aainstitutetext: Institute for Particle Physics Phenomenology, Department of
Physics, Durham University, DH1 3LE, Durham, United Kingdombbinstitutetext:
Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerlandccinstitutetext: PRISMA Cluster of Excellence and Mainz Institute
for Theoretical Physics, Johannes Gutenberg-Universität Mainz, 55099 Mainz,
Germanyddinstitutetext: Faculty of Science, University of Sarajevo, Zmaja od
Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovinaeeinstitutetext:
Theoretical Physics Department, CERN, Geneva, Switzerlandffinstitutetext:
INFN, Sezione di Trieste, Italy

# Maximal Flavour Violation:
a Cabibbo mechanism for leptoquarks

Luca Di Luzio b Javier Fuentes-Martin c, d Admir Greljo e, f Marco Nardecchia
c Sophie Renner<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a mechanism that allows for sizeable flavour violation in quark-
lepton currents, while suppressing flavour changing neutral currents in quark-
quark and lepton-lepton sectors. The mechanism is applied to the recently
proposed “$4321$” renormalizable model, which can accommodate the current
experimental anomalies in $B$-meson decays, both in charged and neutral
currents, while remaining consistent with all other indirect flavour and
electroweak precision measurements and direct searches at high-$p_{T}$. To
support this claim, we present an exhaustive phenomenological survey of this
fully calculable UV complete model and highlight the rich complementarity
between indirect and direct searches.

††arxiv: 1808.00942

##  1 Introduction

In the recent years the central question of flavour physics beyond the
Standard Model (SM) has been the following: _“How is it possible to reconcile
TeV-scale new physics (NP) (as suggested e.g. by naturalness) with the absence
of indirect signals in flavour changing neutral currents (FCNC)?”._ One
possible answer was given by the principle of Minimal Flavour Violation
DAmbrosio:2002vsn , which allowed for exciting NP at ATLAS and CMS while
predicting less room for serendipity at LHCb. Somewhat unexpectedly, we are
faced with the fact that experimental data seem to rather suggest the opposite
situation. In fact, a coherent pattern of SM deviations in semileptonic
$B$-decays, which goes under the widely accepted name of “flavour anomalies",
keeps building up since $2012$ Lees:2012xj ; Lees:2013uzd ; Aaij:2013qta ;
Aaij:2014ora ; Aaij:2015yra ; Aaij:2015oid ; Huschle:2015rga ; Sato:2016svk ;
Hirose:2016wfn ; Hirose:2017dxl ; Aaij:2017vbb ; Aaij:2017deq . Were these
anomalies due to NP, they would certainly imply a shift of paradigm in flavour
physics.

A unified explanation of the whole set of anomalous data minimally requires:
$i)$ a NP contribution in $b\to s\mu\mu$ neutral currents that interferes
destructively with the SM and $ii)$ a NP contribution in charged currents that
enhances the decay rates of $b\to c\tau\nu$ transitions. Despite many models
being proposed so far for the combined explanation of the anomalies (see
Bhattacharya:2014wla ; Alonso:2015sja ; Greljo:2015mma ; Calibbi:2015kma ;
Bauer:2015knc ; Fajfer:2015ycq ; Barbieri:2015yvd ; Das:2016vkr ;
Boucenna:2016wpr ; Boucenna:2016qad ; Becirevic:2016yqi ; Hiller:2016kry ;
Bhattacharya:2016mcc ; Barbieri:2016las ; Becirevic:2016oho ; Bordone:2017anc
; Megias:2017ove ; Crivellin:2017zlb ; Cai:2017wry ; Altmannshofer:2017poe ;
Dorsner:2017ufx ; Buttazzo:2017ixm ; Assad:2017iib ; DiLuzio:2017vat ;
Calibbi:2017qbu ; Bordone:2017bld ; Choudhury:2017ijp ; Barbieri:2017tuq ;
Sannino:2017utc ; Blanke:2018sro ; Greljo:2018tuh ; Marzocca:2018wcf ;
Asadi:2018wea ; Greljo:2018ogz ; Robinson:2018gza ; Azatov:2018knx ;
Bordone:2018nbg ; Becirevic:2018afm ; Kumar:2018kmr ; Trifinopoulos:2018rna ;
Azatov:2018kzb  for an incomplete list), it is fair to say that the majority
of these works suffer from various issues: neglect of key observables (both at
low energy and high-$p_{T}$), missing UV completion, breakdown of the
perturbative expansion, unnatural and tuned values of the parameters, etc. The
difficulties in constructing a viable and coherent NP interpretation of the
flavour anomalies (both in charged and neutral currents) are due to the
simultaneous presence of the following aspects of the phenomenological
situation:

  1. 1.

the NP contribution in $b\to c\tau\nu$ needs to be very large, since it must
compete with a SM tree-level process;

  2. 2.

there is an absence of NP signals in direct searches at the LHC;

  3. 3.

there are very severe constraints from flavour observables in pure hadronic
channels, most notably in $\Delta F=2$ transitions;

  4. 4.

there are very severe constraints from flavour observables in pure leptonic
channels, most notably in processes violating lepton universality and lepton
flavour.

Since the first point clearly contrasts with the remaining ones, finding a
coherent NP framework to explain all these facts remains a non-trivial
challenge. However, the points above are also suggesting in a (qualitative)
way their own solutions. Indeed a viable NP scenario should:

  1. 1.

contain a leptoquark with large flavour violating couplings in order to
trigger the anomalous semileptonic decays in charged currents;

  2. 2.

only introduce new states that are _heavy_ enough to escape direct detection;

  3. 3.

have a protecting flavour symmetry in the purely quark sector, such as a
$U(2)$ acting on the first two families of quarks;

  4. 4.

have a protecting flavour symmetry in the purely lepton sector, such as
$U(1)_{e}\times U(1)_{\mu}\times U(1)_{\tau}$.

Does a model with such properties exist? In this paper we are going to present
a phenomenological attempt to answer this question, by exploring a specific
limit of the “4321 model” introduced in Ref. DiLuzio:2017vat . Here, 4321
stands for the gauge structure of the model, which is invariant under the
local group $SU(4)\times SU(3)^{\prime}\times SU(2)_{L}\times U(1)^{\prime}$.
The symmetry breaking down to the SM delivers a TeV-scale vector leptoquark,
$U_{\mu}\sim(\mathbf{3},\mathbf{1},1/3)$, with the most favourable quantum
numbers in order to mediate the flavour anomalies, as inferred from recent
simplified-model analyses Buttazzo:2017ixm ; Kumar:2018kmr .

While the aspects that we are going to discuss will be exemplified in the
context of the 4321 model (the detailed phenomenological analysis of this
model is in fact one of the main goals of this paper), we believe that the
mechanism presented here should be a welcome ingredient for any extension
aiming at a consistent description of the whole set of anomalies. This
ingredient is nothing but a generalisation of the well-known Cabibbo mixing
Cabibbo:1963yz  to the leptoquark sector. The up- and down-quark sectors in
the SM, when taken in isolation, preserve their own $U(1)^{3}$ family
symmetry. It is only the simultaneous presence of up and down Yukawa matrices
that provides a flavour violating misalignment of the size of the Cabibbo
angle. Our proposal follows in close analogy: quarks and leptons in isolation
preserve their own original symmetries, while flavour violation is a product
of the _collective breaking_ coming from the two sectors. The misalignment
between the second and third family of quark and lepton doublets,
$\theta_{LQ}$, is the generalisation of the Cabibbo angle, $\theta_{C}$. As a
consequence, tree-level neutral currents are (practically) absent and all the
relevant flavour violating interactions only involve the exchange of the
leptoquark. The individual (assumed) larger symmetries in the quark and lepton
sectors guarantee enough flavour protection from low-energy indirect probes,
while a sizeable $\theta_{LQ}$ allows for large effects in the desired $b\to
c\tau\nu$ transitions at tree level. Crucially, a large 3-2 leptoquark
transition allows the scale of NP to be raised and relaxes in turn the bounds
from LHC direct searches. This approach differs from those scenarios in which
the NP is aligned along the third generation and the 3-2 transitions are
obtained via an $\mathcal{O}(V_{cb})$ rotation. In the latter case, the
flavour suppression in the NP amplitude has to be compensated either by a
lower value of the NP scale or by large couplings arising from non-
perturbative dynamics. In both cases, one is faced with very serious
challenges both from precisely measured $Z$-pole observables and $\tau$ decays
Feruglio:2016gvd ; Feruglio:2017rjo  and direct searches (see e.g.
Faroughy:2016osc ; Greljo:2017vvb ).

The connection between low- and high-energy phenomenology in the 4321 model
goes even further. In fact, the large flavour breaking between second and
third generation in the leptoquark sector is responsible for $\Delta F=2$
quark transitions at the one-loop level, for which a GIM-like mechanism is at
work: a sufficient suppression of $B_{s}$ and $D$ mixing is guaranteed by the
lightness of the lepton partners present in the radiative amplitudes. We hence
obtain upper bounds on heavy lepton partners from indirect searches and lower
bounds from direct searches at the LHC. Remarkably, in a large part of the
parameter space this mass window is very narrow: low-energy probes are
suggesting a clear target for direct searches at high-$p_{T}$. The role of the
heavy lepton partners in the 4321 model recall in a sense the charm prediction
from kaon meson mixing in the SM Glashow:1970gm ; Gaillard:1974hs .

The paper is structured as follows: in Sec. 2 we introduce the main elements
of the 4321 model and in Sec. 3 we discuss the leptoquark Cabibbo mechanism
making use of symmetry arguments and analogies with the SM. In Sec. 4 we
collect the main observables relevant for the low-energy phenomenology,
including the flavour anomalies and the relevant constraints from indirect
searches. In Sec. 5 we present the status of direct searches, and show that a
large breaking in the 3-2 sector is needed to lift the NP scale in order to
escape direct detection. In Sec. 6 we summarize our main predictions and
conclude. A thorough discussion of several theoretical aspects of the 4321
model is deferred to App. A.

##  2 The $4321$ model

In this section we summarise the main features of the 4321 model presented in
DiLuzio:2017vat  (see also Diaz:2017lit ). Further details are provided in
App. A. The goal of the model’s construction is to generate a coupling of the
vector leptoquark $U\sim(\mathbf{3},\mathbf{1},2/3)$ mainly to _left-handed_
SM fermions. This allows $i)$ to match with the model-independent fits to
$B$-anomalies Buttazzo:2017ixm ; Kumar:2018kmr  and $ii)$ to tame strong
constraints from chirality-enhanced meson decays into lepton pairs (for an
updated analysis see Ref. Smirnov:2018ske ). To this end we consider the gauge
group $\mathcal{G}_{\rm 4321}\equiv SU(4)\times SU(3)^{\prime}\times
SU(2)_{L}\times U(1)^{\prime}$, which extends the SM group $\mathcal{G}_{\rm
321}\equiv SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ by means of an extra
$SU(4)$ factor. The embedding of colour and hypercharge into $\mathcal{G}_{\rm
4321}$ is defined as $SU(3)_{c}=\left(SU(3)_{4}\times
SU(3)^{\prime}\right)_{\rm diag}$ and $Y=\sqrt{2/3}\,T^{15}+Y^{\prime}$, with
$SU(3)_{4}\subset SU(4)$ and $T^{15}$ being one of the generators of
$SU(4)$.111For a complete list of $SU(4)$ generators see App. A.10. Apart from
the SM gauge fields, the gauge boson spectrum comprises three new massive
vectors belonging to $\mathcal{G}_{\rm 4321}/\mathcal{G}_{\rm 321}$ and
transforming under $\mathcal{G}_{\rm 321}$ as
$U\sim(\mathbf{3},\mathbf{1},2/3)$, $g^{\prime}\sim(\mathbf{8},\mathbf{1},0)$
and $Z^{\prime}\sim(\mathbf{1},\mathbf{1},0)$. Their definition in terms of
the $\mathcal{G}_{\rm 4321}$ gauge fields, as well as their masses, are given
in App. A.4.

An important point to be stressed is that the three massive vectors are
connected by gauge symmetry breaking and it is not possible to parametrically
decouple the $g^{\prime}$ (hereafter called “coloron”) and the $Z^{\prime}$
from the leptoquark mass scale. In App. A.5 we show that this feature persists
also in non-minimal scalar sectors responsible for $\mathcal{G}_{\rm 4321}$
breaking. Moreover, the peculiar embedding of the SM into $\mathcal{G}_{\rm
4321}$ allows for suppressed coupling of the $Z^{\prime}$ and coloron to light
quarks (cf. Sec. A.7). That is not the case in more standard Pati Salam
Pati:1974yy  embeddings such as in Calibbi:2017qbu , where the $Z^{\prime}$
has unsuppressed $\mathcal{O}(g_{s})$ couplings to valence quarks.

The matter content of the model is summarised in Table 1, where we have
emphasised with a grey background the states added on top of the SM-like
fields. The new gauge bosons receive a TeV-scale mass induced by the vacuum
expectation value (VEV) of three scalar multiplets:
$\Omega_{1}\sim\left(\mathbf{\overline{4}},\mathbf{1},\mathbf{1},-1/2\right)$,
$\Omega_{3}\sim\left(\mathbf{\overline{4}},\mathbf{3},\mathbf{1},1/6\right)$
and $\Omega_{15}\sim\left(\mathbf{15},\mathbf{1},\mathbf{1},0\right)$,
responsible for the breaking of $\mathcal{G}_{\rm 4321}\to\mathcal{G}_{\rm
321}$. While only $\Omega_{3}$ would suffice for the breaking, the role of the
other fields is of phenomenological nature as discussed below. By means of a
suitable scalar potential (analysed in App. A.1) it is possible to achieve a
VEV configuration ensuring the proper $\mathcal{G}_{\rm
4321}\to\mathcal{G}_{\rm 321}$ breaking. After removing the linear
combinations corresponding to the would-be Goldstone bosons (GB), the massive
scalar spectrum featuring the radial modes is detailed in App. A.2. The final
breaking of $\mathcal{G}_{\rm 321}$ is obtained via the Higgs doublet field
transforming as $H\sim(\mathbf{1},\mathbf{1},\mathbf{2},1/2)$.

Field | $SU(4)$ | $SU(3)^{\prime}$ | $SU(2)_{L}$ | $U(1)^{\prime}$ | $U(1)_{B^{\prime}}$ | $U(1)_{L^{\prime}}$
---|---|---|---|---|---|---
$q^{\prime i}_{L}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{2}$ | $1/6$ | $1/3$ | 0
$u^{\prime i}_{R}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | $2/3$ | $1/3$ | 0
$d^{\prime i}_{R}$ | $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{1}$ | $-1/3$ | $1/3$ | 0
$\ell^{\prime i}_{L}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $-1/2$ | 0 | $1$
$e^{\prime i}_{R}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-1$ | 0 | $1$
$\Psi^{i}_{L}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 0 | $1/4$ | $1/4$
$\Psi^{i}_{R}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | 0 | $1/4$ | $1/4$
$H$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | 1/2 | 0 | 0
$\Omega_{1}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $-1/2$ | $-1/4$ | $3/4$
$\Omega_{3}$ | $\mathbf{\overline{4}}$ | $\mathbf{3}$ | $\mathbf{1}$ | $1/6$ | $1/12$ | $-1/4$
$\Omega_{15}$ | $\mathbf{15}$ | $\mathbf{1}$ | $\mathbf{1}$ | 0 | $0$ | $0$
Table 1: Field content of the 4321 model. The index $i=1,2,3$ runs over
generations, while $U(1)_{B^{\prime}}$ and $U(1)_{L^{\prime}}$ are accidental
global symmetries (see text for further clarifications). Particles added to
the SM matter content are shown on a grey background.

The would-be SM fermion fields, denoted with a prime, are singlets of $SU(4)$
and are charged under the $SU(3)^{\prime}\times SU(2)_{L}\times U(1)^{\prime}$
subgroup with SM-like charges. Like in the SM, they come in three copies of
flavour. Being $SU(4)$ singlets, they do not couple to the vector leptoquark
directly. In order to induce the required leptoquark interactions to SM
fermions, we introduce three vector-like heavy fermions that mix with the SM-
like fermions once $\Omega_{1,3}$ acquire a VEV (cf. also Fig. 1). The vector-
like fermions transform under $\mathcal{G}_{\rm 4321}$ as
$\Psi_{L,R}=(Q^{\prime}_{L,R},L^{\prime}_{L,R})^{T}\sim(\mathbf{4},\mathbf{1},\mathbf{2},0)$,
with $Q^{\prime}_{L,R}\sim(\mathbf{3},\mathbf{2},1/6)$ and
$L^{\prime}_{L,R}\sim(\mathbf{1},\mathbf{2},-1/2)$ when decomposed under
$\mathcal{G}_{\rm 321}$. The vector-like masses of $Q^{\prime}$ and
$L^{\prime}$ are split by the VEV of $\Omega_{15}$. The mixing among the left-
handed SM-like and vector-like fermions is described by the Yukawa Lagrangian
$\mathcal{L}_{Y}=\mathcal{L}_{\rm SM-like}+\mathcal{L}_{\rm mix}$, with

$\displaystyle\mathcal{L}_{\rm SM-like}$
$\displaystyle=-\overline{q}^{\prime}_{L}\,Y_{d}\,Hd^{\prime}_{R}-\overline{q}^{\prime}_{L}\,Y_{u}\,\tilde{H}u^{\prime}_{R}-\overline{\ell}^{\prime}_{L}\,Y_{e}\,He^{\prime}_{R}+\text{h.c.}\,,$
(1) $\displaystyle\mathcal{L}_{\rm mix}$
$\displaystyle=-\overline{q}^{\prime}_{L}\,\lambda_{q}\,\Omega_{3}^{T}\Psi_{R}-\overline{\ell}^{\prime}_{L}\,\lambda_{\ell}\,\Omega_{1}^{T}\Psi_{R}-\overline{\Psi}_{L}\left(M+\lambda_{15}\,\Omega_{15}\right)\Psi_{R}+\text{h.c.}\,.$
(2)

Here, $\tilde{H}=i\sigma_{2}H^{*}$ and $Y_{u,d,e}$, $\lambda_{q,\ell,15}$, $M$
are $3\times 3$ flavour matrices. The flavour structure of the 4321 model will
be discussed in detail in Sec. 3.

The full Lagrangian (including also the scalar potential in Eq. (A.1)) is
invariant under the accidental global symmetries $U(1)_{B^{\prime}}$ and
$U(1)_{L^{\prime}}$, whose action on the matter fields is displayed in the
last two columns of Table 1.222Note that these global symmetries are anomalous
under $SU(2)_{L}\times U(1)^{\prime}$. The VEVs of $\Omega_{3}$ and
$\Omega_{1}$ break spontaneously both the gauge and the global symmetries,
leaving unbroken two new global $U(1)$s:
$B=B^{\prime}+\frac{1}{\sqrt{6}}\,T^{15}$ and
$L=L^{\prime}-\sqrt{\frac{3}{2}}\,T^{15}$, which for the SM eigenstates
correspond respectively to ordinary baryon and lepton number. These symmetries
protect proton stability, make neutrinos massless and prevent the appearance
of massless state related to the spontaneous breaking of $U(1)_{B^{\prime}}$
and $U(1)_{L^{\prime}}$. Non-zero neutrino masses can be achieved by
introducing an explicit breaking of $U(1)_{L^{\prime}}$, e.g. via a $d=5$
effective operator $\ell^{\prime}\ell^{\prime}HH/\Lambda_{\not{L}}$, where the
effective scale of lepton number violation, $\Lambda_{\not{L}}$, is well above
the TeV scale. In contrast, recent proposals which address the anomalies based
on a non-minimal Pati-Salam extension with gauged $B-L$ broken at the TeV,
such as e.g. Calibbi:2017qbu ; Bordone:2018nbg , generically predict too large
neutrino masses. The latter either require a strong fine-tuning in the Yukawa
structure or a very specific (untuned) realisation of the neutrino mass matrix
by the inverse seesaw mechanism Perez:2013osa ; Greljo:2018tuh .

##  3 Cabibbo mechanism for leptoquarks

Our goal is to introduce the flavour structure required by the anomalies in
the quark-lepton transitions, while simultaneously suppressing the most
dangerous quark-quark and lepton-lepton flavour violating operators.333For a
partially related discussion in the context of the neutral current anomalies,
see Guadagnoli:2018ojc . This step can be neatly understood in terms of the
global symmetries of the Yukawa Lagrangian.

Let us first consider the $\mathcal{L}_{\rm mix}\to 0$ limit. The surviving
term in Eq. (1) corresponds to the SM Yukawa Lagrangian. Exploiting the
$U(3)^{5}$ invariance of the kinetic term of the SM-like fields we choose,
without loss of generality, a basis where $Y_{d}=\hat{Y}_{d}$,
$Y_{u}=V^{\dagger}\,\hat{Y}_{u}$ and $Y_{e}=\hat{Y}_{e}$ (a hat denotes a
diagonal matrix with positive eigenvalues and $V$ is the CKM matrix). For
later convenience, we recall some well-known features of the SM quark Yukawa
sector. In the $Y_{u}\to 0$ limit, the term
$\overline{q}^{\prime}_{L}\hat{Y}_{d}\tilde{H}d_{R}$ leaves invariant the
subgroup $U(1)_{d}\times U(1)_{s}\times U(1)_{b}$, thus implying the absence
of flavour violation in the down sector. Similarly, for $Y_{d}\to 0$ we are
left with $\overline{q}^{\prime}_{L}V^{\dagger}\,\hat{Y}_{u}\tilde{H}u_{R}$ in
the up sector. Reabsorbing $V$ into $q^{\prime}$ bears no physical effects and
the subgroup $U(1)_{u}\times U(1)_{c}\times U(1)_{t}$ is left unbroken. If
both $Y_{u}$ and $Y_{d}$ are present, the two $U(1)^{3}$ are not independent
any more due to the $SU(2)_{L}$ gauge symmetry that forces the transformations
of the left-handed down and up fields to be the same. The intersection of the
two subgroups yields444Here $U(1)_{d+u}$ stands for the simultaneous
transformation $d\to e^{i\theta}d$ and $u\to e^{i\theta}u$, where
$e^{i\theta}$ is an element of $U(1)_{d+u}$. The generalisation to non-abelian
factors, which is employed later on, follows in analogy.

$\displaystyle(U(1)_{d}\times U(1)_{s}\times U(1)_{b})\cap(U(1)_{u}\times
U(1)_{c}\times U(1)_{t})\supseteq$ $\displaystyle U(1)_{d+u}\times
U(1)_{s+c}\times U(1)_{b+t}\xrightarrow[]{V\neq\mathbb{1}}U(1)_{B}\,,$ (3)

where the last step of breaking is due to the CKM mixing and $U(1)_{B}$ is the
baryon number. The consequences of this _collective breaking_ are: $i)$ No
tree-level FCNC are generated. These are forbidden by the two $U(1)^{3}$
symmetries in isolation, either in the up or in the down sector. $ii)$ Flavour
changing charged currents are generated by the misalignment between the up and
down sectors, which is parametrised by the CKM matrix $V$. In the unitary
gauge, the physical effects of flavour violation are fully encoded in the
coupling of the $W$ boson to the up and down quark fields.

Let us consider now the pattern of global symmetries when $\mathcal{L}_{\rm
mix}\neq 0$. The role of the scalar representations $\Omega_{i}$ in
$\mathcal{L}_{\rm mix}$ is the following:

  * •

$\langle\Omega_{3}\rangle$ mixes the would-be SM state $q^{\prime}_{L}$ with
$Q^{\prime}_{L}\subset\Psi_{L}$. In this way the SM quark doublet enters into
the $SU(4)$ representation $\Psi_{L}$ and feels the leptoquark interaction.

  * •

$\langle\Omega_{1}\rangle$ mixes the would-be SM state $\ell^{\prime}_{L}$
with $L^{\prime}_{L}\subset\Psi_{L}$. In this way the SM lepton doublet enters
into the $SU(4)$ representation $\Psi_{L}$ and feels the leptoquark
interaction.

  * •

$\langle\Omega_{15}\rangle$ splits the bare masses of quark and lepton
partners. We can hence effectively trade $M$ and
$\lambda_{15}\langle\Omega_{15}\rangle$ for $M_{Q}$ and $M_{L}$.

Without loss of generality, we use the $U(3)^{7}$ symmetry of the fermionic
kinetic term to pick up the following basis:

$\displaystyle\mathcal{L}_{\rm SM-like}$
$\displaystyle=-\overline{q}^{\prime}_{L}V^{\dagger}\hat{Y}_{u}u^{\prime}_{R}\,\tilde{H}-\overline{q}^{\prime}_{L}\hat{Y}_{d}d^{\prime}_{R}\,H-\overline{\ell}^{\prime}_{L}\hat{Y}_{e}e^{\prime}_{R}\,H+\text{h.c.}\,,$
(4) $\displaystyle\mathcal{L}_{\rm mix}$
$\displaystyle=-\overline{q}^{\prime}_{L}\lambda_{q}\Psi_{R}\,\Omega_{3}-\overline{\ell}^{\prime}_{L}\lambda_{\ell}\Psi_{R}\,\Omega_{1}-\overline{\Psi}_{L}(\hat{M}+\lambda_{15}\Omega_{15})\Psi_{R}+\text{h.c.}\,,$
(5)

where $\lambda_{q}$, $\lambda_{\ell}$ and $\lambda_{15}$ are matrices in
flavour space. If the latter were generic, we would expect large flavour
violating effects both in quark and lepton processes. We are going to argue
that, _assuming_ the following flavour structure:

$\displaystyle\begin{aligned}
\lambda_{q}&=\hat{\lambda}_{q}\equiv\text{diag}\left(\lambda^{q}_{12},\lambda^{q}_{12},\lambda^{q}_{3}\right)\,,\\\
\lambda_{\ell}&=\hat{\lambda}_{\ell}\,W^{\dagger}\equiv\text{diag}\left(\lambda^{\ell}_{1},\lambda^{\ell}_{2},\lambda^{\ell}_{3}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{LQ}&-\sin\theta_{LQ}\\\
0&\sin\theta_{LQ}&\cos\theta_{LQ}\end{array}\right)\,,\\\
\lambda_{15}&\propto\hat{M}\propto\mathbb{1}\,,\end{aligned}$ (6)

provides a good starting point to comply with flavour constraints. Later on we
will comment about the plausibility of our assumptions, but for the moment let
us inspect the physical consequences of Eq. (6).

Mimicking the pure SM discussion, we examine the surviving global symmetries
of $\mathcal{L}_{\rm mix}$ in either of the limits $\lambda_{\ell}\to 0$ or
$\lambda_{q}\to 0$. In the former case $\mathcal{L}_{\rm mix}$ is invariant
under the action of the global symmetry group $\mathcal{G}_{Q}\equiv
U(2)_{q^{\prime}+\Psi}\times U(1)_{q^{\prime}_{3}+\Psi_{3}}$, with the non-
abelian factor acting on the first and second generation. Basically, we are
promoting the approximate $U(2)_{q^{\prime}}$ of the SM (emerging in the limit
where only $(Y_{u,d})_{33}\neq 0$) to be also a symmetry of the NP. This
guarantees in turn:

  * •

the absence of tree-level FCNC for down quarks (note that $Y_{d}$ and
$\lambda_{q}$ are diagonal in the same basis). Such a _down alignment_
mechanism was already introduced in Ref. DiLuzio:2017vat .

  * •

a strong suppression of tree-level FCNC for up quarks. This suppression is
guaranteed by the underlying $U(2)$ symmetry and the physical effects are
proportional to the small breaking induced by the SM-like Yukawa $Y_{u}$ via
the CKM. We will show in Sec. 4 that this protection is crucial in order to
pass the bounds from $D$-$\overline{D}$ mixing.

We continue with the discussion of the lepton sector when $\lambda_{q}\to 0$.
In this limit $\mathcal{L}_{\rm mix}$ has a $U(1)^{3}$ symmetry which is just
the generalisation of the accidental symmetries of the SM in the lepton
sector. To show this let us reabsorb $W$ in a redefinition of the field
$\Psi$, via $\tilde{\Psi}\equiv W^{\dagger}\Psi$. With such a redefinition
$\mathcal{L}_{\rm mix}$ reads

$\mathcal{L}_{\rm mix}(\lambda_{q}\to
0)=-\overline{\ell}^{\prime}_{L}\hat{\lambda}_{\ell}\tilde{\Psi}_{R}\,\Omega_{1}-\overline{\tilde{\Psi}}_{L}(\hat{M}+\hat{\lambda}_{15}\Omega_{15})\tilde{\Psi}_{R}+\text{h.c.}\,.$
(7)

Since everything is diagonal, the global symmetry is identified as
$\mathcal{G}_{L}=U(1)_{\ell^{\prime}_{1}+\tilde{\Psi}_{1}}\times
U(1)_{\ell^{\prime}_{2}+\tilde{\Psi}_{2}}\times
U(1)_{\ell^{\prime}_{3}+\tilde{\Psi}_{3}}$. The limit $\lambda_{q}\to 0$ thus
implies:

  * •

the absence of tree-level FCNC for (charged) leptons. Note indeed that there
exists a basis where $Y_{e}$ and $\lambda_{\ell}$ are simultaneously diagonal.

  * •

that the $W$ matrix is unphysical.

Let us consider now the case where both $\lambda_{q}$ and $\lambda_{\ell}$ are
simultaneously present in $\mathcal{L}_{\rm mix}$. The symmetries in the quark
($\mathcal{G}_{Q}$) and lepton ($\mathcal{G}_{L}$) sectors are not independent
due to the presence of the underlying $SU(4)$ gauge symmetry which locks
together the transformations of the $Q$ and $L$ fields. The intersection of
the two groups yields

$\mathcal{G}_{Q}\cap\mathcal{G}_{L}\supseteq
U(1)_{q^{\prime}_{1}+\ell^{\prime}_{1}+\Psi_{1}}\times
U(1)_{q^{\prime}_{2}+\ell^{\prime}_{2}+\Psi_{2}}\times
U(1)_{q^{\prime}_{3}+\ell^{\prime}_{3}+\Psi_{3}}\xrightarrow[]{W\neq\mathbb{1}}U(1)_{q^{\prime}_{1}+\ell^{\prime}_{1}+\Psi_{1}}\times
U(1)_{q^{\prime}+\ell^{\prime}+\Psi}\,,$ (8)

where the last step of breaking is a consequence of the specific structure of
the $W$ matrix in Eq. (4) featuring only 3-2 mixing. The unbroken groups
correspond to the quantum number of the first family of quarks and leptons,
$U(1)_{q^{\prime}_{1}+\ell^{\prime}_{1}+\Psi_{1}}$, and to the total fermion
number $U(1)_{q^{\prime}+\ell^{\prime}+\Psi}$, namely the simultaneous re-
phasing of all the fermion fields in $\mathcal{L}_{\rm mix}$. The latter is
nothing but $3B^{\prime}+L^{\prime}$ (cf. Table 1), which in combination with
with $T^{15}$ yields ordinary baryon and lepton number after
$\mathcal{G}_{4321}$ breaking.

To simplify our analysis even more we can set the coupling
$\lambda^{\ell}_{1}$ to zero, thus implying a further enhancement of the
symmetry: $U(1)_{q^{\prime}_{1}+\ell^{\prime}_{1}+\Psi_{1}}\to
U(1)_{q^{\prime}_{1}+\Psi_{1}}\times U(1)_{\ell^{\prime}_{1}+\Psi_{1}}$ which
forbids flavour violating transitions involving either down quark or electron
fields. On the other hand, we can still have a large mixing between the second
and third family of quarks and leptons, whose misalignment is parametrised by
the matrix $W$. Such an effect appears in the coupling of $U_{\mu}$ with
quarks and leptons, in complete analogy with the flavour violation involving
the $W^{\pm}$ boson and the quark doublet in the SM. Working e.g. in the basis
$\Psi_{L}=(Q^{\prime}_{L},L^{\prime}_{L})^{T}=(Q_{L},WL_{L})^{T}$, the
interaction of $U_{\mu}$ with quarks and leptons can be readily extracted from
the covariant derivative:

$i\overline{\Psi}_{L}\gamma^{\mu}D_{\mu}\Psi_{L}\supset\frac{g_{4}}{\sqrt{2}}U_{\mu}\,\overline{Q}_{L}\gamma^{\mu}\left(\begin{array}[]{ccc}1&0&0\\\
0&\cos\theta_{LQ}&\sin\theta_{LQ}\\\
0&-\sin\theta_{LQ}&\cos\theta_{LQ}\end{array}\right)L_{L}\,.$ (9)

In the same way that the Cabibbo angle $\theta_{C}$ represents the
misalignment between the up and down quarks of the first two families within
an $SU(2)_{L}$ doublet, here $\theta_{LQ}$ represent the misalignment between
the quark and lepton fields of the second and third generation within an
$SU(4)$ quadruplet. Note, however, that the states $Q_{L}$ and $L_{L}$ have to
be projected along the light SM mass eigenstates, since the breaking induced
by $\langle\Omega_{3}\rangle$ and $\langle\Omega_{1}\rangle$ redirects part of
the SM quark and lepton doublets into $\Psi_{L}$. The net effect is given by
(cf. App. A.7)

$\frac{g_{4}}{\sqrt{2}}\beta_{ij}U_{\mu}\,\overline{q}^{i}_{L}\gamma^{\mu}\ell^{j}_{L}\,,$
(10)

where $\beta$ is a $3\times 3$ matrix describing the flavour structure of the
leptoquark interactions with the light SM mass eigenstates:

$\beta=\textrm{diag}(s_{q_{12}},s_{q_{12}},s_{q_{3}})\,W\,\textrm{diag}(0,s_{\ell_{2}},s_{\ell_{3}})=\left(\begin{array}[]{ccc}0&0&0\\\
0&c_{\theta_{LQ}}s_{q_{12}}s_{\ell_{2}}&s_{\theta_{LQ}}s_{q_{12}}s_{\ell_{3}}\\\
0&-s_{\theta_{LQ}}s_{q_{3}}s_{\ell_{2}}&c_{\theta_{LQ}}s_{q_{3}}s_{\ell_{3}}\end{array}\right)\,.$
(11)

The definitions of the mixing angles in terms of the fundamental parameters of
the Yukawa Lagrangian are given in App. A.6.

A crucial aspect that breaks the analogy with the SM is however the following:
while the global symmetries in the Yukawa sector of the SM are accidental, in
our phenomenological limit the symmetry groups $\mathcal{G}_{Q}$,
$\mathcal{G}_{L}$ and their relative orientation parametrised by $W$ have been
assumed. This clearly calls for a UV understanding in terms of some flavour
dynamics above the scale of $\mathcal{G}_{4321}$ breaking. On the other hand,
since the symmetries that we imposed for phenomenological reasons are nothing
but a generalisation of the accidental and approximate symmetries already
present in the SM, the possibility to create a link between the flavour
structure of the SM and $\mathcal{G}_{Q,L}$ is well motivated, and proposals
such as those in Refs. Bordone:2017bld ; Bordone:2018nbg  might play a role in
achieving this goal. It appears instead more difficult to provide flavour
dynamics responsible for the misalignment induced by $W$, since a large 3-2
misalignment points to flavour-breaking spurions beyond those of the SM
Yukawas. This notwithstanding, our phenomenological limit turns out to be
robust against higher-order effects and is not tuned. It also allows us to
identify the most important observables and understand suppressions or
enhancements directly in terms of the symmetries of the fundamental
Lagrangian. Another difference with respect to the SM is the presence of
radial modes contained in the scalar fields $\Omega_{i}$ which can mediate
flavour violation beyond that induced by the massive vectors. It can be shown,
however, (see Sec. 4) that flavour violating effects mediated by the radial
modes are phenomenologically under control.

$\mathcal{G}_{321}$ |  $\mathcal{G}_{4321}$
---|---
$\theta_{C}$ |  $\theta_{LQ}$
$V$ |  $W$
$W_{\mu}$ |  $U_{\mu}$
$q_{L}=\left(\begin{array}[]{c}u_{L}\\\ Vd_{L}\end{array}\right)$ |  $\Psi_{L}=\left(\begin{array}[]{c}Q_{L}\\\ WL_{L}\end{array}\right)$
$Y_{u}$, $Y_{d}$ |  $\lambda_{q}$, $\lambda_{\ell}$
$SU(2)_{L}$ |  $SU(4)$
$U(1)_{u}\times U(1)_{c}\times U(1)_{t}$ |  $U(2)_{q^{\prime}+\Psi}\times U(1)_{q^{\prime}_{3}+\Psi_{3}}$
$U(1)_{d}\times U(1)_{s}\times U(1)_{b}$ |  $U(1)_{\ell^{\prime}_{1}+\tilde{\Psi}_{1}}\times U(1)_{\ell^{\prime}_{2}+\tilde{\Psi}_{2}}\times U(1)_{\ell^{\prime}_{3}+\tilde{\Psi}_{3}}$
$U(1)_{B}$ |  $U(1)_{q^{\prime}_{1}+\ell^{\prime}_{1}+\Psi_{1}}\times U(1)_{q^{\prime}+\ell^{\prime}+\Psi}$
$u\to d\,\textrm{ tree level}$ |  $Q\to L\,\textrm{ tree level}$
$u_{i}\to u_{j}\,\textrm{ loop level}$ |  $Q_{i}\to Q_{j}\,\textrm{ loop level}$
$d_{i}\to d_{j}\,\textrm{ loop level}$ |  $L_{i}\to L_{j}\,\textrm{ loop level}$
Table 2: Analogies between the SM and the 4321 model.

We conclude this section by summarizing the main features of the _Cabibbo
mechanism for leptoquarks_ advocated above (cf. also Table 2 for a SM
analogy):

  * •

We have found a mechanism that allows for large flavour violation in semi-
leptonic decays in the $3$-$2$ sector, as required by the flavour anomalies.

  * •

Tree-level FCNC involving down quarks and charged leptons are absent.

  * •

Tree-level FCNC in the up sector are protected by the small
$U(2)_{q^{\prime}}$ breaking of the SM Yukawas.

  * •

FCNC not protected by the $U(2)_{q^{\prime}}$ symmetry (both in up and down
sectors) are induced at one loop. While flavour changing processes involving
electrons and down quarks are forbidden, the leptoquark contributes at one
loop to $B_{s}$ and $D$ mixing, as well as lepton flavour violating (LFV)
processes such as $\tau\to\mu\gamma$ and other EW observables. In Sec. 4 we
show that these bounds can be satisfied, also thanks to an extra dynamical
GIM-like suppression provided by the lepton partners running in the loop. One-
loop effects due to the exchange of the coloron, $Z^{\prime}$ and scalar
radial modes are also under control.

  * •

We can now match the UV-complete 4321 model with the simplified-model analysis
performed in Buttazzo:2017ixm . Most importantly, since the theory is fully
calculable, we are also able to provide precise predictions in $\Delta F=2$
and LFV observables.

All these aspects will be addressed in a quantitative way in the next section.

##  4 Low-energy phenomenology

The scope of this section is to discuss the main low-energy observables of the
4321 model, together with the relevant constraints coming from electroweak
precision tests and FCNC. Let us start by outlining the main interactions of
the new vectors with the SM fermions, described in terms of mixing angles
between the would-be SM fermions and their vector-like partners. The flavour
structure of our model, defined by our assumptions in Eq. (6), is such that
(up to CKM rotations) each SM family mixes with only one fermion partner, see
Fig. 1 for illustration.

Figure 1: Interactions of the SM fermions with the heavy vectors induced by
the fermion mixing.

The only non-trivial source of flavour breaking is found in the $W$ matrix,
introduced in the previous section, which is responsible for a misalignment
between quarks and leptons in the leptoquark interactions. The resulting
vector leptoquark interactions with SM fermions closely follow those
introduced in Buttazzo:2017ixm , which were shown to provide a successful
explanation of the $b\to s\ell\ell$ and $R(D^{(*)})$ anomalies. We write these
interactions in the mass basis in a similar fashion555In this section we show
only the interactions of the new gauge bosons with the SM fermions for
illustration. Full expressions, including also the couplings to vector-like
fermions, can be found in App. A.7.

$\displaystyle\mathcal{L}_{U}$
$\displaystyle\supset\frac{g_{4}}{\sqrt{2}}\,U_{\mu}\left[\beta_{ij}\,\overline{q}^{i}\gamma^{\mu}\ell^{j}+h.c.\right]\,,$
(12)

with

$\displaystyle\beta$
$\displaystyle=\begin{pmatrix}s_{q_{1}}\,s_{\ell_{1}}&0&0\\\
0&c_{\theta_{LQ}}\,s_{q_{2}}\,s_{\ell_{2}}&s_{\theta_{LQ}}\,s_{q_{2}}\,s_{\ell_{3}}\\\
0&-s_{\theta_{LQ}}\,s_{q_{3}}\,s_{\ell_{2}}&c_{\theta_{LQ}}\,s_{q_{3}}\,s_{\ell_{3}}\end{pmatrix}\,,$
$\displaystyle q^{i}$ $\displaystyle=\begin{pmatrix}V^{*}_{ji}\,u_{L}^{j}\\\
d_{L}^{i}\end{pmatrix}\,,\,$ $\displaystyle\ell^{i}$
$\displaystyle=\begin{pmatrix}\nu_{L}^{i}\\\ e_{L}^{i}\end{pmatrix}\,.$ (13)

and $V$ the CKM matrix. The interactions of these new gauge bosons with SM
fermions read

$\displaystyle\begin{aligned} \mathcal{L}_{g^{\prime}}&\supset
g_{s}\,\frac{g_{4}}{g_{3}}\,g^{\prime
a}_{\mu}\left[\kappa_{q}^{ij}\,\overline{q}^{i}\gamma^{\mu}T^{a}q^{j}+\kappa_{u}^{ij}\,\overline{u}_{R}^{i}\gamma^{\mu}T^{a}u_{R}^{j}+\kappa_{d}^{ij}\,\overline{d}_{R}^{i}\gamma^{\mu}T^{a}d_{R}^{j}\right]\,,\\\
\mathcal{L}_{Z^{\prime}}&\supset\frac{g_{Y}}{2\sqrt{6}}\,\frac{g_{4}}{g_{1}}\,Z^{\prime}_{\mu}\left[\xi_{q}^{ij}\,\overline{q}^{i}\gamma^{\mu}q^{j}+\xi_{u}^{ij}\,\overline{u}_{R}^{i}\gamma^{\mu}u_{R}^{j}+\xi_{d}^{ij}\,\overline{d}_{R}^{i}\gamma^{\mu}d_{R}^{j}-3\,\xi_{\ell}^{ij}\,\overline{\ell}^{i}\gamma^{\mu}\ell^{j}-3\,\xi_{e}^{ij}\,\overline{e}_{R}^{i}\gamma^{\mu}e_{R}^{j}\right]\,,\end{aligned}$
(14)

with

$\displaystyle\begin{aligned}
\kappa_{q}&\approx\begin{pmatrix}s_{q_{1}}^{2}&0&0\\\ 0&s_{q_{2}}^{2}&0\\\
0&0&s_{q_{3}}^{2}\\\
\end{pmatrix}-\frac{g_{3}^{2}}{g_{4}^{2}}\,\mathbb{1}\,,&\qquad&&\kappa_{u}\approx\kappa_{d}&\approx-\frac{g_{3}^{2}}{g_{4}^{2}}\,\mathbb{1}\,,\\\
\xi_{q}&\approx\begin{pmatrix}s_{q_{1}}^{2}&0&0\\\ 0&s_{q_{2}}^{2}&0\\\
0&0&s_{q_{3}}^{2}\\\
\end{pmatrix}-\frac{2\,g_{1}^{2}}{3\,g_{4}^{2}}\,\mathbb{1}\,,&&&\xi_{u}\approx\xi_{d}&\approx-\frac{2\,g_{1}^{2}}{3\,g_{4}^{2}}\,\mathbb{1}\,,\\\
\xi_{\ell}&\approx\begin{pmatrix}s_{\ell_{1}}^{2}&0&0\\\
0&s_{\ell_{2}}^{2}&0\\\ 0&0&s_{\ell_{3}}^{2}\\\
\end{pmatrix}-\frac{2\,g_{1}^{2}}{3\,g_{4}^{2}}\,\mathbb{1}\,,&&&\xi_{e}&\approx-\frac{2\,g_{1}^{2}}{3\,g_{4}^{2}}\,\mathbb{1}\,.\end{aligned}$
(15)

Note that the $W$ mixing matrix cancels by unitary in the neutral current
sector and hence it does not enter in the $Z^{\prime}$ and $g^{\prime}$
interactions. This situation is completely analogous to the SM, in which the
CKM cancels in the $\gamma$ and $Z$ interactions. Also note that the assumed
down-aligned flavour structure implies no tree-level FCNC in the down-quark
and charged-lepton sectors mediated by these extra gauge bosons. In the case
where $s_{q_{1}}\neq s_{q_{2}}\neq s_{q_{3}}$, FCNC in the up sector
proportional to the CKM matrix elements are induced. These transitions yield
potentially dangerous contributions in $\Delta C=2$ observables. Assuming
$\theta_{q_{1}}=\theta_{q_{2}}\equiv\theta_{q_{12}}$ ensures an additional
$U(2)$-like protection of the FCNC in the up sector. As we show in Sec. 4.3.2,
this extra protection plays a crucial role in keeping the effects in
$D-\overline{D}$ mixing under control. An even larger protection against FCNCs
can be achieved when $\theta_{q_{12}}=\theta_{q_{3}}$, which we denote as
full-alignment limit. In this limit the flavour matrices in Eq. (15) become
proportional to the identity, yielding, as with the $W$ matrix, a unitarity
cancellation of the CKM matrix in the up sector and thus resulting in a
complete absence of tree-level FCNC mediated by the $g^{\prime}$ and the
$Z^{\prime}$. As we show in Secs. and 4.3.2 and 5.3, this latter limit is
disfavoured by low-energy and high-$p_{T}$ data.

The relevant low-energy phenomenology of the model is described in terms of
the fermion mixing angles: $\theta_{q_{i}}$ and $\theta_{\ell_{i}}$, the $W$
matrix, the ratios of fermion masses to the leptoquark mass, and the following
combinations of gauge couplings and vector masses

$\displaystyle C_{U}$ $\displaystyle=\frac{g_{4}^{2}v^{2}}{4M_{U}^{2}}\,,$
$\displaystyle C_{Z^{\prime}}$
$\displaystyle=\frac{g_{Y}^{2}}{24\,g_{1}^{2}}\,\frac{g_{4}^{2}v^{2}}{4\,M_{Z^{\prime}}^{2}}\,,$
$\displaystyle C_{g^{\prime}}$
$\displaystyle=\frac{g_{s}^{2}}{g_{3}^{2}}\,\frac{g_{4}^{2}v^{2}}{4\,M_{G^{\prime}}^{2}}\,,$
(16)

which measure the strength of the new gauge boson interactions relative to the
weak interactions. In the limit $g_{4}\gg g_{1,3}$, in which we are working,
we have $g_{Y}\approx g_{1}$ and $g_{s}\approx g_{3}$. Moreover, in the
phenomenological limit $v_{3}\gg v_{1}\gg v_{15}$, the following approximate
relation among vector masses holds (see App. A.4):

$\displaystyle
M_{g^{\prime}}:M_{U}:M_{Z^{\prime}}\approx\sqrt{2}:1:\frac{1}{\sqrt{2}}~{},$
(17)

while for the NP scale constants we find:

$\displaystyle
C_{g^{\prime}}:C_{U}:C_{Z^{\prime}}\approx\frac{1}{2}:1:\frac{1}{12}~{}.$ (18)

In what follows, we describe the main low-energy constraints on these model
parameters.

###  4.1 Constraints on fermion mixing

The fermion mass mixing induced by Eqs. (1) and (2) is the essential
ingredient in our construction. While the full fermion mass diagonalization is
discussed in App. A.6, here we give a simplified discussion and comment on the
main constraints on the mixing angles. To a good approximation, this mixing is
such that each family of the SM fermions mixes with a single vector-like
family. We introduce the following notation,

$\displaystyle\epsilon^{i}_{u(d)}=\frac{Y_{u(d)}^{i}\,v}{\sqrt{2}\hat{M}_{Q}}~{},\qquad
x_{q_{i}}=~{}\frac{\lambda_{q}^{i}\,v_{3}}{\sqrt{2}\hat{M}_{Q}}~{},$ (19)

where $i=1,2,3$ is the family index, and analogously for the lepton sector.
The quark and lepton mixing angles, expanded in small $\epsilon^{i}_{x}$, are
given by

$\displaystyle\tan\theta_{q_{i},\ell_{i}}\approx
x_{q_{i},\ell_{i}}~{},\qquad\tan\theta_{u^{i}_{R}(d^{i}_{R})}\approx\frac{m_{u^{i}(d^{i})}}{m_{U^{i}(D^{i})}}\,\tan\theta_{q_{i}}\,,\qquad\tan\theta_{e^{i}_{R}}\approx\frac{m_{e^{i}}}{m_{E^{i}}}\,\tan\theta_{\ell_{i}}~{}.$
(20)

The physical masses are instead given by

$\displaystyle\begin{aligned}
m_{u_{i}(d_{i})}&\approx\frac{Y_{u(d)}^{i}\,v}{\sqrt{2}}\cos\theta_{q_{i}}~{},&\qquad&&M_{U_{i}(D_{i})}&\approx\frac{\hat{M}_{Q}}{\cos\theta_{q_{i}}}~{},\\\
m_{e_{i}}&\approx\frac{Y_{e}^{i}\,v}{\sqrt{2}}\cos\theta_{\ell_{i}}~{},&&&M_{E_{i}(N_{i})}&\approx\frac{\hat{M}_{L}}{\cos\theta_{\ell_{i}}}~{}.\end{aligned}$
(21)

Note that large left-handed mixing angles of the third generation quarks and
leptons are required by the $R(D^{(*)})$ anomaly (cf. Eq. (24)). There are a
few subtleties regarding the top quark mixing due to its large mass. After
electroweak symmetry breaking, contributions to electroweak precision tests
are generated, setting important limits on the right-handed top mixing. In
particular, $Z\to b_{L}\overline{b}_{L}$ decay and the $\rho$ parameter, both
induced at one-loop, set upper limits of $\tan\theta_{u^{3}_{R}}\lesssim 0.4$
and $\tan\theta_{u^{3}_{R}}\lesssim 0.15$, respectively (for more details see
Ref. Fajfer:2013wca ). As a consequence, the two charged components of the
doublet are almost degenerate ($M_{T}\approx M_{B}$) since the relative mass
difference,
$M_{U_{i}}/M_{D_{i}}-1\sim\frac{1}{2}(\tan^{2}\theta_{u^{i}_{R}}-\tan^{2}\theta_{d^{i}_{R}})$.
In addition, setting $\sin\theta_{q_{3}}=0.8$, the second relation in Eq. (20)
implies a lower limit $M_{T}\gtrsim 1.7$ TeV. The maximal size of the mixing
angles is also limited by the perturbativity of the Yukawa couplings (cf. Sec.
4.5). For example, setting $\sin\theta_{q_{3}}=0.8$ implies $y_{t}\approx 1.7$
(see Eq. (21)). Similarly, large values for $\lambda^{3}_{q}$ and
$\lambda^{3}_{\ell}$ are also required to keep these angles maximal.

###  4.2 Semileptonic processes

A key element of the Cabibbo mechanism introduced in Sec. 3 is that NP effects
in flavour-violating semileptonic transitions are expected to be maximal. In
particular, the relative misalignment in flavour space between quark and
leptons, parametrised by the $W$ matrix, is responsible for sizeable 3-2
transitions mediated by the leptoquark. In what follows, we describe the main
NP effects in this sector, paying particular attention to the anomalies in
$b\to c\tau\nu$ and $b\to s\mu\mu$ transitions.

####  4.2.1 Charged currents

Current measurements of the $R(D^{(*)})=\mathcal{B}(B\to
D^{(*)}\tau\nu)/\mathcal{B}(B\to D^{(*)}\ell\nu)$ $(\ell=e,\mu)$ ratios
performed by BaBar Lees:2012xj ; Lees:2013uzd , Belle Huschle:2015rga ;
Sato:2016svk ; Hirose:2016wfn ; Hirose:2017dxl  and LHCb Aaij:2015yra ;
Aaij:2017uff ; Aaij:2017deq  point to a large deviation away from lepton
flavour universality (LFU). We define possible NP contributions to these LFU
ratios as

$\displaystyle\Delta R_{D^{(*)}}=\frac{R(D^{(*)})_{\rm exp}}{R(D^{(*)})_{\rm
SM}}-1\,.$ (22)

Effects in these observables are induced in our model by the tree-level
exchange of $U_{\mu}$. Since $U_{\mu}$ only couples to SM fermions of left-
handed chirality (see Eq. (12)), the NP effect has the same structure as the
SM one mediated by the $W_{\mu}^{\pm}$. As a result, our model predicts the
same NP contributions to $R(D)$ and $R(D^{*})$, compatible with current
experimental data. Using the HFLAV experimental average for the $R(D^{(*)})$
ratios Amhis:2016xyh  (summer 2018), taking the arithmetic average of latest
SM predictions for these observables Bigi:2016mdz ; Bernlochner:2017jka ;
Bigi:2017jbd ; Jaiswal:2017rve , and assuming $\Delta R_{D}=\Delta R_{D^{*}}$
(as predicted by the model) we find: $\Delta R_{D^{(*)}}=0.217\pm 0.053$. The
model contribution to this observable (taking only the leading interference
contribution) reads

$\displaystyle\Delta R_{D^{(*)}}\approx
2\,C_{U}\,\beta_{b\tau}\left(\beta_{b\tau}+\beta_{s\tau}\,\frac{V_{cs}}{V_{cb}}\right)\,.$
(23)

In the limit $\beta_{b\tau}|V_{ts}|\ll\beta_{s\tau}$, we can neglect the first
term in the equation above. This allows us to derive the following approximate
expression

$\displaystyle\Delta R_{D^{(*)}}\approx
0.2\left(\frac{2~{}\textrm{TeV}}{M_{U}}\right)^{2}\left(\frac{g_{4}}{3.5}\right)^{2}\sin(2\theta_{LQ})\left(\frac{s_{\ell_{3}}}{0.8}\right)^{2}\left(\frac{s_{q_{3}}}{0.8}\right)\left(\frac{s_{q_{2}}}{0.3}\right)~{},$
(24)

which is helpful in order to understand the parametric dependence: a
successful explanation of the $R(D^{(*)})$ anomaly requires large mixing
angles with third-generation fermions. Moreover, setting $\theta_{LQ}=\pi/4$
and the third family mixing and $g_{4}$ nearly to the maximum value compatible
with perturbativity, the NP contribution to $R(D^{(*)})$ is fixed in terms of
$s_{q_{2}}$ and the NP scale, see blue contours in Fig. 4. Interesting
constraints on the value of $s_{q_{2}}$ arise from $\Delta F=2$ observables
and high-$p_{T}$ searches for vector-like partners, which are addressed
respectively in Secs. 4.3.2 and 5.3.

An interesting remark is that the $W$ matrix introduces an additional source
of $U(2)_{q}$ breaking other than that discussed in Buttazzo:2017ixm . As a
result, we predict a NP enhancement that is different for $b\to c$ and $b\to
u$ transitions. In particular, we have that

$\displaystyle\Delta_{b\to c}:\Delta_{b\to
u}=\left(\beta_{b\tau}+\beta_{s\tau}\frac{V_{cs}}{V_{cb}}\right):\left(\beta_{b\tau}+\beta_{s\tau}\frac{V_{us}}{V_{ub}}\right)\,,$
(25)

where $\Delta_{b\to c(u)}$ is the ratio of the NP amplitude over the SM one
for the $b\to c(u)\tau\nu$ transitions. The previous formula, in the
phenomenological limit of the parameters that we are considering
($\beta_{b\tau}\,|V_{cb}|\ll\beta_{s\tau}$), reduces to

$\displaystyle\Delta_{b\to c}:\Delta_{b\to u}\approx
1:\frac{V_{us}V_{cb}}{V_{ub}V_{cs}}\approx 1:1+2.5\,i\,.$ (26)

While the real part of the NP contribution shows the same universal
enhancement, relatively large non-interfering effects in $b\to u$ transitions
are predicted in our model, which could allow differentiation of this solution
from the one in Buttazzo:2017ixm . So far, the only experimental measurement
of $b\to u\tau\nu$ transitions is $\mathcal{B}(B\to\tau\nu)$. The modification
of $\mathcal{B}(B\to\tau\nu)$ compared to the SM is dictated by $\Delta_{b\to
u}$, while $\Delta_{b\to c}$ is fixed by $\Delta{R_{D^{(*)}}}$. Using Eq.(26)
we can derive the following prediction:

$\Delta\mathcal{B}(B\to\tau\nu)=\frac{\mathcal{B}(B\to\tau\nu)_{\rm
exp}}{\mathcal{B}(B\to\tau\nu)_{\rm SM}}-1=\Delta
R_{D^{(*)}}+1.8\,\left(\Delta R_{D^{(*)}}\right)^{2}\,,$ (27)

which for $\Delta R_{D^{(*)}}=0.22$ yields
$\Delta\mathcal{B}(B\to\tau\nu)=0.31$. Remarkably, using the PDG value
Patrignani:2016xqp  for the experimental input and the UTFit value
UTfit2016WEB  for the SM prediction, we have
$\Delta\mathcal{B}(B\to\tau\nu)=0.35\pm 0.31$, which supports the model
prediction, but still has a very large error. Future improvements of the
sensitivity could be used to test this prediction and possibly discriminate
among different sources of $U(2)_{q}$ breaking.

####  4.2.2 Neutral currents

The effective Hamiltonian describing $b\rightarrow s\ell\ell$ transitions
reads

$\displaystyle\mathcal{H}_{\mbox{\scriptsize
eff}}=-\frac{4G_{F}}{\sqrt{2}}\,\frac{\alpha}{4\pi}\,V_{tb}V_{ts}^{*}\sum_{i}\left(C_{i}^{\ell}\mathcal{O}^{\ell}_{i}+C_{i}^{\prime\ell}\mathcal{O}^{\prime\ell}_{i}\right),$
(28)

with

$\displaystyle\mathcal{O}^{\ell}_{9}$
$\displaystyle=\left(\overline{s}\gamma_{\mu}P_{L}b\right)\left(\overline{\ell}\gamma^{\mu}\ell\right)\,,$
$\displaystyle\mathcal{O}^{\ell}_{10}$
$\displaystyle=\left(\overline{s}\gamma_{\mu}P_{L}b\right)\left(\overline{\ell}\gamma^{\mu}\gamma_{5}\ell\right)\,,$
(29)

where we ignore the scalar and chirality-flipped (primed) operators which
receive negligible contributions in our model and are thus irrelevant for the
present discussion. Due to the assumed down-aligned flavoured structure, only
the leptoquark mediates tree-level contributions to $b\to s\ell\ell$
transitions. Since the leptoquark only couples to left-handed SM fields, NP
contributions to these transitions are of the form (we define
$C_{i}=C_{i}^{\rm SM}+\Delta C_{i}$)

$\displaystyle\left.\Delta C_{9}^{\mu\mu}\right|_{\rm tree}=-\left.\Delta
C_{10}^{\mu\mu}\right|_{\rm
tree}=-\frac{2\pi}{\alpha\,V_{tb}V_{ts}^{*}}\,C_{U}\,\beta_{s\mu}\,\beta_{b\mu}^{*}\,.$
(30)

As recently put forward in Crivellin:2018yvo , given the large values of the
$\beta_{s\tau}$ leptoquark coupling required in our setup to explain the
$R(D^{(*)})$ anomaly, one-loop log-enhanced contributions to these Wilson
coefficients at the scale of the bottom mass can be sizeable. The most
relevant of such contributions is given by a photon penguin with a $\tau$ in
the loop, see Fig. 2.

Figure 2: EFT diagrams contributing to $b\to s\ell\ell$ transitions after
$U_{\mu}$ has been integrated out.

This yields a contribution only to $C_{9}$ that is universal for all leptons.
We find ($\ell=e,\mu,\tau$)

$\displaystyle\begin{aligned} \left.\Delta C_{9}^{\ell\ell}\right|_{\rm
loop}(m_{b}^{2})&\approx-\frac{1}{V_{tb}V_{ts}^{*}}\,\frac{2}{3}\,C_{U}\left(\beta_{s\tau}\,\beta_{b\tau}^{*}\,\log
x_{b}+\beta_{sE_{2}}\,\beta_{bE_{2}}^{*}\,\log
x_{E_{2}}+\beta_{sE_{3}}\,\beta_{bE_{3}}^{*}\,\log x_{E_{3}}\right)\\\
&\approx-\frac{1}{V_{tb}V_{ts}^{*}}\,\frac{2}{3}\,C_{U}\,\frac{1}{2}\sin
2\theta_{LQ}\,s_{q_{2}}\,s_{q_{3}}\left(s_{\tau}^{2}\,\log
x_{b}+c_{\tau}^{2}\log x_{E_{3}}-\log x_{E_{2}}\right)\,,\end{aligned}$ (31)

with $x_{\alpha}=m_{\alpha}^{2}/M_{U}^{2}$. In the computation we neglected
fermion mixing in the muon sector (i.e. we took $s_{\ell_{2}}\approx 0$),
which amounts to a very small correction. Our result is in agreement with the
one in Crivellin:2018yvo , however in our setup we also have computed the
contributions involving the vector-like leptons $E_{2,3}$.

The violations of LFU measured in the $R_{K}$ Aaij:2014ora  and $R_{K^{*}}$
Aaij:2017vbb  ratios (as well as
$Q_{4,5}=P^{\mu\prime}_{4,5}-P^{e\prime}_{4,5}$ Wehle:2016yoi ) fix the (non-
universal) tree-level contribution. Combining the experimental measurements of
LFU observables in $b\to s\ell\ell$ transitions yields the following preferred
region for the non-universal NP effect Capdevila:2017bsm  (see also
Ciuchini:2017mik ; Altmannshofer:2017yso ; Geng:2017svp ; DAmico:2017mtc ;
Alok:2017sui ; Hiller:2017bzc )

$\displaystyle\left.\Delta C_{9}^{\mu\mu}\right|_{\rm tree}=-\left.\Delta
C_{10}^{\mu\mu}\right|_{\rm tree}=-0.66\pm 0.18\,.$ (32)

This value can be perfectly accommodated by fixing $s_{\ell_{2}}$ in terms of
the remainaing parameters. Taking typical values for the other model
parameters, we find that $s_{\ell_{2}}\sim\mathcal{O}(0.1)$ is required in
order to fit the $R_{K^{(*)}}$ anomaly.

Concerning the one-loop contribution, we can connect its value to the NP shift
in $\Delta R_{D^{(*)}}$ in the $\beta_{b\tau}|V_{ts}|\ll\beta_{s\tau}$ limit.
In this limit the following approximate relation holds

$\displaystyle\left.\Delta C_{9}^{\ell\ell}\right|_{\rm
loop}(m_{b}^{2})\approx\frac{1}{3}\,\Delta R_{D^{(*)}}\left(\log
x_{b}-\frac{1}{s_{\tau}^{2}}\,\log x_{E_{2}}\right)\,.$ (33)

For $s_{\tau}=0.8$, $M_{U}=2.5$ TeV, $M_{E}=850$ GeV and $\Delta
R_{D^{(*)}}=0.1$, we find $\left.\Delta C_{9}^{\ell\ell}\right|_{\rm
loop}(m_{b}^{2})\approx-0.3$. The presence of this universal contribution
predicts a further enhancement of $P_{5}^{\prime}$ DescotesGenon:2012zf ,
beyond the one given by the tree-level effect. As shown in Crivellin:2018yvo ,
this prediction is in good agreement with current data.

The assumed flavour structure also implies large NP effects mediated by the
leptoquark in $b\to s\tau\tau$ transitions,

$\displaystyle\Delta C_{9}^{\tau\tau}=-\Delta
C_{10}^{\tau\tau}=-\frac{2\pi}{\alpha\,V_{tb}V_{ts}^{*}}\,C_{U}\,\beta_{s\tau}\,\beta_{b\tau}^{*}\,.$
(34)

For typical values of the model parameters, this contribution is
$\mathcal{O}(100)$ larger than the corresponding NP effect in the $\mu$
channel. Such large NP effects are compatible with current experimental data
and provide an interesting smoking-gun signature that can be tested by future
experiments such as Belle II, see e.g. Alonso:2015sja ; Crivellin:2017zlb ;
Buttazzo:2017ixm ; Capdevila:2017iqn .

Concerning NP contributions to $b\to s\nu\nu$, again here the flavour
structure of the model forbids tree-level contributions mediated by the
$Z^{\prime}$. Moreover, being an $SU(2)_{L}$ singlet, the leptoquark does not
contribute to these transitions at tree level. The leading effects to these
observables thus arise at one loop from $U$ and $W$ boxes and $U$ penguins
with a tree-level $Z$ or $Z^{\prime}$. In contrast to the $b\to s\ell\ell$
case, the contributing penguin diagrams do not have large log-enhancements
and/or are mass-suppressed, thanks to the additional suppression from the $Z$
mass. As a result, we find the model contributions to $b\to s\nu\nu$ to be
well below the current experimental limits.

####  4.2.3 Lepton Flavour Violating transitions

The protection from our flavour structure (aligned to the charged-lepton
sector), forbids tree-level lepton flavour violation (LFV) mediated by the
$Z^{\prime}$. As a result, the dominant LFV effects are mediated by the
leptoquark and hence they necessarily involve semileptonic processes for the
tree-level effects (see Sec. 4.4 for a discussion on one-loop induced LFV
transitions). Further assuming $s_{\ell_{1}}=0$ implies no NP effects in the
electron sector, and offers an additional protection from dangerously large
LFV effects in the $\mu-e$ sector such as in $K_{L}\to\mu e$. Small departures
from charged-lepton alignment and/or $s_{\ell_{1}}=0$ are possible. However,
for simplicity, in the following discussion we only consider this limit (the
possible departures are not connected to the anomalies and hence they are more
model dependent). In this case, the leptoquark contributes to the following
LFV transitions at tree-level:

  * i)

$\bm{\tau\to\mu ss.}$ This is the most promising LFV channel, since it is
enhanced in the large $\beta_{s\tau}$ limit, of phenomenological interest for
the $R(D^{(*)})$ anomaly. The most relevant observable involving this
transition is $\tau\to\mu\,\phi$, for which we find the following expression
for the branching fraction

$\displaystyle\mathcal{B}(\tau\to\mu\,\phi)$
$\displaystyle=\frac{1}{\Gamma_{\tau}}\frac{f_{\phi}^{2}\,m_{\tau}^{3}}{32\pi
v^{4}}\,C_{U}^{2}\,|\beta_{s\mu}\,\beta_{s\tau}^{*}|^{2}\left(1-\eta_{\phi}\right)^{2}\left(1+2\eta_{\phi}\right)\,,$
(35)

with $f_{\phi}\approx 225$ MeV and $\eta_{\phi}\equiv
m_{\phi}^{2}/m_{\tau}^{2}\approx 0.33$. In the large $\beta_{s\tau}$ limit (or
equivalently $|V_{cb}|\ll s_{q_{2}}$ and $\theta_{\rm LQ}\approx\pi/4$) , the
following approximate expression holds

$\displaystyle\mathcal{B}(\tau\to\mu\,\phi)$ $\displaystyle\approx 4.7\times
10^{-9}\,\left(\frac{0.8}{s_{q_{3}}}\right)^{2}\left(\frac{s_{q_{2}}}{0.3}\right)^{2}\left(\frac{\Delta
R_{K^{(*)}}}{0.3}\right)\left(\frac{\Delta R_{D^{(*)}}}{0.2}\right)\,.$ (36)

This is to be compared with the current $90\%$ CL experimental limit by the
Belle Collaboration Miyazaki:2011xe : $\mathcal{B}(\tau\to\mu\,\phi)<8.4\times
10^{-8}$. Our model prediction is found to lie well below the current
experimental sensitivity for the range of model parameters considered here. As
emphasised in Kumar:2018kmr , bounds from this observable can arise in the
very large $\beta_{s\tau}$ limit (i.e. $\beta_{b\tau}<\beta_{s\tau}$), which
in our model yields to the following upper bound: $s_{q_{2}}\lesssim
1.6\,s_{q_{3}}$. However such extreme values of $s_{q_{2}}$ are largely
incompatible, in our model, with other low-energy observables as well as with
direct searches (see discussion in Secs. and 4.3.2 and 5.3) and hence are not
considered.

  * ii)

$\bm{b\to s\tau\mu.}$ These transitions are parametrised by (see Eq. (29) for
the definition of the Wilson coefficients)

$\displaystyle\begin{aligned}
C_{9}^{\tau\mu}=-C_{10}^{\tau\mu}=-\frac{2\pi}{\alpha\,V_{tb}V_{ts}^{*}}\,C_{U}\,\beta_{s\mu}\,\beta_{b\tau}^{*}\,,\qquad
C_{9}^{\mu\tau}=-C_{10}^{\mu\tau}=-\frac{2\pi}{\alpha\,V_{tb}V_{ts}^{*}}\,C_{U}\,\beta_{s\tau}\,\beta_{b\mu}^{*}\,.\end{aligned}$
(37)

Taking the explicit expression for $\beta$ in Eq. (13), we find
$C_{i}^{\tau\mu}=-\,C_{i}^{\mu\tau}~{}(i=9,\,10)$ when $\theta_{LQ}=\pi/4$.
Using the expressions in Crivellin:2015era  (see also Becirevic:2016zri ), we
derive the following limits in the large $\beta_{s\tau}$ limit,

$\displaystyle\begin{aligned} \mathcal{B}(B\to K\tau^{\pm}\mu^{\mp})&\approx
2.0\,\times 10^{-6}\left(\frac{\Delta
R_{K^{(*)}}}{0.3}\right)\left(\frac{\Delta
R_{D^{(}*)}}{0.2}\right)\,,\\\\[2.0pt] \mathcal{B}(B\to
K^{*}\tau^{\pm}\mu^{\mp})&\approx 3.9\,\times 10^{-6}\left(\frac{\Delta
R_{K^{(*)}}}{0.3}\right)\left(\frac{\Delta
R_{D^{(}*)}}{0.2}\right)\,.\end{aligned}$ (38)

Experimental results are only available for the $K^{+}$ channel. The
experimental limit at $90\%$ CL from the BaBar Collaboration reads Lees:2012zz
: $\mathcal{B}(B^{+}\to K^{+}\tau^{\pm}\mu^{\mp})<4.8\times 10^{-5}$.

Another interesting observable in this category is
$\mathcal{B}(B_{s}\to\tau^{\pm}\mu^{\mp})$, whose expression in terms of the
Wilson coefficients reads

$\displaystyle\mathcal{B}(B_{s}\to\tau^{\pm}\mu^{\mp})$
$\displaystyle=\frac{\tau_{B_{s}}\,m_{\tau}^{2}\,m_{B_{s}}f_{B_{s}}^{2}}{32\pi^{3}}\,\alpha^{2}G_{F}^{2}\left|V_{tb}V_{ts}^{*}\right|^{2}\left(1-\frac{m_{\tau}^{2}}{m_{B_{s}}^{2}}\right)^{2}\left(\left|\mathcal{C}_{9}^{\tau\mu}\right|^{2}+\left|\mathcal{C}_{10}^{\tau\mu}\right|^{2}\right)\,.$
(39)

Again, in the large $\beta_{s\tau}$ limit we can write the model prediction in
terms of the NP effect in $R_{K^{(*)}}$ and $R(D^{(*)})$;

$\displaystyle\begin{aligned} \mathcal{B}(B_{s}\to\tau^{\pm}\mu^{\mp})&\approx
2\,\times 10^{-6}\left(\frac{\Delta R_{K^{(*)}}}{0.3}\right)\left(\frac{\Delta
R_{D^{(*)}}}{0.2}\right)\,.\end{aligned}$ (40)

However, no experimental measurement of this observable is currently
available.

  * iii)

$\bm{b\overline{b}\to\tau\mu.}$ In contrast to the case of $\tau\to\mu ss$
transitions, the transitions in this category are suppressed in the large
$\beta_{s\tau}$ limit and are therefore less interesting. The only measured
observables in this category are $\mathcal{B}(\Upsilon(nS)\to\tau\mu)$
($n=1,2,3$) Love:2008ys ; Lees:2010jk . The model predictions for these
observables are found to lie far below the current experimental sensitivity.

###  4.3 Hadronic processes

The most important constraints in this category arise from $\Delta F=2$
transitions. As anticipated in Sec. 3, the assumed flavour structure offers a
protection from the stringent limits set on these transitions. In particular,
the down-alignment hypothesis implies no tree-level contributions to meson
mixing observables in the down-quark sector mediated by the $Z^{\prime}$ and
$g^{\prime}$.666As shown in Bordone:2018nbg , deviations from this hypothesis
are possible, and could even be welcome, if we allow for CP violating
couplings (see also DiLuzio:2017fdq ). For simplicity we restrict ourselves
here to the down-aligned scenario. Furthermore, the $U(2)_{q}$ symmetry
arising from setting $\theta_{q_{1}}=\theta_{q_{2}}\equiv\theta_{q_{12}}$ is
enough to keep the tree-level contributions to $D-\overline{D}$ mixing under
control. As a result we find that the dominant NP contribution to these
observables arises from loops mediated by the leptoquark, and is proportional
to the $W$ matrix. This has two important implications:

  * i)

The assumption that $W$ rotates only second- and third-generation fermion
partners, required to maximise the NP contribution to $R(D^{(*)})$, implies no
NP contributions to $B_{d}-\overline{B}_{d}$ or $K-\overline{K}$ mixing at one
loop.

  * ii)

Unitarity of the $W$ matrix provides a GIM-like protection similar to that in
the SM arising from CKM unitarity.

In what follows we detail the model contributions to $B_{s}$ and $D$ mixing.

Figure 3: Leptoquark mediated one-loop diagrams contributing to
$B_{s}-\overline{B}_{s}$ mixing. The symbol $\mathcal{E}$ denotes a six-
dimensional vector containing SM charged-leptons and their partners.

####  4.3.1 $B_{s}-\overline{B}_{s}$ mixing

The leading NP contribution to the mixing amplitude is given by the leptoquark
box diagrams shown in Fig. 3. The resulting leptoquark contribution follows a
very similar structure as that of the SM with a $W_{\mu}^{\pm}$ boson (see
e.g. Branco:1999fs ). Defining NP contributions to the $B_{s}$ meson-anti-
meson mass difference, $\Delta M_{s}$, as $C_{bs}^{LL}\equiv\Delta
M_{s}/\Delta M_{s}^{\rm SM}-1$, we find

$\displaystyle
C_{bs}^{LL}=-\frac{g_{4}^{2}}{64\pi^{2}}\,C_{U}\,\frac{1}{(V_{tb}V_{ts}^{*})^{2}\,R_{\rm
SM}^{\rm
loop}}\sum_{\alpha,\beta}\lambda^{B}_{\alpha}\lambda^{B}_{\beta}\,F(x_{\alpha},x_{\beta})\,,$
(41)

with $\alpha$ and $\beta$ running over all the leptons, including the vector-
like partners, and where $R^{\rm loop}_{\rm
SM}=\sqrt{2}\,G_{F}\,m^{2}_{W}\,\hat{\eta}_{B}\,S_{0}(x_{t})/16\pi^{2}=1.34\times
10^{-3}$, with $S_{0}(x_{t})\approx 2.37$ being the Inami-Lim function
Inami:1980fz . In this expression $F(x_{\alpha},x_{\beta})$ is a loop function
defined as

$\displaystyle F(x_{\alpha},x_{\beta})$
$\displaystyle=\frac{1}{(1-x_{\alpha})(1-x_{\beta})}\left(\frac{7x_{\alpha}x_{\beta}}{4}-1\right)$
$\displaystyle+\frac{x^{2}_{\alpha}\log
x_{\alpha}}{(x_{\beta}-x_{\alpha})(1-x_{\alpha})^{2}}\left(1-2x_{\beta}+\frac{x_{\alpha}x_{\beta}}{4}\right)$
$\displaystyle+\frac{x^{2}_{\beta}\log
x_{\beta}}{(x_{\alpha}-x_{\beta})(1-x_{\beta})^{2}}\left(1-2x_{\alpha}+\frac{x_{\alpha}x_{\beta}}{4}\right)\,,$
(42)

with $x_{\alpha}=m_{\alpha}^{2}/M_{U}^{2}$ and
$\lambda_{\alpha}^{B}=\beta_{b\alpha}\,\beta_{s\alpha}^{*}$, where $\beta$
denote the leptoquark couplings to left-handed fermions given in Eq. (130).
The explicit form of $\lambda_{\alpha}^{B}$ in terms of fermion mixing angles
reads

$\displaystyle\lambda_{\alpha}^{B}$
$\displaystyle=\frac{1}{2}\,\sin\,2\theta_{LQ}\sin\theta_{q_{3}}\sin\theta_{q_{12}}\left(\sin^{2}\theta_{\ell_{3}}\,\delta_{\alpha
3}+\cos^{2}\theta_{\ell_{3}}\,\delta_{\alpha
6}-\sin^{2}\theta_{\ell_{2}}\,\delta_{\alpha
2}-\cos^{2}\theta_{\ell_{2}}\,\delta_{\alpha 5}\right)\,.$ (43)

Note that, analogously to the SM case, the flavour parameter
$\lambda_{\alpha}^{B}$ has the key property
$\sum_{\alpha}\lambda_{\alpha}^{B}=0$, related to the unitarity of the flavour
rotation matrices (and to the assumed down-aligned flavour structure). This
property, similarly to the GIM-mechanism in the SM, is essential to render the
loop finite and is required to derive the expression in Eq. (41). As a result
of this GIM-like protection, we find that the leptoquark contribution to
$C_{bs}^{LL}$ receives an additional mass suppression proportional to
$M_{L}^{2}/M_{U}^{2}$ with respect to the naive dimensional analysis
expectation with generic leptoquark couplings and no vector-like
fermions.777This GIM-like behaviour has been qualitatively noticed also in a
different model presented in Ref. Calibbi:2017qbu . On the other hand, models
that address the $R(D^{(*)})$ anomaly with scalar leptoquarks do not exhibit
this suppression, see Eq. (5.18) in Marzocca:2018wcf . In particular, we find
that the NP contribution to $\Delta M_{s}$ follows the approximate scaling

$\displaystyle C_{bs}^{LL}\sim\Delta R_{D^{(*)}}^{2}\,M_{L}^{2}\,,$ (44)

and therefore it is completely controlled by $M_{L}$, for fixed $R(D^{(*)})$
anomaly and leptoquark gauge coupling. This scaling is made manifest in Fig. 4
where we show the constraints arising from the leptoquark contribution to
$C_{bs}^{LL}$ in the $M_{U}-s_{q_{12}}$ plane, together with the preferred
region for $R(D^{(*)})$, and for different values of $M_{L}$. The experimental
limit on $C_{bs}^{LL}$ is obtained using the SM determination in
Artuso:2015swg ; Lenz:2011ti ; Lenz:2006hd 888A recent lattice QCD simulation
from the Fermilab/MILC collaboration Bazavov:2016nty  finds a larger central
value (and a smaller error) for the non-perturbative parameter
$f_{B_{s}}\sqrt{\hat{B}}$ entering the determination of $\Delta M_{s}$. That
would imply a 1.8$\,\sigma$ tension with respect to the SM and translates into
very stringent limits for purely left-handed NP contributions featuring real
couplings DiLuzio:2017fdq . Given the fact that the new lattice result has not
been confirmed yet by other collaborations, we conservatively use the pre-2016
determination in Artuso:2015swg .  and the experimental measurement from
Amhis:2016xyh . We have

$\displaystyle C_{bs}^{LL}=1.03\pm 0.15\,.$ (45)

Figure 4: Constraints from $\Delta M_{s}$ at $95\%$ CL using the 2015 SM
prediction Artuso:2015swg , for different values of the vector-like lepton
mass parameter $M_{L}$. The $1\sigma$ and $2\sigma$ preferred regions by the
$R(D^{(*)})$ anomaly are shown in dark and light blue, respectively. We use as
input for the model parameters: $g_{4}=3.5$, $s_{q_{3}}=0.8$,
$s_{\ell_{3}}=0.8$, $v_{3}=1.75\,v_{1}$ and $\lambda_{15}=2.5$.

The radial excitation arising from the linear combination of $\Omega_{1}$ and
$\Omega_{3}$ (see Apps. A.1–A.2) could also potentially yield dangerous NP
contributions not protected by the $U(2)_{q}$ symmetry. These contributions
depend on other parameters (masses and couplings) that are not directly
connected to the anomalies and are therefore more model dependent. Moreover,
in the phenomenological limit $v_{3}\gg v_{1}$ we find the coupling of the
radial mode to be suppressed by $\cot\beta_{T}=v_{1}/v_{3}$ (see App.
A.3).999Note that in this phenomenological limit purely leptonic transitions
mediated by the radial excitations would receive additional $\tan\beta_{T}$
enhancements. However, we find the bounds from this sector to be significantly
smaller and thus they do not pose any relevant constraint on these effects
(see Sec. 4.4). As an estimate of the size of such contributions, we compute
the box diagrams with two radial modes (similar to the ones in Fig. 3 but with
the leptoquark replaced by the radial excitations). Recasting the result in
Bertolini:1990if  for the up squark box we find in our model

$\displaystyle C_{bs}^{LL}$
$\displaystyle=\frac{1}{(V_{tb}V_{ts}^{*})^{2}\,R_{SM}^{\rm
loop}}\,\frac{G_{F}^{2}\,C_{U}^{2}\,M_{3}^{4}}{128\,\pi^{2}}\,t_{\beta_{T}}^{-4}\,s_{q_{2}}^{2}s_{q_{3}}^{2}\,\left\\{\frac{1}{m_{L_{2}}^{2}}\,\left[G^{\prime}(x_{T_{R}\,L_{2}},x_{T_{R}\,L_{2}},1)\right.\right.$
(46)
$\displaystyle\quad\left.\left.-G^{\prime}(x_{T_{R}\,L_{2}},x_{T_{R}\,L_{2}},x_{L_{3}\,L_{2}})\right]+\frac{1}{m_{L_{3}}^{2}}\,\left[G^{\prime}(x_{T_{R}\,L_{3}},x_{T_{R}\,L_{3}},1)-G^{\prime}(x_{T_{R}\,L_{3}},x_{T_{R}\,L_{3}},x_{L_{2}\,L_{3}})\right]\right\\}\,,$

with $x_{ab}=m_{a}^{2}/m_{b}^{2}$ and the loop function $G^{\prime}$ defined
in Bertolini:1990if . Assuming typical values for the model parameters, we
estimate that values as small as $\tan\beta_{T}\gtrsim 1.75$ are enough to
keep this radial-mode contribution to $C_{bs}^{LL}$ to be below $1\%$ and
therefore small enough to be ignored. Mixed contributions involving both the
leptoquark and the radial mode are present as well. Assuming similar sizes for
the loop functions and including the $\cot\beta_{T}=1/1.75$ suppression in the
radial-mode coupling, we find such contribution to be also sufficiently
suppressed to be neglected.

####  4.3.2 $D-\overline{D}$ mixing

Following the analysis from UTfit UTfit2018 ; Carrasco:2014uya , the
constraint obtained from $D-\overline{D}$ transitions can be expressed in
terms of bounds on the Wilson coefficients of the four-fermion effective
Hamiltonian

$\displaystyle\mathcal{H}_{\rm eff}^{\Delta C=2}\supset
C_{1}^{D}\left(\overline{c}_{L}\gamma_{\mu}u_{L}\right)^{2}\,.$ (47)

The latest constraints on $C_{1}^{D}$ from UTFit read UTfit2018

$\displaystyle\begin{aligned} \mathrm{Re}\,(C_{1}^{D})&=\left(0.3\pm
1.4\right)\times 10^{-7}~{}\mathrm{TeV}^{-2}\,,\\\\[5.0pt]
\mathrm{Im}\left(C_{1}^{D}\right)&=(-0.03\pm 0.46)\times
10^{-8}\;\textrm{TeV}^{-2}\,.\end{aligned}$ (48)

In our model, NP effects are induced in both the real and the imaginary parts
of $C_{1}^{D}$. Also, in contrast to the $B_{s}$ mixing case, the model yields
contributions both at tree level and at one loop. In what follows we describe
both contributions.

Tree level. The $Z^{\prime}$ and $g^{\prime}$ mediate tree-level contributions
to the $D-\overline{D}$ amplitude proportional to the CKM matrix elements.
These are given by

$\displaystyle\begin{aligned} \left.C_{1}^{D}\right|_{\rm
tree}&=\frac{4G_{F}}{\sqrt{2}}\left(C_{Z^{\prime}}+\frac{C_{g^{\prime}}}{3}\right)\,(V_{ub}^{*}\,V_{cb})^{2}\,\left(\sin^{2}\theta_{q_{3}}+\sin^{2}\theta_{q_{2}}\frac{V_{us}^{*}\,V_{cs}}{V_{ub}^{*}\,V_{cb}}+\sin^{2}\theta_{q_{1}}\frac{V_{ud}^{*}\,V_{cd}}{V_{ub}^{*}\,V_{cb}}\right)^{2}\,,\end{aligned}$
(49)

with $C_{Z^{\prime}}$ and $C_{g^{\prime}}$ defined in Eq. (16). Setting
$\theta_{q_{1}}=\theta_{q_{2}}\equiv\theta_{q_{12}}$ and using CKM unitarity,
the expression above simplifies into

$\displaystyle\begin{aligned} \left.C_{1}^{D}\right|_{\rm
tree}&=\frac{4G_{F}}{\sqrt{2}}\left(C_{Z^{\prime}}+\frac{C_{g^{\prime}}}{3}\right)\,(V_{ub}^{*}\,V_{cb})^{2}\,\left(\sin^{2}\theta_{q_{3}}-\sin^{2}\theta_{q_{12}}\right)^{2}\,.\end{aligned}$
(50)

This assumption on the mixing angles ensures a $U(2)$-like protection,
rendering the tree-level contribution to $C_{1}^{D}$ sufficiently small to
pass the stringent constraints from $D-\overline{D}$ mixing. In particular, we
find that for values of the NP scale compatible with an explanation of the
$R(D^{(*)})$ anomaly, the tree-level contributions to both the real and the
imaginary parts of $C_{1}^{D}$ are $\approx 10^{-9}~{}\mathrm{TeV}^{-2}$, and
are thus compatible with the present bounds. It is also interesting to note
from Eq. (50) that in the full-alignment limit, corresponding to
$\theta_{q_{12}}=\theta_{q_{3}}$, the tree-level contribution to $C_{1}^{D}$
would completely vanish by unitarity, as expected from the discussion at the
beginning of this section.

Figure 5: Leptoquark mediated one-loop diagrams contributing to
$D-\overline{D}$ mixing. The symbol $\mathcal{N}$ denotes a six-dimensional
vector containing both SM neutrinos and their partners.

One loop. In this case, the computation of the loop effects is technically
more challenging than in the previous section, since now also the $g^{\prime}$
and $Z^{\prime}$ mediate NP contributions at one loop. However, it is
important to note that thanks to the flavour structure of the model all these
additional contributions are protected by the same $U(2)_{q}$ symmetry that
protected the tree-level contribution (i.e. they are proportional to
$(V_{ub}^{*}V_{cb})^{2}$), and therefore they are much smaller than the
(already small) tree-level effect. As in the $B_{s}$-mixing case, we find that
the dominant contributions to $C_{1}^{D}$ arise from loop diagrams involving
the leptoquark (see Fig. 5), which are not protected by the $U(2)_{q}$
symmetry, and that these effects are proportional to the $W$ matrix.
Neglecting corrections of $\mathcal{O}((V_{ub}^{*}V_{cb})^{2})$, we find

$\displaystyle\left.C_{1}^{D}\right|_{\rm
loop}\approx-\frac{4G_{F}}{\sqrt{2}}\frac{g_{4}^{2}}{64\pi^{2}}\,C_{U}\sum_{\alpha,\beta}\lambda_{\alpha}^{D}\lambda_{\beta}^{D}\,F(x_{\alpha},x_{\beta})\,,$
(51)

where the loop function is defined as in Eq. (4.3.1), and
$\lambda_{\alpha}^{D}=V_{ci}V_{uj}^{*}\,\beta_{i\alpha}\,\beta_{j\alpha}^{*}$
with $\beta$ the leptoquark coupling to fermions. Keeping only the
$U(2)_{q}$-violating contributions, $\lambda_{\alpha}^{D}$ can be written in
terms of CKM matrix elements and fermion mixing angles as

$\displaystyle\begin{aligned} \lambda_{\alpha}^{D}&\approx
V_{cs}V_{us}^{*}\;s_{q_{12}}^{2}\left[c_{LQ}^{2}\,s_{\ell_{2}}^{2}\,\delta_{\alpha
2}+c_{LQ}^{2}\,c_{\ell_{2}}^{2}\,\delta_{\alpha
5}+s_{LQ}^{2}\,s_{\ell_{3}}^{2}\,\delta_{\alpha
3}+s_{LQ}^{2}\,c_{\ell_{3}}^{2}\,\delta_{\alpha 6}-\delta_{\alpha 4}\right]\\\
&\quad-(V_{cb}V_{us}^{*}+V_{cs}V_{ub}^{*})\;s_{q_{12}}s_{q_{3}}\,c_{LQ}\,s_{LQ}\left[\,s_{\ell_{2}}^{2}\,\delta_{\alpha
2}+c_{\ell_{2}}^{2}\,\delta_{\alpha 5}-s_{\ell_{3}}^{2}\,\delta_{\alpha
3}-c_{\ell_{3}}^{2}\,\delta_{\alpha 6}\right]\\\
&\quad+V_{cb}V_{ub}^{*}\;s_{q_{3}}^{2}\left[c_{LQ}^{2}\,s_{\ell_{2}}^{2}\,\delta_{\alpha
2}+c_{LQ}^{2}\,c_{\ell_{2}}^{2}\,\delta_{\alpha
5}+s_{LQ}^{2}\,s_{\ell_{3}}^{2}\,\delta_{\alpha
3}+s_{LQ}^{2}\,c_{\ell_{3}}^{2}\,\delta_{\alpha 6}-\delta_{\alpha
4}\right]\,.\end{aligned}$ (52)

Also in this case, the GIM-like protection encoded in
$\sum_{\alpha}\lambda_{\alpha}^{D}\approx 0$ ensures an additional suppression
of the box contributions. More precisely, we find the following approximate
scaling connecting the NP effect in $R(D^{(*)})$ with the one in
$D-\overline{D}$ mixing

$\displaystyle\left.C_{1}^{D}\right|_{\rm loop}$ $\displaystyle\sim
s_{q_{12}}^{2}\,\Delta R_{D^{(*)}}^{2}\,M_{L}^{2}\,.$ (53)

Interestingly, this different scaling results in an upper limit on the maximum
allowed value for $s_{q_{12}}$. This is shown in Fig. 6, where we plot the
constraints from $D-\overline{D}$ mixing (both for the real and imaginary
contributions) together with the preferred region by $R(D^{(*)})$. In the
low-$s_{q_{12}}$ region of the left figure there is a small violation of the
scaling in Eq. (53). This violation is due to the tree-level contribution in
Eq. (50), which for the real part plays a marginal role.

Finally, concerning the contribution from the scalar radial modes, similarly
to the case of $B_{s}$ mixing we find that these receive $\cot\beta_{T}$
suppressions in the phenomenological limit $v_{3}>v_{1}$, making their effect
sufficiently small to be neglected.

Figure 6: Constraints from $D-\overline{D}$ mixing, for different values of
the bare vector-like lepton mass parameter $M_{L}$. In the left bounds arising
from $\mbox{Im}\,(C^{D}_{1})$ and in the right those arising from
$\mbox{Re}\,(C^{D}_{1})$ at $95\%$ CL (see text for more details). The
$1\sigma$ and $2\sigma$ preferred regions by the $R(D^{(*)}$ anomaly are shown
in dark and light blue, respectively. We use as input for the model
parameters: $g_{4}=3.5$, $s_{q_{3}}=0.8$, $s_{\ell_{3}}=0.8$,
$v_{3}=1.75\,v_{1}$ and $\lambda_{15}=2.5$.

###  4.4 Leptonic processes

The fully leptonic transitions play a less important role in the low-energy
phenomenology than hadronic processes. As already mentioned, the assumption of
flavour alignment in the charged-lepton sector forbids tree-level LFV
transitions mediated by the $Z^{\prime}$. The leading effects are therefore
those mediated by the leptoquark at one loop, and are completely controlled by
the $W$ matrix. The assumed structure for this matrix, i.e.
$\theta_{LQ}=\pi/4$, chosen to maximise the NP contribution in $R(D^{(*)})$,
implies no NP contributions to fully leptonic LFV transitions involving
electrons (even for $s_{\ell_{1}}\neq 0$). Furthermore, the loop suppression,
together with the additional suppression coming from the mixing angle of the
muon, $s_{\ell_{2}}\approx 0.1$, are sufficient to render the model
contributions to $\tau\to 3\mu$ and $\tau\to\mu\gamma$ well below the current
experimental sensitivity. Purely-leptonic and electroweak operators generated
by the renormalisation-group running of the semi-leptonic operators from the
mass scale of the leptoquark down to the electroweak scale Feruglio:2016gvd ;
Feruglio:2017rjo , are already taken into account in the global fits of
Buttazzo:2017ixm  and in the limit of large 3-2 mixing studied in this paper
they are even less important.

###  4.5 Perturbativity

Figure 7: Left: Sample one-loop RGE evolution for the benchmark point:
$g_{4}=3.5$ and $\lambda_{q}=\lambda_{\ell}=\lambda_{15}=2.5$ and $Y_{t}=1.7$
at the matching scale $\mu=2.5$ TeV. Other (subleading) couplings are not
shown in the plot. Right: same for $g_{4}=3.5$, $\lambda_{q}=2.1$,
$\lambda_{\ell}=2.0$, $\lambda_{15}=2.3$ and $Y_{t}=1.7$.

The fit of the $R(D^{(*)})$ anomaly (cf. Eq. (24)) requires simultaneously
large $g_{4}$ and mixing angles $s_{q_{3}}$ and $s_{\ell_{3}}$, which
translate to sizeable third generation Yukawa couplings $\lambda_{q}$ and
$\lambda_{\ell}$, thus pushing the model close to the boundary of the
perturbative domain. When assessing the issue of perturbativity, there are two
conceptually different questions that one could address: the first (more
conservative) is to which extent low-energy observables are calculable in
perturbation theory and the second (more ambitious) is up to which energy the
model can be extrapolated in the UV before entering the strongly coupled
regime. Regarding the convergence of the perturbative expansion at low energy,
the most important coupling is $g_{4}$, which for typical benchmarks is $\sim
3$. This is still within the limits imposed by standard perturbativity
criteria: e.g. the beta function criterium of Goertz:2015nkp
$\beta_{g_{4}}/g_{4}<1$ yields $g_{4}<4\pi/\sqrt{10}\approx 4$, while
perturbative unitarity of leptoquark-mediated $2\to 2$ fermion scattering
amplitudes requires $g_{4}<\sqrt{8\pi}\approx 5$ DiLuzio:2017chi ;
DiLuzio:2016sur . Remarkably, the phenomenological requirement of a large
$g_{4}$ coupling in the IR does not prevent extrapolation of the theory in the
UV, thanks to the (one-loop) asymptotic freedom of the $SU(4)$ gauge factor.
Following the $g_{4}$ evolution from the UV to the IR the theory flows towards
the confining phase, until the running is frozen by the spontaneous breaking
of $SU(4)$ via the Higgs mechanism.

From the point of view of the UV extrapolation, the problematic couplings are
actually the Yukawas, which are required to be large in order to generate
sizeable mixings between the third generation SM fields and their vector-like
partners. To investigate their effects we have computed the one-loop
renormalisation group equations (RGEs) of the 4321 model (which are reported
for completeness in App. A.9). In Fig. 7 we show the RGE evolution for two
typical benchmark points which are compatible with low-energy and high-$p_{T}$
observables and which yield a $13\%$ (left panel) and $10\%$ (right panel)
contribution to $\Delta R_{D^{(*)}}$. Depending on the initial values of the
33 components of the $\lambda_{q,\ell,15}$ and $Y_{u}$ matrices, the theory
can be extrapolated in the UV for several decades of energy before hitting a
Landau pole. These figures also clearly give an idea of the tension between
the need to give a sizeable contribution to $\Delta R_{D^{(*)}}$ and that of
extrapolating the 4321 model in the UV.

##  5 High-$p_{T}$ signatures

In this section we survey the main high-$p_{T}$ signatures of the $4321$ model
in $pp$ collisions at the LHC. After reviewing the main features of the
resonances spectrum in Sec. 5.1, we describe the leading decay channels in
Sec. 5.2. In Sec. 5.3, we derive the exclusion limits from the coloron
searches in $t\overline{t}$ and $jj$ final states, $Z^{\prime}$ searches in
$\tau^{+}\tau^{-}$ and vector leptoquark searches. Finally, we highlight the
non-standard phenomenology of the vector-like lepton (and vector-like quarks)
as the most novel aspect of the high-$p_{T}$ discussion. The upshot of this
section is that the $4321$ model predicts a vastly richer set of high-$p_{T}$
signatures than the simplified dynamical model of a vector leptoquark
introduced in Buttazzo:2017ixm .

###  5.1 Resonances spectrum

The $4321$ model predicts a plethora of new resonances around the TeV scale
that are potential targets for direct searches with the ATLAS and CMS
experiments. In this section we discuss the spectrum of new resonances and
their couplings, focusing on the parameter space of the model preferred by the
flavour anomalies and consistent with other low-energy data.

The starting point is the low-energy fit to the charged current anomalies in
$R_{D^{(*)}}$. In the limit $s_{q_{2}}\gg V_{cb}$, the following approximate
formula can be derived,

$\Delta R_{D^{(*)}}\approx
0.2\left(\frac{2~{}\textrm{TeV}}{M_{U}}\right)^{2}\left(\frac{g_{4}}{3.5}\right)^{2}\sin(2\theta_{LQ})\left(\frac{s_{\ell_{3}}}{0.8}\right)^{2}\left(\frac{s_{q_{3}}}{0.8}\right)\left(\frac{s_{q_{2}}}{0.3}\right)~{}.$
(54)

To explain $R_{D^{(*)}}$, one needs (i) a rather low
$\mathcal{G}_{4321}\to\mathcal{G}_{321}$ breaking scale, $M_{U}/g_{4}\sim$
$\mathcal{O}$(TeV), (ii) large leptoquark flavour violation controlled by
$\theta_{LQ}$ and (iii) sizable fermion mass mixings. Requiring, in addition,
the couplings of the model, $g_{4}$, $\lambda_{q}$ and $\lambda_{\ell}$, to be
perturbative, sets an upper limit on the masses of new vectors and fermions.

The spectrum of the new scalar resonances depends on the details of the scalar
potential (see App. A.2), which introduces extra free parameters that are less
directly related to the flavour anomalies. In the following, we focus on the
fermionic and vector resonances, postponing the discussion of the radial
scalar excitations to Sec. 5.3.7.

#### New vectors

Applying Eq. (54) to the perturbative parameter space of the $4321$ model, the
implied mass scale of the new vectors $g^{\prime}_{\mu}$, $U_{\mu}$ and
$Z^{\prime}_{\mu}$ is in the interesting range for direct searches at the LHC.
Setting $\theta_{LQ}=\pi/4$ and maximising the left-handed fermion mixings for
the third family, the spectrum can be further moved up by increasing
$s_{q_{2}}$ and $g_{4}$ – eventually limited by phenomenology (see e.g. Eq.
(53)) and perturbativity, respectively. In the motivated limit, $v_{15}\ll
v_{1}\ll v_{3}$ (for the minimisation of the scalar potential see App. A.1),
and $g_{1},g_{3}\ll g_{4}$, the spectrum of the new vectors approximately
follows the pattern
$m_{g^{\prime}}:m_{U_{1}}:m_{Z^{\prime}}\approx\sqrt{2}:1:\frac{1}{\sqrt{2}}$.
A typical benchmark point is illustrated in Fig. 8 (left panel).

The structure of the $Vf\overline{f}$ interactions is discussed in length in
App. A.7. Here we highlight the key aspects for the high-$p_{T}$ searches. The
fermion mass mixing in the right-handed sector is neglected for the purposes
of this discussion (the largest mixing being $s_{u^{3}_{R}}\lesssim 0.1$). All
the $Vf_{R}\overline{f}_{R}$ interactions are practically flavour diagonal,
except for the leptoquark couplings to fermionic partners described by the $W$
matrix. The couplings to right-handed SM fermions are suppressed.

In contrast, the fermion mass mixing in the left-handed sector plays a major
role. These interactions are worked out in Eqs. (126) to (139). To illustrate
the main implications, in Fig. 8 (right panel) we show the normalized
$Vf_{L}\overline{f}_{L}$ couplings for $Z^{\prime}$ and $g^{\prime}$ as a
function of $\sin\theta_{L}$, valid for any of the left-handed mixing angles.
Solid, dotted and dashed lines represent couplings to light-light, light-heavy
and heavy-heavy combinations, where labels light and heavy denote a SM fermion
and its partner, respectively. Red color is for $g^{\prime}$ couplings
($\mathcal{C}_{g^{\prime}}$) normalized as
$\mathcal{L}\supset\mathcal{C}_{g^{\prime}}\,\frac{g_{4}g_{s}}{g_{3}}\,\overline{\psi}_{q}T^{a}\gamma^{\mu}P_{L}\psi_{q}\,g^{\prime
a}_{\mu}$, while blue is for $Z^{\prime}$ couplings
($\mathcal{C}_{Z^{\prime}}$) normalized as
$\mathcal{L}\supset\mathcal{C}_{Z^{\prime}}\,\frac{\sqrt{3}g_{4}g_{Y}}{6\sqrt{2}g_{1}}\,\left(\overline{\psi}_{q}\gamma^{\mu}P_{L}\psi_{q}-3\,\overline{\psi}_{\ell}\gamma^{\mu}P_{L}\psi_{\ell}\right)\,Z^{\prime}_{\mu}$
. It is worth noting that sizable couplings to SM fermions are generated only
for large mixing angles. In practice, the third family mixings, $s_{q_{3}}$
and $s_{\ell_{3}}$, typically control the decay channels of new resonances,
while $s_{q_{2}}$ ($=s_{q_{1}}$) is relevant for their production mechanisms
in $pp$ collisions.

Figure 8: (Left panel) A typical spectrum of new vectors and fermions. The
benchmark point is: $g_{4}=3.5$, $v_{3}=1.2,\,v_{1}=0.66$ TeV and $v_{15}=0.3$
TeV, which fixes the masses of $g^{\prime}_{\mu}$, $U_{\mu}$ and
$Z^{\prime}_{\mu}$, while $M_{Q}=1.6$ TeV, $M_{L}=0.85$ TeV, $s_{q_{3}}=0.79$,
$s_{\ell_{3}}=0.81$ and $s_{q_{2}}=0.3$, which sets the fermionic masses.
(Right panel) Normalized $Vf_{L}\overline{f}_{L}$ couplings of the
$g^{\prime}$ (red) and $Z^{\prime}$ (blue) to left-handed fermions as a
function of the $\sin\theta_{L}$. Solid, dotted and dashed lines are for the
light-light, light-heavy and heavy-heavy combinations, respectively. The
coupling normalizations are, $\frac{g_{4}g_{s}}{g_{3}}$ for $g^{\prime}$ to
quarks, and $\frac{\sqrt{3}g_{4}g_{Y}}{6\sqrt{2}g_{1}}$
($\frac{\sqrt{3}g_{4}g_{Y}}{-2\sqrt{2}g_{1}}$) for the $Z^{\prime}$ to quarks
(leptons).

#### New fermions

The main features of the fermion spectrum are controlled by the fermion mass
mixing constraints discussed in Sec. 4.1. Relevant facts for the high-$p_{T}$
discussion are the following: $i)$ the components of an $SU(2)_{L}$ doublet
are practically degenerate, $ii)$ partners of the first two families are close
in mass, $iii)$ a partner of the third SM family is always heavier than the
partners of the first two, and $iv)$ lepton partners are typically lighter
than quark partners as required by consistency with loop-induced $\Delta F=2$
observables, see Sec. 4.3.

Consistency with tree-level $\Delta C=2$ transitions requires
$s_{q_{1}}=s_{q_{2}}\equiv s_{q_{12}}$ as discussed in Sec. 4.3.2. One the one
hand, sizeable $s_{q_{2}}$ boosts the NP contribution to $R(D^{(*)})$. On the
other hand, $s_{q_{1}}$ cannot be too large since it leads to an increased
production cross section of new vector bosons in $pp$ collisions due to
couplings to valence quarks. For example, for $s_{q_{12}}\lesssim 0.4$, the
formula for the light quark partner’s mass,
$M_{U^{i}(D^{i})}\approx{\hat{M}_{Q}}$, holds at $10\%$ level, see Eq. (21),
while for the third family quark partner,
$M_{C(U)}/M_{T}\approx\cos\theta_{q_{3}}.$ Note that perturbativity of
$\lambda_{q}^{i}$, together with the requirement of fitting the $R(D^{(*)})$
anomaly, implies an upper limit on the $\hat{M}_{Q}$ to be not far above $\sim
1$ TeV, see Eq. (19). Similar arguments hold for the lepton partners since
$s_{\ell_{3}}$ is almost maximal, while $s_{\ell_{1}},s_{\ell_{2}}$ are rather
small.

The typical spectrum of new fermions is illustrated in Fig. 8 (left panel) and
will serve as a benchmark in the following discussion.

###  5.2 Decay channels

The rich spectrum of new resonances, together with the peculiar structure of
$Vf\overline{f}$ interactions, leads to an interesting decay phenomenology in
the $4321$ model. For example, cascade decays involving particles in Fig. 8
(left panel) are possible, predicting spectacular signatures in the detector.
Let us survey the main decay modes of each new state separately.

####  5.2.1 Vector decays

The dominant decay modes of the vector bosons are $1\to 2$ processes induced
by the $Vf\overline{f}$ couplings listed in App. A.7. We start with the
Lagrangian,

$\mathcal{L}\supset\,\overline{\chi}^{i}\gamma^{\mu}\,(g_{L}P_{L}+g_{R}P_{R})\,\mathcal{C}^{ijk}\psi^{j}\,V^{k}_{\mu}+\textrm{h.c.}~{},$
(55)

where $\chi$ and $\psi$ are Dirac fermions with color indices $i$ and $j$ and
masses $m_{\chi}$ and $m_{\psi}$, respectively, while $\mathcal{C}^{ijk}$ is
the color tensor and $k$ is the color index of vector $V_{\mu}$ with mass
$M_{V}$. Chiral projectors in the spinor space are
$P_{R/L}=(1\pm\gamma^{5})/2$. The general formula for
$V\to\chi\overline{\psi}$ partial decay width, following from this Lagrangian,
is ($M_{V}>m_{\chi}+m_{\psi}$)

$\Gamma(V\to\chi\overline{\psi})=N_{\mathcal{C}}\frac{M_{V}}{48\pi}\sqrt{\lambda_{1}\left(\frac{m_{\chi}}{M_{V}},\frac{m_{\psi}}{M_{V}}\right)}\left[\left(|g_{L}|^{2}+|g_{R}|^{2}\right)\lambda_{2}\left(\frac{m_{\chi}}{M_{V}},\frac{m_{\psi}}{M_{V}}\right)+12\frac{m_{\chi}m_{\psi}}{M^{2}_{V}}\mathcal{R}(g_{L}g^{*}_{R})\right]~{},~{}$
(56)

where $N_{\mathcal{C}}$ is the color factor and

$\begin{split}\lambda_{1}(x_{1},x_{2})&=1-2\,(x_{1}^{2}+x_{2}^{2}+x_{1}^{2}x_{2}^{2})+x_{1}^{4}+x_{2}^{4}~{},\\\
\lambda_{2}(x_{1},x_{2})&=3\,(1-x_{1}^{2}-x_{2}^{2})-\lambda_{1}(x_{1},x_{2})~{}.\end{split}$
(57)

For the $Z^{\prime}$ boson, $\mathcal{C}^{ijk}\to\delta^{ij}$ and $k$ index is
trivial for quarks (or all indices trivial for leptons). The color factor is
$N_{\mathcal{C}}=3\,(1)$ for $Z^{\prime}$ decays to quarks (leptons). For
$g^{\prime}$ instead $\mathcal{C}^{ija}\to T^{a}_{ij}$, where $a=1,...,8$, and
the color factor for decays to quarks is $N_{\mathcal{C}}=1/2$. Finally, for
the $U_{\mu}$ leptoquark, $\mathcal{C}^{ijk}\to\delta^{ik}$, $j$ index is
trivial, and the colour factor for decay to quark and lepton is
$N_{\mathcal{C}}=1$. The same formula Eq. (56) also applies in the case of
$\psi\equiv\chi$ and real vector field $V_{\mu}^{\dagger}\equiv V_{\mu}$,
provided that $g_{L},g_{R}$ are real and there is no ’+ h.c.’ term in Eq.
(55).

Using these formulas, we calculate the total width and the leading branching
ratios for $g^{\prime}$, $U$, and $Z^{\prime}$ vector bosons. The results for
the benchmark spectrum from Fig. 8 (left panel) are shown in Table 3. In this
context, the most relevant parameters are the two largest mixing angles
$s_{q_{3}}$ and $s_{\ell_{3}}$, as well as, the masses of the vector-like
fermions.

Particle | Decay mode |  $\mathcal{B}$ (BP) |  $\Gamma/M$ (BP)
---|---|---|---
$U_{\mu}$ |  $q_{3}\ell_{3}$ $=t\nu,b\tau$ | $\sim 0.3$ | $12\%$
$q_{3}L_{2}$ $=tN_{2},bE_{2}$ | $\sim 0.3$
$q_{3}L_{3}$ $=tN_{3},bE_{3}$ | $\sim 0.1$
$q_{1}L_{1}$ $=jN_{1},jE_{1}$ | $\sim 0.1$
$Q_{2}\ell_{3}$ $=C\nu,S\tau$ | $\sim 0.1$
$g^{\prime}_{\mu}$ |  $q_{3}q_{3}$ $=tt,bb$ | $\sim 0.6$ | $10\%$
$q_{1}Q_{1}$ $=jU,jD$ | $\sim 0.2$
$q_{2}Q_{2}$ $=jC,jS$ | $\sim 0.2$
$Z^{\prime}_{\mu}$ |  $L_{1}L_{1}$ $=N_{1}N_{1},E_{1}E_{1}$ | $\sim 0.4$ | $44\%$
$L_{2}L_{2}$ $=N_{2}N_{2},E_{2}E_{2}$ | $\sim 0.4$
$\ell_{3}\ell_{3}$ $=\tau\tau,\nu\nu$ | $\sim 0.1$
Table 3: Relevant decay channels of new vectors $U_{\mu}$, $g^{\prime}_{\mu}$
and $Z^{\prime}_{\mu}$. Branching ratios ($\mathcal{B}$) are calculated for
the benchmark point (BP) corresponding to the spectrum shown in Fig. 8 (left
panel). The last column shows the total decay width to mass ratio ($\Gamma/M$)
for the BP.

$g^{\prime}$: The coloron will decay most of the time to a pair of third
family SM quarks, $t\overline{t}$ or $b\overline{b}$. It could, in principle,
decay also to vector-like quark partners if these are kinematically
accessible. For the benchmark point, sizeable decays are into light - heavy
combination. The exclusion limits on the coloron from $pp\to jj$ and $pp\to
t\overline{t}$ searches are explored in more detail in Sec. 5.3.1.

$U_{\mu}$: The vector leptoquark is expected to decay to $t\nu$ and $b\tau$
final states. Decay modes involving light - heavy combinations are also
relevant if kinematically allowed. Examples include $U_{\mu}\to tN_{2}$ and
$U_{\mu}\to bE_{2}$ decays.

$Z^{\prime}$: Decays of the $Z^{\prime}$ boson are typically into a pair of
third family SM leptons, $\tau\tau$ and $\nu_{\tau}\nu_{\tau}$, as well as,
heavy vector-like lepton partners, which are required to be relatively light
by $\Delta F=2$ constraints as already discussed in Sec. 4.3. It is worth
noting that, for the benchmark point, $Z^{\prime}$ has a rather large total
decay width $\Gamma/M\sim 40\%$ (unlike $g^{\prime},U$ with $\Gamma/M\sim
10\%$) signalling that the model is at the edge of perturbativity. The extra
decay modes to heavy lepton partners are welcome to avoid the bounds from
$Z^{\prime}\to\tau\tau$ as discussed in Sec. 5.3.2. However, $Z^{\prime}$
assisted production becomes the dominant production mechanism for heavy lepton
partners, as discussed in Sec. 5.3.4.

|  |
---|---|---
(a) | (b) | (c)

Figure 9: (Top panel) Representative Feynman diagrams for dominant vector-like fermion decays. (Bottom panel) Phase space suppression in a fermion decay to three fermions of which one is massive and two massless.  Particle | Decay mode
---|---
$N_{1}$, $E_{1}$ |  $j(t\nu)$, $j(b\tau)$
$N_{2}$ | $t(t\nu),t(b\tau)$
$j(t\nu),j(b\tau)$
$E_{2}$ | $b(t\nu),b(b\tau)$
$j(t\nu),j(b\tau)$
$U$ | $N_{1}(t\nu),N_{1}(b\tau)$
$j(tt),j(bb),j(\tau\tau),j(\nu\nu)$
$D$ | $E_{1}(t\nu),E_{1}(b\tau)$
$j(tt),j(bb),j(\tau\tau),j(\nu\nu)$
$C$ | $N_{2}(t\nu),N_{2}(b\tau)$
$\nu(t\nu),\nu(b\tau)$
$j(tt),j(bb),j(\tau\tau),j(\nu\nu)$
$S$ | $E_{2}(t\nu),E_{2}(b\tau)$
$\tau(t\nu),\tau(b\tau)$
$j(tt),j(bb),j(\tau\tau),j(\nu\nu)$
Table 4: Leading decay modes of vector-like fermion partners of the first and
second family. The corresponding Feynman diagrams are shown in Fig. 9 (top
panel). See Sec. 5.2.2 for more details.

####  5.2.2 Fermion decays

SM-like Yukawa interactions in Eq. (1) induce a vector-like fermion decay to
its SM partner and a Higgs, $W$ or $Z$ since the heavy fermion mass eigenstate
has a projection over the $q^{\prime}_{L}$ or $\ell^{\prime}_{L}$ states.
Working in the SM unbroken phase, the partial decay width for $L^{a}_{1}\to
H^{a}e_{R}$ is

$\Gamma(L^{a}_{1}\to
H^{a}e_{R})=\frac{M_{L}}{16\pi}\left(\frac{m_{e}}{v}t_{\ell_{1}}\right)^{2}~{},$
(58)

where $a=1,2$ denotes the component of an $SU(2)_{L}$ doublet and
$t_{\ell_{1}}\equiv\tan\theta_{\ell_{1}}$. Analogous formulae hold for the
other fermions. Being suppressed by the SM fermion mass squared, this decay
channel is negligible for the fermion partners of the first and second family.
Even for the charm quark partner, we find
$\mathcal{B}(C\to\tilde{H}^{0}c)<10^{-7}$ in the interesting parameter
range.101010This is in contrast to the decays of $(T,B)$ due to the large top
quark mass. The predictions for the branching ratios are $\mathcal{B}(T\to
ht)\approx\mathcal{B}(T\to Zt)\approx 0.5$ and $\mathcal{B}(B\to Wt)\approx
1$. Recent dedicated experimental searches exclude $M_{B}<1.35$ TeV
Aaboud:2018uek  and $M_{T}<1.3$ TeV Aaboud:2018xuw . These are below the
indicated limits from electroweak precision observables discussed in Sec. 4.1.
That is, the collider searches for the third family partners are less relevant
for the spectrum on Fig. 8 (left panel).

In addition, a vector-like fermion decay to a SM fermion and a radial scalar
excitation is, in principle, possible via Eq. (2). The precise details depend
on the scalar potential, however, we expect scalar modes to be heavy enough
such that on-shell $1\to 2$ decay is kinematically forbidden.

The dominant decay modes of the first and second family vector-like fermion
partners are $1\to 3$ processes induced via an off-shell $g^{\prime}$,
$U_{\mu}$ or $Z^{\prime}_{\mu}$ mediator exchanged at tree-level. Typically, a
heavy fermion will decay to three SM fermions of which (at least) two are
third generation, or it will decay to another vector-like partner and two SM
fermions (see representative Feynman diagrams in Fig. 9 (top panel)). To a
good approximation, we can integrate out heavy vectors and work with the
following effective Lagrangian,

$\mathcal{L}\supset\frac{1}{\Lambda_{\textrm{eff}}^{2}}\mathcal{C}^{ijkl}(\overline{\Psi}^{i}_{L}\gamma^{\mu}\psi^{j}_{L})(\overline{\chi}^{k}_{L}\gamma_{\mu}\chi_{L}^{l})+\textrm{h.c.}~{},$
(59)

where $\mathcal{C}^{ijkl}$ is the colour tensor, $\Psi$ is the decaying
fermion (in this case $Q$ or $L$) with color index $i$ and mass $M_{\Psi}$,
while $\psi$ is a massive/massless final state fermion with color index $j$
and mass $m_{\psi}$. Also, $\chi$ and $\overline{\chi}$ are two massless SM
fermions with colour indices $k$ and $l$, respectively. We consider fermions
to be triplets or singlets of colour. The partial decay width following from
this Lagrangian is

$\Gamma(\Psi\to\psi\chi\overline{\chi})=N_{\mathcal{C}}\,\frac{1}{\Lambda_{\textrm{eff}}^{4}}\,\frac{M_{\Psi}^{5}}{1536\pi^{3}}\,F\left(\frac{m^{2}_{\psi}}{m^{2}_{\Psi}}\right)$
(60)

where $N_{\mathcal{C}}$ is the color factor depending on the
$\mathcal{C}^{ijkl}$. For example, for $Q_{1}^{a}\to
L_{1}^{a}\ell_{3}^{b}q_{3}^{b}$, the $\mathcal{C}^{ijkl}\to\delta^{il}$ with
indices $j,k$ trivial and $N_{\mathcal{C}}=1$. Another example is
$L_{1}^{a}\to q_{1}^{a}q_{3}^{b}\ell_{3}^{b}$, where
$\mathcal{C}^{ijkl}\to\delta^{jk}$ with indices $i,l$ trivial and
$N_{\mathcal{C}}=3$. (Here, $SU(2)_{L}$ indices $a,b$ are fixed and not summed
over.) The phase space suppression for $\Psi\to\psi\chi\chi$ decay with
massive $\psi$ and massless $\chi,\overline{\chi}$ is vanRitbergen:1999fi

$F(x)=1-8x-12x^{2}\ln(x)+8x^{3}-x^{4}~{}.$ (61)

This function is plotted in Fig. 9 (bottom panel), showing rather large
suppression factors for sizable $m_{\psi}/M_{\Psi}$. Using these relations, we
calculated the partial decay widths and identified the leading vector-like
fermion decay modes in Table 4. The precise branching ratios depend strongly
on the benchmark point. For example, for the selected BP, diagrams (a) and (b)
from Fig. 9 (top panel) lead to rates of similar sizes, which is, however,
highly sensitive on the $M_{L}/M_{Q}$ ratio, see Fig. 9 (bottom panel). It is
also interesting to note that these resonances are rather narrow,
$\Gamma/M\sim\mathcal{O}(10^{-4})$.

Loop-induced $1\to 2$ decays can, in principle, compete with tree-level $1\to
3$ decays. An example in the $4321$ model is $C\to t\gamma$ with the dipole
operator generated by the heavy neutral lepton and vector letoquark in the
loop. These decays are typically sub-leading in the relevant parameter space
due to an extra suppression from the electroweak gauge coupling.

To sum up, the $4321$ model predicts drastically different signatures of light
vector-like fermion partners from those currently being searched for by
experiments (see e.g. Ref. Sirunyan:2017lzl ).

###  5.3 Collider constraints

In this section we investigate the most stringent current LHC limits on the
$4321$ model, and propose novel (exotic) collider signatures for future
searches. As a recap, the core implication of the $R(D^{(*)})$ anomaly is $i)$
a relatively light vector leptoquark and $ii)$ relatively light vector-like
leptons – to simultaneously pass the bounds from $\Delta F=2$ transitions.

By the model construction, the accompanying vector resonances are close in
mass to the leptoquark, and are resonantly produced in $pp$ collisions. The
strongest collider constraints are due to an $s$-channel coloron (or
$Z^{\prime}$) decaying to a pair of third family SM femions, see Fig. 10. Such
final state has $i)$ a large branching ratio and $ii)$ a simple topology.
Although these topologies have been extensively exploited by experiments, a
simple interpretation in terms of a narrow-width resonance fails to capture
the effect, and a slight complication arises in properly including finite
width and interference effects. By performing a dedicated recast of the
existing dijet and $t\overline{t}$ searches, we show how to consistently
extract bounds on the model’s parameter space.

|
---|---
(a) | (b)
Figure 10: Feynman diagrams for the coloron and $Z^{\prime}$ searches.

An essential ingredient of the $4321$ model is the existence of heavy SM
fermion partners with masses below the vector boson spectrum – with peculiar
new decay channels leading to exotic final states with multiple jets and/or
leptons – a distinct smoking gun signature of the model. Here we provide a
catalog of promising topologies and estimate their potential future impact.

####  5.3.1 Coloron searches in $t\overline{t}$ and $b\overline{b}$ final
states

Figure 11: (Top panel) Coloron contribution to the $pp\to jj$ (left panel) and
$pp\to\overline{t}t$ (right panel) invariant mass spectrum for two
representative benchmark points. (Bottom panel) Coloron exclusion limits in
the mass-total width plane for $jj$ and $t\overline{t}$ for several
representative $s_{q_{2}}$ benchmarks.

The dominant production mechanism of the colour octet $g^{\prime}$ in $pp$
collisions is resonant production from a quark-antiquark pair,
$q\overline{q}\to g^{\prime}$. There is no tree-level coupling between a
single $g^{\prime}$ and a $gg$ pair, see App. A.8. Due to the flavour
structure of the model, the couplings to light quarks are suppressed, however
the PDF enhancement of valence quarks relative to third generation quarks in
the proton ensures that this channel is nevertheless dominant. The interesting
regimes of the model are when the width is rather large (but still calculable)
or the resonance is narrow but rather heavy.

Existing analyses which are most sensitive to the coloron are an ATLAS
$t\overline{t}$ invariant mass measurement Aaboud:2018eqg , an ATLAS dijet
resonance search Aaboud:2017yvp , and an ATLAS dijet resonance search with one
or two jets identified as $b$ jets Aaboud:2018tqo . The relevant Feynman
diagrams are shown in Fig. 10; the largest contribution to the dijet process
is through production of a left-handed $b\overline{b}$ pair.

We calculate the model predictions for the $t\overline{t}$ process using
Madgraph5_aMC@NLO Alwall:2014hca , implementing the coloron and its
interactions in FeynRules Alloul:2013bka , and using the default NNPDF2.3
leading order PDF set Ball:2012cx . The representative benchmark examples are
shown in Fig. 11 (top right panel). We use the measured unfolded, parton-level
invariant mass distribution, which allows direct comparison to parton level
predictions, and involves a cut of $p_{T}>500$ GeV for the leading top quark,
and $p_{T}>350$ GeV for the second leading top quark. Exclusion regions are
then calculated from the measured $t\overline{t}$ invariant mass spectrum
Aaboud:2018eqg  requiring $\Delta\chi^{2}>6.2$. The excluded regions found in
this way, for 3 different values of $s_{q_{2}}$, are shown in green in Fig. 11
(bottom panel). As shown in Fig. 8 (right panel), the coloron coupling to
left-handed valence quarks depends on $s_{q_{2}}$ and leads to the reduced
coloron production for $s_{q_{2}}\sim 0.3$.

Additionally, exclusion regions are calculated from an ATLAS dijet resonance
search Aaboud:2017yvp . The search involves dijet events with $m_{jj}>1.1$
TeV, for which the transverse momentum of the leading (subleading) jet is
greater than 440 (60) GeV, and the rapidity difference between the jets is
less than 0.6. Cross sections differential in the invariant mass are
calculated for the process $pp\to b\overline{b}$ using MSTW PDF sets
Martin:2009iq , including the effects of interference between the SM and
coloron-mediated diagrams. Note that $g^{\prime}\to b\overline{b}$ is by far
the dominant coloron dijet decay.) We estimated the signal acceptance in the
relevant invariant mass region to be about $35\%$.

Following an ATLAS method, we determine whether bumps could be seen in the
total invariant mass spectrum by fitting the background with a curve $f$
defined as

$f(z)=p_{1}(1-z)^{p_{2}}z^{p_{3}}z^{p_{4}\log(z)},$ (62)

where $z=m_{jj}/\sqrt{s}$.111111In Ref. Aaboud:2017yvp , the analysis in fact
made use of a novel fit method with a sliding window, such that in each
section of the spectrum defined by the window, a new three-parameter fit was
made. However, they compared both methods and found compatible results between
this sliding window method and the traditional global four-parameter fit
described here, so we use the four-parameter fit model for simplicity. In each
case, this parameterised curve is binned and added to the binned new physics
contribution, and the $\chi^{2}$ calculated by comparison with the ATLAS
measured data, assuming poissonian errors on the data. Each of the curve
parameters $p_{i}$ is allowed to vary independently to minimise the $\chi^{2}$
value, and this minimum $\chi^{2}$ is used to determine whether the coloron
parameter point is ruled out if $\Delta\chi^{2}>6.2$. The resulting exclusion
regions are shown in red in Fig. 11 (bottom left panel), for three different
$s_{q_{2}}$ values.

A recent ATLAS search Aaboud:2018tqo  looks for bumps, in a very similar way,
in the invariant mass spectrum of dijet events for which one or both of the
leading jets pass $b$-tagging requirements. Since our coloron-mediated dijet
signal is made up almost entirely of $b\overline{b}$ pair production events,
this is clearly an important search. In the “high-mass region” for which the
invariant mass of the dijet pair is $m_{jj}>1.2$ TeV, the analysis requires
that the transverse momentum of the leading (subleading) jet is greater than
430 (80) GeV. Both leading jets are additionally required to have
pseudorapidity $|\eta|<2.0$, and the rapidity difference between them is
required to be less than 0.8. We use the fitting method described above (Eq.
(62)) to extract exclusion regions from the measured invariant mass spectrum
requiring $\geq 1$ $b$-tag.121212The $\geq 1$ $b$-tag selection was chosen
rather than the 2 $b$ tag selection because the signal efficiency becomes very
small for the 2 $b$-tag selection. The $b$-tag efficiency for the signal
events is taken from Fig. 2 (a) of Aaboud:2018tqo . The exclusion regions
found in this way are shown in blue in Fig. 11 (bottom right panel), for three
different $s_{q_{2}}$ values.

Some discussion of the different shapes and reaches of the $t\overline{t}$ and
dijet exclusions shown in Fig. 11 (bottom panel) is in order. The green
$t\overline{t}$ regions exclude even large widths, because the $t\overline{t}$
predictions exist for the SM. This means that even for large widths, when the
signal is spread over many bins, the discrepancy from the SM can still be
apparent. The sensitivity falls off sharply around coloron masses of 3 TeV,
because the spectrum is only measured up to $m_{t\overline{t}}=3$ TeV, and in
the last bin the error on the data is already rather large. By contrast, for
the dijet bump hunts, the SM background must be simply fitted to the data. So
if the coloron has a very large width, such that its effects are spread over
many bins, then the signal can be hidden within the background fit, and the
bump hunt is no longer sensitive. This is why the red and blue dijet regions
do not reach to such large widths as the green $t\overline{t}$ regions. The
thin red lines represent dijet limits when fixing the background to the SM-
only fitted value, rather than profiling. Indeed, these show similar behaviour
to $t\overline{t}$ exclusions.

For low coloron masses, the blue region found from the bump hunt with $b$-tags
reaches larger widths than the red region found from the bump hunt without
$b$-tags. This is because the $b$-tag requirement increases the signal over
background ratio for dijet invariant masses below around 2.5 TeV. This
advantage disappears for larger coloron masses because the signal $b$-tagging
efficiency decreases for higher invariant masses. Finally, we would like to
point out that a different $b$-tagging (and misidentification) operating point
choice might be more optimal for our signal. In particular, one might try a
tighter $b$-tagging requirement with rather severe background rejection rate.

####  5.3.2 $Z^{\prime}$ search in $\tau^{+}\tau^{-}$ final state

Production of high-$p_{T}$ $\tau^{+}\tau^{-}$ pairs in $pp$ collisions (e.g.
Aaboud:2017sjh ) has been identified as a generic signature of models
addressing $R(D^{(*)})$ anomalies Faroughy:2016osc . The $4321$ model is not
an exception, as the effect comes from an on-shell $Z^{\prime}$ boson, i.e.
$pp\to Z^{\prime}\to\tau^{+}\tau^{-}$.

Let us, for a moment, assume that the $Z^{\prime}$ exclusively decays to SM
fermions (unlike in the chosen benchmark). For large $s_{q_{3}}\approx
s_{\ell_{3}}$ (and small $s_{q_{2}}$) the $Z^{\prime}$ decay width is
saturated by decays to $b\overline{b}$, $t\overline{t}$, $\nu\overline{\nu}$
and $\tau^{+}\tau^{-}$, with a branching ratio
$\mathcal{B}(Z^{\prime}\to\tau^{+}\tau^{-})\sim 3/8$. The total $Z^{\prime}$
decay width, for $g_{4}\sim 3$, is at the level of $\Gamma/M\sim 10\%$. For a
small $s_{q_{2}}$, the dominant production mechanism is from
$b_{L}\overline{b}_{L}\to Z^{\prime}$, followed by $u_{R}\overline{u}_{R}$
fusion (a factor of $\sim 4$ smaller). Increasing $s_{q_{2}}$ leads to
sizeable increase in the production cross section from the valence quarks,
$\overline{u}_{L}u_{L}$ and $\overline{d}_{L}d_{L}$. The observed upper limit
on the narrow resonance
$\sigma\times\mathcal{B}(Z^{\prime}\to\tau^{+}\tau^{-})$ is about $\lesssim
10$ fb for the $Z^{\prime}$ masses in the 1 - 3 TeV range (see Figure 7 (c) in
the latest ATLAS search done at 13 TeV with 36 fb-1 Aaboud:2017sjh ). If we
assume (conservatively) that the $Z^{\prime}$ is produced exclusively from
bottom-bottom fusion, without even considering extra $u\overline{u}$ and
$d\overline{d}$ channels, these constraints imply that $m_{Z}^{\prime}\gtrsim
1.8$ TeV. This illustrates the tension with the present data if the
$Z^{\prime}$ exclusively decays to SM fermions.

On the other hand, the reference benchmark point easily avoids the
$\tau^{+}\tau^{-}$ bound due to extra open decay channels to vector-like
lepton partners (see Table 3). Interestingly enough, these states are also
required to be light for the consistency with $\Delta F=2$ observables. The
extra decay channels ensure $i)$ a diluted branching ratio to
$\tau^{+}\tau^{-}$ and $ii)$ a large total decay width ($\Gamma/M\sim 40\%$)
which reduces the effectiveness of the search (see Fig. [4] in Ref.
Faroughy:2016osc ). Finally, we note that the limits from
$Z^{\prime}\to\mu^{+}\mu^{-}$ decay are irrelevant due to the small
$s_{\ell_{2}}\sim 0.1$.

####  5.3.3 Leptoquark signatures

Vector leptoquarks are copiously produced in pairs via QCD interactions. For
more details on their phenomenology at hadron colliders, we refer the reader
to Sec. 2.2 of Ref. Dorsner:2018ynv . We compute the leptoquark pair
production cross-section at LO in QCD in $pp$ collisions at 13 TeV, using the
FeynRules model implementation of Ref. Dorsner:2018ynv  with MadGraph5_aMC@NLO
(see also Ref. Blumlein:1996qp ). The results are shown in Fig. 13 with a
solid black line. It is worth noting the fast drop of the cross section with
the leptoquark mass.

Vector leptoquark decays to $t\nu$ or $b\tau$ final states with large
branching ratios. (Other relevant decays are listed in Table 3.) A dedicated
analysis targeting the simplified dynamical model of Ref. Buttazzo:2017ixm
has recently been performed by CMS Sirunyan:2018kzh , excluding pair-produced
leptoquarks with masses $m_{U_{1}}<1.53$ TeV, under the assumption of
$\mathcal{B}(U_{1}\to t\nu)=\mathcal{B}(U_{1}\to b\tau)=0.5$. This limit is
also shown in Fig. 13 (grey region). For the benchmark point in Table 3, this
bound is slightly relaxed due to somewhat smaller branching ratio. The first
lesson is that direct bounds on leptoquarks cannot compete with those
indirectly inferred from e.g. coloron exclusions.

In fact, as the experimental searches are moving forward, the dominant
mechanism for on-shell leptoquark production will instead become $g\,q_{1}\to
L_{1}\,U_{\mu}$, see Fig. 12. As shown in Fig. 13 (red dashed line), the cross
section for the single leptoquark production in association with a vector-like
lepton $E_{1}$ dominates over the leptoquark pair production for large
$M_{U}$. In this calculation, we fix $M_{L}=0.85$ and
$g_{4}s_{q_{2}}=M_{U}/(2~{}$TeV), as indicated by $R(D^{(*)})$ anomaly. The
present excluded mass reach from the CMS search Sirunyan:2018kzh  is already
nearing the point where this channel becomes dominant, and suggests
reconsideration of the working strategy to search for our leptoquark.

In addition to extending the scope to a novel production channel, we also
suggest searching in new decay modes as listed in Table 3. For example, there
is a significant branching fraction to $U_{\mu}\to q_{3}L_{2}\to
q_{3}(q_{3}\ell_{3}q_{3})$ – clearly calling for a dedicated experimental
analysis.

|
---|---
(a) | (b)
Figure 12: Representative Feynman diagrams for (a) vector leptoquark pair
production and (b) vector leptoquark production in association with an $L_{1}$
lepton partner.  Figure 13: Vector leptoquark ($U_{\mu}$) production cross
section (in fb) at the 13 TeV LHC. The solid black line shows the leptoquark
pair production cross section via QCD interactions. The dashed red line
represents $pp\to E_{1}+U_{\mu}$ production cross section, fixing
$g_{4}s_{q_{2}}=M_{U}/2$ TeV to fit the $R(D^{(*)})$ anomaly, and with the
$E_{1}$ lepton partner mass set to $M_{L}=0.85$ TeV. The grey region is
excluded by the CMS leptoquark search in the $(t\nu)(t\nu)$ final state,
assuming $\mathcal{B}(U_{\mu}\to t\nu)=0.5$ Sirunyan:2018kzh .

####  5.3.4 Vector-like lepton production

Figure 14: Vector-like lepton pair production cross section in $pp$ collisions
at 13 TeV as a function of the lepton mass for several benchmark points.

Vector-like leptons are pair produced in proton-proton collisions via
electroweak interactions. In addition, a sizeable contribution to the total
cross section comes from $s$-channel $Z^{\prime}$ exchange. A quantitative
estimate of this effect is illustrated in Fig. 14, where we plot the total
cross section for $pp\to EE$, $pp\to EN$ and $pp\to NN$ (where $E_{1}$ is the
first generation charged lepton partner and $N_{1}$ is the first generation
neutrino partner) as a function of the vector-like lepton mass $M_{L}$ for
several motivated benchmark points. The cross sections were calculated using
MadGraph5_aMC@NLO (with the $Z^{\prime}$ and vector-like leptons implemented
using FeynRules), including both electroweak production and the
$Z^{\prime}$-assisted process. At each vector-like lepton mass, the
$Z^{\prime}$ width was recalculated, taking into account all the kinematically
accessible final states.

Let us analyse Fig. 14 in more detail. While the charged current process is
fixed by the electroweak interactions, it is important to notice that the
neutral current processes $i)$ receive increased contribution for large
$s_{q_{2}}$, and that $ii)$ the cross section exhibits a plateau for
$M_{L}<M_{Z^{\prime}}/2$, that is when $Z^{\prime}\to L\overline{L}$ is
kinematically open. Neutral current processes are basically dominated by the
$Z^{\prime}$-assisted production in the interesting range of parameters. As
discussed in Sec. 5.3.6, these processes lead to distinct collider signatures
which already set an upper limit on the total production of $\mathcal{O}(10)$
fb. Cross sections below $\lesssim 10$ fb are obtained for relatively heavy
$\hat{M}_{L}\gtrsim 0.8$ TeV and relatively small mixing $s_{q_{2}}\lesssim
0.4$.131313We also note that $pp\to L_{1}\overline{L}_{1}$ can be induced via
vector leptoquark exchanged in t-channel, e.g. $d\overline{d}\to
E_{1}\overline{E}_{1}$ and $u\overline{u}\to N_{1}\overline{N}_{1}$. The cross
section due to this diagram scales with $s_{q_{2}}^{4}$ and is relevant only
for large $s_{q_{2}}$. We have checked that for $s_{q_{2}}\lesssim 0.4$, the
$Z^{\prime}$-assisted production dominates. Having the $Z^{\prime}$ mass below
$2\hat{M}_{L}$ also helps to reduce the yield, however, this scenario is
disfavoured by $Z^{\prime}\to\tau^{+}\tau^{-}$ searches, see Sec. 5.3.2.

####  5.3.5 Vector-like quark production

The QCD induced cross section for $pp\to Q\overline{Q}$ is completely
determined by $m_{Q}$, and it is dominated by the gluon fusion subprocess,
$gg\to Q\overline{Q}$, and the sub-leading quark fusion, $q\overline{q}\to
Q\overline{Q}$. Specific to this model is an extra contribution to the quark
fusion subprocess, $q\overline{q}\to g^{\prime}/Z^{\prime}\to Q\overline{Q}$,
which depends on the $g^{\prime}/Z^{\prime}$ masses and interactions. Due to
the flavour structure of the model, $g^{\prime}$ and $Z^{\prime}$ could decay
to a pair of heavy partners of same flavour, or to a heavy-light combination.

To investigate the importance of the $g^{\prime}$ assisted production, we plot
the total $pp\to Q\overline{Q}$ and $pp\to Q\overline{q},q\overline{Q}$ cross
sections as a function of $m_{Q}$ in Fig. 15 for several benchmark
$g^{\prime}$ masses and $s_{q_{2}}$ mixing. We do not include the
$Z^{\prime}$-mediated process here as it is highly subdominant to the coloron-
mediated process. Again, the cross sections were calculated using
MadGraph5_aMC@NLO (the $g^{\prime}$ and vector-like quarks were implemented
using FeynRules), with the coloron width varying as a function of the vector-
like quark mass.

Let us discuss the main implications of Fig. 15. The $Q\overline{Q}$
production (left panel) is dominated by the $g^{\prime}$ diagram, and shows a
plateau for $2\hat{M}_{Q}<M_{g^{\prime}}$, i.e. when $g^{\prime}\to
Q\overline{Q}$ decay is kinematically opened, while it drops fast for larger
$\hat{M}_{Q}$. On the contrary, single production of a vector-like quark in
association with a light quark (right panel) increases when the $g^{\prime}\to
Q\overline{Q}$ decays is forbidden, due to the jump in
$\mathcal{B}(g^{\prime}\to Qq)$. The benchmark point from Fig. 8 (left panel)
has suppressed cross section for pair production as this process has a more
constraining signature, see Sec 5.3.6.

Figure 15: (Left panel)Vector-like quark pair production cross section in $pp$
collisions at 13 TeV as a function of the quark mass for several benchmark
points. (Right panel) Same for the single vector-like quark production in
association with a light quark.

####  5.3.6 Multi-leptons plus multi-jets

As shown in Figs. 14 and 15, vector-like fermions are produced with sizeable
cross sections in the interesting parameter range ($1-10$ fb), and decay
dominantly to three SM fermions of the third family, $t$, $b$, $\tau$ and
$\nu_{\tau}$. Thus, the signature in the detector contains multiple jets and
leptons and is rich with $b$-tags, hadronic $\tau$-tags, etc. While the
extraction of precise limits requires a dedicated experimental analysis, we
estimate the potential sensitivity in the current and near-future datasets, by
comparing with the existing R-parity conserving (RPC) and R-parity violating
(RPV) supersymmetry (SUSY) searches.

Using 36 fb-1 of $13$ TeV $pp$ collision data, the ATLAS collaboration has
searched for signatures involving multiple b-jets, high missing transverse
momentum and either (at least) three isolated leptons, or two isolated same-
sign leptons Aaboud:2017dmy . Following this general selection, the upper
limits are set on the signal regions based on the number of $b$-jets, jets,
leptons and $E_{T}^{\textrm{miss}}$, which are then interpreted in terms of
simplified SUSY benchmarks. As an example, pair production of gluinos, each
decaying to a top pair and a neutralino, can be qualitatively compared to our
$pp\to N_{2}N_{2}\to(tt\nu)(tt\nu)$. Interpreting naively the exclusion
limits, that is, neglecting any differences in acceptances between our model
and the SUSY benchmarks, we conclude that the signal rate for this process is
$\lesssim 5$ fb. This search is already starting to probe the interesting
parameter space, see Fig. 14 (top right panel). Another relevant RPC example
involves pair production of stops, each decaying to $t$, $W^{\pm}$ and
neutralino, and sets an upper limit on the cross section $\lesssim 10$ fb.
Finally, the limit from RPV searches on gluino pair production, where each
decays to $tbj$, implies an upper limit of $\lesssim 15$ fb.

In addition to these final states, the $4321$ model predicts even more exotic
multi-lepton plus multi-jet signatures due to cascade decays among particles
shown in Fig. 8 (left panel). An example of such process is illustrated in
Fig. 16. In this example, a pair of vector-like quarks is created by an
$s$-channel coloron, and one of them decays to vector-like lepton which
eventually decays to three SM fermions. The final state contains
$3q_{3}+5\ell_{3}$, or $5q_{3}+3\ell_{3}$, where $q_{3}=t,b$ and
$\ell_{3}=\nu_{\tau},\tau$.

Figure 16: An example of the cascade decay process at the LHC leading to
heavy-flavoured multi-lepton + multi-jet final state signature.

To sum up, the $4321$ model predicts a plethora of novel signatures and calls
for a dedicated experimental effort.

####  5.3.7 Scalars

The radial modes of the Higgs fields which break the 4321 symmetry to the SM
group may also be produced at the LHC. They consist of a singlet, a colour
triplet and a colour octet, as described in Sec. A.2. A scalar singlet at the
TeV scale is not easy to find at the LHC, so here we focus on the coloured
states.

The colour octet can be pair produced via QCD and via an $s$-channel coloron,
and will, if kinematically allowed, decay with close to 100% branching ratio
to a vector-like quark and a SM quark. It may also decay to a $\overline{t}t$
pair, but this coupling is suppressed by the right handed top-vector like
quark mixing. The colour octet’s mass is very dependent on parameters of the
scalar sector but can easily be as heavy as $2-3$ TeV. A very similar model
has been investigated in Ref. Bai:2018jsr , and the pair production cross-
section of the colour octet scalar was found to be $O(10^{-2})$ fb for a 2 TeV
octet scalar. Depending on the width of the coloron and the mass of the
scalar, the coloron-assisted production may be larger than this, but will not
be more than $10^{-1}$ fb at 2 TeV. This is more than an order of magnitude
smaller than the vector-like quark pair production cross section investigated
in Sec. 5.3.5 above, and will produce a similar signature with extra jets.

The colour triplet will also be pair produced via QCD and coloron-assisted
production, and will decay to a vector-like quark and a SM lepton, or a
vector-like lepton and a SM quark. The signature will appear as vector-like
quark pair production with extra leptons (or missing energy),vector-like
lepton pair production with extra jets, or a combination of single vector-like
lepton and vector-like quark single production with extra leptons and jets.
Again, the mass depends on the scalar parameters, but can be as large as a few
TeV. The QCD pair production cross-section for this state can be found in Ref.
Diaz:2017lit , and is again $O(10^{-2})$ fb at 2 TeV, while the coloron
assisted production is parameter-dependent but no larger than $O(10^{-1})$ fb.
This is therefore subdominant to other vector-like lepton and vector-like
quark production modes in the model.

##  6 Conclusions

The evidence for flavour anomalies in semi-leptonic $B$ decays is growing with
time. Although they might eventually disappear, they represent at the moment
one of the few hints of NP, and deserve exploration. From a beyond the SM
perspective they are certainly unexpected: the common expectation was that NP
in flavour had to appear first in rare processes where the SM contribution is
suppressed, while the anomalous data in $b\to c\tau\nu$ charged currents
suggest a sizeable deviation in a channel where the SM contributes at tree
level. Even allowing for the most conservative estimate of the scale of NP
based on perturbative unitarity arguments DiLuzio:2017chi , something has to
happen below 9 TeV in order to unitarize the four-fermion operator responsible
for the NP contribution to $b\to c\tau\nu$. How is it possible to reconcile
this with a plethora of long-standing indirect probes (flavour, electroweak,
etc.) and the impressively growing amount of LHC data at high-$p_{T}$?

The main challenge, from a model building point of view, is to find a model
which can offer a coherent explanation for the anomalies in both charged and
neutral currents, while remaining compatible with all other constraints. A
non-trivial point when addressing this question is to show that this is
actually possible in a UV complete construction which allows for a connection
between the anomalies and other observables. This is what we have achieved in
this paper, in the context of the 4321 model introduced in Ref.
DiLuzio:2017vat .

The main virtue of the model is that it is renormalizable and fully
calculable. It is however fair to say that the phenomenological constraints
push the model in the direction of largish couplings, although still in the
perturbative domain. Depending on the size of the anomaly in charged currents,
the model can be extrapolated in the UV over several decades of energy.

The $4321$ model belongs to a class of $B$-anomalies solutions which involve
new dynamics in purely left-handed currents (see e.g. Buttazzo:2017ixm ). The
effect in $R(D^{(*)})$ is obtained either from $i)$ a pure third family
interaction via the CKM, or $ii)$ a large direct 3-2 flavour violation. The
second option points to a larger effective NP scale (by a factor of order
$1/\sqrt{V_{cb}}\approx 5$) alleviating the problems with indirect constraints
in precisely measured $Z$-pole observables and leptonic $\tau$ decays
Feruglio:2016gvd ; Feruglio:2017rjo , as well as direct searches at the LHC
(see e.g. Faroughy:2016osc ; Greljo:2017vvb ); but introduces a potential
problem with FCNC due to the $SU(2)_{L}$ nature of left-handed interactions.
In the context of our model, one has not only to address tree-level $\Delta
F=2$ effects due to $g^{\prime}$ and $Z^{\prime}$, but also a large 3-2
leptoquark transition which feeds in at one loop in $\Delta F=2$ observables.

A crucial phenomenological ingredient of this construction in order to make
the size of the anomalies compatible with indirect and direct constraints is a
generalisation of the Cabibbo-GIM mechanism of the SM, which allows a large
3-2 leptoquark transition while suppressing FCNC at tree level and at one
loop. We have computed these and shown that they are under control as long as
lepton partners, which play a crucial role in suppressing the contribution to
the box diagrams via a GIM-like mechanism, are light enough. The latter are
predicted to be the lowest-lying states of the NP spectrum (cf. Fig. 8) and
hence represent a clear target for LHC. This is a nice example of the
complementarity between indirect and direct searches, which is possible only
thanks to the fact that the model is UV complete and hence the observables are
calculable.

The $4321$ model predicts a very rich phenomenology at high-$p_{T}$. While the
low-energy phenomenology can be matched, to a large extent, to the simplified