

For other particles and/or other scenarios there is more than one decay mode
and the branching ratios depend on the values of $f$ and/or $\kappa$.

In Figs. 3,4 we show the dominating branching ratios of the heavy quark
partners $d_{H},u_{H}$, respectively, in the _Fermion Universality/Light
$\ell_{H}$_ models which show identical results in this regard. As before, we
show curves as functions of both $\kappa$ and $f$. For both up- and down-type
heavy quark partners, the decay into a heavy $W_{H}$ boson and a quark is the
most important decay with a branching ratio of nearly 60 % — whenever it is
kinematically allowed. They are followed by decays into $Z_{H}q$ of order 30 %
and to $A_{H}q$ of order 10 %. A small variation with $f$ becomes visible
which is caused by a subdominant dependence of the respective coupling
constants on $v/f$ (see e.g. Hubisz:2005tx ). This dependence differs between
up- and down-type quarks and thus the variation with $f$ differs for these two
flavors. Note that very small values of $\kappa_{q}\lesssim 0.5$ lead to
$m(q_{H})<m(W_{H}),m(Z_{H})$ and thus forbids decays
$q_{H}\rightarrow(W/Z)_{H}+X$. All $q_{H}$ therefore decay to the light
$A_{H}$ in this region of parameter space.

Note that due to the overall mass degeneracy and the identical quantum numbers
within the _Fermion Universality_ model, the decay signatures of all other
heavy fermions, except for the $T^{\pm}$, are identical after replacing the
corresponding up- and down-type components of the respective SU(2) doublets.
For example, the branching ratio for $\nu_{eH}\rightarrow W_{H}e$ is identical
to the branching ratio $u_{H}\rightarrow W_{H}d$, see Figs. 3-5.

Next, we discuss the decays of the heavy gauge bosons $W_{H}$ and $Z_{H}$ for
the _Fermion Universality_ model in Fig. 7 and for the _Light $\ell_{H}$_
model in Fig. 8. We only show results depending on $\kappa$ as there is no $f$
dependence for the two standard benchmark values $\kappa=1.0,2.0$ which we
considered. In case of _Fermion Universality_ , the decay $V_{H}\rightarrow
f_{H}f^{\prime}$ into a heavy fermion partner is only allowed for
$\kappa\lesssim 0.5$ and in this region decays into heavy quarks dominate. For
larger values of $\kappa$, the only available decays are $W_{H}\rightarrow
WA_{H}$ and $Z_{H}\rightarrow hA_{H}$. In the _Light $\ell_{H}$_ scenario,
this picture changes by construction: the $\ell_{H}$ are fixed to light masses
and thus for $\kappa_{q}\gtrsim 0.5$ both heavy gauge bosons decay to 50 %
into $\ell_{H}\ell$ and $\nu_{H}\nu$. Again, for smaller values of
$\kappa_{q}$ decays into $q_{H}$ are kinematically accessible and have a
dominant branching ratio. The branching ratio curve for the benchmark scenario
_Heavy $q_{H}$_ corresponds to the one for _Fermion Universality_ with the
only exception that the decay $V_{H}\rightarrow q_{H}q$ disappears for
$\kappa<0.5$ and the branching ratios for the other modes scale up
accordingly.

In Fig. 9 we show the branching ratios of the heavy top partner $T^{+}$ (note
that $T^{-}$ always decays to $tA_{H}$ as listed above) in the _Light Top_
benchmark, i.e. for $R=1.0$. As $T^{+}$ is a $T$-parity even particle it must
decay into pairs of $T$-odd particles or purely into SM particles. This
results in four main decay scenarios. The SM decays follow mainly the pattern
of a $SU(2)_{L}$ singlet top partner (cf. e.g. Reuter:2014iya ) of 50 %
branching ratio into $bW^{+}$ and equally a quarter into $th$ and $tZ$. This
is only slightly modified by the only accessible $T$-odd particle decay,
namely roughly 15 % branching ratio into $T^{-}A_{H}$. The changes the top-
like decay into $bW^{+}$ into nearly 45 % branching ratio, while $th$ and $tZ$
have roughly 20 % branching ratio each. These branching ratios have no
dependence on $\kappa$ and only little dependence on $f$ which originates from
the $f$-dependence of the $T^{\pm}$ and $A_{H}$ masses.

We finish the discussion with the branching ratios of the $A_{H}$ in the _TPV_
scenario shown in Fig. 10, which only depend on $f$. The information shown in
this figure has been taken from a detailed calculation performed in Ref.
Freitas:2008mq . One observes that for $f>1200\,\mathrm{GeV}$, decays into on-
shell Standard Model gauge boson pairs dominate. For smaller values of $f$,
the $A_{H}$ mass drops below 180 GeV, the partial decay widths into gauge
bosons decrease due to kinematic suppression and the loop-induced decays into
Standard Model leptons become equally relevant. For $f\lesssim
900\,\mathrm{GeV}$, $A_{H}$ decays predominantly into SM quark pairs.

###  5.3 Expected Final State Topologies and Correspondence to Supersymmetric
Searches

In this section we combine the information of the preceding one with the list
of dominant production processes given in Sec. 4.2 in order to find the
following expected final state signatures. Comparing them to the specialized
analyses of the experimental collaborations for supersymmetry, we can make the
following classification of the signatures and their applicability to the LHT
model:

  * •

In general – if $T$-parity is conserved – all $T$-odd particles produce decay
chains with a stable $A_{H}$ as the lightest $T$-odd particle at the end. This
particle is experimentally invisible and thus produces missing transverse
momentum $\not{E}_{T}$in the event. This is in close analogy to $R$-parity
conserving supersymmetry which produces decay chains with the lightest
neutralino at the end which similarly produces $\not{E}_{T}$. Therefore, many
searches looking for $R$-parity conserving supersymmetry require $\not{E}_{T}$
in the event and thus are sensitive to our model.

  * •

Final states with heavy gauge bosons $W_{H},Z_{H}$ behave differently in the
main benchmark cases. In the _Fermion Universality_ model, where $W_{H}$
decays produce $W$ bosons which either contribute with further jets in their
hadronic decays or with further hard leptons in their leptonic decays, the
heavy $Z_{H}$ adds Higgs bosons in the final state which mainly lead to
additional $b$-jets in the event. This final state topology is thus similar to
supersymmetric electroweakino production
$\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{0}_{2}+\tilde{\chi}^{\pm}_{1}\tilde{\chi}^{\pm}_{1}+\tilde{\chi}^{0}_{2}\tilde{\chi}^{0}_{2}$
with a Wino-like chargino and a Higgsino-like neutralino.

In the _Light Leptons_ model, the heavy gauge bosons almost always decay into
a lepton and the corresponding heavy lepton partner which itself always decays
into a lepton and $A_{H}$. This behavior corresponds to a supersymmetry model
with very light scalar leptons for which there exist specific signal regions
in experimental searches for electroweakinos.

  * •

Final states with heavy $q_{H}$ always produce quarks and $A_{H}$ in their
decays and hence result in final states with jets and missing transverse
momentum. In most cases these decays produce further heavy gauge bosons
$V_{H}$ which, as explained above, add more leptons, $b$-jets or normal jets
to the event. This topology is very similar to supersymmetric scalar quark
production with either direct decays into the lightest supersymmetric particle
or with decay chains producing further neutralinos and/or charginos in the
final state.

  * •

Final states with the $T$-odd $T^{-}$ produce final states with SM tops and
missing transverse momentum, a typical signature of natural supersymmetry with
a light scalar top.

  * •

Final states with the $T$-even $T^{+}$ not necessarily produce missing
transverse momentum but instead decay top-like into $bW^{+}$, hence are
expected to affect SM top measurements, or decay into top + Higgs/gauge boson
final states which is a typical feature of models with an extended quark
sector. Since processes involving $T^{+}$ have a reduced production cross
section, see our earlier discussion, and since our searches mostly focus on
SUSY-like final states, we do not expect these particles to be of great
relevance for our results.

  * •

If $T$-parity is violated by small couplings, we still expect the same
production and decay topologies as in the $T$-parity conserving case which
typically produce 2 $A_{H}$ and the same hard final state objects which we
listed in the previous discussion. However, as now each of these decays into
pairs of Standard Model particles, many more final state topologies appear.
Especially if $f\gtrsim 1.2\,\mathrm{TeV}$ we expect four Standard Model
vector bosons in the final state and as each of these can decay hadronically
or leptonically, a plethora of possible final state exists with various
combinations of additional jets and leptons. These can be covered by analyses
which target very large final state multiplicities for which the Standard
Model background is very small. Furthermore, as both $Z$ and $W$ have sizable
decay rates into final states with neutrinos, the final states may even have a
significant amount of missing transverse momentum and thus may still be
covered by the same supersymmetry-based analysis strategies as mentioned for
the $T$-parity conserving case.

All in all we expect various final states which are very similar to those
expected in typical supersymmetric models and we expect that this model can be
strongly constrained by applying LHC searches originally designed to find
supersymmetric particles. Even though theoretically expected, some of these
topologies not necessarily will result in a large enough signal event rate to
produce a sensible bound and/or various topologies appear simultaneously and
it is difficult to say _a priori_ which of these topologies is expected to
result in the strongest sensitivity. Fortunately, as many of these searches
are implemented in the tool CheckMATE, we expect this tool to perform very
well in our scenarios and determine the respectively strongest bounds for each
benchmark case conveniently. However, not all conceivable topologies mentioned
here have a matching analysis implemented in CheckMATE and therefore most
sensitive topologies determined below might not necessarily correspond to what
we theoretically expect at this stage. Our previous discussion should hence be
understood as a more general summary of interesting LHC topologies worthwhile
investigating at the Large Hadron Collider out of which we cover a large
fraction with our following CheckMATE analysis.

CM identifier |  Final State |  Designed for | Ref.
---|---|---|---
atlas_conf_2016_096 |  $\not{E}_{T}$ \+ 2-3 $\ell$ |  $\tilde{\chi}^{\pm},\tilde{\chi}^{0},\tilde{\ell}$ | ATLAS:2016uwq
atlas_conf_2016_054 |  $\not{E}_{T}$ \+ 1 $\ell$ \+ (b)-j |  $\tilde{q},\tilde{g}$ | ATLAS:2016lsr
atlas_conf_2017_022 |  $\not{E}_{T}$ \+ 0 $\ell$ \+ 2-6 j |  $\tilde{q},\tilde{g}$ | ATLAS:2017cjl
atlas_conf_2017_039 |  $\not{E}_{T}$ \+ 2-3 $\ell$ |  $\tilde{\chi}^{\pm},\tilde{\chi}^{0},\tilde{\ell}$ | ATLAS:2017uun
Table 3: Small summary of all $\sqrt{s}=13\,\mathrm{TeV}$ analyses which
appear in the discussion of our results. More details, also on other tested
analyses, are given in Tab. 5 in the appendix.

##  6 Collider Results from CheckMATE

We now discuss the results of our collider analysis performed with CheckMATE.
Exclusion lines in the $\kappa$-$f$–plane for all $3\times 2\times 2$
scenarios are shown in Figs. 14-22. For each case, we choose two ways to
present our results. On the respective plots in the left column we show the
total exclusion line determined by CheckMATE from LHC analyses at 8 TeV and 13
TeV, respectively. The 8 TeV results allow direct comparison to earlier
studies, e.g. in Reuter:2012sd ; Reuter:2013iya . Drawing them in the same
plot with the updated 13 TeV results illustrates how the increased energy and
the higher integrated luminosities significantly improve the sensitivity on
the Little Higgs Model with fully or nearly conserved $T$-parity. In the
discussion in the main text of this section we focus on the update from the
current results at $\sqrt{s}=13\,\mathrm{TeV}$ and will not discuss the
outdated results at 8 TeV center-of-mass energy. In the same set of plots we
also show mass contours of the most relevant particles to understand the
bounds. These are

  * •

the heavy gauge boson mass $Z_{H}$ ($=W_{H}$),

  * •

the heavy quarks $q_{H}$ for all models except _Heavy $q_{H}$_,

  * •

the heavy leptons $\ell_{H}$ for the model _Heavy $q_{H}$_,

  * •

the $T$-odd heavy top partner $T^{-}$ mass for _Light $T^{\pm}$_ benchmarks
and

  * •

the heavy photon mass $A_{H}$ for _TPV_ models.

To keep the plots readable we do not show all contours in all plots. With the
exception of $T^{\pm}$ whose mass values are only meaningful in the _Light
$T^{\pm}$_ scenario, all plots with same heavy fermion sector scenario (see
Table 2) have the same particle spectrum and therefore, each iso-mass contour
can be understood to appear in all other plots of the same main benchmark
scenario.

Alongside the above results we show a second plot each for all benchmark
scenario where we focus on the experimental signature(s) which lead to the
overall bound. For each benchmark study, we show the respective CheckMATE
analyses which cover the excluded region at $\sqrt{s}=13\,\mathrm{TeV}$. The
names in the legend correspond to the CheckMATE analysis identifiers and we
provide a small summary of their respective covered topologies in Tab. 3 for
convenience. Note that regions with small $\kappa$ and small $f$ are typically
covered by many more LHC analyses but we only show the minimal set of analyses
sufficient to cover the entire excluded region. A full list of all CheckMATE
analyses that we considered for this study can be found in Tab. 5 in the
appendix B.

Figure 11: Results for scenario (_Fermion Universality_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPC_)

Figure 12: Results for scenario (_Fermion Universality_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPC_)

Figure 13: Results for scenario (_Fermion Universality_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPV_)

Figure 14: Results for scenario (_Fermion Universality_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPV_)

###  6.1 Fermion Universality

We start with a discussion of the _Fermion Universality_ model in which the
heavy fermion Yukawa couplings are set to be equal,
$\kappa_{q}=\kappa_{\ell}$, and thus features a degenerate spectrum of heavy
quarks and heavy leptons.

####  $T$-parity conserved and heavy $T^{\pm}$

In Fig. 14 we start with the subscenario of conserved $T$-parity and with the
heavy top sector decoupled. The excluded parameter spaces can be separated
into two main regions:

  * •

For large $f\geq 1\,\mathrm{TeV}$, the exclusion line depends both on $\kappa$
and $f$ and runs nearly parallel to the iso-mass contours of the heavy quarks.
It thus nearly follows the inequality $f\times\kappa<f\kappa_{\text{max}}$
with $f\kappa_{\text{max}}\approx 1.5\,\mathrm{TeV}$ at
$\sqrt{s}=8\,\mathrm{TeV}$ and $\approx 2\,\mathrm{TeV}$ at
$\sqrt{s}=13\,\mathrm{TeV}$. The most sensitive analysis looks for at least
two hard jets and a large amount of missing transverse momentum, a topology
which in this region appears through heavy quark pair production with each
heavy quark decaying into a quark, an invisible heavy photon and possible
additional particles via more complicated casscades in the decay,
$q_{h}q_{H}\rightarrow qqA_{H}A_{H}+X$. The expected event rate for this QCD-
induced process mainly depends on the mass of the heavy quarks and thus
explains why the bound runs nearly parallel to the $q_{H}$ iso-mass contours.
Still, $t$-channel heavy vector bosons also have a small effect on the
production cross section and thus the bound drops slightly faster with higher
$f$, i.e. with larger $m(V_{H})$, than the $m(q_{H}$) iso-mass contour. The
results translate into a bound on $m_{q_{H}}$ of $\geq 3\,\mathrm{TeV}$ for
$f\approx 1\,\mathrm{TeV}$ which decreases to $m_{q_{H}}>2\,\mathrm{TeV}$ for
$f\gtrsim 3\,\mathrm{TeV}$.

  * •

For smaller values of $f$, the bound becomes nearly independent of the
specific values of $f$ or $\kappa_{q}$ and absolutely excludes
$f>900\,\mathrm{GeV}$. For large enough values of $\kappa$, the heavy quarks
are not created abundantly enough and hence we are only sensitive to the
electroweak production of heavy gauge bosons $V_{H}$ whose mass is indepedent
of $\kappa_{q}$. The given limit can then be interpreted as an absolute mass
bound $m_{Z_{H}}=m_{W_{H}}\gtrsim 600\,\mathrm{GeV}$. Even though their mass
is $\kappa$-independent, the bound still becomes stronger for increasing value
of $\kappa_{q}$. This is — see our discussion in Sec. 5.1 — due to
$\kappa_{q}$ affecting the mass of the heavy quarks who in turn interfere
destructively with their contribution to the total $V_{H}V_{H}$ production
cross section. Thus the weakest bound $f>800\,\mathrm{GeV}$ appears for
$\kappa_{q}\approx 2.5$ and improves to $f\gtrsim 950\,\mathrm{GeV}$ for
$\kappa\gtrsim 5.0$.

Interestingly, even though the main production channel has changed, the most
sensitive study is the same multijet analysis as before. The required topology
is created from hadronically decaying $W$-bosons in $W_{H}\rightarrow WA_{H}$
and from $b$-jets in the decay to a Higgs boson of $Z_{H}\rightarrow hA_{H}$.

####  $T$-parity conserved and light $T^{\pm}$

To see how the sensitivity to the heavy top partners compares to the previous
bound, we show below in Fig. 14 the results of the same model, but now with
$R=1.0$ and thus including processes which involve the production of heavy
$T^{\pm}$. Note that for fixed $R$, the mass of the $T^{\pm}$ only depends on
$f$ which is why for large $f$ these particles are not experimentally
accessible. Thus, we get the same bound on $(f\kappa_{\text{max}})$ as
explained for the previous benchmark.

However, if the $T^{\pm}$ are kinematically accessible they play an important
role for the overall bound. For our special case with $R=1.0$, we observe that
the absolute bound on $f$ increases to $f\geq 1.3\,\mathrm{TeV}$ and becomes
entirely $\kappa$ independent as the $T^{\pm}$ production modes, as opposed to
the $V_{H}$ modes discussed before, do not depend on the heavy quark sector.
Again, we observe the search for multijets plus missing transverse momentum to
be most sensitive for the bound.777Note that by the time this work was
completed, the restricted set of analyses implemented in CheckMATE contained
the updated multijet results with an integrated luminosity of 36 fb-1, but had
only searches for scalar tops implemented which use data from 13.3 fb-1. This
may explain why we observe multijet final states to be most sensitive even
though in Sec. 5.3 we expected heavy top partners to produce distinct decay
signatures which mimic scalar top decays in natural supersymmetry.

Clearly, the precise value of the lower limit on $f$ depends on the mass of
the heavy top partner particles which implicitly depends on the value of $R$.
We emphasize here that the choice $R=1.0$ just serves as a benchmark case and
any other $R$ value would directly affect the bound, see Eq. (20), in either
direction. We chose $R=1$ here for the reason that it is rather special as it
minimizes the LHT contributions to the EWPO, cf. Sec. 3. Our more general
conclusion from this benchmark study is thus that searches for $V_{H}$ and for
$T^{\pm}$ can yield competetive absolute lower bounds on $f$, and while the
bound derived from $V_{H}$ production is nearly independent of the chosen
benchmark, the presence of light top partners may put further constraints on
the model.

####  $T$-parity violated

In Figs. 14, 14 we show the results in case we include the anomaly-mediated
decays of the heavy photon $A_{H}$ into vector boson or lepton pairs, both
without (Fig. 14) and including (Fig. 14) the heavy top sector. We again split
the discussion into the two main parameter regions already discussed before:

  * •

We again observe a $\kappa$-dependent bound for large values of $f$ which
follows the iso-mass contour of the heavy quarks. However, compared to the
$T$-parity conserving case the bound is now slightly weaker, $m_{q_{h}}\geq
2.5\,\mathrm{TeV}$ for $f\approx 1\,\mathrm{TeV}$ and $m_{q_{h}}\geq
1.5\,\mathrm{TeV}$ for $f\approx 3\,\mathrm{TeV}$. There are two analyses with
nearly identical sensitivity in this region, namely the already discussed
zero-lepton–multijet plus $\not{E}_{T}$ analysis and the related multijet
analysis which requires one lepton in the final state. The fact that their
sensitivity is fairly similar can be qualitatively understood from the fact
that we expect many additional final state gauge bosons which produce
additional leptons and/or jets. Thus, both multijet studies with and without
leptons become sensitive and we get an overall similar signal event rate in
the respective signal regions of these two studies. In fact, as the branching
ratio to $WW$ increases for smaller $f$, see Fig. 10, and as $W$-bosons
produce on average more charged leptons than $Z$-bosons, we expect analyses
which require a final state lepton to become slightly more sensitive for
smaller $f$ — a feature which we exactly observe in our results in Fig. 14, on
the right hand side.

At first, it appears unexpected that the bound is not significantly weakened,
even though the originally invisible $A_{H}$ now decays into Standard Model
particles and thus appears to remove crucial missing transverse momentum from
the event. However, one should bear in mind that we expect four additional
boosted gauge bosons, two from each $A_{H}$, in the final state. Thus we
expect to pass the $\not{E}_{T}$ constraints if at least one of these decays
into neutrinos. Even though on average the branching ratio $V\rightarrow\nu+X$
is only around 25 %, as we have four gauge bosons the probability of having an
$A_{H}A_{H}$ pair decaying into at least one neutrino and thus producing
$\not{E}_{T}$ is above 70 %. This reduces the $\not{E}_{T}$ cut acceptance
slightly but not drastically compared to the $T$-parity conserving case.
Furthermore, we get the same visible final state objects as in the $T$-parity
conserving case, together with additional boosted particles from the gauge
boson decays which may even improve the final state acceptance. It thus can be
understood why the sensitivity does not drop significantly if $T$-parity
violation is considered.

  * •

Similarly to before, for a symmetry breaking scale $f$ of the order 1 TeV we
observe a $\kappa$ independent bound. Interestingly, the bound has even
improved after turning on $T$-parity violation and excludes $f\gtrsim
1\,\mathrm{TeV}$ for $\kappa\approx 1.5$ and $f\gtrsim 1100\,\mathrm{GeV}$ for
$\kappa\approx 4.0$. To understand why the limit becomes stronger one needs to
look at the analysis coverage map on the right of Fig. 14. We see that the
bound derived from the multijet analysis, which was most sensitive in the
$T$-parity conserving case, slightly weakened. This can be understood with the
same arguments as given before for the large-$f$ region. However, we also
observe that the sensitivity is now dominated by electroweakino-motivated
searches, more specifically by analyses which look for final state leptons and
missing transverse momentum. A more detailed look in the results of that
analysis reveals that it is in fact the signal region SR-Slep-e which produces
the bound. This signal region requires 3 high-$p_{T}$ charged leptons which do
not originate from a leptonically decaying $W$-$Z$-pair and a significant
amount of missing transverse momentum. Interestingly, such a signature could
not be reached in the previous $T$-parity conserving benchmark case, because
the most important topology $pp\rightarrow W_{H}W_{H}\rightarrow WWA_{H}A_{H}$
only produces two leptons. Including $T$-parity violation, we can get a third,
highly energetic lepton if one of the four final state gauge bosons is a
leptonically decaying $W$. Furthermore, since this signal region has no
constraints on the final state jet multiplicity, the decays of the other three
gauge bosons is irrelevant. As such, a large signal event rate is expected for
this analysis if $T$-parity is violated.

If the top partners are kinematically accessible, see Fig. 14, the absolute
bound on $f$ only increases slightly by about 100 GeV. The electroweak search
stays the most sensitive analysis for this model. The resulting bounds
increase as more events from the topology
$pp>T^{-}\bar{T}^{-}\rightarrow(bW)(bW)WWVV$ are expected. Again, the impact
on the bound depends on the precise value of $R$ and we only show one example
here which illustrates that the details of the heavy top partner sector are
relevant for the overall LHC limit.

Interestingly, the multijet analysis does not seem to get a significant
contribution from the presence of the $T^{\pm}$ even though it did in the
previous case when $T$-parity was conserved, cf. Figs. 18, 18. To understand
this behavior one needs to consider the details of the experimental search:
this analysis tries to cover various hierarchies and decay topologies that can
appear in the supersymmetric squark-gluino $\tilde{g},\tilde{q}$ sector and
defines many signal regions which target different jet multiplicities.
Different mass scales in the supersymmetric sector are taken into account by
gradually increasing the requirements on the sum of jet $p_{T}$ in the event
as well as the total amount of $\not{E}_{T}$, more specifically by using cuts
which require minimum values for the ratio $\not{E}_{T}$/$\sum($jet $p_{T})$.
In supersymmetry, jet multiplicity, total hadronic energy and missing
transverse momentum increase simultaneously as heavier particles on average
produce longer decay chains and give more momentum to the visible jets and the
invisible neutralino and thus a cut on $\not{E}_{T}$/$\sum($jet $p_{T})$ has a
good signal acceptance in supersymmetry.

However, such a cut is disadvantageous for our most important topology
$T^{\pm}\rightarrow tA_{H}$ if $A_{H}$ decays via TPV: the additional decay of
$A_{H}$ into gauge bosons is expected to produce a significantly larger amout
of jets and hadronic energy while reducing the amount of missing transverse
momentum, resulting in a large drop in the signal acceptance. Therefore adding
the $T^{\pm}$ to the experimentally accessible spectrum hardly increases the
amount of signal events in this case and the bound only improves little.

Figure 15: Results for scenario (_Heavy $q_{H}$_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPC_)

Figure 16: Results for scenario (_Heavy $q_{H}$_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPC_)

Figure 17: Results for scenario (_Heavy $q_{H}$_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPV_)

Figure 18: Results for scenario (_Heavy $q_{H}$_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPV_)

###  6.2 Heavy $q_{H}$

We continue with the discussion of the results for the _Heavy $q_{H}$_
scenario which fixes $\kappa_{q}$ to 3.0 and thus effectively decouples the
$q_{H}$ from the experimental reach. The results for all subscenarios
(with/without $T$-parity violation and ex-/including the heavy top partner
sector) are shown in Figs. 18-18. The plots show the same information as in
the previous section 6.1, however note that the ordinate is now chosen to be
the free parameter $\kappa_{\ell}$ and the iso-mass contours are given for the
$\ell_{H}$ instead of the $q_{H}$ now.888As the mass of the $\ell_{H}$ and
$q_{H}$ are identical for $\kappa_{q}=\kappa_{\ell}$, see Eq. (26), the iso-
mass contours for $\ell_{H}$ appear at the same position as those for $q_{H}$
in the previous benchmark.

To understand how the bounds change compared to the previous benchmark
scenario, it is worth repeating the two main phenomenological consequences of
this benchmark case:

  1. 1.

$q_{H}\rightarrow qV_{H}$ topologies are replaced by $\ell_{H}\rightarrow\ell
V_{H}$. Multijet final states are thus replaced by multilepton final states.
As the production cross section for $\ell_{H}\ell_{H}$ is 2 to 3 orders of
magnitude smaller than the corresponding cross section for $q_{H}q_{H}$, we
expect a far weaker sensitivity in the heavy fermion dominated region (i.e.
large $f$, small $\kappa$).

  2. 2.

$\sigma(pp\rightarrow V_{H}V_{H})$ was dependent on $\kappa$ but is
independent of $\kappa_{\ell}$ as no contributions from $t$-channel $q_{H}$
exist in this benchmark case. Thus we expect the bounds produced from $V_{H}$
pair production to be entirely $\kappa_{\ell}$ independent and very similar to
the case $\kappa=3.0$ of the previous benchmark.

With these pieces of information in mind, the results in Figs. 18-18 compare
straightforwardly to the bounds of the earlier benchmark scenario in Figs.
14-14:

  * •

For $f\approx 1\,\mathrm{TeV}$, vector boson production and potential heavy
top partner production are the most sensitive channels and they produce
$\kappa_{\ell}$ independent bounds of $f\gtrsim 950\,\mathrm{GeV}$ (TPC, no
$T^{\pm}$), $f\gtrsim 1350\,\mathrm{GeV}$ (TPC, with $T^{\pm}$), $f\gtrsim
1100\,\mathrm{GeV}$ (TPV, no $T^{\pm}$) and $f\gtrsim 1200\,\mathrm{GeV}$
(TPC, with $T^{\pm}$). The bounds correspond to those for the previous
benchmark for large values of $\kappa\gtrsim 4.0$. The most dominant
topologies also do not change: we observe multijet final states to be the most
sensitive ones in case $T$-parity is conserved while multilepton final states
become more important if $T$-parity is violated.

  * •

For $\kappa_{\ell}\lesssim 0.5$, the mass of the $\ell_{H}$ drops below the
mass of the heavy vector bosons and thus decays of type
$V_{H}\rightarrow\ell_{H}\ell$ can happen, see Fig. 7. The boosted final-state
leptons of this decay can be observed via a multilepton analysis as can be
seen in the right of Fig. 18. This significantly improves the sensitivity and
improves the bound on $f$ to up to 1.9 TeV. As the branching ratios depend on
$\kappa$, this bound is now slighly dependent on $\kappa$.

  * •

The “$f\kappa_{\text{max}}$”-bound which we were able to set in the previous
benchmark almost disappears for this scenario where the $q_{H}$ are decoupled.
The expected event rates from $\ell_{H}\ell_{H}$ pair production are so small
that no feasible bound can be set from this topology in case of $T$-parity
conservation, even with the newest $\sqrt{s}=13\,\mathrm{TeV}$ results. It is
only in the case of $T$-parity violation that we can observe an exclusion for
very small values of $\kappa$ which follows the $m(\ell_{H})=1\,\mathrm{TeV}$
mass contour, caused by a slight increase of the expected multilepton event
rates from leptonic gauge boson decays, see our discussion above.

All in all we observe that the presence or absence of the $q_{H}$ partner
particles plays a very important role for determining the LHC limits in the
low $\kappa$ region, i.e. for $\kappa\lesssim 1.5$. However, the heavy gauge
boson sector also puts very important constraints on $f$ and as the collider
phenomenology of this sector is almost, but not completely, independent of the
heavy fermion sector, the absolute bounds on $f$ are very robust against
choices for the heavy quark sector. In fact, they tend to become stronger as
the presence of light $q_{H}$ decreases the $V_{H}V_{H}$ production cross
section.

Figure 19: Results for scenario (_Light $\ell_{H}$_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPC_)

Figure 20: Results for scenario (_Light $\ell_{H}$_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPC_)

Figure 21: Results for scenario (_Light $\ell_{H}$_)$\times$(_Heavy
$T^{\pm}$_)$\times$(_TPV_)

Figure 22: Results for scenario (_Light $\ell_{H}$_)$\times$(_Light
$T^{\pm}$_)$\times$(_TPV_)

###  6.3 Light $\ell_{H}$

In our third main benchmark scenario we again scan $\kappa_{q}$ and thereby
the mass of the heavy quarks. However, the degeneracy with the heavy lepton
sector is now lifted by fixing $\kappa_{\ell}=0.2$. Since we know from the
results of the previous benchmark that no bound can be set if we search for
the direct production of $\ell_{H}$ alone, we _only_ consider the effects of
the light $\ell_{H}$ with respect to the branching ratio of the heavy gauge
bosons. Note again from our results in Sec. 5.3 that while in the _Fermion
Universality_ model we dominantly expect bosonic decays $Z_{H}\rightarrow
hA_{H},W_{H}\rightarrow WA_{H}$, the _Light Leptons_ benchmark mainly produces
leptonic decays
$Z_{H}\rightarrow\ell_{H}\ell,W_{H}\rightarrow\ell_{H}\nu/\nu_{H}\ell$ with
subsequent $\ell_{H}/\nu_{H}\rightarrow\ell/\nu A_{H}$.

In Figs. 22-22 we show the results of this benchmark, again for all four
subscenarios.

  * •

As in the _Fermion Universality_ scenario, we observe two main regions of
exclusion which intersect at $f\approx 1.6\,\mathrm{TeV}$ and
$\kappa_{q}\approx 1.2$.

  * •

For small $\kappa_{q}$ and large $f$, we again observe a $q_{H}$ dominated
bound similar to the one seen in the _Fermion Universality_ scenario. The
analysis coverage map reveals that for $\kappa_{q}>0.5$, the bound is set by a
multilepton analysis. The already mentioned 3$\ell$ signal region is very
sensitive to the final state topology $q_{H}q_{H}\rightarrow
qqW_{H}W_{H}\rightarrow qq\ell\ell\ell\ell A_{H}A_{H}$ with one of the leptons
not being identified and the other three leptons being highly boosted due to
the large $q_{H}-W_{H}$ mass splitting. For $\kappa_{q}<0.5$, the heavy vector
bosons start predominantly decaying into hadronic final states — see Fig. 8 —
in which case multijet final states start becoming more sensitive and
reproduce the same bound as in the _Fermion Universality_ scenario.

  * •

The $q_{H}$ dominated bound is again insensitive to the presence of the heavy
top partners. Furthermore, it again slightly weakens in the presence of
$T$-parity violation as the most sensitive final state stays identical but the
$\not{E}_{T}$ cut efficiency drops due to the $A_{H}$ decaying.

  * •

For larger values of $\kappa_{q}$, we again observe a nearly
$\kappa_{q}$-independent absolute bound on $f$. This bound is again produced
from direct production of heavy vector bosons and shows a small $\kappa_{q}$
dependence due to the cross section dependence of this parameter, see our
discussion before. Compared to the _Fermion Universality_ scenario, the limit
has become tremendously stronger due to the presence of light $\ell_{H}$ and
improves to $f\gtrsim 1.6\,\mathrm{TeV}$ for $\kappa_{q}\approx 1.5$ and to
$f\gtrsim 2\,\mathrm{TeV}$ for $\kappa_{q}\gtrsim 5.0$. As the analysis
coverage map on the right of Fig. 22 shows, the vector-boson dominated region
is now tested by the multilepton analysis which identifies the boosted leptons
from the $V_{H}\rightarrow\ell\ell/\nu A_{H}$ decays. As this final state has
small Standard Model background contamination — most importantly since the
leptons do not originate from $W$ or $Z$ decays — it produces a very clean
signal and thus leads to a very strong exclusion.

  * •

In this scenario, the presence of the heavy top partners does not improve the
bound derived from heavy vector boson production at all: the bound derived
from $T^{-}$ production — see the _Fermion Universality_ benchmark discussion
— is only sensitive to scales $f\lesssim 1350\,\mathrm{GeV}$ and thus cannot
compete with the much stronger bound set from the vector boson sector.
Furthermore, the multilepton final state produced from the $V_{H}$ decays do
not get any contributions from any of the expected $T^{-}$ decays. The limit
is therefore unaffected.

  * •

As the final state leptons from the $V_{H}$ decays already produce a very
clean signal, a possible decay of the $A_{H}$ induced by $T$-parity violation
only results in a smaller $\not{E}_{T}$ cut efficiency as explained before.
Thus, we only observe that the bound is slightly weakened in models with
$T$-parity violation.

To summarize the results of this benchmark, we observe that a lighter
$\ell_{H}$ sector changes the decay patterns of the heavy vector bosons and
this globally leads to a significant improvement on the bounds. This
improvement even overcomes possible contributions from the heavy top partner
sector and is only slightly weakened by the presence of $T$-parity violation.
Therefore we again conclude that the lower limits on $f$ derived in the
_Fermion Universality_ benchmark from searches for heavy vector bosons are
very robust regarding changes in the heavy fermion sector.

Note that for this benchmark we chose a specific value of $\kappa_{\ell}$ and
thus in fact only analyzed the impact of light $\ell_{H}$ for a particular
assumption for their masses. It is thus worthwhile discussing how changing
$\kappa_{\ell}$ would affect our results:

  * •

In our benchmark, the branching ratio $V_{H}\rightarrow\ell_{H}\ell$ was
nearly 100 %. Clearly, the partial decay width $V_{H}\rightarrow\ell_{H}\ell$
depends on the $\ell_{H}$ mass and thus the leptonic branching ratio may drop
if we increase the heavy lepton mass. The resulting bounds would then
gradually shift from those derived in the _Light $\ell_{H}$_ to those in the
_Fermion Universality_ benchmark.

  * •

The kinematic configuration of the $V_{H}\rightarrow\ell\ell^{(\prime)}A_{H}$
decay depends on the mass of the intermediate on-shell $\ell_{H}$. Changing
the mass results in different expected energy distributions for the signal
leptons and can therefore affect the signal acceptance after applying the cuts
in analysis atlas_conf_2017_039. However, as the mass splitting $V_{H}-A_{H}$
is of order 750 GeV for $f\approx 1.5\,\mathrm{TeV}$ and is independent of the
benchmark model, the final state leptons are always expected to be high-
energetic enough to pass the constraints.

###  6.4 Prospects for $\sqrt{s}=14\,\mathrm{TeV}$

CM identifier |  Final State |  Designed for | Ref.
---|---|---|---
atlas_2014_010_hl_3l |  $\not{E}_{T}$ \+ 3 $\ell$ |  $\tilde{\chi}^{\pm},\tilde{\chi}^{0}$ | ATL-PHYS-PUB-2014-010
atlas_phys_2014_010_sq_hl |  $\not{E}_{T}$ \+ 0 $\ell$ \+ 2-6 j |  $\tilde{q},\tilde{g}$ | ATL-PHYS-PUB-2014-010
dilepton_hl |  $\not{E}_{T}$ \+ 2 $\ell$ |  $\tilde{\chi}^{\pm},\tilde{\ell}$ | dileptonhl
Table 4: Small summary of all $\sqrt{s}=14\,\mathrm{TeV}$ analyses which
appear in the discussion of our results. More details, also on other tested
analyses, are given in Tab. 5 in the appendix.

As we observed in our results, the update from a center-of-mass energy of
$\sqrt{s}=8\,\mathrm{TeV}$ to $\sqrt{s}=13\,\mathrm{TeV}$ and the increase of
integrated luminosity between LHC Run 1 and Run 2 yielded significantly
stronger bounds for all of the considered benchmark scenarios. In that
context, the interesting question arises to which extent the sensitivity is
expected to further improve at a high luminosity LHC running at
$\sqrt{s}=14\,\mathrm{TeV}$. For that purpose, we used the ATLAS high
luminosity studies implemented in CheckMATE to determine the expected bounds
at very high statistics, $\int\mathcal{L}=3000\,\mathrm{fb}^{-1}$. This gives
a rough estimate for the overall sensitivity range of the Large Hadron
Collider to the Littlest Higgs Model in general. The corresponding cross
sections are shown in Figs. 32,32 in the appendix A.

Again, all analyses which have been used by this study are listed in Tab. 5 in
the appendix and we provide a shortened version in Tab. 4 which only lists
those analyses which appear in our discussion of the most sensitive analyses.
As one can see in the full table in Tab. 5, at this stage the list of high
luminosity analyses is very limited as only few official, experimental and
some phenomenological high performance studies have been implemented so far.
These cover the most important topologies, i.e. missing transverse momentum
with either a monojet, multijet or multileptons final state, however these old
experimental studies use far fewer, less optimized signal regions compared to
their counterparts at lower center-of-mass energies. Hence, our results should
only be understood as rough approximations and much more sophisticated
studies, especially on the experimental side, would be required to get results
which are qualitatively at the same level as our earlier, detailed re-
interpretation of existing experimental data.

Since the number of tested topologies is fairly small and is not expected to
cover all the various final states we discussed before, we do not consider the
full set of benchmark models introduced previously at this stage. Instead, we
concentrate on the results for _TPC_ $\times$ _Heavy $T^{\pm}$_ for the three
scenarios _Fermion Universality_ , _Heavy $q_{H}$_ and _Light $\ell_{H}$_.
These give a good overview to the general expected sensitivity at high
statistics. As can be seen from the results discussed above, the macroscopic
structure of the excluded parameter areas are very similar for cases with and
without $T$-parity violation and with the heavy top partners included or not.
Hence, one can apply the phenomenological discussions of the previous sections
to appoximately determine the excluded areas for the other benchmark cases
which we do not explicitly discuss in the following.

Figure 23: Expected results at
$\sqrt{s}=14\,\mathrm{TeV},\int\mathcal{L}=3000\,\mathrm{fb}^{-1}$ for
scenario (_Fermion Universality_)$\times$(_Heavy $T^{\pm}$_)$\times$(_TPC_).

Figure 24: Expected results at
$\sqrt{s}=14\,\mathrm{TeV},\int\mathcal{L}=3000\,\mathrm{fb}^{-1}$ for
scenario (_Heavy $q_{H}$_)$\times$(_Heavy $T^{\pm}$_)$\times$(_TPC_).

Figure 25: Expected results at
$\sqrt{s}=14\,\mathrm{TeV},\int\mathcal{L}=3000\,\mathrm{fb}^{-1}$ for
scenario (_Light $\ell_{H}$_)$\times$(_Heavy $T^{\pm}$_)$\times$(_TPC_).

The results of our scans are shown in Figs. 25-25. Note the extended $f$-axis
range compared to Figs. 14-22 to better illustrate the even higher $f$-reach
at high center-of-mass energies. The figures on the left column again show the
overall expected experimental reach at 14 TeV and compare to current results
from 13 TeV data which corresponds to the results discussed in the previous
sections. The figures in the right column, similarly to before, show the most
sensitive analyses in different regions of parameter space. Fluctuations in
the contours originate from sizable statistical uncertainties in our Monte
Carlo description999We use the same sample size as in our previous studies,
however due to the 100-fold integrated luminosity the statistical uncertainty
near the boundary increases approximately by a factor of 10., however do not
affect the qualitative description of the overall bound.

In general, the structure of the bounds is kept, i.e. there is a (nearly)
$\kappa$-independent bound for small $f$ and larger values of $\kappa$ while
there is a bound which follows the iso-mass contours for large values of $f$.

  * •

In the _Fermion Universality_ scenario, the $q_{H}$ mass bound for large
values of $f$ increases by 1 to 1.5 TeV and excludes heavy quarks with masses
$m(q_{H})\gtrsim 4\,\mathrm{TeV}$ for $f\approx 2\,\mathrm{TeV}$ and
$m(q_{H})\gtrsim 3\,\mathrm{TeV}$ for $f\approx 4\,\mathrm{TeV}$. As before,
this bound originates from the high luminosity version of a multijet plus
$\not{E}_{T}$ search designed to find heavy squarks or gluinos in
supersymmetry. The $V_{H}$ dominated bound for large values of $\kappa$ probes
heavy vector boson masses of order 1 TeV. Compared to the previous result
determined at 13 TeV, the most sensitive analysis is now quoted to be the
multilepton instead of the multijet final state. To reduce the contamination
from pileup which is expected to become an important issue for the high
luminosity LHC, the multijet final states require the scalar sum of the
transverse momenta of all reconstructed objects to exceed 3 TeV. In the
$V_{H}$ dominated region, the expected signal $V_{H}\rightarrow
A_{H}V,V\rightarrow\text{hadrons}$ with $m(V_{H})\approx 1\,\mathrm{TeV}$
typically does not pass this constraint and for example requires a boosted
final state due to a high $p_{T}$ jet from initial-state radiation (ISR) whose
requirement significantly reduces the expected event rate.

  * •

The _Heavy $q_{H}$_ scenario at $\sqrt{s}=14\,\mathrm{TeV}$ does not
significantly improve the $\ell_{H}$-induced bound for small values of
$\kappa$. We expect a weak bound which follows the $\ell_{H}$ mass contour and
excludes masses of order $m(\ell_{H})\approx 1-1.5\,\mathrm{TeV}$. This bound
originates from an extrapolated search for dilepton final states. This is
however only a minor improvement to the bound which can be set already from
today’s result. As in the previous benchmark scenario, the $V_{H}$ produces a
$\kappa$-independent bound of $m(V_{H})\gtrsim 1\,\mathrm{TeV}$.

  * •

Lastly, the bound in the _Light $\ell_{H}$_ scenario only improves little
compared to the current 13 TeV results. In the large $f$ region, the most
sensitive analysis channel at LHC Run 2 is a multijet final state with one
additional lepton which has a particularly small Standard Model contamination.
Unfortunately, we do not have a high luminosity version of this analysis
available and can only consider final states with many jets but no final state
lepton. As the characteristic feature of the the _Light $\ell_{H}$_ scenario
is the appearance of at least one lepton in all relevant final state decay
chains, we lose sensitivity due to our restricted amount of available
analyses. For larger values of $\kappa$, the bound on $m(V_{H})$ only
increases by about 100 GeV, determined from a search which requires two
leptons in the final state. This analysis is designed to target either of the
two supersymmetric topologies
$\tilde{\ell}\tilde{\ell}\rightarrow\ell\ell\tilde{\chi}\tilde{\chi}$ or
$\tilde{\chi}^{+}\tilde{\chi}^{-}\rightarrow WW\tilde{\chi}\chi$ followed by
leptonic $W$ decays. Though some of the final states produced by our benchmark
scenario pass the constraints set for these particular topologies, none of the
signal regions are specifically designed for our topology. Thus, again our
bound does not represent the full sensitivity which can be expected from the
high luminosity LHC but significant aditional effort would be required to
determine the necessary experimental predictions for our desired topologies.

##  7 Comparison of LHC limits with Bounds from Electroweak Precision
Observables

Figure 26: Combined results for scenario (_Fermion
Universality_)$\times$(_TPC_), left: _Heavy $T^{\pm}$_, right: $\times$(_Light
$T^{\pm}$_)

Figure 27: Combined results for scenario (_Heavy $q_{H}$_)$\times$(_TPC_),
left: _Heavy $T^{\pm}$_, right: $\times$(_Light $T^{\pm}$_)

Figure 28: Combined results for scenario (_Light $\ell_{H}$_)$\times$(_TPC_),
left: _Heavy $T^{\pm}$_, right: $\times$(_Light $T^{\pm}$_)

In the previous section we discussed the bounds which can be put on various
benchmark scenarios of the Littlest Higgs Model with $T$-parity (and its
possible violation). As explained in Sec. 2, an appealing property of this
model is its considerably small amount of fine tuning in the Higgs sector.
Moreover, not only do the null results of searches for these new $T$-odd
particles set bounds on this model but also, see Sec. 3, electroweak precision
observables (EWPO) put tight constraints on $f$ and $\kappa$. In the following
we want to combine these three pieces of information, putting a particular
focus on the relevance of the newest LHC results for the total combined bound
on the model.

In Figs. 28-28 we show compilations of bounds from electroweak precision
observables, see Sec. 3, the amount of fine tuning in the Higgs sector
according to Eq. (31) and the 8 and 13 TeV LHC bounds discussed in Sec. 6. We
only show results for the case of $T$-parity conservation as electroweak
precision observables are not affected by the presence of $T$-parity violating
operators and the respective TPV collider bounds are very similar, see our
results of the previous section. In each figure we show the results for _Heavy
$T^{\pm}$_ scenario, i.e. $R=0.2$, and _Light $T^{\pm}$_ scenario, i.e.
$R=1.0$. Note that the choice of this parameter has an important impact on the
fine tuning measure $\Delta$.

In general, we observe that LHC results produce an absolute lower bound on $f$
for large $\kappa$ and a lower bound which approximately follows
$f\cdot\kappa$ for small $\kappa$. Electroweak precision data, however, tend
to produce upper bounds which approximately follow the ratio $f/\kappa$.
Therefore, we have two very complementary bounds which together exclude a
considerably large region of parameter space. This complementarity mostly
originates from the opposite dependence of the respective bounds on $\kappa$
and $R$: the collider data produce stronger bounds for lighter particles and
therefore show their largest sensitivity for small values of $\kappa$ and/or
$R=1.0$. Loop corrections to precision observables, however, increase if the
corresponding coupling constants increase and therefore show their strictest
bounds for _large_ values of $\kappa$ and $R=0.2$.101010Note that the free
parameter $R$ defines the Yukawa coupling $\lambda_{2}$ via Eq. (18) which
_increases_ if $R$ _decreases_.

We now move the general discussion to some indiviual results of particular
benchmark models:

  * •

In the case of _Lepton Universality_ , we observe that the updated collider
results from 13 TeV are only relevant in the regions dominated by $q_{H}$ and
$T^{\pm}$ production. Most importantly, bounds derived from $V_{H}V_{H}$
production only cover the region $f<1\,\mathrm{TeV},\kappa>2$ and are not
competitive with the limits from electroweak precision data which cover the
same region in the _Light $T^{\pm}$_ scenario and an even much larger region
$f<1.3\,\mathrm{TeV},\kappa>1.5$ in the case of _Heavy $T^{\pm}$_.

In the case of _Heavy $T^{\pm}$_, the combined bound from electroweak
precision observables and $q_{H}$ production excludes symmetry breaking scales
$f$ below 1.3 TeV, independent of $\kappa$, and by that requires a fine tuning
below 0.5 %. If the heavy top partners $T^{\pm}$ are lighter, the EWPO bounds
weaken. However, at the same time the collider bounds increase, resulting in
approximately the same bound of $f>1.3\,\mathrm{TeV}$ as before which however
corresponds to a slightly smaller fine tuning of approximately 0.6 %. Judging
from the two benchmark scenarios for $T^{\pm}$, we conclude that the
combination of electroweak precision data and newest LHC results does not
allow for values of $f<1.3\,\mathrm{TeV}$ for values of $R\in[0.2,1.0]$. As
the EWPO bounds become stronger for heavier $T^{\pm}$ and the collider result
becomes stricter for lighter $T^{\pm}$, the lower bound on $f$ should become
even stricter for any value of $R$ outside this range.

  * •

The combined results of the _Heavy $q_{H}$_ scenario show a similar
complementarity effect as in the previous model: whilst the LHC results are
significantly weakened if the heavy quarks are decoupled, the bounds from
electroweak precision observables become even stricter due to their dependence
on $\kappa^{2}$, see Sec. 3, and thus become stronger if $\kappa_{q}=3.0$ is
fixed. Here, the bounds implicitly depend on the value of $R$ and exclude
values of $f$ below 1.5 TeV for $R=1.0$ (_Light $T^{\pm}$_), and values below
2 TeV for $R=0.2$ (_Heavy $T^{\pm}$_). Even in the case of _Light $T^{\pm}$_
the LHC result cannot compete. Still, the bounds are already very close to the
EWPO limit such that we again conclude that any other value of $R$ should not
produce a significantly weaker but potentially an even stronger bound on $f$
if the mass of the $T^{\pm}$ is chosen even lighter. Note that for very small
values of $\kappa_{\ell}$, the LHC bound derived from
$V_{H}\rightarrow\ell_{H}\ell$ pushes the lower bound on $f$ by a few hundred
GeV, but not considerably. The minimal allowed fine tuning is around 0.5 % for
the _Light $T^{\pm}$_ scenario and reduces to approximately 0.25 % for the
_Heavy $T^{\pm}$_ scenario.

  * •

For the _Light $\ell_{H}$_ scenario, the complementarity between LHC and EWPO
results appears in the opposite direction as before: Due to the small value of
$\kappa_{\ell}$, electroweak precision observables are slightly weaker than in
the previous benchmark cases. However, at the same time the collider bounds
improve significantly due to the very distinctive decay topology which
produces sevaral hard leptons, see our discussion in the previous section. In
this benchmark, the lower bound $f>1.7\,\mathrm{TeV}$ originates solely from
the collider result and is independent of the details of the heavy top partner
sector. It is only the region with large values of $f\gtrsim
1.8\,\mathrm{TeV},\kappa\gtrsim 2.5$ where the EWPO bound may become more
relevant — depending on the chosen value of $R$. The minimal allowed fine
tuning is around 0.35 % in the _Heavy $T^{\pm}$_ and 0.4 % in the _Light
$T^{\pm}$_ scenario, respectively.

All in all, we observe that without taking the LHC data into account, fine
tuning above 1 % would still be allowed in regions with light $q_{H}$ and
light $T^{\pm}$. These regions, however, are nowadays testable at collider
experiments and results from the first LCH run at 8 TeV already pushed the
fine-tuning to the sub-percent level. Using the updated results acquired
during the $\sqrt{s}=13\,\mathrm{TeV}$ period, limits derived from the Large
Hadron Collider become more and more severe. Though the precise position of
the total bound depends on the details of the heavy fermion sector, the heavy
top partner masses, and the presence or absence of $T$-parity violation, we
observe that due to their complementary behavior regarding the EWPO bounds,
values of $f$ below 1.3 TeV and fine-tuning above 0.6 % seems to be excluded
by now. Within our considered benchmark scenarios we observe that _Fermion
Universality_ is the most weakly constrained model. However, the newest 13 TeV
results show a significant improvement already when put in comparison with the
earlier 8 TeV bounds. Furthermore, our approximate future sensitivity study in
Sec.6.4 gives us reason to expect an even further improvement by LHC results
in the near and far future, putting the Littlest Higgs Model with $T$ parity
more and more to the test.

##  8 Summary

In this study we reinterpreted null results from LHC searches for physics
beyond the Standard Model in the context of the Littest Higgs Model with
conserved and broken $T$-parity. This model is an elegant implementation of
global collective symmetry breaking combined with a discrete symmetry to
explain the natural lightness of the Higgs boson as a (pseudo-)Nambu-Goldstone
boson. Bounds on the symmetry-breaking scale $f$ from data until 2013 were
still as low as roughly 600 GeV. This model predicts heavy partners for the
Standard Model quarks $q_{H}$, leptons $\ell_{H}$, gauge bosons
$W_{H},Z_{H},A_{H}$ and special partners for the top quark $T^{\pm}$. The mass
hierarchies and the presence of the discrete $T$-parity result in a model
which shares many phenomenlogical similarities with supersymmetric extensions
of the Standard Model, most importantly it features a stable, invisible
$A_{H}$ if $T$-parity is conserved, similar to the lightest neutralino in
supersymmetry with conserved $R$-parity.

Using the degrees of freedom for the full theory, we defined a set of
benchmark scenarios which make different assumptions about the mass
hierarchies in the heavy fermion sector, the masses of the heavy top partners
and the possible presence of small $T$-parity violating operators. By making
use of the collider phenomenology tool CheckMATE, we systematically analyzed
all relevant topologies at the LHC and derived bounds for all benchmark
scenarios, excluding those regions which would have predicted a signal in any
of the many considered search channels. We also give rough estimates for the
bounds expected from a high luminosity LHC running with
$\sqrt{s}=14\,\mathrm{TeV}$ and $3\,\mathrm{ab}^{-1}$ of integreated
luminosity.

Our results show that $q_{H}$ pair production, $V_{H}$ pair production and
$T^{-}$ pair production, respectively, produce strong bounds in the model
parameter space due to null results in searches dedicated for squarks and
electroweakinos in supersymmetry. Most importantly, searches which require a
large amount of hard jets and a significant amount of missing transverse
momentum produce the strongest results in regions where $q_{H}$ and $T^{-}$
production is important whilst searches for final states with multilepton and
missing energy become more relevant as soon as heavy vector boson production
is the dominant channel. Color-neutral heavy leptons are mostly irrelevant for
the LHC, unless they are light enough to appear in decay topologies like
$V_{H}\rightarrow\ell_{H}\ell$ in which case they are again largely
constrained by searches for multileptons and missing energy. Allowing for a
small amount of $T$-parity violation surprisingly only has a minor impact on
the result if compared to the case where $T$-parity is exactly conserved. This
can by explained by the fact that in the case of $T$-parity violation via
anomalous WZW-terms, $A_{H}$ decays predominantly into the Standard Model
gauge bosons whose leptonic decays can produce the required missing energy
plus additional hard particles which improve the signal-to-background ratio.

As the masses of the particles $q_{H},\ell_{H},$ $V_{H}$ and $T^{\pm}$ depend
differently on the Yukawa-like parameters $\kappa_{q},\kappa_{\ell}$ and $R$,
precise LHC bounds depend on the particular values of these three parameters.
On the other hand, all particle masses grow linearly with the symmetry
breaking scale $f$ and we conclude that LHC results from the
$\sqrt{s}=13\,\mathrm{TeV}$ run exclude any value of $f$ below 950 GeV at 95%
confidence level. The weakest bound appears in a scenario where only the heavy
gauge bosons are kinematically accessible and all Yukawa parameters are such
that the other particles are decoupled from LHC observability.

Our LHC bounds are complementary to those derived from electroweak precision
observables as the former constrain light particles with small Yukawa
couplings while the latter put limits on sizable contributions from large
Yukawa couplings. This complementarity strongly removes the dependence of the
bound on the details of the heavy fermion sector as weaker limits from the LHC
are compensated by corresponding stronger bounds from EWPO and vice versa. All
in all, from our benchmark results we conclude that the symmetry breaking
scale $f$ must be larger than 1.3 TeV and the fine tuning cannot be better
than 0.4 %. Even stronger bounds are possible if more details about the heavy
fermion sector are known and these limits can easily be derived from our
exhaustive set of results for the various benchmark scenarios. This
constitutes an improvement of more than 700 GeV compared to LHC run 1 data.

Though the Littlest Higgs model with $T$-parity has been constrained much
stronger by LHC run 2 data, it is still a rather natural solution to the
shortcomings of the electroweak and scalar sector, and we will need full high-
luminosity data from the LHC to decide whether naturalness is actually an
issue of the electroweak sector or not. A qualitative improvement of all
bounds on the model, particularly in the Higgs sector and the heavy lepton
sector, might need the running of a high-energy lepton collider (or a hadron
collider at much higher energy).

## Acknowledgments

The authors like to thank Marco Tonini for helping in the validation of our
results. SYS would like to thank Pedro Schwaller for clarifications regarding
the $T$-parity violating decays. JRR likes to thank Maxim Perelstein for
helpful discussions. DD and JRR acknowledge funding and support from the
Collaborative Research Unit (SFB) 676 of the Deutsche Forschungsgemeinschaft
(DFG), projects B1 and B11.

##  Appendix A Supplementary Figures for the Collider Analysis

In this section we provide additional figures which are useful to better
understand and/or reproduce our results but which are not necessarily needed
for the discussion of the main text.

This includes the cross sections for a center-of-mass energy of 8 TeV in Fig.
32 for the Fermion Universality/Light $\ell_{H}$ \+ Light $T^{\pm}$ scenario
as a function of the Little Higgs scale $f$ and $\kappa_{q}$, respectively. In
Fig. 32 we show the 8 TeV cross sections for the Heavy $q_{H}$ \+ Light
$T^{\pm}$ scenario. The lower figures, Fig. 32 and Fig. 32, show the cross
sections for the same scenarios, but now for 14 TeV full LHC center-of-mass
energies.

Figure 29: Same as Fig. 2 for $\sqrt{s}=8\,\mathrm{TeV}$.

Figure 30: Same as Fig. 32 for benchmark model _Heavy $q_{H}$_+_Light
$T^{\pm}$_.

Figure 31: Same as Fig. 2 for $\sqrt{s}=14\,\mathrm{TeV}$.

Figure 32: Same as Fig. 32 for benchmark model _Heavy $q_{H}$_+_Light
$T^{\pm}$_.

##  Appendix B Full List of CheckMATE Analyses

CheckMATE identifier |  Search designed for | #SR | $L_{\text{int}}$ | Ref.
---|---|---|---|---
$\sqrt{s}=8\,\mathrm{TeV}$
atlas_1308_1841 |  New phenomena in final states with large jet multiplicities and $\not{E}_{T}$ | 13 | 20.3 | Aad:2013wta
atlas_1308_2631 |  Direct $\tilde{t}/\tilde{b}$ pair production in final states with $\not{E}_{T}$ and two $b$-jets | 6 | 20.1 | Aad:2013ija
atlas_1402_7029 |  Direct production of $\tilde{\chi}^{\pm}/\tilde{\chi}^{0}$ in events with 3 $\ell$ and $\not{E}_{T}$ | 20 | 20.3 | Aad:2014nua
atlas_1403_4853 |  Direct $\tilde{t}$ pair production in final states with 2$\ell$ | 12 | 20.3 | Aad:2014qaa
atlas_1403_5222 |  Direct $\tilde{t}$ pair production in events with a $Z$, $b$-jets and $\not{E}_{T}$ | 5 | 20.3 | Aad:2014mha
atlas_1404_2500 |  Supersymmetry in final states with jets and 2 SS $\ell$ or 3$\ell$ | 5 | 20.3 | Aad:2014pda
atlas_1405_7875 |  Search for $\tilde{q}$ and $\tilde{g}$ in final states with jets and $\not{E}_{T}$ | 15 | 20.3 | Aad:2014wea
atlas_1407_0583 |  $\tilde{t}$ pair production in final states with 1 isol. $\ell$, jets and $\not{E}_{T}$ | 27 | 20.3 | Aad:2014kra
atlas_1407_0608 |  Pair produced 3rd gen. squarks decaying via $c$ or compressed scenarios | 3 | 20.3 | Aad:2014nra
atlas_1411_1559 |  New phenomena in events with a photon and $\not{E}_{T}$ | 1 | 20.3 | Aad:2014tda
atlas_1501_07110 |  Direct production of $\tilde{\chi}^{\pm}/\tilde{\chi}^{0}$ decaying into a Higgs boson | 12 | 20.3 | Aad:2015jqa
atlas_1502_01518 |  New phenena in final states with an energetic jet and large $\not{E}_{T}$ | 9 | 20.3 | Aad:2015zva
atlas_1503_03290 |  Supersymmetry in events with an SFOS $\ell$ pair, jets and large $\not{E}_{T}$ | 1 | 20.3 | Aad:2015wqa
atlas_1506_08616 |  Direct Pair production third generation squarks | 12 | 20.0 | Aad:2015pfx
atlas_conf_2012_104 |  Supersymmetry in final states with jets, 1 isolated lepton and $\not{E}_{T}$ | 2 | 5.8 | ATLAS:2012tna
atlas_conf_2013_024 |  Directo $\tilde{t}$ pair production in the all-hadronic $t\bar{t}$ \+ $\not{E}_{T}$ final state | 3 | 20.5 | ATLAS:2013cma
atlas_conf_2013_049 |  Direct $\tilde{\ell}/\tilde{\chi}^{\pm}$ production in final states with 2 OS $\ell$, no jets and $\not{E}_{T}$ | 9 | 20.3 | TheATLAScollaboration:2013hha
atlas_conf_2013_061 |  Strongly produced Supersymmetric particles with $\geq 3$ $b$-jets and $\not{E}_{T}$ | 9 | 20.1 | TheATLAScollaboration:2013tha
atlas_conf_2013_089 |  Strongly produced Supersymmetric particles decaying into 2 leptons | 12 | 20.3 | TheATLAScollaboration:2013via
atlas_conf_2015_004 |  Invisibly decaying Higgs bosons produced in vector boson fusion | 1 | 20.3 | ATLAS:2015yda
atlas_conf_2012_147 |  New phenomena in monojets plus $\not{E}_{T}$ | 4 | 10.0 | ATLAS:2012zim
atlas_conf_2013_035 |  Direct production of $\tilde{\chi}^{\pm}/\tilde{\chi}^{0}$ in events with 3 leptons and $\not{E}_{T}$ | 6 | 20.7 | ATLAS:2013rla
atlas_conf_2013_037 |  Direct $\tilde{t}$ pair production in final states with 1 isolated $\ell$, jets and $\not{E}_{T}$ | 6 | 20.7 | ATLAS:2013pla
atlas_conf_2013_047 |  $\tilde{q}$ and $\tilde{g}$ in final states with jets and $\not{E}_{T}$ | 10 | 20.3 | TheATLAScollaboration:2013fha
cms_1303_2985 |  Supersymmetry in hadronic final states with $b$-jets and $\not{E}_{T}$ using $\alpha_{T}$ | 59 | 11.7 | Chatrchyan:2013mys
cms_1408_3583 |  Dark Matter, Extra Dimensions and Unparticles in monojet events | 7 | 19.7 | Khachatryan:2014rra
cms_1502_06031 |  New Physics in events with 2$\ell$, jets and $\not{E}_{T}$ | 6 | 19.4 | Khachatryan:2015lwa
cms_1504_03198 |  Dark Matter produced in association with $t\bar{t}$ in final states with 1$\ell$ | 1 | 19.7 | Khachatryan:2015nua
cms_sus_13_016 |  Supersymmetry in events with 2 OS $\ell$, many jets, $b$-jets and large $\not{E}_{T}$ | 1 | 19.5 | CMS:2013ija
cms_exo_14_014 |  Heavy Majorana neutrinos in events with SS dileptons and jets | 16 | 19.7 | Khachatryan:2016olu
$\sqrt{s}=13\,\mathrm{TeV}$ |
atlas_1602_09058 |  Supersymmetry in final states with jets and two SS leptons or 3 leptons | 4 | 3.2 | Aad:2016tuk
atlas_1604_01306 |  New phenomena in events with a photon and $\not{E}_{T}$ | 1 | 3.2 | Aaboud:2016uro
atlas_1604_07773 |  New phenomena in final states with an energetic jet and large $\not{E}_{T}$ | 13 | 3.2 | Aaboud:2016tnv
atlas_1605_03814 |  $\tilde{q}$ and $\tilde{g}$ in final states with jets and $\not{E}_{T}$ | 7 | 3.2 | Aaboud:2016zdn
atlas_1605_04285 |  Gluinos in events with an isolated lepton, jets and $\not{E}_{T}$ | 7 | 3.3 | Aad:2016qqk
atlas_1605_09318 |  Pair production of $\tilde{g}$ decaying via $\tilde{t}$ or $\tilde{b}$ in events with $b$-jets and $\not{E}_{T}$ | 8 | 3.3 | Aad:2016eki
atlas_1606_03903 |  $\tilde{t}$ in final states with one isolated lepton, jets and $\not{E}_{T}$ | 3 | 3.2 | Aaboud:2016lwz
atlas_1609_01599 |  Measurement of $ttV$ cross sections in multilepton final states | 9 | 3.2 | Aaboud:2016xve
atlas_conf_2015_082 |  Supersymmety in events with leptonically decaying $Z$, jets and $\not{E}_{T}$ | 1 | 3.2 | TheATLAScollaboration:2015nxu
atlas_conf_2016_013 |  Vector-like $t$ pairs or 4 $t$ in final states with leptons and jets | 10 | 3.2 | TheATLAScollaboration:2016gxs
atlas_conf_2016_050 |  $\tilde{t}$ in final states with one isolated lepton, jets and $\not{E}_{T}$ | 5 | 13.3 | ATLAS:2016ljb
atlas_conf_2016_054 |  $\tilde{q}$, $\tilde{g}$ in events with an isolated lepton, jets and $\not{E}_{T}$ | 10 | 14.8 | ATLAS:2016lsr
atlas_conf_2016_076 |  Direct $\tilde{t}$ pair production and DM production in final states with 2$\ell$ | 6 | 13.3 | ATLAS:2016xcm
atlas_conf_2016_078 |  Further searches for $\tilde{q}$ and $\tilde{g}$ in final states with jets and $\not{E}_{T}$ | 13 | 13.3 | ATLAS:2016kts
atlas_conf_2016_096 |  Supersymmetry in events with 2$\ell$ or $3\ell$ and $\not{E}_{T}$ | 8 | 13.3 | ATLAS:2016uwq
atlas_conf_2017_022 |  $\tilde{q}$, $\tilde{g}$ in final states with jets and $\not{E}_{T}$ | 24 | 36.1 | ATLAS:2017cjl
atlas_conf_2017_039 |  Electroweakino production in final states with 2 or 3 leptons | 37 | 36.1 | ATLAS:2017uun
atlas_conf_2017_040 |  Dark Matter or invisibly decaying $h$, produced in associated with a $Z$ | 2 | 36.1 | ATLAS:2017zkz
cms_pas_sus_15_011 |  New physics in final states with an OSSF lepton pair, jets and $\not{E}_{T}$ | 47 | 2.2 | CMS:2015bsf
$\sqrt{s}=14\,\mathrm{TeV}$ |
atlas_phys_pub_2013_011 |  Search for Supersymmetry at the high luminosity LHC ($\tilde{t}$ sector) | 4 | 3000 | ATL-PHYS-PUB-2013-011
atlas_2014_010_hl_3l |  Search for Supersymmetry at the high luminosity LHC ($\tilde{\chi}^{\pm}/\tilde{\chi}^{0}$ sector) | 1 | 3000 | ATL-PHYS-PUB-2014-010
atlas_phys_2014_010_sq_hl |  Search for Supersymmetry at the high luminosity LHC ($\tilde{q}/\tilde{g}$ sector) | 10 | 3000 | ATL-PHYS-PUB-2014-010
dilepton_hl* |  Custom Search for $\tilde{\ell}/\tilde{\chi}^{\pm}$ in final states with 2 leptons and $\not{E}_{T}$ | 9 | 3000 | dileptonhl
atlas_14tev_monojet* |  Custom Search for DM in final states with an energetic jet and $\not{E}_{T}$ | 5 | 3000 | Kim:2015hda
Table 5: Full list of all CheckMATE analyses used for this study. The column
labelled #SR yields the number of signal regions. Entries for the integrated
luminosities $L_{\text{int}}$ are given in fb-1.

Table 5 gives the full list of used CheckMATE analyses. The first column shows
the CheckMATE idenitifer, the second the purpose for which the analysis was
designed for. The last three columns show the number of signal regions in the
corresponding analysis (marked #SR), the integrated luminosity for that
analysis and the reference to the publication or conference notes from the
experimental collaborations. More details on the respective analyses and
corresponding validation material can be found on
http://checkmate.hepforge.org. High luminosity analyses marked with * do not
correspond to official experimental studies but have been implemented by the
CheckMATE collaboration. More information can be found in the respective
references.

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