MITP-21-016

ZU-TH-13/21

Reading the footprints of the $B$-meson flavor anomalies
Claudia Cornellaa, Darius A. Faroughya, Javier Fuentes-Martínb,

Gino Isidoria and Matthias Neuberta,b,c

aPhysik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland
bPRISMA+ Cluster of Excellence & MITP, Johannes Gutenberg University, 55099
Mainz, Germany
cDepartment of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A.

Abstract

> Motivated by the recent LHCb announcement of a $3.1\sigma$ violation of
> lepton-flavor universality in the ratio $R_{K}=\Gamma(B\to
> K\mu^{+}\mu^{-})/\Gamma(B\to Ke^{+}e^{-})$, we present an updated,
> comprehensive analysis of the flavor anomalies seen in both neutral-current
> ($b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$) and charged-current ($b\to
> c\tau\bar{\nu}$) decays of $B$ mesons. Our study starts from a model-
> independent effective field-theory approach and then considers both a
> simplified model and a UV-complete extension of the Standard Model featuring
> a vector leptoquark $U_{1}$ as the main mediator of the anomalies. We show
> that the new LHCb data corroborate the emerging pattern of a new,
> predominantly left-handed, semileptonic current-current interaction with a
> flavor structure respecting a (minimally) broken $U(2)^{5}$ flavor symmetry.
> New aspects of our analysis include a combined analysis of the semileptonic
> operators involving tau leptons, including in particular the important
> constraint from $B_{s}$–$\bar{B}_{s}$ mixing, a systematic study of the
> effects of right-handed leptoquark couplings and of deviations from minimal
> flavor-symmetry breaking, a detailed analysis of various rare $B$-decay
> modes which would provide smoking-gun signatures of this non-standard
> framework (LFV decays, di-tau modes, and $B\to K^{(*)}\nu\bar{\nu}$), and
> finally an updated analysis of collider bounds on the leptoquark mass and
> couplings.

###### Contents

  1. 1 Introduction
  2. 2 EFT analysis of the $B$ anomalies
    1. 2.1 Operator basis and general flavor structure
    2. 2.2 The $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ anomalies
    3. 2.3 The $b\to c\tau\bar{\nu}$ anomalies
    4. 2.4 Combined analysis of the semileptonic couplings involving $\tau$ leptons
  3. 3 The simplified $U_{1}$ model
    1. 3.1 Low-energy fit in the simplified model
    2. 3.2 Constraints from high-$p_{T}$ observables
  4. 4 UV completion and further predictions
    1. 4.1 UV-insensitive low-energy observables
    2. 4.2 UV-sensitive low-energy observables
    3. 4.3 Collider signatures from the additional TeV-scale states
  5. 5 Conclusions
  6. A The $U(2)^{5}$ symmetry
  7. B Theoretical expressions for flavor observables
    1. B.1 $b\to c(u)\tau\bar{\nu}$ transitions
    2. B.2 $b\to s(d)\ell^{-}\ell^{+}$ transitions
    3. B.3 $\tau$ decays

##  1 Introduction

Since 2012, several deviations from the predictions of the Standard Model (SM)
have been observed in a series of semileptonic decays of $B$ mesons, providing
strong evidence of a new short-distance interaction violating Lepton Flavor
Universality (LFU). The recent result by the LHCb collaboration on the ratio
$R_{K}={\mathcal{B}}(B^{+}\to K^{+}\mu^{+}\mu^{-})/{\mathcal{B}}(B^{+}\to
K^{+}e^{+}e^{-})$ [1] marks an important milestone in the study of these
phenomena, which in the literature are often referred to as the “$B$
anomalies”. The evidence collected so far can naturally be grouped into two
categories, according to the underlying partonic process: $i$) deviations from
$\mu/e$ universality in $b\to s\ell^{+}\ell^{-}$ neutral-current transitions
[2, 3, 4] together with deviations from the SM predictions in observables
involving $\mu^{+}\mu^{-}$ pairs only [5, 6, 7], and $ii$) deviations from
$\tau/\mu$ (and $\tau/e$) universality in $b\to c\ell\bar{\nu}$ charged-
current transitions [8, 9, 10, 11, 12, 13, 14]. With the recent result
reported in [1], for the first time a single observable affected by negligible
theoretical uncertainties exhibits a deviation from the SM exceeding the
$3\sigma$ level. Equally striking is the overall coherence of the picture that
emerges, especially in $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ transitions.
As we shall show in this paper, combining all the $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ observables in a very conservative way, the
significance of the New Physics (NP) hypothesis formulated in 2014–2015 of a
purely left-handed LFU-violating contact interaction has now reached a
significance of 4.6$\sigma$.

Attempts to explain one or both sets of anomalies have stimulated an intense
theoretical activity, which ranges from pure Effective Field Theory (EFT)
approaches to the formulation of motivated ultraviolet (UV) completions of the
SM. In the latter category, models containing a TeV-scale vector leptoquark,
$U_{1}\sim(\mathbf{3},\mathbf{1},2/3)$, as the main mediator are particularly
appealing. Besides addressing both sets of anomalies, such models can connect
them to an underlying theory of flavor. The purpose of this paper is to
reanalyze this class of models, taking into account the new data and,
importantly, distinguishing the robust predictions of the models from those
sensitive to the details of the UV completion, which are still largely
unknown. The goal is to identify a set of observables which could allow us to
validate or disprove these models in the future, possibly answering the
questions that are still open.

Before introducing the class of models we are interested in, it is worth
recalling the key steps that led to their formulation. The first important
observation was the identification of a purely left-handed contact interaction
involving muons as the most natural candidate to explain the $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ data available in 2014 [15]. The link between
$R_{K}$, originally proposed in [Bobeth:2007dw] as a clean NP probe, and the
various anomalies in $b\to s\mu^{+}\mu^{-}$ observables, and in particular the
non-standard behavior of the angular distribution in $B\to
K^{*}\mu^{+}\mu^{-}$ [17, 18, 19], was then established more firmly by a
series of global analyses (see e.g. [20, 21, 22, 23, 24, 25]). A second
important step was the hypothesis that such a weak contact interaction could
be the result of a stronger interaction, which violates lepton flavor and
involves mainly the third-generation fermions [26]. Soon after, it was
realized that the two sets of anomalies (in $b\to s$ and $b\to c$ transitions)
can be linked in an EFT approach involving mainly left-handed fields [27, 28,
29, 30]. Meanwhile, models involving leptoquarks started to emerge as the most
promising candidates for a UV completion of these EFTs, both in the case of
$b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ transitions [15, 31] – for which the
possibility of a leptoquark-induced effect in motivated NP models was proposed
even prior to the existence of the anomalies [32] – and especially in the case
of combined explanations [28, 33, 30, 34, 35, 36, 37], which necessarily
require a lower NP scale. The reason is simple: leptoquarks can contribute at
the tree-level to the semileptonic transitions exhibiting anomalies, while
they contribute only at the loop level to four-quark or four-lepton contact
interactions, which so far do not exhibit significant deviations from the SM.
It also emerged more clearly that the flavor structure of the new interaction
is quite constrained in the case of combined explanations, and naturally
follows the hypothesis of a minimally broken $U(2)^{5}$ flavor symmetry [29,
35] – a hypothesis formulated well before the anomalies appeared [38], which
links them to the origin of the hierarchies observed in the SM Yukawa
couplings. The simplified model proposed in [35] at the end of 2015 already
contained the two key ingredients of the UV models we are interested in, since
it was based on a TeV-scale $U_{1}$ mediator and exhibited a $U(2)^{5}$ flavor
symmetry. At that time other options, such as colorless mediators, were still
open, but it was soon realized that electroweak precision observables and LFU
tests in $\tau$ decays [39, 40], collider bounds [41, 42], and other flavor
observables [30, 43] imply very strong constraints. The detailed EFT analysis
presented in [43] has shown that the $U_{1}$ case is the simplest option at
the level of simplified models.

A massive vector field necessarily requires a UV completion and, as pointed
out in [35], the $U_{1}$ field naturally points toward the $SU(4)$ group
unifying quarks and leptons, which was proposed by Pati and Salam in 1974
[44]. The $SU(4)$ group could either be realized as a global symmetry of some
new strongly-interacting sector [45, 46] or be part of a local symmetry broken
above the electroweak scale to the SM gauge group. The original Pati-Salam
model offers a very elegant possibility, but it does not match the flavor
structure required to explain the anomalies, because the massive $U_{1}$ (and
associated $Z^{\prime}$) arising from the spontaneous symmetry breaking ${\rm
PS}\to{\rm SM}$ is flavor blind. An interesting proposal to overcome this
problem has been put forward in [47, 48], following the idea laid out in [49]
that color could appear as a diagonal subgroup of a larger $SU(3+N)\times
SU(3)$ local symmetry valid at high energies. In the model of [47], a suitable
mixing between the SM fermions and new heavy fermions allows one to adjust the
effective $U_{1}$ couplings so as to obtain the desired flavor structure.
Alternative UV models based on variations of the original Pati-Salam model
have been proposed in [50, 51]. A more structural way of addressing the flavor
structure of the model is the idea of implementing Pati-Salam unification in a
flavor non-universal manner, originally proposed in [52] and further developed
in [53, 54, 55]. This proposal is quite appealing, since third-family quark-
lepton unification close to the electroweak scale is phenomenologically
allowed, and the setup naturally accommodates an accidental $U(2)^{5}$ global
flavor symmetry at the TeV scale [52]. Remarkably, this setup can also provide
a coherent description of neutrino masses [55]. Moreover, its rich field
content and the three-scale structure behind the origin of the flavor
hierarchies finds a simple interpretation in terms of a compact extra
dimension of spacetime, with the flavor index of the four-dimensional fields
being in one-to-one correspondence with the location of four-dimensional
branes along the fifth dimension.

There are different ways of implementing Pati-Salam unification in a flavor
non-universal manner. However, the low-energy dynamics of this class of models
is characterized by a flavor non-universal $SU(4)\times SU(3)\times
SU(2)\times U(1)$ gauge symmetry (the “4321 group”), where third-generation
fermions are charged under $SU(4)$, the light SM fermions are charged under
$SU(3)$, and the $U_{1}$ field acquires mass from the symmetry breaking
$SU(4)\times SU(3)\times SU(2)\times U(1)\to{\rm SM}$ occurring near the TeV
scale. This is the class of UV completions we are most interested in. However,
as we shall discuss, many of the phenomenological conclusions we can derive
from the present data are valid for the wider class of models where the
$U_{1}$ is the main mediator of the anomalies, including in particular the
well motivated case of models of compositeness. The gauge-theory extension
offers the advantage of allowing for a systematic estimate of quantum
corrections [56, 57, 58]. We will thus refer specifically to this option when
discussing UV-sensitive observables.

Following the bottom-up approach that led to the formulation of these models,
the analysis presented in this paper is structured as follows. In Section 2 we
present a model-independent EFT analysis of the $B$ anomalies. The $U_{1}$
hypothesis enters only indirectly, via the selection of the relevant set of
semileptonic operators at the TeV scale. In Section 3 we first present an
analysis of the effective $U_{1}$ couplings to SM fermions, taking into
account all relevant low-energy constraints (including also loop-induced
observables). We then confront the results of the fit to low-energy data with
the collider constraints on the $U_{1}$ leptoquark. In Section 4 we present
predictions for a series of low-energy observables, which in the future could
allow us to resolve those model-building aspects that are currently still
open. To do so, we go beyond the simplified model, analyzing the effects of
vector-like fermions. We further analyze high–$p_{T}$ constraints on the heavy
TeV-scale color-octet boson $G^{\prime}$, which necessarily accompanies the
$U_{1}$ leptoquark in 4321 models. The results presented in this work
represent a substantial step forward in the study of a combined solution of
the $B$-meson flavor anomalies on all the three fronts: the model-independent
EFT analysis, the analysis within a simplified model for a $U_{1}$ leptoquark,
and the phenomenology of the 4321 framework. In addition to important updates
of the experimental inputs on the low-energy side, the main innovative points
of the present analysis with respect to previous studies can be summarized as
follows:

  * •

In Section 2, we present for the first time a combined analysis of the
semileptonic operators involving tau leptons, at the pure EFT level, taking
into account both sets of flavor anomalies, collider constraints and $\Delta
F=2$ bounds.

  * •

In Section 3, we analyze and compare two benchmark scenarios for the right-
handed couplings of the $U_{1}$ leptoquark to the third-generation fermions.
We also relax and validate the hypothesis of minimal breaking of the
$U(2)^{5}$ flavor symmetry for the subleading mixing terms involving first-
generation quarks. Most importantly, we update the analysis of the high-energy
constraints on the $U_{1}$ leptoquark by taking into account ATLAS and CMS
results with full Run-II statistics.

  * •

The analysis in Section 4 takes into account, for the first time, complete NLO
corrections in the leptoquark coupling $\alpha_{4}$, which were computed in
[56, 57, 58] within the non-universal 4321 model. The latter play a key role
in the predictions we obtain for $B$-meson mixing and the rare decay $B\to
K\nu\bar{\nu}$. The ${\mathcal{O}}(\alpha_{4})$ corrections are also
implemented for the first time in the analysis of the $G^{\prime}$ constraints
from $pp\to$ dijet and $pp\to t\bar{t}$, which turn out to be the most
relevant constraints on the overall mass scale of the model, once recent ATLAS
and CMS results are taken into account.

##  2 EFT analysis of the $B$ anomalies

###  2.1 Operator basis and general flavor structure

The goal of this section is to provide a general analysis of the flavor
anomalies in terms of semileptonic four-fermion operators. We start by
analyzing the two sets of anomalies separately and then discuss the
consequences of a combined analysis within the EFT.

Rather than considering all possible dimension-6 operators that can describe a
single set of measurements, we focus on the operators which have been
identified in previous studies as the relevant set necessary for a combined
explanation of both anomalies, once all constraints (including high-$p_{T}$
data, electroweak precision tests and other flavor observables) are taken into
account [43]. In practice, this set coincides with the operators generated at
the tree-level by the exchange of a spin-1 $SU(2)_{L}$-singlet leptoquark
$U_{1}$, i.e. (a contraction of color and $SU(2)_{L}$ indices between fermions
inside parenthesis is implied)

$\displaystyle\begin{aligned}
{\mathcal{O}}^{ij\alpha\beta}_{LL}&=(\bar{q}_{L}^{\,i}\gamma_{\mu}\ell_{L}^{\alpha})(\bar{\ell}_{L}^{\beta}\gamma^{\mu}q_{L}^{\,j})\\!\\!\\!&=\leavevmode\nobreak\
&\frac{1}{2}\left[Q_{lq}^{(1)}+Q_{lq}^{(3)}\right]^{\beta\alpha ij},\\\
{\mathcal{O}}^{ij\alpha\beta}_{LR}&=(\bar{q}_{L}^{\,i}\gamma_{\mu}\ell_{L}^{\alpha})(\bar{e}_{R}^{\beta}\gamma^{\mu}d_{R}^{\,j})\\!\\!\\!&=\leavevmode\nobreak\
&-2\,\big{[}Q^{\dagger}_{ledq}\big{]}^{\beta\alpha ij}\,,\\\\[2.84526pt]
{\mathcal{O}}^{ij\alpha\beta}_{RR}&=(\bar{d}_{R}^{\,i}\gamma_{\mu}e_{R}^{\alpha})(\bar{e}_{R}^{\beta}\gamma^{\mu}d_{R}^{\,j})\\!\\!\\!&=\leavevmode\nobreak\
&\big{[}Q_{ed}\big{]}^{\beta\alpha ij}\,,\end{aligned}$ (2.1)

where $Q_{lq}^{(1,3)}$, $Q_{ledq}$ and $Q_{ed}$ are defined as in the so-
called Warsaw basis [59] of dimension-6 SMEFT operators built out of SM
fields. We normalize the effective Lagrangian describing the NP contributions
as

${\mathcal{L}}^{\rm NP}_{\rm
EFT}=-\frac{2}{v^{2}}\left[{\mathcal{C}}_{LL}^{ij\alpha\beta}\,{\mathcal{O}}^{ij\alpha\beta}_{LL}+{\mathcal{C}}_{RR}^{ij\alpha\beta}\,{\mathcal{O}}^{ij\alpha\beta}_{RR}+\left({\mathcal{C}}_{LR}^{ij\alpha\beta}\,{\mathcal{O}}^{ij\alpha\beta}_{LR}+{\rm
h.c.}\right)\right],$ (2.2)

where $v=(\sqrt{2}\,G_{F})^{-1/2}\approx 246$ GeV, and the Wilson coefficients
are inversely proportional to the square of the NP scale $\Lambda$. The
coefficients of the hermitian operators ${\mathcal{O}}_{LL}$ and
${\mathcal{O}}_{RR}$ satisfy the relations

$\displaystyle{\mathcal{C}}_{LL}^{ji\beta\alpha}$
$\displaystyle=\big{(}{\mathcal{C}}_{LL}^{ij\alpha\beta}\big{)}^{*}\,,$
$\displaystyle{\mathcal{C}}_{RR}^{ji\beta\alpha}$
$\displaystyle=\big{(}{\mathcal{C}}_{RR}^{ij\alpha\beta}\big{)}^{*}\,.$ (2.3)

We assume that these coefficients respect an approximate $U(2)^{5}$ flavor
symmetry, with non-negligible breaking terms only in the left-handed quark and
lepton sectors (see Appendix A). This assumption implies that the leading
couplings in ${\mathcal{L}}^{\rm NP}_{\rm EFT}$ are those with third-
generation indices, while all other couplings are suppressed. More
specifically, we make the following two assumptions:

  * •

Wilson coefficients of operators containing first- or second-generation right-
handed fields are negligibly small, i.e.
${\mathcal{C}}_{RR}^{ij\alpha\beta}\approx 0$ unless $i=j=3$ and
$\alpha=\beta=\tau$, and ${\mathcal{C}}_{LR}^{ij\alpha\beta}\approx 0$ unless
$j=3$ and $\beta=\tau$.

  * •

Wilson coefficients associated with second-generation left-handed particles
are suppressed (relative to those for third-generation particles) by factors
of $\epsilon_{q},\epsilon_{\ell}\sim 10^{-1}$ for each second-generation quark
or lepton, e.g.
${\mathcal{C}}_{LL}^{23\tau\tau}\sim\epsilon_{q}\,{\mathcal{C}}_{LL}^{33\tau\tau}$,
${\mathcal{C}}_{LL}^{23\mu\mu}\sim\epsilon_{q}\,\epsilon_{\ell}^{2}\,{\mathcal{C}}_{LL}^{33\tau\tau}$
etc., and a further suppression arises in the case of operators involving
first-generation fields.

As explicitly indicated by the labels, the flavor basis of the lepton fields
is taken to be the charged-lepton mass basis. The flavor basis of the quark
fields is identified with the mass basis of the down-type quarks. However, we
use numerical indices to stress that this choice is a model-dependent
assumption (the implications of a possible small misalignment are briefly
discussed in Section 4). Note that a change of basis from down-type to up-type
quarks would not invalidate the scaling discussed above, and would suggest
that –at least in the quark sector– first-generation indices bring an
additional $\epsilon_{q}$ suppression compared to second-generation indices.
The flavor structure specified by the two assumptions stated above is the
rationale behind the combined explanation of the two sets of anomalies and
their possible connection to the dynamics underlying the structure of the SM
Yukawa matrices. As we shall show, these scaling rules are clearly supported
by the present data.

As pointed out in [60], one can obtain the same set of relevant operators and
flavor structure by starting from the full set of SMEFT operators and imposing
the assumption of a minimally-broken $U(2)^{5}$ flavor symmetry, without any
hypothesis about the mediator. The only relevant difference under this more
general hypothesis is that the operators $Q_{lq}^{(1)}$ and $Q_{lq}^{(3)}$ can
appear in a different linear combination than in (2.1). An EFT analysis
leaving their coefficients as free parameters has been performed in [43],
where it was shown that the combination orthogonal to ${\mathcal{O}}_{LL}$ is
tightly constrained by data on $b\to s\bar{\nu}\nu$ transitions and
electroweak precision tests (at least for the leading flavor structures).
Since this combination is not generated by $U_{1}$ tree-level exchange, and
ignoring it does not lead to a qualitative change in the description of the
two sets of anomalies, we shall not consider it further in this section. We
will, however, come back to this term in Section 4, when discussing the
effects generated by the exchange of the $U_{1}$ leptoquark beyond tree level.

###  2.2 The $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ anomalies

In $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ transitions ($\ell=e,\mu$), the NP
effects induced by ${\mathcal{L}}^{\rm NP}_{\rm EFT}$ in (2.2) amount to a
modification of the Wilson coefficients already present in the SM below the
electroweak scale. The latter are usually normalized as [61]

${\mathcal{L}}_{b\to s\hskip
0.85358pt\ell^{+}\ell^{-}}=\frac{4G_{F}}{\sqrt{2}}\,V_{ts}^{*}V_{tb}\,\sum_{i}\,{\mathcal{C}}_{i}^{\ell}\,{\mathcal{O}}_{i}^{\ell}\,,$
(2.4)

where $V_{ij}$ denote the elements of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix and the relevant semileptonic operators are defined as

${\mathcal{O}}^{\ell}_{9}=\frac{\alpha}{4\pi}\,(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}\gamma^{\mu}\ell)\,,\qquad{\mathcal{O}}^{\ell}_{10}=\frac{\alpha}{4\pi}\,(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)\,.$
(2.5)

The Wilson coefficients of operators involving a right-handed quark current
are, by assumption, negligibly small in our approach. A key prediction of the
SM is that the Wilson coefficients of these operators are lepton-flavor
universal. In order to analyze in general terms NP effects that violate this
prediction but preserve the SM operator basis, it is convenient to distinguish
LFU-breaking contributions from universal NP corrections. We choose to define
the latter using the Wilson coefficients of the electron modes as reference,
i.e.

$\Delta{\mathcal{C}}_{i}^{U}\equiv{\mathcal{C}}^{e}_{i}-{\mathcal{C}}^{\rm
SM}_{i}\,,$ (2.6)

such that the LFU-breaking terms can be defined as

$\Delta{\mathcal{C}}_{i}^{\mu}\equiv{\mathcal{C}}_{i}^{\mu}-{\mathcal{C}}_{i}^{e}={\mathcal{C}}^{\mu}_{i}-({\mathcal{C}}^{\rm
SM}_{i}+\Delta{\mathcal{C}}_{i}^{U})\,.$ (2.7)

In general terms, this amounts to introducing four complex NP parameters for
$i=9,10$.

From a tree-level matching with ${\mathcal{L}}^{\rm NP}_{\rm EFT}$ in (2.2),
it is straightforward to derive the expressions for the LFU-breaking
contributions in our setup. We find

$\Delta{\mathcal{C}}_{9}^{\mu}=-\Delta{\mathcal{C}}_{10}^{\mu}\equiv\Delta{\mathcal{C}}_{L}^{\mu}\,,$
(2.8)

where

$\Delta{\mathcal{C}}_{L}^{\mu}=-\frac{2\pi}{\alpha
V_{ts}^{*}V_{tb}}\,\big{[}{\mathcal{C}}_{LL}^{23\mu\mu}-{\mathcal{C}}_{LL}^{23ee}\big{]}\approx-\frac{2\pi}{\alpha
V_{ts}^{*}V_{tb}}\,{\mathcal{C}}_{LL}^{23\mu\mu}\,.$ (2.9)

The last relation follows from the assumption that
$|{\mathcal{C}}_{LL}^{23ee}|\ll|{\mathcal{C}}_{LL}^{23\mu\mu}|$, which is a
key hypothesis of our framework. The multiplicative correction of
${\mathcal{C}}_{LL}^{23\mu\mu}$ due to renormalization-group (RG) evolution
from the NP scale $\Lambda$ to the electroweak scale is at the percent level
and can be safely neglected. As a result, the LFU-violating corrections
$\Delta{\mathcal{C}}_{9,10}^{\mu}$ are scale independent to good accuracy. On
the other hand, since
$|{\mathcal{C}}_{LL}^{23\tau\tau}|\gg|{\mathcal{C}}_{LL}^{23\mu\mu}|$, the
mixing of ${\mathcal{O}}_{LL}^{23\tau\tau}$ into operators containing light
leptons is an important RG effect. These loop-induced contributions are
responsible for the flavor-universal corrections
$\Delta{\mathcal{C}}_{i}^{U}$, which are sizable in the case of
${\mathcal{C}}_{9}$ only. One finds [62]

$\displaystyle\Delta{\mathcal{C}}_{9}^{U}(m_{b})$
$\displaystyle=\frac{1}{V_{ts}^{*}V_{tb}}\,\frac{2}{3}\sum_{\ell=e,\mu,\tau}{\mathcal{C}}_{LL}^{23\ell\ell}(\Lambda)\,\ln\frac{\Lambda^{2}}{m_{b}^{2}}\,\approx\frac{1}{V_{ts}^{*}V_{tb}}\,\frac{2}{3}\,{\mathcal{C}}_{LL}^{23\tau\tau}(\Lambda)\,\ln\frac{\Lambda^{2}}{m_{b}^{2}}\,,$
(2.10) $\displaystyle\Delta{\mathcal{C}}_{10}^{U}(m_{b})$
$\displaystyle\approx 0\,.$

The smallness of $\Delta{\mathcal{C}}_{10}^{U}$ is a dynamical feature of the
setup we are considering. Since flavor-violating effects at the high scale are
encoded in semileptonic operators only, the loop-induced flavor-violating
couplings of the $Z$ boson are suppressed by the square of the tau-lepton
Yukawa coupling. The result for $\Delta{\mathcal{C}}_{9}^{U}$ reported above
is obtained in the leading logarithmic approximation. The resummed RG
contribution, which is included in our numerical analysis, leads to a relative
decrease of the effect by about 10% for $\Lambda=2\,\mathrm{TeV}$.

Within our framework, the final expressions of ${\mathcal{C}}_{9,10}^{\ell}$
including the effects described by ${\mathcal{L}}^{\rm NP}_{\rm EFT}$ can thus
be written in terms of two independent parameters
($\Delta{\mathcal{C}}^{\mu}_{L}$ and $\Delta{\mathcal{C}}^{U}_{9}$) rather
than four, which we further assume to be real after factoring out the CKM
matrix elements, as in the SM case:111A justification of this assumption is
given in Section 4.2 in terms of the UV completion of the model.

$\displaystyle\begin{aligned}
&{\mathcal{C}}_{9}^{e}={\mathcal{C}}_{9,\mathrm{SM}}+\Delta{\mathcal{C}}_{9}^{U}\,,&&{\mathcal{C}}^{e}_{10}={\mathcal{C}}_{10,\mathrm{SM}}\,,\\\
&{\mathcal{C}}^{\mu}_{9}={\mathcal{C}}_{9,\mathrm{SM}}+\Delta{\mathcal{C}}_{9}^{U}+\Delta{\mathcal{C}}_{L}^{\mu}\,,&\quad&{\mathcal{C}}^{\mu}_{10}={\mathcal{C}}_{10,\mathrm{SM}}-\Delta{\mathcal{C}}_{L}^{\mu}\,.\end{aligned}$
(2.11)

It is worth mentioning that the relation (2.8) among the LFU-breaking terms as
well as the absence of operators with right-handed quark currents in the
effective Lagrangian (2.2) are direct consequences of the assumption of a
minimally-broken $U(2)^{5}$ flavor symmetry [38, 60]. On the other hand, as
mentioned earlier, the relations (2.11) and, in particular, the absence of a
universal correction to ${\mathcal{C}}_{10}$, are consequences of the
dynamical assumptions we are making. An additional specific feature of the
dynamical model we are considering is the smallness of the coefficient of the
scalar operator $(\bar{s}_{L}b_{R})\,(\bar{\mu}_{R}\mu_{L})$; the minimally-
broken $U(2)^{5}$ flavor symmetry alone, while implying a strong suppression
for this operator, would not forbid a relevant contribution to the helicity-
suppressed $B_{s}\to\mu^{+}\mu^{-}$ rate [63, 60].

Observable | Experiment | SM
---|---|---
$R_{K^{*}}^{[0.045,1.1]}$ |  $0.66^{+0.11}_{-0.07}\pm 0.03$ [3] |  $0.906\pm 0.028$ [64]
$R_{K^{*}}^{[1.1,6.0]}$ |  $0.69^{+0.11}_{-0.07}\pm 0.05$ [3] |  $1.00\pm 0.01$ [64]
$R_{K^{\phantom{*}}}^{[1.1,6.0]}$ |  $0.846^{+0.042+0.013}_{-0.039-0.012}$ [1] |  $1.00\pm 0.01$ [64]
$\mathcal{B}(B_{s}\to\mu^{+}\mu^{-})$ |  $(2.85^{+0.32}_{-0.31})\times 10^{-9}$ [65, 66, 67] |  $(3.66\pm 0.14)\times 10^{-9}$ [68]
Table 2.1:  Experimental results and SM predictions for the clean observables
in $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ decays.

##### Data analysis.

In order to extract the values of the NP Wilson coefficients from data, we
distinguish two sets of observables:

  * •

Clean observables. This set contains the observables with high sensitivity to
short-distance dynamics and small, controllable theoretical uncertainties.
Among the quantities measured so far, we include in this category only the
LFU-testing ratios measured by LHCb, $R_{K}$ [1] and $R_{K^{*}}$ [3] (defined
in (B.11)) in all $q^{2}$ bins, and the branching ratio
${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$, where we perform our own combination
of ATLAS [65], CMS [66] and the recent LHCb [67] measurements.222More
precisely, each of the two-dimensional profile likelihoods provided by the
experimental collaborations is fitted to a two-dimensional variable-width
Gaussian [69], which are then summed and fitted to a new variable-width
Gaussian. We note, however, that CMS and ATLAS measurements do not include the
recent LHCb update of the ratio of $B_{s}^{0}$ and $B^{0}$ fragmentation
fractions $f_{s}/f_{d}$ [70], which is expected to modify the reported
measurements slightly. Within the SM, the theory uncertainty on $R_{K}$ and
$R_{K^{*}}$ is only due to QED effects and does not exceed $1\%$ in $R_{K}$
and in the high-$q^{2}$ bin of $R_{K^{*}}$ [64, 71]. The theory uncertainty on
${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$, which is due to a combination of
parametric uncertainties (the leading contribution) and QCD corrections that
are difficult to estimate, amounts to about $4\%$ [68]. In both cases, these
uncertainties are well below the current experimental precision, as shown in
Table 1.

  * •

Other $b\to s\mu^{+}\mu^{-}$ observables. This set includes all the other
relevant observables, namely the differential branching ratios and angular
distributions of the semileptonic decays $B\to K^{(*)}\mu^{+}\mu^{-}$ [72, 73,
74, 75, 76, 77, 78, 79], $B_{s}\to\phi\mu^{+}\mu^{-}$ [72, 80, 70] and
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ [81, 82], including the recent LHCb
experimental updates. These observables retain some sensitivity to short-
distance dynamics, but are afflicted by sizable theoretical uncertainties, in
particular due to the possibility of $c\bar{c}$ re-scattering effects in the
final state, which are difficult to quantify reliably.

The clean observables are sensitive to $\Delta{\mathcal{C}}_{9}^{\mu}$ and
$\Delta{\mathcal{C}}_{10}^{\mu}$ via $R_{K}$ and $R_{K^{*}}$ (in the limit
where we neglect small threshold effects in the low-$q^{2}$ bin of
$R_{K^{*}}$), and to $\Delta{\mathcal{C}}_{10}^{U}$ via the
$B_{s}\to\mu^{+}\mu^{-}$ branching ratio. Explicit phenomenological
expressions of the observables in terms of the Wilson coefficients are
collected in Appendix B. According to (2.11), in our setup these observables
are described by the single NP parameter $\Delta C_{L}^{\mu}$. Importantly,
however, the data allow us to check the consistency of this hypothesis. In the
left panel of Figure 2.1, we show the result of a fit to the clean observables
(blue contour lines) performed under the assumption that
$\Delta{\mathcal{C}}_{10}^{U}=0$. We also show separately the parameter
regions preferred by $R_{K}$ and $R_{K^{*}}$ (purple regions) and
${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ (orange bands), as well as the line
where $\Delta{\mathcal{C}}_{9}^{\mu}=-\Delta{\mathcal{C}}_{10}^{\mu}$. The
inclusion of a non-vanishing $\Delta{\mathcal{C}}_{10}^{U}$ parameter would
amount to a translation of the ${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$ bands
in the vertical direction. The plot shows that the two fit regions overlap in
a region compatible with
$\Delta{\mathcal{C}}^{\mu}_{10}=-\Delta{\mathcal{C}}^{\mu}_{9}$, without the
need for an extra shift due to a non-zero $\Delta{\mathcal{C}}_{10}^{U}$ (or a
non-vanishing Wilson coefficient for the scalar operator). This observation
provides a strong consistency check that all NP effects in the clean
observables can be described by a single parameter. More quantitatively, the
$p$-value of the fit to the clean observables only, assuming
$\Delta{\mathcal{C}}^{\mu}_{10}=-\Delta{\mathcal{C}}^{\mu}_{9}$ (single-
parameter fit) is 12%. The significance of the NP hypothesis we are
considering with respect to the SM (again based on a single-parameter fit) is
4.6$\sigma$. It should be stressed that this estimate of the significance is a
conservative one, because it does not include the contributions of the other
$b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ observables. Including them would
further reinforce the NP hypothesis, but at the expense of introducing larger
hadronic uncertainties.

Figure 2.1:  EFT constraints from the $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$
anomalies. Left: Results of the two-dimensional fit
$\Delta{\mathcal{C}}^{\mu}_{9}$ vs. $\Delta{\mathcal{C}}^{\mu}_{10}$ using
clean observables only ($1\sigma$, $2\sigma$ and $3\sigma$ intervals). Also
shown are the $1\sigma$ and $2\sigma$ intervals from $R_{K^{(*)}}$ and
${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})$, the latter under the hypothesis
$\Delta{\mathcal{C}}_{10}^{U}=0$. Right: Results of the two-dimensional fit
$\Delta{\mathcal{C}}^{\mu}_{9}=-\Delta{\mathcal{C}}^{\mu}_{10}$ vs.
$\Delta{\mathcal{C}}_{9}^{U}$ using all $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ observables. The vertical band shows the result
using clean observables only ($1\sigma$ interval), while the ellipse denote
the contribution of all the other observables, estimated using Flavio
($1\sigma$ interval). The upper and right axes show the corresponding
constraint on the high-scale EFT coefficients (see main text).

Besides the clean observables, whose contribution is considered separately in
the fit described above, the effect of the other $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ observables is taken into account using the public
code Flavio [83]. For these observables, we follow the same prescription as in
[84], including only $q^{2}$-bins below the $J/\psi$ resonance (extending up
to $6\leavevmode\nobreak\ \mathrm{GeV}^{2}$) and bins above the $\psi(2S)$
resonance that are at least $4\leavevmode\nobreak\ \mathrm{GeV}^{2}$ wide. In
the right panel of Figure 2.1, we show the result of a combined fit of all
observables in terms of $\Delta{\mathcal{C}}_{9}^{\mu}$ (which equals
$-\Delta{\mathcal{C}}_{10}^{\mu}$ in our framework) and
$\Delta{\mathcal{C}}_{9}^{U}$. As can be seen from this figure, once the
strong constraint on $\Delta{\mathcal{C}}_{9}^{\mu}$ arising from the clean
observables is implemented (orange band), the other $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ observables (purple region) can be used to
constrain ${\mathcal{C}}_{9}^{U}$, which is found to differ from zero by more
than $2\sigma$ (see also [85]). In the same plot, we also show the results
interpreted as constraints on the high-scale Wilson coefficients
${\mathcal{C}}_{LL}^{23\mu\mu}$ and ${\mathcal{C}}_{LL}^{23\tau\tau}$,which
follow from (2.9) and (2.10). In the latter case, the contribution from RG
evolution is estimated setting $\Lambda=2$ TeV. As can be seen, the hierarchy
of these Wilson coefficients is perfectly compatible with the scaling rule
${\mathcal{C}}_{LL}^{23\mu\mu}\sim\epsilon_{\ell}^{2}\,{\mathcal{C}}_{LL}^{23\tau\tau}$
discussed in Section 2.1.

Observable | Experiment | SM
---|---|---
{$R_{D}$, $R_{D^{*}}$} |  $\begin{matrix}\\{0.337(30),0.298(14)\\}\\\\[-2.0pt] \rho=-0.42\\\ \end{matrix}$ [86] |  $\\{0.299(3),0.258(5)\\}$ [87]
${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$ |  $1.09(24)\times 10^{-4}$ [88] |  $0.812(54)\times 10^{-4}$ [89]
Table 2.2:  Experimental results and SM predictions for $b\to
c(u)\tau\bar{\nu}$ decays. In the first entry, we provide the present combined
experimental average, with $\rho$ denoting the correlation among the two
observables.

###  2.3 The $b\to c\tau\bar{\nu}$ anomalies

In $b\to c\tau\bar{\nu}$ and $b\to u\tau\bar{\nu}$ charged-current
transitions, the NP effects induced by the effective Lagrangian
${\mathcal{L}}^{\rm NP}_{\rm EFT}$ in (2.2) not only amount to a simple
rescaling of the SM contribution but also introduce new (scalar-current)
operators not present in the SM. More precisely, we find that in our approach
the low-energy effective Lagrangian for these transitions takes the form

$\displaystyle{\mathcal{L}}_{b\to
u_{i}\tau\bar{\nu}}=-\frac{4G_{F}}{\sqrt{2}}\,\sum_{i=1,2}\,\bigg{[}$
$\displaystyle\Big{(}V_{ib}+\sum_{k=1}^{3}\,V_{ik}\,{\mathcal{C}}_{LL}^{k3\tau\tau}\Big{)}(\bar{u}^{i}_{L}\gamma_{\mu}b_{L})(\bar{\tau}_{L}\gamma^{\mu}\nu_{L})$
(2.12)
$\displaystyle-2\,\sum_{k=1}^{3}\,V_{ik}\,{\mathcal{C}}_{LR}^{k3\tau\tau}\,(\bar{u}^{i}_{L}b_{R})(\bar{\tau}_{R}\,\nu_{L})\bigg{]}\,.$

We recall that the flavor basis for the NP operators is the down-quark and
charged-lepton mass basis, i.e.

$\displaystyle q_{L}^{i}=\begin{pmatrix}V_{ji}^{*}\,u^{j}_{L}\\\
d_{L}^{i}\end{pmatrix},\qquad\ell_{L}^{i}=\begin{pmatrix}\nu^{i}_{L}\\\
e_{L}^{i}\end{pmatrix}.$ (2.13)

This implies that different Wilson coefficients contribute to a given $b\to
u^{i}$ transition. This is conveniently taken into account defining new
coefficients (for $i=1,2$)

${\mathcal{C}}_{LL}^{u_{i}}\equiv{\mathcal{C}}^{33\tau\tau}_{LL}\left[1+\frac{{\mathcal{C}}^{23\tau\tau}_{LL}}{{\mathcal{C}}^{33\tau\tau}_{LL}}\,\frac{V_{is}}{V_{ib}}\left(1+\frac{{\mathcal{C}}^{13\tau\tau}_{LL}}{{\mathcal{C}}^{23\tau\tau}_{LL}}\,\frac{V_{ud}}{V_{is}}\right)\right],$
(2.14)

and similarly for ${\mathcal{C}}_{LR}^{u_{i}}$ with $LL\to LR$ everywhere,
such that ${\mathcal{C}}_{LL}^{u_{i}}$ and ${\mathcal{C}}_{LR}^{u_{i}}$ are
the only two effective combinations appearing in $b\to u_{i}$ transitions.
With these definitions, the low-energy effective Lagrangian reads

${\mathcal{L}}_{b\to
u_{i}\tau\nu}=-\frac{4G_{F}}{\sqrt{2}}\,\sum_{i=1,2}\,V_{ib}\,\bigg{[}\Big{(}1+{\mathcal{C}}_{LL}^{u_{i}}\Big{)}(\bar{u}^{i}_{L}\gamma_{\mu}b_{L})(\bar{\tau}_{L}\gamma^{\mu}\nu_{L})-2\,{\mathcal{C}}_{LR}^{u_{i}}\,(\bar{u}^{i}_{L}b_{R})(\bar{\tau}_{R}\,\nu_{L})\bigg{]}\,.$
(2.15)

Under the generic assumption
$|{\mathcal{C}}^{13\tau\tau}_{LL,LR}|\ll|{\mathcal{C}}^{23\tau\tau}_{LL,LR}|$
stated in Section 2.1, the hierarchy of the relevant CKM matrix elements
implies that the coefficients ${\mathcal{C}}^{13\tau\tau}_{LL,LR}$ play a
negligible role in $b\to c\tau\bar{\nu}$ transitions, while they might be
relevant in $b\to u\tau\bar{\nu}$ decays. Hence, in this generic case the NP
effects in the two processes are not strictly correlated. Under the stronger
assumption of a minimal breaking of the $U(2)^{5}$ flavor symmetry see
Appendix A) we however expect that

$\frac{{\mathcal{C}}^{13\tau\tau}_{LL}}{{\mathcal{C}}^{23\tau\tau}_{LL}}=\frac{{\mathcal{C}}^{13\tau\tau}_{LR}}{{\mathcal{C}}^{23\tau\tau}_{LR}}=\frac{V_{td}^{*}}{V^{*}_{ts}}\,.$
(2.16)

This implies a complete correlation of the NP effects in $b\to c\tau\bar{\nu}$
and $b\to u\tau\bar{\nu}$ transitions, i.e.

$\frac{{\mathcal{C}}^{23\tau\tau}_{LL}}{{\mathcal{C}}^{33\tau\tau}_{LL}}\,\frac{V_{is}}{V_{ib}}\left(1+\frac{{\mathcal{C}}^{13\tau\tau}_{LL}}{{\mathcal{C}}^{23\tau\tau}_{LL}}\,\frac{V_{id}}{V_{is}}\right)=-\frac{{\mathcal{C}}^{23\tau\tau}_{LL}}{{\mathcal{C}}^{33\tau\tau}_{LL}}\,\frac{V_{tb}^{*}}{V_{ts}^{*}}\,,$
(2.17)

and similarly for the LR coefficients. This in turn implies that

$\displaystyle{\mathcal{C}}_{LL}^{u}$
$\displaystyle={\mathcal{C}}_{LL}^{c}\,,$
$\displaystyle{\mathcal{C}}_{LR}^{u}$
$\displaystyle={\mathcal{C}}_{LR}^{c}\,.$ (2.18)

Figure 2.2:  EFT constraints from the $b\to c\tau\bar{\nu}$ anomalies. The
solid blue ellipses denote the $1\sigma$, $2\sigma$ and $3\sigma$ intervals of
the two-dimensional fit to $R_{D}$ and $R_{D^{*}}$ in the
${\mathcal{C}}_{LL}^{c}$– ${\mathcal{C}}_{LR}^{c}$ plane (coefficients
evaluated at $\Lambda=2\leavevmode\nobreak\ \mathrm{TeV}$). The dashed
contours denote the fit results taking also the constraint from
${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$ into account, under the hypothesis of
minimal $U(2)^{5}$ breaking (i.e. for
${\mathcal{C}}_{LL}^{u}={\mathcal{C}}_{LL}^{c}$,
${\mathcal{C}}_{LR}^{u}={\mathcal{C}}_{LR}^{c}$). The bands correspond to
$1\sigma$ regions. The red lines show the benchmark scenarios we consider in
our analysis.

##### Data analysis.

At present, the observables constraining NP effects in $b\to c\tau\bar{\nu}$
transitions are the LFU ratios $R_{D}$ and $R_{D^{*}}$ (defined in (B.4)),
which can be used to probe the NP coefficients ${\mathcal{C}}_{LL}^{c}$ and
${\mathcal{C}}_{LR}^{c}$. In the situation where we correlate $b\to
c\tau\bar{\nu}$ and $b\to u\tau\bar{\nu}$ transitions via the flavor-symmetry
assumption in (2.18), also the branching ratio for the decay
$B^{-}\to\tau\bar{\nu}$ plays an important role. The data used for the fit are
reported in Table 2.2, while the explicit expressions of the various
observables in terms of the Wilson coefficients are collected in Appendix B.

In Figure 2.2, we show the allowed regions for the coefficients
${\mathcal{C}}_{LL}^{c}$ and ${\mathcal{C}}_{LR}^{c}$ obtained from the
measurements of $R_{D}$, $R_{D^{*}}$ and
${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$ (colored bands), where in the latter
case we assume the validity of the relations (2.18). The solid contour lines
show the result of a fit to $R_{D}$ and $R_{D^{*}}$ only, while the dashed
lines refer to a fit including also ${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$
under the hypothesis of minimal $U(2)^{5}$ breaking. In the first case, which
is more conservative, the significance of the NP hypothesis compared to the SM
case (two-parameter fit) is 3.2$\sigma$. In the figure, we report the results
in terms of the effective coefficients ${\mathcal{C}}_{LL,LR}^{c}$ evaluated
at the high scale $\Lambda=2$ TeV. While RG evolution effects are negligible
for ${\mathcal{C}}_{LL}^{c}$, the mixed-chirality coefficients
${\mathcal{C}}_{LR}^{c}$ exhibits a sizable scale variation due to QCD
corrections.

In Figure 2.2, we also show as red lines the relations
${\mathcal{C}}_{LR}^{c}=0$ and
${\mathcal{C}}_{LR}^{c}=-{\mathcal{C}}_{LL}^{c}$. The latter is consistent
with the expectation
$|{\mathcal{C}}^{i3\tau\tau}_{LR}|=|{\mathcal{C}}^{i3\tau\tau}_{LL}|$ that, in
turn, is a natural benchmark for models in which these two coefficients are
generated by the tree-level exchange of a Pati-Salam-like massive leptoquark
(see Section 3 for a detailed discussion). From the figure we draw the
following conclusions:

  * •

Without the inclusion of ${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$, the present
data are compatible with both ${\mathcal{C}}_{LR}^{c}=0$ and
${\mathcal{C}}_{LR}^{c}=-{\mathcal{C}}_{LL}^{c}$, and likewise with any
intermediate case.

  * •

The inclusion of ${\mathcal{B}}(B^{-}\to\tau\bar{\nu})$ under the hypothesis
of minimal $U(2)^{5}$ breaking is perfectly consistent with the other
constraints. However, it slightly disfavors (by less than $2\sigma$) a
scenario where $|{\mathcal{C}}_{LR}^{c}|\approx|{\mathcal{C}}_{LL}^{c}|$.

In our analysis below we study the cases ${\mathcal{C}}^{i3\tau\tau}_{LR}=0$
and ${\mathcal{C}}^{i3\tau\tau}_{LR}=-{\mathcal{C}}^{i3\tau\tau}_{LL}$ as two
representative benchmark scenarios. To keep the discussion general, we allow
in both cases for non-minimal, subleading $U(2)^{5}$-breaking terms, which
modify the relation (2.16), and provide an a posteriori validation of it.

###  2.4 Combined analysis of the semileptonic couplings involving $\tau$
leptons

We will now study the overall consistency of the EFT description of the two
sets of anomalies by focusing on the couplings
${\mathcal{C}}^{33\tau\tau}_{LL}$ and ${\mathcal{C}}^{23\tau\tau}_{LL}$
involving tau leptons. In the approximation where the very small contribution
to ${\mathcal{C}}_{LL}^{c}$ proportional to ${\mathcal{C}}^{13\tau\tau}_{LL}$
in (2.14) is neglected, the observables $R_{D}$ and $R_{D^{*}}$ are sensitive
to NP effects described by both ${\mathcal{C}}^{33\tau\tau}_{LL}$ and
${\mathcal{C}}^{23\tau\tau}_{LL}$. The blue band in Figure 2.3 shows the
allowed $1\sigma$ and $2\sigma$ regions in the
${\mathcal{C}}^{33\tau\tau}_{LL}$–$\,{\mathcal{C}}^{23\tau\tau}_{LL}$ plane in
the two benchmark scenarios defined above, in which the corresponding mixed-
chirality coefficients ${\mathcal{C}}^{33\tau\tau}_{LR}$ and
${\mathcal{C}}^{23\tau\tau}_{LR}$ are fixed. The $b\to s\mu^{+}\mu^{-}$
observables, on the other hand, are sensitive to NP effects parameterized by
${\mathcal{C}}^{23\tau\tau}_{LL}$ alone, after we marginalize over
${\mathcal{C}}^{33\mu\mu}_{LL}$ (see the right panel of Figure 2.1). The
corresponding allowed region (at $1\sigma$ and $2\sigma$) is shown by the
horizontal orange bands.

The EFT approach considered so far does not allow us to take into account in a
precise way all constraints on the couplings ${\mathcal{C}}^{33\tau\tau}_{LL}$
and ${\mathcal{C}}^{23\tau\tau}_{LL}$ derived from other observables not
directly related to the flavor anomalies. These will be analyzed in a more
systematic way in the next section. However, we show in Figure 2.3 in a semi-
quantitative way the three most relevant constraints. They arise from high-
energy measurements of the $\tau^{+}\tau^{-}$ production cross section at the
LHC, which can be affected in the presence of the four-fermion contact
interactions in (2.2), from LFU tests in $\tau$ decays, and from precision
studies of the $B_{s}$–$\bar{B}_{s}$ mixing amplitude.

Figure 2.3:  Combined analysis of the coefficients
${\mathcal{C}}_{LL}^{33\tau\tau}$ and ${\mathcal{C}}_{LL}^{23\tau\tau}$ in the
two benchmark scenarios defined by ${\mathcal{C}}_{LR}^{i3\tau\tau}=0$ (left
panel) and ${\mathcal{C}}_{LR}^{i3\tau\tau}=-{\mathcal{C}}_{LL}^{i3\tau\tau}$
(right panel). The blue bands denote the $1\sigma$ and $2\sigma$ regions
preferred by $b\to c\tau\bar{\nu}$ data, while the gray bands show the
exclusion regions derived from $\sigma(pp\to\tau^{+}\tau^{-}+X)$. The
preferred values of ${\mathcal{C}}^{23\tau\tau}_{LL}$ derived from $b\to
s\mu^{+}\mu^{-}$ data (at $1\sigma$ and $2\sigma$) are indicated by the
horizontal orange bands. The dashed red lines provide a qualitative indication
of the bound from $B_{s}$–$\bar{B}_{s}$ mixing (see text for more details).
The gray lines indicate reference values of the ratio
${\mathcal{C}}^{23\tau\tau}_{LL}/{\mathcal{C}}^{33\tau\tau}_{LL}\sim\epsilon_{q}$.

The bound from modifications of the high-$p_{T}$ tails in
$pp\to\tau^{+}\tau^{-}+X$ processes, whose detailed derivation is postponed to
Section 3, is only weakly sensitive to the details of the UV completion of the
EFT. At the energies accessible at the LHC, the effect of the heavy (multi-TeV
scale) mediators is expected to be well described by the contact interactions.
We include in our estimates the contributions from $b\bar{b}$-, $b\bar{s}$\-
and $s\bar{b}$-initiated scattering processes. In the benchmark scenario with
${\mathcal{C}}_{LR}^{i3\tau\tau}=0$ (left panel) we also set
${\mathcal{C}}_{RR}^{i3\tau\tau}=0$, while in the scenario with
${\mathcal{C}}_{LR}^{i3\tau\tau}=-{\mathcal{C}}_{LL}^{i3\tau\tau}$ (right
panel) we take
${\mathcal{C}}_{RR}^{33\tau\tau}=-{\mathcal{C}}_{LR}^{33\tau\tau}={\mathcal{C}}_{LL}^{33\tau\tau}$,
as expected from a UV completion with a Pati-Salam-like $U_{1}$ leptoquark.

Figure 2.4:  Different NP contributions to $B_{s}$–$\bar{B}_{s}$ mixing in the
EFT. The red vertices denote the insertion of
${\mathcal{O}}^{32\tau\tau}_{LL}$ from (2.2), while the blue vertex indicates
generic short-distance contributions probing, in general, all heavy degrees of
freedom present in the UV completion of the EFT. The first diagram contributes
at dimension-8 order in SMEFT power counting, but the fact that it is
quadratically divergent indicates that the blue vertex must contain a
corresponding contribution with the same coupling structure.

The EFT treatment of $B_{s}$–$\bar{B}_{s}$ mixing is more model dependent. In
general, we define the short-distance contributions to the effective
Lagrangian for the $\Delta F=2$ mixing amplitudes as

${\mathcal{L}}^{\Delta
F=2}=-{\mathcal{C}}_{bs}\left(\bar{b}_{L}\gamma_{\mu}s_{L}\right)^{2}-{\mathcal{C}}_{bd}\left(\bar{b}_{L}\gamma_{\mu}d_{L}\right)^{2}-{\mathcal{C}}_{uc}\left(\bar{u}_{L}\gamma_{\mu}c_{L}\right)^{2}+\text{h.c.}\,,$
(2.19)

where for later convenience we also write the four-quark operators needed for
the description of the $B_{d}$ –$\bar{B}_{d}$ and $D$–$\bar{D}$ mixing
amplitudes. The SM contribution to $B_{s}$–$\bar{B}_{s}$ mixing is given by

${\mathcal{C}}^{\rm
SM}_{bs}=\frac{G_{F}^{2}m_{W}^{2}}{4\pi^{2}}\,\big{(}V_{tb}^{*}V_{ts}\big{)}^{2}\,S_{0}(x_{t})\,,$
(2.20)

with $x_{t}=m_{t}^{2}/m_{W}^{2}$ and $S_{0}(x_{t})\approx 2.37$ [61]. Without
reference to a concrete UV completion, there is no model-independent relation
between the Wilson coefficient ${\mathcal{C}}_{bs}$ and the Wilson
coefficients entering the $\Delta F=1$ effective Lagrangian in (2.2). Note, in
particular, that loop diagrams such as the first graph in Figure 2.4,
containing two insertions of dimension-6 SMEFT operators, contribute only at
dimension-8 in the SMEFT power counting. However, naive dimensional analysis
shows that these loop graphs are quadratically UV-divergent, and therefore
there must unavoidably exist a corresponding short-distance contribution to
the Wilson coefficient of the dimension-6 operator
$(\bar{b}_{L}\gamma_{\mu}s_{L})^{2}$ with the same coupling structure, as
shown by the second graph in Figure 2.4. We thus expect that (with $i=s,d$)

$\frac{{\mathcal{C}}_{bi}^{\rm NP}}{{\mathcal{C}}_{bi}^{\rm
SM}}=\frac{\Lambda_{bi}^{2}}{2m_{W}^{2}\,S_{0}(x_{t})}\left[\frac{\left({\mathcal{C}}^{i3\tau\tau}_{LL}\right)^{*}}{V_{tb}^{*}V_{ti}}\right]^{2}+\dots\,,\qquad{\mathcal{C}}_{uc}^{\rm
NP}=\frac{G_{F}^{2}\Lambda_{uc}^{2}}{8\pi^{2}}\,\sum_{i,j=1}^{3}\,\big{(}V_{ui}V_{cj}^{*}\,{\mathcal{C}}_{LL}^{ij\tau\tau}\big{)}^{2}+\dots\,,$
(2.21)

where $\Lambda_{ij}$ are combinations of mass parameters associated with new
heavy particles in the UV theory, and the dots represent possible other UV
contributions associated with different coupling parameters. Depending on the
details of the model, the scale $\Lambda_{bs}$ can be lighter than the mass of
the mediator responsible for the $\Delta F=1$ contact interactions shown in
(2.2).

For instance, in the class of models analyzed in [90, 58], in which the
$U_{1}$ leptoquark is a massive gauge boson, the scale $\Lambda_{bs}$ relevant
to $B_{s}$–$\bar{B}_{s}$ mixing is related to the mass $M_{L}$ of the vector-
like leptons, which are responsible for the 2-3 flavor mixing in the $U_{1}$
couplings. More precisely, for $M_{L}\ll M_{U}$ one finds that
$\Lambda_{bs}\approx\,M_{L}$ [58]. The fact that in these models
$\Lambda_{bs}$ is associated with the mass of a colorless particle (a vector-
like lepton) is a highly non-trivial feature, which allows for a relatively
low value $\Lambda_{bs}\lesssim 1$ TeV without conflicting with current bounds
from direct searches at the LHC (see Section 4 for more details).

In the case of $B_{s}$–$\bar{B}_{s}$ and $B_{d}$ –$\bar{B}_{d}$ mixing, there
exist stringent constraints on both the magnitude and the phase of the mixing
amplitudes. It is then convenient to define (with $i=s,d$)

$\delta(\Delta m_{B_{i}})=\frac{\Delta m_{B_{i}}-\Delta m^{\rm
SM}_{B_{i}}}{\Delta m^{\rm SM}_{B_{i}}}=\left|1+\frac{{\mathcal{C}}^{\rm
NP}_{bi}}{{\mathcal{C}}_{bi}^{\rm
SM}}\right|-1\,,\qquad\delta(\phi_{B_{i}})=\arg\left(1+\frac{{\mathcal{C}}^{\rm
NP}_{bi}}{{\mathcal{C}}_{bi}^{\rm SM}}\right).$ (2.22)

The horizontal red line in Figure 2.3 corresponds to $\delta(\Delta
m_{B_{s}})=0.1$, i.e. to a 10% correction to the magnitude of $\Delta
m_{B_{s}}$, under the assumption that $\Lambda_{bs}=1$ TeV. Without entering
into model-dependent considerations, we note that the absence of direct
signals of new physics at the energy frontier implies that is very hard to
conceive explicit models with $\Lambda_{bs}$ much below 1 TeV. This is why the
region above the red line should be considered as disfavored by
$B_{s}$–$\bar{B}_{s}$ mixing, barring cancellations between the contribution
shown explicitly in (2.21) and additional contributions related to different
coupling parameters.

##### Discussion.

From the two plots in Figure 2.3 we draw the following conclusions:

  * •

Overall there is a good consistency between the values of
${\mathcal{C}}^{i3\tau\tau}_{LL,LR}$ necessary to fit $b\to c\tau\bar{\nu}$
data, the value of ${\mathcal{C}}^{23\tau\tau}_{LL}$ indicated by $b\to
s\hskip 0.85358pt\ell^{+}\ell^{-}$ data, and the flavor scaling assumed in
Section 2.1 (gray lines in Figure 2.3).

  * •

The bound from $pp\to\tau^{+}\tau^{-}$ prevents a solution of the $b\to
c\tau\bar{\nu}$ anomalies with vanishing ${\mathcal{C}}^{23\tau\tau}_{LL,LR}$,
while $B_{s}$–$\bar{B}_{s}$ mixing tends to favor the smallest possible value
of $|{\mathcal{C}}^{23\tau\tau}_{LL}|$.

  * •

While the benchmark scenario with ${\mathcal{C}}^{i3\tau\tau}_{LR}=0$ appears
to be favored by the $b\to c\tau\bar{\nu}$ data alone (see Figure 2.2), the
$B_{s}$–$\bar{B}_{s}$ mixing bound is more stringent in this case. Hence, with
present data it is still useful to consider both benchmark scenarios.

##  3 The simplified $U_{1}$ model

In this section we analyze the case where the effective operators in (2.2) are
generated by the tree-level exchange of a
$U_{1}^{\mu}\sim(\mathbf{3},\mathbf{1})_{2/3}$ leptoquark. At the expense of
introducing some model dependence, this assumption allows to consider a wider
class of observables in terms of a reduced number of free parameters (the
effective leptoquark couplings), and it simplifies the interpretation of
available data in the class of UV completions we are interested in. Note that
introducing a new heavy vector boson necessarily requires an additional new-
physics sector, which gives mass to this particle. Here we consider a
simplified model, focusing only on the $U_{1}$ as the dominant source of new
flavor-changing interactions. In Section 4, we will then explore a concrete
example of a consistent UV completion.

The most general Lagrangian for a $U_{1}$ vector leptoquark coupling to SM
particles is given by

$\displaystyle\begin{aligned}
\mathcal{L}_{U}=&-\frac{1}{2}\,U_{\mu\nu}^{\dagger}\,U^{\mu\nu}+M_{U}^{2}\,U_{\mu}^{\dagger}\,U^{\mu}-ig_{s}\,(1-\kappa_{c})\,U_{\mu}^{\dagger}\,T^{a}U_{\nu}\,G^{\mu\nu,a}\\\
&-\frac{2i}{3}\,g_{Y}\,(1-\kappa_{Y})\,U_{\mu}^{\dagger}\,U_{\nu}\,B^{\mu\nu}+\frac{g_{U}}{\sqrt{2}}\,(U^{\mu}J^{U}_{\mu}+\mathrm{h.c.})\,,\end{aligned}$
(3.1)

where $U_{\mu\nu}=D_{\mu}U_{\nu}-D_{\nu}U_{\mu}\,,$ with
$D_{\mu}=\partial_{\mu}-ig_{s}\,G_{\mu}^{a}T^{a}-i\,\frac{2}{3}g_{Y}B_{\mu}$.
Here $G_{\mu}^{a}$ ($a=1,\dots,8$) and $B_{\mu}$ denote the $SU(3)_{c}$ and
$U(1)_{Y}$ gauge bosons, $g_{s}$ and $g_{Y}$ are the corresponding gauge
couplings, and $T^{a}$ are the generators of $SU(3)_{c}$. In models in which
the vector leptoquark has a gauge origin, $\kappa_{c}=\kappa_{Y}=0$, while
this is not necessarily the case in models in which the $U_{1}$ arises as a
bound state from a strongly-coupled sector. The interaction of the $U_{1}$
with the SM fermions involves the currents

$J^{U}_{\mu}=\beta_{L}^{i\alpha}\,(\bar{q}_{L}^{\,i}\gamma_{\mu}\ell_{L}^{\alpha})+\beta_{R}^{i\alpha}\,(\bar{d}_{R}^{\,i}\gamma_{\mu}e_{R}^{\alpha})\,,$
(3.2)

where the couplings $\beta_{L}$ and $\beta_{R}$ are complex $3\times 3$
matrices in flavor space. Following our hypothesis on the flavor structure of
the theory, we can write

$\displaystyle\beta_{L}=\begin{pmatrix}0&0&\beta_{L}^{d\tau}\\\\[5.0pt]
0&\beta_{L}^{s\mu}&\beta_{L}^{s\tau}\\\\[5.0pt]
0&\beta_{L}^{b\mu}&1\end{pmatrix}\,,\qquad\qquad\beta_{R}=\begin{pmatrix}0&0&0\\\\[5.0pt]
0&0&0\\\\[5.0pt] 0&0&\beta_{R}^{b\tau}\end{pmatrix}\,,$ (3.3)

with $|\beta_{L}^{d\tau,s\mu}|\ll|\beta_{L}^{s\tau,b\mu}|\ll 1$ and
$\beta_{R}^{b\tau}=\mathcal{O}(1)$. The normalization of $g_{U}$ is chosen
such that $\beta_{L}^{b\tau}=1$. The null entries in (3.3) should be
understood as small terms which have a negligible impact on the observables we
analyze. It is worth stressing that this structure is a direct consequence of
the hypothesis of a $U(2)^{5}$ flavor symmetry with sizable breaking only
along the $U(2)_{q}$ direction.333As far as right-handed mixing is concerned,
this symmetry hypothesis alone implies that the natural size of the largest
off-diagonal entries is
$\beta_{R}^{s\tau}\sim(m_{s}/m_{b})\,\beta_{L}^{s\tau}$ and
$\beta_{R}^{b\mu}\sim(m_{\mu}/m_{\tau})\,\beta_{L}^{s\tau}$, well below the
corresponding left-handed entries. Under the stronger assumption of a single
spurion transforming as doublet of $U(2)_{q}\in U(2)^{5}$, we further expect
$\beta_{L}^{d\tau}/\beta_{L}^{s\tau}=V^{*}_{td}/V^{*}_{ts}$ (see Appendix A).

By integrating out the vector leptoquark at tree level, we obtain the
following matching conditions for the effective operators introduced in
Section 2.1:

${\mathcal{C}}_{LL}^{ij\alpha\beta}=C_{U}\beta_{L}^{i\alpha}(\beta_{L}^{j\beta})^{*}\,,\qquad{\mathcal{C}}_{LR}^{ij\alpha\beta}=C_{U}\beta_{L}^{i\alpha}(\beta_{R}^{j\beta})^{*}\,,\qquad{\mathcal{C}}_{RR}^{ij\alpha\beta}=C_{U}\beta_{R}^{i\alpha}(\beta_{R}^{j\beta})^{*}\,,$
(3.4)

where $C_{U}\equiv g_{U}^{2}v^{2}/(4M_{U}^{2})$.

###  3.1 Low-energy fit in the simplified model

In this section, we perform a fit to the $U_{1}$ model parameters described
above with the value of $\beta_{R}^{b\tau}$ fixed to one of the two reference
values $\beta_{R}^{b\tau}=0$ and $\beta_{R}^{b\tau}=-1$, corresponding to the
two benchmark scenarios discussed in Section 2.4. We recall that in models
with third-family Pati-Salam unification, and in absence of a mixing of the SM
fermions with exotic fermions, one expects $|\beta_{R}^{b\tau}|=1$ (see
Section 4). A value $|\beta_{R}^{b\tau}|\ll 1$ can be obtained, for instance,
in models where the $U_{1}$ is a composite state. The condition
$|\beta_{R}^{b\tau}|=1$ does not fix the phase of $\beta_{R}^{b\tau}$. In the
large-$\beta_{R}^{b\tau}$ scenario we set $\beta_{R}^{b\tau}=-1$ in order to
maximize the constructive interference of left-handed and right-handed
contributions in the charged-current anomalies [52]. For similar reasons, we
assume $\beta_{L}^{s\tau}$ to be real (in the down-quark mass basis). In the
absence of observables providing stringent constraints on the corresponding
phases, we also assume $\beta_{L}^{b\mu}$ and $\beta_{L}^{s\mu}$ to be real.
Note, however, that we treat $\beta_{L}^{d\tau}$ as a complex parameter,
because its phase plays an important role in $B_{d}$ –$\bar{B}_{d}$ mixing.
The observables entering the fit (in addition to those discussed in Section 2
and collected in Tables 2.1 and 2.2), together with their SM predictions and
experimental values, are reported in Table 3.1. These include LFU tests in
$\tau$ decays, encoded in the ratios
$\left(g_{\tau}/g_{e,\mu}\right)_{\ell,\pi,K}$ defined in (B.17), $B$ decays
based on the $b\to s\tau^{+}\tau^{-}$ and $b\to s\tau^{\pm}\mu^{\mp}$
transitions, the LFV tau decays $\tau\to\mu\gamma$ and $\tau\to\mu\phi$, and
$\Delta F=2$ amplitudes. The choice of these observables is motivated by their
potential in constraining the fit parameters.

Observable | Experiment/constraint | SM prediction | Theory expr.
---|---|---|---
$(g_{\tau}/g_{e,\mu})_{\ell,\pi,K}$ |  $1.0012\pm 0.0012$ [91] | $1$ | (B.18)
$\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})$ |  $(-0.8\pm 3.5)\times 10^{-3}$ [92] |  $(7.73\pm 0.49)\times 10^{-7}$ [93] | (B.13)
$\mathcal{B}(B^{+}\to K^{+}\tau^{+}\tau^{-})$ |  $(1.35\pm 0.70)\times 10^{-3}$ [94] |  $(1.4\pm 0.2)\times 10^{-7}$ [95] | (B.2)
$\mathcal{B}(B_{s}\to\tau^{\pm}\mu^{\mp})$ |  $<4.2\times 10^{-5}\leavevmode\nobreak\ (95\%\,\mathrm{CL})$ [96] | $0$ | (B.16)
$\mathcal{B}(B^{+}\to K^{+}\tau^{+}\mu^{-})$ |  $<3.3\times 10^{-5}\leavevmode\nobreak\ (95\%\,\mathrm{CL})$ [97] | $0$ | (B.2)
$\mathcal{B}(\tau\to\mu\gamma)$ |  $<5.2\times 10^{-8}\leavevmode\nobreak\ (95\%\,\mathrm{CL})$ [91] | $0$ | (B.23)
$\mathcal{B}(\tau\to\mu\phi)$ |  $<1.0\times 10^{-7}\leavevmode\nobreak\ (95\%\,\mathrm{CL})$ [98] | $0$ | (B.24)
$\delta(\Delta m_{B_{s}})$ |  $0.0\pm 0.1$ [*] | $0$ |
$\delta(\Delta m_{B_{d}})$ |  $0.0\pm 0.1$ [*] | $0$ |
$\delta(\phi_{d})\,[^{o}]$ |  $-1.0\pm 0.9$ [99, 89] | $0$ | (2.21)–(2.22)
${\rm Im}({\mathcal{C}}_{uc}^{\rm NP})\leavevmode\nobreak\ [{\rm GeV}^{-2}]$ |  $(-0.03\pm 0.46)\times 10^{-14}$ [100, 99] | $0$ |
${\rm Re}({\mathcal{C}}_{uc}^{\rm NP})\leavevmode\nobreak\ [{\rm GeV}^{-2}]$ |  $(0.3\pm 1.4)\times 10^{-13}$ [100, 99] | $0$ |
Table 3.1:  Low-energy observables included in the fit of the $U_{1}$
couplings (in addition to the observables in Tables 2.1 and 2.2). The entries
marked with a [*] denote our constraint imposed on the magnitude of the
$\Delta F=2$ amplitudes (see text for further explanation).

The fit results are shown in Figure 3.1 and summarized in Table 3.2. Such
results represent a refined version of the analysis presented in [95], with
updated inputs and a series of relevant differences, as listed below.

  * •

Contrary to [95], we analyze two scenarios with $\beta_{R}^{b\tau}=0$ or
$\beta_{R}^{b\tau}=-1$. Moreover, in both cases we analyze the impact of
relaxing the minimally-broken $U(2)^{5}$ relation
$\beta_{L}^{d\tau}/\beta_{L}^{s\tau}={V_{td}^{\ast}}/{V_{ts}^{\ast}}$ on the
subleading coupling $\beta_{L}^{d\tau}$.

  * •

We impose a smooth cutoff on large values of $|\beta_{L}^{s\tau}|$ and
$|\beta_{L}^{b\mu}|$ via a Gaussian suppression factor with $\sigma=0.05$ for
$|\beta_{L}^{s\tau,b\mu}|>0.2$.

  * •

We include constraints from the $\Delta F=2$ mixing amplitudes as reported in
Table 3.1. Since these amplitudes depend on the UV completion of the model, we
implement the corresponding constraints in a “mild” way in order to minimize
the model-dependence. In practice, we use the estimates derived in (2.21) and
evaluate them setting
$\Lambda_{bs}=\Lambda_{bd}=\Lambda_{uc}=1\leavevmode\nobreak\ {\rm TeV}$.
Moreover, in order to avoid a possible bias from UV-sensitive observables, we
require the contributions to $|\delta(\Delta m_{B_{s,d}})|$ thus estimated not
to exceed $10\%$ (at $1\sigma$), which is in line with the present error on
$|\delta(\Delta m_{B_{s,d}})|$ from global CKM fits [99].

  * •

We implement the one-loop contribution to $\tau\to\mu\gamma$ according to the
complete result presented in [58]. As explained in [95], this observable
remains largely insensitive to the UV completion of the model (see the
discussion in Section 4).

Figure 3.1:  Two-dimensional constraints on the $U_{1}$ couplings obtained by including the low-energy inputs in Tables 2.1, 2.2 and 3.1. The two colors correspond to the benchmarks $\beta_{R}^{b\tau}=0$ (orange) and $\beta_{R}^{b\tau}=-1$ (purple), with $\beta_{L}^{d\tau}$ unconstrained. For each benchmark, the darker and lighter regions denote $\Delta\chi^{2}\leq 2.30$ ($1\sigma$) and $\Delta\chi^{2}\leq 6.18$ ($2\sigma$), respectively. In the $C_{U}$–$\beta_{L}^{s\tau}$ plot (upper left), we also show dashed contour lines corresponding to $\delta(\Delta m_{B_{s}})=0.1$ for different values of $\Lambda_{bs}$ in (2.21). In the $\beta_{L}^{s\tau}$–$\beta_{L}^{d\tau}$ plot (lower left) the red solid line corresponds to the $U(2)^{5}$ relation $\beta_{L}^{d\tau}/\beta_{L}^{s\tau}=V^{*}_{td}/V^{*}_{ts}$. Scenario | $\boldsymbol{\Delta\chi^{2}}$ | Parameter | best fit | $\boldsymbol{1\sigma}$
---|---|---|---|---
no RH currents ($\beta_{R}^{b\tau}=0$) | min. $U(2)^{5}$ breaking ($\beta_{L}^{d\tau}=V_{td}^{\ast}/V_{ts}^{\ast}\,\beta_{L}^{s\tau}$) | $58.6$ | $C_{U}$ | $0.010$ | $\left[0.007,0.017\right]$
$\beta_{L}^{b\mu}$ | ${-0.15}$ | ${\left[-0.26,-0.02\right]}$
$\beta_{L}^{s\tau}$ | $0.19$ | $\left[0.10,0.25\right]$
$\beta_{L}^{s\mu}$ | $0.014$ | $\left[0.004,0.14\right]$
$\beta_{L}^{d\tau}$ free | $59.5$ | $C_{U}$ | $0.011$ | $\left[0.007,0.018\right]$
$\beta_{L}^{b\mu}$ | $-0.14$ | $\left[-0.25,-0.02\right]$
$\beta_{L}^{s\tau}$ | $0.19$ | $\left[0.11,0.24\right]$
$\beta_{L}^{s\mu}$ | $0.013$ | $\left[0.005,0.125\right]$
$\left|V_{ts}^{\ast}/V_{td}^{\ast}\,\beta_{L}^{d\tau}\right|$ | $0.11$ | $\left[0.04,0.24\right]$
$\mathrm{arg}\left(V_{ts}^{\ast}/V_{td}^{\ast}\,\beta_{L}^{d\tau}\right)$ | $0.4\,\pi$ | $\left[0.0,0.5\right]\pi$
max. RH currents ($\beta_{R}^{b\tau}=-1$) | min. $U(2)^{5}$ breaking ($\beta_{L}^{d\tau}=V_{td}^{\ast}/V_{ts}^{\ast}\,\beta_{L}^{s\tau}$) | $54.0$ | $C_{U}$ | $0.004$ | $\left[0.002,0.006\right]$
$\beta_{L}^{b\mu}$ | $-0.21$ | $\left[-0.26,-0.16\right]$
$\beta_{L}^{s\tau}$ | $0.21$ | $\left[0.12,0.26\right]$
$\beta_{L}^{s\mu}$ | $0.03$ | $\left[0.01,0.04\right]$
$\beta_{L}^{d\tau}$ free | $56.7$ | $C_{U}$ | $0.005$ | $\left[0.004,0.007\right]$
$\beta_{L}^{b\mu}$ | $-0.21$ | $\left[-0.25,-0.14\right]$
$\beta_{L}^{s\tau}$ | $0.21$ | $\left[0.15,0.26\right]$
$\beta_{L}^{s\mu}$ | $0.02$ | $\left[0.01,0.07\right]$
$\left|V_{ts}^{\ast}/V_{td}^{\ast}\,\beta_{L}^{d\tau}\right|$ | $0.24$ | $\left[0.13,0.34\right]$
$\mathrm{arg}\left(V_{ts}^{\ast}/V_{td}^{\ast}\,\beta_{L}^{d\tau}\right)$ | $-0.6\,\pi$ | $\left[-0.7,-0.5\right]\pi$
Table 3.2:  Fit results for $\beta_{R}^{b\tau}=0$ and $\beta_{R}^{b\tau}=-1$,
with and without the additional assumption of minimal $U(2)^{5}$ breaking. For
each scenario we report the $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm SM}$, the
best-fit values and the $1\sigma$ confidence intervals (marginalizing over the
other parameters) for all fit parameters.

Figure 3.2:  Preferred $1\sigma$ and $2\sigma$ regions for the ratios $\delta
R_{D^{(\ast)}}$ and $\delta R_{K}^{[1.1,6]}$ resulting from the low-energy fit
for $\beta_{R}^{b\tau}=0$ (orange) and $\beta_{R}^{b\tau}=-1$ (purple). Note
that in both cases $\delta R_{K}^{\ast}\approx\delta R_{K}$ to a very good
approximation. The colored error bars show the current experimental
measurements at $1\sigma$.

##### Discussion.

From Table 3.2 and the plots in Figure 3.1 we draw the following conclusions:

  * •

All four scenarios provide a very good fit to the data. Taking into account
that the NP fits have 4 or 6 parameters, depending on whether we leave
$\beta_{L}^{d\tau}$ free to vary in magnitude and phase, the $\Delta\chi^{2}$
values reported in Table 3.2 indicate a strong significance of the NP
hypothesis (well above the $5\sigma$ level) compared to the SM in all four
cases.

  * •

A different way to illustrate the previous statement is offered by Figure 3.2,
where we show the $1\sigma$ and $2\sigma$ preferred regions for the LFU-
violating observables

$\delta R_{K^{(*)}}=\frac{R_{K^{(*)}}-R^{\rm SM}_{K^{(*)}}}{R^{\rm
SM}_{K^{(*)}}}\,,\qquad\delta R_{D^{(*)}}=\frac{R_{D^{(*)}}-R^{\rm
SM}_{D^{(*)}}}{R^{\rm SM}_{D^{(*)}}}\,,\qquad$ (3.5)

obtained for the two reference options $\beta_{R}^{b\tau}=0$ with minimal
$U(2)^{5}$ breaking, and $\beta_{R}^{b\tau}=-1$ with generic
$\beta_{L}^{d\tau}$, corresponding to the two benchmark scenarios analyzed in
Section 2. As can be seen, the overall agreement with data is very good in
both cases. The $\beta_{R}^{b\tau}=-1$ case is slightly disfavored (as
indicated by the lower $\Delta\chi^{2}$ values in Table 3.2), because of the
lower prediction of $R_{D^{*}}$ compared to data.

  * •

As found in previous studies [43, 95], there is a strong correlation between
$C_{U}$ and $\beta_{L}^{s\tau}$ (upper left plot in Figure 3.1). This is a
consequence of $\delta R_{D^{(*)}}$, which fixes the product of $C_{U}$ and
$\beta_{L}^{s\tau}$.

  * •

The value of $\delta R_{K^{(*)}}$ fixes the size and the sign of the product
of $\beta_{L}^{b\mu}$ and $\beta_{L}^{s\mu}$. The individual signs of the two
couplings are not determined, and we choose $\beta_{L}^{b\mu}<0$ and
$\beta_{L}^{s\mu}>0$ when displaying the fit results. The magnitudes of the
$\beta_{L}^{i\ell}$ parameters are consistent with the expected scaling rules
$|\beta_{L}^{s\tau}|\sim\epsilon_{q}\sim 0.1$,
$|\beta_{L}^{b\mu}|\sim\epsilon_{\ell}\sim 0.1$ and
$|\beta_{L}^{s\mu}|\sim\epsilon_{q}\hskip 0.85358pt\epsilon_{\ell}\sim
10^{-2}$.

  * •

As already found in the EFT analysis, the $\Delta F=2$ constraint is
particularly relevant, especially in the scenario with only left-handed
couplings. Depending on the value of the effective mass parameter
$\Lambda_{bs}$ in (2.21), it cuts out a significant region parameter space
above the dashed contour lines shown in the upper left plot in Figure 3.1. For
the relatively low value $\Lambda_{bs}=1$ TeV adopted in our analysis, the
effect of imposing this cut is still relatively mild, but larger values would
give rise to much tighter cuts. As mentioned earlier, the value of 1 TeV is
motivated by the fact that in the specific UV completion we have in mind the
parameter $\Lambda_{bs}$ is associated with the mass of a heavy vector-like
lepton (see Section 4 for more details).

  * •

The minimal $U(2)^{5}$-breaking relation
$\beta_{L}^{d\tau}/\beta_{L}^{s\tau}=V^{*}_{td}/V^{*}_{ts}$ is well supported
by data. This becomes evident when one compares the $\Delta\chi^{2}$ values in
Table 3.2 obtained with and without imposing this hypothesis. However, as
shown by the lower plots in Figure 3.1, ${\mathcal{O}}(1)$ deviations in both
magnitude and phase are still allowed (and slightly favored in the
$\beta_{R}^{b\tau}=-1$ case).

Figure 3.3: LHC constraints for the $U_{1}$ vector leptoquark for the
benchmark scenarios with $\beta_{R}^{b\tau}=0$ (left) and
$\beta_{R}^{b\tau}=-1$ (right). The $1\sigma$ and $2\sigma$ regions obtained
from the fit to low-energy data are also shown.

###  3.2 Constraints from high-$p_{T}$ observables

Having discussed the most relevant low-energy constraints, we now turn our
attention to the bounds from collider (“high-$p_{T}$”) physics. We focus here
on the constraints on the $U_{1}$ leptoquark that can be derived within the
simplified model defined by the Lagrangian in (3.1), and postpone the
discussion of effects of possible additional TeV-scale states to Section 4.3.
As we did for the low-energy fit, we consider the two reference benchmark
scenarios $\beta_{R}^{b\tau}=0$ and $\beta_{R}^{b\tau}=-1$, and we assume the
same $U(2)$-inspired scaling rules for the $\beta_{L}^{i\alpha}$ couplings
discussed at the beginning of this section and supported by the low-energy
fit.

Leptoquark pair-production cross sections at the LHC are dominated by QCD
dynamics [101, 102, 48, 103] (figure 3.4 a) and thus are largely independent
of the leptoquark couplings to fermions. Nevertheless, a certain model
dependence is still retained in the form of non-minimal couplings to gluons,
parameterized in the Lagrangian in (3.1) by the quantity $\kappa_{c}$. In
models where the vector leptoquark has a gauge origin, this non-minimal
coupling is absent ($\kappa_{c}=0$), allowing for robust theory predictions
for the pair-production cross section. As a consequence, the largest model
dependence for this type of searches arises through the leptoquark branching
fractions to its different decay channels [104]. The flavor structure emerging
from our analysis of the $B$-meson anomalies suggests that the dominant decay
channels are those involving pairs of third-generation fermions, namely
$U_{1}\to b\tau^{+}$ and $U_{1}\to t\bar{\nu}$, with branching fractions that
depend on the value of $\beta_{R}^{b\tau}$. For the benchmark scenarios
considered here, the largest cross section is obtained for $pp\to
U_{1}^{*}U_{1}\to b\tau t\nu$. The CMS collaboration has performed a dedicated
search for this channel using $137\leavevmode\nobreak\ \mathrm{fb}^{-1}$ of
$13\leavevmode\nobreak\ \mathrm{TeV}$ data [105]. The corresponding exclusion
regions (obtained for $\kappa_{c}=0$) are shown in Figure 3.3 for both
benchmark scenarios, together with the $1\sigma$ and $2\sigma$ regions
obtained from the low-energy fit. We also show the projected limits for the
high-luminosity phase of the LHC (HL-LHC with $3\,\mathrm{ab}^{-1}$ of
integrated luminosity) under the assumption that no NP signal is detected and
that statistical and systematic uncertainties scale with the square root of
the luminosity. As can be seen, these searches offer only a relatively small
coverage of the parameter space favored by the low-energy fit. Other direct
searches, such as single-leptoquark production from quark-gluon scattering
[106, 107, 108, 102] (see figure 3.4 b) or resonant production via lepton-
quark fusion [109, 110, 111] (exploiting the recently determined lepton PDFs
from photon splitting [112]) will play a crucial role in the event of a
discovery, but are currently not competitive with other high-$p_{T}$ searches.

Another interesting collider constraint is obtained by searching for
modifications of the high-$p_{T}$ tail in the dilepton invariant mass
distribution in the Drell-Yan process $pp\to\tau^{+}\tau^{-}+X$ induced by
$t$-channel $U_{1}$ exchange [41, 113, 103, 114] (see figure 3.4
c).444Analogous limits from $pp\to\mu\tau$ [103, 115] and
$pp\to\mu^{+}\mu^{-}$ [42] do not provide competitive bounds because of the
flavor suppression of the light-lepton couplings, though they might play a
relevant role in the future in the event of discovery. Similarly, limits
derived from $pp\to\tau\bar{\nu}$ [116] are found to be weaker due to the
smallness of $V_{cs}^{*}\,\beta_{L}^{s\tau}$ and $V_{cb}^{*}$ compared to the
dominant third-generation couplings. The dominant production mechanism for
this channel is via a $b\bar{b}$ initial state, while contributions from
$b\bar{s}$\- and $s\bar{s}$-initiated processes are subdominant due to the
underlying flavor structure of the leptoquark couplings. Stringent limits from
$pp\to\tau^{+}\tau^{-}+X$ data can be obtained by recasting the ATLAS analysis
in [117] with $139\leavevmode\nobreak\ \mathrm{fb}^{-1}$ of $13$ TeV data,
following the same recasting procedure described in [103]. As shown in Figure
3.3, high-$p_{T}$ lepton tails provide important constraints on the parameter
space preferred by the low-energy fit, especially for $\beta_{R}^{b\tau}=-1$,
where the limit is about two times stronger than in the $\beta_{R}^{b\tau}=0$
scenario. However, for both benchmark scenarios a large region of the
parameter space still remains viable. Together with the present bounds, we
also show the projected limits for the HL-LHC, again assuming a naive
luminosity scaling of the uncertainties. Interestingly, we find that the
preferred $1\sigma$ and $2\sigma$ regions for both benchmarks are completely
within the reach of the HL-LHC. We stress that this sensitivity projections do
not consider possible improvements in these searches, e.g. due to a finer and
more extended binning of the transverse mass, which will be available when
more events are collected, or by searching for $b$-tagged jets in the final
state (see e.g. [118, 119]).

Figure 3.4:  Representative Feynman diagrams for vector leptoquark pair
production (a), single-leptoquark production (b), and $t$-channel Drell-Yan
production (c).

##  4 UV completion and further predictions

Any UV completion for a TeV-scale $U_{1}$ field coupled to SM fermions via the
simplified Lagrangian in (3.1) requires additional TeV-scale heavy vectors and
vector-like fermions. New scalar degrees of freedom (radial modes associated
to the symmetry breaking sector) are also expected to arise, but their masses
can be pushed in the ${\mathcal{O}}(10\leavevmode\nobreak\ {\rm TeV})$ domain,
yielding negligible impact on collider phenomenology and low-energy
observables. In all models based on the 4321 gauge group [47],

$\mathcal{G}_{4321}\equiv SU(4)\times SU(3)^{\prime}\times SU(2)_{L}\times
U(1)_{X}\,,$ (4.1)

the additional heavy vectors feature a color octet,
$G^{\prime}\sim(\boldsymbol{8},\boldsymbol{1},0)$, and a SM-singlet,
$Z^{\prime}\sim(\boldsymbol{1},\boldsymbol{1},0)$. As discussed in [103], this
choice is unavoidable in any UV completion (including composite models) by the
closure of the $SU(4)$ algebra hosting the $U_{1}$ generators and by the
requirement of flavor non-universality. The vector-like fermions, transforming
as doublets of $SU(2)_{L}$, are a key ingredient to generate a non-trivial
flavor structure for the $U_{1}$ couplings to left-handed fermions, yielding
flavor off-diagonal couplings such as $\beta_{L}^{s\tau}$ and
$\beta_{L}^{b\mu}$. As shown in [90], the vector-like fermion mixing can be
realized in such a way that sizable off-diagonal entries are generated in the
$U_{1}$ couplings while preserving a flavor-diagonal structure in the
$G^{\prime}$ and $Z^{\prime}$ (tree-level) couplings. The underlying mechanism
resembles the up-down (or $SU(2)_{L}$) CKM flavor misalignment but now in
quark-lepton (or $SU(4)$) space.

The flavor non-universal but flavor-diagonal nature of $G^{\prime}$ and
$Z^{\prime}$ couplings holds in a specific $SU(2)_{L}$ basis for quarks and
leptons. The strong constraints from down-type $\Delta F=2$ observables and
LFV charged lepton decays, which would receive tree-level contributions if
off-diagonal couplings of the $G^{\prime}$ and $Z^{\prime}$ were present, are
the main reason why we adopt the down-quark and charged-lepton mass eigenstate
basis in (2.13) as interaction basis, i.e the basis defining the different
4321 charges of the SM fermions. In such basis, all fields couple with the
same strength, up to group factors, to third-generation fermions. The $U_{1}$
has sizable off-diagonal couplings to the light families controlled by the
mixing with vector-like fermions, while $G^{\prime}$ and $Z^{\prime}$ couple
(almost) universally to light fermions via small couplings suppressed by
$g_{s}^{2}/g_{U}^{2}$ and $g^{2}_{Y}/g_{U}^{2}$ compared to the third-
generation case [95].

In this section, we analyze the predictions of this set up for a series of
low- and high-energy observables. We can group them in three categories: i)
low-energy observables insensitive to the details of the UV dynamics; ii) UV-
sensitive low-energy observables; iii) high-$p_{T}$ observables related to the
additional heavy states predicted by the UV completion.

###  4.1 UV-insensitive low-energy observables

Figure 4.1:  Preferred $1\sigma$ and $2\sigma$ regions for LFV processes
resulting from the low-energy fit in the $U_{1}$ simplified model for
$\beta_{R}^{b\tau}=0$ (orange) and $\beta_{R}^{b\tau}=-1$ (purple). The gray
bands show the $95\%$ CL experimental exclusion limits. Future projections
(rescaled to $95\%$ CL) are denoted by dashed lines. In the $x-$axis of the
left plot only the more stringent exclusion band from
$B_{s}\to\tau^{-}\mu^{+}$ is shown.

Figure 4.2:  Preferred $1\sigma$ and $2\sigma$ regions for di-tau final states
resulting from the low-energy fit in the $U_{1}$ simplified model for
$\beta_{R}^{b\tau}=0$ (orange) and $\beta_{R}^{b\tau}=-1$ (purple). The gray
bands show the $95\%$ CL experimental exclusion limits. Future projections
(rescaled to $95\%$ CL) are denoted by dashed lines. The red line in the lower
plot shows the expected relation between
$\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})$ and
$\mathcal{B}(B\to\tau^{+}\tau^{-})$ for
$\beta_{L}^{d\tau}/\beta_{L}^{s\tau}=V^{*}_{td}/V^{*}_{ts}$ (minimally broken
$U(2)^{5}$).

Several low-energy observables can be predicted directly using the results of
the $U_{1}$ simplified model fit. The most interesting ones are the Lepton
Flavor Violating (LFV) decays $B^{+}\to K^{+}\tau^{+}\mu^{-}$,
$B_{s}\to\tau^{-}\mu^{+}$, $\tau\to\mu\gamma$ and $\tau\to\mu\phi$ (Figure
4.1), and the rare $B_{(s)}$ decays into di-tau final states
$B_{(s)}\to\tau^{+}\tau^{-}$ and $B\to K\tau^{+}\tau^{-}$ (see Figure 4.2).
With the exception of $\tau\to\mu\gamma$, all these observables are dominated
by the tree-level contribution from the $U_{1}$ exchange. The radiative decay
$\tau\to\mu\gamma$ is generated at the loop level, but for
$\beta_{R}^{b\tau}=-1$ it is completely dominated by the contribution due to
$b$-quark running inside the loop (see Section 4.2), and is therefore
insensitive to the UV completion of the model. The explicit expressions of the
amplitudes in terms of the $U_{1}$ couplings are reported in the Appendix B.
It is worth recalling that we used the present bounds on all these observables
but $B\to\tau^{+}\tau^{-}$ (which does not yield significant constraints on
the parameter space) to determine the best fit points. As a consequence,
Figure 4.1 and Figure 4.2 do not show unbiased predictions, but rather
expected ranges for the observables, taking into account all the available
information on the allowed parameter space of the model. As can be seen from
these plots, the case with large $|\beta_{R}^{b\tau}|$ leads to predictions
for $B_{s}\to\tau^{+}\tau^{-}$, $B\to K\tau^{+}\tau^{-}$,
$B_{s}\to\tau^{-}\mu^{+}$, and $\tau\to\mu\gamma$ close to present bounds. In
the dilepton modes, this is because of the chiral-enhancement of the
corresponding hadronic matrix element while for $\tau\to\mu\gamma$, this
occurs because of the large loop-induced amplitude. This observable is not
shown for $\beta_{R}^{b\tau}=0$ since in this case it is well below present
bounds and sensitive to the details of the UV completion, see Section 4.2.

We recall that the two benchmarks we have chosen for $\beta_{R}^{b\tau}$ are
representative of two extreme options. In gauge models, the naïve expectation
is $|\beta_{R}^{b\tau}|=1$, but with a suitable mixing with right-handed
vector-like fermions it is possible to achieve smaller values. A possible
detection of $B_{s}\to\tau^{-}\mu^{+}$ would therefore represent a key
ingredient to determine the size of $|\beta_{R}^{b\tau}|$. An important role
is also played by $\tau\to\mu\phi$, which is a very promising LFV observable
for the LHC and Belle II experiments, and which does not lie far from present
bounds in the pure left-handed case [120, 121]. In the right panel of Figure
4.2 we show the predictions of ${\mathcal{B}}(B\to\tau^{+}\tau^{-})$ vs.
${\mathcal{B}}(B_{s}\to\tau^{+}\tau^{-})$. The ratio of these two rates is
controlled by $\beta_{L}^{d\tau}/\beta_{L}^{s\tau}$, and would therefore be an
ideal probe to test the assumption of minimal breaking of the $U(2)^{5}$
flavor symmetry in the 1–3 sector, in the scenario with sizable right-handed
couplings. In the pure left-handed case, subleading $U(2)^{5}$ breaking terms
could be tested via the LFU ratio
$R_{\pi}={\mathcal{B}}(B^{+}\to\pi^{+}\mu^{+}\mu^{-})/{\mathcal{B}}(B^{+}\to\pi^{+}e^{+}e^{-})$
[60, 122], which would allow us to probe $\beta_{L}^{d\mu}/\beta_{L}^{s\mu}$.

Significant improvements in the experimental searches for all the processes
shown in Figure 4.1 and Figure 4.2 are expected from future Belle II and LHCb
data [123, 124]. In particular, with $5\,\mathrm{ab}^{-1}$ Belle II expects to
set the bounds $\mathcal{B}(B^{+}\to K^{+}\tau^{+}\tau^{-})<6.5\times 10^{-5}$
($90\%$ CL) and $\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})<8.1\times 10^{-4}$
($90\%$ CL) in the absence of any signals555Access to $B_{s}$ at Belle II
requires dedicated runs at the $\Upsilon(5S)$ resonance. The limit we report
assumes an integrated luminosity of $5\,\mathrm{ab}^{-1}$ [123]., while LHCb
should reach $\mathcal{B}(B_{s}\to\tau^{+}\tau^{-})<5\times 10^{-4}$ ($95\%$
CL) by the end of the Upgrade II, thus probing the preferred parameter space
for the $|\beta_{R}^{b\tau}|=1$ case almost entirely. For the LFV processes,
Belle II is expected to reach a sensitivity of $1-2\times 10^{-9}$ ($90\%$ CL)
for both $\mathcal{B}(\tau\to\mu\gamma)$ and $\mathcal{B}(\tau\to\mu\phi)$
with $50\,\mathrm{ab}^{-1}$, and LHCb should achieve
$\mathcal{B}(B_{s}\to\tau\mu)<3\times 10^{-6}$ ($90\%$ CL) at the end of
Upgrade II.

###  4.2 UV-sensitive low-energy observables

We denote as “UV-sensitive” the observables receiving contribution from
dimension-six operators other than those introduced in Section 2.1. In order
to correlate them to the observables discussed so far (i.e. semileptonic
processes involving charged leptons) we need a complete model. Within this
category, we concentrate here on three classes of rare processes which are
particularly interesting given the strong suppression within the SM and the
stringent experimental constraints: meson-antimeson mixing,
${\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})$, and $\tau\to\mu\gamma$.

In all these observables we can identify a $U_{1}$-induced loop contribution,
which is unavoidable in our setup, and additional contributions due to the
other mediators. The latter are controlled by additional parameters, unrelated
to those introduced so far, such as the mixing angles controlling the
misalignment of the interaction basis from the down-quark and charged-lepton
mass eigenstate basis in (2.13). As discussed in Refs. [95, 55], such
misalignment is strongly constrained, especially in the quark sector. In the
following, we concentrate mainly on the $U_{1}$ loop-induced amplitudes, which
can be considered the irreducible contributions to these rare processes
barring fine-tuned cancellations with tree-level mediated $G^{\prime}$ and/or
$Z^{\prime}$ amplitudes. We will discuss in more detail this point in the case
of $\Delta F=2$ observables.

A systematic analysis of the loop-induced amplitudes in the non-universal 4321
model has been presented in [58]. A key role in evaluating the $U_{1}$ loop-
induced contributions is provided by the additional vector-like fermions. To
take their effect into account, we modify the $J^{U}_{\mu}$ current in (3.2)
as follows

$J^{U}_{\mu}\to
J^{U}_{\mu}+\beta_{L}^{Q\alpha}\,(\bar{Q}\gamma_{\mu}\ell_{L}^{\alpha})+\beta_{L}^{iL}\,(\bar{q}_{L}^{i}\gamma_{\mu}L)\leavevmode\nobreak\
,$ (4.2)

where $Q$ ($L$) denote vector-like fermions with left-handed quark (lepton)
quantum numbers that mix with the third and second generation left-handed
chiral quarks (leptons). For the sake of simplicity, and given we focus mainly
on $3\to 2$ transitions in the quark sector, we assume a single family of
vector-like fermions. In order to take advantage of the calculations presented
in [58], we match the notation for the $U_{1}$ couplings to fermions used
therein to the one adopted in the present paper via

$\displaystyle\begin{aligned}
\beta_{L}^{b\tau}&=W_{11}\,,&\beta_{L}^{s\tau}&=-s_{Q}W_{21}\,,&\beta_{L}^{b\mu}&=-s_{L}W_{12}\,,&\beta_{L}^{s\mu}&=s_{Q}s_{L}W_{22}\,,\\\
\beta_{L}^{Q\tau}&=c_{Q}W_{21}\,,&\beta_{L}^{bL}&=c_{L}W_{12}\,,&\beta_{L}^{Q\mu}&=-c_{Q}s_{L}W_{22}\,,&\beta_{L}^{sL}&=-s_{Q}c_{L}W_{22}\,.\end{aligned}$
(4.3)

Here $s_{L(Q)}$ and $c_{L(Q)}$ denote sine and cosine of the mixing angles of
the vector-like fermions with the light (second-generation) chiral fermions,
whereas $W_{ij}$ are the elements of the complex $2\times 2$ unitary matrix
describing the mixing among vector-like fermions and third-generation chiral
fermions. The natural expectation is $|s_{L(Q)}|\ll 1$, $|c_{L(Q)}|\approx 1$,
and $W_{ij}={\mathcal{O}}(1)$.666Note that in (4.3) we have not redefined the
value of $g_{U}$ to absorb the difference of $\beta_{L}^{b\tau}$ from unity.
The mixing matrix $W_{ij}$ is real in the limit where the mass matrix of the
vector-like fermions and the mixing between chiral and vector-like fermions
are aligned in phase. We work under this hypothesis, which justifies having
treated the $\beta_{L}^{i\ell}$ corresponding to $3\leftrightarrow 2$ mixing
as real couplings.

In the following we discuss the three classes of UV sensitive low-energy
observables using the results obtained in [58]. While these results have been
obtained in the non-universal 4321 model, they can easily be extended to the
other realizations of 4321 models (i.e. 4321 models with different charge
assignments for the SM fermions [47]) provided the only relevant flavor mixing
terms are those defined in (4.2).

#### $\Delta F=2$

Figure 4.3:  Upper left: $1\sigma$ and $2\sigma$ regions in the ($\delta
R_{D^{*}}$, $\delta(\Delta m_{B_{s}})$) plane preferred by the low-energy fit,
with $\Lambda_{bs}=1\,\mathrm{TeV}$. Upper right: $\delta(\Delta m_{B_{s}})$
as a function of $\delta R_{D^{*}}$ in the UV complete model for different
values of $M_{L}$, fixing $M_{U}=4$ TeV and $\beta_{L}^{s\tau}=0.15$. Lower
plots: Preferred $1\sigma$ and $2\sigma$ regions for $\delta(\Delta
m_{B_{s}})$ as a function of the vector-like lepton mass for
$M_{U}=4\,\mathrm{TeV}$. As in the other plots, orange and purple correspond
to the benchmarks $\beta_{R}^{b\tau}=0$ and $\beta_{R}^{b\tau}=-1$.

We start the discussion of the $\Delta F=2$ amplitudes by addressing the
possible tree-level contributions from the additional massive gauge bosons. In
the limit where the small and flavor-universal couplings to light families are
neglected, the tree-level exchange of $G^{\prime}$ and $Z^{\prime}$ fields in
non-universal 4321 models leads to four-quark contact interactions among
third-generation fermions. The relevant Lagrangian for these interactions
reads

$\displaystyle{\mathcal{L}}^{(4q)}_{\rm eff}$
$\displaystyle=-\frac{2}{v^{2}}\,\left[C_{G^{\prime}}\left(\bar{q}_{L}^{3}\gamma^{\mu}\,T^{a}q^{3}_{L}+\bar{q}_{R}^{3}\gamma_{\mu}\,T^{a}q_{R}^{3}\right)^{2}+\frac{C_{Z^{\prime}}}{24}\left(\bar{q}_{L}^{3}\gamma^{\mu}q^{3}_{L}+\bar{q}_{R}^{3}\gamma_{\mu}q_{R}^{3}\right)^{2}\right],$
(4.4)

where we have defined $q_{R}^{3}\equiv(u_{R}^{3}\;d_{R}^{3})^{\intercal}$ and
$C_{G^{\prime},Z^{\prime}}=g_{4}^{2}\,v^{2}/(4\,M_{G^{\prime},Z^{\prime}}^{2})+\mathcal{O}(g_{s,Y}^{2}/g_{4}^{2})$,
with $g_{4}=g_{U}$ denoting the $SU(4)$ coupling. Note that the right-handed
currents could have a different overall weight (or even be absent) in other
4321 models.

In the limit where we assume perfect alignment of the right-handed quarks to
their mass basis and of the left-handed quarks to the down-quark mass basis,
there are no contributions to down-type flavor-changing amplitudes. However,
contributions to $D$–$\bar{D}$ mixing are still present. Using the notation
introduced in (2.19) we get

${\mathcal{C}}_{uc}^{\rm NP-
tree}=\frac{2}{v^{2}}\,\left(\frac{C_{G^{\prime}}}{3}+\frac{C_{Z^{\prime}}}{24}\right)\left(V_{ub}^{*}V_{cb}\right)^{2}\approx
1.2\times 10^{-15}\,{\rm
GeV}^{-2}\times\left(\frac{C_{U}}{0.01}\right)\left(\frac{V_{ub}^{*}V_{cb}}{1.0\times
10^{-4}}\right)^{2}\leavevmode\nobreak\ ,$ (4.5)

where, in the second equality, we have taken the limit of degenerate heavy-
vector masses $M_{G^{\prime}}=M_{Z^{\prime}}=M_{U}$, for which
$C_{G^{\prime}}\approx C_{Z^{\prime}}\approx C_{U}$ up to terms of
${\mathcal{O}}(g_{s,Y}^{2}/g_{U}^{2})$. Given the limits reported in Tables
3.1, this contribution falls below the current experimental sensitivity on
$D$–$\bar{D}$ mixing for both the real and the imaginary part of the
amplitude.

On the other hand, if we allow for a misalignment from the right-handed-quark
or down-quark mass bases, a potentially sizable contribution appears in
$B_{(s)}$–$\bar{B}_{(s)}$ mixing. Considering a misalignment only in the left-
handed sector, and denoting by $L_{d}$ the unitary matrix connecting the 4321
interaction basis to the down-quark mass basis
($q^{i}_{L}=(L_{d})_{id_{i}}\,q^{d_{i}}_{L}$, see Appendix A), in the case of
$B_{s}$ mixing we get

$\frac{{\mathcal{C}}_{bs}^{\rm NP-tree}}{{\mathcal{C}}_{bs}^{\rm
SM}}=\frac{16\pi^{2}v^{2}}{m_{W}^{2}\,S_{0}(x_{t})}\left(\frac{C_{G^{\prime}}}{3}+\frac{C_{Z^{\prime}}}{24}\right)\left[\frac{\left(L_{d}\right)^{*}_{3b}(L_{d})_{3s}}{V_{tb}^{*}V_{ts}}\right]^{2}\approx
2.3\times\left(\frac{C_{U}}{0.01}\right)\left[\frac{\left(L_{d}\right)^{*}_{3b}(L_{d})_{3s}}{V_{tb}^{*}V_{ts}}\right]^{2}\leavevmode\nobreak\
,$ (4.6)

where, once more, we took the limit of degenerate heavy-vector masses in the
second equality. For this contribution to give $\delta(m_{B_{s}})<0.1$, the
left-handed down-quark rotation needs to satisfy $|(L_{d})_{3s}|\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}0.2|V_{ts}|$.
This does not require a strong tuning, since the natural size of the possible
misalignment is ${\mathcal{O}}(|V_{ts}|)$. In models with minimal breaking of
the $U(2)^{5}$ flavor symmetry, the term between square brackets in (4.6) is
real [60] and yields a positive contribution to $B_{s}$ mixing. Contributions
of arbitrary sign can be obtained only by relaxing the assumption of minimal
$U(2)^{5}$ breaking (either in the left-handed sector or in the right-handed
sector). However, conceiving models where this tree-level contribution cancels
significantly against the $U_{1}$-induced loop-induced contribution (which is
also positive, as we show below) requires a tuning of magnitude and phase of
the mixing terms for both the $B_{s}$ and the $B_{d}$ amplitudes, which is
rather unnatural. This is why we prefer to adopt the assumption of small
misalignment from the down-type basis, i.e. $|(L_{d})_{3s}|\ll|V_{ts}|$, which
is a more motivated choice from the model-building point of view.

Having analyzed the tree-level contribution to $\Delta F=2$ amplitudes, we now
discuss the $U_{1}$ loop-induced amplitudes. For simplicity, we focus on
$B_{s}-\bar{B}_{s}$ mixing. Following the analysis of [58], the (finite) loop
contribution to ${\mathcal{C}}_{bs}$, taking into account both light and heavy
leptons, is

${\mathcal{C}}_{bs}^{\rm NP-
loop}=\frac{2C_{U}}{v^{2}}\,\frac{g_{4}^{2}}{16\pi^{2}}\,\left[\left(\beta_{L}^{sL}\right)^{*}\beta_{L}^{bL}\right]^{2}\,F_{\Delta
F=2}(x_{L})\leavevmode\nobreak\ ,$ (4.7)

where $x_{L}=M_{L}^{2}/M_{U}^{2}$, with $M_{L}$ being the vector-like lepton
mass, and the expression for the loop function reads [58]

$\displaystyle F_{\Delta F=2}(x)$
$\displaystyle=\frac{x\,(x+4)(x+1)}{8(x-1)^{2}}\left(\frac{1}{2}+\frac{x\ln
x}{1-x^{2}}\right)\,.$ (4.8)

For the sake of completeness we kept a complex notation for
$\beta^{i\ell}_{L}$ although, as anticipated, they are real in the limit where
we introduce a single family of vector-like leptons.

Normalizing to the SM contribution, and taking into account the relations
among the different $\beta^{i\ell}_{L}$ in the limit $|s_{L(Q)}|\ll 1$, we can
write

$\frac{{\mathcal{C}}_{bs}^{\rm NP-tree}}{{\mathcal{C}}_{bs}^{\rm
SM}}\approx\frac{4M_{U}^{2}}{m_{W}^{2}\,S_{0}(x_{t})}C_{U}^{2}\left[\frac{\left(\beta_{L}^{s\tau}\right)^{*}}{V_{tb}^{*}V_{ts}}\right]^{2}\,F_{\Delta
F=2}(x_{L})\leavevmode\nobreak\ .$ (4.9)

Using the expansion $F_{\Delta F=2}(x)=x/4+\mathcal{O}(x^{2})$, the effective
mass combination $\Lambda_{bs}$ introduced in (2.21) can be identified with
$\Lambda_{bs}=\sqrt{2}\,M_{L}$ in the small $x_{L}$ limit. We stress that,
rather than the value of $M_{L}$, what controls the overall size of the
$\Delta F=2$ amplitude is the component of the vector-like mass that breaks
the $SU(4)$ custodial symmetry and is responsible for the flavor mixing of the
$SU(4)$-charged fermions, namely $\Delta M_{L}\propto W_{12}M_{L}$ [58]. In
other words, the $\Delta F=2$ amplitude is finite in the limit of heavy
vector-like masses. However, for $M_{L}\to\infty$ the effective mixing
$\beta_{L}^{s\tau}$ vanishes.

In Figure 4.3, we show the expected value of $\delta(\Delta m_{B_{s}})$ as a
function of $\delta R_{D^{*}}$ and of the vector-like mass using the results
of the low-energy fit. In both cases, we observe an approximate quadratic
dependence, which can be easily understood from (4.9). The quadratic
dependence from $\delta R_{D^{*}}$ is closely connected to the quadratic
dependence from $|\beta_{L}^{s\tau}|$ that, to a large extent, controls the
size of $\delta R_{D^{*}}$. The quadratic dependence from $M_{L}$ is due to
the behavior of $F_{\Delta F=2}(x)$ at small $x$. From these plots it is clear
that, in absence of tuned scenarios allowing a partial cancellation between
tree and loop amplitudes, the $\Delta F=2$ bounds require a vector-like lepton
with mass not far from 1 TeV.

#### $B\to K^{(*)}\nu\bar{\nu}$

We now analyze $b\to s\nu\bar{\nu}$ transitions. As already mentioned in
Section 2, the $U_{1}$ leptoquark does not contribute to these transitions at
the tree-level. These are however generated once we include loop corrections.
Moreover, in the complete UV model, we also have contributions from tree-level
exchange of the $Z^{\prime}$, making $b\to s\nu\bar{\nu}$ transitions a clear
example of UV-sensitive low-energy observable.
We define the relevant Lagrangian as

$\displaystyle{\mathcal{L}}_{b\to
s\nu\bar{\nu}}=-\frac{4G_{F}}{\sqrt{2}}\,V_{ts}^{*}V_{tb}\,{\mathcal{C}}_{\nu}^{\alpha\beta}\,(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\nu}_{L}^{\alpha}\gamma^{\mu}\nu_{L}^{\beta})\,.$
(4.10)

Neglecting effects suppressed by light quark masses, the lepton-flavor-
conserving and -universal SM contribution reads

$\displaystyle{\mathcal{C}}_{\nu,\,{\rm
SM}}=\frac{\alpha_{W}}{2\pi}\,X_{t}\,,$ (4.11)

where $X_{t}=1.48\pm 0.01$ [125], and $\alpha_{W}=g_{L}^{2}/(4\pi)$ with
$g_{L}$ being the $SU(2)_{L}$ coupling. Due to the underlying $U(2)^{5}$
flavor structure, NP effects are dominant in the Wilson coefficient involving
the third family, while the other flavor combinations receive negligible
contributions. Namely, we have ($\ell,\ell^{\prime}=e,\,\mu$)

$\displaystyle{\mathcal{C}}_{\nu}^{\tau\tau}$
$\displaystyle={\mathcal{C}}_{\nu,\,{\rm SM}}+{\mathcal{C}}_{\nu,\,{\rm
NP}}^{\tau\tau}\,,$ $\displaystyle{\mathcal{C}}_{\nu}^{\ell\ell}$
$\displaystyle\approx{\mathcal{C}}_{\nu,\,{\rm SM}}\,,$
$\displaystyle{\mathcal{C}}_{\nu}^{\tau\ell}$
$\displaystyle\approx{\mathcal{C}}_{\nu}^{\ell\tau}\approx{\mathcal{C}}_{\nu}^{\ell\ell^{\prime}}\approx
0\,.$ (4.12)

With these definitions, the NP correction to the $B\to K^{(*)}\nu\bar{\nu}$
branching ratio reads

$\displaystyle\frac{{\mathcal{B}}(B\to
K^{(*)}\nu\bar{\nu})}{{\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})_{\rm
SM}}\approx\frac{2}{3}+\frac{1}{3}\,\left|\frac{{\mathcal{C}}_{\nu,{\rm
NP}}^{\tau\tau}+{\mathcal{C}}_{\nu,{\rm SM}}}{{\mathcal{C}}_{\nu,{\rm
SM}}}\right|^{2}\,.$ (4.13)

We further split the NP effects into $Z^{\prime}$-mediated and $U_{1}$ loop-
induced contributions as follows:

$\displaystyle{\mathcal{C}}_{\nu,\,{\rm NP}}^{\tau\tau}$
$\displaystyle={\mathcal{C}}_{\nu,\,Z^{\prime}}^{\tau\tau}+{\mathcal{C}}_{\nu,\,U}^{\tau\tau}\,.$
(4.14)

At NLO accuracy, we have [56, 58]

$\displaystyle\begin{aligned}
\mathcal{C}_{\nu,\,U}^{\tau\tau}&\approx\mathcal{C}_{\nu,\,U}^{\rm
RGE}+\frac{\alpha_{4}}{4\pi}\,C_{U}\left[-2\,\frac{(L_{d})^{*}_{3s}\,(L_{d})_{3b}}{V_{ts}^{*}V_{tb}}-\frac{\beta_{L}^{sL}\beta_{L}^{bL\,*}}{V_{ts}^{*}V_{tb}}\,F_{\Delta
Q=1}(x_{L})\right]\,,\\\
\mathcal{C}_{\nu,\,Z^{\prime}}^{\tau\tau}&\approx-\,\frac{C_{U}}{4x_{Z^{\prime}}}\left[\frac{(L_{d})^{*}_{3s}\,(L_{d})_{3b}}{V_{ts}^{*}V_{tb}}\left(1+\frac{3}{8}\,\frac{\alpha_{4}}{4\pi}\right)-\frac{\alpha_{4}}{4\pi}\,\frac{\beta_{L}^{sL}\beta_{L}^{bL\,*}}{V_{ts}^{*}V_{tb}}\,G_{\Delta
Q=1}(x_{L},x_{Z^{\prime}},x_{R})\right]\,,\end{aligned}$ (4.15)

where the loop functions are given by,777For concreteness we choose the model
I scenario in [58] when writing $G_{\Delta Q=1}$. As shown in this reference,
similar predictions are obtained in the model II scenario.

$\displaystyle F_{\Delta Q=1}(x)$
$\displaystyle\approx\frac{2x}{1-x}+\frac{2x\ln x}{(x-1)^{2}}\,,$
$\displaystyle G_{\Delta Q=1}(x_{1},x_{2},x_{3})$
$\displaystyle\approx\frac{5}{4}\,x_{1}+\frac{x_{1}}{2}\left(x_{2}-\frac{3}{2}\right)\left(\ln
x_{3}-\frac{5}{2}\right)\,,$ (4.16)

and we defined $x_{Z^{\prime}}\equiv M_{Z^{\prime}}^{2}/M_{U}^{2}$,
$x_{L}\equiv M_{L}^{2}/M_{U}^{2}$ and $x_{R}\equiv M_{R}^{2}/M_{U}^{2}$, with
$M_{R}$ being a scale associated to new scalar degrees of freedom. The
coefficient $\mathcal{C}_{\nu,\,U}^{\rm RGE}$ encodes the RGE-induced
contribution from the tree-level leptoquark-mediated operator
$\mathcal{O}_{LL}^{23\tau\tau}$. Using DsixTools [126] and setting
$\Lambda=2\leavevmode\nobreak\ \mathrm{TeV}$, we find

$\displaystyle\mathcal{C}_{\nu,\,U}^{\rm RGE}$
$\displaystyle=-0.047\,\frac{{\mathcal{C}}_{LL}^{23\tau\tau}(\Lambda)}{V_{ts}^{*}V_{tb}}=-0.047\,C_{U}\,\frac{\beta_{s\tau}}{V_{ts}^{*}V_{tb}}\,.$
(4.17)

Out of the different contributions, those proportional to $(L_{d})_{3s}^{*}$
are tightly constrained by $B_{s}$-mixing (see discussion above) and can
therefore induce at most a $\pm 3\%$ correction to ${\mathcal{B}}(B\to
K^{(*)}\nu\bar{\nu})$. The other contributions, on the other hand, can be
sizable, yielding up to $\mathcal{O}(1)$ corrections to the SM value.
Moreover, the sign of this correction is unambiguously connected to the NP
effect in ${\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})$. More precisely, noting
that $\beta_{L}^{sL}\beta_{L}^{bL\,*}\approx-\beta_{L}^{s\tau}$, an
enhancement in the $R_{D^{(*)}}$ ratio unavoidably yields an enhancement also
in ${\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})$. Note that this enhancement does
not depend directly on $\beta_{R}^{b\tau}$; however, the value of
$\beta_{R}^{b\tau}$ indirectly influences the effect via the extraction of
$C_{U}$ from the fit of $R_{D^{(*)}}$ (see Figure 3.1). This is why the NP
impact in ${\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})$ shown in Figure 4.4 is
smaller for $\beta_{R}^{b\tau}=-1$.

Figure 4.4: Model predictions (best $1\sigma$ and $2\sigma$ fit regions) for
the $B\to K^{(*)}\nu\bar{\nu}$ branching ratio as a function of $M_{L}$,
setting $(L_{d})_{3s}^{*}=0$, $M_{U}=4$ TeV and $g_{4}=3$, and varying the
scale of the scalar degrees of freedom and the $Z^{\prime}$ mass in the
$M_{R}=[1,2\pi]\,M_{U}$ and $M_{Z^{\prime}}=[0.5,1]\,M_{U}$ ranges. As in the
other plots, orange and purple correspond to the benchmarks
$\beta_{R}^{b\tau}=0$ and $\beta_{R}^{b\tau}=-1$.

In Figure 4.4, we show the model predictions for ${\mathcal{B}}(B\to
K^{(*)}\nu\bar{\nu})$ as function of $M_{L}$ for a fixed value of $g_{4}=3$
and $M_{U}=4$ TeV. Current experimental limits $\mathcal{B}(B\to
K\nu\bar{\nu})/\mathcal{B}(B\to K\nu\bar{\nu})_{\rm SM}=2.4\pm 0.9$ [127, 128,
129, 130] and $\mathcal{B}(B\to K^{*}\nu\bar{\nu})/\mathcal{B}(B\to
K^{*}\nu\bar{\nu})_{\rm SM}<3.2\;(95\%\,\mathrm{CL})$ [128] are still far from
our model expectations. However, the Belle II Collaboration will measure
${\mathcal{B}}(B\to K^{(*)}\nu\bar{\nu})$ to an accuracy of 10% of its SM
value [123], thus probing the entire relevant parameter space of the model.

#### $\tau\to\mu\gamma$

The matching contribution to the dipole operator is given by

$\displaystyle\begin{aligned}
{\mathcal{C}}_{e\gamma}^{\mu\tau}(\Lambda)&=-\frac{C_{U}}{16\pi^{2}}\left[\frac{y_{\tau}(\Lambda)}{2}\beta_{L}^{Q\tau}\beta_{L}^{Q\mu\ast}\,\big{(}G_{1}(x_{Q})-2G_{2}(x_{Q})\big{)}+\frac{3\,y_{b}(\Lambda)}{2}\beta_{R}^{b\tau}\beta_{L}^{b\mu\ast}\right]\,,\\\
{\mathcal{C}}_{e\gamma}^{\tau\mu\ast}(\Lambda)&=-\frac{C_{U}}{16\pi^{2}}\,\frac{y_{\tau}(\Lambda)}{2}\beta_{L}^{Q\mu\ast}\beta_{L}^{Q\tau}\,\big{(}G_{1}(x_{Q})-2G_{2}(x_{Q})\big{)}\,,\end{aligned}$
(4.18)

where $x_{Q}=M_{Q}^{2}/M_{U}^{2}$ and the functions $G_{1,2}(x)$ are given by
[58]

$\displaystyle G_{1}(x)$ $\displaystyle=x\left[\frac{2-5x}{2(x-1)^{4}}\,\ln
x-\frac{4-13x+3x^{2}}{4(x-1)^{3}}\right]\,,$ $\displaystyle G_{2}(x)$
$\displaystyle=x\left[\frac{4x-1}{2(x-1)^{4}}\,x\ln
x+\frac{2-5x-3x^{2}}{4(x-1)^{3}}\right]\,.$ (4.19)

Using these results, and given the size of the various $\beta^{i\ell}_{L}$
couplings, it is easy to verify that the leading contribution is the one
proportional to $y_{b}$, which is independent from the mass of the vector-like
fermions. An even larger contribution to the decay rate is the one generated
by the RGE contribution of ${\mathcal{O}}^{33\mu\tau}_{LR}(m_{\tau})$ in
(B.20). This is why we considered ${\mathcal{B}}(\tau\to\mu\gamma)$ among the
UV-insensitive observables in Section 4.1, at least in the case of large
$\beta_{R}^{b\tau}$ (which is the only case where it can reach values close to
present bounds).

###  4.3 Collider signatures from the additional TeV-scale states

Figure 4.5: Collider constraints from resonant $G^{\prime}$ production with
dijet (green) and $t\bar{t}$ (blue) final states for the benchmark couplings
$\kappa_{q}^{ii}=-g_{s}(M_{G}^{\prime})^{2}/g_{G^{\prime}}^{2}$ (solid) and
$\kappa_{q}^{ii}=0$ (dashed), and $\kappa_{R}=1$ in both cases (see text for
details). We mark in gray those values of the width which are below the sum of
$t\bar{t}$ and dijet partial widths for $g_{G^{\prime}}=3$.

##### Coloron searches.

 The most general Lagrangian for a coloron coupling to SM particles is
given by

$\displaystyle\mathcal{L}_{G^{\prime}}$
$\displaystyle=-\frac{1}{4}\,G_{\mu\nu}^{\prime\,a}\,G^{\prime\,a\,\mu\nu}+\frac{1}{2}\,M_{G^{\prime}}^{2}\,G_{\mu}^{\prime\,a}\,G^{\prime\,a\,\mu}+\frac{1}{2}\,\kappa_{G^{\prime}}\,G_{\mu\nu}^{\prime\,a}\,G^{a\,\mu\nu}+g_{s}\,\tilde{\kappa}_{G^{\prime}}f_{abc}\,G_{\mu}^{\prime\,a}\,G_{\nu}^{\prime\,b}\,G^{c\,\mu\nu}$
$\displaystyle\quad+g_{G^{\prime}}\,G^{\prime\,a\,\mu}\,(\kappa_{q}^{ij}\,\bar{q}^{i}_{L}\,T^{a}\,\gamma_{\mu}\,q_{L}^{j}+\kappa_{u}^{ij}\,\bar{u}^{i}_{R}\,T^{a}\,\gamma_{\mu}\,u_{R}^{j}+\kappa_{d}^{ij}\,\bar{d}^{i}_{R}\,T^{a}\,\gamma_{\mu}\,d_{R}^{j})\,.$
(4.20)

In models where the coloron has a gauge origin, such as the ones discussed
here, we have $\kappa_{G}=\tilde{\kappa}_{G}=0$. This is an important feature,
since it implies that coloron couplings to two gluons are absent at the tree-
level, thus effectively reducing the coloron production cross-section at the
LHC. Concerning the coloron couplings to fermions, as already discussed in
Section 4.2, off-diagonal couplings are strongly constrained by meson-
antimeson mixing and thus do not play a relevant role at high-$p_{T}$.
Furthermore, in all UV completions aimed at the explanation of the
$B$-anomalies, the following relations are satisfied

$\displaystyle g_{G^{\prime}}$ $\displaystyle\approx K_{\rm NLO}\,g_{U}\,,$
$\displaystyle\kappa_{q}^{33}$ $\displaystyle=1\,,$
$\displaystyle\kappa_{u,d}^{33}$ $\displaystyle=\kappa_{R}\,,$
$\displaystyle\kappa_{q}^{ii}$
$\displaystyle=s_{Q}^{2}-\frac{g_{s}^{2}}{g_{G^{\prime}}^{2}}\,,$
$\displaystyle\kappa_{u,d}^{ii}$
$\displaystyle=-\frac{g_{s}^{2}}{g_{G^{\prime}}^{2}}\,,$ (4.21)

with $i=1,2$ and $g_{s}$ evaluated at the coloron mass scale
($g_{s}(M_{G^{\prime}})\approx 1$). In this expression, $s_{Q}$ parametrizes
possible contributions from vector-like fermion mixing, assuming the latter
enter in an $U(2)$-invariant way, namely that they are the same (up to small
corrections) for first and second families. This condition is necessary to
avoid the strong constraints from $D$–$\bar{D}$ mixing [90]. Since $s_{Q}$ is
connected to the value of $\beta_{L}^{s\tau}$ (c.f. (4.3)), the constraint
from the low-energy fit naturally leads to a significant suppression of the
$\kappa_{q}^{ii}$ couplings. To illustrate this effect, we consider two
benchmark values for these couplings in our analysis: $\kappa_{q}^{ii}=0$ and
$\kappa_{q}^{ii}=-g_{s}^{2}/g_{U}^{2}$. The value of $\kappa_{R}$ can vary
depending on the UV completion. For concreteness, here we take $\kappa_{R}=1$,
which is common to all models with third-family quark-lepton unification.
Finally, in UV models where $U_{1}$ and $G^{\prime}$ have a common origin, we
have $g_{G^{\prime}}\approx g_{U}$. In order to account for NLO corrections to
coloron on-shell production, in (4.21) we modified this relation by adding the
$K$-factor [56, 57]

$\displaystyle K_{\rm
NLO}\approx\left(1+2.65\,\frac{\alpha_{4}}{4\pi}+8.92\,\frac{\alpha_{s}}{4\pi}\right)^{-1/2}\,,$
(4.22)

which amounts to an $\mathcal{O}(10\%)$ reduction of $g_{G^{\prime}}$ for
$g_{U}=3$.

Given the coupling structure in (4.21), the most effective coloron searches
consist in resonant production with $t\bar{t}$ and $b\bar{b}$ final states.
Since the coloron couples to two gluons only at the loop level, the dominant
production channel is via quark fusion, i.e. $q\bar{q}\to G^{\prime}$ (see
Figure 4.6). Even though the couplings to light quarks are suppressed by
$g_{s}^{2}/g_{G^{\prime}}^{2}$, the PDF enhancement of valence quarks relative
to third generation quarks ensures that production via valence quarks is
nevertheless dominant. As a result, for fixed coloron width, the cross section
for $pp\to G^{\prime}\to t\bar{t}/b\bar{b}$ has only a mild dependence on
$g_{G^{\prime}}$, except for extreme values of this coupling, where the
$b\bar{b}$ initiated process starts to dominate. For concreteness, in what
follows we fix $g_{G^{\prime}}=3$, noting that the limits we derive will not
be significantly affected for lower values of $g_{G^{\prime}}$ (up to a
trivial rescaling of the width). The partial width of the coloron to SM
particles for $g_{G^{\prime}}=3$ is $\Gamma_{G^{\prime}\to
q\bar{q}}/M_{G^{\prime}}\approx 0.24$. In realistic UV models, the total width
could be larger if the decay channel to vector-like quarks becomes
kinematically available. Therefore, we leave the total width of the coloron as
a free parameter in our analysis.

For the inclusive dijet final state, we use the recent CMS analysis in [131]
with $137\leavevmode\nobreak\ \mathrm{fb}^{-1}$ of $13$ TeV data. This
analysis targets both narrow and broad resonances with widths up to $55\%$ of
the resonance mass, which makes it particularly suitable for the model
discussed here. We calculate the model predictions for the dijet process with
Madgraph5_aMC@NLO [132] using the default NNPDF23LO PDF set and the coloron
UFO model presented in [103], publicly available in the Feynrules [133] model
database (https://feynrules.irmp.ucl.ac.be/wiki/LeptoQuark). In order to set
limits, we compute for different values of the total width the $\sigma(pp\to
G^{\prime}\to jj)$ cross-section subject to the same rapidity cuts
$|\eta(j_{1})|,|\eta(j_{2})|<2.5$ and $|\Delta\eta(j_{1},j_{2})|<1.1$ used in
the experimental analysis. The resulting cross-sections are then confronted
directly with the $95\%$ CL limits for a $q\bar{q}$-initiated spin-1 resonance
provided by CMS in Ref.[131], Figure 10.

The extracted limits are shown in Figure 4.5 (green region). As can be seen,
despite the large-width coverage of the search, the limits significantly
weaken for larger values of the coloron width. Taking as a reference
$\Gamma_{G^{\prime}}/M_{G^{\prime}}\approx 0.24$, an exclusion limit is found
at $M_{G^{\prime}}\approx 3.3$ TeV for $\kappa_{q}^{ii}=-g_{s}^{2}/g_{U}^{2}$
and $M_{G^{\prime}}\approx 2.7$ TeV for $\kappa_{q}^{ii}=0$. Since our dijet
signal is made almost entirely of $b\bar{b}$ pairs, improved limits could be
obtained if at least one of the final-state jets is $b$-tagged. In this work
we have not extracted limits from $b$-tagged dijet searches, because these
usually focus on narrow resonances and the improvement on the limits over the
inclusive dijet searches is not substantial [90]. Moreover, the traditional
$b$-taggers used to identify very heavy resonances decaying to $b$-jets suffer
from a fast decline in the $b$-tagging efficiency with increasing jet
transverse momentum. Recently, new $b$-taggers based on deep neural networks,
like DL1r [134], DeepCSV [135] and DeepJet [136], have shown very promising
results in identifying boosted heavy flavor jets. These novel techniques may
be useful in the near future to improve searches for broad colorons in the
$b$-tagged dijet invariant mass spectrum.

Figure 4.6:  Feynman diagrams describing the main processes for $G^{\prime}$,
$Z^{\prime}$, and vector-like lepton production at the LHC.

Much stronger coloron limits are found in the $pp\to G^{\prime}\to t\bar{t}$
channel. The precise measurements of the differential cross-sections for
boosted top-quarks from $t\bar{t}$ production are powerful probes for heavy
resonances, the invariant mass distribution $m_{t\bar{t}}$ in particular.
Following the same procedure as in [103, 90], we use the unfolded parton-level
data of the normalized $m_{t\bar{t}}$ distribution provided by the
experimental collaborations to set limits on the coloron mass and width. A
novelty with respect to these previous studies is that in addition to the
ATLAS search with $36.1\leavevmode\nobreak\ \mathrm{fb}^{-1}$ [137], a new
search by CMS with $35.9\leavevmode\nobreak\ \mathrm{fb}^{-1}$ [138] has
recently been released. Below we present results combining the two searches
for a total luminosity of $72$ fb-1, under the assumption that the
measurements from each experiment (and the invariant mass bins within) are
completely uncorrelated. While both experimental searches are very similar,
one key difference between the two unfolded data sets is their energy reach:
the data provided by ATLAS extends up to 3 TeV in the invariant mass
distribution, while the data provided by CMS reaches 4 TeV. The latter is
therefore more relevant to set limits on the coloron. As discussed in [103],
in order to obtain good agreement between the SM leading-order predictions
from MC event generators and the experimental data, it is necessary to exclude
the lower invariant mass bins from our analysis. We find very good agreement
with the MC predictions by CMS (ATLAS) for bins above $m_{t\bar{t}}>1250$ GeV
($m_{t\bar{t}}>1200$ GeV). After computing the normalized $m_{t\bar{t}}$
distributions subject to partonic cuts888$m_{t\bar{t}}>800$ GeV,
$p_{T}(t)>400$ GeV, $|\eta(t)|<2.4$ for CMS, and $p_{T}(t_{1})>500$ GeV,
$p_{T}(t_{2})>350$ GeV for ATLAS. for a range of coloron masses and widths, we
performed a combined fit to the ATLAS and CMS data sets. Excluded regions in
the mass-width plane are shown in Figure 4.5 (light blue region) together with
the dijet bounds discussed above. As shown there, in spite of the lower
statistics, $t\bar{t}$ provides much stronger limits in the entire relevant
region in parameter space, making this a key channel to test the $U_{1}$
solution of the anomalies. The largest energy reach of the CMS search compared
to the one of ATLAS gives rise to limits that are stronger than those
presented in [103, 90] by about $500$ GeV. As a result, we find coloron
exclusion limits of $M_{G^{\prime}}\approx 4.0$ TeV for
$\kappa_{q}^{ii}=-g_{s}^{2}/g_{U}^{2}$ and $M_{G^{\prime}}\approx 3.5$ TeV for
$\kappa_{q}^{ii}=0$, for a nominal width of
$\Gamma_{G^{\prime}}/M_{G^{\prime}}\approx 0.24$.

#####  $Z^{\prime}$ searches.

 The dominant search channel for the $Z^{\prime}$ at hadron colliders is
via the Drell-Yan process $pp\to Z^{\prime}\to\tau^{+}\tau^{-}$, see Figure
4.6 b.999Because of the coupling structure of the $Z^{\prime}$, decay channels
into dijets and $t\bar{t}$ pairs are suppressed by about a factor of 3
compared to $pp\to Z^{\prime}\to\tau^{+}\tau^{-}$ and are therefore less
competitive [103]. Contributions to the production of the $Z^{\prime}$ from
initial light quarks are suppressed by the coupling ratio $(g_{Y}/g_{4})^{2}$
and can therefore be neglected in a first approximation. On the other hand,
due to the underlying gauge structure of the model, the contributions from the
$b\bar{b}$–initiated process are about 4 times smaller than those from the
$U_{1}$ leptoquark when the masses of both mediators are equal and outside the
region of resonant $Z^{\prime}$ production. As a result and as shown in [103],
limits on the 4321 model from $Z^{\prime}$ searches are less competitive than
those obtained from the $U_{1}$ and $G^{\prime}$ gauge bosons. We checked this
explicitly by matching the $Z^{\prime}$ to the effective operators in (2.1)
for which limits on the Wilson coefficients from $pp\to\tau^{+}\tau^{-}$ are
extracted. For example, if we assume equal left-handed and right-handed
couplings, a 100% branching ratio into di-taus and a coupling satisfying
$g_{Z^{\prime}}\approx g_{U}\geq 3$, the limits from the effective operators
translate to an exclusion limit for the mass around $M_{Z^{\prime}}\gtrsim 3$
TeV. Note that using the EFT limits is already a good approximation as long as
the $Z^{\prime}$ mass ranges around $\sim 3$ TeV. The resulting bounds for the
$Z^{\prime}$ are generically weaker than the $U_{1}$ limits displayed in
Figure. 3.3. Hence, the presence of the $Z^{\prime}$ is not expected to modify
the exclusion limits discussed there.

##### Searches for vector-like fermions.

 The phenomenological consequences of heavy vector-like quarks and leptons
are of fundamental importance in characterizing the UV completion. Besides
their indirect effects at low-energies, these heavy states lead to a rich
array of signatures at hadron colliders [90]. Of particular importance are the
vector-like leptons $L$. In order to comply with the stringent limits from
$\Delta F=2$ transitions at low energies, the mass of $L$ must be relatively
light, around $M_{L}\sim 1$ TeV, well within the energy reach of the LHC. The
main production mechanisms at the LHC are the resonant and electroweak pair
production modes $q\bar{q}\to\gamma/Z^{*}/Z^{\prime}\to L\bar{L}$ and the
associated single production mode $q^{i}g\to L^{i}U_{1}$. The resonant
production of $L\bar{L}$ from the decay of an on-shell $Z^{\prime}$ (Figure
4.6 c) is a promising search channel for the vector-like lepton, provided such
channel is kinematically available. The heavy leptons can decay either into a
Higgs boson via the 2-body process $L^{i}\to h\ell^{i}$, or via the 3-body
process $L^{i}\to q^{i}\ell^{j}\bar{q}^{k}$ induced by the vector leptoquark
mediator, depending on the UV completion. In either case, one expects high-
multiplicity final states composed of $\tau$-leptons, $b$-jets, top-quarks and
missing energy from neutrinos. In a similar way, vector-like quarks $Q$ can be
searched at the LHC. These states are pair produced purely from QCD
interactions via gluon fusion, or through the decay of the coloron $pp\to
G^{\prime}\to Q\bar{Q}$ (if kinematically allowed). Once produced, these heavy
fermions decay via $Q^{i}\to hq^{i}$, $Q^{i}\to\ell^{i}q^{j}\bar{\ell}^{k}$,
$Q^{i}\to\ell^{i}q^{j}\bar{L}^{k}$, leading to a large amount of third-
generation SM fermions in the final state. The vector-like quarks, however,
are in principle allowed to have larger masses than the vector-like leptons,
and can lie beyond the current reach of heavy fermion searches at the LHC. To
this date, no dedicated search by the LHC experimental collaborations for
vector-like quarks or leptons with the characteristics described above is
available. ATLAS and CMS have recently released searches for ‘excited’ leptons
$\ell^{\star}$ [139, 140] in the 3-body decay channel $\ell^{\star}\to\ell
q\bar{q}$ through a contact interaction. Unfortunately, these typically target
first and second generation decay products, and the excited leptons are singly
produced $pp\to\ell^{\star}\bar{\ell}$ (single vector-like lepton production
$pp\to L\bar{\ell}$ in the 4321 models is suppressed by small mixing angles).
Existing searches for SUSY by ATLAS [141, 142, 143] and CMS [144, 145], in
particular stop searches, can be reinterpreted to set limits on the heavy
vector-like leptons and quarks. A naive comparison of $\sigma(pp\to L\bar{L})$
with the limits on the signal cross-sections from [141] indicates that search
strategies with multi-lepton and multi-jet final states are starting to become
sensitive to vector-like leptons with masses around $M_{L}\approx 0.8$ TeV
[90]. These experimental analyses are however not completely optimized for our
vector-like lepton signatures and are challenging to recast. We therefore
leave for future work a dedicated collider analysis of vector-like fermions in
the 4321 model.

##  5 Conclusions

The recent LHCb result on $R_{K}$ [1] marks an important milestone in the
study of the $B$ anomalies, putting on a firmer statistical basis the evidence
of LFU violation in $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ decays. As we
have shown, a conservative combination of $b\to s\hskip
0.85358pt\ell^{+}\ell^{-}$ observables leads to a 4.6$\sigma$ significance for
the motivated NP hypothesis of a purely left-handed LFU-violating contact
interaction.

The evidence of LFU in $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ observables,
taken alone, does not provide a clear indication of the scale of NP. As shown
in [146], this could be well above the TeV scale and, in practice, not
accessible via direct searches. The situation is different if we combine the
evidence of LFU violation in $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$
transitions with the less significant, but still tantalizing hints of LFU
violation in $b\to c\tau\bar{\nu}$ decays. This is the path we have followed
in this work.

Besides the phenomenological interest of dealing with a TeV-scale model of NP,
the attempt of combining the two anomalies has two strong theoretical
motivations: a possible connection to the flavor problem, for instance along
the lines proposed in [52, 55, 147], and to the Higgs hierarchy problem, as
discussed in [45, 46, 54, 147].

In this work we have presented a detailed phenomenological analysis of a
combined solution of the $B$ anomalies based on the hypothesis of a TeV-scale
vector leptoquark $U_{1}$ as leading mediator, which previous studies have
singled out as the most viable option. We have analyzed the problem on three
levels: i) a pure EFT approach, where the $U_{1}$ hypothesis enters only in
the selection of the relevant set of semileptonic operators; ii) a simplified
model, which has allowed us to analyze in general terms the leading signatures
of the $U_{1}$ boson at high energies; iii) a complete UV scenario based on
the non-universal 4321 gauge group, which has allowed us to address some
important UV-sensitive observables at both low and high energies.

The main conclusions of our analysis can be summarized as follows:

  * •

A combined explanation of both $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ and
$b\to c\tau\bar{\nu}$ anomalies in terms of a massive $U_{1}$ is perfectly
consistent with all available data. Interestingly, the low-energy data imply a
preferred parameter region for mass and coupling of the $U_{1}$ that can be
probed entirely within the high-luminosity phase of the LHC, as shown in
Figure 3.3.

  * •

The above conclusion is driven by the $b\to c\tau\bar{\nu}$ anomalies and
holds irrespective of various unknown parameters, such as the strength of
right-handed leptoquark couplings or the size of subleading
$U(2)^{5}$-breaking terms. As we have pointed out, the latter could be
determined by a series of measurements of low-energy observables. As already
found in previous studies, a general expectation is a large enhancement over
the SM of the $B$-meson decay rates into di-tau final states. Moreover, lepton
flavor-violating $B$ decays into $\tau^{\pm}\mu^{\mp}$ final states are
expected to be within the experimental reach. However, contrary to previous
studies, we have shown that the predictions for both these classes of rare
processes are subject to large uncertainties due to model parameters which are
still poorly constrained. The predictions presented in Section 4.1 could allow
one to determine these parameters using future data.

  * •

$B_{s}$–$\bar{B}_{s}$ mixing poses a severe constraint on the UV completion of
the simplified model. In the 4321 framework, this constraint can be satisfied
in a natural way only if the vector-like leptons, which are a necessary
ingredient of the model, have masses below 1.5–2 TeV (depending on the other
model parameters).

  * •

In the 4321 framework, the $B\to K^{(*)}\nu\bar{\nu}$ decay rates are expected
to be significantly enhanced over their SM values, as shown in Figure 4.4.

  * •

As shown in Figure 4.5, in the 4321 framework the most severe constraint on
the overall scale of the model is represented by the bound on the coloron mass
from $pp\to t\bar{t}$. Present data indicate $M_{G^{\prime}}\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}4$ TeV,
close to the bounds of the perturbative regime. We therefore await with great
interest the improvement of the experimental analyses of $pp\to t\bar{t}$ with
the full LHC run-II data set.

In conclusion, following the path of a combined explanation of the two sets of
flavor anomalies in neutral-current and charged-current decays of $B$ mesons,
besides being a very interesting possibility from the theoretical point of
view, has the virtue of defining a concrete NP framework, which can be
confirmed or excluded in the next few years thanks to a series of well-defined
signatures both at low and high energies.

### Note Added

While this project was under completion, a few papers containing updated EFT
analysis of the $b\to s\hskip 0.85358pt\ell^{+}\ell^{-}$ anomalies [148, 149,
150] and of the simplified $U_{1}$ model [114, 151] appeared. Our results in
Section 2.2 are compatible with those obtained by these authors.

### Acknowledgements

We thank Riccardo Barbieri for useful discussions. This project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme under grant agreement 833280
(FLAY), and by the Swiss National Science Foundation (SNF) under contract
200021-175940. The work of J.F. and M.N. was supported by the Cluster of
Excellence Precision Physics, Fundamental Interactions, and Structure of
Matter (PRISMA+, EXC 2118/1) funded by the German Research Foundation (DFG)
within the German Excellence Strategy (Project ID 39083149) and by the BMBF-
Project 05H2018 - Belle II. M.N. thanks the particle theory group at Zurich
University and the Pauli Center for hospitality during a sabbatical stay.

##  Appendix A The $U(2)^{5}$ symmetry

The largest global flavor symmetry commuting with the gauge symmetry of the SM
Lagrangian is $U(3)^{5}$ [152]. The $U(2)^{5}$ flavor symmetry is the subgroup
of the $U(3)^{5}$ global symmetry that, by construction, distinguishes the
first two generations of fermions from the third one [38, 153, 154]. For each
set of SM fermions with the same gauge quantum numbers, the first two
generations form a doublet of a given $U(2)$ subgroup, whereas the third one
transforms as a singlet. The five independent flavor doublets are denoted
$Q,L,U,D,E$ and the flavor symmetry decomposes as

$\displaystyle U(2)^{5}=U(2)_{Q}\otimes U(2)_{L}\otimes U(2)_{U}\otimes
U(2)_{D}\otimes U(2)_{E}\leavevmode\nobreak\ .$ (A.1)

In the limit of unbroken $U(2)^{5}$, only third-generation fermions can have
non-vanishing Yukawa couplings, which is an excellent first-order
approximation for the SM Lagrangian.

A set of symmetry-breaking terms sufficient to reproduce the complete
structure of the SM Yukawa couplings is

$\displaystyle\begin{aligned}
V_{q}&\sim\left(\boldsymbol{2},\boldsymbol{1},\boldsymbol{1},\boldsymbol{1},\boldsymbol{1}\right)\,,&V_{\ell}&\sim\left(\boldsymbol{1},\boldsymbol{2},\boldsymbol{1},\boldsymbol{1},\boldsymbol{1}\right)\,,\\\
\Delta_{u(d)}&\sim\left(\boldsymbol{2},\boldsymbol{1},\boldsymbol{\bar{2}}(\boldsymbol{1}),\boldsymbol{1}(\boldsymbol{\bar{2}}),\boldsymbol{1}\right)\,,&\Delta_{e}&\sim\left(\boldsymbol{1},\boldsymbol{2},\boldsymbol{1},\boldsymbol{1},\boldsymbol{\bar{2}}\right)\,.\end{aligned}$
(A.2)

In the quark sector, this is the minimal set of terms with different
$U(2)^{5}$ transformation properties necessary to describe all quark masses
and the off-diagonal entries of the CKM matrix [38]. In the lepton sector, the
absence of mixing (for vanishing neutrino masses) does not allow us to
conclude that $V_{\ell}$ is strictly necessary; however, it is natural to
introduce such a breaking term to maintain a quark–lepton symmetric structure
[154]. Given that the $U(2)^{5}$ symmetric limit is a good approximation to
the SM Lagrangian, all the $U(2)^{5}$-breaking terms can be chosen to be small
in size.

By construction, $V_{q,\ell}$ are complex two-vectors and $\Delta_{e,u,d}$ are
complex $2\times 2$ matrices. In terms of these spurions, we can express the
Yukawa couplings as

$\displaystyle
Y_{e}=y_{\tau}\left(\begin{matrix}\Delta_{e}&x_{\tau}V_{\ell}\\\
0&1\end{matrix}\right),$ $\displaystyle
Y_{u}=y_{t}\left(\begin{matrix}\Delta_{u}&x_{t}V_{q}\\\
0&1\end{matrix}\right),$ $\displaystyle
Y_{d}=y_{b}\left(\begin{matrix}\Delta_{d}&x_{b}V_{q}\\\
0&1\end{matrix}\right),$ (A.3)

where $y_{\tau,t,b}$ and $x_{\tau,t,b}$ are free complex parameters expected
to be of order $\mathcal{O}(1)$. The observed mass hierarchies and the
smallness of the CKM mixing angles are attributed to the size (and structure)
of the $U(2)^{5}$-breaking spurions. The latter cannot be determined
completely, but are constrained requiring no tuning in the ${\mathcal{O}}(1)$
parameters. In particular, from the 2–3 mixing in the CKM matrix we deduce