aainstitutetext: Physik-Institut, Universität Zürich, Winterthurerstrasse 190,
CH-8057 Zürich, Switzerlandbbinstitutetext: Paul Scherrer Institut, CH–5232
Villigen PSI, Switzerlandccinstitutetext: Laboratoire de Physique Théorique et
Hautes Energies (LPTHE), UMR 7589, Sorbonne Université et CNRS, 4 place
Jussieu, 75252 Paris Cedex 05, Franceddinstitutetext: Max Planck Institute for
Physics, Föhringer Ring 6, 80805 München, Germany

# Explaining the hints for lepton flavour universality violation with three
$S_{2}$ leptoquark generations

Andreas Crivellin c , Benjamin Fuks d  and  Luc Schnell
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Leptoquarks are prime candidates for explaining the intriguing hints for
lepton flavour universality violation. In particular, the $SU(2)_{L}$ doublet
of scalar leptoquarks $S_{2}$ is capable of providing an explanation for the
tensions between the measurements and the Standard Model predictions in
$(g-2)_{\mu}$, $b\to s\ell^{+}\ell^{-}$ and $b\to c\tau\nu$ processes, as well
as in non-resonant di-electron production. However, in the minimal setup with
a single leptoquark generation, a common explanation for all these issues is
not possible as this would lead to unacceptably large charged lepton flavour
violation. We therefore propose a model with three generations of $S_{2}$,
each coupling exclusively to a single lepton flavour, i.e. a model extending
the Standard Model particle content by an electroquark, a muoquark and a
tauquark. We show that after taking into account other constraints, such as
those originating from electroweak precision observables and $\Delta F=2$
processes, it is possible to provide a combined explanation for all these
hints of lepton flavour universality violation. Moreover, we find that the
presence of the tauquark can generate a dimension-six ${\cal O}_{9}^{U}$
operator via off-shell photon penguin diagrams, which, together with the
muoquark contribution, further improves the global fit to $b\to
s\ell^{+}\ell^{-}$ data.

††preprint:  MPP-2022-30, PSI-PR-22-07, ZU-TH 08/22

##  1 Introduction

The Standard Model (SM) of particle physics has been tested extensively and
confirmed within the last decades, culminating in the discovery of its last
missing piece, the Higgs boson, in 2012 at the LHC ATLAS:2012yve ; CMS:2012qbp
. Nonetheless, it is evident that the SM cannot be the ultimate theory of
nature, as it cannot for example account for dark matter or non-vanishing
neutrino masses. However, these phenomena can be accounted for by new physics
(NP), which could a priori lie within a wide energy range (i.e. from the keV
to the scale of Grand Unification), and is therefore not necessarily within
the reach of current or even future colliders.

Fortunately, in recent years exciting hints for lepton flavour universality
violation (LFUV) beyond the SM have been accumulated (see e.g. Refs.
Fischer:2021sqw ; Crivellin:2021sff  for recent reviews), and an explanation
of these anomalies requires TeV-scale new physics. This makes a discovery at
the LHC possible, and an observation at future colliders even guaranteed. In
particular, measurements of $b\to s\ell^{+}\ell^{-}$ CMS:2014xfa ;
Aaij:2015oid ; Abdesselam:2016llu ; Aaij:2017vbb ; Aaij:2019wad ; Aaij:2020nrf
; LHCb:2021trn ; BELLE:2019xld ; Belle:2019oag ; LHCb:2016ykl ; LHCb:2021zwz ;
LHCb:2020zud  and $b\to c\tau\nu$ Lees:2012xj ; Lees:2013uzd ; Aaij:2015yra ;
Aaij:2017deq ; Aaij:2017uff ; Abdesselam:2019dgh  observables deviate from
their corresponding SM predictions by more than $7\,\sigma$
Altmannshofer:2021qrr ; Geng:2021nhg ; Alguero:2021anc ; Hurth:2021nsi ;
Kowalska:2019ley ; Ciuchini:2021smi ; DAmico:2017mtc ; Arbey:2019duh ;
Kumar:2019nfv  (this tension being reduced to $4\\!-\\!5\sigma$ when including
only LFUV observables Altmannshofer:2021qrr ; Geng:2021nhg ; Alguero:2021anc ;
Hurth:2021nsi ; Isidori:2021vtc ) and by more than $3\,\sigma$ HFLAV:2019otj
respectively. Moreover, measurements of the anomalous magnetic moment of the
muon $(g-2)_{\mu}$ Muong-2:2006rrc ; Muong-2:2021ojo  are known to deviate
from the SM predictions by more than $4\,\sigma$ when following the community
consensus Aoyama:2020ynm . The latter is based on the results of Refs.
Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva ;
Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019gzf
; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ;
Bijnens:2019ghy ; Colangelo:2019uex ; Blum:2019ugy ; Colangelo:2014qya  that
do not include the recent lattice findings of the Budapest-Marseilles-
Wuppertal collaboration (BMWc) for the hadronic vacuum polarisation (HVP)
contributions Borsanyi:2020mff . Whereas they render the SM predictions for
$a_{\mu}$ compatible with data, the BMWc results are in tension with the HVP
value determined from $e^{+}e^{-}\to$ hadrons data Davier:2017zfy ;
Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019gzf ; Davier:2019can
; Keshavarzi:2019abf . Furthermore, HVP also enters global electroweak (EW)
fits Passera:2008jk , and its (indirect) determination leads to a value
smaller than that obtained by the BMWc Haller:2018nnx . Therefore, making use
of the BMWc predictions for the HVP contributions to $a_{\mu}$ would increase
the tension originating from the EW fits Crivellin:2020zul ;
Keshavarzi:2020bfy , and we opted for omitting it until the situation gets
clarified.

Prime examples that can address most, if not even all, LFUV anomalies are
models featuring leptoquarks (LQs), hypothetical new particles with common
couplings to quarks and leptons. LQs were originally proposed in the Pati-
Salam model Pati:1974yy  and in Grand Unified Theories (GUTs) Georgi:1974sy ;
Dimopoulos:1980hn ; Senjanovic:1982ex ; Frampton:1989fu ; Witten:1985xc , and
it has been shown that the anomalies in $b\to s\ell^{+}\ell^{-}$
Alonso:2015sja ; Calibbi:2015kma ; Hiller:2016kry ; Bhattacharya:2016mcc ;
Buttazzo:2017ixm ; Barbieri:2015yvd ; Barbieri:2016las ; Calibbi:2017qbu ;
Crivellin:2017dsk ; Bordone:2018nbg ; Kumar:2018kmr ; Crivellin:2018yvo ;
Crivellin:2019szf ; Cornella:2019hct ; Bordone:2019uzc ; Bernigaud:2019bfy ;
Aebischer:2018acj ; Fuentes-Martin:2019ign ; Popov:2019tyc ; Fajfer:2015ycq ;
Blanke:2018sro ; deMedeirosVarzielas:2019lgb ; deMedeirosVarzielas:2015yxm ;
Crivellin:2019dwb ; Saad:2020ihm ; Saad:2020ucl ; Gherardi:2020qhc ;
DaRold:2020bib ; Greljo:2021xmg , $R(D^{(*)})$ Alonso:2015sja ;
Calibbi:2015kma ; Fajfer:2015ycq ; Bhattacharya:2016mcc ; Buttazzo:2017ixm ;
Barbieri:2015yvd ; Barbieri:2016las ; Calibbi:2017qbu ; Bordone:2017bld ;
Bordone:2018nbg ; Kumar:2018kmr ; Biswas:2018snp ; Crivellin:2018yvo ;
Blanke:2018sro ; Heeck:2018ntp ; deMedeirosVarzielas:2019lgb ;
Cornella:2019hct ; Bordone:2019uzc ; Sahoo:2015wya ; Chen:2016dip ;
Dey:2017ede ; Becirevic:2017jtw ; Chauhan:2017ndd ; Becirevic:2018afm ;
Popov:2019tyc ; Fajfer:2012jt ; Deshpande:2012rr ; Freytsis:2015qca ;
Bauer:2015knc ; Li:2016vvp ; Zhu:2016xdg ; Popov:2016fzr ; Deshpand:2016cpw ;
Becirevic:2016oho ; Cai:2017wry ; Altmannshofer:2017poe ; Kamali:2018fhr ;
Mandal:2018kau ; Azatov:2018knx ; Wei:2018vmk ; Angelescu:2018tyl ;
Kim:2018oih ; Aydemir:2019ynb ; Crivellin:2019qnh ; Yan:2019hpm ;
Crivellin:2017zlb ; Marzocca:2018wcf ; Fuentes-Martin:2020pww ;
Bigaran:2019bqv ; Crivellin:2019dwb ; Saad:2020ihm ; Dev:2020qet ;
Saad:2020ucl ; Altmannshofer:2020axr ; Fuentes-Martin:2020bnh ;
Gherardi:2020qhc ; DaRold:2020bib , $b\to c\tau\nu$ Alonso:2015sja ;
Calibbi:2015kma ; Fajfer:2015ycq ; Bhattacharya:2016mcc ; Buttazzo:2017ixm ;
Barbieri:2015yvd ; Barbieri:2016las ; Calibbi:2017qbu ; Bordone:2017bld ;
Bordone:2018nbg ; Kumar:2018kmr ; Biswas:2018snp ; Crivellin:2018yvo ;
Blanke:2018sro ; Heeck:2018ntp ; deMedeirosVarzielas:2019lgb ;
Cornella:2019hct ; Bordone:2019uzc ; Sahoo:2015wya ; Chen:2016dip ;
Dey:2017ede ; Becirevic:2017jtw ; Chauhan:2017ndd ; Becirevic:2018afm ;
Popov:2019tyc ; Fajfer:2012jt ; Deshpande:2012rr ; Freytsis:2015qca ;
Bauer:2015knc ; Li:2016vvp ; Zhu:2016xdg ; Popov:2016fzr ; Deshpand:2016cpw ;
Becirevic:2016oho ; Cai:2017wry ; Altmannshofer:2017poe ; Kamali:2018fhr ;
Mandal:2018kau ; Azatov:2018knx ; Wei:2018vmk ; Angelescu:2018tyl ;
Kim:2018oih ; Aydemir:2019ynb ; Crivellin:2019qnh ; Yan:2019hpm ;
Crivellin:2017zlb ; Marzocca:2018wcf ; Bigaran:2019bqv ; Crivellin:2019dwb ;
Saad:2020ihm ; Dev:2020qet ; Saad:2020ucl ; Altmannshofer:2020axr ; Fuentes-
Martin:2020bnh ; Gherardi:2020qhc ; DaRold:2020bib ; Endo:2021lhi ;
Belanger:2021smw ; Lee:2021jdr , and $(g-2)_{\mu}$ Bauer:2015knc ;
Djouadi:1989md ; Chakraverty:2001yg ; Cheung:2001ip ; Popov:2016fzr ;
Chen:2016dip ; Biggio:2016wyy ; Davidson:1993qk ; Couture:1995he ;
Mahanta:2001yc ; Queiroz:2014pra ; ColuccioLeskow:2016dox ; Chen:2017hir ;
Das:2016vkr ; Crivellin:2017zlb ; Cai:2017wry ; Crivellin:2018qmi ;
Kowalska:2018ulj ; Dorsner:2019itg ; Crivellin:2019dwb ; DelleRose:2020qak ;
Saad:2020ihm ; Bigaran:2020jil ; Dorsner:2020aaz ; Fuentes-Martin:2020bnh ;
Gherardi:2020qhc ; Babu:2020hun ; Crivellin:2020tsz ; Marzocca:2021azj ;
Wang:2021uqz ; Perez:2021ddi  can all be explained by them. Moreover, LQs can
provide an explanation for the CMS excess in non-resonant di-electron
production Crivellin:2021egp ; Crivellin:2021bkd .

In this context, models including a $SU(2)_{L}$ leptoquark doublet $S_{2}$
that transforms under the SM gauge group $SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$ as $\left(\mathbf{3},\mathbf{2},\frac{7}{6}\right)$, from now on
called $\Phi_{2}$ (in the literature also called $R_{2}$), are particularly
interesting. This LQ can restore the agreement between theory and data for
$b\to s\ell^{+}\ell^{-}$ observables via a $W$-box contribution
Becirevic:2017jtw , for $b\to c\tau\nu$ observables and the excess
Crivellin:2021rbf  in CMS di-lepton data CMS:2021ctt  via tree-level
contributions Crivellin:2021egp ; Crivellin:2021bkd , and for $(g-2)_{\mu}$
via an $m_{t}/m_{\mu}$ chirally enhanced effect. Furthermore, the tension in
the difference of the $B\to D^{*}\ell\nu$ forward-backward asymmetries
($\Delta A_{\rm FB}$) Belle:2018ezy ; Bobeth:2021lya  can be softened
Crivellin:2020mjs , and the global EW fit can be improved through the
generation of a shift in the $W$-boson mass predictions Crivellin:2020ukd ,
where a constructive effect is currently preferred deBlas:2021wap , via LQ
interactions with the Higgs field Crivellin:2021ejk .

However, a combined explanation of the flavour anomalies in the minimal setup
with a single leptoquark $S_{2}$ is not possible as this would lead to
unacceptably large charged lepton flavour violation (LFV). In order to avoid
this, we propose to extend the SM by three generations of leptoquarks, like
the three generations of squarks in minimal $R$-parity violating supersymmetry
Barger:1989rk ; Barbier:2004ez . In this setup each LQ flavour couples to the
corresponding lepton flavour, i.e. the electroquark $\Phi_{2,e}$ coupling to
electrons, the muoquark $\Phi_{2,\mu}$ to muons and the tauquark
$\Phi_{2,\tau}$ to tau leptons. While this does not introduce additional
degrees of freedom concerning the couplings to fermions, it decouples the LQ
interactions with the different lepton generations from each others, rendering
joint explanations of the hints for LFUV possible.

We proceed by defining our three-generation model of $S_{2}$ in Section 2,
then discuss the most relevant observables in Section 3. Section 4 contains
our phenomenological analysis, including a statistical analysis, and we
conclude in Section 5.

##  2 Setup

Minimal extensions of the SM with a single $SU(2)_{L}$ doublet scalar LQ
$\Phi_{2}$ have been studied in detail in the literature Becirevic:2017jtw ;
Popov:2019tyc ; Angelescu:2021lln ; Angelescu:2018tyl ; Crivellin:2021egp ;
Crivellin:2021bkd ; He:2021yck ; Bigaran:2020jil ; Bigaran:2021kmn ;
deMedeirosVarzielas:2015yxm ; Sahoo:2015wya ; Kamali:2018fhr ;
Chakraverty:2001yg ; Cheung:2001ip ; Queiroz:2014pra ; ColuccioLeskow:2016dox
; Chen:2017hir ; Kowalska:2018ulj ; Dorsner:2020aaz ; Crivellin:2020tsz ;
Iguro:2020keo . In such models, the most general LQ Lagrangian reads
Crivellin:2021ejk

$\displaystyle\mathcal{L}_{\text{LQ}}$
$\displaystyle=\bigg{(}Y_{ij}^{RL}\bar{u}_{i}\left[\Phi_{2}\cdot
L_{j}\right]+Y_{ij}^{LR}\left[\bar{Q}_{i}e_{j}\Phi_{2}\right]+\text{
H.c.}\bigg{)}-\bigg{(}M^{2}+Y^{H(1)}\Big{[}H^{\dagger}H\Big{]}\bigg{)}\Phi_{2}^{\dagger}\Phi_{2}$
(1)
$\displaystyle\phantom{===}-Y^{H(3)}\Big{[}H\cdot\Phi_{2}\Big{]}^{\dagger}\Big{[}H\cdot\Phi_{2}\Big{]}+\mathcal{L}_{4\Phi}\,,$

where $Q_{i}\sim\left(\mathbf{3},\mathbf{2},\frac{1}{6}\right)$,
$L_{i}\sim\left(\mathbf{1},\mathbf{2},-\frac{1}{2}\right)$,
$e_{i}\sim\left(\mathbf{1},\mathbf{1},-1\right)$,
$u_{i}\sim\left(\mathbf{3},\mathbf{1},\frac{2}{3}\right)$ and
$H\sim\left(\mathbf{1},\mathbf{2},\frac{1}{2}\right)$ are the usual SM fields,
$i,j=1,2,3$ are flavour indices and the dot stands for the invariant product
of two fields lying in the fundamental representation of $SU(2)_{L}$.
Considering the version of the above Lagrangian after EW symmetry breaking, we
refer to the SM fermions by their usual names $q\in\\{u,c,t,d,s,b\\}$,
$\ell\in\\{e,\mu,\tau\\}$ and
$\nu_{\ell}\in\\{\nu_{e},\nu_{\mu},\nu_{\tau}\\}$. Moreover, the Lagrangian
term $\mathcal{L}_{4\Phi}$ contains the LQ quartic interactions
Crivellin:2021ejk  that are not relevant for our analysis.

The couplings $Y^{LR}_{ij}$ and $Y^{RL}_{ij}$ are a priori arbitrary complex
matrices in the flavour space, leading in general to charged lepton flavour
violation. As outlined in the introduction, explaining anomalies related to
different lepton generations at the same time leads to unacceptably large
effects in charged lepton flavour violating observables. However, one can
avoid this by assigning a lepton flavour number to $\Phi_{2}$. In fact, in
Refs. Davighi:2020qqa ; Greljo:2021npi  a $L_{\mu}-L_{\tau}$ symmetry
He:1990pn ; Foot:1990mn ; He:1991qd  was used to impose that $\Phi_{2}$ only
interacts with second-generation leptons. However, this model can at most
address muon-related anomalies, and it might be considered unnatural to give
the muon such a special treatment because the tau lepton is even more massive.

Therefore, we propose to introduce three generations of leptoquarks
$\Phi_{2}$, so that the field $\Phi_{2,\ell}$ now carries a generation index
$\ell=1,2,3$, or equivalently and interchangeably $\ell=e,\mu,\tau$. In
addition, we require that $\Phi_{2,\ell}$ only interacts with the lepton
flavour $\ell$. While we remain agnostic about the specific underlying
mechanism that enforces this, it could for instance again be achieved via a
$L_{\mu}-L_{\tau}$ symmetry by assigning the charges $0$, $1$ and $-1$ to
$\Phi_{2,e}$, $\Phi_{2,\mu}$ and $\Phi_{2,\tau}$, respectively. In addition,
the assignment of lepton flavours to LQs automatically avoids proton decay to
all orders in perturbation theory, as it forbids di-quark couplings, in case
this coupling would be allowed by the other quantum numbers.

In this setup, the LQ interaction Lagrangian reads

$\displaystyle\mathcal{L}_{\text{LQ}}$
$\displaystyle=\sum_{\ell}\bigg{(}Y_{i\ell}^{RL}\bar{u}_{i}\left[\Phi_{2,\ell}\cdot
L_{\ell}\right]+Y_{i\ell}^{LR}\left[\bar{Q}_{i}e_{\ell}\Phi_{2,\ell}\right]+\text{
H.c.}\bigg{)}$ (2)
$\displaystyle\phantom{===}-\bigg{(}M_{\ell}^{2}+Y^{H(1)}_{\ell}\Big{[}H^{\dagger}H\Big{]}\bigg{)}\Phi_{2,\ell}^{\dagger}\Phi_{2,\ell}-Y_{\ell}^{H(3)}\Big{[}H\cdot\Phi_{2,\ell}\Big{]}^{\dagger}\Big{[}H\cdot\Phi_{2,\ell}\Big{]}+\mathcal{L}_{4\Phi}\,.$

Comparing Eq. (2) to Eq. (1), it is apparent that we do not introduce any
additional degrees of freedom in the LQ couplings to fermions compared to the
minimal model with a single $\Phi_{2}$.

Once the Higgs doublet $H$ acquires its vacuum expectation value $v$, the
Yukawa terms generate mass matrices for quarks and leptons. Here we assume
that the lepton Yukawa matrices are diagonal in the basis of Eq. (2) such that
no charged LFV is induced by EW symmetry breaking. This means that lepton
flavour is only broken by the tiny neutrino masses, with negligible
consequences in our phenomenological analysis. Concerning quarks, we choose to
work in the down-type quark basis, so that CKM matrix elements only appear in
couplings involving left-handed up-type quarks. We therefore define

$\hat{Y}^{LR}_{i\ell}\equiv V^{\rm CKM}_{ij}Y^{LR}_{j\ell}$ (3)

for brevity.

##  3 Observables

In the following, we discuss the most relevant observables allowing to test
and constrain our model. For the low-energy precision and flavour observables
we match the full LQ theory onto the weak effective theory (WET) whose
Lagrangian is generically written as

$\mathcal{L}_{\text{eff}}=\sum_{i}\mathcal{C}_{i}\mathcal{O}_{i}\,.$ (4)

We refer to the manual of flavio Straub:2018kue , a package that we employ in
our phenomenological analysis, for a precise definition of the operators.

###  3.1 $R_{D^{(*)}}$ anomalies

We consider the ratios

$\displaystyle R_{D^{(*)}}=\left.\frac{\text{Br}(B\to
D^{(*)}\tau\bar{\nu})}{\text{Br}(B\to
D^{(*)}\ell\bar{\nu})}\right|_{\ell\in\\{e,\mu\\}}\,,$ (5)

whose current experimental averages read

$\displaystyle R^{\text{exp}}_{D}$
$\displaystyle=0.346(31)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lees:2013uzd, Huschle:2015rga,
Abdesselam:2019dgh}{\@@citephrase{(}}{\@@citephrase{)}}}}\qquad\text{and}\qquad
R^{\text{exp}}_{D^{*}}$
$\displaystyle=0.296(16)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Lees:2013uzd, Huschle:2015rga, Abdesselam:2019dgh,
Hirose:2016wfn}{\@@citephrase{(}}{\@@citephrase{)}}}}\,.$ (6)

These values are compared to our theoretical predictions whose SM component is
provided by flavio MILC:2015uhg ; Na:2015kha ; Fajfer:2012vx ;
FlavourLatticeAveragingGroup:2019iem ; Iguro:2020keo ,

$\displaystyle R^{\text{SM}}_{D}$
$\displaystyle=0.297(8)\qquad\text{and}\qquad R^{\text{SM}}_{D^{*}}$
$\displaystyle=0.245(8)\,,$ (7)

although more accurate predictions have been calculated in the meantime
HFLAV:2019otj ; Bigi:2016mdz ; Gambino:2019sif ; Bordone:2019vic . At low
energy, the new physics contributions are described by the WET operators

$\displaystyle\left(\mathcal{O}_{S_{L}}\right)_{bc\tau\nu_{\tau}}=$
$\displaystyle\ -\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{23}\left(\bar{c}P_{L}b\right)\left(\bar{\tau}P_{L}\nu_{\tau}\right)\,,$
(8) $\displaystyle\left(\mathcal{O}_{T}\right)_{bc\tau\nu_{\tau}}=$
$\displaystyle\ -\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{23}\left(\bar{c}\sigma^{\mu\nu}P_{L}b\right)\left(\bar{\tau}\sigma_{\mu\nu}P_{L}\nu_{\tau}\right)\,,$

where $P_{L}$ and $P_{R}$ (for further reference) are the usual chirality
projectors and $G_{F}$ is the Fermi constant. The corresponding Wilson
coefficients are evaluated at the matching scale,

$\left(\mathcal{C}_{S_{L}}\right)^{\text{LQ}}_{bc\tau\nu_{\tau}}=4\left(\mathcal{C}_{T}\right)^{\text{LQ}}_{bc\tau\nu_{\tau}}=-\frac{\sqrt{2}}{4G_{F}V^{\rm
CKM}_{23}}\left(\frac{Y^{RL}_{2\tau}Y^{LR*}_{3\tau}}{2M_{\tau}^{2}}\right)\,.$
(9)

While WET renormalisation group (RG) running is accounted for by flavio, we
additionally multiply the predictions by appropriate correction factors to
account for RG running from $M_{\tau}$ to the electroweak scale $M_{W}$ in the
SM Effective Field Theory (SMEFT). These factors are determined using the
Python package wilson Aebischer:2018bkb 111Whereas QCD matching corrections
exist Aebischer:2018acj , we do not include them in our analysis as the
package wilson is restricted to one-loop RG evolution., and are equal to 0.97
and 0.93 for $\mathcal{C}_{S_{L}}$ and $\mathcal{C}_{T}$ respectively. They
modify the ratio between $\mathcal{C}_{S_{L}}$ and $\mathcal{C}_{T}$ at the
low scale, and are thus relevant in our analysis.

###  3.2 Rare $B$ decays to kaons ($b\to s\ell^{+}\ell^{-}$)

($\mathcal{C}_{9}$-1)

($\mathcal{C}_{9}$-2)

($\mathcal{C}_{7}$-1)

($\mathcal{C}_{7}$-2)

($\mathcal{C}_{8}$)

Figure 1: Feynman diagrams depicting the one-loop contributions of
$\Phi_{2,\tau}$ to the Wilson coefficients
$\left(\mathcal{C}_{9}\right)_{bs\ell\ell}$,
$\left(\mathcal{C}_{7}\right)_{bs}$ and $\left(\mathcal{C}_{8}\right)_{bs}$,
that play an important role in the global fit to $b\to s\ell^{+}\ell^{-}$
data.

The LHCb and Belle collaborations determined ratios
$R^{[q_{1}^{2},q_{2}^{2}]}_{K^{(*)}}$ of $B$-meson branching fractions into
muons and electrons for different intervals $q_{1}^{2}\leq q^{2}\leq
q_{2}^{2}$, the SM predicting values close to one in all energy bins
Bordone:2016gaq ; Isidori:2020acz ; Serra:2016ivr ; Capdevila:2017ert ;
Bharucha:2015bzk ; Jager:2014rwa . In our analysis, we use the measurements
selected in the package smelli Aebischer:2018iyb  for the $B^{+}\to
K^{(*)}\ell^{+}\ell^{-}$ and $B_{0}\to K^{(*)}\ell^{+}\ell^{-}$ decays,

$\displaystyle R_{K}^{\text{exp}[1.1,6.0]}$
$\displaystyle=0.846^{+0.042}_{-0.039}{}^{+0.013}_{-0.012}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb:2021trn}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(10) $\displaystyle R_{K_{s}^{0}}^{\text{exp}[1.1,6.0]}$
$\displaystyle=0.66^{+0.20}_{-0.14}{}^{+0.02}_{-0.04}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb:2021lvy}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
$\displaystyle R_{K^{*+}}^{\text{exp}[0.045,6.0]}$
$\displaystyle=0.70^{+0.18}_{-0.13}{}^{+0.03}_{-0.04}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb:2021lvy}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$

that we complement with the complete set of observables related to rare
$B$-decay into kaons included both in the programs flavio and smelli.

After integrating out all leptoquarks in our model, the relevant effective
operators for $b\to s\ell^{+}\ell^{-}$ decays are given by

$\displaystyle\begin{aligned}
\left(\mathcal{O}_{7}\right)_{bs}&=\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}\frac{e}{16\pi^{2}}m_{b}\left(\bar{s}\sigma^{\mu\nu}P_{R}b\right)F_{\mu\nu}\,,\\\
\left(\mathcal{O}_{8}\right)_{bs}&=\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}\frac{g_{s}}{16\pi^{2}}m_{b}\left(\bar{s}\sigma^{\mu\nu}T^{a}P_{R}b\right)G^{a}_{\mu\nu}\,,\\\
\left(\mathcal{O}_{9}\right)_{bs\ell\ell}&=\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma^{\mu}P_{L}b\right)\left(\bar{\ell}\gamma_{\mu}\ell\right)\,,\\\
\left(\mathcal{O}_{10}\right)_{bs\ell\ell}&=\frac{4G_{F}}{\sqrt{2}}V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma^{\mu}P_{L}b\right)\left(\bar{\ell}\gamma_{\mu}\gamma_{5}\ell\right)\,,\end{aligned}$
(11)

where $e$ stands for the electromagnetic coupling constant, and $F_{\mu\nu}$
and $G_{\mu\nu}$ are the photon and gluon field strength tensors. While the
anomalies in $b\to s\ell^{+}\ell^{-}$ observables mainly point towards new
physics interacting with muons, the presence of a tauquark $\Phi_{2,\tau}$ in
the model yields a flavour-universal contribution to $\mathcal{C}_{9}$ via the
off-shell photon penguin diagrams shown in the first line of Figure 1. Calling
it $\mathcal{C}_{9}^{U}$, it is given in the leading-logarithmic approximation
by

$\displaystyle\left(\mathcal{C}_{9}^{U}\right)^{\text{LQ}}=\frac{\sqrt{2}}{4G_{F}}\frac{1}{V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}}\left(\frac{Y^{LR}_{2\tau}Y^{LR*}_{3\tau}\log\left(\frac{M_{\tau}^{2}}{m_{b}^{2}}\right)}{3M_{\tau}^{2}}\right)\,.$
(12)

Additionally, the $\mathcal{C}_{7}$ and $\mathcal{C}_{8}$ Wilson coefficients
are extracted from the diagrams shown in the second line of Figure 1, and read

$\left(\mathcal{C}_{7}\right)^{\text{LQ}}_{bs}=\frac{8}{3}\left(\mathcal{C}_{8}\right)^{\text{LQ}}_{bs}=\frac{\sqrt{2}}{4G_{F}}\frac{1}{V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}}\left(\frac{Y^{LR}_{2\tau}Y^{LR*}_{3\tau}}{9M_{\tau}^{2}}\right)\,,$
(13)

following flavio’s sign conventions for covariant derivatives (see Appendix
A.3 of Ref. Aebischer:2018iyb ). While $\mathcal{C}_{8}$ does not contribute
to the $b\to s\ell^{+}\ell^{-}$ observables directly, it mixes into
$\mathcal{C}_{7}$ when RG running from the scale $M_{\tau}$ to $m_{b}$ is
accounted for,

$\begin{pmatrix}\left(\mathcal{C}_{7}\right)_{bs}\\\
\left(\mathcal{C}_{8}\right)_{bs}\end{pmatrix}_{\text{4.2
GeV}}=\begin{pmatrix}0.49&0.12\\\
0.0055&0.54\end{pmatrix}\begin{pmatrix}\left(\mathcal{C}_{7}\right)_{bs}\\\
\left(\mathcal{C}_{8}\right)_{bs}\end{pmatrix}_{\text{1.7 TeV}}\,.$ (14)

In addition to these tauquark effects in $b\to s\ell^{+}\ell^{-}$, we get
well-known contributions from the muoquark $\Phi_{2,\mu}$. Writing
$\mathcal{C}_{9}^{\ell}\equiv\left(\mathcal{C}_{9}\right)_{bs\ell\ell}$ and
$\mathcal{C}_{10}^{\ell}\equiv\left(\mathcal{C}_{10}\right)_{bs\ell\ell}$ to
simplify the notation (for further reference), the tree-level matching to the
WET gives

$\displaystyle\left(\mathcal{C}_{9}^{\mu}\right)^{\text{LQ}}$
$\displaystyle=\left(\mathcal{C}_{10}^{\mu}\right)^{\text{LQ}}=-\frac{\sqrt{2}}{4G_{F}}\frac{1}{V^{\rm
CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}}\frac{16\pi^{2}}{e^{2}}\left(\frac{Y^{LR}_{2\mu}Y^{LR*}_{3\mu}}{4M_{\mu}^{2}}\right)\,.$
(15)

whereas the one-loop contributions Becirevic:2017jtw  are

$\displaystyle\left(\mathcal{C}^{\mu}_{9}\right)^{\text{LQ}}=-\left(\mathcal{C}^{\mu}_{10}\right)^{\text{LQ}}=\sum^{3}_{j,k=1}\frac{V^{\rm
CKM}_{j3}\big{(}V^{\rm CKM}_{k2}\big{)}^{*}}{V^{\rm CKM}_{33}\big{(}V^{\rm
CKM}_{32}\big{)}^{*}}Y^{RL}_{k\mu}Y^{RL*}_{j\mu}\mathcal{F}_{1}(x_{j},x_{k},x_{\mu})\,,$
(16)

with $x_{\mu}=M_{\mu}^{2}/m_{W}^{2}$, $x_{i}=m_{u_{i}}^{2}/m_{W}^{2}$ and
Crivellin:2020mjs

$\mathcal{F}_{1}(x,y,z)=\frac{\sqrt{xy}}{8e^{2}}\Bigg{[}\frac{y(y\\!-\\!4)\log
y}{(y\\!-\\!1)(x\\!-\\!y)(y\\!-\\!z)}+\frac{x(x\\!-\\!4)\log
x}{(x\\!-\\!1)(y\\!-\\!x)(x\\!-\\!z)}-\frac{z(z\\!-\\!4)\log
z}{(z\\!-\\!1)(z\\!-\\!x)(z\\!-\\!y)}\Bigg{]}\,.$ (17)

###  3.3 $B$ and $D$ decays into tau leptons ($b\to s\tau^{+}\tau^{-}$ and
$D_{s}\to\tau\nu$)

The structure of the tauquark couplings that yield contributions to $b\to
c\tau\nu$ and $b\to s\tau^{+}\tau^{-}$ decay observables also leads to effects
in the branching ratios $\text{Br}(B\to K^{(*)}\tau^{+}\tau^{-})$,
$\text{Br}(B_{s}\to\tau^{+}\tau^{-})$ and $\text{Br}(D_{s}\to\tau\nu)$. In the
SM, their values are given by $1.7(2)\times 10^{-7}$, $7.8(3)\times 10^{-7}$
and $0.0532(4)$ according to flavio.

The LQ effects from our model in $B\to K^{(*)}\tau^{+}\tau^{-}$ and
$B_{s}\to\tau^{+}\tau^{-}$ decay observables are embedded in the WET
coefficients of Eqs. (12) and (13). In contrast, the $D_{s}\to\tau\nu$ decay
is sensitive to the $\left({\cal
O}_{S_{L}}\right)^{\text{LQ}}_{sc\tau\nu_{\tau}}$ and $\left({\cal
O}_{T}\right)^{\text{LQ}}_{sc\tau\nu_{\tau}}$ operators, whose associated
Wilson coefficients can be derived from Eq. (9) by replacing the quark index 3
with 2. We finally have

$\text{Br}(B_{s}\to\tau^{+}\tau^{-})=\left|1+\left(\mathcal{C}^{\tau}_{10}\right)^{\text{LQ}}/\left(\mathcal{C}^{\tau}_{10}\right)^{\rm
SM}\right|^{2}\,,$ (18)

while for $\text{Br}(B\to K^{(*)}\tau^{+}\tau^{-})$ we refer the reader to
Ref. Capdevila:2017iqn .

###  3.4 Difference in forward-backward asymmetries $\Delta A_{\text{FB}}$

Ref. Bobeth:2021lya  unveiled a tension of about $4\sigma$ between the SM
predictions and data for the difference between the forward-backward asymmetry
originating from $\bar{B}\to D^{*}\mu\bar{\nu}$ decays and that originating
from $\bar{B}\to D^{*}e\bar{\nu}$ decays,

$\displaystyle\Delta
A_{\text{FB}}=A_{\text{FB}}^{(\mu)}-A_{\text{FB}}^{(e)}\,.$ (19)

In their fit based on Belle data, the authors of Ref. Bobeth:2021lya  found an
experimental value of

$\displaystyle\Delta
A^{\text{exp}}_{\text{FB}}=0.0349(89)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Belle:2018ezy}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(20)

which has to be compared to SM predictions exhibiting a negative value

$\displaystyle\Delta A^{\text{SM}}_{\text{FB}}=-0.0045(3)\,,$ (21)

as obtained using flavio.

Leptoquark solutions to this $\Delta A_{\text{FB}}$ anomaly have been
discussed in Ref. Carvunis:2021dss . While models with an $S_{2}$ leptoquark
cannot describe the data as well as models with an $S_{1}$ leptoquark, they
can still significantly improve the fit. The relevant WET coefficients are the
ones shown in Eq. (9), once we replace tau leptons $\tau$ with muons $\mu$.

###  3.5 Anomalous magnetic moments and electric dipole moments of leptons

Measurements of charged lepton anomalous magnetic moments ($a_{\ell}$) and
electric dipole moments (EDMs) are both highly sensitive probes of new
physics. The effects of $\Phi_{2}$ on the predictions for these observables
are all related to the operator ${\cal O}_{7}$. Whereas we could in principle
consider all three generations of leptons, we ignore tau leptons as the
existing bounds are not constraining. We should however keep in mind that a
polarised beam option at Belle II Liptak:2021opc  offers a possibility for a
measurement of $a_{\tau}$ that could be sensitive to various new physics
models Bernabeu:2007rr ; Bernabeu:2008ii ; Crivellin:2021spu , including ours.

The current averaged values for the muon and electron anomalous magnetic
moments read

$a_{e}^{\text{exp}}=115\ 965\ 218.091(26)\times
10^{-11}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ParticleDataGroup:2020ssz}{\@@citephrase{(}}{\@@citephrase{)}}}}\
\ \text{and}\ \ a_{\mu}^{\text{exp}}=116\ 592\ 061(41)\times
10^{-11}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Muong-2:2021ojo}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(22)

where the latter measurement includes Run 1 data from the Fermilab Muon $g-2$
experiment. The corresponding SM predictions are given by

$\displaystyle a_{e}^{\text{SM, Cs}}$ $\displaystyle=115\ 965\
218.161(23)\times 10^{-11}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Parker:2018vye}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(23) $\displaystyle a_{e}^{\text{SM, Rb}}$ $\displaystyle=115\ 965\
218.0252(95)\times 10^{-11}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Morel:2020dww}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
$\displaystyle a_{\mu}^{\text{SM}}$ $\displaystyle=116\ 591\ 810(43)\times
10^{-11}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aoyama:2020ynm}{\@@citephrase{(}}{\@@citephrase{)}}}}\,.$

For $a_{e}$ we quote two sets of contradicting predictions, that are
respectively based on the measurement of the fine-structure constant
$\alpha^{-1}$ in Cs and in Rb atoms. While the disagreement between the
experimental determination and the SM prediction for $a_{e}$ remains unclear,
the deviation in $a_{\mu}$ is better established and currently amounts to
$4.2\sigma$. It is widely known as the $(g-2)_{\mu}$ anomaly. In our numerical
analysis, we use the package flavio but consider both theoretical
determinations of $a_{e}$ stated in Eq. (23) individually. In contrast, the
measurements of the lepton EDMs have so far all yielded null results.
Currently, the most stringent exclusion limits read

$d_{e}^{\text{exp}}<0.11\times 10^{-28}\text{ e
cm}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ParticleDataGroup:2020ssz}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,\qquad
d_{\mu}^{\text{exp}}<1.8\times 10^{-19}\text{ e
cm}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{ParticleDataGroup:2020ssz}{\@@citephrase{(}}{\@@citephrase{)}}}}\,.$
(24)

While the latter limit is currently not constraining, it is expected to be
significantly improved in future experiments Adelmann:2021udj ; Aiba:2021bxe .

As already mentioned, the sole relevant WET operator governing charged
lepton’s $g-2$ and EDMs is the ${\cal O}_{7}$ operator, defined by

$\left(\mathcal{O}_{7}\right)_{\ell\ell}=\frac{4G_{F}}{\sqrt{2}}\frac{e}{16\pi^{2}}m_{\ell}\left(\bar{\ell}\sigma^{\mu\nu}P_{R}\ell\right)F_{\mu\nu}\,.$
(25)

Starting from our model featuring three generations of $\Phi_{2}$ LQs, the
associated Wilson coefficient is obtained after integrating out the heavy LQs
Crivellin:2020mjs ; Aebischer:2021uvt ; Bigaran:2021kmn ; Dorsner:2016wpm ,

$\left(\mathcal{C}_{7}\right)^{\text{LQ}}_{\ell\ell}\approx\frac{\sqrt{2}}{4G_{F}m_{\ell}}\\!\Bigg{[}\\!-\\!\frac{N_{c}m_{\ell}}{8M_{\ell}^{2}}\sum_{i=1}^{3}\left(\left|Y_{i\ell}^{LR}\right|^{2}\\!+\\!\left|Y_{i\ell}^{RL}\right|^{2}\right)\\!+\\!\frac{N_{c}}{12M_{\ell}^{2}}\sum_{i=1}^{3}m_{u_{i}}\hat{Y}_{i\ell}^{LR}Y_{i\ell}^{RL*}\mathcal{E}_{1}\left(x_{i}\right)\Bigg{]}\,,$
(26)

with $x_{i}=m_{u_{i}}^{2}/M_{\ell}^{2}$ and $\mathcal{E}_{1}(x)\equiv
1+4\log(x)$. The dominant contribution is the one included in the second term
for $i=3$, i.e. the contribution enhanced by the large value of the top mass
$m_{t}$. Predictions for $a_{\ell}$ and $d_{\ell}$ are then given by

$a_{\ell}^{\text{LQ}}=\frac{G_{F}m_{\ell}^{2}}{\sqrt{2}\pi^{2}}\
\text{Re}\big{\\{}\left(\mathcal{C}_{7}\right)_{\ell\ell}\big{\\}}\qquad\text{and}\qquad\left|d_{\ell}^{\text{LQ}}\right|=\frac{eG_{F}m_{\ell}}{2\sqrt{2}\pi^{2}}\
\Big{|}\text{Im}\big{\\{}\left(\mathcal{C}_{7}\right)_{\ell\ell}\big{\\}}\Big{|}\,,$
(27)

with the dominant contribution being the one proportional to the real and
imaginary part of $\hat{Y}_{3\ell}^{LR}Y_{3\ell}^{RL*}$ respectively. In
addition, the same combination of LQ Yukawa matrix elements contributes to a
radiative shift in the charged lepton masses Bigaran:2021kmn ,

$m^{\text{LQ}}_{\ell}\approx-\frac{m_{t}N_{c}}{16\pi^{2}}\mathcal{E}_{3}\left(\frac{\mu^{2}}{M_{\ell}^{2}},\frac{m_{t}^{2}}{M_{\ell}^{2}}\right)\hat{Y}^{LR}_{3\ell}Y^{RL*}_{3\ell}\
\ \text{with}\ \
\mathcal{E}_{3}\left(x,y\right)=\frac{1}{\epsilon}\\!+\\!1\\!+\\!\log\left(x\right)\\!+\\!y\log\left(y\right)\,,$
(28)

when we restrict ourselves to the contribution enhanced by the top mass.

###  3.6 Parity violation observables

By measuring parity-violating interactions between electrons and nucleons, the
nucleon weak charges can be determined. Currently, the best measurement of the
weak charge of the proton, $Q_{w}^{p}$, comes from the $Q_{\rm weak}$
experiment Qweak:2014xey ; Carlini:2019ksi  at Jefferson Lab,

$Q_{w}^{\text{exp}}(p)=0.0704(47)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Qweak:2018tjf,Crivellin:2021bkd}{\@@citephrase{(}}{\@@citephrase{)}}}}.$
(29)

In addition, the most precise results from atomic parity violation experiments
were obtained for ${}^{133}\text{Cs}$ atoms Wood:1997zq ; Guena:2004sq ,

$Q_{w}^{\text{exp}}\left({}^{133}\text{Cs}\right)=-72.94(43)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Cadeddu:2021dqx}{\@@citephrase{(}}{\@@citephrase{)}}}}.$
(30)

These weak charge measurements can be used to constrain the necessarily chiral
quark-electron interactions induced by LQs. The corresponding effects in
parity violation (PV) observables have already been studied in detail in Ref.
Crivellin:2021bkd . The relevant WET operators are

$\displaystyle\left(\mathcal{O}_{V}^{LR}\right)_{qqee}$
$\displaystyle=\frac{4G_{F}}{\sqrt{2}}\left(\bar{q}\gamma^{\mu}P_{L}q\right)\left(\bar{e}\gamma_{\mu}P_{R}e\right)\,,$
(31) $\displaystyle\left(\mathcal{O}_{V}^{LR}\right)_{eeqq}$
$\displaystyle=\frac{4G_{F}}{\sqrt{2}}\left(\bar{e}\gamma^{\mu}P_{L}e\right)\left(\bar{q}\gamma_{\mu}P_{R}q\right)\,,$

for $q=u,d$. Integrating out the heavy LQ fields in our model, we can derive
the corresponding Wilson coefficients,

$\displaystyle\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{uuee}$
$\displaystyle=\frac{-\sqrt{2}}{4G_{F}}\frac{\left|\hat{Y}_{1e}^{LR}\right|^{2}}{2M_{e}^{2}}\,,\qquad$
$\displaystyle\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{eeuu}$
$\displaystyle=\frac{-\sqrt{2}}{4G_{F}}\frac{\left|Y_{1e}^{RL}\right|^{2}}{2M_{e}^{2}}\,,$
(32) $\displaystyle\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{ddee}$
$\displaystyle=\frac{-\sqrt{2}}{4G_{F}}\frac{\left|Y_{1e}^{LR}\right|^{2}}{2M_{e}^{2}}\,,\qquad$
$\displaystyle\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{eedd}$
$\displaystyle=0\,.$

There, for our numerical analysis we use Crivellin:2021egp

$\displaystyle Q_{w}$
$\displaystyle=-2\Bigg{[}Z\left(2\mathcal{C}_{1u}^{e}+\mathcal{C}_{1d}^{e}\right)+N\left(\mathcal{C}_{1u}^{e}+2\mathcal{C}_{1d}^{e}\right)\Bigg{]}\,,$
(33)

where $Z$ and $N$ are the atomic number and the number of neutrons in a
nucleus respectively, and where we have defined

$\mathcal{C}_{1q}^{e}=\mathcal{C}_{1q}^{e,\text{SM}}+\mathcal{C}_{1q}^{e,\text{LQ}}\,.$
(34)

In this last relation,

$\mathcal{C}_{1u}^{e,\text{SM}}=-0.1888\,,\
\mathcal{C}_{1d}^{e,\text{SM}}=0.3419~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Erler:2013xha,
ParticleDataGroup:2020ssz}{\@@citephrase{(}}{\@@citephrase{)}}}}\ \
\text{and}\ \
\mathcal{C}_{1q}^{e,\text{LQ}}=\Big{[}\left(\mathcal{C}^{LR}_{V}\right)^{\text{LQ}}_{qqee}\\!-\\!\left(\mathcal{C}^{LR}_{V}\right)^{\text{LQ}}_{eeqq}\Big{]}\,.$
(35)

We recall that for protons $Z=1$ and $N=0$, whereas for ${}^{133}\text{Cs}$
atoms we have $Z=55$ and $N=78$. In order to extract constraints on our model,
we finally build a likelihood functions

$\displaystyle-2\log\mathcal{L}=\frac{\left(Q_{w}^{\text{exp}}-Q_{w}\right)^{2}}{\sigma^{2}}\,,$
(36)

where $\sigma$ stands for the experimental resolution.

###  3.7 $Z$-boson decays into leptons and neutrinos

The Lagrangian describing the interaction of the $Z$ boson with left-handed
and right-handed leptons can be generically written as

$\delta\mathcal{L}_{\text{eff}}^{Z}=\frac{g}{c_{w}}\sum_{\ell}\bar{\ell}\gamma^{\mu}\big{[}g_{\ell_{L}}P_{L}+g_{\ell_{R}}P_{R}\big{]}\ell
Z_{\mu}+\frac{g}{c_{w}}\sum_{\ell}\bar{\nu}_{\ell}\gamma^{\mu}\big{[}g_{\nu_{L}}P_{L}\big{]}\nu_{\ell}Z_{\mu}\,,$
(37)

where $g$ is the weak coupling constant and $c_{w}=\cos\theta_{w}$ is the
cosine of the electroweak mixing angle $\theta_{w}$. SM predictions for the
charged lepton sector lead to

$g_{\ell_{L}}^{\text{SM}}=-0.26919(20)\qquad\text{and}\qquad
g_{\ell_{R}}^{\text{SM}}=0.23208(17)\,,$ (38)

which can be compared with measurements at LEP ALEPH:2005ab ,

$\displaystyle g_{e_{L}}^{\text{exp}}$ $\displaystyle=-0.26963(30)\,,\qquad$
$\displaystyle g_{e_{R}}^{\text{exp}}$ $\displaystyle=0.23148(29)\,,$ (39)
$\displaystyle g_{\mu_{L}}^{\text{exp}}$ $\displaystyle=-0.2689(11)\,,\qquad$
$\displaystyle g_{\mu_{R}}^{\text{exp}}$ $\displaystyle=0.2323(13)\,,$
$\displaystyle g_{\tau_{L}}^{\text{exp}}$
$\displaystyle=-0.26930(58)\,,\qquad$ $\displaystyle
g_{\tau_{R}}^{\text{exp}}$ $\displaystyle=0.23274(62)\,.$

In our numerical analysis, we include the impact of our model on those
couplings through the likelihood function

$-2\log\mathcal{L}\\!=\\!\sum_{\ell_{A},\ell^{\prime}_{B}}\Bigg{[}\Big{(}g_{\ell_{A}}^{\text{SM}}+\text{Re}\big{\\{}g^{\text{LQ}}_{\ell_{A}}\big{\\}}-g_{\ell_{A}}^{\text{exp}}\Big{)}\Big{(}V^{-1}\Big{)}_{\ell_{A}\ell^{\prime}_{B}}\Big{(}g_{\ell^{\prime}_{B}}^{\text{SM}}+\text{Re}\big{\\{}g^{\text{LQ}}_{\ell^{\prime}_{B}}\big{\\}}-g_{\ell^{\prime}_{B}}^{\text{exp}}\Big{)}\Bigg{]}\,,$
(40)

where $\ell,\ell^{\prime}\in\\{e,\mu,\tau\\}$, $A,B\in\\{L,R\\}$ and $V$
stands for the covariance matrix including the experimental uncertainties as
well as the correlations among the measurements. The expressions for
$g^{\text{LQ}}_{\ell_{A}}$ quantifying the LQ contributions to the $Z$
couplings are given in Appendix A.

The $Z$-boson coupling to neutrinos can be extracted from the LEP measurement
of the effective number of neutrino generations ALEPH:2005ab

$N_{\nu}^{\text{exp}}=2.9840(82)\,.$ (41)

This last measurement can be numerically confronted to predictions from our
model through the likelihood function

$-2\log\mathcal{L}=\frac{\left(N_{\nu}-N_{\nu}^{\text{exp}}\right)^{2}}{\sigma^{2}}\qquad\text{with}\qquad
N_{\nu}=\sum_{i}\left(1+\frac{\text{Re}\\{g^{\text{LQ}}_{\nu_{\ell,L}}\\}}{g_{\nu_{L}}^{\text{SM}}}\right)^{2}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Arnan:2019olv}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(42)

where $\sigma$ refers to the experimental uncertainty. In this expression, we
consider the SM $Z$-boson coupling value that has been measured at LEP
ALEPH:2005ab ,

$g^{\text{SM}}_{\nu_{L}}=0.50199(19)\,,$ (43)

and the expression for $g^{\text{LQ}}_{\nu_{\ell,L}}$ given in Appendix A.

###  3.8 $\Delta F=2$ meson mixing observables

Measurements of $K^{0}-\bar{K}^{0}$, $B_{d}-\bar{B}_{d}$, $B_{s}-\bar{B}_{s}$
and $D^{0}-\bar{D}^{0}$ mixing parameters allow for the extraction of
constraints on our LQ model. We use in our analysis three meson mass
differences and a $D$-meson mixing parameter for which the associated
measurements,

$\displaystyle\Delta M_{K^{0}}$ $\displaystyle=-3.484(6)\times 10^{-15}\text{
GeV}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aoki:2021kgd}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(44) $\displaystyle\Delta M_{B_{d}}$ $\displaystyle=-3.33(1)\times
10^{-13}\text{ GeV}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{HFLAV:2019otj}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(45) $\displaystyle\Delta M_{B_{s}}$ $\displaystyle=-1.168(1)\times
10^{-11}\text{ GeV}~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{HFLAV:2019otj}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(46) $\displaystyle x_{D^{0}}$
$\displaystyle=0.0035(15)~{}\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bona:2017gut}{\@@citephrase{(}}{\@@citephrase{)}}}}\,,$
(47)

are in good agreement with SM theory predictions. For $K^{0}$, $B_{d}$ and
$B_{s}$ mixing the sign of $\Delta M$ is known, while this is not the case for
$D^{0}$ mixing. Moreover, we further include the UTfit UTfit:2006onp ;
UTfit:2007eik  likelihood for the phase $\phi_{B_{s}}$, that restricts the new
physics contributions to the complex phase inherent to $B_{s}-\bar{B}_{s}$
mixing, as well as the $D^{0}-\bar{D}^{0}$ mixing phase $\Phi_{12}$ whose
UTfit value is

$\Phi_{12}=\left(0.045\pm 1.335\right)^{\circ}\,.$ (48)

The relevant four-fermion operators affecting the $\Delta F=2$ observables
related to $D^{0}-\bar{D}^{0}$ mixing are

$\displaystyle\left(\mathcal{O}_{V}^{AB}\right)_{ucuc}$
$\displaystyle=\left(\bar{c}\gamma^{\mu}P_{A}u\right)\left(\bar{c}\gamma_{\mu}P_{B}u\right)\,,\qquad$
(49)

with $A,B\in\\{L,R\\}$, whilst those relevant for other meson mixings are
obtained via straight-forward exchanges of the quark flavours in the
expression of the operators (49). At leading order, the Wilson coefficients
associated with these operators are Crivellin:2021lix

$\displaystyle\left(\mathcal{C}_{V}^{LL}\right)^{\text{LQ}}_{ucuc}$
$\displaystyle=\frac{-1}{128{\pi^{2}}}\sum\limits_{\ell}\frac{\left(\hat{Y}_{1\ell}^{LR}\right)^{2}\left(\hat{Y}_{2\ell}^{LR*}\right)^{2}}{M_{\ell}^{2}}\,,$
(50) $\displaystyle\left(\mathcal{C}_{V}^{RR}\right)^{\text{LQ}}_{ucuc}$
$\displaystyle=\frac{-1}{64{\pi^{2}}}\sum\limits_{\ell}\frac{\left(Y_{1\ell}^{RL}\right)^{2}\left(Y_{2\ell}^{RL*}\right)^{2}}{M_{\ell}^{2}}\,,$
$\displaystyle\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{ucuc}$
$\displaystyle=\frac{1}{32{\pi^{2}}}\sum\limits_{\ell}\Bigg{(}\frac{\hat{Y}_{1\ell}^{LR}\hat{Y}_{2\ell}^{LR*}Y_{1\ell}^{RL}Y_{2\ell}^{RL*}}{M_{\ell}^{2}}\Bigg{)}\,,$

whereas for $B$ and $K$ meson mixing, the relevant Wilson coefficients read

$\left(\mathcal{C}_{V}^{LL}\right)^{\text{LQ}}_{d_{i}d_{j}d_{i}d_{j}}=\frac{-1}{128{\pi^{2}}}\sum\limits_{\ell}\frac{\left(Y_{j\ell}^{LR}\right)^{2}\left(Y_{i\ell}^{LR*}\right)^{2}}{M_{\ell}^{2}}\,,\
\
\left(\mathcal{C}_{V}^{RR}\right)^{\text{LQ}}_{d_{i}d_{j}d_{i}d_{j}}=\left(\mathcal{C}_{V}^{LR}\right)^{\text{LQ}}_{d_{i}d_{j}d_{i}d_{j}}=0\,,$
(51)

for generation indices $(i,j)=(2,1)$, $(3,1)$ and $(3,2)$.

###  3.9 Drell-Yan di-lepton searches at the LHC

(SM)

(LQ)

Figure 2: The leading-order SM and LQ diagrams (drawn with TikZ-Feynman
Ellis:2016jkw ) that contribute to di-lepton production
$pp\to\ell^{+}\ell^{-}$ at the LHC. While the SM contribution occurs via the
$s$-channel exchange of a (virtual) photon or $Z$-boson, the LQ diagram on the
right features a $t$-channel exchange and receives an enhancement at high di-
lepton invariant masses.

As illustrated by the neutral-current Feynman diagrams shown in Figure 2, LQ
exchanges can importantly contribute to Drell-Yan (DY) production at the LHC.
Due to the relative energy enhancement of the LQ $t$-channel partonic
amplitude relative to the SM one, measuring high-energy tails in
$pp\to\ell^{+}\ell^{-}$ and $pp\to\ell\nu$ processes at the LHC can
potentially offer competitive bounds on the LQ couplings to fermions, even
when parton density suppression is taken into account.

The CMS analysis of non-resonant di-lepton production CMS:2021ctt  provided
measurements for inclusive cross-section ratios in $n\\!=\\!9$ bins in the di-
lepton invariant mass $m_{\ell\ell}$ included in the $[200,3500]$ GeV range
(with $\ell=e,\mu$),

$R_{\mu\mu/ee,n}\equiv\dfrac{\int_{\text{bin
n}}\frac{d\sigma(pp\to\mu^{+}\mu^{-})}{dm_{\mu\mu}}dm_{\mu\mu}}{\int_{\text{bin
n}}\frac{d\sigma(pp\to e^{+}e^{-})}{dm_{ee}}dm_{ee}}\,.$ (52)

In the CMS publication, the results are normalised to SM predictions obtained
from Monte-Carlo (MC) simulations,

$R_{\mu\mu/ee,n}^{\text{data}}\ /\ R_{\mu\mu/ee,n}^{\text{MC}}\,.$ (53)

In this double ratio, many uncertainties cancel Greljo:2017vvb . Furthermore,
$R_{\mu\mu/ee}$ was normalised to unity in a combined bin ranging from 200 to
400 GeV to correct for the different detector sensitivity to electrons and
muons. The CMS collaboration measured an excess in the di-electron channel, so
that the ratios $R_{\mu\mu/ee,n}$ are smaller than one for large
$m_{\ell\ell}$ values. This clearly points towards another source of LFUV.

LQ contributions that could explain this excess have been studied in Refs.
Crivellin:2021bkd ; Crivellin:2021egp ; Crivellin:2021rbf . Here we largely
follow the analysis presented in the addendum to Ref. Crivellin:2021egp , but
simulated the cross-section with the full dependence on the LQ propagator. We
determined

$\int_{\text{bin
n}}\frac{d\sigma^{\text{SM(+LQ)}}(pp\to\ell^{+}\ell^{-})}{dm_{\ell\ell}}dm_{\ell\ell}\,,$
(54)

at leading order using MadGraph5_aMC@NLO Alwall:2014hca  version 3.2.0 with
the UFO Degrande:2011ua  model lqnlo_v5222Available from https://www.uni-
muenster.de/Physik.TP/research/kulesza/leptoquarks.html. Borschensky:2020hot ;
Borschensky:2021jyk  and the PDF set NNPDF40_nlo_as_01180 Ball:2021leu .

On the other hand, the ATLAS collaboration also performed measurements of the
high-$m_{\ell\ell}$ tail in di-lepton production, that we can thus use as an
additional probe for our model. For electrons and muons we recast their non-
resonant analysis presented in Ref. ATLAS:2020yat . There, a parametric
background-model function is fitted to the di-lepton invariant mass
distribution in the control regions $m_{ee}\in$ [280 GeV, 2200 GeV] and
$m_{\mu\mu}\in$ [310 GeV, 2070 GeV] for electrons and muons, respectively, and
then extrapolated to the signal regions (SRs) in which $m_{ee}\in$ [2200 GeV,
6000 GeV] and $m_{\mu\mu}\in$ [2070 GeV, 6000 GeV]. SM predictions for the
expected number of events in the SRs are next compared with measurements to
derive bounds on any new physics interfering constructively with SM DY
production. Interestingly, the ATLAS collaboration has also found slightly
more di-electron events and less di-muon events than expected.

For the tau-lepton channel, we recast the ATLAS search for heavy Higgs bosons
decaying into two tau leptons that has been presented in Ref. ATLAS:2020zms .
We consider events where both taus decay hadronically
($\tau_{\text{had}}\tau_{\text{had}}$), but we carry out a $b$-jet inclusive
analysis.

Finally, charged-current DY processes can also yield strong constraints on our
model. In particular, limits from the process $pp\to\tau\nu$ are directly
sensitive to the $\Phi_{2,\tau}$ coupling combination shown in Eq. (9). Bounds
on the $S_{2}$ couplings to taus were presented in Ref. Jaffredo:2021ymt
based on the latest 139 fb-1 mono-tau search at the LHC ATLAS:2021bjk . The
re-interpretation of the results of this analysis includes all contributing
Feynman diagrams, and in particular those featuring a LQ propagator. The
latter are found to be important for LQ masses in the TeV range, which
corresponds to a configuration for which an effective field theory description
is no longer valid. Limits derived from this $pp\to\tau\nu$ analysis have been
found less constraining than the ones originating from the
$pp\to\tau^{+}\tau^{-}$ analysis. We therefore impose the former as a sharp
cut on the LQ parameter space, without deriving in detail a full likelihood
function.

More information on all considered analyses and how we use them to constrain
our model can be found in Appendix B.

###  3.10 Single-resonant leptoquark production (SRP)

In Ref. Buonocore:2020erb , limits on LQ Yukawa couplings are derived based on
LQ single-resonant production $\ell q\to\Phi_{2,\ell}\to\ell q$, where the
initial-state lepton $\ell$ originates from the lepton density in the proton.
The authors include $e+q_{\text{light}}$ and $\mu+q_{\text{light}}$ final
states in their analysis, where $q_{\text{light}}\in\\{u,d,c,s\\}$, and they
further assume a minimal model containing a single $SU(2)_{L}$ LQ singlet that
couples exclusively to one quark and one lepton generation. In our numerical
analysis we generalise this setup and implement SRP exclusion limits as sharp
cuts in the LQ parameter space. This makes sure that current SRP bounds are
not violated through a choice of large enough LQ masses.

Since the $\Phi_{2,\ell}$ states have multiple $SU(2)_{L}$ components and they
potentially couple to multiple quark generations, we need to recast the limits
of Ref. Buonocore:2020erb  accordingly. In the minimal model and assuming
$M_{\ell}\gg m_{\ell},m_{q}$, the cross section for the process $\ell
q_{i}\to{\Phi_{2,\ell}}\to\ell q_{\text{light}}$ is proportional to
$\left|\lambda^{q}_{i\ell}\right|^{2}$, where $\lambda^{q}_{i\ell}$ for
$q=u,d$ are the Yukawa couplings to the $i^{\text{th}}$ quark generation and
the lepton flavour $\ell$. Comparing this to our model, we define the
effective couplings

$\displaystyle\left|\lambda_{i\ell}^{u,\text{eff}}\right|^{2}$
$\displaystyle\equiv\left(\left|\hat{Y}^{LR}_{i\ell}\right|^{2}+\left|Y^{RL}_{i\ell}\right|^{2}\right)\dfrac{\sum_{j=1}^{2}\left|\hat{Y}^{LR}_{j\ell}\right|^{2}+\left|Y^{RL}_{j\ell}\right|^{2}}{\sum_{j=1}^{3}\left|\hat{Y}^{LR}_{j\ell}\right|^{2}+\left|Y^{RL}_{j\ell}\right|^{2}}\,,$
(55) $\displaystyle\left|\lambda_{i\ell}^{d,\text{eff}}\right|^{2}$
$\displaystyle\equiv\left|Y^{LR}_{i\ell}\right|^{2}\dfrac{\sum_{j=1}^{2}\left|Y^{LR}_{j\ell}\right|^{2}}{\sum_{j=1}^{3}\left|Y^{LR}_{j\ell}\right|^{2}+\left|Y^{RL}_{j\ell}\right|^{2}}\,,$

such that $\sigma\sim\left|\lambda_{i\ell}^{q,\text{eff}}\right|^{2}$. Using
the limits $\left|\lambda^{q,\text{lim}}_{i\ell}\right|$ derived in Ref.
Buonocore:2020erb , the effective couplings of our model must satisfy

$\sum_{i=1}^{2}\left|\dfrac{\lambda_{i\ell}^{u,\text{eff}}}{\lambda_{i\ell}^{u,\text{lim}}}\right|^{2}+\left|\dfrac{\lambda_{i\ell}^{d,\text{eff}}}{\lambda_{i\ell}^{d,\text{lim}}}\right|^{2}<1\,,$
(56)

for $\ell=e,\mu$. Here we assume that the mass difference between the
$SU(2)_{L}$ components of $\Phi_{2,\ell}$ is negligible, which holds for $v\ll
M_{\ell}$.

###  3.11 Leptoquark pair production (PP)

| Coupling
---|---
|  $Y^{RL}_{1e}$ |  $Y^{RL}_{2e}$ |  $Y^{RL}_{3e}$ |  $Y^{RL}_{1\mu}$ |  $Y^{RL}_{2\mu}$ |  $Y^{RL}_{3\mu}$ |  $Y^{RL}_{3\tau}$
Limit [GeV] | $1790$ | $1760$ | $1480$ | $1730$ | $1690$ | $1470$ | 1440
Analysis | $e+q_{\text{light}}$ | $e+c$ | $e+t$ | $\mu+q_{\text{light}}$ | $\mu+c$ | $\mu+t$ | $\tau+t$
Reference | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020xov  | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020xov  | ATLAS:2021oiz
$\beta_{\text{eff}}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | 1.0 | 1.0

|  $Y^{LR}_{1e}$ |  $Y^{LR}_{2e}$ |  $Y^{LR}_{3e}$ |  $Y^{LR}_{1\mu}$ |  $Y^{LR}_{2\mu}$ |  $Y^{LR}_{3\mu}$ |  $Y^{LR}_{3\tau}$
---|---|---|---|---|---|---|---
Limit [GeV] | $1910$ | $1790$ | $1740$ | $1850$ | $1730$ | $1720$ | 1440
Analysis | $e+q_{\text{light}}$ | $e+q_{\text{light}}$ | $e+b$ | $\mu+q_{\text{light}}$ | $\mu+q_{\text{light}}$ | $\mu+b$ | $\tau+t$
Reference | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2020dsk  | ATLAS:2021oiz
$\beta_{\text{eff}}$ | $1.9$ | $1.0$ | $1.0$ | $1.9$ | $1.0$ | 1.0 | 1.0

Table 1: 95% C.L. limits on the LQ masses $M_{\ell}$ derived from the ATLAS
pair production analyses targeting the processes
$pp\to\Phi_{2,\ell}\Phi_{2,\ell}\to\ell q\ell q$. Only the couplings indicated
in the table header are assumed to be non-zero. We state the ATLAS analysis
that yields the most stringent limits for the respective scenario, and
additionally indicate the corresponding $\beta_{\text{eff}}$ value. The light
quarks $q_{\text{light}}$ consist of $u,d$ and $s$ quarks. No analysis exists
for leptoquark states coupling to $\tau$ and one of the $u$, $d$, $c$ or $s$
quarks.

The ATLAS and CMS collaborations have published several analyses targeting the
production of a pair of scalar LQs decaying into a specific $\ell+q$ or
$\nu+q$ final state. Since the exclusion limits based on the former decay mode
are more constraining and since
$\beta\equiv\text{Br}\left(\Phi_{2,\ell}\to\ell+q\right)\geq 0.5$ in general,
we solely focus on the $\ell q\ell q$ class of final states in our numerical
analysis. We derive mass limits for the specific coupling structure that we
consider, and make sure that the LQ masses $M_{\ell}$ lie well above these
bounds.

In analogy to Section 3.10, we make the choice of recasting exclusion limits
obtained by the ATLAS collaboration, enforcing the fact that $\Phi_{2,\ell}$
multiplets have two $SU(2)_{L}$ components and that $\beta$ can be different
from 1 for a specific choice of $\ell$ and $q$ flavours. Again assuming that
the mass difference between the $SU(2)_{L}$ components of $\Phi_{2,\ell}$ is
negligible, we introduce an “effective” $\beta_{\text{eff}}$ parameter,

$\beta^{2}_{\text{eff}}\equiv\left(\dfrac{\sum_{i\in
U}\left|\hat{Y}^{LR}_{i\ell}\right|^{2}+\left|Y^{RL}_{i\ell}\right|^{2}}{\sum_{i=1}^{3}\left|\hat{Y}^{LR}_{i\ell}\right|^{2}+\left|Y^{RL}_{i\ell}\right|^{2}}\right)^{2}\\\
+\left(\dfrac{\sum_{i\in
D}\left|Y^{LR}_{i\ell}\right|^{2}}{\sum_{i=1}^{3}\left|Y^{LR}_{i\ell}\right|^{2}+\left|Y^{RL}_{i\ell}\right|^{2}}\right)^{2}\,,$
(57)

for an analysis that would effectively include up-quark generations
$U\subset\left\\{1,2,3\right\\}$, down-quark generations
$D\subset\left\\{1,2,3\right\\}$ and charged-lepton generations
$\ell=e,\mu,\tau$. Considering the $\beta$-dependence of the limits given in
Refs. ATLAS:2019qpq ; ATLAS:2021oiz ; ATLAS:2020dsk ; ATLAS:2020xov , we
obtain the LQ mass limits presented in Table 1.

###  3.12 Oblique corrections

The effect of vacuum-polarization amplitudes of EW gauge bosons can be
parametrized by the oblique Peskin-Takeuchi parameters $S$, $T$ and $U$
Peskin:1990zt . The authors of Ref. Ellis:2018gqa  performed a global fit to
electroweak precision observables including LEP ALEPH:2005ab , Tevatron
CDF:2013dpa  and LHC ATLAS:2019fyu  data. While the fit is compatible with SM
predictions, it can be improved via a positive contribution to $\Delta T$.

The oblique corrections from LQs were studied extensively in Ref.
Crivellin:2020ukd . Their contributions to the Peskin-Takeuchi parameters read

$S^{\text{LQ}}\approx-\frac{7N_{c}v^{2}}{36\pi}\sum_{\ell}\frac{Y_{\ell}^{H(3)}}{M_{\ell}^{2}}\qquad\text{and}\qquad
T^{\text{LQ}}\approx+\frac{N_{c}v^{2}}{96\pi^{2}\alpha}\sum_{\ell}\left(\frac{Y_{\ell}^{H(3)}}{M_{\ell}}\right)^{2}\,.$
(58)

##  4 Phenomenological Analysis

In this section we perform a global analysis of our model with three
generations of the $\Phi_{2}$ leptoquark. For this we construct full
likelihood functions for the LQ parameters whenever possible
AbdusSalam:2020rdj . Implementing the formulas of Section 3, we use for the
numerical analysis the software packages flavio Straub:2018kue  and smelli
Aebischer:2018iyb . In order to maximize the log likelihood
$-2\log\mathcal{L}$, we employ the optimize.fmin algorithm of scipy
Virtanen:2019joe .

Figure 3: Preferred parameter space region providing an explanation for the
$R_{D^{(*)}}$ anomalies (orange), shown in the
$Y^{LR}_{3\tau}$–$Y^{RL}_{2\tau}$ plane. Our results assume that
$Y^{LR}_{3\tau}$ is real and we profile over the phase of $Y^{RL}_{2\tau}$. We
overlay to this contour the region favoured by $Z\tau^{+}\tau^{-}$ and
$Z\bar{\nu}\nu$ data (blue), as well as that favoured by the di-tau ATLAS DY
analysis (red) and that excluded by the mono-tau search (hatched). Our results
exhibit a mild tension.

###  4.1 Tauquark

As pointed out in Ref. Angelescu:2021lln , we can explain the $R_{D}$ and
$R_{D^{*}}$ anomalies with the product of couplings
$Y^{LR*}_{3\tau}Y^{RL}_{2\tau}$, given the presence of a large complex phase.
This phase avoids interference with the SM, at the price that the couplings
need to be large and the LQ mass needs to be low. We thus set $M_{\tau}=1.7$
TeV, which is compatible with the limits originating from LQ pair production
given in Table 1. In fact, such a choice leads to a LQ pair-production cross
section that is a factor 4 smaller than the corresponding bound. At the same
time, a non-zero $Y^{LR}_{3\tau}$ value leads to an $m_{t}^{2}/m_{Z}^{2}$
enhancement in the $Z\tau^{+}\tau^{-}$ and $Z\bar{\nu}\nu$ couplings, and our
non-zero $Y^{LR}_{2\tau}$ value contributes significantly, due to the second
generation quarks involved, to non-resonant $pp\to\tau^{+}\tau^{-}$ and
$pp\to\tau\nu$ production.

Figure 4: $\mathcal{C}_{9}^{U}\approx 36\times\mathcal{C}_{7}(\mu_{b})$ as a
function of $Y^{LR}_{2\tau}Y^{LR*}_{3\tau}$ for $M_{\tau}=1.7$ TeV. The
contour lines indicate predicted values for the $B_{s}\to\tau^{+}\tau^{-}$
branching ratio, and in the grey region the effect in $B_{s}-\bar{B}_{s}$
mixing is higher than 20% of the SM predictions.

We display in Figure 3 the preferred parameter space region that provides an
explanation for the $R_{D^{(*)}}$ anomalies (orange), the results being
projected in the $Y_{3\tau}^{LR}$–$Y_{2\tau}^{RL}$ plane. As can be seen, this
leads to strong bounds on the size of the tauquark couplings to fermions. In
order to derive those constraints, we have profiled over the relative complex
phase of the $Y^{LR*}_{3\tau}$ and $Y^{RL}_{2\tau}$ couplings, which we
attribute to $Y^{RL}_{2\tau}$ (i.e. we assume that $Y_{3\tau}^{LR}$ is real).
Despite of this new source of CP violation, EDM bounds are not (yet)
constraining (see Appendix C). The figure also shows the parameter space
region favoured by $Z\to\tau^{+}\tau^{-}$ and $Z\to\bar{\nu}\nu$ coupling data
(blue) as well as by neutral-current and charged-current DY measurements at
the LHC in the di-tau (red) and mono-tau (hatched) channel. While the
$R_{D^{(*)}}$ anomalies can be partially explained, there is a mild tension
with the EW fit (i.e. with $Z\to\tau^{+}\tau^{-}$ and $Z\to\bar{\nu}\nu$
data), and the DY di-tau bounds derived from recent measurements achieved by
the ATLAS collaboration. This leads to a combined likelihood difference

$-2\Delta\log\mathcal{L}\equiv
2\log\mathcal{L}_{\text{SM}}-2\log\mathcal{L}=-8.34\,,$ (59)

which corresponds to a pull of $2.1\sigma$ for three degrees of freedom
(d.o.f.).

As calculated in Section 3.2, the loop-diagrams in Figure 1 induce lepton
flavour violation universal effects in $b\to s\ell^{+}\ell^{-}$ decays (via
${\cal O}_{9}^{U}$ and ${\cal O}_{7}$ operators) that are proportional to the
product of couplings $Y^{LR}_{2\tau}Y^{LR*}_{3\tau}$. While these
contributions cannot account for the $R_{K^{(*)}}$ anomalies, they are capable
of explaining (partially) $P_{5}^{\prime}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$
data. Moreover, as shown in the next subsection, this effect further improves
the fit to $b\to s\ell^{+}\ell^{-}$ data once the LFUV effects originating
from the presence of the muoquark are included. On the other hand, the size of
$Y^{LR}_{2\tau}Y^{LR*}_{3\tau}$ is limited by $B_{s}-\bar{B}_{s}$ mixing.
While $Y^{LR*}_{3\tau}$ was already constrained by $Z\tau^{+}\tau-$ and
$Z\bar{\nu}\nu$ data (see above), a non-zero value of $Y^{LR}_{2\tau}$
additionally contributes to $D^{0}-\bar{D}^{0}$ mixing. However, for
$|Y^{LR}_{2\tau}|\approx 1/2Y^{LR*}_{3\tau}$, $B_{s}-\bar{B}_{s}$ mixing
provides the leading constraint and the corresponding bounds, together with
their impact on the Wilson coefficients $\mathcal{C}_{9}^{U}$ and
$\mathcal{C}_{7}$ are shown in Figure 4.

###  4.2 Muoquark

Figure 5: Top: Bounds on the $\Phi_{2,\tau}$ couplings $Y^{LR}_{3\tau}$ and
$Y^{LR}_{2\tau}$ derived from the combined fit of $\Phi_{2,\tau}$ and
$\Phi_{2,\mu}$ effects to $b\to s\ell^{+}\ell^{-}$ observables (orange). Our
results are obtained by profiling over the $\Phi_{2,\mu}$ contributions
depending on the $Y^{RL}_{3\mu}$ coupling. The coupling $Y_{2\tau}^{LR}$ is
tightly constrained by $D^{0}-\bar{D}^{0}$ mixing (green), and large
$Y^{LR}_{3\tau}$ values worsen the fit to $B_{s}-\bar{B}_{s}$ mixing (green as
well) and leads to tension with $Z\ell^{+}\ell^{-}$ and $Z\bar{\nu}\nu$ data
(blue). The figure also includes predictions for the
$B_{s}\to\tau^{+}\tau^{-}$ branching ratio. Bottom: The same solution
presented in terms of the product $Y^{LR}_{2\tau}Y^{LR*}_{3\tau}$ and the
muoquark coupling $Y^{RL}_{3\mu}$. We profile over the ratio
$Y^{LR*}_{3\tau}/Y^{LR}_{2\tau}$ that is relevant for the bounds originating
from $D^{0}-\bar{D}^{0}$ mixing, as well as from the fit to
$Z\tau^{+}\tau^{-}$ and $Z\bar{\nu}\nu$ data.

Regarding $b\to s\ell^{+}\ell^{-}$, an excellent fit to data can be obtained
in new physics scenarios featuring an LFU $\mathcal{C}_{9}^{U}$ effect in
addition to a LFUV violating effect
$\mathcal{C}_{9}^{\mu}=-\mathcal{C}_{10}^{\mu}$ Alguero:2018nvb ;
Alguero:2019ptt . In fact, this scenario even constitutes the two-dimensional
one which gives the best fit to data Alguero:2021anc . This is precisely the
setup realised in the considered model with three generations of LQs, in which
the photon penguin contribution induced by the presence of the tauquark is
combined with the $W$-box contribution involving the muoquark, so that
$\mathcal{C}^{\mu}_{9}=-\mathcal{C}^{\mu}_{10}$. We checked that the tree-
level effect giving rise to $\mathcal{C}_{9}^{\mu}=+\mathcal{C}_{10}^{\mu}$
does not improve the fit, such that $Y^{LR*}_{3\mu}Y^{LR}_{2\mu}$ must be
small. Furthermore, also the preferred value of
$\mathcal{C}_{9}^{e}=+\mathcal{C}_{10}^{e}$ that is induced by the
electroquark is consistent with zero once added to the other LQ contributions.
As before, we choose the muoquark mass to be as low as possible while still
satisfying the limits from LQ pair-production searches at the LHC by a clear
margin. We hence fix $M_{\mu}=2$ TeV.

In Figure 5 we focus on the three couplings $Y^{LR}_{3\tau}$, $Y^{LR}_{2\tau}$
and $Y^{RL}_{3\mu}$ of the tauquark $\Phi_{2,\tau}$ and the muoquark
$\Phi_{2,\mu}$, assuming them to be real. While these couplings are
constrained by $B_{s}-\bar{B}_{s}$ mixing, $D^{0}-\bar{D}^{0}$ mixing, as well
as by $Z\tau^{+}\tau^{-}$ and $Z\bar{\nu}\nu$ coupling measurements, we can
still significantly improve the fit to $b\to s\ell^{+}\ell^{-}$ data. In fact,
a likelihood value $-2\Delta\log{\cal L}=-70.4$, corresponding to a combined
pull of $7.9\sigma$ for three d.o.f., can be reached. Whereas the $W$-box
contribution leading to $\mathcal{C}_{9}^{\mu}=-\mathcal{C}_{10}^{\mu}$ via a
$Y^{RL}_{3\mu}$ coupling is in tension with $Z\mu^{+}\mu^{-}$ and
$Z\bar{\nu}\nu$ Angelescu:2021lln  data, this tension is reduced once the
contribution from the tauquark discussed in Section 4.1 is accounted for.
Therefore, the combined effect of $\Phi_{2,\tau}$ and $\Phi_{2,\mu}$
leptoquarks does not only result in a better fit to $b\to s\ell^{+}\ell^{-}$
data, but also weakens the bounds from EW precision observables as a smaller
coupling $Y^{RL}_{3\mu}$ suffices.

Figure 6: Simultaneous solution to the $b\to s\ell^{+}\ell^{-}$ (orange) and
$\left(g-2\right)_{\mu}$ (light blue) anomalies, as obtained by combining the
effects of the couplings $Y^{LR}_{3\tau}$, $Y^{LR}_{2\tau}$, $Y^{RL}_{3\mu}$
and $Y^{LR}_{3\mu}$. We additionally show the relative radiative correction to
the muon mass (black lines), that are derived by only taking into account only
the finite part of the involved loop function.

We now include $(g-2)_{\mu}$ in our analysis. To provide an explanation to
this anomaly, we need the combined effect of non-zero $Y^{RL}_{3\mu}$ and
$Y^{LR}_{3\mu}$ couplings so that we can get the desired $m_{t}/m_{\mu}$
enhancement. Adding therefore $Y^{LR}_{3\mu}$ as a free parameter, but
profiling over $Y_{3\tau}^{LR}$ and $Y_{2\tau}^{LR}$, we obtain the results of
Figure 6. While we have assumed that all couplings are real, they can
generally be complex. As consequence, a large muon EDM can be generated
Crivellin:2018qmi . Our results additionally show that a chirally enhanced
explanation for the $(g-2)_{\mu}$ anomaly leads to large radiative
contributions to the muon mass as well as to the $h\to\mu\mu$ branching ratio
Crivellin:2021rbq . While in order to measure the latter percent-level effect
a future precision determination, like at FCC-hh FCC:2018vvp , would be
necessary Crivellin:2020tsz ; Crivellin:2020mjs , the contribution to the muon
mass can be of order one (see the dashed isolines in Figure 6). However, this
effect is not physical. It can be absorbed in a re-definition of the muon mass
and thus only be bounded by fine-tuning arguments requiring the absence of
large accidental cancellations. The combined log-likelihood difference is
further decreased to $-87.5$, corresponding to a SM pull of $8.7\sigma$ for
four d.o.f.

Finally, as shown in Ref. Carvunis:2021dss , the presence of a muoquark in the
model can only weaken, but not fully explain, the anomaly in $\Delta
A_{\text{FB}}$. In this case, a product of couplings
$Y^{RL}_{2\mu}Y^{LR*}_{3\mu}\approx-1.4$ would lead to a pull of $1.8\sigma$
for two d.o.f. .

###  4.3 Electroquark

Figure 7: Parameter space region leading to a possible explanation for the
excess in di-electrons found by the CMS collaboration (light blue). The
solution via real couplings $Y^{LR}_{1e}$ and $Y_{1e}^{RL}$ prefers values of
$\mathcal{O}(0.6)$, which also yields an improved fit to PV observables
Crivellin:2021bkd . The limit from the electron EDM measurement additionally
places a strong constraint on the complex phases of the couplings.  Figure 8:
Potential effect of the $\Phi_{2,e}$ leptoquark on $(g-2)_{e}$ via the
couplings $Y^{LR}_{3e}$ and $Y^{RL}_{3e}$. We show the preferred parameter
space regions for the two contradicting SM determinations of
$\left(g-2\right)_{e}$ based on Rb (blue) and Cs (orange) atoms. In the
preferred parameter space regions, sizeable loop corrections to the electron
mass are generated, and the EDM constraint places strong limits on the complex
phases of these couplings.

In the electron sector we aim to explain the CMS excess found in non-resonant
di-electron production. In order to satisfy the bounds on the electroquark
mass coming from its pair production at the LHC, $M_{e}$ should be of at least
2.1 TeV. However, we also have to consider SRP limits. The latter severely
restrict the electroquark couplings to light quarks, which need to be large
enough to yield a sizeable effect in non-resonant di-electron production to
explain the deviations considered. Importantly, SRP involves on-shell LQs,
while the corresponding effect on the non-resonant di-electron spectrum occurs
from $t$-channel exchange (see the diagram on the right in Figure 2).
Therefore, by increasing the electroquark mass, the former bounds can be
avoided independently of the LQ couplings while sizeable contributions to the
latter process are still possible. We find that $M_{e}>3.5$ TeV is sufficient
to avoid any SRP limits without requiring a too large Yukawa couplings when
building an explanation for the non-resonant di-electron signal.

The corresponding preferred region of the parameter space is presented in the
$Y^{LR}_{1e}$–$Y^{RL}_{1e}$ plane in Figure 7. Our results combine the CMS and
ATLAS analyses of non-resonant di-electron production, and are shown in blue.
Interestingly, these couplings can also achieve a slightly improved fit to the
low-energy parity violation data Crivellin:2021bkd . Moreover, for sizeable
$Y^{LR}_{1e}$ values the electron EDM limit places a strong constraint on
$\text{arg}(Y^{LR}_{1e}Y^{RL*}_{1e})$. For the best-fit point in Figure 7, it
has to be smaller than $\mathcal{O}(10^{-5})$.

Finally, the electroquark couplings $Y^{LR}_{3e}$ and $Y^{RL}_{3e}$ could in
principle also lead to a deviation from SM predictions in $(g-2)_{e}$. The
corresponding preferred regions in the parameter space are shown in Figure 8
for the two contradicting determinations of $\alpha$ considered. In this case,
significant radiative corrections to the electron mass arise, which can become
larger than the measured mass itself. Additionally, the same couplings can
also give rise to an electron EDM, placing strong constraints on the complex
phases of the involved couplings. For the best fit values for the product
$Y^{LR}_{3e}Y^{RL*}_{3e}$ (i.e. at the center of the blue and orange regions
in Figure 7), the complex phase needs to be $\lesssim\mathcal{O}(10^{-5})$ in
order to satisfy EDM constraints.

###  4.4 $W$ mass

Finally, we discuss the shift in the predictions for the $W$ mass generated by
the presence of LQs in our model. We show the global fit Ellis:2018gqa  to new
physics contributions in the Peskin-Takeuchi parameters $S$ and $T$ in Figure
9 (for one d.o.f.), together with the effects that our model can yield. The
agreement with data is improved for positive values for $\Delta S$ and $\Delta
T$. As an illustration, the best fit point in our model is given by

$Y_{\ell}^{H(3)}=-1.16~{}~{}~{}(\ell=e,\mu,\tau)\,,$ (60)

where we assume that $Y^{H(3)}_{e}=Y^{H(3)}_{\mu}=Y^{H(3)}_{\tau}$ for
simplicity. The corresponding likelihood difference value is $-1.54$.

Figure 9: Global fit to the (difference of the) Peskin-Takeuchi parameters
$\Delta S$ and $\Delta T$. This fit can be improved via a positive
contribution to both $\Delta S$ and $\Delta T$, which corresponds to negative
couplings $Y^{H(3)}_{\ell}$. We indicate the position of the points
$Y^{H(3)}=\pm 1,\pm 2$ when we assume that
$Y^{H(3)}_{e}=Y^{H(3)}_{\mu}=Y^{H(3)}_{\tau}\equiv Y^{H(3)}$.

##  5 Conclusions

While it is known that a single $SU(2)_{L}$ doublet of scalar leptoquarks can
explain $b\to s\ell^{+}\ell^{-}$ data, $R_{D^{(*)}}$ and $(g-2)_{\mu}$
separately, a common explanation is not possible. This would indeed require
couplings to taus, muons and electrons simultaneously, which would lead to
unacceptably large effects in charged lepton flavour violating processes. In
order to overcome this obstacle, we proposed in this article to extend the SM
by three generations of scalar $SU(2)_{L}$ doublets $\Phi_{2,e}$,
$\Phi_{2,\mu}$ and $\Phi_{2,\tau}$, each of them carrying the corresponding
lepton flavour number. In that way, we can have multiple sources of LFUV while
the individual lepton flavours are still exactly conserved, as in the SM (with
massless neutrinos). As a result, our setup can be distinguished from LQ
scenarios with a single LQ already by considering low energy observables: It
predicts that while effects in observables involving different lepton
generations are possible, no sign of charged lepton flavour violating should
be observed. Furthermore, assigning lepton flavour numbers to LQs
automatically avoids bounds from proton decay experiments, as it forbids di-
quark couplings.

In this setup we showed that:

  * •

The tauquark $\Phi_{2,\tau}$, despite effects in $Z\to\tau^{+}\tau^{-}$,
$Z\to\bar{\nu}\nu$, $D^{0}-\bar{D}^{0}$ mixing and the bounds stemming from
$\tau^{+}\tau^{-}$ searches at the LHC, can (mostly) explain $b\to c\tau\nu$
data in the presence of a complex phase.

  * •

The presence of the tauquark $\Phi_{2,\tau}$ generates a universal operator
${\cal O}_{9}^{U}$ via off-shell photon penguin diagram contributions with tau
leptons running in the loop. While the effect is bounded by
$B_{s}-\bar{B}_{s}$ mixing and crucially depends on $M_{\tau}$ (i.e. the
relative effect in $B_{s}-\bar{B}_{s}$ mixing is enhanced for larger LQ
masses), a relevant contribution is possible.

  * •

An excellent fit to $b\to s\ell^{+}\ell^{-}$ data can be obtained via a
combination of tauquark contributions (generating $\mathcal{C}_{9}^{U}$) and
muoquark $W$-box contribution (generating
$\mathcal{C}^{\mu}_{9}=-\mathcal{C}^{\mu}_{10}$). This leads to a likelihood
difference $-2\Delta\log\mathcal{L}=-70.4$, which corresponds to $7.9\sigma$
for three d.o.f. .

  * •

$\left(g-2\right)_{\mu}$ can be explained via an $m_{t}/m_{\mu}$ enhanced
muoquark contribution.

  * •

The electron excess appearing in the invariant-mass tails of DY di-electron
production at the LHC $pp\to e^{+}e^{-}$ can be accounted for via a
contribution from the electroquark alone, without violating any bounds from
$D^{0}-\bar{D}^{0}$ mixing.

  * •

The EW fit can be improved by a constructive contribution to the $W$-boson
mass.

  * •

A (possible) new physics effect in $(g-2)_{e}$ could be incorporated.

For the latter, the phase of the couplings in general needs to be precisely
tuned in order to avoid the bounds from electron EDM. However, this obstacle
could be resolved in our model by requiring that the lepton masses are
generated at the loop level Greljo:2021npi . Such a radiative mass generation
leads to an automatic phase alignment between the mass term and the dipole
operators such that the electron EDM vanishes Borzumati:1999sp ;
Crivellin:2010gw .

##  Appendix A Leptoquark effects in $Z$-boson couplings

The $\Phi_{2,\ell}$ contributions $g^{\text{LQ}}_{\ell_{A}}$ to the $Z$ boson
couplings to leptons are given by Crivellin:2020mjs ; Arnan:2019olv 333Similar
results for the diquark contribution to $Z\to\ell^{+}\ell^{-}$ have been
obtained in Ref. Djouadi:1989me .

$\begin{split}g^{\text{LQ}}_{\ell_{L}}=&\
N_{c}\frac{Y^{RL}_{3\ell}Y^{RL*}_{3\ell}}{16\pi^{2}}\Bigg{[}\frac{x_{t,\ell}\left(x_{t,\ell}-1-\log
x_{t,\ell}\right)}{2\left(x_{t,\ell}-1\right)^{2}}+\frac{x_{Z,\ell}}{12}\mathcal{G}_{L}(x_{t,\ell},g^{\text{tree}}_{\ell_{L}})\Bigg{]}\\\
&\qquad+x_{Z,\ell}N_{c}\sum_{k=1,2}\frac{Y^{RL}_{k\ell}Y^{RL*}_{k\ell}}{48\pi^{2}}\Bigg{[}\frac{2s_{w}^{2}}{3}\
\left(\log
x_{Z,\ell}-i\pi-\frac{1}{6}\right)+\frac{g^{\text{tree}}_{\ell_{L}}}{6}\Bigg{]}\,,\\\
g^{\text{LQ}}_{\ell_{R}}=&\
N_{c}\frac{\hat{Y}^{LR}_{3\ell}\hat{Y}^{LR*}_{3\ell}}{16\pi^{2}}\Bigg{[}-\frac{x_{t,\ell}\left(x_{t,\ell}-1-\log
x_{t,\ell}\right)}{2\left(x_{t,\ell}-1\right)^{2}}+\frac{x_{Z,\ell}}{12}\mathcal{G}_{R}(x_{t,\ell},g^{\text{tree}}_{\ell_{R}})\Bigg{]}\\\
&\quad+x_{Z,\ell}N_{c}\sum_{k=1,2}\frac{\hat{Y}^{LR}_{k\ell}\hat{Y}^{LR*}_{k\ell}}{48\pi^{2}}\Bigg{[}\left(-\frac{1}{2}+\frac{2s_{w}^{2}}{3}\right)\left(\log
x_{Z,\ell}-i\pi-\frac{1}{6}\right)+\frac{g^{\text{tree}}_{\ell_{R}}}{6}\Bigg{]}\\\
&\quad+x_{Z,\ell}N_{c}\sum_{k=1,2,3}\frac{Y^{RL}_{k\ell}Y^{RL*}_{k\ell}}{48\pi^{2}}\Bigg{[}\left(\frac{1}{2}-\frac{s_{w}^{2}}{3}\right)\left(\log
x_{Z,\ell}-i\pi-\frac{1}{6}\right)+\frac{g^{\text{tree}}_{\ell_{R}}}{6}\Bigg{]}\,,\end{split}$
(61)

where $x_{t,\ell}\equiv m_{t}^{2}/M_{\ell}^{2}$, $x_{Z,\ell}\equiv
m_{Z}^{2}/M_{\ell}^{2}$ and the tree-level SM couplings are
$g^{\text{tree}}_{\ell_{L}}=-\frac{1}{2}+s_{w}^{2}$ and
$g^{\text{tree}}_{\ell_{R}}=s_{w}^{2}$. The functions $\mathcal{G}_{L}$ and
$\mathcal{G}_{R}$ contain the $\mathcal{O}\left(x_{Z}\log x_{t}\right)$ terms
that induce non-negligible corrections when the LQ mass is small. They read

$\begin{split}\mathcal{G}_{L(R)}(x,g_{\ell})=&\
g^{\text{tree}}_{u_{R(L)}}\frac{\left(x-1\right)\left(5x^{2}-7x+8\right)-2\left(x^{3}+2\right)\log
x}{\left(x-1\right)^{4}}\\\ &\
+g^{\text{tree}}_{u_{L(R)}}\frac{\left(x-1\right)\left(x^{2}-5x-2\right)+6x\log
x}{\left(x-1\right)^{4}}\\\ &\
+g_{\ell}\frac{\left(x-1\right)\left(-11x^{2}+7x-2\right)+6x^{3}\log
x}{3\left(x-1\right)^{4}}\,,\end{split}$ (62)

where $g^{\text{tree}}_{u_{L}}=\frac{1}{2}-\frac{2}{3}s_{w}^{2}$ and
$g^{\text{tree}}_{u_{R}}=-\frac{2}{3}s_{w}^{2}$. The neutrino coupling
$g^{\text{LQ}}_{\nu_{\ell,L}}$ can be derived from $g^{\text{LQ}}_{\ell_{L}}$
by replacing the tree-level coupling $g^{\text{tree}}_{\ell_{L}}$ by
$g^{\text{tree}}_{\nu_{\ell,L}}=\frac{1}{2}$.

##  Appendix B Details of the LHC analyses

###  B.1 CMS non-resonant di-lepton production at the LHC

We determined the cross sections in Eq. (54) for each value of the couplings
$Y^{LR}_{i\ell},Y^{RL}_{i\ell}\in\\{0.25,0.5,1.0,2.0\\}$ individually, setting
the other couplings to zero. We implemented a selection on the transverse
momentum and pseudo-rapidity of the electrons (muons), namely $p_{T}>35$ GeV
(53 GeV) and $|\eta|<2.5$ (2.4), which allows us to mimic the lepton candidate
definitions of Ref. Sirunyan:2021khd . By fitting a polynomial in the Yukawa
couplings to the resulting cross sections, we determined the contributions
from the SM alone, the LQs alone, and the LQ-SM interference term for LQ-
fermion couplings set to one. This provides enough information to calculate
the different components of the cross section for general Yukawa coupling
values through a rescaling of the results.

Figure 10: The resulting values $R_{\mu\mu/ee}$ in the nine di-lepton mass
bins. We show the distributions for an electroquark extension of the SM with
$Y^{LR}_{1e}=1.0$ and/or $Y^{RL}_{1e}=1.0$.

Based on these cross sections we derive the ratios
$R^{\text{LQ+SM}}_{\mu\mu/ee,n}\big{/}R^{\text{SM}}_{\mu\mu/ee,n}$. The
results for a specific scenario are shown in Figure 10, overlayed with the
data from Ref. Sirunyan:2021khd . This allows us to build the likelihood
function

$-2\log\mathcal{L}=\sum_{n=1}^{9}\frac{\left(\dfrac{R_{\mu\mu/ee,n}^{\text{data}}}{R_{\mu\mu/ee,n}^{\text{MC}}}-\dfrac{R_{\mu\mu/ee,n}^{\text{SM+LQ}}}{R_{\mu\mu/ee,n}^{\text{SM}}}\right)^{2}}{\sigma_{n}^{2}}\,,$
(63)

where $n$ runs over the nine $m_{\ell\ell}$ bins and $\sigma_{n}$ are the
experimental uncertainties reported in Ref. Sirunyan:2021khd .

###  B.2 ATLAS non-resonant di-lepton production at the LHC

In our analysis, we estimated the individual cross sections in the SRs chosen
by the ATLAS collaboration using MadGraph_aMC@NLO, analogously to the setup
described in Section 3.9. We implemented lepton $p_{T}$ cuts of 30 GeV (30
GeV) and $|\eta|$ cuts of 2.47 (2.5) on electron (muon) candidates. Based on
the resulting cross sections for Yukawa coupling values of one, we extracted
the ratios

$\mu_{\ell}\equiv\dfrac{\int_{\text{SR}}\frac{d\sigma^{\text{LQ+SM}}}{dm_{\ell\ell}}\left(pp\to\ell^{+}\ell^{-}\right)dm_{\ell\ell}}{\int_{\text{SR}}\frac{d\sigma^{\text{SM}}}{dm_{\ell\ell}}\left(pp\to\ell^{+}\ell^{-}\right)dm_{\ell\ell}}$
(64)

with $\ell=e,\mu$ for general Yukawa coupling matrices $Y^{LR}$ and $Y^{RL}$.
We then followed the statistical analysis achieved by the ATLAS collaboration,
and built a likelihood function using a single-bin Poissonian counting-
experiment approach. In the latter, the uncertainties are accounted for as
Gaussian constraints that we profile over ATLAS:2020yat ; Junk:1999kv .

###  B.3 Non-resonant di-tau production at the LHC

Figure 11: Signal ratios $\mu_{\tau}$ for the six $m_{T}^{\text{tot}}$ bins
studied in the considered ATLAS analysis (for $m_{T}$ ranging from 600 to 1500
GeV). We show the contributions of a tauquark $\Phi_{2,\tau}$ with a mass
$M_{\tau}$ = 1.7 TeV and Yukawa couplings $Y^{LR}_{2\tau}=1.5$ and
$Y^{RL}_{2\tau}=1.5$. We distinguish between the contributions from
$Y^{LR}_{2\tau}$ and those from $Y^{RL}_{2\tau}$, taking the two couplings
individually, and include as well results for a setup with both of them
included.

We focus on the production of a pair of tau leptons at the LHC when the di-tau
system has a large invariant mass. We consider di-tau events populating the
signal region of the ATLAS analysis of Ref. ATLAS:2020yat , and focus on the
six highest bins in the total transverse mass $m_{T}^{\text{tot}}$ bins
defined by

$\displaystyle
m_{T}^{\text{tot}}\equiv\sqrt{\left(p_{T}^{\tau_{1}}+p_{T}^{\tau_{2}}+E_{T}^{\text{miss}}\right)^{2}-\left(\mathbf{p}_{T}^{\tau_{1}}+\mathbf{p}_{T}^{\tau_{1}}+\mathbf{E}_{T}^{\text{miss}}\right)^{2}}\,.$
(65)

In this expression, $\mathbf{p}_{T}^{\tau_{1}}$ and
$\mathbf{p}_{T}^{\tau_{2}}$ are the transverse momenta of the visible daughter
particles originating from the two hadronically-decaying taus, and
$\mathbf{E}_{T}^{\text{miss}}$ is the total missing transverse momentum vector
stemming from the unresolved particles including the daughter neutrinos. In
our notation, $p_{T}^{\tau_{1}}$, $p_{T}^{\tau_{2}}$ and $E_{T}^{\text{miss}}$
are the respective moduli of the different two-vectors. The corresponding
measurements are shown in Figure 11 for $m_{T}^{\rm tot}$ bins defined by
endpoints in $\\{600,700,800,900,1000,1150,1500\\}$ GeV.

In our analysis, we generated partonic $pp\to\tau^{+}\tau^{-}$ events using
MadGraph_aMC@NLO, analogously to what has been done in Section 3.9 but with
loose cuts on the tau leptons ($p_{T}>65$ GeV, $\left|\eta\right|<2.5$) and on
the invariant mass of the di-tau system $m_{\tau\tau}>200$ GeV. We then
showered these events with Pythia version 8.306 Sjostrand:2014zea  and
analysed the resulting hadronic events using MadAnalysis 5 Conte:2012fm ;
Conte:2014zja ; Conte:2018vmg  and a FastJet-based detector simulation
Cacciari:2011ma ; Araz:2020lnp . At the reconstructed level, we imposed
$p_{T}>160$ GeV and $|\eta|<2.5$ for each hadronic tau and finally carried out
the binning in $m_{T}^{\text{tot}}$.

This yielded

$\displaystyle\mu_{\tau,n}\equiv\dfrac{\int_{\text{bin
n}}\frac{d\sigma^{\text{LQ+SM}}}{dm^{\text{tot}}_{T}}\left(pp\to\tau_{\text{had}}^{+}\tau_{\text{had}}^{-}\right)dm^{\text{tot}}_{T}}{\int_{\text{bin
n}}\frac{d\sigma^{\text{SM}}}{dm^{\text{tot}}_{T}}\left(pp\to\tau_{\text{had}}^{+}\tau_{\text{had}}^{-}\right)dm^{\text{tot}}_{T}}\times
R_{n}\,,$ (66)

for the $n=1,\dots,6$ bins indicated above, and for general Yukawa coupling
matrices $Y^{LR}$ and $Y^{RL}$. The factor $R_{n}\in[0,1]$ accounts for the
additional SM backgrounds originating from multijet production and the
charged-current process $pp\to W\to\tau\nu$ that are small but not negligible
compared to the background from the Drell-Yan process
$pp\to(Z/\gamma)^{\ast}\to\tau^{+}\tau^{-}$ ATLAS:2020zms . The resulting
signal ratios for a specific scenario are given in Figure 11. Finally, we
again built a likelihood using a multi-bin Poissonian counting-experiment
approach with Gaussian constraints for the uncertainties.

##  Appendix C Electric dipole moments of hadrons

| Coupling
---|---
| $X_{12}$ | $X_{22}$ | $X_{13}$ | $X_{23}$
Fundamental limits | $<9\times 10^{-4}$ | – | $<7\times 10^{-5}$ | –
Measurement | $d_{n}$ | – | $d_{n}$ | –
Strict limits | $4\times 10^{-4}$ | 0.5 (3) | $3\times 10^{-5}$ | $3\times 10^{-2}(0.2)$
Measurement | $d_{\text{Hg}}$ | $d_{\text{Hg}}$ | $d_{\text{Hg}}$ | $d_{\text{Hg}}$
Table 2: Limits on the LQ couplings to fermions coming from EDM measurements
in hadrons. We only show the constraints that are more stringent than the ones
originating from lepton EDM measurements.

The experiments targeting the observation of EDMs in neutrons
Pendlebury:2015lrz ; Baker:2006ts ; nEDM:2020crw  and Hg atoms Griffith:2009zz
; Graner:2016ses  have so far yielded null results. For our model, the most
relevant resulting upper limits (at 90% C.L.) are given by

$\displaystyle d_{n}$ $\displaystyle<1.8\times 10^{-26}\text{e
cm}\qquad\text{and}\qquad d_{\text{Hg}}$ $\displaystyle<6.3\times
10^{-30}\text{e cm}\,.$ (67)

Ref. Dekens:2018bci  presented a detailed study of the LQ effects in the EDMs
of hadrons. In Table 2 we list the subset of their limits that are more
stringent than the ones coming from the lepton EDMs, extracting the bounds on
the quantity

$X_{i\ell}\equiv\text{Im}\left[\frac{\hat{Y}^{LR}_{i\ell}Y^{RL*}_{i\ell}}{M_{\ell}^{2}}\right]\,,$
(68)

for $i,\ell=1,2,3$. The authors of Ref. Dekens:2018bci  presented two sets of
limits. The first one is based on a “pessimistic” approach (fundamental
limits), where all matrix elements are varied within their admissible
theoretical ranges without assuming any probability distribution. This method
is called the range-fit method that is suitable to rule out models. The second
set of limits is based on a more “optimistic” approach (strict limits), where
the theoretical uncertainties are neglected and the central values of the
hadronic, nuclear, and atomic matrix elements are assumed to be true. The real
constraints are likely to lie between these two extremes. The table
additionally includes, in parentheses, the limits that can be obtained after
neglecting the contributions from charm tensor charge.

Whereas the strictest limit for $X_{23}$ in Table 2 would rule out the
$\Phi_{2,\tau}$ solution to $R_{D^{(*)}}$, the authors of Ref. Dekens:2018bci
noted that due to the large theoretical uncertainties inherent to the
derivation of this limit it is too early to draw strong conclusions regarding
the viability of the $\Phi_{2,\tau}$ solution. It will however be interesting
to see whether next-generation $d_{n}$ or $d_{\text{Hg}}$ experiments will be
able to detect EDM signals.

###### Acknowledgements.

The work of A.C. is supported by a Professorship Grant (PP00P2_176884) of the
National Science Foundation. We thank Yuta Takahashi and Arne Christoph
Reimers for useful discussions regarding the PP limits, Ulrich Haisch for
pointing out the EDM constraints and his help regarding the MadGraph_aMC@NLO
simulations as well as Diego Guadagnoli, Alexandre Carvunis and Claudio Andrea
Manzari for sharing the flavio implementations of $\Delta A_{\rm FB}$ and
$R_{K_{s}^{0}}$ with us.

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