#  DESY 13-080 Strong Signatures
of Right-Handed Compositeness

Michele Redi1<EMAIL_ADDRESS>Veronica Sanz2,3
<EMAIL_ADDRESS>Maikel de Vries4<EMAIL_ADDRESS>Andreas
Weiler4<EMAIL_ADDRESS>

1 _INFN, Sezione di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy_
2 _Deptartment of Physics and Astronomy, York University,_
_4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada_
3 _Deptartment of Physics and Astronomy, University of Sussex,_
_Sussex House, Brighton BN1 9QH, United Kingdom_
4 _DESY, Notkestrasse 85, D-22607 Hamburg, Germany_

###### Abstract

Right-handed light quarks could be significantly composite, yet compatible
with experimental searches at the LHC and precision tests on Standard Model
couplings. In these scenarios, that are motivated by flavor physics, one
expects large cross sections for the production of new resonances coupled to
light quarks. We study experimental strong signatures of right-handed
compositeness at the LHC, and constrain the parameter space of these models
with recent results by ATLAS and CMS. We show that the LHC sensitivity could
be significantly improved if dedicated searches were performed, in particular
in multi-jet signals.

##  1 Introduction

Modern realizations of composite Higgs models rely on the hypothesis of
partial compositeness, each SM state has a heavy partner with equal quantum
numbers under the SM symmetries, see [1, 2] and references therein. Until
recently most studies focused on the so called “anarchic scenario” where the
SM light quarks are mostly elementary and the top largely composite [3]. This
hypothesis hides strong coupling effects from flavor and electroweak
observables but also eliminates the typical collider signatures of
compositeness.

In references [4, 5, 6, 7] it was shown that a different philosophy is
possible within the partial compositeness paradigm, where one chirality of SM
light quarks has large compositeness. These scenarios are in fact strongly
motivated by flavor physics. Assuming universal couplings for either left-
handed or right-handed fermions allows to realize the hypothesis of Minimal
Flavor Violation (MFV) [9] in strongly coupled theories, solving the flavor
problem of composite Higgs models [8]. Here the compositeness of the up quark
cannot be small, it being determined by the one of the top. Generalisations
allowing to split the third generation can also be considered [10].

In this note we will focus on the phenomenologically attractive scenario of
composite right-handed quarks that is weakly constrained by precision
electroweak tests allowing a large degree of compositeness, see [11] for a
recent discussion. We will study in detail the collider phenomenology
extending and updating the results in [5]. The experimental signatures are
dramatically different from the ones of the widely studied anarchic models
[12]. There the fact that the proton constituents are elementary makes it
difficult to produce the new states at the LHC. If right-handed up and down
quarks are composite instead, the couplings to the strong sector will be
large. This implies larger production cross sections for the heavy states that
can be tested with present LHC data.

The typical collider signatures of our scenario are jet final states. In
particular we derive a strong bound on gluon resonances from the latest dijet
searches at LHC. The phenomenology of heavy fermions depends on the chirality
of the associated SM particles. Partners of left-handed quarks can be singly
produced through electroweak interactions with large cross sections already at
the 8 TeV LHC. This places a stringent and rather model independent bound that
can be extracted from an ATLAS search [13]. Partners of right-handed quarks
are instead more difficult to produce and lead to final states with up to six
jets and no missing energy. We find that present multi-jet LHC searches,
tailored for supersymmetric scenarios, are mostly insensitive to this
signature even in the R-parity violation case. Bounds could be here
significantly improved with dedicated searches and we suggest some
possibilities that could be explored by the experimental collaborations.

The paper is organized as follows: In section 2 we review the model and
discuss the relevant features of right-handed compositeness. We emphasize in
particular the importance of chromomagnetic interactions. In section 3 we
discuss the phenomenology of the color octet. The relevant experimental
searches will be discussed and limits on the octet mass extracted. In sections
4 and 5 the collider signatures of heavy quark partners will be discussed.
Available searches will be analyzed and dedicated search strategies will be
proposed in section 6. We conclude in section 7. In appendix A the model used
in our simulations is presented and in appendix B the $p_{T}$ distribution in
single production of heavy quark partners is discussed.

##  2 Composite Light Quarks

Within the framework of partial compositeness SM fields mix with states of the
composite sector of equal quantum numbers under the SM symmetries, see [5] for
a detailed discussion. All the new states are classified according to
representations of the composite sector global symmetry. We will make the
minimal assumption that this contains $SU(3)_{c}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{X}$. The SM Yukawa couplings are schematically given by

$y_{SM}=\sin\phi_{L}\cdot Y\cdot\sin\phi_{R},$ (2.1)

where $\sin\phi_{L,R}$ are the mixings matrices of left and right chiralities
of the SM quarks with the composite states. The coupling $Y$, in general a
matrix, has a typical strength that characterizes the composite sector. For
simplicity we will often assume this to be equal to the coupling of spin-1
resonances $g_{\rho}$ but it should be kept in mind that these are in
principle independent parameters.

The standard assumption, naturally realized in Randall-Sundrum scenarios, is
that the degree of compositeness is controlled by the mass of the SM states.
Within this logic the light generations are practically elementary and couple
only through mixing of the SM gauge fields. This property makes the new states
experimentally well hidden both from direct and indirect searches. It was
pointed out however that at least the right-handed chiralities of the light
generations could be composite [14, 6, 5]. In this case the effects of
compositeness are more visible at LHC because the proton constituents are
strongly coupled to the composite states. Despite the large degree of
compositeness, corrections to precisions observables measured at LEP are small
and can be compatible with experimental bounds555Modified Higgs couplings
could also be obtained. See reference [15] for the discussion of Higgs
precision phenomenology in models with composite right- handed quarks and
reference [16] for related work.. This perhaps counterintuitive possibility is
in fact quite naturally realized if the right-handed quarks couple to singlets
of the custodial symmetry. Moreover this possibility is automatic in scenarios
that realize the MFV hypothesis [5] because a flavor symmetry relates the
compositeness of the up quark to the one of the top that is necessarily large.

Contrary to anarchic scenarios, composite light quarks have striking
experimental signatures that could be seen at LHC. Among the new states we
will consider the lightest partners of the up and down quarks. For the right-
handed quarks we assume that these are singlets of $SU(2)_{L}\times SU(2)_{R}$
while left-handed quarks will be associated to bi-doublets. For the up sector
we have,

$L_{U}={\bf(2,2)_{\frac{2}{3}}}=\begin{pmatrix}U&U_{\frac{5}{3}}\\\
D&U_{\frac{2}{3}}\end{pmatrix}\,,\quad\qquad\tilde{U}={\bf(1,1)_{\frac{2}{3}}}.$
(2.2)

The full model can be found in the appendix.

Figure 1: _Above couplings of the color octet to SM quarks and their heavy
partners. Below couplings to electroweak gauge bosons._

Of the composite spin one states only the gluon partner, a massive color octet
vector, will be included. We assume that the color octet couples as a gauge
field with strength $g_{\rho}$. Electroweak resonances will not be studied
here but we expect the rough features to be similar. The mixing with SM quarks
generates the trilinear couplings of the heavy gluon of figure 1 with
strengths

$\displaystyle X_{R}^{qq}$
$\displaystyle=g_{s}\left(\sin^{2}\phi_{Rq}\cot\theta-\cos^{2}\phi_{Rq}\tan\theta\right)$
$\displaystyle X_{R}^{qQ}$
$\displaystyle=g_{s}\frac{\sin\phi_{Rq}\cos\phi_{Rq}}{\sin\theta\cos\theta}$
$\displaystyle X_{R}^{QQ}$
$\displaystyle=g_{s}\left(\cos^{2}\phi_{Rq}\cot\theta-\sin^{2}\phi_{Rq}\tan\theta\right).$
(2.3)

where $\tan\theta=g_{s}/g_{\rho}$. We denote by $q$ ($Q$) a light (heavy)
quark. Analogous formulas hold for the left-handed chiralities. We will be
interested in the situation where the right-handed up and down quarks are
significantly composite. Strictly in MFV models
$\sin\phi_{Ru}=\sin\phi_{Rt}>\lambda_{t}/g_{\rho}$ but this can be relaxed in
more general constructions based on $SU(2)$ flavor symmetries [10]. The SM
right-handed quarks can couple to gluon resonances with a trilinear coupling
$qq\rho$ as large as $g_{\rho}\sin^{2}\phi_{Rq}$. Moreover the partners of
right-handed quarks can be produced and decay through the heavy-light vertex
in figure 1. On the other hand the vertex with left-handed partners is
negligible because the compositeness of left-handed light quarks is extremely
small.

For electroweak interactions the situation is exactly reverted, see figure 1.
In the limit of zero quark masses in the up sector the relevant vertices are

$\displaystyle
Y_{uD}=Y_{uU_{\frac{5}{3}}}=\frac{g}{\sqrt{2}}\frac{Y_{U}\,v}{\sqrt{2}m_{Q}}\sin\phi_{Ru}$
$\displaystyle
Y_{uU}=-Y_{uU_{\frac{2}{3}}}=\frac{g}{2\cos\theta_{W}}\frac{Y_{U}\,v}{\sqrt{2}m_{Q}}\sin\phi_{Ru}$
(2.4)

where $v=246$ GeV and $Y_{U}$ is the up sector fermionic coupling, see
appendix A. These interactions allow to singly produce the partners of left-
handed quarks. Higgs interactions are also generated but we will not study
them here, for more information see [17].

The last important ingredient in our analysis will be the chromomagnetic
operator

$\mathcal{L}_{\mathrm{chromo}}^{\mathrm{SM}}=\kappa\frac{g_{s}}{m_{Q}}\,\bar{U}_{L}\sigma_{\mu\nu}T^{a}u_{R}G_{\mu\nu}^{a}+\mathrm{h.c.}$
(2.5)

This dimension five operator is relevant in our analysis because it controls
the decay of the right-handed partners in the region $m_{\rho}>m_{Q}$ where
the decay into $Q\to\rho\,q$ is kinematically forbidden. It is generated by
loops of the strong sector fields with a size (see appendix A)

$\kappa\sim\frac{g_{\rho}^{2}}{16\pi^{2}}\frac{m_{Q}^{2}}{m_{\rho}^{2}}\sin\phi_{uR}.$
(2.6)

Let us briefly comment on the scenario where left-handed quarks are strongly
composite. Here precision electroweak tests, in particular modified coupling
to the $Z$, strongly disfavours large compositeness. One finds [5],

$\sin\phi_{Lq}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{\lambda_{t}}{2\,g_{\rho}}\left(\frac{m_{\rho}}{\rm
3\,TeV}\right).$ (2.7)

Repeating the analysis above implies that cross sections not larger than in
the anarchic scenario will be obtained, at least for the scales and couplings
that we expect in composite models that address the hierarchy problem. In fact
due to the opposite sign of the two contributions in eq. (2) the couplings may
even turn out to be smaller. In what follows we will only consider the
scenario with composite right-handed quarks.

###  2.1 Simulations

In this paper we will study the phenomenology of the gluon resonance, partners
of left-handed quarks $\bf{(2,2)_{2/3}}$ and partners of right-handed quarks,
$\bf{1_{2/3}}$ and $\bf{1_{-1/3}}$. We focus on the first generation partners
whose mass is however equal to the one of the top partners under the MFV
hypothesis. The searches are very sensitive to the spectrum of the new states.
We will mostly work under the assumption that the fermionic scale $m_{Q}$ is
smaller than $m_{\rho}$. This hypothesis appears to be necessary for the
theory to be natural, given that spin one particles lighter than $2$ TeV are
disfavoured. On the other hand new vectorial fermions are the most relevant
from the naturalness point of view, have weaker direct bounds.

In our simulations we generate event samples with MadGraph5 [18], using a
model666The FeynRules implementation of the right-handed partial compositeness
model is available upon request by the authors. generated with Feynrules 1.6
[19]. The parton level events are passed to Pythia 6.4 [20] to simulate the
effects of parton showering, and then to Delphes 2.0 [21] or ATLFAST [22] for
a fast detector simulation. We use the default CMS and ATLAS parameters for
Delphes depending on what experimental analysis we are comparing with, and
reconstruct jets with the anti-$k_{T}$ algorithm [23] using $0.5$ and $0.7$
for the jet cone radius respectively. These simulated events are then analyzed
using the experimental analyses, providing a method to interpret the relevant
experimental searches in terms of our model.

##  3 Color Octet

Among possible spin-1 resonances we will focus on the gluon partner, a color
octet with mass $m_{\rho}$. The experimental searches of dijets and $t\bar{t}$
by CMS and ATLAS imply important bounds on the parameter space of our scenario
that we derive in this section. Constraints on spin one resonances from flavor
physics are not necessarily negligible, even if MFV is realized, as certain
operators (in particular $(\bar{q}_{L}y_{u}y_{u}^{\dagger}q_{L})^{2}$) are
generated at tree level [10]. Nevertheless, these bounds are more model
dependent (for example they could be avoided in extensions of MFV) and we will
not include them here (see however [11]).

Figure 2: _The heavy color octet is dominantly produced from a quark anti-
quark pair and then decays into any kinematically accessible combination of
light and heavy quarks._

###  3.1 Octet Phenomenology

The color octet can be produced through the Drell-Yan process
$q\bar{q}\to\rho$ of figure 2. Through the coupling with light quarks (2), it
can be copiously produced at LHC if $\sin\phi_{Ru}$ is sufficiently large. No
gluon fusion is possible due to gauge invariance.

The decay of the $\rho$ will play an important role in the phenomenology. The
decay into SM right-handed quarks is equal for all generations while only the
one into $t_{L}$ is relevant for left-handed quarks. If the heavy fermions are
lighter than the color octet the decay into a single heavy and one SM fermion
or two heavy fermions (for $2m_{Q}<m_{\rho}$) will be possible. Since the
couplings to the composite states are large this can affect strongly the
phenomenology.

The decay modes are displayed in figure 2. Analytic formulas for the partial
widths read

$\displaystyle\Gamma(\rho\to q\bar{q})$
$\displaystyle=\frac{\alpha_{s}}{12}m_{\rho}\left[\left(X_{L}^{qq}\right)^{2}+\left(X_{R}^{qq}\right)^{2}\right]$
$\displaystyle\Gamma(\rho\to q\bar{Q},\,Q\bar{q})$
$\displaystyle=\frac{\alpha_{s}}{12}m_{\rho}\left(1-\frac{m_{Q}^{2}}{m_{\rho}^{2}}\right)\left(1-\frac{m_{Q}^{2}}{2m_{\rho}^{2}}-\frac{m_{Q}^{4}}{2m_{\rho}^{4}}\right)\left[\left(X_{L}^{qQ}\right)^{2}+\left(X_{R}^{qQ}\right)^{2}\right]$
$\displaystyle\Gamma(\rho\to Q\bar{Q})$
$\displaystyle=\frac{\alpha_{s}}{12}m_{\rho}\sqrt{1-\frac{4m_{Q}^{2}}{m_{\rho}^{2}}}\left[\left(1-\frac{m_{Q}^{2}}{m_{\rho}^{2}}\right)\left[\left(X_{L}^{QQ}\right)^{2}+\left(X_{R}^{QQ}\right)^{2}\right]+6\frac{m_{Q}^{2}}{m_{\rho}^{2}}X_{L}^{QQ}X_{R}^{QQ}\right].$
(3.1)

in the limit $m_{q}\ll m_{Q}$. $X_{L/R}$’s are the couplings as defined in
equation (2). As shown in figure 3 the width of the color octet changes
drastically when the decay modes to one or two heavy fermions open up. In the
last case the resonance is very broad.

Figure 3: _Typical behavior of the color octet width as a function of the
octet mass, for $g_{\rho}=3$. The width increases significantly as the decay
channels to the quark partners open up, these qualitative features hold
independently of the mixings._

###  3.2 Compositeness Bounds

We start our analysis considering compositeness bounds reported by ATLAS and
CMS, see also [5, 24]. In the large $m_{\rho}$ limit we can integrate out the
color octet and replace it with an effective four fermion operator. Such an
operator produces dijets with an angular distribution different from the QCD
that allows to distinguish it from the background. The experiments in
particular place a bound on the effective operators with light quarks,

$c_{LL}\,(\bar{q}_{L}\gamma^{\mu}q_{L})^{2}+c_{RR}\,(\bar{q}_{R}\gamma^{\mu}q_{R})^{2}+2c_{LR}\,(\bar{q}_{L}\gamma^{\mu}q_{L})(\bar{q}_{R}\gamma_{\mu}q_{R}).$
(3.2)

that can be recast in our scenario. Recent experimental results on the angular
distributions of dijet final states by both ATLAS [25] and CMS [26] imply

$\displaystyle c_{LL,RR}^{(+)}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.10\,\mathrm{TeV}^{-2}$
$\displaystyle\quad\mathrm{ATLAS}$ $\displaystyle
c_{LL,RR}^{(-)}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.06\,\mathrm{TeV}^{-2}$
$\displaystyle\quad\mathrm{CMS}.$ (3.3)

The $\pm$ superscript refers to the sign of the coefficient, and the ATLAS
analysis only considers the case of destructive interference. CMS provides an
exclusion for both signs of the coefficient and the most constraining one is
used. Note that the operators with heavy quarks (such as $u\bar{u}c\bar{c}$)
are expected to be less relevant at LHC since dijet production requires a
quark-antiquark initial state in that case, a process suppressed by the
protons PDFs. Integrating out the heavy color octet one generates the four
fermion operator [5]

$\frac{g_{\rho}^{2}}{6m_{\rho}^{2}}\sin^{4}\phi_{L,Rq}\left(\bar{q}_{L,Rq}\gamma^{\mu}q_{L,Rq}\right)^{2}.$
(3.4)

Using the strongest bound reported by CMS and the coefficient in equation
(3.4) we derive

$\sin^{2}\phi_{Ru}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{\hskip
0.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}\frac{0.6}{g_{\rho}}\left(\frac{m_{\rho}}{\mathrm{TeV}}\right).$
(3.5)

Both constraints by ATLAS and CMS are displayed in figure 4. The compositeness
of right-handed down quarks is slightly less constrained due to the
predominance of up quarks in the proton.

Note that the sign of coefficient obtained integrating out heavy vectors (a
similar conclusion holds for scalars) is fixed and corresponds to the most
constrained sign in the CMS analysis. Hence, it is useful that the experiments
report the bound for both signs of the operator.

###  3.3 Resonance Searches

Figure 4: _Exclusion plot for a color octet with $g_{\rho}=3$. In blue, region
excluded by compositeness bounds. In red (ATLAS) and green (CMS) exclusion
from direct production. The different regions correspond to $95$% confidence
level exclusion for two hypothetical scenarios where the quark partners are
light ($m_{Q}=1$ TeV, solid contour) or heavy ($m_{Q}=2$ TeV, dashed
contour)._

If the resonance is sufficiently light it can be produced in $pp$ collisions
and then decay into jets. The natural search strategy is to look for a bump in
the invariant mass distribution of dijets. We emphasize that this search is of
a very different nature compared to compositeness bounds that rely on the
angular distribution of dijets to distinguish new physics effects from the
enormous QCD background. While the effective operator bound is limited by the
energy of the accelerator, the on-shell production, when kinematically
accessible, is limited by statistics.

We use the most recent analyses of ATLAS [27] and CMS [28, 29] based on $8$
TeV data. The experiments provide a limit on $\sigma(pp\to\rho)\times
BR(\rho\to qq)$ of resonances coupled to light quarks that can be applied to
our scenario. We follow the procedure given by ATLAS for a Gaussian resonance
with a particular width ranging from zero to 15%. Roughly the same strategy is
applied to the CMS search, which provides limits on $\sigma\times
BR\times\epsilon$. Our bounds are conservative as we explicitly take into
account the width of the resonance. Both ATLAS and CMS perform a search for a
relatively narrow resonance through a bump hunter algorithm. If the width of
the the resonance exceeds a certain threshold the bump hunter search is
invalidated and hence we discard the limit whenever the color octet width is
above $15$%. In the $m_{\rho}$ and mixing angle plane, a grid of points is
generated for which $\sigma\times Br$ are computed, and the efficiencies of
the experimental cuts are analyzed. The results are compared to the
experimental limit, that only depends on the resonance mass, and are then
interpolated to form exclusion regions.

The limits for a resonance with $g_{\rho}=3$ are presented in figure 4. The
blue region corresponds to the bound on the effective four fermion operators
discussed in the previous section. The exclusion due to the on-shell
production is given by the red and green regions. This exclusion limit depends
strongly on the fermionic spectrum, because of two reasons. One is the
increase in the width of the resonance possibly invalidating the search. This
becomes particularly relevant when the decay into two heavy partners is
kinematically accessible, see figure 3. Moreover when other channels open up
the signal strength is reduced since only the decay into SM quarks will
generate a bump in the invariant mass distribution of the two leading jets.
For this reason the region with $m_{Q}=1$ TeV is weakly constrained777We do
not include here the partners of left-handed down quarks that would further
increase the width if the decay into two heavy quarks is kinematically
accessible. . Note also that model independently the region of high
compositeness is not constrained because the width is in this case always too
large.

###  3.4 $t\bar{t}$ Searches

Figure 5: _Ratio of $\sigma(pp\to\rho\to t{\bar{t}})$ in MFV models compared
to the experimental benchmark Randall-Sundrum model. The blue region
corresponds to a width greater than $0.2\;m_{\rho}$ where the experimental
bounds are not be applicable. In this comparison the assumption is made that
the decay to heavy fermions is kinematically forbidden._

In anarchic scenarios gluon resonances are strongly coupled to the third
generation and decay mostly into top quarks. For example in Randall-Sundrum
scenarios as considered in reference [30] one finds that the branching of
heavy gluons into top right is almost $100$%. To connect with our
parametrization this model roughly corresponds to $g_{\rho}=5$,
$\sin\phi_{Rq}\approx 0$ for the light quarks and $\sin\phi_{Rt}=1$ for the
top quark. A strong bound on gluon resonances is obtained through searches of
resonances that decay into $t\bar{t}$ pairs. Exclusion limits for this
benchmark point have been reported in the searches from ATLAS [31] and CMS
[32]. In the case of the Randall-Sundrum benchmark the heavy gluon resonance
is excluded below $1.5$ TeV at $95$% confidence level.

In models that realize MFV, or more generally models with composite light
quarks, the situation is different both for the production and decay of the
heavy gluon, and one may obtain an even stronger bound. In these models the
decay into third generation is typically not dominant. This depletion of the
signal is however easily compensated by the increased production cross
section. To get an idea of the bounds in this case, we can estimate
$\sigma(pp\to\rho)\times BR(\rho\to t\bar{t})$ by rescaling the couplings of
the anarchic scenario888For simplicity we assume equal compositeness of up and
down type right-handed quarks. The result is then approximately independent of
PDFs.. The numerical result is presented in figure 5. We see that the cross
section in $t\bar{t}$ is typically larger than in anarchic scenarios. As a
consequence slightly stronger bound will apply.

Figure 6: _Constraints from $t\bar{t}$ searches by ATLAS (red) and CMS
(green). The exclusion limits correspond to $95$% confidence level for two
hypothetical scenarios where the quark partners are light ($m_{Q}=1$ TeV,
solid contour) and where they are heavy ($m_{Q}=2$ TeV, dashed contour)._

One caveat must be considered: similarly to the dijet searches, the
experimental bound is obtained by looking for bumps in the invariant mass
spectrum of $t\bar{t}$ pairs. This procedure depends on the width of the
resonance and becomes inefficient for large widths. In anarchic scenarios the
resonances are relatively broad. In the example of [30] the width is below
$20$% of the mass. The width can be larger in the MFV scenario due to
multiplicity factors and the decay to heavy quark partners, as explained in
the section above. The blue region in figure 5 corresponds to a width greater
than $0.2\;m_{\rho}$. In this region the experimental bound must be
reconsidered. This region is however excluded by compositeness bounds
discussed in the previous section. To compare the limits with the dijet
searches also an exclusion plot in the $(m_{\rho},\sin\phi)$ plane is provided
in figure 6. These exclusion plots have been obtained in a similar fashion as
for the dijet limits from the previous section, including a careful treatment
of the width of the heavy partners possibly invalidating the $t\bar{t}$
search.

We should mention that in extensions of the MFV scenario based on $SU(2)$
rather than $SU(3)$ flavor symmetries the compositeness of the third
generation can be different from the first two [10]. Those scenarios are
attractive phenomenologically as the light generations can be mostly
elementary, avoiding compositeness bounds but with the same virtues as MFV for
what concerns flavor. In this case the phenomenology of heavy gluons will be
similar to anarchic scenarios.

###  3.5 Combined Bounds

Summarizing the direct limits on the color octet depend heavily on the
fermionic spectrum. We differentiate two scenarios, one with light fermionic
partners, $1\,{\rm TeV}<m_{Q}<2\,{\rm TeV}$ and one with heavier partners
$m_{Q}>2\,{\rm TeV}$. In the first we find,

$m_{\rho}>1500\textrm{ GeV}$ (3.6)

at $95$% confidence level. For the heavy scenario we find the constraint

$m_{\rho}>2000\textrm{ GeV}$ (3.7)

that is slightly stronger than the bound in anarchic scenarios. This
constraint holds for all values of the mixings $\sin\phi_{Rq}$ and tighter
bounds on the octet mass are obtained for specific mixings. More stringent
bounds can be inferred from from flavor physics and precision tests but these
rely on extra assumptions on the structure of the theory and do not directly
test the hypothesis of large compositeness of the first and second generation.

##  4 Bounds on Left-Handed Quark Partners

We start our study of fermionic partners with the left-handed sector, focusing
in particular on the $\bf{(2,2)_{2/3}}$ colored fermions. These states can of
course be pair produced through strong interactions, see [33] for a study in
anarchic scenarios. As we will see in the next section the exclusion on top
partners of reference [34] can be translated in MFV scenarios into an
exclusion of these states around 600 GeV. Here we derive the bound obtained
from single production through the electroweak vertices of figure 2. Using the
results in [13] we derive a bound significantly stronger than the one on top
partners.

The single production of left-handed partners is dominated by $t$-channel
exchange of electroweak gauge bosons producing a forward jet ($p_{T}\sim
m_{W}$) and a heavy quark. This decays mainly through weak interactions into
jets and $W$, $Z$ or Higgs999We do not include the decay into Higgs in our
analysis. This was recently studied in [17].. The jet has the same flavor of
the mother particle so that only light quark jets are obtained in the final
state.

Figure 7: _Exclusion of left-handed partners of the up quark by ATLAS[34]. The
red dashed (blue solid) line shows the 95 % C.L. observed upper limit on
$Y_{U}\sin\phi_{Ru}$ obtained from the search of charge 2/3 (5/3) heavy
quarks. The regions above the lines are excluded._

In the anarchic scenarios only third generation quarks can be produced in this
way because the coupling is proportional to the degree of compositeness that
is significant only for the third generation. To produce third generation
partners one needs to scatter a $W$ or $Z$ boson and a top, the latter
originating from the splitting of a gluon. This will be perhaps the most
promising channel for the production of heavy fermions at LHC14 [35] but in
the present run suppression from PDFs and low luminosity is too severe for
this process to be dominant. This is different with composite light quarks
since the heavy partners can be directly produced with the proton
constituents. In this case one can produce the left-handed partners through
diagram b) in figure 8 with access to the valence quarks of the proton. A
related aspect is that the width of the resonance is larger than in anarchic
scenarios.

The search of vector like quarks coupled to the first generation was performed
by the ATLAS collaboration based on [36] (see also [37, 38]) and can be
applied to our scenario. We use the most recent results in [13] obtained with
5 fb-1 luminosity and 8 TeV energy. The search constrains directly the
combination $Y_{U}\sin\phi_{Ru}$. The derived exclusion is shown in figure 7
for the charge 2/3 and exotic charge 5/3 states, the latter being the
strongest. Recall that in MFV scenarios there is a constraint,

$Y_{U}\sin\phi_{Ru}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1$ (4.1)

necessary to reproduce the top mass. From this it follows that the left-handed
partners are often excluded up to 2 TeV and always below 1.5 TeV. This can
only be avoided in extensions of MFV where the third generations can be split
[10].

We emphasize that this is an extremely strong bound that pushes the model into
fine tuning territory. In view of the recent discovery of a $125$ GeV
resonance [39] some of the fermions associated to the top should be light if
the theory shall remain natural. Recent analyses have shown that the lightest
top partner should be typically below $1$ TeV in a natural theory [40]. In MFV
scenarios the mass of the top partners is the same as the one of the light
generations, up to mixing effects. Hence, we can translate the bound on the
light generations into a bound on the top partners.

##  5 Bounds on Right-Handed Quark Partners

The phenomenology of partners of right-handed quarks is entirely different as
they cannot be singly produced by electroweak interactions and they mostly
decay into two or three jets leading to multi-jet final states. The majority
of multi-jet searches at LHC, being motivated by supersymmetry, assumes a
large missing energy typically of the order of few hundreds GeV or more. In
our scenario, the missing energy in the event is a consequence of jet
calibration accumulated by all jets, typically below 50 GeV. Therefore, we do
not expect vanilla supersymmetric searches to play a role in constraining the
parameter space of right-handed compositeness. Analysis of the relevant ATLAS
and CMS searches will be done in the next two sections, separated into single
production (through heavy resonances) and double production (both through QCD
and heavy resonances). Dedicated searches that could improve the experimental
reach will be discussed in the section 6. Before analyzing the different
searches at the LHC we first review production modes and decay channels in
detail.

a)  b)  c)  d)

e)  f)  g)  h)

Figure 8: _Fermion production modes: $a)$ chromomagnetic $s$-channel,
$b)\,$-$\,c)$ single production and $d)\,$-$\,h)$ double production._

##### Production Modes:

The heavy fermions associated to the first generation can be singly produced
in association with a quark via a $t$-channel exchange of the color
octet101010Single production via the chromomagnetic interaction (2.5) will be
subdominant under the assumption that the coefficient is loop suppressed..
Double production of heavy fermions proceeds through $s$-channel gluon or
color octet exchange or a $t$-channel color octet or heavy fermion. Both the
production modes with either an $s$-channel or a $t$-channel color octet
dominate. The various production modes are depicted in figure 8. The relevant
production modes can be summarized in associate single production and double
production. For these modes the production cross section as a function of the
color octet mass and the heavy quark mass is given in figure 9.

Figure 9: _On the left total cross section of associated production of a heavy
quark partner at LHC8 obtained with MadGraph[18] for $g_{\rho}=3$,
$\sin\phi_{Ru}=\sin\phi_{Ru}=0.6$. On the right double production of right-
handed quarks partners ($t_{R}$ excluded) through QCD and heavy gluon
exchange._

##### Decay Channels:

The heavy partners of SM right-handed quarks are singlets of $SO(4)$. Due to
this fact they decay almost entirely into jets. The different decay channels
are displayed in figure 10. The chromomagnetic interaction induces a decay to
a gluon and a quark and generates a width

$\Gamma_{\mathrm{chromo}}(Q\to
qg)=\frac{4}{3}\alpha_{s}\kappa^{2}\sin^{2}\phi_{Ru}\,\frac{1}{m_{Q}^{5}}\left|m_{Q}^{2}-m_{q}^{2}\right|^{3},$
(5.1)

and the same for the down type quarks with $\phi_{Ru}\to\phi_{Rd}$. This decay
is induced at one loop and is typically very small, competing with three body
decay mediated by an off-shell $\rho$. An analytical expression for the three
body decay is quite lengthy and therefore we only give the limiting behavior
(with all light quark masses set to zero $m_{q}=m_{q^{\prime}}=0$ and narrow
width approximation for the $\rho$: $\Gamma_{\rho}\ll m_{\rho}$)

$\Gamma_{\textrm{3-body}}^{\rho}(Q\to
qq^{\prime}\bar{q}^{\prime})=\left\\{\begin{aligned}
&\frac{\alpha_{s}^{2}}{72\,\pi}\left[\left(X_{L}^{qQ}\right)^{2}+\left(X_{R}^{qQ}\right)^{2}\right]\sum_{q^{\prime}}\left[\left(X_{L}^{q^{\prime}q^{\prime}}\right)^{2}+\left(X_{R}^{q^{\prime}q^{\prime}}\right)^{2}\right]&\\\
&\times\left[\frac{6m_{\rho}^{4}-3m_{Q}^{2}m_{\rho}^{2}-m_{Q}^{4}}{m_{Q}m_{\rho}^{2}}+\frac{m_{\rho}^{2}(m_{\rho}^{2}-m_{Q}^{2})}{m_{Q}^{3}}\log\frac{m_{\rho}^{2}-m_{Q}^{2}}{m_{\rho}^{2}}\right]&\mathrm{if}\;m_{Q}<m_{\rho}\\\
&\frac{\alpha_{s}}{6}\left(\frac{m_{Q}^{6}-3m_{Q}^{2}m_{\rho}^{4}+2m_{\rho}^{6}}{m_{Q}^{3}m_{\rho}^{2}}\right)\left[\left(X_{L}^{qQ}\right)^{2}+\left(X_{R}^{qQ}\right)^{2}\right]&\mathrm{if}\;m_{Q}\gg
m_{\rho}\end{aligned}\right.$ (5.2)

The full analytic expression including the width of the heavy color octet has
been used for the analyses. This decay suffers from the octet being off-shell
and phase space suppression. Finally a decay to SM quarks plus a longitudinal
W, Z or Higgs [41] is possible

$\Gamma_{\mathrm{2-body}}^{\mathrm{EW}}(Q\to
qH)\approx\frac{1}{4\pi}\frac{m_{q}^{2}}{v^{2}}\frac{\cos^{2}\phi_{Ru}}{\sin^{2}\phi_{Ru}}m_{Q}.$
(5.3)

Figure 10: _Fermion decay channels: two body decay via the chromomagnetic
operators, three body decay via an off-shell color octet and electroweak two
body decay._

In the MFV scenario the electroweak two body decay is entirely negligible for
the first generation as it is suppressed by the light quark mass over the
vacuum expectation value. It can also be subleading for the second while it is
certainly dominant for the third generation. Note that this conclusion does
not hold in the anarchic scenario, in that case $\sin\phi_{Ru}$ is smaller and
the decay through electroweak interactions dominates producing $W,Z,h+jets$
final states.

To avoid model dependence in what follows we only focus on light generation
partners. For single production the situation effectively reduces to this
while for double production this is a conservative assumption and larger cross
sections can often be obtained due to the flavor multiplicity. Because of this
our conclusions can be considered conservative.

The phenomenology and experimental strategies are strongly dependent on
whether the two body or three body decay dominates, since this will result in
either two or three jet final states. One interesting fact is that for
$m_{Q}<m_{\rho}$ two body and three body decay scale in the same way with the
masses. In figure 11 it is shown in what regions of parameter space the two
body or three body decay dominates. One should however keep in mind that other
contributions could exist which possibly spoil this conclusion. Indeed the
decay widths are in any case extremely small and so even normally subleading
effects could be important.

Figure 11: _Relevance of the heavy fermion three body decay compared to the
two body chromomagnetic decay displayed in the $(m_{Q},\sin\phi_{Rq})$ plane
for $m_{\rho}=2.5$ TeV and $g_{\rho}=3$. The thicker contour line indicates
where the two and three body branching fractions are exactly equal to each
other._

##### Bounds on 3rd generation partners

: Before delving into the direct searches of partners of the light generations
let us consider the indirect bounds that can be derived from the top partners.
Third generation partners behave very differently from light ones.
Compositeness of left-handed top and bottom is sizable so even the right-
handed partners decay through electroweak interactions as in the anarchic
scenario. ATLAS places a strong bound on these states [34]. The precise bound
depends on the branching fraction (in the model under consideration the
singlet $\tilde{T}$ decays in $W\,b$, $Z\,t$ and $h\,t$) but in any case it
typically ranges between 500 and 700 GeV. If $\tilde{T}$ is at the bottom of
the spectrum then the branching fractions can be predicted, $BR(\tilde{T}\to
Wb)\simeq 2BR(\tilde{T}\to ht)\simeq 2BR(\tilde{T}\to Zt)$, corresponding to
an exclusion around 600 GeV.

In models that realize MFV this bound translates into a bound on the mass of
light generation right-handed partners. This can only be avoided in extensions
of MFV that allow to split the third generation [10]. However this can only be
done at the price of making the third generation partners heavier than the
first two, at odds with naturalness. With this in mind we proceed to extract
the direct exclusion limits that as we will see are rather weak.

###  5.1 Single Production

If the heavy quark is singly produced in association with a light quark it
then leads, at parton level, to three or four jet final states depending on
whether chromomagnetic or color octet mediated decay prevails.

####  5.1.1 Chromomagnetic Decay Scenario

The topology of the event is a pair of jets (originating from a quark and a
gluon) with the invariant mass of the heavy partner and a third jet from the
spectator quark. The bump hunter search of resonances decaying into dijets
already considered in section 3 looks for features in the invariant mass of
the two leading jets, where leading refers to $p_{T}$ ordering. Therefore this
search will be effective if the heavy fermion is the father of the two leading
jets, a situation that depends on $m_{Q}$ as we now explain.

Let us first discuss the $p_{T}$ distribution of the recoiling jet due to the
$t$-channel production of the heavy quark. One might think that the typical
$p_{T}$ of the recoiling quark is controlled by the mass of the heavy fermion.
In our region of parameter space however this is not true due to parton
distribution function (PDF) suppression. In fact we find that the average
$p_{T}$ is almost independent on $m_{Q}$ being controlled by the total energy.
This can be seen in figure 12 on the left. The solid line is the average
$p_{T}$ of recoiling quark obtained with $m_{\rho}=2$ TeV. For $E_{CM}=8$ TeV
this is around 500 GeV. Moreover this feature persists for different values of
$m_{\rho}$. In fact since as we have seen $m_{\rho}$ cannot be light,
approximating the interaction with an effective operator is always a good
approximation. Changing $m_{\rho}$ simply rescales the cross section.
Quantitatively a good approximation to the cross section is given by

$\displaystyle\frac{d\sigma}{d|p_{T}|}$ $\displaystyle\propto$
$\displaystyle\frac{1}{S}\,\frac{p_{T}^{2}}{m_{\rho}^{4}}\left(p_{T}+\sqrt{m_{Q}^{2}+p_{T}^{2}}\right)\left(\frac{p_{T}^{2}+m_{Q}^{2}+p_{T}\sqrt{m_{Q}^{2}+p_{T}^{2}}}{S}\right)^{-\alpha},$
(5.4)

where $\alpha\sim 3-6$ is a slowly varying function of $\hat{s}$ determined by
the PDFs. This result is derived in appendix B.

Given the $p_{T}$ of the recoiling quark we can derive the $p_{T}$ ordering of
the jets in an event. Neglecting spin effects, the jets from the heavy quark
will be isotropically distributed in their CM frame with $p_{T}\sim m_{Q}/2$.
Boosting to the lab frame one finds $p_{T}^{1,2}\sim\left|p_{T}^{spectator}\pm
m_{Q}/2\right|$. Therefore for large $m_{Q}$ we expect the two jets to be
leading and the opposite for small $m_{Q}$.

This is confirmed by our simulation. In figure 12 on the right we plot the
probability of the spectator quark to be the first, second or third jet in
$p_{T}$. As we increase $m_{Q}$ the spectator quark tends to have the lowest
$p_{T}$. Therefore in this region the standard dijet search will capture the
signal. However, with the production cross sections given in figure 9 no bound
is obtained in our model if we perform a recast. Moreover recent updates of
dijet searches require a cut on the invariant mass of the jet pair to exceed
$1$ TeV so this search is unlikely to produce a bound even in the future. The
situation for $m_{Q}<1.5$ TeV is even less promising as in the case the
spectator quark often gives rise to the first or second jet so the dijet
search will not be efficient. In this case a different ordering of jets should
be considered. Indeed requiring at least three jets in the final state and
looking for bumps in the invariant mass of the second and third jet seems a
promising strategy to reduce the background.

Figure 12: _Plots of the $p_{T}$ spectrum of the three jet final state. On the
left the average $p_{T}$ of the three different jets in the event as a
function of $m_{Q}$ are displayed. On the right the fraction of events with
the spectator jet being the $i^{\mathrm{th}}$ $p_{T}$ jet for LHC8._

####  5.1.2 Three Body Decay Scenario

In this case we have a four jet final state with three jets reconstructing the
mass of the heavy quark. The searches for four jets by ATLAS [42] and CMS [43]
with no missing energy cuts are optimized for pair production of a heavy
resonance, both decaying into two jets. Although these searches share the same
final state, they have a low efficiency to pick up our signal because of the
different topology.

Figure 13: _Plots of the $p_{T}$ spectrum of the four jet final state. On the
left the average $p_{T}$ of the four different jets in the event as a function
of $m_{Q}$ are displayed. On the right the fraction of events with the
spectator jet being the $i^{\mathrm{th}}$ $p_{T}$ jet for LHC8._

Obviously to capture the signal one should study the invariant mass of
trijets. As previous case an important element is the ordering of jets. This
is shown in figure 13 on the right. Qualitatively this is similar to the two
body decay. As intuitive however the recoiling jet is more likely to be the
leading jet. This happens 50% of the times for a fermion with 1 TeV mass.
Therefore in this case a dedicated search pairing the second, third and fourth
jet is expected to be very effective.

###  5.2 Double Production

We have in this case 4 or 6 jet final states at parton level. 5 jets could
also be obtained in certain regions of parameters where 2 body and 3 body
decay are comparable but we will neglect this possibility. In the 4 jets case
two pairs of jets form the same invariant mass equal to the heavy quark mass.
In the 6 jets case two sets of three jets each form the invariant mass of the
heavy quark.

####  5.2.1 Chromomagnetic Decay Scenario

CMS and ATLAS analyzed double dijets final states, where they look for pair
production of a heavy resonance decaying into two jets in [42, 43]. ATLAS only
considers a mass region between $150$ and $350$ GeV, whereas CMS considers a
region from $320$ to $1200$ GeV. Since our interest is mainly in the mass
region up to around $1$ TeV for the heavy quark partners, only the CMS
analysis is considered. This search is expected to be effective for relatively
low partner masses since for high masses the three body decay is favoured, see
figure 11.

The CMS analysis investigates events with at least four jets with $|\eta|<2.5$
and $p_{T}>150$ GeV and then combines the four highest-$p_{T}$ jets into dijet
combinations with $\Delta R_{jj}>0.7$. Then the dijet pair combination with
minimal $\Delta m/m_{\mathrm{avg}}$ is selected, where $\Delta
m=|m_{\mathrm{jj}}^{(1)}-m_{\mathrm{jj}}^{(2)}|$ and
$m_{\mathrm{avg}}=\tfrac{1}{2}(m_{\mathrm{jj}}^{(1)}+m_{\mathrm{jj}}^{(2)})$,
with a maximum $\Delta m/m_{\mathrm{avg}}<0.15$ to suppresses the QCD
background. Then a last requirement is

$\Delta=\sum_{i=1,2}p_{T,i}-m_{\mathrm{avg}}>25\,\mathrm{GeV},$ (5.5)

ensuring a smoothly falling paired dijet mass spectrum. In the absence of any
observed resonances CMS then provides the limits on the folded
$\sigma\times\mathrm{Br}\times\epsilon$ as a function of the resonance mass,
to which our scenario will be compared to obtain limits.

Figure 14: _Constraints from the CMS double dijet search[43] for the double
production combined with chromomagnetic decay scenario for $g_{\rho}=3$ and
$m_{\rho}=2.5$ TeV. Displayed is the limit cross section over the new physics
cross section in the $(m_{Q},\sin\phi_{R})$ plane. The red region is already
excluded at $95$% CL by the current searches, whereas the gray contours give
an indication for the needed increase in sensitivity to exclude further
regions. The peculiar shape, the “island” in particular, is accounted for by
upward fluctuations in the data around $600$ and $800$ GeV._

Our scenario is almost completely equivalent to the coloron model considered
in the CMS analysis only differing in the production modes. Therefore we
expect similar final state topology and the selection criteria to be next to
optimal. To compare with the coloron exclusion limits we generate the dijet
resonances using our FeynRules-MadGraph-Pythia-Delphes chain (also in our case
the width of the resonance is negligible compared to the experimental
resolution). For a set of points in the $(m_{Q},\sin\phi_{R})$ plane we
analyze the efficiencies and obtain a value for
$\sigma\times\mathrm{Br}\times\epsilon$ to be compared to the CMS limit. We
focus on the excluded region and the possible exclusion potential. Therefore
we plot the limiting cross section $\sigma_{\mathrm{lim}}$ divided by the new
physics cross section $\sigma_{\mathrm{NP}}$ of our model which removes the
dependence on branching ratios and acceptances. This gives a good indication
of the increase in sensitivity required to exclude certain regions of
parameter space. The resulting contour plot is given in figure 14. Any region
with $\sigma_{\mathrm{lim}}/\sigma_{\mathrm{NP}}\leq 1$ is excluded by the
current searches, this is the red contour with the thick edge. We conclude
that heavy partners with masses between $320$ and $500$ GeV are excluded,
provided that the chromomagnetic decay dominates.

####  5.2.2 Three Body Decay Scenario

Figure 15: _Distribution of triplets mass $M_{jjj}$ versus the triplet scalar
$p_{T}$ of all 20 triplets in each event for LHC7. For quark partner masses of
$400$ GeV (left) and $1000$ GeV (right), the selection criterion from equation
(5.6) is given by the orange dashed line. These plots give an indication of
the leakage of combinatorial background into the signal region._

In this case we have a six jet final state, where two combinations of three
jets originate from identical mother particles. The search closest to this
topology we are aware of is by CMS [44] where they look for the invariant mass
of three jets in events with at least six jets with $p_{T}>70$ GeV and
$|\eta|<3.0$. Furthermore the total scalar sum of $p_{T}$ is required to be
higher than $900$ GeV for each event. The search aims to capture pair produced
trijet resonances and is interpreted in terms of RPV gluinos decaying into
three jets. The six highest $p_{T}$ jets are combined into all 20 three jet
combinations and in order to reduce both combinatorial and QCD background the
requirement

$M_{jjj}<\sum_{i=1}^{3}p_{T}^{i}-\Delta\,,\quad\left(\Delta=160\;\textrm{GeV}\right)$
(5.6)

for each triplet is imposed. The acceptance is then defined as all the events
with at least one triplet of jets passing this cut. The experiment provides
the $95$% CL limits on $\sigma\times\textrm{Br}$ as a function of the
resonance mass in the range from $280$ GeV up to $1000$ GeV.

Even though our topology shares the same final state as the RPV gluinos
studied in the CMS analysis the kinematics are quite different. One of the
main differences is that the quark partners are produced mostly by the color
octet rather than the gluon. Moreover the gluino decay is modeled by a four
fermion effective interaction, whereas the heavy quark decay proceeds through
an off-shell color octet. Hence, for the analysis to be applicable roughly the
same acceptances for both scenarios should be obtained. Especially the
selection criterion in equation (5.6) should have the same effect on the
combinatorial background and the trijets coming from the decay of the heavy
partner. This is relevant since after this selection criterion a resonance
search in the triplet invariant mass spectrum is performed. In order to
analyze the effect of the cuts, in particular (5.6), our signal has been
simulated and the distribution of events in the trijet mass versus triplet
scalar $p_{T}$ has been plotted in figure 15.

Figure 16: _Acceptance of triplet events for the selection criterion in
equation ( 5.6). The black line shows the acceptance for the triplets
originating from one of the heavy quarks, whereas the gray dashed line shows
the acceptance for the other triplets forming combinatorial background._

The CMS analysis is optimized for the gluino scenario choosing $\Delta=160$
GeV and for triplets originating from the gluino the probability for passing
this selection criterion ranges between $2$% and $13$% depending on the gluino
mass. In figure 16 the acceptance for our signal is plotted, from which one
can see that the acceptances are generally higher, however also the
combinatorial background grows. At high masses we see that the combinatorial
background starts to dominate over the signal, hence the selection as in
equation (5.6) is not efficient for our topology. Therefore no significant
bound can be extracted. However, the search for this final state is
potentially interesting and could be optimized with minor effort for the
topology of right-handed compositeness.

##  6 Dedicated Searches

In the previous section we recasted the existing multi-jet searches of ATLAS
and CMS to set limits on the heavy quark partners. Those limits are rather
weak for the right-handed partners, see for example figure 14, since the
searches are not optimized to the most distinctive topology of the model, the
single production of the heavy quark $Q$ in association with a light jet. In
this section we are going to propose searches that exploit the characteristic
behavior of this production mechanism, namely $pp\to Qq$. This leads to 2+1
(3+1) jets for the two body (three body) decays of the heavy quark partners.

The topology is characterised by at least three hard jets, where some of the
jets reconstruct the mass of the fermion $Q$. The main background arise from
QCD jets and is dominated by the diagrams given in figure 17.

Figure 17: _Typical QCD background events leading to three high- $p_{T}$
jets._

When looking at the dijet searches, a cut on the hardness of the third and
fourth jet aids in reducing the background, but the effect is not drastic. For
example, in 8 TeV QCD samples where the two leading jets have $p_{T}>$ 150
GeV, asking for a third one with $p_{T}>$ ($25$, $70$, $100$ and $150$) GeV
has an efficiency of ($40$, $9$, $4$ and $1$)%. A larger reduction of the
background can be achieved with more sophisticated cuts, for which we provide
details later.

Other studies with some overlap with the single production topology are the
CMS and ATLAS studies of double dijets. In the previous section, we applied
one of these searches to double production $pp\to QQ$ where $Q\to jj$,
concluding that the reach is rather weak. In the single production case the
sensitivity is even lower, as the topology does not resemble the double dijet.
For example, the efficiencies of the signal $pp\to qQ$ where $Q\to 3j$ to this
search for $m_{Q}=$ $1$, $2$ and $3$ TeV is in the range of $1$-$4$%. As in
the case of dijet bump searches, a dedicated search should be carried out. In
the following we discuss the kinematic variables which show a better
discrimination power of signal versus background for this topology. Two
benchmark scenarios are considered which correspond to $m_{\rho}=2500$ GeV,
$g_{\rho}=3$ and $\sin\phi_{R_{u,d}}=0.6$, where the heavy fermion partner
mass equals $m_{Q}=600$ or $1200$ GeV. The relevant kinematic variables are
now discussed.

##### The $H_{T}$ variable:

We use the usual definition of the $H_{T}$ variable

$H_{T}=\sum_{i=\textrm{jets}}p_{T,i}\,.$ (6.1)

In figure 18 (left) one can see that signal has a larger $H_{T}$ distribution
than the background, and it increases with $m_{Q}$. Note that in this plot and
all the following ones, basic cuts on the jets are $p_{T,j}>$ 70 GeV and
$|\eta_{j}|<$ 2.5.

Figure 18: _(Left) The $H_{T}$ distribution for QCD events with
$n_{j}\geqslant 3$ (black line), 2+1 signals with $m_{Q}=$ 600 GeV (purple
distribution) and $m_{Q}=$ 1200 GeV (magenta distribution) in LHC8. (Right)
The $\Delta\phi$ distributions between the two subleading QCD jets (black) and
the dijets from the decaying $Q$ particle, with $m_{Q}=$ 600 (1200) GeV in
blue (red). Both figures are generated at parton level, and in the right
figure truth information is used to identify the jets from the heavy quark. _

##### Angular distribution:

Since the heavy resonance is produced with little boost, one would expect a
symmetric angular distribution among the jets coming from the decaying
particle. In the $Q\to 2j$ case, the daughter jets tend to be produced with
$\Delta\phi=\pi$, whereas in the $Q\to 3j$ case one would expect a
distribution near $\Delta\phi=2\pi/3$.

In the QCD case, though, jets would not have such a preference. In three jet
QCD events, like the ones in figure 17, one would expect a rather symmetric
distribution of jets, more so as we increase the cut on $p_{T}$. This is seen
in figure 19, where as we increase the $p_{T}$ threshold, the distribution is
more and more peaked towards $2\pi/3$, hence the Mercedes configurations. With
the same cut on all jets, this configuration minimizes the overall centre of
mass energy of the three jet system, $M\sim 3p_{T,min}$. For the configuration
where the subleading jets are close and back-to-back with the leading jet, the
minimal mass equals $M\sim 4p_{T,min}$. Here the two subleading jets have
$p_{T}=p_{T,min}$ and the leading jet $p_{T}$ is $2p_{T,min}$ to balance
momentum.

Figure 19: _The angular distribution between the two subleading jets in QCD
events, as a function of the $p_{T}$ cuts on all jets for LHC8. The left plot
is the partonic result and the plot on the right is reco-level._

This is shown in figure 18 (right), where we see that the jets from the $Q$
decay tend to be symmetric, more so as the mass increases and the $Q$ has
smaller boosts. We find similar discriminating features when looking at the
$Q\to 3j$ case, now with the peak at $\Delta\phi=2\pi/3$ for the three jets
from $Q$. Note, though, that figure 18 (right) has been done using parton
level truth events (where the information of the mother particle was known).
When showering, detector effects and combinatorial background is added, the
discriminating power of $\Delta\phi_{jj}$ is greatly reduced.

Figure 20: _(Left)  The invariant mass distribution of the two subleading jets
for QCD events (black) and signal events in the 2+1 topology with $m_{Q}=$ 600
(1200) GeV in blue (red) at LHC8. (Right)  The distribution $\Delta_{24}$ for
QCD events (black) and signal events in the 3+1 topology with $m_{Q}=$ 500
(1000) GeV in blue (red) at LHC8. Both plots are generated at the parton
level._

##### Mass bump reconstruction:

An obvious characteristic of the signal is the presence of a mass bump, if the
right combination of jets was chosen. In the previous section, we showed in
figures 12 and 13 (right panels), that the leading jet tends to be the
spectator jet for low $m_{Q}\lesssim$ 1 TeV, more so for the 3+1 than the 2+1
topology. We then choose in each event the two (three) subleading jets and
form an invariant mass. In figure 20 (left), we plot the invariant mass of the
subleading jets for the 2+1 topology. The QCD distribution is peaked at low
values, whereas there is a peak in the signal at high $m_{jj}$. The peak is
more pronounced (lower combinatorial background) for low $m_{Q}$, but also the
leakage of QCD events in the distribution is larger.

##### Gaps:

Figure 21: _ $H_{T}$ versus $p_{T}^{1}-p_{T}^{3}$ for the 3+1 topology at LHC8
for $m_{Q}=$ 500 GeV (left) and 1 TeV (right). The pink-scatter plot
corresponds to the QCD background. The events are at detector level for LHC8._

One could also exploit the gaps among the jets momenta and invariant masses.
This is specially interesting in the case of 3+1 topologies, on which we focus
in the following. In figure 20 (right) we plot the variable
$\Delta_{24}=\sum_{i=2,3,4}p_{T,i}-m_{234}$, a gap between the $p_{T}$’s and
invariant mass. In the QCD background, the invariant mass and the $p_{T}$ sum
are close to each other, and we expect to be peaked at low values. In the
signal events, the invariant mass tends to be smaller than the scalar sum of
$p_{T}$’s.

The gap between the jets in the event can also be used to discriminate between
signal and background. In figure 21, we plot $H_{T}=\sum_{i=1}^{4}p_{T}^{i}$
versus $\Delta_{13}=p_{T}^{1}-p_{T}^{3}$, the $p_{T}$ difference between the
first and third jet, for $m_{Q}$= 500 GeV (left) and 1 TeV (right). The signal
is characterized by a larger $H_{T}$ and also by a larger hierarchy between
the first and third jet. The differences between QCD and signal are weaker at
low $m_{Q}$, and a harder cut on both variables should be done to keep QCD
under control. Although the two variables are clearly correlated, a modified
ABCD method could be used here to estimate the amount of QCD background
leaking into the signal region.

##### Obtaining $S/B=1$:

We would like to quantify the effect of the cuts on signal and QCD background
using the variables described above. In table 1 we describe the cut-flow of
those variables for the 2+1 case. The 3+1 case behaves very similarly in terms
of signal efficiencies. Note that the QCD background of $n_{j}\gtrsim 3$, 4
jets with $p_{T}>$ 70 GeV and $|\eta_{j}|<$ 2.5 at LHC8 is 3 $\times 10^{4}$
pb and 3 $\times 10^{3}$ pb, respectively. The signal cross section can be
read in figure 9 for specific values of $g_{\rho}$, $\sin\phi_{R_{u,d}}$, and
it typically varies between 1 to 10 pb for $m_{\rho}\lesssim$ 2.5 TeV. To
achieve $S/B\sim 1$, one would need to have a relative suppression of
efficiencies of $10^{2}-10^{4}$. In the table 1, one can see how this can be
achieved by implementing cuts on the variables described above.

Cut-flow |  $m_{Q}=600$ GeV |  $m_{Q}=1200$ GeV
---|---|---
signal | QCD | signal | QCD
$p_{T}$ leading jet $>450$ GeV | 0.51 | 0.0067 | 0.90 | 0.0067
$H_{T}>m_{Q}$ | 0.51 | 0.0067 | 0.80 | 0.0015
$|m_{jj}-m_{Q}|<$ (30, 50) GeV | 0.15 | 0.00037 | 0.11 | 2.5$\times 10^{-5}$
$\Delta\phi_{jj}>1.5$ | 0.045 | 9.9 $\times 10^{-5}$ | 0.060 | $2.1\times 10^{-7}$
Table 1: _Cut-flow demonstrating the effect on signal and background of
cutting on the variables presented in the text. The numbers correspond to the
efficiency to specified set of cumulative cuts. Here $jj$ is the combination
of the two subleading jets. For the background, the final numbers represent
the cut-flow with either $m_{Q}=600$ GeV or $m_{Q}=1200$ GeV._

To produce this cut-flow, we took two benchmark masses, $m_{Q}$= 600 and 1200
GeV, and the 2+1 signature. We chose the 2+1 topology, as it suffers from the
largest background, still interesting $S/B$ can be achieved using these cuts.
Note that we have not truly optimized the cuts to a specific signal, and the
intention of the table is to show that a background reduction in the required
range is possible. Note also that we have not made use of the gap variables in
this cut-flow, which could improve the sensitivity of the search.

Figure 22: _QCD background (black lines) and signal (solid colors) when the
cuts on the table 1 are applied, except the one on $m_{jj}$. See text for
details._

Note that the cutflow table and figure are produced using detector level
events showered with Pythia with MLM matching [45] and simulated with Delphes
[21] with anti-$k_{T}$ jets of $R=0.7$.

In figure 22 we illustrate this cut-flow with a normalized background for 10
fb-1 of luminosity and a signal of $m_{Q}=$ 600 GeV and $\sigma=$ 5 pb. In
this figure, the three black lines correspond to QCD 3 jets with 1.) $p_{T}>$
70 GeV and $|\eta_{j}|<$ 2.5, 2.) $\oplus p_{T}^{leading}>$ 450 GeV and
$H_{T}>m_{Q}$ and 3.) $\oplus\Delta\phi_{jj}>$ 1.5. Similarly, the solid
histograms correspond to the same cuts, applied now in the signal.

At 14 TeV, the production cross section for QCD with $n_{j}\geqslant$ 3 and
$p_{T}>$ 70 GeV, $|\eta_{j}|<$ 2.5, increases by a factor three respect to the
8 TeV run. For the $m_{Q}=$ 600 GeV cut-flow described in table 1, the
efficiencies to pass the cuts increase by a factor $\cal O$(2) from 14 TeV
respect to the 8 TeV case. The $m_{Q}$=1200 GeV is more dramatic, with an
efficiency increase for the QCD case of $\gtrsim\cal O$(10). As we already
mentioned, the cut-flow presented here should be seen as indicative of the
strategy to follow, and it is clear one would need to re-optimize when moving
from the 8 TeV to the 14 TeV run.

##  7 Conclusions

In this paper we have investigated the experimental signatures and bounds of
partially composite Higgs models where right-handed quarks are strongly
composite. This scenario, strongly motivated by flavor physics, was until
recently very weakly constrained experimentally. The situation is rapidly
changing with the LHC results that are progressively carving out significant
regions of parameter space. We presented the most relevant bounds that can be
extracted from the latest LHC data. It is worth to emphasize that the
experimental strategies to test these models at LHC are of a different nature
from the ones of the more studied anarchic scenarios or supersymmetry. In
particular they typically produce jet final states without leptons or missing
energy. For this reason existing analysis are in some cases not optimal and
could be improved with dedicated searches.

One of the most important constraints on right-handed quark compositeness
arises from dijet searches. These place a direct bound on the spin one gluon
resonances. In some regions of parameters these states are excluded up to 3
TeV but the result is strongly sensitive on the fermionic spectrum.

We also derive bounds on the masses of the lightest fermionic partners. These
are particularly relevant given their role for the naturalness of the theory.
In the light of the 125 GeV Higgs discovery some fermions should be lighter
than 1 TeV for a small tuning of the theory [46]. One interesting experimental
feature is that single production of the new fermions dominates the bounds
unlike the case of anarchic scenarios where at present double production
produces the strongest constraints.

We derive an extremely strong bound on the left partners that are excluded up
to 2 TeV in theories that realize MFV. This is obtained from single
electroweak production of partners of the up quark studied by the ATLAS
collaboration. Right-handed quark partners can be singly produced through the
gluon resonances with smaller cross sections and different final states. The
direct bound is much weaker in this case. Overall our study shows that models
that realize MFV are at least as tuned as the anarchic scenarios. This can be
avoided abandoning MFV in favour of theories based $SU(2)$ flavor symmetry
[10] where the light generations can be more elementary than the top, see also
[11] for a related discussion.

We conclude by noting that our scenario motivates more general experimental
searches than the ones presently published. This is already possible with the
existing data with minor modifications of experimental analyses. In particular
our multi-jet signals could be more efficiently captured with a different
ordering of jets. Dijet studies should also be extended to trijets. Let us
also mention that the multi-jet signals originate from quarks, whereas the
background is dominated by high-multiplicity QCD gluons. Therefore, jet
tagging techniques, such as in reference [47], would be valuable to reject the
background. We hope that these efforts will be pursued by the experiments.

Acknowledgments: We thank Georges Azuelos, Andrea Banfi, Gilad Perez, Riccardo
Rattazzi, and Gavin Salam for useful discussions. MR would like to thank the
Galileo Galilei Institute (GGI) in Florence for hospitality while part of this
work was carried out. The work of MR is supported in part by the MIUR-FIRB
grant RBFR12H1MW. The work of VS is supported by the Science Technology and
Facilities Council (STFC) under grant number ST/J000477/1. The work of MdV and
AW was supported in part by the German Science Foundation (DFG) under the
Collaborative Research Center (SFB) 676.

##  Appendix A Right-Handed Composite Model

In this appendix we describe the effective Lagrangian used in our simulations.
This is a simple extension of [41]. We will focus on the quark sector. The
composite states are multiplets of the global symmetry $SU(3)\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$. As described in section 2 we take
the quark partners in the following representations

$\displaystyle\tilde{U}={\bf(1,1)_{\frac{2}{3}}}\qquad$
$\displaystyle\tilde{D}={\bf(1,1)_{-\frac{1}{3}}}$ (A.1) $\displaystyle
L_{U}={\bf(2,2)_{\frac{2}{3}}}=\begin{pmatrix}U&U_{\frac{5}{3}}\\\
D&U_{\frac{2}{3}}\end{pmatrix}\qquad$ $\displaystyle
L_{D}={\bf(2,2)_{-\frac{1}{3}}}=\begin{pmatrix}D_{-\frac{1}{3}}&U\\\
D_{-\frac{4}{3}}&D\end{pmatrix},$

all fundamentals of $SU(3)$. Focusing on the first generation we consider the
following Lagrangian for the composite fermions

$\displaystyle\mathcal{L}_{\mathrm{composite}}=$
$\displaystyle-\frac{1}{4}\rho_{\mu\nu}^{i2}+\frac{m_{\rho}^{i2}}{2}\rho_{\mu}^{i2}+\mathrm{Tr}\left[|D_{\mu}H|^{2}\right]-V(H)$
$\displaystyle+$
$\displaystyle\mathrm{Tr}\left[\bar{L}_{U}(iD\\!\\!\\!\\!{\bf/}-m_{L_{U}})L_{U}\right]+\bar{\tilde{U}}(iD\\!\\!\\!\\!{\bf/}-m_{\tilde{U}})\tilde{U}$
$\displaystyle+$ $\displaystyle
Y_{U}\mathrm{Tr}\left[\bar{L}_{U}\mathcal{H}\right]_{L}U_{R}+\mathrm{h.c.}$
$\displaystyle+$ $\displaystyle\left\\{U\to D\right\\}.$ (A.2)

We only include the composite Yukawas that are relevant for the generation of
the SM flavor structure. Among the spin-1 resonances we consider a massive
octet of $SU(3)$ and assume that interacts as a gauge field. The elementary
Lagrangian is just QCD with massless quarks

$\mathcal{L}_{\mathrm{elementary}}=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\bar{q}_{L}iD\\!\\!\\!{\bf/}q_{L}+\bar{u}_{R}iD\\!\\!\\!{\bf/}u_{R}+\bar{d}_{R}iD\\!\\!\\!{\bf/}d_{R}.$
(A.3)

SM quarks mix with the fermions of equal quantum numbers

$\mathcal{L}_{\mathrm{mixing}}=\Delta_{Lu}\bar{q}_{L}Q_{Ru}+\Delta_{Ru}\bar{q}_{L}Q_{Rd}\,+\Delta_{Ru}\bar{\tilde{U}}_{L}u_{R}+\Delta_{Rd}\,\bar{\tilde{D}}_{L}d_{R}+\mathrm{h.c.}$
(A.4)

where the $Q_{Ru}$ and $Q_{Rd}$ are the doublets contained in $L_{U}$ and
$L_{D}$ respectively. We will assume $\lambda_{Ld}\ll\lambda_{Lu}$. Similarly
the gauging of SM symmetries introduces a linear mixing between the SM and the
composite spin-1 resonances.

Diagonalizing the elementary-composite mixings the Lagrangian in the mass
basis reads

$\displaystyle\mathcal{L}_{\textrm{gauge}}=$
$\displaystyle-\frac{1}{4}G_{\mu\nu}G^{\mu\nu}+\frac{1}{2}\left(D_{\mu}\rho_{\nu}D_{\nu}\rho_{\mu}-D_{\mu}\rho_{\nu}D_{\mu}\rho_{\nu}\right)+\frac{M^{2}}{2\cos^{2}\theta}\rho_{\mu}\rho^{\mu}$
$\displaystyle+\frac{ig_{s}}{2}G_{\mu\nu}\left[\rho_{\mu},\rho_{\nu}\right]+2ig_{s}\cot
2\theta
D_{\mu}\rho_{\nu}\left[\rho_{\mu},\rho_{\nu}\right]+\frac{g_{s}^{2}}{4}\left(\frac{\sin^{4}\theta}{\cos^{2}\theta}+\frac{\cos^{4}\theta}{\sin^{2}\theta}\right)\left[\rho_{\mu},\rho_{\nu}\right]^{2},$
(A.5)

for the bosons and

$\displaystyle\mathcal{L}_{\textrm{fermion}}=$
$\displaystyle\,\bar{q}_{L}iD\\!\\!\\!{\bf/}q_{L}+\bar{Q}_{u}\left(iD\\!\\!\\!{\bf/}-m_{Q_{u}}\right)Q_{u}$
$\displaystyle\,+g_{s}\bar{q}_{L}\left(\sin^{2}\phi_{Lu}\cot\theta-\cos^{2}\phi_{Lu}\tan\theta\right)\rho_{\mu}\gamma^{\mu}q_{L}$
$\displaystyle\,+g_{s}\bar{q}_{L}\left(\frac{\sin\phi_{Lu}\cos\phi_{Lu}}{\sin\theta\cos\theta}\right)\rho_{\mu}\gamma^{\mu}Q_{Lu}+\;\mathrm{h.c.}$
$\displaystyle\,+g_{s}\bar{Q}_{Lu}\left(\cos^{2}\phi_{Lu}\cot\theta-\sin^{2}\phi_{Lu}\tan\theta\right)\rho_{\mu}\gamma^{\mu}Q_{Lu}$
$\displaystyle\,+\left\\{(q_{L},Q_{u})\to(q_{L},Q_{d})\;,(u_{R},U_{L})\;,(d_{R},D_{L})\right\\}.$
(A.6)

for the fermions. In the expressions above $\tan\theta=g_{\mathrm{el}}$,
$g_{\rho}$ $\tan\phi=\Delta/m$ and $g_{s}=g_{\mathrm{el}}\cos\theta$ is the
QCD coupling. This is the final form of the Lagrangian which has been
implemented in FeynRules [19] to study the LHC phenomenology.

The Higgs vacuum expectation value introduces the following mixings in the up
sector

$\mathcal{L}_{\mathrm{int}}^{L}=-\frac{Y_{U}v}{\sqrt{2}}\sin\phi_{Ru}\bar{u}_{R}\left[U+U_{2/3}\right]+\mathrm{h.c.}$
(A.7)

Diagonalizing these terms generates the electroweak interactions of equation
(2) relevant for single production of left-handed partners [35, 36].

Figure 23: One-loop new physics contributions to the chromomagnetic operator
in partially composite models.

Finally in the effective Lagrangian of the strong sector we include the
dimension 5 operator

$\mathcal{L}_{\mathrm{chromo}}=\frac{g_{s}\,\kappa_{0}}{m_{Q}}\,\bar{Q}_{L}\sigma_{\mu\nu}T^{a}q_{R}G_{\mu\nu}^{a}+\mathrm{h.c.}$
(A.8)

The chromomagnetic interaction is generated by loops of the strong sector
fields, see figure 23. The naive estimate is
$\kappa_{0}\sim\frac{g_{\rho}^{2}}{16\pi^{2}}$. We will however be interested
in the region $m_{Q}<m_{\rho}$. In this case the loops generate

$\kappa_{0}\sim\frac{g_{\rho}^{2}}{16\pi^{2}}\frac{m_{Q}^{2}}{m_{\rho}^{2}}.$
(A.9)

Dressing the operator with the mixing the interaction (2.5) is obtained. In
our numerical evaluation we will use the estimate

$\kappa=\frac{N_{c}}{32\pi^{2}}\frac{m_{Q}^{2}}{m_{\rho}^{2}}X_{R}^{QQ}X_{R}^{Qq}.$
(A.10)

The suppression is relevant phenomenologically because it renders two body and
three body decay widths comparable.

##  Appendix B Approximate $p_{T}$ Distribution

In the following, we derive the approximate cross section given in equation
(5.4) for the process $u\,u\to u\,U$. We only include the relevant left-handed
couplings, a good approximation for $g_{\rho}\gg g$ and derive the $t$-channel
expression111111There is a an additional $u$-channel contribution but since
$\Delta y\to 0$ minimizes $\hat{s}$, see (B.3), we typically have $t\approx
u$. for a given $p_{T}$ of the spectator quark. The amplitude squared summed
over initial and final states is proportional to

$\left|\overline{\mathcal{M}_{fi}}\right|^{2}\propto\frac{\hat{s}(\hat{s}-m_{Q}^{2})}{(\hat{t}-m_{\rho}^{2})^{2}}.$
(B.1)

This leads to a dependence on the Mandelstam variables in the cross section
given by

$\frac{d^{3}\sigma}{dy_{3}dy_{4}d|p_{T}|}\propto(f(x_{1})\,f(x_{2}))\,\frac{p_{T}}{S}\,\frac{\hat{s}-m_{Q}^{2}}{(\hat{t}-m_{\rho}^{2})^{2}},$
(B.2)

where $y_{3,4}$ are the rapidities of the daughter particles, $S$ is the
c.o.m. energy and $x_{1,2}$ are the usual partonic momentum fractions carried
by the initial partons. Recalling that

$\displaystyle\hat{s}$
$\displaystyle=m_{Q}^{2}+2\,p_{T}^{2}+2\,p_{T}\sqrt{m_{Q}^{2}+p_{T}^{2}}\cosh\Delta
y$ (B.3) $\displaystyle\hat{t}$
$\displaystyle=-p_{T}\left(p_{T}+\sqrt{m_{Q}^{2}+p_{T}^{2}}\,\exp{(-\Delta
y)}\right)$ (B.4) $\displaystyle\hat{u}$
$\displaystyle=-p_{T}\left(p_{T}+\sqrt{m_{Q}^{2}+p_{T}^{2}}\,\exp{(\Delta
y)}\right),$ (B.5)

we find

$\frac{d^{3}\sigma}{dy_{3}dy_{4}d|p_{T}|}\propto(f(x_{1})\,f(x_{2}))\,\frac{p_{T}^{2}\left(\cosh(\Delta
y)\sqrt{m_{Q}^{2}+p_{T}^{2}}+p_{T}\right)}{36\pi\left(m_{\rho}^{2}+p_{T}\left(e^{-\Delta
y}\sqrt{m_{Q}^{2}+p_{T}^{2}}+p_{T}\right)\right)^{2}}.$ (B.6)

Figure 24: _Comparison between approximate and exact $p_{T}$ distributions for
various values of the fermionic partner and octet masses
$(m_{Q},m_{\rho})=(100,2500);\;(250,2500);\;(500,2500)$ with $g_{\rho}=6$ and
$\sin\phi_{R_{u,d}}=0.6$. The red line uses the full $t$-channel propagator,
whereas the green line assumes a contact interaction. We use constant $\alpha$
in the plots. Naturally, for very large $p_{T}$ the $\rho$ dynamics is
resolved and taking $\alpha$ constant ceases to be a good approximation._

We can now derive a simple approximation for the $p_{T}$ distribution. Since
$\Delta y\to 0$ minimizes $\hat{s}$, we can set $\Delta y=0$ in the following.
The parton luminosities are steeply falling functions of $\hat{s}/S$,
therefore we can approximate the remaining integration by the threshold value
of the parton luminosities which we model as a steeply falling polynomial
$(\hat{s}/S)^{-\alpha}$. We extracted $\alpha$ from the MSTW2008 pdfs [48].
For heavy color octets ($m_{\rho}\gg m_{Q},p_{T}$), we can also ignore the
octet propagator. Combining these approximations we find

$\frac{d\sigma}{d|p_{T}|}\propto\frac{1}{S}\,\frac{p_{T}^{2}}{m_{\rho}^{4}}\left(p_{T}+\sqrt{m_{Q}^{2}+p_{T}^{2}}\right)\left(\frac{p_{T}^{2}+m_{Q}^{2}+p_{T}\sqrt{m_{Q}^{2}+p_{T}^{2}}}{S}\right)^{-\alpha}$
(B.7)

where $\alpha\sim 3-6$ is a slowly varying function of $\hat{s}$ determined by
the parton luminosities. The maximum of the $p_{T}$ distribution is therefore
approximately at $(p_{T})^{\rm
max}\approx\frac{1.5}{\sqrt{4\alpha-6}}m_{Q}\approx\frac{1}{2}\,m_{Q}$. In
figure 24 we compare the above approximation with a parton level simulation
using the full model implementation, validating the result and the
approximations.

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