capbtabboxtable[][] ††institutetext: LPTh, Department of Physics, University
of Jijel,
B. P. 98 Ouled Aissa, 18000 Jijel, Algeria

# NLO QCD corrections to single vector-like top partner production in
association with top quark at the LHC

Mohamed Sadek Zidi<EMAIL_ADDRESS>
###### Abstract

The single vector-like top partner hadroproduction in association with top
quark in the five and the six-flavour schemes at the LHC is investigated. The
inclusive cross section and the differential distributions are evaluated for
many benchmark scenarios at leading and next-to-leading orders, both for
fixed-order and fixed-order matched to parton shower. The associated Higgs
production with a top quark pair, where the final state arises from the decay
of the vector-like top partner into a Higgs boson and a top quark, is
examined.

###### Keywords:

Vector-like quarks, NLO QCD corrections, parton shower, 6F-scheme

##  1 Introduction

The Standard Model (SM) is a successful theory of particle physics. All the
particles predicted by this theory were observed at accelerators, like the
famous Higgs boson which has been predicted long time ago by theory higgs-
mechanism1 ; higgs-mechanism2 , and has been confirmed by ATLAS and CMS
collaborations in 2012 higgs-atlas ; higgs-cms . Despite its successes, there
are still some open questions (Hierarchy problem, neutrinos mass, $\cdots$
etc) which are not yet answered by SM. To resolve these problems, many
extensions of the SM were proposed. Most of these models predict new particles
which are still not yet observed at experiments, as the vector-like quarks for
example of which the single production at the Large Hadron Collider (LHC) is
the main subject of this paper.

Vector-like quarks (VLQs) are spin-half fermions, where their left and right-
handed components transform the same way under the electroweak symmetry group
aguil1 . They are assumed to have the same colour quantum number as the
ordinary quarks, which means that they behave exactly in the same way under
the strong interaction. The vector-like nature of these particles allows to
add their gauge invariant mass term without the need for the Higgs mechanism,
i.e. their masses are not generated by the Yukawa couplings to the Higgs
scalar doublet. The presence of vector-like-quarks is predicted by many models
beyond the Standard Model (BSM) like extra-dimensions, little Higgs, composite
Higgs and grand unification models extra-dim-vlqs ; little-higgs-vlqs ;
composite-higgs-vlqs ; grand-unif-vlqs … etc. They are, mainly, introduced to
treat the problem of naturalness of the electroweak scale and to stabilise the
Higgs mass. VLQs can be treated in model independent way at leading-order
model-indep-vlqs-1 ; model-indep-vlqs-2 ; model-indep-vlqs-3 ; model-indep-
vlqs-4 ; model-indep-vlqs-5 , where they are assumed to interact with SM
quarks via the Yukawa mixing. This approach is based on an effective
Lagrangian with a minimal set of free parameters, which enables us to study
their phenomenology independently of the model model-indep-vlqs-4 . Extension
to NLO QCD corrections of the model independent parameterisation was
introduced in ref. model-indep-vlqs-6 .

These new heavy states can be produced in different manners at the LHC. They
can be produced in pairs, which are dominated by the QCD contribution and they
are model independent. They can be produced singly in association with top
quarks, jets, heavy gauge bosons or the Higgs. In contrast with the fourth
generation of quarks, VLQs are consistent with the existing Higgs data and the
recent LHC measurements. Due to their large mass and the fact that they can be
produced in pairs via the strong interaction, which means relatively large
cross sections, the research for such heavy states at the LHC is relatively
accessible. One of the distinguishing characteristics of VLQs is the existence
of flavour-changing neutral current, which open new particle production
mechanisms and leads to widely diverse final states, which makes the
investigation of these particles at the LHC very promising. The search for
VLQs has been analysed in many publications by ATLAS and CMS collaborations
for LHC runI and runII THt1 ; THt2 ; TZt1 ; TZt2 ; TWb1 ; TWb2 . The analysis
of recent data put strong bounds on the masses of these particles, the lower
limit on the mass of the top quark partner, for example, is set between
$715$-$950\,\text{GeV}$ bounds-vlqs .

In this paper we are interested in the production of a single top quark
partner in association with a top quark in proton-proton collisions at leading
and next-to-leading orders in the five and the six-flavour schemes (5FS and
6FS), and the decay of the top partner into a Higgs boson and a top quark,
which leads to the interesting final state $t\bar{t}H$. We mention that the
analysis of such final state, in SM and beyond, has received more and more
attention by both theorists ref43 ; ref44  and experimentalists ref31 ; ref32
; ref33 ; ref35 ; ref37  in the last few years, especially after the discovery
of the Higgs boson. In section 2, we provide a brief overview of the model
independent parameterisation and its extension to include QCD one loop
corrections. In section 3, we calculate the total cross section and the
differential distributions of the singly produced top partner in association
with top quark in the 5FS. In section 4, we study the same signal but in the
6FS at the LHC and at the future hadron colliders
($\sqrt{s}=100\,\text{TeV}$). We discuss the implications of including top
quark initiated processes at LO and NLO accuracies. In section 5, we analyse
the decay product of the new heavy state to a Higgs boson and a top quark.

##  2 VLQs model independent parameterisation overview

It is very important to have a model independent parameterisation which allows
to study the main features and characteristics of VLQs for large variety of
models. Actually, this was provided in model-indep-vlqs-4 , where the vector-
like quarks are allowed to mix and decay to SM quarks via heavy weak gauge
bosons and the Higgs. The main point of this approach, is that the interaction
of such new heavy states with ordinary quarks is involved through the Yukawa
couplings to the Higgs field doublet. In this strategy, the physics of VLQs is
traced by an ensemble of few independent parameters handling the single
production of these particles and their decay into heavy bosons and ordinary
quarks. This parameterisation was extended to include QCD one loop corrections
in model-indep-vlqs-6 , on which our present work is based.

In this work, we are interested in the single production of the top partner
$T$ in association with the top quark at the LHC, and the investigation of the
associated Higgs with top quark pair final state which arises from the decay
of the $T$ into a Higgs boson and a top quark (i.e. $pp\rightarrow
T\bar{t}+\bar{T}t\rightarrow t\bar{t}H$). We consider many benchmark scenarios
which might lead to such final states, where the $T$ is supposed to interact
with the first or the third generations quarks. The general form of the
Lagrangian, that we have to add to SM to describe those scenarios, can be
expressed as follows:

$\displaystyle\mbox{${\cal L}$}_{\scriptscriptstyle T}$
$\displaystyle=-H[\bar{T}(\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{L}}P_{L}+\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}P_{R})q_{u}+\text{h.c.}]+\frac{g}{2c_{\scriptscriptstyle
W}}[\bar{T}\not{\\!\\!Z}(\kappa^{\scriptscriptstyle{(Z)}}_{\scriptscriptstyle{L}}P_{L}+\kappa^{\scriptscriptstyle{(Z)}}_{\scriptscriptstyle{R}}P_{R})q_{u}+\text{h.c.}]$
$\displaystyle+\frac{g}{\sqrt{2}}[\bar{T}\not{\\!\\!W}(\kappa^{\scriptscriptstyle{(W)}}_{\scriptscriptstyle{L}}P_{L}+\kappa^{\scriptscriptstyle{(W)}}_{\scriptscriptstyle{R}}P_{R})q_{d}+\text{h.c.}]+i\bar{T}\not{\\!\\!D}T-m_{\scriptscriptstyle
T}\bar{T}T.$ (1)

where $q_{u}$ and $q_{d}$ are, respectively, up and down quark fields for a
given generation. $D_{\mu}$ is the covariant derivative,
$m_{\scriptscriptstyle T}$ is the mass of the top partner, $g$ is the weak
coupling constant, $c_{\scriptscriptstyle W}$ is the cosine of the Weinberg
mixing angle and $P_{L/R}$ are the left and right-handed chirality projectors.
The free parameters
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{L/R}}$,
$\kappa^{\scriptscriptstyle{(Z)}}_{\scriptscriptstyle{L/R}}$ and
$\kappa^{\scriptscriptstyle{(W)}}_{\scriptscriptstyle{L/R}}$ characterise the
strength of the mixing between the $T$ and the quarks via the Higgs, the $Z$
and the $W$ bosons, respectively. In term of the branching fractions to SM
quarks and heavy bosons, they are expressed as follows (the original
parameterisation is given in model-indep-vlqs-4 ):

$\displaystyle\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{L}}$
$\displaystyle=\frac{\kappa_{\scriptscriptstyle T}m_{\scriptscriptstyle
T}}{v}\sqrt{\frac{\zeta_{i}\xi_{H}^{T}}{\Gamma_{H}^{T}}},$
$\displaystyle\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}$
$\displaystyle=\frac{\kappa_{\scriptscriptstyle
T}m_{t}}{v}\sqrt{\frac{\zeta_{i}\xi_{H}^{T}}{\Gamma_{H}^{T}}},$
$\displaystyle\kappa^{\scriptscriptstyle{(W)}}_{\scriptscriptstyle{L}}$
$\displaystyle=\kappa_{\scriptscriptstyle
T}\sqrt{\frac{\zeta_{i}\xi_{W}^{T}}{\Gamma_{W}^{T}}},$
$\displaystyle\kappa^{\scriptscriptstyle{(Z)}}_{\scriptscriptstyle{L}}$
$\displaystyle=\kappa_{\scriptscriptstyle
T}\sqrt{\frac{\zeta_{i}\xi_{Z}^{T}}{\Gamma_{Z}^{T}}}.$
$\displaystyle\xi_{V}^{T}$
$\displaystyle=\Gamma_{V}^{T}/(\sum_{\scriptscriptstyle
V^{\prime}=Z,W,H}\Gamma_{V^{\prime}}^{T}),$ $\displaystyle\Gamma_{H}^{T}$
$\displaystyle\approx 1/2,$ $\displaystyle\Gamma_{W}^{T}$
$\displaystyle\approx 1,$ $\displaystyle\Gamma_{Z}^{T}$ $\displaystyle\approx
1/2.$ (2)

where $v$ and $m_{t}$ are, respectively, the Higgs vacuum expectation value
and the top quark mass. The parameter $\zeta_{i}$ is the branching ratio into
the $i^{\text{th}}$ SM quark generation. The parameter $\xi_{V}^{T}$ is the
branching ratio into a $V$-boson (for $V=Z,W,H$). The quantities
$\Gamma_{V}^{T}$ are the kinematic functions of the branching ratio of $T$
into a $V$-boson assuming that the quarks are massless. We mention that
$\kappa^{\scriptscriptstyle{(W)}}_{\scriptscriptstyle{R}}$ and
$\kappa^{\scriptscriptstyle{(Z)}}_{\scriptscriptstyle{R}}$ are assumed to be
negligible in this approach, see model-indep-vlqs-4  for more detail. The
parameter $\kappa_{\scriptscriptstyle T}$ encodes information relative to the
coupling strength to different generations. For mixing with quarks of the
first and the third generations, we fix its value to $0.07$ and $0.1$,
respectively, as suggested in ref. model-indep-vlqs-6 . For all the Standard
Model quarks except the top quark, one can choose only one specific chirality
since the mixing angle of the other one is suppressed by factor of
$m_{q}/m_{T}$. However for the channel $T$-$H$-$t$, this is not a good
approximation due to the top large mass, especially for top partner masses
below $1\,\text{TeV}$. So one has to fix the parameters describing the mixing
as in eq. (2) assuming that the coupling of the sub-leading vertex is
proportional to the same flavour mixing angle.

Finally, the Feynman rules, the ultraviolet counterterms, the reduction
rational terms and the other required ingredients to compute one-loop QCD
radiative corrections are generated automatically by the packages FeynRules,
NLOCT and FeynArts ref231 ; ref232 ; ref4 ; ref5  from the Lagrangian of the
SM and the Lagrangian (1). In this work, we generate a vector-like quark NLO
model, where we include both right and left-handed couplings of $T$-$H$-$t$,
as shown in eq. (2), to take into account the effect of the top quark mass on
the mixing of the top partner with the top quark via the Higgs boson.
Furthermore, we employ the public UFO VLQ NLO model to verify our results (UFO
for Universal FeynRules Output)111The UFO and FeynRules models are available
in http://feynrules.irmp.ucl.ac.be/wiki/NLOModels.. For more detail about the
renormalization of the model, see ref. model-indep-vlqs-6 222In this work, we
suppose that top quark partner interacts with SM quarks via the exchange of
one, two or three heavy bosons, where the right-handed coupling of $T$-$H$-$t$
is included. The reactions $pp\rightarrow T\bar{t}+\bar{T}t$ and
$pp\rightarrow T\bar{t}+\bar{T}t\rightarrow t\bar{t}H$ being examined for the
first time in this model at NLO order in 5FS and 6FS..

##  3 Top partner production in association with top quark at the LHC

Figure 1: Leading-order Feynman diagrams for the single $T$ production in
association with the top quark at proton-proton collisions:
$q\bar{q}\rightarrow T\bar{t}+\bar{T}t$, where $q$ ($\bar{q}$) runs over all
the massless quarks (anti-quarks).

In this section, we compute the inclusive cross section and the differential
distributions of the weak production of a single top partner in association
with a top quark, at leading and next-to-leading orders in proton-proton
collisions. The full NLO QCD cross section is calculated by interfacing the
one-loop QCD corrections and the Born amplitudes after adding all the required
UV counterterms and combining them with the squared amplitude of the real
emission in the FKS subtraction framework fks1 . This can be done
automatically by MadGraph once the UFO VLQ NLO model is implemented, thanks to
the automatic computation tools: FeynRules, NLOCT, FeynArts, MadLoop, MadFKS
and MadGraph ref231 ; ref232 ; ref4 ; ref5 ; madloop ; fks2 ; madgraph . Let’s
call $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$
the benchmark scenario on which the vector like quark $T$ is allowed to mix
with the third quark generations via the gauge weak bosons $Z,W$ and the
Higgs. The lowest order processes, at $\mathcal{O}(\alpha^{2})$, that lead to
a final state with a single top quark partner associated with a top quark are,

$\displaystyle q\bar{q}$ $\displaystyle\rightarrow T\bar{t},$ $\displaystyle
q\bar{q}$ $\displaystyle\rightarrow\bar{T}t.$ (3)

In this scenario, the leading order weak production mechanism occurs by the
exchange of $Z$ and $W$ bosons in the $s$-channel and the $t$-channel,
respectively. We mention that the Higgs exchange in the $s$-channel is absent,
since all the initial state quark flavours are massless. We point out that we
work with the 5FS, where all the SM quarks are taken to be massless except the
top. On top of that, we neglect all mixing between different generations of SM
quarks since they give negligible contribution333We have checked this
numerically at the LO order (hadronic cross section) by employing a UFO VLQ
model, where all the CKM mixing matrix elements of the SM quarks have been
implemented.. The tree level Feynman diagrams describing those processes are
shown in figure 1. The one-loop virtual amplitude, of order
$\mathcal{O}(\alpha^{2}\alpha_{s})$, is obtained by interfacing the weak Born
Feynman diagrams, shown in figure 1, with all their one-loop QCD virtual
correction diagrams with at least one virtual gluon running on the loop, i.e.
we keep all the triangles and the boxes with at least two strong vertices. In
the first and the second lines of figure 2, we depict some of the one-loop
Feynman diagrams. To suppress the ultraviolet divergences of this amplitude,
one has to add all the necessary UV counterterms. This can be done
automatically once the vector-like quarks renormalised model is implemented in
MadGraph. We notice that the interference of the Born Feynman diagrams with
some of the one-loop Feynman diagrams gives vanishing contribution. For
example, all the boxes interfaced with the Born diagram of the same channel
are colour suppressed444The boxes must be included especially for $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$ to insure
IR divergence cancellation., see the first two diagrams on the second line of
figure 2. This is due to the fact that the interference of the tree level
Feynman diagram (without any strong vertex) and the box (with one internal
virtual gluon) is proportional to two closed fermion loops with one strong
vertex for each, which means that each one is proportional to $Tr(T^{a})=0$
($T^{a}$ are the generators of $SU(3)$ in the fundamental representation).
Regarding the triangle loop diagrams without any virtual gluon running on the
loop (see the last two diagrams on the second line of figure 2), their
interference with the $s$-channel Born Feynman diagram is colour suppressed
for the same reasons as exposed above. Nevertheless, the interference with the
$t$-channel Born Feynman diagram is not vanishing. In any case, such type of
diagrams are not included in this calculation, since the VLQs NLO model that
we employ does not include the necessary counterterms to perform mixed NLO QCD
and electroweak corrections.

Figure 2: Some of one-loop and real emission Feynman diagrams associated to
both Born diagrams.

The interference of the Born and the virtual diagrams contains infrared (soft
and/or collinear) divergent terms. For consistency of perturbation theory, one
has to include diagrams with real emission of an extra parton, of order
$\mathcal{O}(\alpha^{2}\alpha_{s})$, at the partonic cross section level. The
real emission processes are listed here,

$\displaystyle q\bar{q}$ $\displaystyle\rightarrow(T\bar{t}+\bar{T}t)g,$
$\displaystyle qg$ $\displaystyle\rightarrow(T\bar{t}+\bar{T}t)q,$
$\displaystyle\bar{q}g$ $\displaystyle\rightarrow(T\bar{t}+\bar{T}t)\bar{q}.$
(4)

Some of their corresponding Feynman diagrams are depicted in the last three
lines of figure 2. The first process in eq. (4) is obtained from the Born
diagrams by emitting an extra gluon by the initial or the final state quarks.
The initial state emission can have soft or collinear singularities since the
emitted and the emitter partons are both massless. However, the final state
emission can only have soft singularities due to the non-vanishing mass of the
heavy quarks ($T$ and $t$). After integrating the squared amplitude of these
diagrams over the gluon momenta, we can show that it has exactly the same soft
divergent terms, with opposite sign, as the one-loop virtual contribution
interfaced with the Born. So, once these two contributions are combined
together, the soft divergences are suppressed, thanks to Kinoshita-Lee-
Nauenberg theorem (KLN) kln1 ; kln2 . Regarding the second and the third
processes of eq. (4), some of their Feynman diagrams are depicted in the last
two lines of figure 2. Only the second diagrams of the fourth and the fifth
lines which are divergent (initial state collinear singularity). The remaining
collinear singularities, after combining the virtual and real squared
amplitudes, are factorized in the NLO parton distribution functions, thanks to
the factorization theorem fact1 ; fact2 . Symbolically, the full NLO QCD cross
section can be written as,

$\displaystyle\sigma^{\scriptscriptstyle\text{NLO}}=\int
d\Phi_{2}d\sigma^{\scriptscriptstyle\text{Born}}$ $\displaystyle+\int
d\Phi_{2}(d\sigma^{\scriptscriptstyle\text{virt}}+d\sigma^{\scriptscriptstyle\text{virt}}_{\scriptscriptstyle\text{sub}})+\int
d\Phi_{2+1}(d\sigma^{\scriptscriptstyle\text{real}}+d\sigma^{\scriptscriptstyle\text{real}}_{\scriptscriptstyle\text{sub}})$
$\displaystyle+\int dx\int
d\Phi_{2}(d\sigma^{\scriptscriptstyle\text{fact}}+d\sigma^{\scriptscriptstyle\text{fact}}_{\scriptscriptstyle\text{sub}}).$
(5)

where $d\Phi_{2}$ and $d\Phi_{2+1}$ are the two and three particle phase space
of the final state; $\sigma^{\scriptscriptstyle\text{virt}}$ is the cross
section of the virtual contribution (interference of the Born with the one
loop diagrams), it is of order $\mathcal{O}(\alpha^{2}\alpha_{s})$ and it is
soft and collinear divergent; $\sigma^{\scriptscriptstyle\text{real}}$ is the
cross section of the real emission, it is of the same order of
$\sigma^{\scriptscriptstyle\text{virt}}$ and has the same soft divergent terms
with opposite sign (squared amplitude of all the real emission diagrams) and
it is also collinear divergent; $\sigma^{\scriptscriptstyle\text{fact}}$ is
the collinear counterterms, it arises from the redefinition of the parton
distribution functions. Each of these three contributions is IR divergent,
however the sum of all of them is finite (factorization and KLN theorems). To
be able to integrate numerically each contribution alone, we have to subtract
the divergences from each one ($\sigma^{\scriptscriptstyle\text{virt, real,
fact}}_{\scriptscriptstyle\text{sub}}$) so that they give no contribution to
the full cross section as shown here,

$\displaystyle\int
d\Phi_{2}d\sigma^{\scriptscriptstyle\text{virt}}_{\scriptscriptstyle\text{sub}}+\int
d\Phi_{2+1}d\sigma^{\scriptscriptstyle\text{real}}_{\scriptscriptstyle\text{sub}}+\int
dx\int
d\Phi_{2}d\sigma^{\scriptscriptstyle\text{fact}}_{\scriptscriptstyle\text{sub}}$
$\displaystyle=0.$ (6)

This can be done by employing a given subtraction method. In this paper, we
use the FKS subtraction method which is automated by the module MadFKS fks2 .
Finally, the total inclusive cross section can be written as

$\displaystyle\sigma(T\bar{t}+\bar{T}t)$
$\displaystyle=\sigma_{\scriptscriptstyle Z}^{\scriptscriptstyle
T\bar{t}+\bar{T}t}+\sigma_{\scriptscriptstyle W}^{\scriptscriptstyle
T\bar{t}+\bar{T}t}+\sigma_{\scriptscriptstyle ZW}^{\scriptscriptstyle
T\bar{t}+\bar{T}t}.$ (7)

where $\sigma_{\scriptscriptstyle Z}^{\scriptscriptstyle T\bar{t}+\bar{T}t}$
and $\sigma_{\scriptscriptstyle W}^{\scriptscriptstyle T\bar{t}+\bar{T}t}$ are
the inclusive (LO or NLO) hadronic cross sections for the singly produced top
quark partner with the top through the exchange of $s$-channel $Z$ and
$t$-channel $W$ bosons, respectively. $\sigma_{\scriptscriptstyle
ZW}^{\scriptscriptstyle T\bar{t}+\bar{T}t}$ is the interference contribution
of the $s$ and $t$-channels Feynman diagrams. It contributes only for the
scenarios $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$ and for
processes initiated by $b$-quarks. For the later one, the inclusion of the
boxes is indispensable to insure IR divergence cancellation.

$m_{\scriptscriptstyle T}$ [GeV] Scenario
$\sigma_{\scriptscriptstyle\text{fLO}}$ [pb]
$\sigma_{\scriptscriptstyle\text{fNLO}}$ [pb] $K$-factor
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{(s)}}$ [pb]
$K$-factor  $500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$
$(9.230\times 10^{-4}){}^{+3.5\%}_{-3.4\%}{}^{+6.3\%}_{-6.3\%}$ $(1.894\times
10^{-2}){}^{+19.8\%}_{-15.3\%}{}^{+1.5\%}_{-1.5\%}$ 20.52 $(1.252\times
10^{-3}){}^{+3.4\%}_{-3.0\%}{}^{+1.9\%}_{-1.9\%}$ 1.36  $1000$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$
$(6.456\times 10^{-5}){}^{+6.9\%}_{-6.2\%}{}^{+6.7\%}_{-6.7\%}$ $(3.053\times
10^{-3}){}^{+23.3\%}_{-17.6\%}{}^{+2.8\%}_{-2.8\%}$ 47.29 $(8.201\times
10^{-5}){}^{+4.1\%}_{-3.9\%}{}^{+2.5\%}_{-2.5\%}$ 1.27  $1500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$
$(8.877\times 10^{-6}){}^{+9.2\%}_{-8.0\%}{}^{+7.9\%}_{-7.9\%}$ $(6.948\times
10^{-4}){}^{+25.5\%}_{-19.0\%}{}^{+4.4\%}_{-4.4\%}$ 78.27 $(1.065\times
10^{-5}){}^{+4.7\%}_{-4.6\%}{}^{+3.5\%}_{-3.5\%}$ 1.20  $2000$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$
$(1.663\times 10^{-6}){}^{+11.0\%}_{-9.4\%}{}^{+9.8\%}_{-9.8\%}$ $(1.846\times
10^{-4}){}^{+27.3\%}_{-20.0\%}{}^{+6.3\%}_{-6.3\%}$ 111.00 $(1.880\times
10^{-6}){}^{+5.2\%}_{-5.1\%}{}^{+5.1\%}_{-5.1\%}$ 1.13  $500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$
$(4.614\times 10^{-4}){}^{+3.5\%}_{-3.4\%}{}^{+6.3\%}_{-6.3\%}$ $(9.452\times
10^{-3}){}^{+19.8\%}_{-15.4\%}{}^{+1.5\%}_{-1.5\%}$ 20.49 $(6.255\times
10^{-4}){}^{+3.4\%}_{-3.0\%}{}^{+1.9\%}_{-1.9\%}$ 1.36  $1000$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$
$(3.230\times 10^{-5}){}^{+6.9\%}_{-6.2\%}{}^{+6.7\%}_{-6.7\%}$ $(1.527\times
10^{-3}){}^{+23.3\%}_{-17.6\%}{}^{+2.8\%}_{-2.8\%}$ 47.28 $(4.106\times
10^{-5}){}^{+4.1\%}_{-3.9\%}{}^{+2.5\%}_{-2.5\%}$ 1.27  $1500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$
$(4.438\times 10^{-6}){}^{+9.2\%}_{-8.0\%}{}^{+7.9\%}_{-7.9\%}$ $(3.475\times
10^{-4}){}^{+25.5\%}_{-18.9\%}{}^{+4.4\%}_{-4.4\%}$ 78.30 $(5.327\times
10^{-6}){}^{+4.6\%}_{-4.5\%}{}^{+3.4\%}_{-3.4\%}$ 1.20  $2000$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$
$(8.316\times 10^{-7}){}^{+11.0\%}_{-9.4\%}{}^{+9.8\%}_{-9.8\%}$ $(9.225\times
10^{-5}){}^{+27.2\%}_{-20.0\%}{}^{+6.3\%}_{-6.3\%}$ 111.93 $(9.386\times
10^{-7}){}^{+5.2\%}_{-5.2\%}{}^{+5.2\%}_{-5.2\%}$ 1.13  $m_{\scriptscriptstyle
T}$ [GeV] Scenario  $\sigma_{\scriptscriptstyle\text{fLO}}$ [pb]
$\sigma_{\scriptscriptstyle\text{fNLO}}$ [pb] $K$-factor
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle(q\bar{q})}$ [pb]
$K$-factor  $500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$
$(1.018\times 10^{-2}){}^{+8.0\%}_{-10.4\%}{}^{+21.9\%}_{-21.9\%}$
$(1.013\times 10^{-2}){}^{+3.0\%}_{-1.4\%}{}^{+5.0\%}_{-5.0\%}$ 0.99
$(1.222\times 10^{-2}){}^{+20.2\%}_{-19.3\%}{}^{+4.6\%}_{-4.6\%}$ 1.20  $1000$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$
$(1.788\times 10^{-3}){}^{+3.1\%}_{-5.0\%}{}^{+32.2\%}_{-32.2\%}$
$(1.716\times 10^{-3}){}^{+3.0\%}_{-1.3\%}{}^{+9.3\%}_{-9.3\%}$ 0.96
$(2.121\times 10^{-3}){}^{+16.2\%}_{-15.7\%}{}^{+8.6\%}_{-8.6\%}$ 1.19  $1500$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$
$(3.907\times 10^{-4}){}^{+0.0\%}_{-1.6\%}{}^{+44.5\%}_{-44.5\%}$
$(3.813\times 10^{-4}){}^{+2.9\%}_{-1.7\%}{}^{+14.7\%}_{-14.7\%}$ 0.98
$(4.768\times 10^{-4}){}^{+13.9\%}_{-13.5\%}{}^{+13.7\%}_{-13.7\%}$ 1.22
$2000$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$
$(9.474\times 10^{-5}){}^{+1.1\%}_{-2.3\%}{}^{+58.3\%}_{-58.3\%}$
$(9.716\times 10^{-5}){}^{+2.7\%}_{-2.0\%}{}^{+21.6\%}_{-21.6\%}$ 1.03
$(1.228\times 10^{-4}){}^{+12.3\%}_{-11.9\%}{}^{+20.2\%}_{-20.2\%}$ 1.30
$500$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,H\\}}$
$(6.786\times 10^{-3}){}^{+8.0\%}_{-10.4\%}{}^{+21.8\%}_{-21.8\%}$
$(6.755\times 10^{-3}){}^{+3.0\%}_{-1.4\%}{}^{+5.0\%}_{-5.0\%}$ 0.99
$(8.153\times 10^{-3}){}^{+20.1\%}_{-19.3\%}{}^{+4.6\%}_{-4.6\%}$ 1.20  $1000$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,H\\}}$
$(1.191\times 10^{-3}){}^{+3.1\%}_{-5.0\%}{}^{+32.0\%}_{-32.0\%}$
$(1.147\times 10^{-3}){}^{+3.0\%}_{-1.3\%}{}^{+9.3\%}_{-9.3\%}$ 0.96
$(1.410\times 10^{-3}){}^{+16.1\%}_{-15.7\%}{}^{+8.6\%}_{-8.6\%}$ 1.18  $1500$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,H\\}}$
$(2.604\times 10^{-4}){}^{+0.0\%}_{-1.6\%}{}^{+44.4\%}_{-44.4\%}$
$(2.554\times 10^{-4}){}^{+2.9\%}_{-1.7\%}{}^{+14.7\%}_{-14.7\%}$ 0.98
$(3.179\times 10^{-4}){}^{+13.9\%}_{-13.5\%}{}^{+13.8\%}_{-13.8\%}$ 1.22
$2000$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,H\\}}$
$(6.329\times 10^{-5}){}^{+1.1\%}_{-2.3\%}{}^{+58.3\%}_{-58.3\%}$
$(6.476\times 10^{-5}){}^{+2.7\%}_{-2.0\%}{}^{+21.6\%}_{-21.6\%}$ 1.02
$(8.188\times 10^{-5}){}^{+12.3\%}_{-11.9\%}{}^{+20.2\%}_{-20.2\%}$ 1.29
$m_{\scriptscriptstyle T}$ [GeV] Scenario
$\sigma_{\scriptscriptstyle\text{fLO}}$ [pb]
$\sigma_{\scriptscriptstyle\text{fNLO}}$ [pb] $K$-factor
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{($q\bar{q}$)}}$
[pb] $K$-factor  $500$ $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$
$(7.139\times 10^{-3}){}^{+7.5\%}_{-9.8\%}{}^{+21.0\%}_{-21.0\%}$
$(1.319\times 10^{-2}){}^{+9.1\%}_{-5.8\%}{}^{+3.2\%}_{-3.2\%}$ 1.85
$(8.661\times 10^{-3}){}^{+19.4\%}_{-18.2\%}{}^{+4.3\%}_{-4.3\%}$ 1.21  $1000$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$
$(1.218\times 10^{-3}){}^{+2.9\%}_{-4.8\%}{}^{+31.7\%}_{-31.7\%}$
$(2.173\times 10^{-3}){}^{+10.3\%}_{-6.7\%}{}^{+6.1\%}_{-6.1\%}$ 1.78
$(1.447\times 10^{-3}){}^{+15.9\%}_{-15.3\%}{}^{+8.4\%}_{-8.4\%}$ 1.19  $1500$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$
$(2.636\times 10^{-4}){}^{+0.0\%}_{-1.5\%}{}^{+44.0\%}_{-44.0\%}$
$(4.875\times 10^{-4}){}^{+11.2\%}_{-7.5\%}{}^{+9.7\%}_{-9.7\%}$ 1.85
$(3.219\times 10^{-4}){}^{+13.7\%}_{-13.3\%}{}^{+13.6\%}_{-13.6\%}$ 1.22
$2000$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$
$(6.373\times 10^{-5}){}^{+1.2\%}_{-2.3\%}{}^{+57.9\%}_{-57.9\%}$
$(1.266\times 10^{-4}){}^{+12.2\%}_{-8.4\%}{}^{+14.0\%}_{-14.0\%}$ 1.99
$(8.269\times 10^{-5}){}^{+12.2\%}_{-11.7\%}{}^{+20.0\%}_{-20.0\%}$ 1.30
$500$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$
$(5.359\times 10^{-3}){}^{+7.5\%}_{-9.8\%}{}^{+21.0\%}_{-21.0\%}$
$(9.885\times 10^{-3}){}^{+9.1\%}_{-5.8\%}{}^{+3.2\%}_{-3.2\%}$ 1.84
$(6.509\times 10^{-3}){}^{+19.4\%}_{-18.2\%}{}^{+4.3\%}_{-4.3\%}$ 1.22  $1000$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$
$(9.132\times 10^{-4}){}^{+2.9\%}_{-4.8\%}{}^{+31.7\%}_{-31.7\%}$
$(1.626\times 10^{-3}){}^{+10.3\%}_{-6.7\%}{}^{+6.1\%}_{-6.1\%}$ 1.78
$(1.082\times 10^{-3}){}^{+15.9\%}_{-15.4\%}{}^{+8.4\%}_{-8.4\%}$ 1.18  $1500$
$\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$
$(1.978\times 10^{-4}){}^{+0.0\%}_{-1.5\%}{}^{+44.0\%}_{-44.0\%}$
$(3.649\times 10^{-4}){}^{+11.3\%}_{-7.5\%}{}^{+9.6\%}_{-9.6\%}$ 1.84
$(2.416\times 10^{-4}){}^{-13.8\%}_{+13.3\%}{}^{+13.5\%}_{-13.5\%}$ 1.22
$2000$ $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$
$(4.787\times 10^{-5}){}^{+1.2\%}_{-2.3\%}{}^{+57.9\%}_{-57.9\%}$
$(9.493\times 10^{-5}){}^{+12.3\%}_{-8.4\%}{}^{+14.0\%}_{-14.0\%}$ 1.98
$(6.199\times 10^{-5}){}^{+12.2\%}_{-11.8\%}{}^{+20.0\%}_{-20.0\%}$ 1.29

Table 1: fLO and fNLO QCD inclusive cross sections for $T\bar{t}+\bar{T}t$
production at $\sqrt{s}=13$ TeV. The results are shown together with their
associated scale (left) and PDF (right) uncertainties. The values of the
external parameters are fixed to: $m_{H}=125\,\text{GeV}$,
$m_{Z}=91.1876\,\text{GeV}$, $m_{t}=173.1\,\text{GeV}$,
$\alpha_{s}(m_{Z})=0.118$ and $G_{F}=1.16637\times 10^{-5}\,\text{GeV}^{-2}$,
where $\alpha_{s}(m_{Z})$ is the strong coupling constant at the $Z$ mass.

We use MadGraph5aMc@NLO version 2.5.4, where the NNPDF 3.0 PDF set is adopted
(we use NNPDF30_lo_as_0118 for LO and NNPDF30_nlo_as_0118 for NLO) lhapdf6 ;
nnpdf . The couplings of the top partner to the third generation quarks are
defined in eq. (2). We use the dynamical scale scheme, where we set the
renormalization ($\mu_{R}$) and factorization ($\mu_{F}$) scales equal to each
other ($\mu_{R}=\mu_{F}=\mu=M_{T}/2$, where $M_{T}$ is the transverse mass of
the final state particles system). To estimate the theoretical uncertainties,
we evaluate the cross section for the three values of the scale $\mu/2$, $\mu$
and $2\mu$. In table 1, we furnish the leading and the next-to-leading fixed
orders (fLO and fNLO) cross sections with their relative scale and PDF
uncertainties, for the production of a top quark partner in association with a
top quark for different values of $m_{\scriptscriptstyle T}$ in six benchmark
scenarios555We mention that the lower limits of the $T$ mass range between
$715$-$950\,\text{GeV}$ according to recent LHC data. Nevertheless, in this
work, we vary it between $500\,\text{GeV}$ and $2000\,\text{GeV}$ to better
study the behaviour of the cross section and to compare the scale
uncertainties for low and high $T$ mass.. We notice that
$\sigma_{\scriptscriptstyle\text{fNLO}}$ is the full fNLO cross section, where
all the real emission processes are taken into account. On the other hand, for
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{(s)}}$ the
$t$-channel gluon-quark (anti-quark) real emission corrections to the
$s$-channel Born Feynman diagrams are omitted, and for
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{($q\bar{q}$)}}$
all the gluon-quark (anti-quark) initiated processes are ignored. The first
thing that we observe, from table 1 and figure 3, is the huge difference
between $\sigma_{\scriptscriptstyle\text{fLO}}$ and
$\sigma_{\scriptscriptstyle\text{fNLO}}$ for the scenarios $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$ which leads
to gigantic $K$-factors (increasing with the $T$ mass). This is due to the
fact that the $t$-channel gluon-quark (anti-quark) Feynman diagrams
contribution (seen as a new Born contribution), which appears exclusively at
NLO calculation, dominates over the $s$-channel tree level and one loop
Feynman diagram contributions. If one omit the real emission $t$-channel
contribution, as shown on the last two columns of the first part of table 1
($\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{($s$)}}$),
the $K$-factors for the different masses are reduced (1.36 for
$m_{\scriptscriptstyle T}=500\,\text{GeV}$ and 1.13 for $m_{\scriptscriptstyle
T}=2000\,\text{GeV}$). As a matter of fact, the squared amplitudes of the
$t$-channel real emission Feynman diagrams, the last two graphs in the fourth
line of figure 2, are proportional to $m_{Z}^{-4}$ (for small $t$-channel $Z$
boson virtuality). On the other hand, the squared amplitudes of the
$s$-channel Feynman diagrams (neglecting the interference contribution) are
proportional to $(\hat{s}-m_{Z}^{2})^{-2}$, where
$\hat{s}\geq(m_{T}+m_{t})^{2}$. On top of that, the $t$-channel contribution
comes from the gluon-quark initiated processes, which make the former
contribution very dominant and almost control the behaviour of the so called
fixed order NLO QCD cross section, see figure 3. One of the major purposes of
NLO calculation is the reduction of the renormalization and factorization
scales dependency of the cross section. From this table and figure 4, we see
that the scale dependence is reduced only if we suppress the $t$-channel
gluon-quark (anti-quark) real emission Feynman diagrams contribution for the
scenarios $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,H\\}}$666If we
suppress the contribution of gluon-quark (anti-quark) real emission processes,
the scale dependency is much more improved as shown on the left side of figure
4 (dotted black curve). Nevertheless, this is not physical since these
processes contribute to the scaling violation of the quark distribution
functions.. If we consider such contribution, the fNLO cross section behaves
on the scale as LO-like cross section i.e. it is monotonically decreasing on
the scale (see the red-dashed curve on the left side of figure 4). So, to
reduce the scale dependency, one has to calculate the higher order corrections
(NNLO for example), which is not the purpose of this paper. Another
alternative to reduce the $K$-factors is to restrict the phase space of the
real emission contribution. For example, by enforcing the transverse momentum
of the jet to be $P_{T}(j)>m_{\scriptscriptstyle
T}^{\scriptscriptstyle(i)}/(3/2+i)$ for $i=1,\cdots 4$, the $K$-factors are
reduced to $1.40$ for $m_{\scriptscriptstyle
T}^{\scriptscriptstyle(1)}=500\,\text{GeV}$, $1.39$ for $m_{\scriptscriptstyle
T}^{\scriptscriptstyle(2)}=1000\,\text{GeV}$, $1.56$ for
$m_{\scriptscriptstyle T}^{\scriptscriptstyle(3)}=1500\,\text{GeV}$ and $1.70$
for $m_{\scriptscriptstyle T}^{\scriptscriptstyle(4)}=2000\,\text{GeV}$. In
fact, these cuts restrict the virtuality of the $Z$ boson (especially
$t$-channel exchange) to be much larger than its mass which makes the
contribution of the real emission (especially gluon-quark initiated processes)
sub-dominant. Regarding the scenarios $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,H\\}}$, all the
partonic processes contributing to them are initiated by bottom quark pairs or
gluon-bottom (anti-bottom). From the second part of this table, we see that
the real and virtual contributions almost cancel together which leads to
$K$-factors nearly equal to one. On the other hand, the scale dependency is
improved especially for relatively low top partner mass for both scenarios
(see the blue solid curve on the right side of figure 4). In what concern the
scenarios $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}$, the full
fNLO cross section is about the double of the fLO one. This is attributed to
the quark-gluon (anti-quark) initiated processes contribution which arises
only at NLO and it is known to increase the $K$-factor. If the later
contribution is suppressed (see the last two columns of the third part of
table 1, the $K$-factors are reduced to be of order 1.2. Concerning the
uncertainties associated to the parton distribution functions, they increase
with the mass of the top partner for both fLO and fNLO cross sections and for
all the benchmark scenarios777For benchmark scenarios where the $W$ boson is
present, the PDF uncertainties are very large compared to the other scenarios.
This is due to the bottom quark PDF which shows the largest difference between
different sets for large Bjorken-$x$ (especially the LO PDF).. This is due to
the fact that, for large Bjorken-$x$, the errors of the different PDF sets
become larger as a result of the lack of data in this region.

Figure 3: Variation of the fLO and fNLO cross sections in term of the top
partner mass for the three benchmark scenarios: $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$, $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$ (at center-
of-mass energy $\sqrt{s}=13\,\text{TeV}$).

Figure 4: Scale variation of the total cross section for the benchmark
scenarios: $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$, $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{0,W,0\\}}$ and $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$
($m_{\scriptscriptstyle T}=500\,\text{GeV}$). We notice that the red-dashed
plot on the left-side follows the Y-axis on the right-side of the panel.

In general, precise differential distributions are helpful to compare theory
with experiment, and to put constraints on the free parameters of the model
under consideration. In figure 5, we show the differential distributions on
the transverse-momentum and the pseudo-rapidity of $T$ and $\bar{t}$, and the
transverse-momentum of the system $T\bar{t}$ for the benchmark scenario $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$. These
distributions are provided for fLO, fNLO, LO matched to parton shower (LO+PS)
and NLO matched to parton shower (NLO+PS) for the two different values of the
top partner mass $m_{T}=500,1000\,\text{GeV}$, where Pythia8 ref41  is used.
We employ MadAnalysis 5 madanalysis  to produce and analyse the distributions.

Figure 5: Differential distributions for the benchmark scenario $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,0\\}}$.

We observe that the parton shower does not affect significantly the fLO and
the fNLO transverse momentum and the rapidity distributions of the individual
particles of the final state ($T$ and $\bar{t}$). Only for the very low
$P_{\scriptscriptstyle T}$ and large $|\eta|$, considerable changes are seen
which are nonetheless still bellow $28\%$. It tends to reduce the differential
cross section at small $P_{\scriptscriptstyle T}$ and slightly increasing it
at high $P_{\scriptscriptstyle T}$ for the single top partner, where the fNLO
and the NLO+PS distributions match better compared to the LO and the LO+PS
distributions in this region as expected (especially for low $T$ mass). The
slight modifications of these distributions are attributed to the PDF
uncertainties in this phase space region. On the other hand, the distributions
of the anti-top quark show a good agreement between the fLO (fNLO) and the
LO+PS (NLO+PS) for both the transverse momentum and the pseudo-rapidity, see
the first four plots of figure 5. By contrast, the spectrum of the NLO+PS
transverse momentum distribution of the system $T\bar{t}$
($P_{\scriptscriptstyle T}(T\bar{t})$) shows a considerable change compared to
fNLO predictions. In fact, the last two plots of figure 5 show the same
$d\sigma/dP_{\scriptscriptstyle T}(T\bar{t})$ spectrum. On the left side, it
is given in bins of $100\,\text{GeV}$ up to $1000\,\text{GeV}$ and, on the
right side, it is provided in bins of $5\,\text{GeV}$ up to $500\,\text{GeV}$.
With the small binning, the fNLO predictions show a diverging behaviour for
small $P_{\scriptscriptstyle T}(T\bar{t})$. This is due to an incomplete
compensation between the logarithmically divergent terms of the virtual and
the real pieces in this region (due to the lack of phase space of the extra
parton). However, with such a small binning the resummation to all orders is
achieved in the NLO+PS approach, where the predictions are stable and well
behaved for low transverse momentum. Since the fNLO and the NLO+PS are
normalised to the same total cross section, one observes that at larger
$P_{\scriptscriptstyle T}$ values NLO+PS predicts larger cross section since
it gives lower predictions at low $P_{T}$. However it is expected that at very
large $P_{\scriptscriptstyle T}$ they should converge as can be seen on the
left plot (modulo the statistical fluctuations).

##  4 Top partner production in association with top quark in 6FS

Figure 6: Leading-order Feynman diagrams for the reaction $pp\rightarrow
T\bar{t}+\bar{T}t$ in 6FS.

In the previous section, we have seen that benchmark scenarios without gauge
boson $t$-channel exchange at the Born level have huge $K$-factors if no cut
is applied on the extra partons of the real emission. To allow for such
exchange (of $Z$ boson), one considers working in the 6FS. Furthermore,
predictions in such scheme should be more reliable than in 5FS for very high
energy regime $\sqrt{s}=100\,\text{TeV}$ (future proton-proton colliders).
Also, 6FS leads to new production mechanisms of the same final state studied
previously with more important cross sections than in 5FS. In this section, we
work in the standard massless 6FS, i.e. we take into account the top quark
initiated processes where the top quark is treated as massless in the initial
and the final states888To take into account the mass effects of the top quark
explicitly, one has to modify the DGLAP evolution equations, extend the
subtraction method to include massive partons in the initial state $\cdots$
etc. Fortunately, neglecting the mass of the initial state heavy quarks, in
our case, is justified. According to the simplified ACOT scheme (s-ACOT), the
mass of the heavy quarks may be neglected for partonic sub-processes initiated
by heavy quarks, see refs. 6f1 ; 6f2 ; 6f3  for more detail.. We suppose that
the top quark partner can mix with quarks of the first or the third
generations via the exchange of the $Z$ boson. In this case, the partonic
processes which lead to the associated top partner with top quark final state
are,

$\displaystyle u\bar{t}\,(\bar{u}t)$ $\displaystyle\rightarrow
T\bar{t}\,(\bar{T}t),$ $\displaystyle q\bar{q}$ $\displaystyle\rightarrow
T\bar{t}+\bar{T}t,$ $\displaystyle t\bar{t}$ $\displaystyle\rightarrow
T\bar{t}+\bar{T}t.$ (8)

where $q\equiv u,d,c,s,b,g$. The corresponding Feynman diagrams are depicted
in figure 6. For the benchmark scenario $\bf
T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$, only the
first process contributes to the Born cross section. It is mediated by
$t$-channel $Z$ boson (first diagram of figure 6). Regarding the benchmark
scenario $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$, the second
(same processes of 5FS) and the third processes contribute to the Born cross
section. The latter one is initiated by top quarks and mediated by the
exchange of $s$ and $t$-channels $Z$ boson (second and third diagrams of
figure 6). The calculation is done within the MadGraph framework by making use
of an UFO renormalised vector-like quark model, where the top quark is
considered as massless. We use the fixed renormalization and factorization
scales $\mu=\mu_{F}=\mu_{R}=m_{T}+m_{t}$, where $m_{t}$ is the physical top
mass. We employ the PDF set NNPDF30_nlo_as_0118_nf_6 for LO and NLO
calculations. In table 2, we give the fLO and fNLO cross sections for
$m_{\scriptscriptstyle T}=1000\,\text{GeV}$ and $m_{\scriptscriptstyle
T}=1500\,\text{GeV}$, in the two benchmark scenarios, at
$\sqrt{s}=13\,\text{TeV}$ and $\sqrt{s}=100\,\text{TeV}$. The fixed next-to-
leading order cross section is provided with cut on the invariant mass of the
top quark pair $M_{t\bar{t}}>2m_{t}$
($\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut1}}$),
and with cuts on the $P_{\scriptscriptstyle T}$ of the jet and
$M_{t\bar{t}}>2m_{t}$
($\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut2}}$)999The
cuts on $P_{\scriptscriptstyle T}$ are chosen to reduce the $K$-factors. At
$\sqrt{s}=13\,\text{TeV}$, we require that $p_{T}(j)>m_{\scriptscriptstyle
T}/2$ for $m_{\scriptscriptstyle T}=1000\,\text{GeV}$ and
$p_{T}(j)>2m_{\scriptscriptstyle T}/5$ for $m_{\scriptscriptstyle
T}=1500\,\text{GeV}$. At $\sqrt{s}=100\,\text{TeV}$, we require that
$p_{T}(j)>2m_{\scriptscriptstyle T}$ for the two mass values.. In figure 7, we
show the variation of the cross section in terms of the top partner mass.

Scenario $m_{\scriptscriptstyle T}$ [GeV] $\sqrt{s}$ [TeV]
$\sigma_{\scriptscriptstyle\text{fLO}}$ [pb]
$\sigma_{\scriptscriptstyle\text{fNLO}}\text{[pb]}$
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut1}}\text{[pb]}$
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut2}}\text{[pb]}$
$K$-factor $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ 1000 13
$(8.566\times 10^{-5}){}^{+2.4\%}_{-1.6\%}{}^{+2.0\%}_{-2.0\%}$ $(3.269\times
10^{-2}){}^{+20.5\%}_{-16.0\%}{}^{+2.4\%}_{-2.4\%}$ $(3.150\times
10^{-2}){}^{+20.6\%}_{-16.1\%}{}^{+1.8\%}_{-1.8\%}$ $(1.219\times
10^{-4}){}^{+23.8\%}_{-19.6\%}{}^{+3.6\%}_{-3.6\%}$ 1.42 100 $(1.296\times
10^{-2}){}^{+45.7\%}_{-39.9\%}{}^{+1.2\%}_{-1.2\%}$ $(5.088\times
10^{0}){}^{+8.5\%}_{-7.7\%}{}^{+0.9\%}_{-0.9\%}$ $(4.997\times
10^{0}){}^{+10.1\%}_{-8.8\%}{}^{+0.9\%}_{-0.9\%}$ $(2.370\times
10^{-2}){}^{+18.1\%}_{-26.2\%}{}^{+1.8\%}_{-1.8\%}$ 1.82 2-8 1500 13
$(1.067\times 10^{-5}){}^{+1.0\%}_{-0.6\%}{}^{+3.3\%}_{-3.3\%}$ $(6.195\times
10^{-3}){}^{+22.1\%}_{-17.0\%}{}^{+2.5\%}_{-2.5\%}$ $(5.754\times
10^{-3}){}^{+22.8\%}_{-17.5\%}{}^{+3.0\%}_{-3.0\%}$ $(1.694\times
10^{-5}){}^{+22.9\%}_{-19.2\%}{}^{+6.8\%}_{-6.8\%}$ 1.59 100 $(5.521\times
10^{-3}){}^{+39.8\%}_{-37.1\%}{}^{+1.6\%}_{-1.6\%}$ $(2.396\times
10^{0}){}^{+9.9\%}_{-8.8\%}{}^{+0.9\%}_{-0.9\%}$ $(2.131\times
10^{0}){}^{+10.5\%}_{-9.2\%}{}^{+0.9\%}_{-0.9\%}$ $(9.669\times
10^{-3}){}^{+11.2\%}_{-18.5\%}{}^{+2.0\%}_{-2.0\%}$ 1.75 $\bf
T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ 1000 13
$(1.097\times 10^{-3}){}^{+18.2\%}_{-25.8\%}{}^{+1.4\%}_{-1.4\%}$
$(4.481\times 10^{-3}){}^{+13.3\%}_{-9.4\%}{}^{+1.2\%}_{-1.2\%}$ $(1.537\times
10^{-3}){}^{+28.0\%}_{-29.9\%}{}^{+1.4\%}_{-1.4\%}$ 1.40 100 $(1.364\times
10^{-1}){}^{+26.2\%}_{-30.9\%}{}^{+1.0\%}_{-1.0\%}$ $(3.811\times
10^{-1}){}^{+8.2\%}_{-6.9\%}{}^{+1.2\%}_{-1.2\%}$ $(1.931\times
10^{-1}){}^{+28.9\%}_{-29.2\%}{}^{+1.0\%}_{-1.0\%}$ 1.42 2-8 1500 13
$(2.643\times 10^{-4}){}^{+11.9\%}_{-18.2\%}{}^{+2.6\%}_{-2.6\%}$
$(9.018\times 10^{-4}){}^{+18.0\%}_{-13.0\%}{}^{+3.0\%}_{-3.0\%}$
$(3.808\times 10^{-4}){}^{+22.4\%}_{-24.1\%}{}^{+2.3\%}_{-2.3\%}$ 1.44 100
$(7.423\times 10^{-2}){}^{+19.9\%}_{-24.1\%}{}^{+1.0\%}_{-1.0\%}$
$(1.820\times 10^{-1}){}^{+5.0\%}_{-5.3\%}{}^{+0.9\%}_{-0.9\%}$ $(1.037\times
10^{-1}){}^{+24.2\%}_{-24.5\%}{}^{+1.0\%}_{-1.0\%}$ 1.40

Table 2: fLO and fNLO cross sections in 6FS. For
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut1}}$, we
apply a cut on the invariant mass $M_{t\bar{t}}>2m_{t}$. For
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut2}}$, we
apply cuts on $p_{T}(j)$ and $M_{t\bar{t}}>2m_{t}$. The $K$-factor, given in
the last column, is defined by the fractions:
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut1}}/\sigma_{\scriptscriptstyle\text{fLO}}$
for $\bf T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$
and
$\sigma_{\scriptscriptstyle\text{fNLO}}^{\scriptscriptstyle\text{cut2}}/\sigma_{\scriptscriptstyle\text{fLO}}$
for $\bf T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$.

(a)  (b)

Figure 7: Variation of the cross section in term of $m_{\scriptscriptstyle T}$
at $\sqrt{s}=13\,\text{TeV}$ and $100\,\text{TeV}$

We observe that, $\sigma_{\scriptscriptstyle\text{fLO}}$ in the scenario $\bf
T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ (the
reaction is allowed only in 6FS) is larger than
$\sigma_{\scriptscriptstyle\text{fLO}}$ in the scenario $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ in both 5FS
and 6FS at $\sqrt{s}=13\,\text{TeV}$ (solid red, black and blue curves in sub-
figure 7a). This shows the dominance of the $t$-channel exchange even if the
luminosity of the top is very small compared to the other quarks at the LHC.
In fact, the virtuality of the $t$-channel exchanged $Z$ boson can be very
small, which explains the increase of the cross section. The same thing is
observed for very high center-of-mass energy, see sub-figure 7b.

We find that, the fLO cross section of $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ in the 6FS
is very close to the cross section calculated in the 5FS at
$\sqrt{s}=13\,\text{TeV}$, see the solid black and blue curves in sub-figure
7a. This is due to the fact that the leading contribution (second process in
eq. (8)) is the same as in the 5FS, and the sub-leading contribution (third
process in eq. (8)) is very small compared to the former one because of the
very small top quark PDF at the LHC. In fact, the slight difference between
the cross sections in the two schemes is due mainly to the top mass which is
neglected in 6FS. We observe also that the initiated top quark sub-processes
contribution increases with the mass of the top partner, see the black dotted
and dashed curves in sub-figure 7a. It represents, respectively, about $7\%$
and $24\%$ of the total cross section for $m_{\scriptscriptstyle
T}=500\,\text{GeV}$ and $m_{\scriptscriptstyle T}=2000\,\text{GeV}$ at
$\sqrt{s}=13\,\text{TeV}$. For very high center-of-mass energy (sub-figure
7b), it becomes the dominant contribution especially for
$m_{\scriptscriptstyle T}$ above $600\,\text{GeV}$. It represents about $41\%$
and $90\%$ of the total cross section for $m_{\scriptscriptstyle
T}=500\,\text{GeV}$ and $m_{\scriptscriptstyle T}=2000\,\text{GeV}$,
respectively. This shows the increase of the top density inside the proton
with the increase of the mass scale and for very high centre-of-mass energy.
It shows also, the dominance of the $t$-channel exchange $Z$ boson
contribution (second diagram of figure 6).

In what concerns the $K$-factors. They are still very large for the benchmark
scenario $\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$ if no cut
is applied. However, if we apply cuts on the invariant mass of the system
$t\bar{t}$ and $P_{\scriptscriptstyle T}$ of the jet, they are significantly
reduced, see the first part of the last column of table 2. Regarding the
scenario $\bf
T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}$, the
$K$-factors (without cut) are not very large like in the previous case (about
$3.41$ at $\sqrt{s}=13\,\text{TeV}$ and $2.45$ at $\sqrt{s}=100\,\text{TeV}$
for $m_{\scriptscriptstyle T}=1500\,\text{GeV}$). If we restrict
$M_{t\bar{t}}>2m_{t}$, they are considerably reduced as shown in the last
column of table 2. Actually, these cuts enforce the virtuality of the
$t$-channel exchanged $Z$ boson for the gluon-quark (anti-quark) real emission
to be larger which reduces their contribution.

##  5 Investigation of the single $T$ decaying into a Higgs and a top quark

Following the benchmark scenarios discussed above, the top quark partner can
decay into one of the third generation SM quarks via one of the three channels
$bW$, $tZ$ or $tH$, where mixing with the first and the second quark
generations is ignored. In this section, we examine the $t\bar{t}H$ final
state obtained from the reaction studied above (in 5FS) through the decay of
the heavy vector-like quark into a top quark and a Higgs boson
($pp\longrightarrow T\bar{t}+\bar{T}t\longrightarrow t\bar{t}H$) at
$\sqrt{s}=13\,\text{TeV}$. We do not consider the full NLO calculation, but we
just match the calculation done in section 3 to parton-shower (pythia8) and
perform the decay by MadSpin ref42  and MadWidth madwidth . This approximation
constitutes a first step towards a more sophisticated treatment, where we are
going to consider the full next-to-leading order calculation for $t\bar{t}H$
production at the LHC for models with vector-like quarks.

The investigation of this final state, in SM and BSM, has received more and
more attention by both theorists and experimentalists in the last few years,
especially after the discovery of the Higgs boson. In the context of
confirming the predicted and the measured properties of the SM Higgs boson,
and aiming to discover BSM physics, ATLAS and CMS collaborations analyse, in
many recent publications, the $t\bar{t}H$ final state for different decay
modes at the LHC runI and runII. For example, in ref31  evidence for the
production of the Higgs boson in association with a top quark pair at ATLAS
were shown. They presented a good agreement with the measured and the
predicted cross sections for different Higgs decay modes and data collected
from LHC runI and runII. Alternative publications by both collaborations, on
the same subject, are given in ref32 ; ref33 ; ref35 ; ref37 . The search for
the single production of top quark partner $T$ decaying into a Higss boson and
top quark was the subject of many publications of ATLAS and CMS
collaborations. In THt2  for example, CMS puts for the first time exclusion
limits on the cross section for $T$ single production at
$\sqrt{s}=13\,\text{TeV}$, where they targeted the decay of the Higgs boson
into a pair of bottom quarks, and the decay of the top quark includes a muon
or an electron. The other possible decay modes ($tZ$ and $bW$) of the singly
produced heavy top quark partner were investigated in TZt1 ; TZt2 ; TWb1 ;
TWb2  for example.

At this stage, we focus on the differential $K$-factors, i.e. the ratio of the
NLO+PS and LO+PS differential cross sections. Let’s consider the differential
distributions of the following observables: the transverse momentum of the
Higgs boson ($P_{\scriptscriptstyle T}(H)$), the transverse momentum of the
top quark ($P_{\scriptscriptstyle T}(t)$), the invariant mass of the final
state system $t\bar{t}H$ ($M_{t\bar{t}H}$), the rapidity of the Higgs
($Y_{\scriptscriptstyle H}$), and the rapidity of the top quark
($Y_{\scriptscriptstyle t}$). The distributions are calculated at LO and NLO
matched to parton shower for the benchmark scenario ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}}$. The
calculation is done at center-of-mass energy $\sqrt{s}=13\,\text{TeV}$ for the
two different mass values $m_{\scriptscriptstyle T}=500\,\text{GeV}$ and
$m_{\scriptscriptstyle T}=800\,\text{GeV}$, where the dynamical scale is used
(the average transverse mass of the final state particles).

(a)  (b)  (c)  (d)  (e)  (f)

Figure 8: Differential distributions of some observables for the $t\bar{t}H$
final state in the benchmark scenario ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,W,H\\}}}$, for
different $m_{\scriptscriptstyle T}$ values, at leading and next-to-leading
orders matched to parton shower.

In figures 8a, 8b, 8c, 8d and 8e, we plot the differential cross sections as
functions of the different kinematic variables, introduced above. The solid
red and blue plots represent the LO+PS distributions and the dotted red and
blue plots represent the NLO+PS distributions. In the lower panels, we display
the differential $K$-factors for each distribution. The first comment is that
all the LO+PS and NLO+PS distributions, for the same mass, have mainly
identical shapes with different magnitudes, where the magnitudes of the shown
observables decrease if the top partner mass increase, as expected101010For
large $P_{\scriptscriptstyle T}(t)$ and $P_{\scriptscriptstyle T}(H)$, it is
not the case. However, in this region, the differential cross sections are
very small and dominated by the statistical fluctuations.. Actually, we have
found that the integrated cross sections for $m_{\scriptscriptstyle
T}=500\,\text{GeV}$ are: $\sigma^{\text{LO+PS}}=2.15\times 10^{-3}\,\text{pb}$
and $\sigma^{\text{NLO+PS}}=3.97\times 10^{-3}\,\text{pb}$, and for
$m_{\scriptscriptstyle T}=800\,\text{GeV}$ are:
$\sigma^{\text{LO+PS}}=5.15\times 10^{-4}\,\text{pb}$ and
$\sigma^{\text{NLO+PS}}=9.25\times 10^{-4}\,\text{pb}$. This implies that the
$K$-factors are of order $1.80$, i.e. they are not changed and they are still
large for this final state. This is due to the contribution of the gluon-quark
(anti-quark) real emission Feynman diagrams as explained in the previous
section. We observe that the differential $K$-factors of the Higgs and the top
quark transverse momentum are relatively flat for $P_{T}$ ranging between
$0$-$250\,\text{GeV}$ and between $0$-$420\,\text{GeV}$ for
$m_{\scriptscriptstyle T}=500\,\text{GeV}$ and $m_{\scriptscriptstyle
T}=800\,\text{GeV}$, respectively. They coincide for $P_{T}$ ranging between
$0$-$250\,\text{GeV}$ for both distributions and for the two masses, where
they are about $1.9$. For higher $P_{T}$’s ($P_{T}>250$ for the smaller mass
and $P_{T}>420$ for the larger mass), the $K$-factors become arbitrary
increasing, i.e. the higher order corrections are much more important than the
leading order in this region. The invariant mass distributions of the final
state system peak at $750\,\text{GeV}$ for $m_{T}=500$, and at
$1150\,\text{GeV}$ for $m_{T}=800\,\text{GeV}$ and decrease very quickly with
increasing $M_{t\bar{t}h}$. The differential $K$-factors are relatively flat
(around 2.0) with some discrepancy before the peaks and for very high
invariant mass. Regarding, the differential $K$-factor for the Higgs rapidity
spectrum, it is flat and sable (nearly 2.0) except for high
$|y_{\scriptscriptstyle H}|$, where it shows a little perturbation. Anyway, in
the regions where the differential $K$-factors are not stable, the
differential distributions are dominated by the PDF uncertainties (and
statistical fluctuations).

Finally, we remind that for the decay of the vector-like top partner to a
Higgs and a top quark, we have taken into account the sub-leading right-handed
coupling $T$-$H$-$t$. The later one has non-negligible effects if the mass of
$T$ is lower than $1000\,\text{GeV}$. In figure 8f, we display the LO+PS
differential cross section as a function of the Higgs rapidity for
$m_{\scriptscriptstyle T}=600\,\text{GeV}$ and $m_{\scriptscriptstyle
T}=1500\,\text{GeV}$, and for both cases
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}=0$ and
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}\neq 0$. In the lower
panel, we plot the ratio of this distribution for
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}=0$ and
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}\neq 0$, i.e.
($d\sigma({\footnotesize{\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}\neq
0}})/dY_{\scriptscriptstyle
H})/(d\sigma({\footnotesize\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}=0})/dY_{\scriptscriptstyle
H}$). This differential fraction is about $1.28$ for $m_{\scriptscriptstyle
T}=600\,\text{GeV}$ and $1.05$ for $m_{\scriptscriptstyle T}=1500\,\text{GeV}$
(for most of the bins), which is in agreement with the integrated cross
section. For the later one, we have found that
$\sigma(\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}\neq
0)=1.14\times 10^{-3}\,\text{pb}$ and
$\sigma(\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}=0)=8.94\times
10^{-4}\,\text{pb}$ for $m_{\scriptscriptstyle T}=600\,\text{GeV}$, and
$\sigma(\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}\neq
0)=5.18\times 10^{-5}\,\text{pb}$ and
$\sigma(\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}=0)=4.94\times
10^{-5}\,\text{pb}$ for $m_{\scriptscriptstyle T}=1500\,\text{GeV}$. Then, the
cross section is about $28\%$ ($5\%$) higher if
$\kappa^{\scriptscriptstyle{(H)}}_{\scriptscriptstyle{R}}$ is included for
$m_{\scriptscriptstyle T}=600\,\text{GeV}$ ($m_{\scriptscriptstyle
T}=1500\,\text{GeV}$).

##  6 Conclusion

In this paper, we have presented the calculation of the QCD next-to-leading
order cross section for the production of a single vector-like top partner in
association with a top quark, for many possible benchmark scenarios, in
proton-proton collisions, in the five-flavour and the six-flavour schemes. The
computation is performed within the MadGraph5_aMC@NLO framework by making use
of 5FS and 6FS UFO NLO vector-like quarks models, which offers all the
required ingredients to do such calculation. We have investigated the fixed-
order and the fixed-order matched to parton shower differential distributions
at LO and NLO accuracies. We have examined the decay of the heavy top partner
into a Higgs boson and a top quark which leads to the interesting Higgs with a
top quark pair final state. This constitutes a first step towards performing
the full next-to-leading order calculation of the production of the associated
top quark pair with the Higgs at the LHC, including all the QCD and the mixed
QCD and electro-weak effects, for models with vector-like quarks.

We have shown, in 5FS, that the NLO $K$-factors for the total cross section
and the differential distributions are very different from one benchmark
scenario to another. For the benchmark scenarios where the heavy top quark
partner interacts with the third generation quarks and the $Z$ boson (and the
Higgs), the $K$-factors are gigantic and increase dramatically with the mass
of the top partner. We have found that this is due mainly to the $t$-channel
gluon-quark (anti-quark) real emission which appears exclusively at NLO order,
and it is seen as a new channel which dominates over the $s$-channel tree
level and one loop Feynman diagram contributions. We have shown that if we
enforce the transverse momentum of the extra partons of the real emission to
be larger than a given fraction of $m_{\scriptscriptstyle T}$, the $K$-factors
are significantly reduced. Regarding the scenarios where the heavy top quark
partner interacts with the third generation quarks and the $W$ boson (and the
Higgs), the $K$-factor is about one however the bottom quark PDF uncertainties
are relatively large. On the other hand, the scenarios where the heavy top
quark partner interacts with the third generation quarks, the $Z$ and $W$
bosons (and the Higgs), the $K$-factor is about two, which is due to the
gluon-quark (anti-quark) contribution.

We have found that the Born cross section, in 6FS, if $T$ is assumed to mix
only with first quark generation (the reaction is allowed only in 6FS) is
larger than the one obtained in the scenario ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}}$ for the
same final state. We have shown that top quark initiated processes
contribution becomes more and more important for high mass scale and for very
high center-of-mass energy (in ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}}$). It
becomes dominant for $T$ mass above $600\,\text{GeV}$ at
$\sqrt{s}=100\,\text{TeV}$. This is explained by the increase of the top quark
density inside the proton in this regime and, the dominance of the $t$-channel
$Z$ boson exchange. In what concern the $K$-factors in 6FS, they are still
large if no cut is applied for the case ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}}$ like in
5FS. However, for ${\bf
T^{\scriptscriptstyle\\{1,0,0\\}}_{\scriptscriptstyle\\{Z,0,0\\}}}$ they are
significantly lower than in ${\bf
T^{\scriptscriptstyle\\{0,0,3\\}}_{\scriptscriptstyle\\{Z,0,0\\}}}$ even if
without phase space restriction especially for very high energy regime.

We mention that we have included the right-handed chirality of the coupling of
the Higgs boson to the top partner and the top quark. We have shown, in the
last section, that it is very important to include such coupling for low top
partner mass. For $m_{\scriptscriptstyle T}=600\,\text{GeV}$, it contributes
with $28\%$ to the LO+PS cross section of the reaction $pp\rightarrow
T\bar{t}+\bar{T}t\rightarrow t\bar{t}H$.

###### Acknowledgements.

I am very thankful to P. Aurenche and J. Ph. Guillet for useful discussions
and precious remarks. I acknowledge the hospitality of LAPTh where a portion
of this work was done.

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