# Beyond-Standard-Model Physics Associated with the Top Quark

Roberto Franceschini
Università degli Studi and INFN Roma Tre, Via della Vasca Navale 84, I-00146,
Roma
email<EMAIL_ADDRESS>
###### Abstract

We review scenarios of physics beyond the Standard Model in which the top
quark plays a special role. Models that aim at the stabilization of the weak
scale are presented together with the specific phenomenology of partner states
that are characteristic of this type of model. Further, we present models of
flavor in which the top quark is singled out as a special flavor among the SM
ones. The flavor and collider phenomenology of these models is broadly
presented. Finally, we discuss the possibility that dark matter interacts
preferably with the top quark flavor and broadly present the dark matter
phenomenology of these scenarios, as well as collider and flavor signals.

###### keywords:

top quark, beyond the standard model, hierarchy problem, flavor, dark matter,
new physics

###### Contents

  1. 1 Introduction
  2. 2 Top quark and BSM related to the Higgs boson and the origin of the weak scale
    1. 2.1 Supersymmetry
      1. 2.1.1 Phenomenology
      2. 2.1.2 Experimental searches
    2. 2.2 Composite and pNGB/Little Higgs
      1. 2.2.1 Models
      2. 2.2.2 Phenomenology
      3. 2.2.3 Experimental searches at colliders
  3. 3 EFT at current and future colliders
  4. 4 Top quark and BSM related to Flavor Dynamics or Dark Matter (or both)
    1. 4.1 Top quark and BSM related to flavor
    2. 4.2 Flavored dark matter models
  5. 5 Conclusions

##  1 Introduction

The top quark is a singular object amidst the fermions of the Standard Model
as it is the heaviest among them. This entails several peculiar properties: in
the domain of QCD it stands out as it is the only quark never to be observed
into a hadron, as its decay is much faster than the hadron formation time;
after the discovery of the Higgs boson, and for all the time before in which
the Higgs mechanism has dominated the landscape of model building for the
electroweak sector of the SM, the top quark stands out as the only one with a
“normal” size coupling with the Higgs boson. This latter property has made the
top quark very interesting both for the question about the origin of the
structure of flavor in the SM and for the origin of the electroweak scale
itself. The special interest about top flavor has to do with its strong
preference to decay into bottom quarks, i.e not involving other flavor
families, which in the $CKM$ picture results in $V_{tb}=1$ up to small
corrections, and its large mass, which can possibly act as a magnifier of the
effects of physics beyond the Higgs boson as origin of flavor. For electroweak
physics the top quark plays a crucial role in that it affects the properties
of the Higgs boson, and by the Higgs mechanism for weak bosons mass
generation, also in the physics of weak gauge bosons: its effect can be seen
in their masses and decay rates, which are sensitive to the strength of the
top quark gauge and Yukawa couplings and to its mass. Deviations of these
properties from the SM predictions can be signs of new physics related to the
top quark. While the importance of the top quark can be appreciated already
from these general facts, the detailed role played by the top quark can be
better understood going closer to explicit new physics models, which will pace
the exposition of the greatest part of the following material.

In sections 2.1 and 2.2 we discuss models in which the top quark plays a
special role for the origin of the electroweak symmetry. The discussion is
further extended in section 3 in a more model independent direction using a
flavor-conserving effective field theory of the top quark sector, which also
allow to discuss prospects for top quark physics at future colliders. In
section 4.1 we attack a different problem, that of the origin of the flavors
of the SM. In section 4.2 we extend the discussion to the possibility that SM
flavor plays a part in the stabilization of the dark matter in a way that
makes the dark matter interact preferably with the top quark flavor and
discuss the phenomenology of dark matter in these scenarios. Finally in
section 5 we offer some conclusions.

Being the subjects list rather large, the discussion is necessarily kept free
from some details, which are available in the provided references. This review
is conceived so that it can also be useful for younger graduate students
seeking an high-level introduction to the subject(s) discussed. Hopefully the
readers can start here their own exploration on topics that would otherwise
require to go through a large stack of literature. References are kept to a
minimum of key works as to encourage the reader to actually study these
selected works.

##  2 Top quark and BSM related to the Higgs boson and the origin of the weak
scale

###  2.1 Supersymmetry

Supersymmetry has been proposed as a space-time symmetry involving fermionic
generators. Unlike in gauge symmetries, this makes possible to involve spin
and momentum in the definition of the symmetry algebra, which, up to
violations of the symmetry itself, would require interactions and masses of
bosonic and fermionic particles to be tightly related. One such relation would
require the electron to be accompanied by exactly mass degenerate states of
spin-0, pretty much the same as Lorentz symmetry of space-time built-in the
Dirac equation implies the existence of exactly mass degenerate anti-particles
of the electron. The absence of any evidence in experiments for spin-0
electron-like state motivates to consider supersymmetry as an approximate
symmetry, broken at some unknown scale so that all the supersymmetric partners
of the SM states are pushed beyond the mass scale presently probed by
experiments.

The mechanism for supersymmetry breaking is a subject for model building,
which is outside of the scope of this review. For our purpose it is key to
recall that the supersymmetry breaking top quark sector has the rather model-
independent tendency to determine the Higgs bosons mass and quartic coupling,
thus leading to the identification of the supersymmetric top scalar quark,
most often called “stop squark”, as the main player setting the Higgs boson
potential. In the Minimal Supersymmetric Standard Model (see [1] for an
extensive review) this is represented by equations for the constraints on the
minimization of the Higgs boson potential

$\displaystyle m_{Z}^{2}$
$\displaystyle=\frac{\left|m_{H_{d}}^{2}-m_{H_{u}}^{2}\right|}{\sqrt{1-\sin^{2}(2\beta)}}-m_{H_{u}}^{2}-m_{H_{d}}^{2}-2\left|\mu\right|^{2}\,,$
$\displaystyle\sin(2\beta)$
$\displaystyle=\frac{2b}{m_{H_{u}}^{2}+m_{H_{d}}^{2}+2\left|\mu\right|^{2}}\,,$

coupled with the 1-loop effect of the top quark and top squark on the bilinear
terms of the 2 Higgs doublets $H_{u}$ and $H_{d}$ . In particular, for the
Higgs doublet $H_{u}$ that interacts with up-type quarks, hence feels the top
quark sector, the RGE equations is

$\displaystyle\frac{d}{d\log Q}m_{H_{u}}^{2}$
$\displaystyle=3X_{t}-6g_{2}\left|M_{2}\right|^{2}-\frac{6}{5}g_{1}^{2}\left|M_{1}\right|^{2}+\frac{3}{5}g_{1}^{2}S\,,$
(1)

where
$X_{t}=2\left|y_{t}\right|^{2}\left(m_{H_{u}}^{2}+m_{Q_{3}}^{2}+m_{\bar{u}_{3}}^{2}\right)+2\left|a_{t}\right|^{2}$,
$M_{1,2}$ are the U(1) and SU(2) gaugino mass terms, and
$S=Tr[Y_{j}m_{\phi_{j}}^{2}]$.

These equations naturally lead to possibility that the supersymmetry breaking
stop masses $m_{Q_{3}}^{2}$ and $m_{\bar{u}_{3}}^{2}$ or a large A-term
$\left|a_{t}\right|$ might induce a large $X_{t}$, which in turn drives
$m_{H_{u}}^{2}<0$ as $\log Q$ diminishes from some high-scale down to the weak
scale. This possibility has made the role of stop squarks a very central one
in supersymmetric models. In essence, the supersymmetric partner of the top
quark is responsible for breaking the electroweak symmetry, by making
$m_{H_{u}}^{2}<0$ hence making the Higgs boson potential unstable at the
origin of the $H_{d},H_{u}$ fields space, and setting the value of the masses
that set the weak scale, e.g. $m_{Z}$ from the above equation or the mass of
the Higgs boson that receives the above mentioned large radiative corrections
from the stop squark.

As a matter of fact, once the Higgs boson was discovered and its mass was
known, a number of works tried to determine the impact of this measurement on
the properties of the stop squark (e.g. see Ref. [2] for the MSSM and some
extensions). In turn, the necessity for peculiar supersymmetry breaking to
accommodate the Higgs mass has spurred investigations on the possible
supersymmetry breaking models that can lead to such peculiar stop squarks (see
e.g. [3, 4, 5, 6, 7, 8] for some examples of supersymmetry breaking models
emerged or re-emerged to address the null searches of supersymmetry and the
Higgs discovery).

####  2.1.1 Phenomenology

The phenomenology of the supersymmetric partners of the top quark is largely
dictated by one feature of the supersymmetric models: the existence of a
conserved quantum number that distinguishes SM states from their
supersymmetric partners. The standard choice for such quantity is called
$R$-parity, a $Z_{2}$ symmetry under which all SM states are even and all
partners states are odd. The conservation of this symmetry implies that
partners states can appear in interaction vertexes only in even number, e.g.
one SM states can interact with two supersymmetric states and it is not
possible for a single supersymmetric state to interact with a pair of SM
states. For particle colliders this implies that the lowest order process to
produce supersymmetric states in collisions is

$SM\,SM\to SUSY\>SUSY,$

and the decay of supersymmetric particles to any number of SM states is
forbidden unless there is at least one supersymmetric particle (or an odd
number of them), e.g.

$SUSY\to SUSY\,SM\,.$

When $R$-parity is exact a most copious production mechanism for stop squarks
at the LHC is

$gg\to\tilde{t}_{i}\tilde{t}_{j}^{*},$ (2)

where we denoted $\tilde{t}_{k}$ for $k=1,2$ the two stop squarks mass
eigenstates 111The definition of mass eigenstate as “stops” assumes that
flavor labels we give in the SM are the same for the partners states. It must
be stressed that the fate of flavor in the supersymmetric partners sector is
largely model dependent and it is possible to use flavor mixing to change the
phenomenology of stop squarks, see e.g. [9]. See [1] for more details on the
gauge and flavor structure of the squark sector.. Other production mechanisms
are possible, e.g. in decays of supersymmetric partners heavier than the stops
or via production of stops in association with other supersymmetric states.

Once produced, the stop squark can decay in a number of possible channels,
depending on which supersymmetric states are lighter than the state
$\tilde{t}_{k}$ at hand. Most studied 2-body decay modes are

$\tilde{t}\to t\chi^{0},\tilde{t}\to b\chi^{+}\,,$ (3)

which feature fermions $\chi$ that are mixtures of supersymmetric partners of
gauge bosons of the electroweak interactions and of the Higgs bosons of the
model. The motivation for the prevalence of these decay modes is that, by the
rules of unbroken supersymmetry, these decays are mediated by couplings given
by gauge and Yukawa couplings of the SM, hence they are pretty much impossible
to switch off unless $m_{\tilde{t}}-m_{\chi}<0$. As a matter of fact the
quantity $m_{\tilde{t}}-m_{\chi}$ plays a major role in determining the stop
phenomenology. When $m_{\tilde{t}}-m_{\chi}\to 0$ it becomes necessary to
consider multi-body processes are also possible and may be phenomenologically
relevant, e.g.

$\tilde{t}\to bW^{+}\chi^{0},\,\tilde{t}\to b\bar{f}f^{{}^{\prime}}\chi^{0},$
(4)

as well as possible flavor violating decays that may be induced at loop level,
such as

$\tilde{t}\to c\chi^{0}\,.$ (5)

In the above discussion the particle $\chi^{0}$ is considered as the lightest
supersymmetric state (LSP), so that, by the conservation of $R$-parity, it is
absolutely stable. As $\chi^{0}$ is not electrically charged and it is color
neutral, pretty much like neutrinos it does not leave directly observables
traces in detectors. For this reason the presence of $\chi^{0}$ can be
detected only as momentum missing in the overall momentum conservation in each
collision. As we cannot reliably measure the fractions of the longitudinal
momentum of the colliding protons taken by the partons initiating the
production of stops, e.g. the gluons entering in eq.(2), and the fraction
taken by the rest of the partons, the longitudinal momentum conservation is
usually not exploited in hadron colliders, therefore the presence of
$\chi^{0}$ is usually sought for as missing transverse momentum, most often
(mis)named missing transverse energy $mET$.

Being an electrically neutral stable particle charged only under
supersymmetric Yukawa and electroweak gauge interactions, $\chi^{0}$ qualifies
as perfect candidate for a WIMP Dark Matter. The possibility to have a Dark
Matter candidate stemming out of supersymmetry has given formidable motivation
to pursue this scenario for the past decades. So much so, that missing
transverse energy searches have becomes synonymous of searches for
supersymmetry. It must be said, however, that the null searches of
supersymmetric particles, as well as WIMP Dark Matter in the mass range
suitable for $\chi^{0}$[10], has put this idea under great pressure lately
[11, 12].

Given these experimental results, and the vast range of possible models for
supersymmetry breaking, it must be recalled that in general it is possible to
have other states than $\chi^{0}$ as lightest supersymmetric particles. For
instance the supersymmetric partner of a neutrino or even top sector squarks.
The latter leads to peculiar phenomena due to the formation of hadrons
containing supersymmetric states[13][14], but these models typically suffer
from quite stringent limits [15, 16, 17]. Therefore the majority of the
searches for supersymmetric states in the top quark sector are carried out in
the $\chi^{0}$ LSP setting.

Wholly alternative phenomenological scenarios for supersymmetric top quark
partners are possible and are actively pursued in experimental searches. The
main possible alternative has to do with the non-conservation of $R$-parity
[18]. With broken $R$-parity all supersymmetric particles can in principle be
produced singly and can decay into just SM states, e.g.

$SM\,SM\to SUSY\>\text{and }SUSY\to SM\,SM\,,$

are now possible processes. In this situation there is no longer an absolutely
stable weak scale particle to purse the idea of Dark Matter as a
WIMP222Alternative DM candidates can be found in these models, see e.g. [19]
for a possible gravitino dark matter scenarios and issues related to this
possibility. and the phenomenology of supersymmetric states linked to the top
quark is now greatly different from the picture given above [20]. For instance
$R$-parity violating couplings, still respecting the full gauge symmetry of
the SM, allow, among other possibilities, the decays

$\tilde{t}\to bs\text{ or }\tilde{t}\to\ell d\,.$

As the final states of stop decay can now be made entirely of SM particles it
is possible to detect stop squarks as resonances, a very powerful signature,
that is not possible to pursue when $\chi^{0}$ is forced to appear among the
decay products. Furthermore these decays, being mediated by $R$-parity
breaking couplings, that need to be small for a number of constraints [18],
can lead to meta-stable supersymmetric states, which can live measurable
lengths in experiments.

####  2.1.2 Experimental searches

In a detailed model it is possible to derive very specific signals from top
sector supersymmetric partners, including both signatures at collider
experiments and as well as low energy precision ones. The latter, however,
turn out to be usually very much dependent on the model considered for low
energy precision experiments [21]. A similar issue exists with early universe
physics, on top of the signals being quite difficult to detect. For this
reason collider searches are the prime way to search for top sector
supersymmetric partners.

Before listing relevant searches it is necessary to clarify a point on their
scope. The above searches are sensitive in principle to _any_ sign of new
physics related to the top quark sector involving mET or some kind of pair
produced resonances. Although the search is optimized for supersymmetric
partners, it can indeed be used to set bounds on other models. The interested
reader can refer for instance to Ref. [22] for an interpretation of the
“supersymmetry searches” in the context of fermionic top partners to be
discussed in later Section 2.2.2.

The searches for top sector supersymmetric partners can be divided into two
main categories:

  * •

searches in large momentum transfer signals, which feature detector objects
(jets, leptons, photons, …) with energy and transverse momentum greater than
the typical SM events;

  * •

searches in low momentum transfer signal, in which the detector objects
arising from top sector supersymmetric partners production are not very
different from that of typical SM events.

The large momentum transfer ones are “classic” searches for new physics, and
were envisaged already at the time of design of the experiments [23, 24].
Currently these searches can probe supersymmetric top partners up to a mass
around 1.2 TeV, although not in full generality. Indeed it is quite hard to
probe in full generality even a model as “minimal” as one having the full
freedom to vary the branching ratios of decays eqs.(3)-(5). For a complete
assessment is then necessary to test very accurately a large number of
searches at once, often relying on a “phenomenological” incarnation of a
sufficiently general supersymmetric model, as studied for instance in Ref.
[25]. The interpretation of these results is quite difficult, as many
constraints on the model are imposed at once, e.g. the top partners states are
required to “fix” the mass of the SM Higgs boson to its measured value by the
dynamics of radiative corrections embodied in eq.(1). This requirement, while
being a sensible one in the context of the specific model, can significantly
alter the conclusion of that study. Therefore it remains difficult to answer
questions as simple as finding the lightest not excluded values of the mass of
stop-like top partners 333One possible answer in the context of [25] is
offered in the supplementary material of that analysis[26]. .

Further difficulties can arise and make nearly impossible to probe
experimentally supersymmetric top partners, e.g when special kinematical
configurations become the typical configuration of top partners decay
products. In these cases the search in low momentum transfer signatures can
help. Indeed, these searches have been developed to overcome the difficulty
that arise in the limit $m_{\tilde{t}}-m_{\chi}\to 0$. The shortcomings of the
large momentum transfer searches can be clearly seen in Figure 1, as the
excluded stop mass for large $m_{\tilde{t}}-m_{\chi}$ is much larger than for
small values of this mass difference. In addition, when the stop-LSP mass gap
is small and the stop becomes lighter, its production and decay cannot be
reliably distinguished from other SM processes, e.g. the SM top quark
production. This observation motivates a zoom inset in the figure to display
how these peculiar cases are covered. The most useful strategies to attack
these difficult signatures have turned out to be the studies of angular
observables and fiducial rates of top-like final states [27, 28, 29, 30, 31].
Especially in angular observables there are modest, but persistent
disagreement between the measurements in the top quark sample [32] and
theoretical predictions. These disagreement are also accompanied by other
disagreements of small entity, but persisting from Run1 LHC through Run2, in
the kinematics of the reconstructed top quarks e.g. in Refs. [33, 34]. The
possibility to see effects of BSM related to the top quark and the precision
in measurements afforded by the LHC and the HL-LHC has motivated the great
improvement of predictions for top quark SM observables, e.g. [35] for a
seamless description of fixed NLO and PS calculations of top quark resonant
and non-resonant rates, [36, 37] for specific NNLO and EW corrections to the
BSM sensitive rates and more in general drawing attention on possibly BSM-
sensitive high energy top quarks (see e.g. [38]) and other production modes
which may be of interest for both SM studies and BSM searches (see e.g. [39,
40]).

Figure 1: Searches for top sector supersymmetric partners in the Stop-LSP mass
plane.

The searches mentioned above, though motivated and sometimes optimized on
supersymmetry searches, are rather general. Thus it is important to stress
that the observation of an excess in one of these “supersymmetry searches”
would not at all prove the supersymmetric nature of the discovered state. A
reliable statement on the supersymmetric nature of the newly discovered object
would require several measurements. For some proposal at the LHC the
interested reader can look for instance at [41]. In general it is believed
that a machine cleaner than a hadron collider, e.g. an $e^{+}e^{-}$ collider,
capable of producing the new particle would be needed to truly confer it the
status of “supersymmetric partner” state of some SM state.

At the time of writing there are no statistically significant and convincing
signs of new physics in searches for new physics, the searches for
supersymmetric top partners being no exception. Despite the absence of signals
for top sector supersymmetric partners these are still believed to one our
best chances to find new physics. Looking at the glass as “half full” one
could even argue that in the minimal model of supersymmetry the relatively
large observed Higgs boson mass requires large loop level corrections from
contributions of the kind of eq.(1). These large loop corrections point
towards a stop squarks mass scale at the TeV or larger, thus perfectly
compatible with the present limits and possibly awaiting us for a next
discovery at one of the next updates of the searches as more data is collected
at the LHC.

As the mass scale of top quark supersymmetric partners is not entirely fixed
it often considered that these particles may be too heavy for the LHC to
discover them. Therefore the discovery reach for these particles is often
considered in the evaluation of the physics case of future particle
accelerators. Projections for a 100 TeV $pp$ collider [42, 43] usually cover a
mass range 5-8 times larger than what can be probed at the LHC, while the
expectation for a high energy lepton collider, such as multi-TeV muon
collider[44, 45, 46, 47, 48], is to probe the existence of top partners up to
the kinematic limits at $\sqrt{s}/2$.

###  2.2 Composite and pNGB/Little Higgs

####  2.2.1 Models

New physics associated to the top quark sector has been motivated also from a
series of model building activities aimed at explaining the origin of the
electroweak scale through the Goldstone boson nature of the agent of its
breaking, resulting in theories of the Higgs boson as a pseudo Nambu-Goldstone
boson. From a low energy effective point of view these theories can be put in
the language of a composite Higgs boson, whose lightness compared to its scale
of compositeness is justified by its goldstonian nature. Models built in this
family are reviewed in Refs. [49, 50, 51, 52] and they all share the need to
enlarge the symmetries of the SM by a new global symmetry, that is broken at
some scale above the TeV to a smaller symmetry, with the associated Nambu-
Goldstone bosons, which will host the yet smaller symmetry group of the SM at
even lower energies. The minimal model of this type [53] that is able to pass
bounds from electroweak precision tests including $Zb\bar{b}$ couplings
assumes an $SO(5)$ global symmetry, broken to $SO(4)\simeq SU(2)\times SU(2)$
which contain the weak interactions gauged $SU(2)$.

The enlargement of the symmetry of the SM motivates appearance of matter
representations in multiplets that are necessarily larger than the usual
doublets and singlets of the SM. In particular, in order to obtain Yukawa
interactions the constructions of pNGB and little Higgs model converges in the
existence of “partner” states for the top quark, the bottom quark and in
principle for all the fermions of the SM. The precise phenomenological
manifestation of the “partner” states is highly model dependent, as it depends
on the choice the new global symmetry group that one has in building this type
of models, the representation of this symmetry group that one chooses for the
new matter and the imagined mechanism to originate the SM fermion masses at
the most microscopic level.

One possible limitation to the model building choices may comes from the
requirement of not introducing large deviations in well known couplings, e.g.
the $Zbb$ couplings [54], still a large set of possibilities exists. For this
review we focus on a unifying feature of many models, that is the presence of
“partner” states directly connected to the SM top quark sector via Yukawa and
gauge interactions with relatively universal decay patterns [55, 56, 57],
although other decay modes and more “exotic” partners may exist including
possible couplings to scalar states accompanying the Higgs boson in some
models [58, 59, 60].

####  2.2.2 Phenomenology

At the core of the experimental tests of the idea of fermion top partners lies
the assumption that the main interaction leading to the decay of these top
partners into SM states is the Yukawa of the top quark, in which the Higgs
boson or longitudinal components of the weak gauge bosons appear. For this
reason the large majority of the searches are presented in terms of exclusions
for branching fractions of the top partners states into the following pairs of
SM states

$T\to tZ,th,Wb,\,$

where $T$ is a charge 2/3 top partner and

$B\to bZ,bh,Wt,$

where $B$ is a charge -1/3 partner of the bottom quark, whose existence is
consequence of the SU(2) weak isospin symmetry that must hold in the theory
that supersedes the SM at high energies. In models with a symmetry larger than
SU(2), e.g. [54][53], it is typical to have further partners states that
appear as necessary to furnish full representations of the larger symmetry. A
much studied case is the state of charge 5/3 that leads to a very
characteristic decay

$X_{5/3}\to W^{+}t\,,$

which in turn gives a characteristic same-sign di-lepton signal [61]. For
little Higgs models the appearance of this type of exotic partners requires
the formulation of somewhat more involved models, but it is definitively a
possibility[58, 62].

####  2.2.3 Experimental searches at colliders

Experimental searches for new states are carried out at the LHC exploiting the
color charge of the top partners in processes such as

$gg\to TT\,,$

that are analogous to previous processes for supersymmetric partners and
depend only on the QCD charge of $T$. Unlike for supersymmetric partners, for
which the conservation of R-parity plays a crucial role, the single production
of top partners

$gq\to q^{\prime}Tb\,,$

is possible in the most minimal models and can in principle lead to a deeper
understanding of the BSM physics, as this process involves directly new
physics couplings for the production of the top partners state [63]. For
instance the rate of single production of top partners states can be a
discriminant with respect to so-called “vector-like” quarks, whose couplings
are not dictated by Goldstone property of the Higgs (see Ref. [64] for a more
in-depth discussion).

A great difference in the search for the partners discussed in this section is
that they can in principle give rise to resonant signals, e.g. in the
invariant mass of an hadronic top and one hadronic Higgs boson in the decay
$T\to th$ and other signals discussed for instance in the search of Ref.[65].

Another consequence of the top partner decaying in purely SM final states is
that even the “heavy” SM particles, such as $t$, $Z$, $W$, $h$, are produced
with significant boost in the majority of the events. This motivates the use
of special experimental techniques for the identification of those detector
objects [66] as for instance in the search of Ref.[67].

The search strategies mentioned above are combined by the experimental
collaborations, that present results in a plane with axes spanning the
possible values of two decays, e.g. if figure 2 an example is shown for $T\to
Ht$ and $T\to Wb$. The underlying assumption of this presentation of the
results is that the top partner does not decay in any BSM states, hence the
branching ratio of $T\to Zt$ is determined by the two branching rations
displayed. The right panel of the same figure shows how the different searches
have different sensitivity to each decay mode and can be patched together to
better exclude top partners of a given mass. For more exotic signals from
$X_{5/3}$ searches are carried out as well, e.g. in Ref. [68]. Results of
searches at LHC collected in figure 2 and newer results [69, 67] on the kinds
of top partners described so far put bounds on the top partners mass at around
1.2 TeV.

Figure 2: Searches for top fermionic partners [70, 71] in the plane $BR(T\to
Ht)$ vs $BR(T\to Wb)$ with the constraint $B(bW)+B(th)+B(tZ)=1$. For
reference, some model-dependent choices of the branching ratios introduced in
Ref.[72] are shown.

As mentioned above it is possible to have larger groups and larger
representations in the symmetry breaking pattern. For instance if the large
global symmetry of which breaking the Higgs is a pNGB is chosen to be $SO(6)$
broken to $SO(5)$ and top quark partners states are chosen to furnish a
$6$-dimensional representation there is one extra top partners state compared
to the case of top quark partners in the $5$ of $SO(5)$ considered for the
minimal model of Ref. [54]. If we call this new top partner state $\Psi_{1}$,
the name signals the fact that it is a singlet under the remnant $SO(5)$
symmetry, we can have new signals from its production via QCD interactions and
decay that do not fit into any of the previously considered categories e.g.

$\Psi_{1}\to th,tZ,t\eta,Wb\;,$

where $\eta$ is an extra pNGB that arises due to the larger number of broken
generators in the breaking $SO(6)\to SO(5)\to SO(4)\simeq SU(2)\times SU(2)$ .

In general the extensions of pNGB models can include possible FCNC of top
quarks with new physics states, e.g. Ref. [73] has considered decays of the SM
top quark that violate flavor

$t\to c\eta$

as a consequence of underlying flavor-changing dynamics in the top partners by
a coupling $Tc\eta$ which would also yield a new possible search channel for a
top partner $T\to c\eta$. Other exotic possibilities are covered in the
literature, e.g. $T\to tg,t\gamma,$ $X^{5/3}\to t\phi^{+}$ and more exotics
ones are presented in Refs. [74, 75, 76] and can in principle lead to new
signals for top quark partners.

##  3 EFT at current and future colliders

The previous sections dealt with explicit models of new physics giving rise to
signals from direct production of particles beyond those of the Standard
Model. As these searches have so far yield no evidence of new physics a
growing interest and motivation have risen for the description of new physics
in Effective Field Theories. The effective character of these theories is due
to the fact that they arise by the removal of heavy states from a theory more
microscopic than the SM and they lead to a set of BSM interactions, that is
usually in overlap with the set generated by other microscopic theories.
Therefore it has been done a great work in identifying the most general sets
of interactions under given assumptions [77, 78], so that new physics studies
can be carried in a “model-independent” fashion, e.g. searching for very
characteristic interactions involving four top quarks [79, 80, 81, 82] or
other four-fermion operators involving top quarks, or other kinds of contact
interactions independently of their microscopic origin.

The plus side of the EFT approach is that it is very comprehensive. The
converse of this comprehensiveness is the possible loss of contact with the
microscopic origin of physics beyond the Standard Model which gives rise to
specific patterns and organization principles for the size of each contact
interaction. Thus it is necessary to strike a balance between a fully general
EFT and a “physically efficacious” effective theory. Where this balance lies
is very much dependent on the amount of data that one can use in constraining
the couplings of the effective interactions, as well as the theoretical
prejudice on what effects are worth being considered, e.g. pure top sector
effects [83, 84, 85, 86, 87, 78], or effects involving EW and Higgs physics as
well [88, 89] or exploring flavor changing effects [90, 91, 92, 93, 94, 95].

As the effect of BSM contact interactions from the EFT affects precision
measurements of SM processes, this enhanced attention towards signals of BSM
associated to top quarks has produced activity on the improvement of the
description of several processes that are either backgrounds or serve as SM
reference on top of which search for signs of BSM, e.g. see recent Ref. [96]
for four-top production, recent $ttV$ results discussed in Refs. [97, 98, 99],
$tth$ results in Ref. [100] and references therein. For an up to date snapshot
of the characterization of the top quark electroweak interactions and possible
BSM in deviations from the SM we refer the reader to Refs. [83, 84, 85, 86,
87]. The upshot of the work is that present measurements, also thanks to the
availability of differential measurements and trustable computations in the
same phase-space regions, can put bounds on generic new physics in the top
quark sector in the TeV ballpark.

The possibility to identify indirect signs of new physics in signatures
related to the top quark has become a commonly used benchmark in the
evaluation of performances of future colliders, especially clean $e^{+}e^{-}$
machines, whose best chance to see new physics in the top sector is through
indirect effects. Works such as [101, 102, 103, 104] have studied the outcome
of analyses to be carried out at future colliders and the interplay between
present and future colliders probes of new physics in top quark effective
field theory. The results are summarized in Figure 3, which shows the
significant improvement that will be attained by the HL-LHC, especially on
single-couplings effects. The figure also shows the strong tightening of the
bounds with the addition of data from future $e^{+}e^{-}$ data at the $Zh$
threshold, the $t\bar{t}$ threshold and above, which will make the global EFT
constraints particularly robust by the removal of possible flat directions in
couplings-space and providing new data in channels that can be probed best at
clean $e^{+}e^{-}$ machines.

Figure 3:  Summary from Ref. [102, 101] of the constraints on contact
interactions involving the top quark. The left panel shows the effect of HL-
LHC compared to present constraints. The right panel shows the effect of
future $e^{+}e^{-}$ machines. The taller (and lighted) bars for each case
represent the looser bounds that are obtained when the coupling of interest is
bound while the others are allowed to float, see [102, 101] and references
therein for details.

##  4 Top quark and BSM related to Flavor Dynamics or Dark Matter (or both)

###  4.1 Top quark and BSM related to flavor

The top quark flavor remains a special one in the SM. Indeed the top quark is
so heavy that one can easily single out the third generation of quarks as a
peculiars source of breaking of the flavor symmetry

$G_{F}=U(3)_{q_{L}}\times U(3)_{u_{R}}\times U(3)_{d_{R}}$

that the SM would enjoy if all quark masses were zero. A hierarchy of breaking
dominated by the third generation can be accommodated easily, thanks to the
freedom about the possible symmetry breaking patterns and possible mechanisms
for breaking the flavor symmetry of the SM that one can consider. In addition,
this way of organizing the breaking of flavor symmetry is most compatible with
experimental bounds. In fact, bounds on first and second generation flavor
changing processes are the most tight, whereas there is a relative lack of
constraints on the third generation. If the sole breaking of the symmetry
$G_{F}$ arises from the Yukawa couplings of the SM, or new sources are aligned
with the Yukawa matrices, the breaking is said to comply with “Minimal Flavor
Violation” (MFV) [105, 106, 107]. In this setting the bounds from flavor
observables are most easily accommodated, but it is not the only possibility
to comply with observations. The fact that the top quark Yukawa coupling is a
possible large source of flavor symmetry breaking motivates to consider BSM
related to the top flavor, but this conclusion holds also in other settings.

A classification of possible states that can couple to quark bilinears charged
under the flavor symmetry, e.g. a new scalar coupled as $\phi_{tu}tu$, has
proven useful in the past to assess the possibility of flavorful signs of new
physics. For a recent listing of the possible states one can read tables on
Ref. [108]. From a phenomenological point of view these models give rise to
transitions in four-quark scatterings that do not conserve the flavor charge.
For instance the scattering

$uu\to tt$

can arise via a $t$-channel exchange of a flavored boson. This can alter the
kinematic of top quark production as well as the net charge of the top quark
sample at hadron colliders. Indeed new flavorful boson of this kind were
advocated in response to TeVatron experiments claiming disagreements between
the SM predictions and measured top quark properties, such as the forward-
backward asymmetry in the production of top quarks [109, 110, 111]. In
addition these new flavored states coupled to the top quark can give rise to
transitions

$f\bar{f}\to t\phi_{tj}u_{j}\,,$

that can be observed quite easily at $e^{+}e^{-}$ colliders in multi-jet final
states, the detailed final state depending on the model-dependent decay of the
flavored state $\phi$.

The possibility that a flavored state connected to the top quark might be
among the lightest new states from the new physics sector has appeared also in
models of gauged flavor symmetries. In these models the flavor symmetry
$G_{F}$ is gauged, as to not have to deal with unobserved massless Goldstone
bosons. For instance Refs. [112, 113] have proposed a new set of states that
would have the notable property to make the $G_{F}$ gauging free from
triangular anomalies by the addition of vector-like new quarks. In this kind
of models the new quarks are charged under the SM flavor symmetry and can be
arranged as to have top-flavor new states to be the lightest ones. Indeed in
these models the masses of the the SM quarks would be explained by a see-saw-
like mechanism in which the lightest SM fermions are mixed with a very heavy
new state, whereas the heaviest SM states are mixed with the lightest of the
new physics states. In this case the SM top quark would be the state coupled
to the lightest of the new physics states, named $t^{\prime}$, possibly
accompanied by a partner state for the bottom quark, named $b^{\prime}$.
Remarkably this type of model gives phenomenological signatures very similar
to those of top partners states of composite and little Higgs, e.g. the
partner states can be produced by strong interactions and decay as

$b^{\prime}\to bh,bZ,tW$

and

$t^{\prime}\to th,tZ,bW\,.$

These ideas also lend themselves to be paired with supersymmetry. Although
supersymmetry is not necessary for the idea of gauged flavor symmetries in
general, these models can provide a setup to originate R-parity breaking with
an underlying structure for the flavor structure of the RPV couplings [114,
115], that for instance would motivate

$\tilde{t}\to bs$

as the main channel to search RPV stops [116].

A solution with a hierarchy of flavored new physics scales inverted with
respect to that of the SM quarks has been proposed also for composite Higgs
models [117, 118, 119, 120], which would otherwise suffer from severe bounds
from high-$p_{T}$ and flavor observables (see e.g. [121, 122, 123]), even in
presence of some degree of model building [124, 125, 126] aimed at keeping all
the new physics at a common low-scale and still survive flavor tests thanks to
a friendly, possibly MFV-like structure, of the flavor origin in the
microscopic completion of the composite Higgs model. As it can be appreciated
in Fig. 4 the top quark sector emerges still as a less constrained one and
further motivates to consider BSM physics related to the top quark, and
possibly exclusively to the top quark or to the third generation of SM
fermions.

Figure 4: Lower bound on the scale of new physics related to the SM fermion
mass generation in a composite Higgs scenario [117] under different
assumptions on the compositeness of SM fermions.

Observables of interests include indirect probes such as electric dipoles
moments (see e.g. [127]), meson oscillations and decays, and in principle rare
$Z$ and Higgs bosons flavor-violating decays which usually receive important
contributions from the top quark sector [117]. In addition, it is possible to
have phenomena more directly related to the top quark such as

$t\to cV,$

where $V=\gamma,Z,g$[128, 129] and deviations from $V_{tb}=1$ in the CKM
matrix [64, 130, 131, 132].

###  4.2 Flavored dark matter models

Given the strength of the bounds from direct searches of dark matter
scattering on heavy nuclei it has become interesting to consider dark matter
models in which the flavor of SM quarks and leptons plays a role, as the
strongest bounds hinge on effective couplings of the dark matter to first and,
to a slightly lesser extent, to second generation quarks and gluons.

Rather interestingly the flavor puzzle of the SM comes equipped with a
symmetry, which, though not exact, can be used to stabilize the dark matter if
it is broken according to Minimal Flavor Violation [133, 134] and even with
more general patterns of flavor symmetry and its breaking [135]. As a dark
matter coupling sensitive to flavor could mediate flavor changing transitions
the option of the MFV structure, or slight departures from it, has been so far
been a main route in model building aimed at removing possible tensions with
flavor observables.

Among the possible flavor structures that the Dark Matter and the SM can
fields can be cast in, for our work here we focus on the possibility that the
top quark flavor has a special role. Explicit models have appeared in the
context of possible explanations of the CDF $A_{FB}$ anomaly [109, 110, 111],
e.g. see the model built in Ref. [136], but the idea stands out on itself even
without anomalies in top quark physics. Indeed if one considers that the
complexity of the SM may be replicated in the sector of dark matter it is
natural to consider multiple species of dark matter, that are “flavors” of
dark matter [137, 138, 139]. These flavors can be separated from our own SM
flavors or can be related to our species of fermions. In case some relation
exists between flavors of the SM and of the dark sector the possibility that
the top quark flavored dark matter is the lightest state is at least as
probable as any other flavor assumption. For example, when Minimal Flavor
Violation is advocated one can explicitly write a mass term for the dark-
flavor fermion multiplet $\chi$ which in general has the form

$\bar{\chi}\left(m_{0}+\varUpsilon(YY)\right)\chi\,,$

where $\varUpsilon$ is a function of combinations of the Yukawa matrices of
the SM that form singlets under the flavor group that is dominated by the
piece proportional to $Y_{u}^{\dagger}Y_{u}$, hence the top quark flavor tends
to be special just from the principle of MFV itself. In a concrete case we can
have interactions of SM fermions $u_{R}^{(i)}$ and mass terms for the dark
matter flavor multiplet $\chi$

$\phi\bar{\chi}\left(g_{0}+g_{1}Y_{u}^{\dagger}Y_{u}\right)u_{R}^{(i)}+h.c.+\bar{\chi}\left(m_{0}+m_{1}Y_{u}^{\dagger}Y_{u}+...\right)\chi\,,$
(6)

where $\phi$ is a suitable representation of $G_{SM}\otimes G_{F}$. In Ref.
[136] for instance
$\phi\sim(\mathbf{3},\mathbf{1},2/3)_{SM}\otimes(\mathbf{1},\mathbf{1},\mathbf{1})_{F}$,
$\chi\sim(\mathbf{1},\mathbf{1},0)_{SM}\otimes(\mathbf{1},\mathbf{3},\mathbf{1})_{F}$
and the Yukawa matrices, as in general in MFV, transform as spurions
$Y_{u}\sim(\mathbf{3},\mathbf{\bar{3}},1)_{F}$ and
$Y_{d}\sim(\mathbf{3},1,\mathbf{\bar{3}})_{F}$. We see that it is possible to
pick $m_{1}$ as to partly cancel the flavor universal $m_{0}$ term, making
$\chi_{t}$ the lightest particle of the $\chi$ multiplet while retaining full
freedom to pick the combinations of $g_{0}$ and $g_{1}$ that corresponds to
the couplings of the mass eigenstates $\chi_{i}$.

In absence of a field $\phi$ one can imagine contact operators to couple the
Dark Matter and the SM flavors $i$ and $j$, e.g. operators of the type

$\left(\bar{\chi}\Gamma_{S}\chi\right)\left(\bar{\psi}^{(i)}\Gamma_{S}\psi^{(j)}\right)$
(7)

for some Lorentz structure $\Gamma_{S}$ have been considered as low energy
remnants of flavored gauge bosons [137] or other heavy scalar and fermion
states charged under a MFV-broken flavor symmetry or in a horizontal symmetry
model [138]. Operators involving the SM Higgs boson, e.g.

$\left(\bar{Q}\chi\right)\left(\chi^{*}Hu\right)$

have also been considered in [133] for a scalar
$\chi\sim(\mathbf{1},\mathbf{1},0)_{SM}\otimes(\mathbf{3},\mathbf{1},\mathbf{1})_{F}$.
A variation of the model of Ref. [137] could lead to top quark flavor being
singled out, the other referred works already consider the third generation,
hence the top quark and/or the bottom quark, as special due to either the MFV
structure or as a result of the horizontal symmetry.

The phenomenology of top flavored dark matter is very rich as it comprises
both possible signals in dark matter searches and in precision flavor
observables as well as in high energy collider searches. Flavor observables
put in general stringent bounds on flavored dark matter models, the case of
top-flavored dark matter being significantly less constrained due to majority
of data belonging to $u,d,s,c,b$ quark systems. Dark matter direct detection
is also in general suppressed because nucleons involved in dark matter
scattering do not contain top flavor, hence the interactions are usually
originated at loop level or via breaking of the flavor alignments, i.e. the
dark matter interacts almost exclusively with top quark flavor, but it may
have a small, though not completely negligible coupling to light flavors. The
existence of such coupling depends on the model. A specific analysis for a
case in which only top quark flavor interacts with the DM in the model eq.(6)
is presented in Ref. [140] for both dark matter direct detection and collider
prospects in a MFV scenario. The annihilation rate for the thermal freeze-out
is set by the scattering

$\chi\chi\to tt$ (8)

mediated by a mediator $\phi$ (other scatterings are discussed in detail for
instance in [141]). In this specific case the direct detection scattering on
nucleons

$\chi\mathcal{N}\to\chi\mathcal{N}$

is mediated by a loop induced couplings of $Z,\gamma$ to $\chi$ from a bubble
loop of $t$ and $\phi$ from eq.(6). Despite the smallness of these couplings
the reach of current and future large exposure experiments, e.g. see [142],
could probe such low level of scattering rates for exposure around 1 ton year,
that means the model can be tested with presently available data [10].

A more recent analysis [143] considered flavor, direct dark matter detection
and collider searches for a model featuring a top-flavored dark matter $\chi$
and a new state $\phi$. In this work a “Dark Minimal Flavor Violation” flavor
structure that extends MFV, but can recover it as a limit, is considered and
allows for a more generic structure in flavor space for the vertex

$\lambda_{ij}\bar{u}_{R}^{(i)}\phi\chi+h.c.\,.$

In this context it is possible to delay the observation of $\chi$ in direct
detection experiments, as new contributions to the direct detection rate
appear compared to the MFV case and it is possible to arrange for
cancellations among scattering amplitudes. It remains an open questions if it
is going to be possible to claim an observation in spite of the so-called
“neutrino fog” that future Xenon experiments [142] face when probing rates so
small that neutrinos from the Sun, supernovae and other natural sources are
expected to contribute an event rate comparable or larger than that of the
dark matter.

In principle it is possible to have $m_{\chi}<m_{t}$ so that the thermal
freeze-out is controlled by other processes than the simple tree-level
exchange of eq.(8). Reference [143] experimented with this possibility in Dark
Minimal Flavor Violations, but it appears in tension with the direct detection
experiments. This conclusion concurs with what can be extrapolated from the
earlier MFV analysis of [136].

The search for models with mediators, that are colored in all models
considered so far, can be carried out very effectively at hadron colliders
searching for signals

$pp\to\phi\phi\to t\chi t\chi\,,$

that very much resemble the search for supersymmetric top partners. Depending
on the model there can be more general combinations of flavors of quarks

$pp\to\phi\phi\to q_{j}\chi q_{i}\chi\,.$

Therefore it is in general useful to consider the whole list of squark
searches to put bounds on this type of models. References [143, 144] reports
bounds in the TeV ballpark which inherit the strengths and weaknesses
discussed for the search of supersymmetric quark partners.

Other possible signals at hadron collider are the

$pp\to t\chi\chi$

scattering, which can arise from interactions such as eq.(7), studied in
[138], or associated production $\phi\chi$, followed by $\phi\to t\chi$
studied for instance in [144].

It is also possible to consider models that go beyond what we have considered
here starting from the notable feature that MFV and some extensions may render
the DM stable. In a model of such “top-philic” dark matter model on can have
[145] scalars that couple to $t\chi$ as well as to light quark bilinears, e.g.
from RPV supersymmetry, so that they mediate scatterings of the type

$q_{i}\bar{q}_{j}\to S_{ij}\to t\chi\,.$

Other potentially interesting signals possible flavored gauge bosons with
couplings $\rho_{ij}q_{i}q_{j}$ can appear, replacing $S_{ij}$ with
$\rho_{ij}$ in the above process. Further signals in this type of models
arise, e.g.

$q_{i}g\to t\rho_{ti}$

possibly followed by $\rho\to\chi t$, and similarly for $S$. A model with a
flavored gauge boson has been studied in [146] with the goal of pinning down
the flavor of light quark that interacts with the top quark and the dark
matter leveraging charm-tagging and lepton charge asymmetry at the LHC.

Though many general issues follow the same path for scalar and fermionic dark
matter it is worth mentioning that references [147, 148] contain a full study
of the case in which the partner and the dark matter are a fermion and a
scalar, respectively, at the converse of most of what we discussed above.
Further studies of top and dark matter related matters can be found in the
context of simplified models building [141, 149, 150].

##  5 Conclusions

The connection between new physics and the top quark sector is well
established and has lead to a large amount of model building and
phenomenological studies. Here we have presented supersymmetric top partners,
motivated by supersymmetry as the symmetry that stabilizes the weak scale, and
top partners states motivated by the possible compositeness and pseudo-Nambu-
Goldstone boson nature of the Higgs boson. The phenomenological relevance of
these incarnations of “BSM in the top quark sector” is tightly tied to the
motivations of the models to which the top partners states belong. As the
models in question are themselves in a “critical” phase at the moment, so is
the situation for this type of new physics in the top quark sector. We say
this in the sense that on one hand we have reached a point at which the
expectation was to have already discovered signs of new physics, especially in
the top quark sector in the mass range explored by current experiments, hence
we should start to dismiss these ideas, while on the other hand we are still
largely convinced of the validity of the arguments that lead to the
formulation of these models. Furthermore no serious alternatives have appeared
in the model building landscape and we still have plenty of evidence for the
existence of physics beyond the Standard Model. Thus one can be lead to
reconsider if the entire motivational construction for these models was
somewhat wrong or at least biased towards a “close-by” and experimentally
friendly solution.

The way out of this crisis, in absence of experimental results changing the
situation, is for everyone to decide. A possibility is to conclude that we
need to update our beliefs about “where” [151] new physics can appear in the
top quark sector and more in general in going beyond the SM. In this sense top
partner searches are a gauge of our progress on testing well established ideas
on new physics.

It should be remarked that the top quark sector remains central also in the
formulation of new physics models that try alternatives to the more well
established ideas, see e.g. Refs. [152, 153] on possible ways the top quark
can lead the way to construct new physics models of a somewhat different kind
that the two mainstream ideas discussed here.

Given the absence of clear signs and directions in model building into which
entrust our hopes for new physics we have discussed the power of general
effective field theory analyses that can be used to search for new physics in
precise SM measurements. These tools have become the weapon of choice in a
post-LHC epoch for the so-called model-independent search of new physics. We
have presented the power of current LHC and future HL-LHC analyses to see
deviations from the SM due to top quark interactions. Overall the LHC has a
chance to see deviation in some more friendly observables for a new physics
scale in the TeV range. In order to secure this result and avoid possible
blind-spots a new particle accelerator is needed, a most popular option being
an $e^{+}e^{-}$ capable of operating at or above the $t\bar{t}$ threshold with
the luminosity to produce around $10^{6}$ top quark pairs.

Other great mysteries beyond the origin of the electroweak scale remain
unsolved in the Standard Model. We have looked at possible solutions of the
flavor puzzle in which the top quark flavor plays a special role. The
phenomenology of models with lowest lying new physics states charged under top
flavor has some similarity with that of top quark partners at colliders, but
there is also the possibility to generate observable flavor violations as
further distinctive experimental signatures.

We have examined the possibility that the top quark may be a key to solve the
mystery of dark matter of the Universe. We have presented scenarios in which
the dark matter interacts predominantly or exclusively with the top quark
flavor, possibly ascribing the stability of the dark matter to the same flavor
structure that makes the top quark flavor special among the SM flavors. Such
possibility appears very well motivated as a way to reduce otherwise
intolerably large couplings of dark matter with lighter generations and
explain the stability of dark matter. The flavor dependence of the couplings
has motivated efforts to build models for the realization of this idea in a
coherent, though maybe still effective, theory of favor of which we have
presented a few instances. We remarked how in these scenarios the dark matter
phenomenology is quite different from other types of thermal dark matter and
we have summarized dedicated analyses that have been carried out to identify
the relevant bounds and constraints. The upshot is that idea can be broadly
tested with current and future direct detection dark matter experiments. At
the same time the new states associated with the dark matter may be observed
on-shell at colliders, which can in principle also probe contact interactions
that originate from off-shell states associated with the dark matter. Low
energy flavor observables can also help to restrict the range of possible
models of flavored dark matter leading to significant constraints both on MFV
and non-MFV scenarios when a thermal relic abundance and a significant
suppression of spin-dependent and spin-independent direct detection rates are
required.

### Acknowledgments

It is a pleasure to thank Kaustubh Agashe for discussions on top quark
partners in composite Higgs models.

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