††institutetext: Tata Institute of Fundamental Research, Homi Bhabha Road,
Colaba, Mumbai 400005, India

# SMEFT predictions for semileptonic processes

Siddhartha Karmakar ID , Amol Dighe ID and Rick S. Gupta ID
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The $SU(2)_{L}\times U(1)_{Y}$ invariance of the Standard Model Effective
Field Theory (SMEFT) predicts multiple restrictions in the space of Wilson
coefficients of $U(1)_{em}$ invariant effective lagrangians such as the Low-
energy Effective Field Theory (LEFT), used for low-energy flavor-physics
observables, or the Higgs Effective Field Theory (HEFT) in unitary gauge,
appropriate for weak-scale observables. In this work, we derive and list all
such predictions for semileptonic operators up to dimension 6. We find that
these predictions can be expressed as 2223 linear relations among the
HEFT/LEFT Wilson coefficients, that are completely independent of any
assumptions about the alignment of the mass and flavor bases. These relations
connect diverse experimental searches such as rare meson decays, high-$p_{T}$
dilepton searches, top decays, $Z$-pole observables, charged lepton flavor
violating observables and non-standard neutrino interaction searches. We
demonstrate how these relations can be used to derive strong indirect
constraints on multiple Wilson coefficients that are currently either weakly
constrained from direct experiments or have no direct bound at all. These
relations also imply, in general, that evidence for new physics in a
particular search channel must be accompanied by correlated anomalies in other
channels.

###### Keywords:

Flavor Physics, SMEFT, HEFT, LEFT, Semi-Leptonic Decays

††preprint: TIFR/TH/24-3

##  1 Introduction

The Standard Model Effective Field Theory (SMEFT) Buchmuller:1985jz ;
Grzadkowski:2010es ; Jenkins:2013zja ; Isidori:2023pyp  is a model-independent
way to incorporate the effects of beyond Standard Model (BSM) physics at low
energies. It modifies the Standard Model SM lagrangian by the addition of all
possible higher dimensional operators respecting the SM symmetries:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{SM}+\frac{1}{\Lambda^{2}}\sum_{i}{{\mathcal{C}}}_{i}^{(6)}{\cal
O}_{i}^{(6)}+\cdots,$ (1)

where $\Lambda$ is the cut-off scale, typically of the order of TeV or higher.
Here, ${\mathcal{O}}_{i}^{(d)}$ represent the $d$-dimensional BSM operators
and ${\mathcal{C}}_{i}^{(d)}$ represent the corresponding Wilson coefficients
(WCs). We assume here that the new physics preserves baryon and lepton numbers
and therefore do not include dimension-5 operators. The ellipsis represents
higher order operators with dimension $>6$.

SMEFT is manifestly invariant under $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$, the SM gauge symmetry. As a consequence, there are specific
relationships among different flavor observables. For instance, the SMEFT
requirement that the up-type and down-type left-handed fermionic fields should
arise from $SU(2)_{L}$ doublets implies relations among flavor observables
probing the up sector and those probing the down sector. In this work, we
initiate a systematic derivation of such relations, beginning with the
semileptonic processes in this article.

In flavor physics, effective field theories (EFT) have long served as a
standard framework to parameterize the effects of heavy new physics. However,
for most flavor physics processes, the experimental energy scale is at or
below the mass of the $b$ quark; this includes weak decays of mesons, neutral
meson mixing, $\tau$ decays, etc. The relevant EFT at these energies is the so
called Low-energy Effective Field Theory (LEFT)111LEFT is sometimes referred
to as weak effective field theory (WET or WEFT) in literature Jenkins:2017jig
; Aebischer:2017gaw ; Aebischer:2017ugx ; London:2021lfn . Buchalla:1995vs ,
which assumes only the $SU(3)_{C}\times U(1)_{em}$ invariance and not the full
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ invariance of SM.

The flavor structure of new physics (NP) can also be probed at higher scales,
for instance, in flavor-violating decays of the $Z,W^{\pm}$, and the Higgs
boson $h$, via flavor-violating production or decay of the top quark $t$, or
by constraining the Drell-Yan processes initiating from a flavor off-diagonal
diquark state. In order to include both high-energy and low-energy
observables, one of course needs to write all possible $SU(3)_{C}\times
U(1)_{em}$ invariant operators, as in LEFT, but terms involving the top quark,
Higgs boson and electroweak bosons also need to be included. An appropriate
framework that can encompass both, low-energy flavor observables as well as
this second class of processes involving heavier SM states, is the so-called
Higgs Effective Field Theory (HEFT) Alonso:2012px ; Buchalla:2013rka ;
Pich:2016lew . This is a more general framework than SMEFT and also includes
scenarios where the EW symmetry is realized non-linearly. In the unitary
gauge, it leads to a lagrangian involving all possible $SU(3)_{C}\times
U(1)_{em}$ invariant operators. Given the HEFT lagrangian, it is possible to
derive the corresponding LEFT lagrangian by simply integrating out the heavier
SM states $W,\,Z,\,h$ and $t$.

Figure 1:  Schematic representation of EFTs above and below the electroweak
scale. UV4f represents the subset of SMEFT where the BSM physics only has
four-fermion operators.

For a given set of processes, a general parametrization of possible BSM
deviations assuming only $SU(3)_{C}\times U(1)_{em}$ invariance gives rise to
many more free parameters up to a given order than the number of SMEFT WCs to
that order. This is simply because the former does not assume the full
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ invariance of SM. This situation
has been schematically presented in Fig. 1 where SMEFT can be seen to be a
subset of the more general HEFT. Within this region satisfying SMEFT
assumptions, the smaller number of free parameters implies relationships among
the WCs of HEFT. These relationships can be thought of as predictions of SMEFT
at a certain order; these predictions can be broken only by violating the
basic underlying assumptions of SMEFT.

An apparent obstacle in deriving these relations is that, while SMEFT is
written in the flavor basis, HEFT or LEFT operators have to be written in the
mass basis if we wish to connect them to physical observables. The equations
connecting HEFT Wilson coefficients in the mass basis to SMEFT Wilson
coefficients in the flavor basis, thus, contain elements of the rotation
matrices of the left-handed and right-handed up-type and down-type fermions,
which cannot be fixed by experiments. We show, however, that only the
measurable elements of the Cabbibo-Kobayashi-Maskawa (CKM) quark-mixing matrix
and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) lepton-mixing matrix appear in
the final relations among HEFT WCs. This allows us to derive the implications
of SMEFT on flavor physics observables in a way that is completely independent
of assumptions about the alignment of the flavor and the mass bases, often
referred to as UV flavor assumptions.

In this work, we consider the 3240 semileptonic four-fermion operators in HEFT
that get contributions from the 1053 SMEFT operators, giving rise to 2187
constraints. In addition, we consider 144 HEFT operators that can contribute
to low-energy flavor observables via the exchange of $Z,W^{\pm}$ and $h$
bosons. In SMEFT, these arise from 108 independent operators, thus implying 36
constraints in the HEFT space. We derive all these 2223 constraints and
express them as analytic relations independent of any UV flavor assumptions.

Some other recent studies have also considered the implications of the
$SU(2)_{L}\times U(1)_{Y}$ invariance of SMEFT on flavor observables
Alonso:2014csa ; Cata:2015lta ; Fuentes-Martin:2020lea ; Bause:2020auq ;
Bause:2020xzj ; Bissmann:2020mfi ; Bause:2021cna ; Bause:2021ihn ;
Bause:2022rrs ; Sun:2023cuf ; Grunwald:2023nli ; Greljo:2023bab ;
Fajfer:2012vx ; Bause:2023mfe ; Chen:2024jlj . To the best of our knowledge,
however, the present work is the first study to comprehensively derive and
list all the 2223 analytic relations relevant for semileptonic processes (see,
however, Ref. Bause:2020auq ; Bause:2021cna ; Bause:2021ihn  where a subset of
the above relations has been presented.) Our approach also makes it clear that
these implications can be obtained and presented in a way that is free from
all UV flavor assumptions. A similar approach has been used to derive SMEFT
predictions in Higgs physics in Ref. gupta ; LHCHiggs .

The SMEFT predictions derived in this work are expressed as linear
relationships among $SU(3)_{C}\times U(1)_{em}$ invariant BSM couplings in the
mass basis. These relationships can be directly translated to exact relations
among experimental observables. As we will see, these relations connect
diverse experimental searches: low-energy flavor observables in different
sectors such as kaon, B-meson, charm and $\tau$-decays, the Drell-Yan process
at high-$p_{T}$, top production and decay channels, $Z$-decays, and searches
for non-standard neutrino interaction. These relationships thus allow us to
utilize experimental limits on a set of well-constrained observables to put
bounds on other, otherwise poorly constrained, observables. Our work
demonstrates that indirect constraints on many WCs — such as those related to
$d\bar{d}\to\nu\bar{\nu}$, $u_{i}\to u_{j}\nu\bar{\nu}$ and top decays —
obtained in the above manner, would surpass direct bounds.

Another crucial implication of these relations among WCs is that, in general
they disallow an isolated non-vanishing WC. This is because a nonzero WC will,
via the linear relations, imply a nonzero value for multiple other WCs. This
indicates that deviations from SM would typically not appear in isolated
channels. For instance, it is known that the observed excess in $B\to K\nu\nu$
branching fraction can be explained by a nonzero WC for the operator involving
the transition $b\to s\nu\nu$. We show that this would imply non-vanishing
values for WCs involving processes such as $b\to c\ell\nu_{\ell}$, $b\to
u\ell\nu_{\ell}$ , $t\to c\mu e$, $t\to u\mu e$, etc.

While the SMEFT predictions we derive are completely independent of UV flavor
assumptions, we find that as far as phenomenological implications are
concerned, the sharpest conclusions can be drawn in an important class of
models where the dominant new physics effects come from four-fermion operators
and not from modifications of $Z,W^{\pm}$ and $h$ couplings. We call these
models ‘UV4f’ models and represent them by the dashed rectangle in Fig. 1.
This is a highly motivated class of UV completions that encompasses a majority
of the models proposed to explain the flavor anomalies. These include minimal
leptoquark models Hiller:2014yaa ; Gripaios:2014tna ;
deMedeirosVarzielas:2015yxm ; Sahoo:2015wya  and many $Z^{\prime}$ models
Altmannshofer:2014cfa ; Bonilla:2017lsq ; Bian:2017rpg ; Alonso:2017uky ;
Cline:2017ihf  proposed in the literature.

The plan of this paper is as follows. In Sec. 2, we present the list of
relevant operators in SMEFT, HEFT and LEFT and provide the relations among the
WCs. We discuss the phenomenological applications of these relations in Sec.
3, where we derive the indirect bounds on WCs associated with left-handed
quarks and leptons. In Sec. 4, we discuss possible directions of NP searches
suggested by the relations among the WCs, given some of the current observed
deviations from SM. We present concluding remarks in Sec. 5. In Appendix A, we
write the HEFT operators used in the text in the $SU(2)_{L}\times U(1)_{Y}$
invariant form, with the electroweak symmetry non-linearly realized, and
compare our list with the previous literature. In Appendix B, we briefly
discuss some details of the SMEFT basis used and the rationale for our choice.
In Appendix C, we present all the analytic relations in terms of semileptonic
LEFT WCs and WCs that modify the $Z$, $W^{\pm}$ and Higgs couplings to
fermions. In Appendix D, we present tables of $90\%$ C.L limits on the LEFT
WCs.

##  2 SMEFT predictions for semileptonic operators

In this section, we present all possible semileptonic operators respecting the
$U(1)_{em}$ symmetry (as the $SU(3)_{C}$ symmetry is always respected, we will
not mention it separately from here on), and derive the analytic relations
among them that are predicted by SMEFT. We consider the following lagrangian
terms at the weak scale:

$\displaystyle\mathcal{L}_{\rm HEFT}$ $\displaystyle\supset\mathcal{L}^{\rm
SM}+\sum_{f,i,j}\left[{{\mathbf{c}}}_{fZ}\right]^{ij}\,(\bar{f}_{i}\gamma^{\mu}\,f_{j})\,Z_{\mu}+\sum_{f_{u},f_{d},i,j}\left[{{\mathbf{c}}}_{f_{u}f_{d}W}\right]^{ij}\,(\bar{f}_{u_{i}}\gamma^{\mu}\,f_{d_{j}})\,W^{+}_{\mu}$
$\displaystyle~{}+\sum\left[{{\mathbf{c}}}_{fh}\right]^{ij}\,(\bar{f}_{i}\,P_{R}\,f_{j})\,h+\frac{1}{\Lambda^{2}}\,\sum_{i}{{\mathbf{c}}}_{i}\,{{\mathbf{o}}}^{4f}_{sl,i}+h.c.,$
(2)

where, in addition to the SM lagrangian $\mathcal{L}^{SM}$ and the term
containing all possible semileptonic four-fermion operators
${{\mathbf{o}}}^{4f}_{sl}$, we also include corrections to the couplings of
$Z$, $W^{\pm}$ and Higgs boson $h$ to fermions.222 We have not considered
four-quark operators and electroweak dipole operators. Although these can
contribute to semileptonic processes, they do not get matched to semileptonic
operators at the tree level. Furthermore, these operators are constrained from
processes which are not semileptonic. The four-quark operators can get
constraints from nonleptonic decays, whereas the dipole operators are bounded
by measurements such as the precise observations of dipole moments of
elementary particles, the $b\to s\gamma$ process etc. This is because the
diagrams with Higgs, $W^{\pm}$ and $Z$ exchange can generate four-fermion
effective operators at the low energies relevant to semileptonic flavor
observables. Here $f\in\\{u_{L},u_{R},d_{L},d_{R},e_{L},e_{R},\nu_{L}\\}$,
$f_{u}$ denotes neutrinos and up-type quarks (both left-handed and right-
handed) whereas $f_{d}$ denotes down-type quarks and charged leptons. A
lagrangian containing all these operators with independent coefficients is
equivalent to the HEFT lagrangian, $\mathcal{L}_{\rm HEFT}$, in the unitary
gauge. This is because, although formally $SU(2)_{L}\times U(1)_{Y}$
invariance is non-linearly realized in the HEFT lagrangian, in the unitary
gauge HEFT reduces to a lagrangian with all possible $U(1)_{em}$-invariant
terms. As we show in Appendix A, our list of operators can be rewritten in an
invariant form with non-linearly realized electroweak symmetry as in Ref.
Buchalla:2012qq . The HEFT basis of Appendix A excludes some redundant
operators that appeared in the HEFT bases presented in earlier literature
(e.g. Buchalla:2012qq ; Cata:2015lta ) and also includes some operators that
were missed in previous work. Further note that in the UV4f scenario discussed
in the Sec. 1, the coupling modifications of $Z,W^{\pm}$ and $h$ are absent,
i.e. the second, third and fourth terms on the RHS of eq. (2) vanish.

The semileptonic four-fermion operators ${{\mathbf{o}}}^{4f}_{sl}$ can be
directly probed by high-energy processes such as the Drell-Yan process
$\bar{q}_{i}q_{j}\to ll$, top production and decay processes, etc. We consider
these operators in Sec. 2.1 and list the dimension-6 (dim-6) SMEFT operators
that contribute to them. We find that the number of HEFT operators is larger
than the number of dim-6 SMEFT operators, which results in SMEFT predictions
for these HEFT WCs. These predictions are in the form of linear relations
among the HEFT WCs; we explicitly derive these relations in Sec. 2.1.

Next we consider the corrections to the SM couplings of $Z,W^{\pm}$ and $h$ to
fermions, indicated by the second, third and fourth terms in the RHS of eq.
(2). Although our reason for inclusion of these operators is that they
contribute to low-energy semileptonic processes via $Z,W^{\pm}$ and $h$
exchange, these couplings can be probed independently by studying decays of
the $Z,W^{\pm}$ and $h$. We list the SMEFT operators contributing to these in
Sec. 2.2. We find that, while the number of SMEFT operators is the same as the
number of HEFT operators for $h$ coupling corrections, the number of
contributing SMEFT operators in the case of gauge boson coupling corrections
is smaller. This results in relations among the corrections to $Z$ and
$W^{\pm}$ couplings; we derive these in Sec. 2.2.

Finally, in Sec. 2.3 we rewrite the analytic relations derived in Sec. 2.1 and
Sec. 2.2 in terms of WCs at the low scale relevant for most of the important
flavor observables, such as those connected to meson mixing, rare decays, etc.
The lagrangian relevant at these scales is the sum of the LEFT neutral-current
and charged-current lagrangians333Note that, to distinguish different EFTs, we
denote the Wilson coefficients by ‘${{\mathcal{C}}}$’ for SMEFT, by
‘${{\mathbf{c}}}$’ for HEFT and by ‘$C$’ for LEFT. The corresponding operators
are denoted by ‘${\mathcal{O}}$’, ‘${\mathbf{o}}$’ and ’$O$’ respectively.

$\displaystyle\mathcal{L}_{\textrm{LEFT}}^{\rm NC}$
$\displaystyle=\mathcal{L}^{\rm
NC}_{\textrm{SM}}+\frac{4G_{F}}{\sqrt{2}}\sum_{i}^{\textrm{NC
only}}\,{{C}}_{i}\,O_{i}^{\rm NC}~{},$ (3)
$\displaystyle\textrm{and}\quad\mathcal{L}_{\textrm{LEFT}}^{\rm CC}$
$\displaystyle=\mathcal{L}^{\rm
CC}_{\textrm{SM}}+\frac{4G_{F}}{\sqrt{2}}\sum_{i}^{\textrm{CC
only}}\,{{C}}_{i}\,O_{i}^{\rm CC}~{},$ (4)

where the first terms on the RHS arise from the first term in eq. (2) by
integrating out $Z,W^{\pm}$ and $h$, assuming SM couplings.444 A loop factor
of $e^{2}/(16\pi^{2})$ is usually included for the NC Lagrangian for LEFT in
literature Aebischer:2015fzz ; Bause:2020auq . In our convention, we have not
included this factor in order to have uniformity in our analytic relations to
be presented later. Here ‘${\rm NC}$’ and ‘${\rm CC}$’ stand for neutral-
current and charged-current, respectively. In order to obtain the SMEFT
predictions for relations among the LEFT WCs, we need to match the LEFT
coefficients above to the HEFT ones including the effect of $Z$, $W^{\pm}$,
and $h$ exchange diagrams. These matching relations can then be inverted to
write the HEFT WCs and the relations among them in terms of the LEFT ones. We
carry out this procedure in Sec. 2.3.

Vector operators $LLLL$
---
|  NC | Count
$[{{\mathbf{c}}}_{e_{L}d_{L}}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{d}_{L}^{i}\gamma^{\mu}d_{L}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{u}_{L}^{i}\gamma^{\mu}u_{L}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\alpha\beta ij}$ |  $(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{d}_{L}^{i}\gamma^{\mu}d_{L}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{\nu uLL}^{V}]^{\alpha\beta ij}$ |  $(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{u}_{L}^{i}\gamma^{\mu}u_{L}^{j})$ | 81 (45)
|  CC |
$[{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{u}_{L}^{i}\gamma^{\mu}d_{L}^{j})$ | 162 (81)
Vector operators $RRRR$
|  NC | Count
$[{{\mathbf{c}}}_{edRR}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{euRR}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 81 (45)

Vector operators $LLRR$
---
|  NC | Count
$[{{\mathbf{c}}}_{edLR}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{euLR}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}e_{L}^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{\nu dLR}^{V}]^{\alpha\beta ij}$ |  $(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{\nu uLR}^{V}]^{\alpha\beta ij}$ |  $(\bar{\nu}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 81 (45)
|  CC |
$[{{\mathbf{c}}}_{LR}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{L}^{\alpha}\gamma_{\mu}\nu_{L}^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 162 (81)
Vector operators $RRLL$
|  NC | Count
$[{{\mathbf{c}}}_{edRL}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{d}_{L}^{i}\gamma^{\mu}d_{L}^{j})$ | 81 (45)
$[{{\mathbf{c}}}_{euRL}^{V}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{u}_{L}^{i}\gamma^{\mu}u_{L}^{j})$ | 81 (45)

Scalar operators with $d_{R}$
---
|  NC | Count
$[{{\mathbf{c}}}_{ed,RLLR}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,e_{L}^{\beta})(\bar{d}_{L}^{i}\,d_{R}^{j})$ | 162 (81)
$[{{\mathbf{c}}}_{ed,RLRL}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,e_{L}^{\beta})(\bar{d}_{R}^{i}\,d_{L}^{j})$ | 162 (81)
|  CC |
$[{{\mathbf{c}}}_{RLLR}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,\nu_{L}^{\beta})(\bar{u}_{L}^{i}\,d_{R}^{j})$ | 162 (81)
Scalar operators with $u_{R}$
|  NC | Count
$[{{\mathbf{c}}}_{eu,RLLR}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,e_{L}^{\beta})(\bar{u}_{L}^{i}\,u_{R}^{j})$ | 162 (81)
$[{{\mathbf{c}}}_{eu,RLRL}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,e_{L}^{\beta})(\bar{u}_{R}^{i}\,u_{L}^{j})$ | 162 (81)
|  CC |
$[{{\mathbf{c}}}_{RLRL}^{S}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\,\nu_{L}^{\beta})(\bar{u}_{R}^{i}\,d_{L}^{j})$ | 162 (81)

Tensor operators with $d_{R}$
---
|  NC | Count
$[{{\mathbf{c}}}_{ed,RLLR}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}e_{L}^{\beta})(\bar{d}_{L}^{i}\sigma^{\mu\nu}d_{R}^{j})$ | 162 (81)
$[{{\mathbf{c}}}_{ed,RLRL}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}e_{L}^{\beta})(\bar{d}_{R}^{i}\sigma^{\mu\nu}d_{L}^{j})$ | 162 (81)
|  CC |
$[{{\mathbf{c}}}_{RLLR}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}\nu_{L}^{\beta})(\bar{u}_{L}^{i}\sigma^{\mu\nu}d_{R}^{j})$ | 162 (81)
Tensor operators with $u_{R}$
|  NC | Count
$[{{\mathbf{c}}}_{eu,RLLR}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}e_{L}^{\beta})(\bar{u}_{L}^{i}\sigma^{\mu\nu}u_{R}^{j})$ | 162(81)
$[{{\mathbf{c}}}_{eu,RLRL}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}e_{L}^{\beta})(\bar{u}_{R}^{i}\sigma^{\mu\nu}u_{L}^{j})$ | 162(81)
|  CC |
$[{{\mathbf{c}}}_{RLRL}^{T}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\sigma_{\mu\nu}\nu_{L}^{\beta})(\bar{u}_{R}^{i}\sigma^{\mu\nu}d_{L}^{j})$ | 162(81)

Table 1:  Semileptonic operators in HEFT. Here ${{\mathbf{c}}}$’s are the WCs
for the corresponding operators in the flavor basis. The indices
$\alpha,\beta$ denote lepton families and $i,j$ denote quark families. NC and
CC correspond to neutral-current and charged-current operators. Count denotes
the number of independent operators; the number inside the brackets is the
number of independent operators if all WCs were real. Note that for vector CC
operators as well as all the scalar and tensor operators we have not
explicitly listed their hermitian conjugates but included them in the count.

Vector operators $LLLL$
---
|  Operator | Count
$[{{\mathcal{C}}}_{{\ell}q}^{(1)}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}\gamma_{\mu}l^{\beta})(\bar{q}^{i}\gamma^{\mu}q^{j})$ | 81 (45)
$[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}\gamma_{\mu}\tau^{I}l^{\beta})(\bar{q}^{i}\gamma^{\mu}\tau^{I}q^{j})$ | 81 (45)
Vector operators $RRRR$
|  Operator | Count
$[{{\mathcal{C}}}_{ed}]^{\alpha\beta ij}$ |  $(\bar{e}^{\alpha}\gamma_{\mu}e^{\beta})(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 81 (45)
$[{{\mathcal{C}}}_{eu}]^{\alpha\beta ij}$ |  $(\bar{e}^{\alpha}\gamma_{\mu}e^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 81 (45)

Vector operators $LLRR$
---
|  Operator | Count
$[{{\mathcal{C}}}_{{\ell}d}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}\gamma_{\mu}l^{\beta})(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})$ | 81 (45)
$[{{\mathcal{C}}}_{{\ell}u}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}\gamma_{\mu}l^{\beta})(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 81 (45)
Vector operators $RRLL$
|  Operator | Count
$[{{\mathcal{C}}}_{eq}]^{\alpha\beta ij}$ |  $(\bar{e}_{R}^{\alpha}\gamma_{\mu}e_{R}^{\beta})(\bar{q}^{i}\gamma^{\mu}q^{j})$ | 81 (45)
|  |

Scalar operators with $d_{R}$
---
|  Operator | Count
$[{{\mathcal{C}}}_{{\ell}edq}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}_{a}\,e_{R}^{\beta})(\bar{d}_{R}^{i}\,q^{j}_{a})$ | 162 (81)

Scalar operators with $u_{R}$
---
|  Operator | Count
$[{{\mathcal{C}}}_{{\ell}equ}^{(1)}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}_{a}\,e_{R}^{\beta})\epsilon_{ab}(\bar{q}^{i}_{b}\,u_{R}^{j})$ | 162 (81)

Tensor operators
---
|  Operator | Count
$[{{\mathcal{C}}}_{{\ell}equ}^{(3)}]^{\alpha\beta ij}$ |  $(\bar{l}^{\alpha}_{a}\sigma_{\mu\nu}e_{R}^{\beta})\epsilon_{ab}(\bar{q}^{i}_{b}\sigma^{\mu\nu}u_{R}^{j})$ | 162 (81)

Table 2:  Semileptonic operators in SMEFT. Here ${{\mathcal{C}}}$’s are the
WCs for the corresponding operators in the flavor basis. The indices
$\alpha,\beta$ denote lepton families and $i,j$ denote quark families. Here
$l=(\nu_{L},e_{L})^{T}$, $q=(u_{L},d_{L})^{T}$, $\tau^{I}$ are the Pauli
matrices and $\epsilon_{ab}$ is the $(2\times 2)$ anti-symmetric matrix with
$\epsilon_{12}=1$. Count denotes the number of independent operators; the
number inside the brackets is the number of independent operators if all WCs
were real. Note that for the scalar and tensor operators we have not
explicitly listed their hermitian conjugates but included them in the count.

###  2.1 Predictions for semileptonic operators at high energies

We begin our analysis with the 3240 (1674) semileptonic four-fermion
operators555For non-hermitian operators, we consider the operator and its
hermitian conjugate as two distinct operators, as one can treat
$(O+O^{\dagger})$ and $i(O-O^{\dagger})$ as two separate operators with real
Wilson coefficients. present in HEFT (see Table 1), where the number within
the parenthesis denotes the number of independent operators if the WCs of all
these operators were real. Note that each entry in Table 1 represents multiple
operators corresponding to different possible values for the family indices.
The first entry $[{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta ij}$, for instance,
represents 81 (45) operators, since the indices $\alpha$, $\beta$ denote three
lepton families and the indices $i$, $j$ denote three quark families. In Table
2, we list the 1053 (558) semileptonic four-fermion operators in SMEFT which
would give rise to the above HEFT operators.

The operators in Table 1 and 2 are divided into categories based on their
Lorentz structure and the chiralities of the fields involved. In the
following, we discuss the mapping between SMEFT and HEFT operators and the
resulting SMEFT predictions for each of these categories.

#### LLLL vector operators:

In this category, there are 486 (261) independent operators in HEFT, as shown
in Table 1, which correspond to the 162 (90) SMEFT operators shown in Table 2.
The SMEFT operators, when expanded in the unitary gauge, give the following
contributions to the HEFT Wilson coefficients:

$\displaystyle[{{\mathbf{c}}}_{\nu uLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=([{{\mathcal{C}}}_{{\ell}q}^{(1)}]^{\alpha\beta
ij}+[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta
ij})~{},\quad[{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta
ij}=([{{\mathcal{C}}}_{{\ell}q}^{(1)}]^{\alpha\beta
ij}-[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta ij}),$ (5)
$\displaystyle[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=([{{\mathcal{C}}}_{{\ell}q}^{(1)}]^{\alpha\beta
ij}-[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta
ij})~{},\quad[{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta
ij}=([{{\mathcal{C}}}_{{\ell}q}^{(1)}]^{\alpha\beta
ij}+[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta ij})~{},$ (6)
$\displaystyle[{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta ij}$
$\displaystyle=2\,[{{\mathcal{C}}}_{{\ell}q}^{(3)}]^{\alpha\beta ij}~{},$ (7)

where we have written both the SMEFT and HEFT WCs in the flavor basis. One can
easily read off the 324 (171) SMEFT predictions implied by the above
equations:

$\displaystyle[{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\alpha\beta ij}~{},$ (8)
$\displaystyle[{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathbf{c}}}_{\nu uLL}^{V}]^{\alpha\beta ij}~{},$ (9)
$\displaystyle\,[{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta
ij}-[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\alpha\beta ij}~{}.$ (10)

These predictions are in the flavor basis. We would like to have the relations
in terms of HEFT operators in the mass basis for later matching with the LEFT
operators and with the observables. This can be achieved by the use of unitary
matrices $S_{L,R}$ and $K_{L,R}$ for quarks and leptons, respectively. The
fields are transformed as

$\displaystyle u_{L}^{i}$
$\displaystyle\rightarrow(S_{L}^{u})^{ij}u_{L}^{j}~{},\quad
d_{L}^{i}\rightarrow(S_{L}^{d})^{ij}d_{L}^{j}~{},\quad
u_{R}^{i}\rightarrow(S_{R}^{u})^{ij}u_{R}^{j}~{},\quad
d_{R}^{i}\rightarrow(S_{R}^{d})^{ij}d_{L}^{j}~{},$ (11) $\displaystyle
e_{L}^{\alpha}$
$\displaystyle\rightarrow(K_{L}^{e})^{\alpha\beta}e_{L}^{\beta}~{},\quad\nu_{L}^{\alpha}\to(K_{L}^{\nu})^{\alpha\beta}\nu_{L}^{\beta}~{},\quad
e_{R}^{\alpha}\to(K_{R}^{e})^{\alpha\beta}e_{R}^{\beta}~{}.$ (12)

The relation in eq. (8) gets transformed in the mass basis as666 A hat on top
of the HEFT WC indicates that it is in the mass basis, otherwise it is in the
flavor basis.

$\displaystyle(K_{L}^{e})^{\alpha\rho}\,(S_{L}^{u})^{ik}\,[\hat{{\mathbf{c}}}_{euLL}^{V}]^{\rho\sigma
kl}\,(S_{L}^{u\dagger})^{{\ell}j}(K_{L}^{e\dagger})^{\sigma\beta}$
$\displaystyle=(K_{L}^{\nu})^{\alpha\rho}\,(S_{L}^{d})^{ik}\,[\hat{{\mathbf{c}}}_{\nu
dLL}^{V}]^{\rho\sigma
kl}\,(S_{L}^{d\dagger})^{{\ell}j}(K_{L}^{\nu\dagger})^{\sigma\beta}.$ (13)

Suppressing the lepton and quark family indices, the above equation can be
rewritten in a compact form as

$\displaystyle
K_{L}^{e}\,S_{L}^{u}\,\hat{{\mathbf{c}}}_{euLL}^{V}\,S_{L}^{u\dagger}\,K_{L}^{e\dagger}$
$\displaystyle=K_{L}^{\nu}\,S_{L}^{d}\,\hat{{\mathbf{c}}}_{\nu
dLL}^{V}\,S_{L}^{d\dagger}\,K_{L}^{\nu\dagger}~{},$ (14)

where the matrices $S$ and $K$ carry only quark and lepton indices,
respectively. This relation may be further expressed as

$\displaystyle V^{\dagger}\,\hat{{\mathbf{c}}}_{euLL}^{V}\,V$
$\displaystyle=U^{\dagger}\,\hat{{\mathbf{c}}}_{\nu dLL}^{V}\,U~{},$ (15)

using the CKM and PMNS matrices

$\displaystyle V$ $\displaystyle\equiv
V_{CKM}=S_{L}^{u\dagger}\,S_{L}^{d}~{}~{}\textrm{and}\quad U\equiv
U_{PMNS}=K_{L}^{\nu\dagger}\,K_{L}^{e}~{}.$ (16)

Following similar steps, we can rewrite the relations from eqs. (9) and (10)
in the mass basis as

$\displaystyle V\,\hat{{\mathbf{c}}}_{edLL}^{V}\,V^{\dagger}$
$\displaystyle=U^{\dagger}\,\hat{{\mathbf{c}}}_{\nu uLL}^{V}\,U~{},$ (17)
$\displaystyle V^{\dagger}\,\hat{{\mathbf{c}}}_{LL}^{V}\,U$
$\displaystyle=\hat{{\mathbf{c}}}_{edLL}^{V}-U^{\dagger}\,\hat{{\mathbf{c}}}_{\nu
dLL}^{V}\,U~{}.$ (18)

Note that the final SMEFT predictions, i.e. the relations among the HEFT WCs
shown in eqs. (15), (17) and (18), involve only the physically measurable CKM
and PMNS matrices. This makes the relations completely independent of any UV
flavor assumption. The relations in eqs. (15) and (17) were derived previously
in Ref. Bissmann:2020mfi .

#### LLRR vector operators:

Similar to the previous case, in this category there are 486 (261) independent
HEFT operators and 162 (90) independent SMEFT operators, as shown in Table 1
and 2, respectively. The HEFT WCs can be written in terms of the SMEFT ones as
follows:

$\displaystyle[{{\mathbf{c}}}_{\nu uLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\,[{{\mathcal{C}}}_{{\ell}u}]^{\alpha\beta
ij},\quad[{{\mathbf{c}}}_{euLR}^{V}]^{\alpha\beta
ij}=\,[{{\mathcal{C}}}_{{\ell}u}]^{\alpha\beta ij},$ (19)
$\displaystyle[{{\mathbf{c}}}_{\nu dLR}^{V}]^{\alpha\beta ij}$
$\displaystyle=\,[{{\mathcal{C}}}_{{\ell}d}]^{\alpha\beta
ij}~{},\quad[{{\mathbf{c}}}_{edLR}^{V}]^{\alpha\beta
ij}=\,[{{\mathcal{C}}}_{{\ell}d}]^{\alpha\beta
ij}~{},\quad[{{\mathbf{c}}}_{LR}^{V}]^{\alpha\beta ij}=0~{}.$ (20)

Thus, here we get 324 (171) relations among the HEFT coefficients. In the
flavor basis and the mass basis, these relations are

$\displaystyle{{\mathbf{c}}}_{euLR}^{V}={{\mathbf{c}}}_{\nu uLR}^{V}\quad$
$\displaystyle\Rightarrow\quad\hat{{\mathbf{c}}}_{euLR}^{V}=U^{\dagger}\,\hat{{\mathbf{c}}}_{\nu
uLR}^{V}\,U~{},$ (21)
$\displaystyle{{\mathbf{c}}}_{edLR}^{V}={{\mathbf{c}}}_{\nu dLR}^{V}\quad$
$\displaystyle\Rightarrow\quad\hat{{\mathbf{c}}}_{edLR}^{V}=U^{\dagger}\,\hat{{\mathbf{c}}}_{\nu
dLR}^{V}\,U~{},$ (22) $\displaystyle{{\mathbf{c}}}_{LR}^{V}=0\quad$
$\displaystyle\Rightarrow\quad\hat{{\mathbf{c}}}_{LR}^{V}=0~{}.$ (23)

Note that in this category, only right-handed quark fields appear and the
rotation matrices for the right-handed quarks cancel out in the relations when
translated to the mass basis. As a result, there is no CKM matrix in these
relations and only the PMNS matrix $U$ appears for the leptons. The relations
above show that the charged-current HEFT WCs vanish for this category of
operators. This is because in SMEFT, as noted in Burgess:2021ylu ;
Cata:2015lta , right-handed quarks cannot participate in charged-current
semileptonic processes at dimension 6 level due to hypercharge conservation.

Category | Analytic relations | Count
---|---|---
$LLLL$ | $V^{\dagger}_{ik}\,[\hat{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta kl}\,V_{{\ell}j}=U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu dLL}^{V}]^{\rho\sigma ij}\,U_{\sigma\beta}$ | 81 (45)
$V_{ik}\,[\hat{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\beta kl}\,V^{\dagger}_{{\ell}j}=U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu uLL}^{V}]^{\rho\sigma ij}\,U_{\sigma\beta}$ | 81 (45)
$V^{\dagger}_{ik}\,[\hat{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta kj}=[\hat{{\mathbf{c}}}_{edLL}^{V}]^{\alpha\rho ij}\,U^{\dagger}_{\rho\beta}-U^{\dagger}_{\alpha\sigma}\,[{{\mathbf{c}}}_{\nu dLL}^{V}]^{\sigma\beta ij}$ | 162 (81)
$RRRR$ | No relations |
$LLRR$ | $[\hat{{\mathbf{c}}}_{edLR}^{V}]^{\alpha\beta ij}=U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu dLR}^{V}]^{\rho\sigma ij}\,U_{\rho\beta}$ | 81 (45)
$[\hat{{\mathbf{c}}}_{euLR}^{V}]^{\alpha\beta ij}=U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu uLR}^{V}]^{\rho\sigma ij}\,U_{\rho\beta}$ | 81 (45)
$[\hat{{\mathbf{c}}}_{LR}^{V}]^{\alpha\beta ij}=0$ | 162 (81)
$RRLL$ | $[\hat{{\mathbf{c}}}_{edRL}^{V}]^{\alpha\beta ij}=V^{\dagger}_{ik}\,[\hat{{\mathbf{c}}}_{euRL}^{V}]^{\rho\sigma kl}\,V_{lj}$ | 81 (45)
Scalar ($d_{R}$) | $V_{ik}\,[\hat{{\mathbf{c}}}_{ed,RLLR}^{S}]^{\alpha\beta kj}=[\hat{{\mathbf{c}}}_{RLLR}^{S}]^{\alpha\rho ij}\,U_{\rho\beta}$ | 162 (81)
$[\hat{{\mathbf{c}}}_{ed,RLRL}^{S}]^{\alpha\beta ij}=0$ | 162 (81)
Scalar ($u_{R}$) | $[\hat{{\mathbf{c}}}_{eu,RLRL}^{S}]^{\alpha\beta ik}\,V_{kj}=-[\hat{{\mathbf{c}}}_{RLRL}^{S}]^{\alpha\rho ij}\,U_{\rho\beta}$ | 162 (81)
$[\hat{{\mathbf{c}}}_{eu,RLLR}^{S}]^{\alpha\beta ij}=0$ | 162 (81)
Tensor ($d_{R}$) | $[\hat{{\mathbf{c}}}_{ed,\,\textrm{all}}^{T}]^{\alpha\beta ij}=0$ | 324 (162)
$[\hat{{\mathbf{c}}}_{RLLR}^{T}]^{\alpha\beta ij}=0$ | 162 (81)
Tensor ($u_{R}$) | $[\hat{{\mathbf{c}}}_{eu,RLRL}^{T}]^{\alpha\beta ik}\,V_{kj}=-[\hat{{\mathbf{c}}}_{RLRL}^{T}]^{\alpha\rho ij}\,U_{\rho\beta}$ | 162 (81)
$[\hat{{\mathbf{c}}}_{eu,RLLR}^{T}]^{\alpha\beta ij}=0$ | 162 (81)
$Z$ and $W^{\pm}$ | $[\hat{{\mathbf{c}}}_{ud_{L}W}]^{ij}=\frac{1}{\sqrt{2}}\cos\theta_{w}\,([\hat{{\mathbf{c}}}_{u_{L}Z}]^{ik}\,V_{kj}-V_{ik}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{kj})$ | 18 (9)
$[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\rho}\,U_{\rho\beta}=\frac{1}{\sqrt{2}}\cos\theta_{w}\,([\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}-U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,U_{\sigma\beta})$ | 18 (9)

Table 3:  Linear relations among the HEFT semileptonic WCs in the mass basis
predicted by the SMEFT. Summation over repeated indices is implicit. Count
denotes the number of independent operators; the number inside the brackets is
the number of independent operators if all WCs are real.

#### RRRR vector operators:

Right-handed fermions are not charged under $SU(2)_{L}$. Thus even in SMEFT,
the up-type and down-type right-handed fields can appear independently in
neutral-current semileptonic operators, as in HEFT. This makes the number of
neutral-current operators of RRRR type in HEFT and SMEFT to be the same as
shown in Tables 1 and 2, respectively. Furthermore, in the absence of right-
handed neutrinos, there are no charged-current operators either in HEFT or in
SMEFT in this category. As a result, in this category, there are no relations
among the HEFT coefficients.

#### RRLL vector operators:

In the case of vector operators involving right-handed leptons and left-handed
quarks, there are 162 (90) independent operators in HEFT and 81 (45) in SMEFT,
respectively. This results in 81 (45) relations among the HEFT WCs. The
mapping between HEFT and SMEFT WCs in the flavor basis and the resulting
relations in the mass basis for this category are

$\displaystyle{{\mathbf{c}}}_{ed}^{V}=\,{{\mathcal{C}}}_{eq}~{},$
$\displaystyle\quad{{\mathbf{c}}}_{eu}^{V}=\,{{\mathcal{C}}}_{eq}~{}\quad\Rightarrow\quad\hat{{\mathbf{c}}}_{edRL}^{V}=V^{\dagger}\,\hat{{\mathbf{c}}}_{euRL}^{V}\,V~{}.$
(24)

Note that the PMNS matrix does not appear in these relations as only right-
handed electrons are involved and the corresponding flavor rotations cancel
out. Furthermore, there are no charged-current operators in this category as
there is no right-handed neutrino in SM.

#### Scalar operators:

There are 486 (243) scalar semileptonic operators with right-handed down-type
quarks and 486 (243) operators with right-handed up-type quarks in HEFT. In
SMEFT, there are 162 (90) operators for each of these scenarios. We find 324
(153) relations among the HEFT coefficients for each scenario. Mapping of
these operators between HEFT and SMEFT in the flavor basis gives

$\displaystyle[{{\mathbf{c}}}_{ed,RLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathcal{C}}}_{{\ell}edq}]^{\beta\alpha ji*}~{},$
$\displaystyle[{{\mathbf{c}}}_{eu,RLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=0~{},$ (25)
$\displaystyle[{{\mathbf{c}}}_{ed,RLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=0~{},$
$\displaystyle[{{\mathbf{c}}}_{eu,RLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=-[{{\mathcal{C}}}_{{\ell}equ}]^{\beta\alpha ji*}~{},$ (26)
$\displaystyle[{{\mathbf{c}}}_{RLLR}^{S}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathcal{C}}}_{{\ell}edq}]^{\beta\alpha ji\,*}~{},$
$\displaystyle[{{\mathbf{c}}}_{RLRL}^{S}]^{\alpha\beta ij}$
$\displaystyle=[{{\mathcal{C}}}_{{\ell}equ}]^{\beta\alpha ji\,*}~{}.$ (27)

From the above equations, we get the following relations among the HEFT WCs in
the mass basis:

$\displaystyle V\,\hat{{\mathbf{c}}}_{ed,RLLR}^{S}$
$\displaystyle=\hat{{\mathbf{c}}}_{RLLR}^{S}\,U~{},$
$\displaystyle\hat{{\mathbf{c}}}_{eu,RLRL}^{S}\,V$
$\displaystyle=-\hat{{\mathbf{c}}}_{RLRL}^{S}\,U~{},$ (28)
$\displaystyle\hat{{\mathbf{c}}}_{ed,RLRL}^{S}$ $\displaystyle=0~{},$
$\displaystyle\hat{{\mathbf{c}}}_{eu,RLLR}^{S}$ $\displaystyle=0~{}.$ (29)

Note that both the above relations in eq. (2.27) represent relations among
neutral-current scalar operators (on the LHS) and charged-current scalar
operators (on the RHS). The WCs in eq. (2.28) vanish777 Note that the
vanishing of these HEFT WCs correspond to the relations $C_{S}=-C_{P}$ and
$C_{S}^{\prime}=C_{P}^{\prime}$ in the conventional LEFT for the UV4f models
as noted in Alonso:2014csa ; Cata:2015lta . since the corresponding SMEFT
operators would not satisfy $U(1)_{Y}$.

#### Tensor operators:

There is no tensor operator with right-handed down-type quarks in SMEFT as
these operators cannot conserve $U(1)_{Y}$ hypercharge. Thus, all the tensor
operators with right-handed down-type quarks in HEFT get zero contribution
from SMEFT. As a result, SMEFT predicts 486 (243) constraints on such HEFT
WCs:

$\displaystyle\hat{{\mathbf{c}}}_{ed,RLLR}^{T}$
$\displaystyle=0~{},\quad\hat{{\mathbf{c}}}_{ed,RLRL}^{T}=0~{},\quad\hat{{\mathbf{c}}}_{RLLR}^{T}=0~{}.$
(30)

For the case of tensor operators with right-handed up-type quarks, the mapping
and relations are exactly the same as the scalar operators:

$\displaystyle\hat{{\mathbf{c}}}_{eu,RLRL}^{T}\,V$
$\displaystyle=-\hat{{\mathbf{c}}}_{RLRL}^{T}\,U~{},\quad\hat{{\mathbf{c}}}_{eu,RLLR}^{T}=0~{}.$
(31)

The reason for the vanishing of the WCs in the last equation is again that the
corresponding SMEFT operators would not preserve the $U(1)_{Y}$ hypercharge
symmetry. See also references Alonso:2014csa ; Cata:2015lta .

In Table 3, we present all the relations among the HEFT WCs corresponding to
four-fermion semileptonic operators, which would be predicted by SMEFT. We
express these relations in the mass basis and explicitly put the indices for
the quark and the lepton families.

###  2.2 Predictions for the couplings of $Z$, $W^{\pm}$ and $h$ to fermions

HEFT
---
$LL$ quarks
| Operator | Count
$[{{\mathbf{c}}}_{u_{L}Z}]^{ij}$ | $(\bar{u}_{L}^{i}\gamma^{\mu}u_{L}^{j})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{d_{L}Z}]^{ij}$ | $(\bar{d}_{L}^{i}\gamma^{\mu}d_{L}^{j})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{ud_{L}W}]^{ij}$ | $(\bar{u}_{L}^{i}\gamma^{\mu}d_{L}^{j})\,W_{\mu}^{+}$ | 18(9)
$RR$ quarks
| Operator | Count
$[{{\mathbf{c}}}_{u_{R}Z}]^{ij}$ | $(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{d_{R}Z}]^{ij}$ | $(\bar{d}_{R}^{i}\gamma^{\mu}d_{R}^{j})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{ud_{R}W}]^{ij}$ | $(\bar{u}_{R}^{i}\gamma^{\mu}d_{R}^{j})\,W_{\mu}^{+}$ | 18(9)
$LL$ leptons
| Operator | Count
$[{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}$ | $(\bar{\nu}_{L}^{\alpha}\gamma^{\mu}\nu_{L}^{\beta})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}$ | $(\bar{e}_{L}^{\alpha}\gamma^{\mu}e_{L}^{\beta})\,Z_{\mu}$ | 9(6)
$[{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}$ | $(\bar{e}_{L}^{\alpha}\gamma^{\mu}\nu_{L}^{\beta})\,W_{\mu}^{+}$ | 18(9)
$RR$ leptons
| Operator | Count
$[{{\mathbf{c}}}_{e_{R}Z}]^{\alpha\beta}$ | $(\bar{e}_{R}^{\alpha}\gamma^{\mu}e_{R}^{\beta})\,Z_{\mu}$ | 9(6)
Scalar operators
| Operator | Count
$[{{\mathbf{c}}}_{eh}]^{\alpha\beta}$ | $(\bar{e}_{L}^{\alpha}\,e_{R}^{\beta})\,h$ | 9(6)
$[{{\mathbf{c}}}_{dh}]^{ij}$ | $(\bar{d}_{L}^{i}\,d_{R}^{j})\,h$ | 9(6)
$[{{\mathbf{c}}}_{uh}]^{ij}$ | $(\bar{u}_{L}^{i}\,e_{R}^{j})\,h$ | 9(6)

SMEFT
---
$LL$ quarks
| Operator | Count
$[{{\mathcal{C}}}_{Hq}^{(1)}]^{ij}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}H)(\bar{q}^{i}\gamma^{\mu}q^{j})$ | 9(6)
$[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}\,\tau^{I}H)(\bar{q}^{i}\gamma^{\mu}\,\tau^{I}q^{j})$ | 9(6)
$RR$ quarks
| Operator | Count
$[{{\mathcal{C}}}_{Hu}]^{ij}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}H)(\bar{u}_{R}^{i}\gamma^{\mu}u_{R}^{j})$ | 9(6)
$[{{\mathcal{C}}}_{Hd}]^{ij}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}\,\tau^{I}H)(\bar{d}_{R}^{i}\gamma^{\mu}\,\tau^{I}d_{R}^{j})$ | 9(6)
$[{{\mathcal{C}}}_{Hud}]^{ij}$ | $(\widetilde{H}^{\dagger}\overleftrightarrow{D}_{\mu}\,H)(\bar{u}_{R}^{i}\gamma^{\mu}\,d_{R}^{j})$ | 18(9)
$LL$ leptons
| Operator | Count
$[{{\mathcal{C}}}_{Hl}^{(1)}]^{\alpha\beta}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}H)(\bar{l}^{\alpha}\gamma^{\mu}l^{\beta})$ | 9(6)
$[{{\mathcal{C}}}_{Hl}^{(3)}]^{\alpha\beta}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}\,\tau^{I}H)(\bar{l}^{\alpha}\gamma^{\mu}\,\tau^{I}l^{\beta})$ | 9(6)
$RR$ leptons
| Operator | Count
$[{{\mathcal{C}}}_{He}]^{\alpha\beta}$ | $(H^{\dagger}\overleftrightarrow{D}_{\mu}H)(\bar{e}_{R}^{\alpha}\gamma^{\mu}e_{R}^{\beta})$ | 9(6)
Scalar quarks
| Operator | Count
$[{{\mathcal{C}}}_{eH}]^{\alpha\beta}$ | $(H^{\dagger}\,H)\,(\bar{l}^{\alpha}\,e_{R}^{\beta}H)$ | 9(6)
$[{{\mathcal{C}}}_{dH}]^{ij}$ | $(H^{\dagger}\,H)\,(\bar{q}^{i}\,d_{R}^{j}H)$ | 9(6)
$[{{\mathcal{C}}}_{uH}]^{ij}$ | $(H^{\dagger}\,H)\,(\bar{q}^{i}\,u_{R}^{j}\widetilde{H})$ | 9(6)

Table 4:  Left column: HEFT operators representing the couplings of $Z$,
$W^{\pm}$ and $h$ with fermions. Right column: SMEFT operators contributing to
the corresponding HEFT operators (following notations of Grzadkowski:2010es ).
Count denotes the number of independent operators; the number inside the
brackets is the number of independent operators if all WCs were real. The
SMEFT basis in which these operators are written is defined in Appendix B.

In addition to quarks and leptons, HEFT also involves $Z,W^{\pm}$ and $h$
bosons as degrees of freedom. The BSM couplings of these bosons to the
fermions appear as HEFT WCs, as shown in eq. (2). These WCs contribute to low-
energy semileptonic processes via $Z,W^{\pm}$ and $h$ exchange diagrams. They
can also be probed independently by studying decays of $Z,W^{\pm}$, and $h$.
However, when the BSM couplings of these bosons to fermions are parameterized
in terms of SMEFT WCs, the number of independent WCs are less than the total
number of relevant HEFT WCs. Thus, SMEFT predicts relations among the
corresponding HEFT WCs, as earlier. In this section, we derive these
relations.

In Table 4, we show the 144 (87) HEFT operators and 108 (69) SMEFT operators
that give rise to $Z,W^{\pm}$ and $h$ couplings to fermions. Once again, the
HEFT operators have been presented in the unitary gauge as $U(1)_{em}$
invariant terms. These operators can be rewritten in an $SU(2)_{L}\times
U(1)_{Y}$ invariant form where this symmetry is non-linearly realized, as
shown in Appendix A. Note that the SMEFT basis we have used is not the
commonly used Warsaw basis. The details of our basis and our rationale for it
have been presented in Appendix B.

While the number of dimension-6 SMEFT and HEFT operators are the same for the
coupling with $h$, the number of HEFT operators contributing to $Z$ and
$W^{\pm}$ coupling deviations to left-handed quarks or leptons is 36 (21) and
it exceeds the number of contributing SMEFT operators, 18 (12). This implies
18 (12) relations among the HEFT WCs. The expressions for the HEFT WCs for
these operators in terms of the SMEFT ones can be written in the flavor basis
as

$\displaystyle[{{\mathbf{c}}}_{u_{L}Z}]^{ij}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hq}^{(1)}]^{ij}-[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij}),$
$\displaystyle[{{\mathbf{c}}}_{\nu_{L}Z}]^{\alpha\beta}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hl}^{(1)}]^{\alpha\beta}-[{{\mathcal{C}}}_{Hl}^{(3)}]^{\alpha\beta})~{},$
(32) $\displaystyle[{{\mathbf{c}}}_{d_{L}Z}]^{ij}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hq}^{(1)}]^{ij}+[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij})~{},$
$\displaystyle[{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hl}^{(1)}]^{\alpha\beta}+[{{\mathcal{C}}}_{Hl}^{(3)}]^{\alpha\beta})~{},$
(33) $\displaystyle[{{\mathbf{c}}}_{ud_{L}W}]^{ij}$
$\displaystyle=\eta_{LW}\,[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij}~{},$
$\displaystyle[{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}$
$\displaystyle=\eta_{LW}\,[{{\mathcal{C}}}_{Hq}^{(3)}]^{\alpha\beta}~{}.$ (34)

Here $\eta_{LZ}=-g/(2\,\cos\theta_{W})$, where $\theta_{W}$ is the Weinberg
angle, and $\eta_{LW}=g/(\sqrt{2})$. These expressions can then be used to
derive the relations among the HEFT WCs:

$\displaystyle[\hat{{\mathbf{c}}}_{ud_{L}W}]^{ij}$
$\displaystyle=\frac{1}{\sqrt{2}}\cos\theta_{w}\,([\hat{{\mathbf{c}}}_{u_{L}Z}]^{ik}\,V_{kj}-V_{ik}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{kj})~{},$
(35)
$\displaystyle[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\rho}\,U_{\rho\beta}$
$\displaystyle=\frac{1}{\sqrt{2}}\cos\theta_{w}\,([\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}-U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,U_{\sigma\beta})~{}.$
(36)

These relations are also shown in the last two rows of Table 3. Once again,
the relations in the mass basis contain only the physically measurable CKM and
PMNS matrices.

The relations in eq. (35) and (36) should be independent of the choice of the
SMEFT basis. In the Warsaw basis, the additional operators ${\cal
O}_{T}=(H^{\dagger}\,\overleftrightarrow{D}H)^{2}$ and ${\cal
O}_{WB}=gg^{\prime}(H^{\dagger}\tau^{I}\,H)\,W_{\mu\nu}^{I}\,B^{\mu\nu}$ would
contribute to the couplings of gauge bosons to the fermions by affecting their
mass and kinetic terms. However, their contributions in the above two
relations cancel out. This can be more transparently seen in the SMEFT basis
that we use, where these two operators are traded for two other operators
which do not affect the gauge boson couplings (see Appendix B).

###  2.3 Predictions for semileptonic operators at low energies

In the previous two subsections, we discussed the predictions of SMEFT at high
energies, i.e., relations among HEFT WCs above the EW scale. Now we consider
the SMEFT predictions for the low-energy observables where the relevant
effective field theory is LEFT. The forms of LEFT operators are the same as in
Table 1 apart from the fact that the operators involving the top quark would
not be included in the LEFT lagrangian. We will now rewrite the relations
derived in the previous two subsections in terms of the LEFT WCs. In order to
carry this out, we need the matching relations between the WCs of HEFT and
LEFT operators. For instance, for operators of the $LLLL$ category, the
matching relations in the flavor basis888In our notation, $\tilde{{C}}$
corresponds to LEFT coefficient in the flavor basis and ${{C}}$ in the mass
basis. are

$\displaystyle[\tilde{{C}}_{{\ell}qLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{{\ell}qLL}^{V}]^{\alpha\beta
ij}+k_{{\ell}_{L}}\,[{{\mathbf{c}}}_{q_{L}Z}]^{ij}\,\delta_{\alpha\beta}+k_{q_{L}}\,[{{\mathbf{c}}}_{l_{L}Z}]^{\alpha\beta}\,\delta_{ij}~{},$
(37) $\displaystyle[\tilde{{C}}_{LL}^{V}]^{\alpha\beta ij}$
$\displaystyle=\omega\,[{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta ij}+k_{e\nu
W}\,[{{\mathbf{c}}}_{ud_{L}W}]^{ij}\,\delta_{\alpha\beta}+k_{udW}\,[{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}\,\delta_{ij}~{},$
(38)

where $l\in\\{\nu,e\\}$, $q\in\\{u,d\\}$, $\omega=v^{2}/(2\,\Lambda^{2})$, and
the $k$ coefficients are

$\displaystyle k_{f_{L}}$
$\displaystyle=\frac{2\cos\theta_{w}}{g}(T^{3}_{f}-Q_{f}\sin^{2}\theta_{w})~{},\quad\textrm{and}~{}~{}k_{e\nu
W}=k_{udW}=\frac{\sqrt{2}}{g}~{},$ (39)

with $f_{L}\in\\{\nu_{L},e_{L},u_{L},d_{L}\\}$.

In this work, we have not considered effects of the renormalization group (RG)
running of the LEFT coefficients from the weak scale to the scale of the
relevant experiments. These would need to be included using the RG equations
of Ref. Jenkins:2013zja  for a more precise phenomenological treatment.

Using matching relations like eq. (37) and eq. (38), we can now rewrite the
relations of Table 3, which were written in terms of the HEFT WCs, the LEFT
WCs, and the BSM couplings of $Z$ and $W^{\pm}$. For the $LLLL$ operators, for
example, these relations become

$\displaystyle V_{ik}\left[[{{C}}_{edLL}^{V}]^{\alpha\beta
kl}-\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{kl}\,\delta_{\alpha\beta}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{kl}\right)\right]V^{\dagger}_{{\ell}j}$
$\displaystyle~{}~{}=U^{\dagger}_{\alpha\rho}\left[[{{C}}_{\nu
uLL}^{V}]^{\rho\sigma
ij}-\chi~{}\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{u_{L}Z}]^{ij}\,\delta_{\rho\sigma}+k_{u_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]U_{\sigma\beta},$
(40) $\displaystyle V^{\dagger}_{ik}\left[[{{C}}_{euLL}^{V}]^{\alpha\beta
kl}-\chi~{}\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{u_{L}Z}]^{kl}\,\delta_{\alpha\beta}+k_{u_{L}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{kl}\right)\right]V_{{\ell}j}$
$\displaystyle~{}~{}=U^{\dagger}_{\alpha\rho}\left[[{{C}}_{\nu
dLL}^{V}]^{\rho\sigma
ij}-\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\rho\sigma}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]U_{\sigma\beta}~{},$
(41) $\displaystyle V^{\dagger}_{ik}\,\left[[{{C}}_{LL}^{V}]^{\alpha\beta
kj}-\chi\left(k_{e\nu
W}\,[\hat{{\mathbf{c}}}_{ud_{L}W}]^{kj}\,\delta_{\alpha\beta}+[k_{udW}]^{kj}\,[\hat{{\mathbf{c}}}_{e\nu_{L}W}]^{\alpha\beta}\right)\right]$
$\displaystyle~{}~{}=\left[[{{C}}_{edLL}^{V}]^{\alpha\rho
ij}-\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\alpha\rho}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\rho}\,\delta_{ij}\right)\right]\,U^{\dagger}_{\rho\beta}$
$\displaystyle~{}~{}~{}-U^{\dagger}_{\alpha\sigma}\,\left[[{{C}}_{\nu
dLL}^{V}]^{\sigma\beta
ij}-\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{ij}\,\delta_{\sigma\beta}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\sigma\beta}\,\delta_{ij}\right)\right]~{},$
(42)

where, for WCs involving top quark, we have defined999This convention is used
for the purpose of giving eqs. (40) to (42) a unified form for all quarks.
This is an exception to our normal convention, where we use ‘$C$’ only for
LEFT WCs. $[{{C}}_{{\ell}qLL}^{V}]^{\alpha\beta
ij}\equiv\omega[\hat{{\mathbf{c}}}_{{\ell}qLL}^{V}]^{\alpha\beta ij}$ and
$[{{C}}_{LL}^{V}]^{\alpha\beta
ij}\equiv\omega[\hat{{\mathbf{c}}}_{LL}^{V}]^{\alpha\beta ij}$. In eq. (42),
$\chi=0$ ($\chi=1$) if the respective four-fermion operator contains (does not
contain) the top quark. The introduction of $\chi$ ensures that the HEFT WCs
are replaced by LEFT ones for all the four-fermion operators not containing
the top quark. The relations for the WCs in the other categories can be
similarly derived and have been presented in Appendix C.

We now mention two important scenarios where the SMEFT predictions derived in
this section can be simplified. First, note that apart from neutrino physics
experiments, it is impossible to distinguish the different flavors of
neutrinos in observables. These observables thus depend on combinations of WCs
with neutrino flavor indices summed over and are independent of the basis used
for neutrinos. In particular, we can choose to work in a basis aligned to the
charged-lepton flavor basis. This amounts to substituting $U=1$ in all the
SMEFT predictions, whether it is for HEFT WCs in Table 5 or for LEFT WCs such
as those in eqs. (40-42) or in Appendix C. Secondly, in the UV4f scenario
where there are no modifications to $Z,W^{\pm}$ and $h$ couplings with respect
to SM, the matching equations in eq. (38) get simplified and we can obtain
SMEFT predictions involving LEFT WCs simply by substituting
$[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$ by $[{{C}}]^{\alpha\beta ij}$ in Table
5. This scenario becomes more relevant in the phenomenological applications of
the SMEFT predictions that we present in the following section.

##  3 SMEFT-predicted constraints on new physics

In this section, we will show how the SMEFT predictions derived in Sec. 2 can
be used to obtain bounds on the LEFT Wilson coefficients $[{{C}}]^{\alpha\beta
ij}$. We utilize the fact that the SMEFT predictions give analytic equations
that can connect strongly constrained WCs to poorly constrained ones, thus
allowing us to extract stronger bounds on the latter. In this section, we
restrict ourselves to UV4f models, where the UV physics generates only four-
fermionic operators in SMEFT, so that the operators discussed in Sec. 2.2 are
absent. While a more general analysis using the constraints on $Z$ and
$W^{\pm}$ couplings (see Ref. Efrati:2015eaa ) is possible, our primary aim
here is to illustrate the power of the SMEFT predictions and thus we focus on
the very well-motivated UV4f scenario. As discussed in the previous section,
in this scenario we can use the relations in Table 3 by simply replacing
$[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$ by $[{{C}}]^{\alpha\beta ij}$.
Furthermore, as explained at the end of the previous section, the observables
in this section will be insensitive to the flavor of neutrinos, and hence we
can take $U\to 1$ in the SMEFT predictions.

We further restrict ourselves to the operators involving only left-handed
quarks and leptons (i.e. $LLLL$ discussed in Sec. 2.3) as these provide
leading corrections with respect to SM.101010 While low-energy flavor
observables get interference level corrections from both $RRLL$ and $LLLL$
operators, as far as high $p_{T}$ observables are concerned, only $LLLL$
operator contributions can interfere with SM contribution if fermion masses
are neglected. The relations amongst $LLLL$ operators in UV4f models are given
by

$\displaystyle[{{C}}_{euLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=V_{ik}\,[{{C}}_{\nu dLL}^{V}]^{\alpha\beta
kl}V^{\dagger}_{{\ell}j}~{},$ (43)
$\displaystyle[{{C}}_{edLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=V^{\dagger}_{ik}\,[{{C}}_{\nu uLL}^{V}]^{\alpha\beta
kl}V_{{\ell}j}~{},$ (44) $\displaystyle[{{C}}_{LL}^{V}]^{\alpha\beta ij}$
$\displaystyle=V_{ik}\,([{{C}}_{edLL}^{V}]^{\alpha\beta kj}-[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta kj})~{},$ (45)

as can be obtained from eqs. (40 –42). Recall that RG effects have been
ignored in deriving the above relations, and as discussed in Sec. 2.3, the WCs
in the above equations that involve the top quark have been defined as
$[{{C}}]^{\alpha\beta ij}\equiv\omega\,[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$.
These WCs can be constrained using data from top production and decays. All
other WCs in the above equation are the standard LEFT WCs of eqs. (3) and (4).

Note that eqs. (43-45) involve 486 (261) WCs $[{{C}}]^{\alpha\beta ij}$, which
arise from 162 (90) SMEFT coefficients. These three equations therefore
correspond to 324 (171) relations among the WCs. Note that in several earlier
analyses (e.g. Fuentes-Martin:2020lea ; Bause:2020auq ) the WCs have been
assumed to be real. This is of course valid for the WCs of Hermitian
operators, i.e where $\alpha=\beta$ and $i=j$. However, as eqs. (43–45) show,
all the WCs are related linearly with complex coefficients (i.e, combinations
of CKM matrix elements) which makes it inconsistent for all of them to be
real. Note that even if all the WCs of SMEFT in the UV scale are real in the
flavor basis, phases will appear in $[{{C}}]^{\alpha\beta ij}$ through CKM
elements while matching. We in our analysis consider complex values for all
the WCs of non-Hermitian operators.

In the rest of this section, we focus on deriving bounds on the WCs from
semileptonic processes. To start with, in Sec. 3.1 to Sec. 3.3 we consider
only processes involving muon and muon neutrinos, i.e $\alpha=\beta=2$. This
is because many of the direct bounds from the muon channel are quite stringent
compared to those from the electron or tau channel. The terms in eqs. (43) and
(44) contain only neutral current WCs. On the other hand, in eq. (45) charged
current WCs are expressed in terms of neutral current WCs. Based on these
relations, in Sec. 3.1 and 3.2 we obtain indirect bounds on neutral current
WCs appearing in eqs. (43) and (44) respectively. In Sec. 3.3 we discuss about
the indirect bounds for charged current WCs. In Sec. 3.4, we further indicate
how these relations may be used in conjunction with constraints on lepton
flavor violating decays to constrain Wilson coefficients involving other
lepton families.

###  3.1 Bounds on neutral-current WCs involving $(\nu d)$ and $(eu)$

There are 6 complex and 6 real neutral-current WCs in eq. (43) with
$\alpha=\beta=2$. These WCs correspond to operators either with neutrinos and
down-type quarks ($\nu dLL$), or with charged leptons and up-type quarks
($euLL$). We first discuss direct bounds on these WCs. We consider both low-
energy observables, such as rare decays, as well as high-energy observables,
such as the high-$p_{T}$ Drell-Yan process, top decays, etc. While the former
can directly bound the LEFT WCs $[{{C}}]^{\alpha\beta ij}$, the latter can
directly bound only the high energy HEFT WCs,
$[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$. As we are considering UV4f models
here, however, the bounds on $[\hat{{\mathbf{c}}}]^{\alpha\beta ij}$ can be
converted to bounds on $[{{C}}]^{\alpha\beta ij}$ in a straightforward way by
keeping only the first term in the matching relations, eq. (37) and eq. (38).

Direct bounds on the WCs $[{{C}}_{\nu dLL}^{V}]^{2212}$, $[{{C}}_{\nu
dLL}^{V}]^{2213}$ and $[{{C}}_{\nu dLL}^{V}]^{2223}$ are obtained from rare
decays of $K$ and $B$ mesons. For $[{{C}}_{\nu dLL}^{V}]^{2212}$, we have used
the recent measurement of the branching ratio of $K^{+}\to\pi^{+}\nu\nu$ in
the NA62 experiment NA62:2022qes . For $[{{C}}_{\nu dLL}^{V}]^{2213}$ we take
the $90\%$ upper bounds on the branching ratios of the decay modes
$B^{+}\to\rho^{+}\,\nu\,\nu$ and $B^{+}\to\pi^{+}\,\nu\,\nu$
ParticleDataGroup:2022pth . For $[{{C}}_{\nu dLL}^{V}]^{2223}$, we include the
recent measurement of $B^{+}\to K^{+}\nu\,\nu$ branching ratio in Belle-
II:2023esi  along with the $90\%$ upper bound on the branching ratio of
$B^{+}\to K^{*+}\,\nu\,\nu$ ParticleDataGroup:2022pth . The theoretical values
for the discussed mesonic decay modes are calculated using the package
‘flavio’ Straub:2018kue . The bound on $[{{C}}_{\nu dLL}^{V}]^{2211}$ is
obtained from constraints111111The bounds presented in Farzan:2017xzy  are for
the vector and axial vector WCs. We convert these to bounds on operators in
our basis by adding the $1\sigma$ ranges in quadrature. on non-standard
interactions of neutrinos in atmospheric and accelerator neutrino experiments
Farzan:2017xzy ; Escrihuela:2011cf . These bounds are shown in the top panels
of Fig. 2. For the WCs $[{{C}}_{\nu dLL}^{V}]^{2222}$ and $[{{C}}_{\nu
dLL}^{V}]^{2233}$, there are no direct bounds available.

Figure 2: Direct bounds on the complex WCs ${{C}}_{\nu dLL}^{V}$ (top panels)
and ${{C}}_{euLL}^{V}$ (middle panels). The cyan color represents bounds from
rare meson decays, orange represents bounds from high-$pT$ dimuon searches
while purple represents bounds from top productions and decays. The WCs shown
in the bottom panels are real due to the hermiticity of the corresponding
operators. Note that the bottom panel uses the symmetric log scale. See
Appendix D for numerical values of the bounds.

The direct bounds for WCs containing up-type quarks and charged leptons are
obtained from rare decays, high-$p_{T}$ dilepton searches as well as top
production and decays. The WC $[{{C}}_{euLL}^{V}]^{2212}$ gets constraints
from rare decays of $D$ meson Fuentes-Martin:2020lea . For
$[{{C}}_{euLL}^{V}]^{2211}$, $[{{C}}_{euLL}^{V}]^{2212}$ and
$[{{C}}_{euLL}^{V}]^{2222}$, strong bounds are obtained from high-$p_{T}$
dimuon searches at the LHC. In the UV4f scenario and with the approximation of
negligible RG effects, these bounds can be taken to be bounds on the LEFT WCs.
We use CMS data for the dimuon mode CMS:2021ctt  and the package ‘HighPT’
Allwicher:2022mcg ; Allwicher:2022gkm  which provides bounds on SMEFT WCs. In
order to convert these into bounds on isolated LEFT WCs, we turn on those
linear combinations of SMEFT WCs which make that particular LEFT WC nonzero,
and leave other dimuon modes unaffected. Bounds on WCs involving top quark
(e.g. $[{{C}}_{euLL}^{V}]^{2213}$, $[{{C}}_{euLL}^{V}]^{2213}$ and
$[{{C}}_{euLL}^{V}]^{2233}$ ) are obtained from data on top production and
decays Afik:2021jjh . These direct bounds are shown in Fig. 2.

Note that, in order to obtain the direct bounds in Fig. 2, we have only
bounded the individual contribution of the relevant $LLLL$ operator with
$\alpha=\beta=2$ and ignored possible contributions from other operators.
Under some very reasonable assumptions, however, including these contributions
would not significantly alter the bounds we have obtained. First of all, as
far as the dineutrino decay modes are concerned, the experiments cannot
distinguish between different neutrino flavors. To extract bounds on the
$\bar{\nu}_{\mu}\nu_{\mu}$ mode, we assume that there are no large
cancellations between the interference contributions of the different neutrino
flavor modes. Also, for low energy observables a linear combination of $LLLL$
WCs and WCs of other vector operators in Table 1 enter the interference term
in EFT corrections. In the cases where measurements are sensitive to the
interference term, there can in principle be flat directions where the bounds
obtained here get weakened, but this would again require a fine-tuned
cancellation between the interference terms of the $LLLL$ and other vector
operators; we assume such cancellations are absent. Finally, there are
operators in Table 1, such as the scalar and tensor operators, that give
contributions proportional to the square of their WCs but the inclusion of
such positive definite terms would only strengthen our bounds. Thus, under
these assumptions, the direct bounds discussed here hold also in the presence
of other operator contributions.

Figure 3: Direct bounds from low-energy (cyan) and high-$p_{T}$ (orange)
processes, along with the indirect (green) bounds on the complex WCs
$[{{C}}_{euLL}^{V}]^{2213}$ and $[{{C}}_{euLL}^{V}]^{2223}$ and on the real
WCs $[{{C}}_{\nu dLL}^{V}]^{2211}$, $[{{C}}_{\nu dLL}^{V}]^{2222}$ and
$[{{C}}_{euLL}^{V}]^{2222}$. The input parameters used are the four complex
WCs $[{{C}}_{\nu dLL}^{V}]^{2212}$, $[{{C}}_{\nu dLL}^{V}]^{2213}$,
$[{{C}}_{\nu dLL}^{V}]^{2223}$ and $[{{C}}_{euLL}^{V}]^{2212}$ and one real WC
$[{{C}}_{euLL}^{V}]^{2211}$. Note that the bottom panel uses the symmetric log
scale. See Appendix D for numerical values of the bounds.

Now we turn to the indirect bounds obtained by using the SMEFT predictions.
Counting the real and imaginary parts of the WCs separately, eq. (43) involves
a total of 18 parameters, connected by 9 linear relations. Our goal is to find
indirect bounds on WCs that are weakly bound or have no direct bound, with the
help of these relations. To this end, we first choose the 9 parameters which
have the most stringent bounds:

$\displaystyle\textrm{Re}\left([{{C}}_{\nu
dLL}^{V}]^{2212}\right),~{}\textrm{Im}\left([{{C}}_{\nu
dLL}^{V}]^{2212}\right),~{}\textrm{Re}\left([{{C}}_{\nu
dLL}^{V}]^{2213}\right),~{}\textrm{Im}\left([{{C}}_{\nu
dLL}^{V}]^{2213}\right),$ $\displaystyle\textrm{Re}\left([{{C}}_{\nu
dLL}^{V}]^{2223}\right),\textrm{Im}\left([{{C}}_{\nu
dLL}^{V}]^{2223}\right),~{}\textrm{Re}\left([{{C}}_{euLL}^{V}]^{2212}\right),~{}\textrm{Im}\left([{{C}}_{euLL}^{V}]^{2212}\right)~{},$
(46)

and the real WC $[{{C}}_{euLL}^{V}]^{2211}$. The remaining 9 parameters can
then be written in terms of these using eq. (43), and indirect bounds on them
may be obtained. In Fig. 3 we show the resultant indirect bounds on these
parameters. For the complex WCs $[{{C}}_{euLL}^{V}]^{2213}$ and
$[{{C}}_{euLL}^{V}]^{2223}$, the region of intersection between the indirect
and the direct bounds can put a tighter constraint on the preferred values.
These WCs correspond to single top production along with two leptons or top
decays via $t\to c\ell\ell$ and $t\to u\ell\ell$ channels. It may be noticed
that the constraints on the imaginary part of $[{{C}}_{euLL}^{V}]^{2223}$ are
strong, making $[{{C}}_{euLL}^{V}]^{2223}$ appear almost as a real WC. This
feature may be understood as follows. Eq (43) implies

$\displaystyle[{{C}}_{\nu
dLL}^{V}]^{2233}=|V_{tb}|^{2}[{{C}}_{euLL}^{V}]^{2233}+V_{cb}^{*}\,V_{tb}\,[{{C}}_{euLL}^{V}]^{2223}+\mathcal{O}(\lambda^{3})~{}.$
(47)

Here $\lambda=\sin(\theta_{c})$ where $\theta_{c}\sim 0.227$ is the Cabbibo
angle. Since $[{{C}}_{euLL}^{V}]^{2233}$ and $[{{C}}_{\nu dLL}^{V}]^{2233}$
are real and $V_{cb}^{*}V_{tb}$ is real up to $\mathcal{O}(\lambda^{3})$, the
only imaginary quantity appearing in this equation is
$\textrm{Im}([{{C}}_{euLL}^{V}]^{2223})$; hence it is strongly constrained.

As far as the real WCs are concerned, for $[{{C}}_{\nu dLL}^{V}]^{2211}$ we
get a better constraint than the available direct bound which may be tested in
experiments studying matter effects on neutrino oscillations. At the same
time, $[{{C}}_{\nu dLL}^{V}]^{2222}$, which has no direct bound, now gets
bounded. For $[{{C}}_{euLL}^{V}]^{2222}$, the indirect bound is slightly worse
than the direct bound. For the other two, viz. $[{{C}}_{\nu dLL}^{V}]^{2233}$
and $[{{C}}_{\nu dLL}^{V}]^{2233}$, the indirect bounds are much worse than
the direct bounds.

Similar relations have been explored in literature in order to put indirect
bounds on various EFT coefficients, albeit for a smaller subset of WCs, with
some UV flavor assumptions, or by neglecting CKM elements. In Bause:2020auq ;
Bause:2021cna ; Bause:2021ihn , similar bounds have been calculated assuming
the WCs to be real and neglecting terms in eqs. (43-45) having CKM elements
that are higher order in $\lambda$. The indirect bounds obtained on the real
WCs in Bause:2020auq  become weaker when all the CKM matrix elements are
inserted.

Note that our choice of the 9 input parameters need not have been the best one
for finding the best indirect bounds on any parameter. A different set of 9
input parameters could be optimum. Indeed the best bounds may be obtained by
using all the available direct bounds in a combined fit. Since the primary aim
of this paper is to illustrate the utility of the linear relations in
obtaining indirect bounds, we leave the detailed analysis for future work.

Figure 4: The top panels and the bottom left panel show direct bounds from
meson decays (cyan) for the complex WCs $[{{C}}_{edLL}^{V}]^{2212}$,
$[{{C}}_{edLL}^{V}]^{2213}$, $[{{C}}_{edLL}^{V}]^{2223}$. The orange
background in these three panels indicates that the parameter space of these
complex WCs displayed in this figure is allowed by the high-$p_{T}$ dimuon
searches, and only constrained by meson decays. The bottom right panel shows
the constraints from high-$p_{T}$ dimuon searches (orange) on the real WCs
$[{{C}}_{edLL}^{V}]^{2211}$, $[{{C}}_{edLL}^{V}]^{2222}$ and
$[{{C}}_{edLL}^{V}]^{2233}$. Note that the bottom-right panel uses the
symmetric log scale. See Appendix D for numerical values of the bounds.

###  3.2 Bounds on neutral-current WCs involving $(ed)$ and $(\nu u)$

In this section, we perform a similar analysis as in Sec. 3.1 for neutral-
current WCs involving the muon family, using the relation in eq. (44). The WCs
involved correspond to the operators containing either charged leptons and
down-type quarks $(edLL)$, or neutrinos and up-type quarks $(\nu uLL)$.

The bounds on $(edLL)$ WCs are typically stronger since they involve charged
muon. The WCs $[{{C}}_{edLL}^{V}]^{2212}$, $[{{C}}_{edLL}^{V}]^{2213}$ and
$[{{C}}_{edLL}^{V}]^{2223}$ get direct bounds from rare decays of $K$ and $B$
mesons. Bound on the absolute value of $[{{C}}_{edLL}^{V}]^{2212}$ is provided
in Bause:2020auq ; we convert this to bounds on the real and the imaginary
parts of this WC by taking into account all possible values for its phase. For
$[{{C}}_{edLL}^{V}]^{2213}$, we obtain the bound from the branching ratio
measurement of $B^{0}\to\mu^{+}\,\mu^{-}$ ParticleDataGroup:2022pth . For the
real and the imaginary parts of $[{{C}}_{edLL}^{V}]^{2223}$, we use a combined
fit to the observables $\mathcal{B}(B^{(+,0)}\to
K^{(+,0)}\,\mu^{+}\,\mu^{-})$, $\mathcal{B}(B^{(+,0)}\to
K^{*\,(+,0)}\,\mu^{+}\,\mu^{-})$, $R_{K^{(*)}}$,
$\mathcal{B}(B_{s}\to\mu^{+}\,\mu^{-})$, as well as the angular observables
$P_{5}^{\prime}$ and $F_{L}$ in $B^{0}\to K^{*0}\,\mu^{+}\,\mu^{-}$. The
high-$p_{T}$ dimuon searches give bounds on the three real WCs
$[{{C}}_{edLL}^{V}]^{2211}$, $[{{C}}_{edLL}^{V}]^{2222}$ and
$[{{C}}_{edLL}^{V}]^{2233}$. We show these bounds in Fig. 4. Among the $(\nu
uLL)$ WCs, only a weak bound is available on $[{{C}}_{\nu uLL}^{V}]^{2211}$
from constraints on non-standard interactions of neutrinos in atmospheric
neutrino experiments Farzan:2017xzy . Once again, while these bounds are on
the individual contributions of the respective operators, inclusion of other
operators would not significantly alter them given the assumptions stated in
Sec. 3.1.

Figure 5: Indirect bounds (green) on the complex WCs $[{{C}}_{\nu
uLL}^{V}]^{2212}$, $[{{C}}_{\nu uLL}^{V}]^{2213}$, $[{{C}}_{\nu
uLL}^{V}]^{2223}$, and the real WCs $[{{C}}_{\nu uLL}^{V}]^{2211}$,
$[{{C}}_{\nu uLL}^{V}]^{2222}$, $[{{C}}_{\nu uLL}^{V}]^{2233}$. The direct
bound is available only for the real WC $[{{C}}_{\nu uLL}^{V}]^{2211}$ (shown
in cyan). Note that the bottom-right panel uses the symmetric log scale. See
Appendix D for numerical values of the bounds.

Counting the real and imaginary parts of the WCs separately, eq. (43) involves
a total of 18 parameters, connected by 9 linear relations. In order to get
stronger bounds on the $(\nu uLL)$ WCs, we take the 9 parameters corresponding
to the $(edLL)$ WCs as inputs and derive the indirect bounds using this
relation. These bounds have been shown in Fig. 5. It can be seen that the
complex WCs $[{{C}}_{\nu uLL}^{V}]^{2212}$, $[{{C}}_{\nu uLL}^{V}]^{2213}$,
$[{{C}}_{\nu uLL}^{V}]^{2223}$ and the real WCs $[{{C}}_{\nu uLL}^{V}]^{2222}$
and $[{{C}}_{\nu uLL}^{V}]^{2233}$, which do not have any direct bounds, get
indirect constraints. The first among these WCs would contribute to the
invisible decay widths of $D$ mesons while the next two would contribute to
the semileptonic top decays, $t\to u\nu\nu$ and $t\to c\nu\nu$. The indirect
bound also improves the constraints on $[{{C}}_{\nu uLL}^{V}]^{2211}$
significantly. This indirect bound would be important for constraining models
with neutrino non-standard interactions (NSI) Farzan:2017xzy  and can be
tested in precision neutrino oscillation experiments.

Note that the indirect constraints suggest that $[{{C}}_{\nu uLL}^{V}]^{2212}$
is almost real. This can be understood by looking at the leading-order
contributions to $[{{C}}_{\nu uLL}^{V}]^{2212}$ in eq. (44):

$\displaystyle[{{C}}_{\nu uLL}^{V}]^{2212}$
$\displaystyle=V_{ud}\,V_{cs}^{*}[{{C}}_{edLL}^{V}]^{2212}+V_{ud}\,V_{cd}^{*}[{{C}}_{edLL}^{V}]^{2211}+V_{us}\,V_{cs}^{*}\,[{{C}}_{edLL}^{V}]^{2222}$
$\displaystyle~{}+V_{us}\,V_{cd}^{*}\,[{{C}}_{edLL}^{V\,*}]^{2212}+\mathcal{O}(\lambda^{3})~{}.$
(48)

In the above equation, all the CKM coefficients are real up to ${\cal
O}(\lambda^{3})$. The WCs $[{{C}}_{edLL}^{V}]^{2211}$ and
$[{{C}}_{edLL}^{V}]^{2222}$ are real, while $\textrm{Im}([{{C}}_{\nu
uLL}^{V}]^{2212})$ has strong constraints of $\mathcal{O}(0.02)$. Therefore,
the imaginary part of the left-hand side, i.e. $\textrm{Im}\left([{{C}}_{\nu
uLL}^{V}]^{2212}\right)$, is strongly constrained.

###  3.3 Bounds on charged-current WCs

Eq. (45) allows us to express charged-current WCs as combinations of neutral-
current WCs. Restricting to the muon family of lepton, i.e $\alpha=\beta=2$,
there are 9 charged-current WCs on the left-hand side of eq. (45); all of them
can be complex in general. All these charged-current WCs would get indirectly
constrained due to the bounds on the neutral-current WCs. In this section, we
first show the direct bounds for the 9 charged-current WCs from mesonic decays
and from high-$p_{T}$ monolepton searches. Later, we compare these bounds with
the ones derived indirectly using eq. (45).

Figure 6: Direct bounds on the charged-current WCs from meson decays (cyan)
and from high-$p_{T}$ mono-muon searches (orange). Note that there are no
direct constraints on the WCs associated with charged current decays of top
quark. See Appendix D for numerical values of the bounds.

For the WCs $[{{C}}_{LL}^{V}]^{2211}$, $[{{C}}_{LL}^{V}]^{2212}$,
$[{{C}}_{LL}^{V}]^{2213}$, $[{{C}}_{LL}^{V}]^{2221}$,
$[{{C}}_{LL}^{V}]^{2222}$ and $[{{C}}_{LL}^{V}]^{2223}$, we obtain direct
bounds using the branching ratios ParticleDataGroup:2022pth  of the decay
modes $\pi^{+}\to\mu^{+}\nu$, $K^{+}\to\pi\mu^{+}\nu$, $B^{+}\to\pi^{0}l\nu$
$D^{+}\to\mu^{+}\nu$, $D_{s}\to\mu\nu$ and $B^{+}\to Dl\nu$, respectively.
However, stronger bounds can be obtained for these WCs from high-$p_{T}$
monolepton searches. In order to do this121212In Allwicher:2022gkm , bounds on
SMEFT coefficients are provided using high-$p_{T}$ single lepton and dilepton
searches. However, no combination of SMEFT coefficients can map to a single
charged-current LEFT coefficient without generating other LEFT coefficient
that can contribute to the same single charged-lepton final state mode.
Therefore we calculate these bounds independently, we generate bin-wise events
in MadGraph Alwall:2014hca . Note that the charged-current NP would not change
the shape of the $q^{2}$ dependence from the SM prediction, since the relevant
charged-current operators in SM and NP are identical. We use the results from
the ATLAS analysis in Ref. ATLAS:2019lsy , and incorporate the effect of their
cuts by using a re-scaling factor on our generated events such that they
reproduce the ATLAS data for SM. We then perform a $\chi^{2}$ fit for the
isolated charged-current WC to obtain bound on the NP WC. These direct bounds
obtained from the meson decays (cyan) and from the high-$p_{T}$ mono-muon
searches (orange) are shown in Fig. 6.

Figure 7: Direct bounds (orange) on the charged-current WCs from high-$p_{T}$
mono-muon searches along with the indirect bounds (green) obtained using eq.
(45). Note that the quantities in the bottom panels have no direct bounds. See
Appendix D for numerical values of the bounds.

In order to obtain indirect bounds, we use the best available bounds (direct
or indirect) for the neutral-current WCs appearing on the right-hand side of
eq. (45). These indirect bounds (green) along with the best available direct
bounds (orange) are shown in Fig. 7. The figure shows that this method
provides constraints on $[{{C}}_{LL}^{V}]^{2231}$, $[{{C}}_{LL}^{V}]^{2232}$
and $[{{C}}_{LL}^{V}]^{2233}$, where no direct bounds were available. For
$[{{C}}_{LL}^{V}]^{2221}$, the indirect constraints are significantly stronger
than the direct bounds. These WCs would contribute to branching ratios of
semileptonic decays of top quark and D meson decays, viz. $D\to\pi\mu\nu$,
etc.

In addition, the imaginary parts of $[{{C}}_{LL}^{V}]^{2211}$,
$[{{C}}_{LL}^{V}]^{2212}$, $[{{C}}_{LL}^{V}]^{2222}$ and
$[{{C}}_{LL}^{V}]^{2223}$ are constrained more strongly. These WCs would
contribute to branching ratios of meson decays, viz. $K\to\pi\mu\nu$, $B\to
D\mu\nu$, etc. The reason for strong indirect constraints on the imaginary
parts of these four WCs may be understood using eq. (45). For example,
$[{{C}}_{LL}^{V}]^{2211}$ may be written using eq. (45) as

$\displaystyle[{{C}}_{LL}^{V}]^{2211}$
$\displaystyle=V_{ud}\,([{{C}}_{edLL}^{V}]^{2211}-[{{C}}_{\nu
dLL}^{V}]^{2211})+V_{us}\,([{{C}}_{edLL}^{V}]^{2212}-[{{C}}_{\nu
dLL}^{V}]^{2212})+\mathcal{O}(\lambda^{3})~{}.$ (49)

The CKM coefficients appearing on the right-hand side of the above equation
are real up to $\mathcal{O}(\lambda^{3})$. The WCs $[{{C}}_{edLL}^{V}]^{2211}$
and $[{{C}}_{\nu dLL}^{V}]^{2211}$ are real. Furthermore, the bounds on the
imaginary parts of $[{{C}}_{edLL}^{V}]^{2212}$ and $[{{C}}_{\nu
dLL}^{V}]^{2212}$ are of $\mathcal{O}(0.02)$ and these WCs appear with a CKM
coefficient of $\mathcal{O}(\lambda)$. Thus, the imaginary part of the WC on
the left-hand side, i.e. $[{{C}}_{LL}^{V}]^{2211}$, is strongly constrained.
Using similar arguments, we can show that the WCs $[{{C}}_{LL}^{V}]^{2212}$,
$[{{C}}_{LL}^{V}]^{2222}$ and $[{{C}}_{LL}^{V}]^{2233}$ are expected to be
dominantly real.

###  3.4 Predictions for lepton flavor violating observables

So far we have considered the relations only among WCs involving one lepton
family i.e. muon. In this subsection, we expand our discussion to SMEFT
predictions that include all lepton families, while remaining in the UV4f
scenario. These relations will relate diverse reaction channels like rare
decays of $B$, $D$ and $K$ mesons as well as lepton flavor violating (LFV)
processes such as $\tau\rightarrow{\ell}\,q_{i}\,q_{j}$ and
${\ell}\,N\to{\ell}^{\prime}\,N$. Focusing once again on UV4f models, we shall
indicate the methodology by one example and present a set of relevant
processes in Table 5.

Eq. $\downarrow$ | LHS WC |  RHS WCs |  Transitions | Processes
---|---|---|---|---
(43) | $[{{C}}_{euLL}^{V}]^{{\ell}311}$ |  $[{{C}}_{\nu dLL}^{V}]^{{\ell}311}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}312}$ |  $\tau\to u\,u\,{\ell}$ $s\to d\,\nu\,\nu$ |  LFV $\tau$ decay, $K\to\pi\nu\nu$
$[{{C}}_{euLL}^{V}]^{{\ell}{\ell}^{\prime}11}$ |  $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}11}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}12}$ |  ${\ell}\,u\to{\ell}^{\prime}\,u$ $s\to d\,\nu\,\nu$ |  LFV ${\ell}\,N\to{\ell}^{\prime}\,N$ $K\to\pi\nu\nu$
$[{{C}}_{euLL}^{V}]^{{\ell}312}$ |  $[{{C}}_{\nu dLL}^{V}]^{{\ell}312}$ |  $c\,u\to\tau\,{\ell}$ $s\to d\,\nu\,\nu$ |  LFV $D$ decays, $K\to\pi\nu\nu$ Bause:2020auq
$[{{C}}_{euLL}^{V}]^{{\ell}{\ell}^{\prime}i3}$ |  $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}13}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}23}$ |  $t\to u_{i}\,{\ell}\,{\ell}^{\prime}$ $b\to d\,\nu\,\nu$ $b\to s\,\nu\,\nu$ |  LFV top decay, $B$ decays to dineutrinos Bause:2020auq
(44) | $[{{C}}_{edLL}^{V}]^{{\ell}3ij}$ |  $[{{C}}_{\nu uLL}^{V}]^{{\ell}312}$ |  $\tau\to d\,d\,{\ell}$ $c\to u\,\nu\,\nu$ |  LFV $\tau$ decay, $D$ decay to dineutrinos
$[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}ij}$ |  $[{{C}}_{\nu uLL}^{V}]^{{\ell}{\ell}^{\prime}12}$ |  ${\ell}\,d\to{\ell}^{\prime}\,d$ $c\to u\,\nu\,\nu$ |  LFV ${\ell}\,N\to{\ell}^{\prime}\,N$ $D$ decay to dineutrinos
$[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}i3}$ |  $[{{C}}_{\nu uLL}^{V}]^{{\ell}{\ell}^{\prime}13}$ $[{{C}}_{\nu uLL}^{V}]^{{\ell}{\ell}^{\prime}23}$ |  $b\to d_{i}\,{\ell}\,{\ell}^{\prime}$ $t\to u\,\nu\,\nu$ $t\to c\,\nu\,\nu$ |  LFV B decay, top decays to dineutrinos
(45) | $[{{C}}_{LL}^{V}]^{{\ell}311}$ |  $[{{C}}_{edLL}^{V}]^{{\ell}311}$ $[{{C}}_{edLL}^{V}]^{{\ell}312}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}311}$ |  $\tau\to u\,d\,\nu$ $\tau\to d\,d\,{\ell}$ $\tau\to d\,s\,{\ell}$ $s\to d\nu\,\nu$ |  CC decay of $\tau$ LFV $\tau$ decay, $K\to\pi\,\nu\,\nu$
$[{{C}}_{LL}^{V}]^{{\ell}{\ell}^{\prime}11}$ |  $[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}11}$ $[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}12}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}11}$ |  ${\ell}\to u\,d\,\nu$ ${\ell}\,d\to{\ell}^{\prime}\,d$ $s\to d\,{\ell}\,{\ell}^{\prime}$ $s\to d\nu\,\nu$ |  LFV ${\ell}\,N\to{\ell}^{\prime}\,N$ $K\to\pi\,{\ell}\,{\ell}^{\prime}$ $K\to\pi\,\nu\,\nu$
$[{{C}}_{LL}^{V}]^{{\ell}{\ell}^{\prime}i3}$ |  $[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}13}$ $[{{C}}_{edLL}^{V}]^{{\ell}{\ell}^{\prime}23}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}13}$ $[{{C}}_{\nu dLL}^{V}]^{{\ell}{\ell}^{\prime}23}$ |  $b\to u_{i}\,{\ell}\,\nu$ $b\to d_{i}\,{\ell}\,{\ell}^{\prime}$ $b\to d_{i}\,\nu\,\nu$ |  CC decay of $B$ meson, LFV $B$ decays, $B$ decays to dineutrinos Bause:2021ihn
Table 5: Correlations among different WCs involving all lepton families,
derived from eqs. (43-45). The second column shows the WC appearing on the
left hand side of these equations, whereas the third column contains the WCs
appearing on the right-hand side of those equations with large CKM
coefficients, with values $\mathcal{O}(\lambda)$ or more.

From eq. (43), we get the following relation among the LEFT WCs:

$\displaystyle[{{C}}_{euLL}^{V}]^{{\ell}311}$
$\displaystyle=|V_{ud}|^{2}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}311}+(V_{ud}^{*}\,V_{cd}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}312}+\mbox{c.c.})+(V_{ud}^{*}\,V_{td}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}313}+\mbox{c.c.})$
$\displaystyle~{}+|V_{cd}|^{2}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}322}+(V_{cd}^{*}\,V_{td}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}323}+\mbox{c.c.})+|V_{td}|^{2}\,[{{C}}_{\nu
dLL}^{V}]^{{\ell}333}~{},$ (50)

where $\ell=1~{}{\rm or}~{}2$. Among the CKM coefficients in this equation,
the leading ones are $|V_{ud}|^{2}\sim\mathcal{O}(1)$ and
$|V_{ud}^{*}V_{cd}|\sim\mathcal{O}(\lambda)$. All the other coefficients are
$\mathcal{O}(\lambda^{2})$ or smaller. Therefore, at the leading order, this
equation connects the three WCs $[{{C}}_{euLL}^{V}]^{{\ell}311}$, $[{{C}}_{\nu
dLL}^{V}]^{{\ell}311}$ and $[{{C}}_{\nu dLL}^{V}]^{{\ell}312}$. Hence the new
physics WCs contributing to LFV tau decays and $K\to\pi\,\nu\,\nu$ are related
to each other.

Further relations involving other lepton and quark families are given in Table
5. Some similar relations have been presented in Fuentes-Martin:2020lea . Note
that such discussions in earlier literature often assume some flavor structure
for the quark sector. We emphasize again that in our discussion, the
implications presented in this section are independent of any NP flavor
structure assumption for the quarks.

So far, we have discussed observables that are insensitive to the flavor of
neutrinos. Neutrino experiments that are sensitive to neutrino flavor can
probe the neutrino non-standard interactions (NSI) generated by the operators
in Table 1 containing neutrinos. The predictions in eq. (43)-eq. (45) in UV4f
models (or the more general predictions in Table 3) would then imply
constraints on NSI from charged LFV. We discuss this in more detail in Sec.
4.3.

##  4 SMEFT-predicted evidence for new physics

In this section, we discuss how to use the SMEFT predictions derived in Sec. 2
in the event that measurements provide evidence for certain new physics WCs to
be nonzero. We will show that, given the SMEFT predictions derived in this
work, it is in general not consistent to assume a single non-zero WC to
explain an excess in a certain channel.131313For some operators, such as the
$RRRR$ vector operators, there are no constraints implied by SMEFT. For this
category of operators, therefore, we can have a single non-zero WC. In fact,
we will show that for certain operators a non-zero WC must be accompanied by
multiple other WCs that are non-vanishing. This would imply that the observed
excess must be accompanied by correlated excesses in many other channels. This
is because the SMEFT predictions in Table 5 are linear equations involving
multiple WCs, implying that it is not possible for only one of these
coefficients to be nonzero.

For example, consider the situation where an observed deviation from SM in a
particular channel indicates that one of the $LLLL$ LEFT WCs is non-vanishing.
In SMEFT, this LEFT coefficient might arise either from a four-fermion
operator or an operator inducing an off-diagonal $W$ or $Z$ coupling to
fermions.

The former situation is realized in the UV4f models, where we can use SMEFT
predictions in eq. (43). These are 6 linear equations involving 12 (possibly)
complex WCs when $\alpha=\beta$. If one of these WCs, (say $C_{1}$) is found
to be nonzero, we can write these equations in a form where $C_{7}$ to
$C_{12}$ are expressed as linear combinations of $C_{1}$ to $C_{6}$. Then, as
long as the coefficient of $C_{1}$ is nonzero in all these equations (as is
generically observed to be the case), all the 6 coefficients $C_{7}$ to
$C_{12}$ also have to be nonzero. For one of them to be vanishing, we will
need one of the other coefficients, $C_{2}$ to $C_{6}$, to be nonzero in order
to cancel the $C_{1}$ contribution. Thus, the nonvanishing nature of $C_{1}$
necessarily implies that overall at least 7 WCs are nonvanishing in principle.
Of course, depending on the CKM coefficients, the magnitudes of these
coefficients may be small or large.

When $\alpha\neq\beta$, eq. (43) gives 9 linear equations, therefore one
nonzero WC among these will imply at least 10 of the WCs of the type ($\nu d$)
or ($eu$) nonvanishing. As eq. (44) is completely decoupled from eq. (43), it
is of course still consistent for all the WCs appearing in it to vanish. The
charged current WCs in eq. (45), however, cannot all vanish and one can use
similar arguments to conclude that at least 3 of them must be nonzero whether
or not $\alpha$ equals $\beta$.

Similarly, from eq (44), for $\alpha=\beta$ $(\alpha\neq\beta)$ we have 6 (9)
linear relations. These imply that, if one of the WCs of the kind ($ed$) or
($\nu u$) is found to be nonzero, then a total of at least 7 (10) WCs of these
kinds should be nonzero in principle. Again, by eq. (45) a non-zero neutral-
current WC will lead to at least 3 non-vanishing charged-current WCs. The CKM
coefficients will guide us regarding which of these WCs are likely to have
larger magnitudes. Thus, these relations direct us toward specific decay
channels where deviation from SM is expected to be present.

In the latter situation, i.e. when the LEFT operators arise from modifications
of $W/Z$ couplings, the low-energy pattern of deviations is very different.
For example, if one of the $Z$ coupling to down quarks gets BSM corrections,
the penultimate row of Table 3 would imply at least three $W$-coupling
modifications. Alternatively, if all the $W$ couplings are to be at their SM
value, this would imply modifications of at least 10 of the 18 $Z$ couplings
to up and down-type quarks. Once the $W$ and $Z$ are integrated out, each
$W$-coupling modification will induce 3 non-vanishing semileptonic LEFT WCs,
and each $Z$-coupling modification will induce 6 non-vanishing semileptonic
LEFT WCs. Studying the pattern of BSM deviations can, therefore, help pinpoint
the underlying UV physics. We shall not consider this scenario further in this
section.

###  4.1 Implications of the measured excess in $B\to K\nu\nu$

In the recent measurement of $B\to K\nu\nu$ at Belle II Belle-II:2023esi , the
observed branching ratio has $3.5\sigma$ excess over the SM value. If this
excess were to be explained in terms of the LEFT coefficients $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23}$, the required values of these WCs in various
scenarios are shown in Fig. 8. In the first scenario, we assume that new
physics turns on a lepton flavor universal (LFU) combination of WCs whereas in
the second (i.e. LFUV) and third (i.e. LFV) scenarios, we assume that a single
WC is turned on with $\alpha=\beta$ and $\alpha\neq\beta$, respectively.

Figure 8: Preferred parameter region at $90\%$ C.L. for $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23}$ in order to explain the observed excess in $B\to
K\nu\nu$ branching ratio. The left panel shows lepton flavor universal (LFU)
scenario, where $[{{C}}_{\nu dLL}^{V}]^{\alpha\beta 23}$ is nonzero and equal
for all $\alpha=\beta\in\\{e\,,\mu\,,\tau\\}$. The middle panel shows lepton
flavor nonuniversal (LFUV) scenario where $[{{C}}_{\nu dLL}^{V}]^{\alpha\beta
23}$ is nonzero only for one value of $\alpha=\beta$. The right panel depicts
the LFV scenario with $\alpha\neq\beta$ and only one $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23}$ nonzero.

From this figure, it is clear that the coefficient $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23}$ is non-vanishing at $90\%$ C.L. for all scenarios
considered. As discussed earlier, a nonzero $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23}$ will indicate at least seven (ten) non-vanishing
WCs appearing in eq. (43) for $\alpha=\beta$ $(\alpha\neq\beta)$.

For example, in the LFUV (LFV) scenarios, eq. (43) corresponds to 27 (54)
equations of the form

$\displaystyle[{{C}}_{euLL}^{V}]^{\alpha\beta ij}$
$\displaystyle=V_{i2}\,[{{C}}_{\nu dLL}^{V}]^{\alpha\beta
23}V^{\dagger}_{3j}+...$ (51)

in UV4f models. Since the CKM coefficients $V_{cs}V_{tb}^{*}$ and
$V_{us}V_{tb}^{*}$, which are $\mathcal{O}(1)$ and $\mathcal{O}(\lambda)$
respectively, are significant, it is expected that in the absence of any
cancellation coming from other $[{{C}}_{\nu dLL}^{V}]^{\alpha\beta ij}$
elements, the WCs $[{{C}}_{euLL}^{V}]^{\alpha\beta 13}$ and
$[{{C}}_{euLL}^{V}]^{\alpha\beta 23}$ will have significant nonzero values.
Thus the modes $t\to c\,e^{\alpha}\,e^{\beta}$ and $t\to
ue^{\alpha}\,e^{\beta}$ will be the ones where there can be potential new
physics. Currently the bounds on these coefficients are
$|[{{C}}_{euLL}^{V}]^{\alpha\beta 13}|<0.003$ and
$|[{{C}}_{euLL}^{V}]^{\alpha\beta 23}|<0.02$, respectively. Exploration of
these modes further may lead to discovery of further anomalies in these two
channels. These processes will also test the solution of $B\to K\nu\nu$
anomaly in terms of $[{{C}}_{\nu dLL}^{V}]^{\alpha\beta 23}$. This
demonstrates that the semileptonic neutral-current top decays will be strong
probes of the origin of the $B\to K\nu\nu$ anomaly in the context of SMEFT.

Eq. (45), in this LFUV (LFV) scenario, gives the 9 (18) equations of the form

$\displaystyle[{{C}}_{LL}^{V}]^{\alpha\beta i3}$
$\displaystyle=V_{i2}\,([{{C}}_{edLL}^{V}]^{\alpha\beta 23}-[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta 23})~{}.$ (52)

Since the CKM coefficients $V_{cs}$ and $V_{us}$, which are $\mathcal{O}(1)$
and $\mathcal{O}(\lambda)$, respectively, are significant, it is expected that
in the absence of any cancellation coming from other $[{{C}}_{\nu
dLL}^{V}]^{\alpha\beta ij}$ or $[{{C}}_{edLL}^{V}]^{\alpha\beta ij}$ elements,
the WCs $[{{C}}_{LL}^{V}]^{\alpha\beta 23}$ and $[{{C}}_{LL}^{V}]^{\alpha\beta
13}$ will have significant nonzero values. Thus, charged-current semileptonic
$B$ meson decays would also be sensitive probes of the origin of $B\to
K\nu\nu$ anomaly.

Similar discussions can be found in Bause:2023mfe ; Chen:2024jlj . In
Bause:2023mfe , relations among the WCs in the $LLRR$ category, as shown in
Table 3, have been used to relate $b_{R}\to s_{R}\tau\tau$ and $b_{R}\to
s_{R}\nu\nu$. These relations, as discussed in Bause:2023mfe , predict excess
branching fractions for the modes $B\to K^{(*)}\tau\tau$, $B_{s}\to\tau\tau$,
etc. In Chen:2024jlj , matching relations have been derived among the SMEFT
and LEFT WCs, assuming up-alignment. These have been then used to obtain the
effects of the observed excess in $B\to K\nu\nu$ on other processes, namely,
$B\to D^{(*)}\ell\nu_{\ell}$, $B\to K^{(*)}\ell^{+}\ell^{-}$,
$B_{q}\to\tau\nu_{\tau}$, $B_{s}\to\tau^{+}\tau^{-}$,
$D_{s}\to\tau^{+}\nu_{\tau}$, etc.

###  4.2 Implications of the $R(D^{(*)})$ anomalies

One possible explanation of multiple anomalies observed in the $b\to
c\tau\bar{\nu}$ channels, such as $R(D)$, $R(D^{*})$ and $R(J/\psi)$, is to
have nonzero values for the LEFT WC $[{{C}}_{LL}^{V}]^{3323}$. We show the
preferred range of this WC at $90\%$ C.L. in Fig. 9. Note that this preferred
range does not include the point $[{{C}}_{LL}^{V}]^{3323}=0$.

Figure 9: Preferred parameter region at $90\%$ C.L. for the WC
$[{{C}}_{LL}^{V}]^{3323}$.

From eq. (45), we can write $[{{C}}_{LL}^{V}]^{3323}$ in terms of the neutral-
current WCs as

$\displaystyle[{{C}}_{LL}^{V}]^{3323}$
$\displaystyle=V_{cd}\left[[\hat{{C}}_{edLL}^{V}]^{3313}-[{{C}}_{\nu
dLL}^{V}]^{3313}\right]+V_{cs}\left[[\hat{{C}}_{edLL}^{V}]^{3323}-[{{C}}_{\nu
dLL}^{V}]^{3323}\right]$
$\displaystyle~{}+V_{cb}\left[[\hat{{C}}_{edLL}^{V}]^{3333}-[{{C}}_{\nu
dLL}^{V}]^{3333}\right]~{}.$ (53)

Since $[{{C}}_{LL}^{V}]^{3323}\neq 0$, it suggests that at least one WC
appearing on the right-hand side of eq. (53) has to be nonzero. Relevant
interesting modes could be of the type $b\to d\tau\tau$, $b\to s\tau\tau$,
$b\to d\nu\nu$ and $b\to s\nu\nu$ which suggests that the NP can manifest in
observables related to processes such as $B\to\tau\tau$, $B_{s}\to\tau\tau$,
$B\to K^{(*)}\tau\tau$, $B\to K^{(*)}\nu\nu$, etc.

###  4.3 Implications of the violation of SMEFT predictions

In this subsection, we consider a scenario where many anomalies have been
observed and multiple LEFT coefficients must have non-zero values to explain
them. According to our results, these LEFT WCs must obey the SMEFT predictions
of Table 5. We now discuss what an observation of a violation of these
predictions would imply.

First of all, if low-energy measurements indicate a violation of the UV4f
predictions in eq. (43–45), it may only mean that the UV model is not in the
UV4f category, but still maps to SMEFT when heavier degrees of freedom are
integrated out. It would only indicate that we are outside the UV4f region of
Fig. 1, and not necessarily outside the SMEFT region. We must then check
whether or not the more general predictions Table 3 are obeyed. This would
require looking for deviations in $W$ and $Z$ decays and/or high-$p_{T}$
Drell-Yan data.141414If we use only the high-$p_{T}$ data to test our
predictions, we can directly use Table 3 and thus test the validity of our
predictions without taking into account $Z$ and $W^{\pm}$ decays.

If the violation of the predictions persists at the level of Table 3 (or the
equations in Appendix. C), it would imply that one of the assumptions used in
deriving these predictions is incorrect. Note, first of all, that we have only
included dim-6 operators in our analysis. Inclusion of dimension-8 (dim-8)
operators will result in a violation of these predictions at ${\cal
O}(v^{4}/\Lambda^{4})$. For instance, the dim-8 operator

$\left[{\cal O}_{{\ell}q3}\right]^{\alpha\beta
ij}=(\bar{l}^{\alpha}\gamma_{\mu}l^{\beta})(\bar{q}^{i}\gamma^{\mu}\tau^{I}q^{j})(H^{\dagger}\tau^{I}H)$
(54)

will break the equality in the first row of Table 3, as follows:

$V^{\dagger}_{ik}\,[\hat{{\mathbf{c}}}_{euLL}^{V}]^{\alpha\beta
kl}\,V_{{\ell}j}-U^{\dagger}_{\alpha\rho}\,[\hat{{\mathbf{c}}}_{\nu
dLL}^{V}]^{\rho\sigma ij}\,U_{\sigma\beta}\sim v^{4}/\Lambda^{4}\left[{\cal
C}_{{\ell}q3}\right]~{}.$ (55)

Similarly, other operators at dim-8 or higher order will introduce a breaking
of the other predictions in Table 3. Such effects are, however, higher order
in the SMEFT expansion parameter $v^{2}/\Lambda^{2}$, and are thus expected to
be small.

If larger, ${\cal O}(1)$ violations of the predictions are observed, it would
indicate something more radical, namely, that one of the assumptions of SMEFT
itself is violated and we lie outside the SMEFT region of Fig. 1. This would
be the case if (i) the scale of new physics is below the weak scale, (ii)
there is heavy new physics that does not decouple because it gets a large
fraction of its mass from the electroweak vacuum expectation value
Banta:2021dek ; Cohen:2020xca , or (iii) the observed 125 GeV scalar, $h$, is
not a part of the SU(2) doublet that breaks the electroweak symmetry
Falkowski:2019tft ; Cohen:2020xca ; Banta:2021dek ; Cata:2015lta .

As an example, consider the case of neutrino NSI that are induced by operators
containing neutrinos in Table 1. As mentioned in Sec. 3.4, for a given choice
of the quark flavor indices, eqs. (43–45) (or the equations in Table 3), imply
relations between the NSI and the stringently constrained lepton flavor
violating operators Proceedings:2019qno . These predictions can, however, be
evaded in new physics scenarios where dim-8 operators become important. For
instance, if the leading contribution to the NSI is from dim-8 (and not dim-6)
operators like

$\left[{\cal O}_{l3q}\right]^{\alpha\beta
ij}=(\tilde{H}^{\dagger}\tau^{I}\tilde{H})(\bar{l}^{\alpha}\gamma_{\mu}\tau^{I}l^{\beta})(\bar{q}^{i}\gamma^{\mu}q^{j})~{},$
(56)

new physics affects only the neutrino and not the charged-lepton sector. Even
in this case, however, dim-6 charged-lepton flavor-violating effects will be
generated at loop level Ardu:2022pzk . A more natural way of decoupling these
two sectors is if the new physics scale is below the electroweak scale (see,
e.g. Ref. Farzan:2019xor ).

##  5 Concluding remarks

In this work, we have systematically derived the consequences of the
$SU(2)_{L}\times U(1)_{Y}$ invariance of SMEFT on semileptonic flavor
observables. These consequences arise from the fact that a complete
parametrization of BSM deviations in flavor physics observables can be only
achieved by writing a lagrangian that respects $U(1)_{em}$ and not the full
symmetry of SMEFT. For instance, while the left-handed up and down type
fermions form $SU(2)_{L}$ doublets and always appear together in SMEFT
operators, as far as flavor observables are concerned, searches in the up and
down sectors are completely independent. Therefore, BSM deviations in these
channels must be parameterized by independent operators.

To be more precise, while the most general $U(1)_{em}$ invariant lagrangian
has 3240 independent semileptonic four-fermion operators (see Table 1) and
another set of 144 operators that contribute to semileptonic processes via
$Z,W^{\pm}$ and $h$ exchange (see Table 4), the number of dim-6 SMEFT
operators in these categories are 1053 (see Table 2) and 108 (see Table 4),
respectively. This then results in 2223 constraints in the space of WCs of the
$U(1)_{em}$ invariant lagrangian that can be thought of as predictions of
SMEFT at the dim-6 level. One of the main results of this work is the
derivation of these 2223 constraints. We present these constraints as linear
relations among the WCs of the $U(1)_{em}$ invariant lagrangian, in Table 3.
These relations are a succinct expression of the consequences of the
$SU(2)_{L}\times U(1)_{Y}$ invariance of SMEFT for semileptonic operators.
They are completely independent of UV flavor assumptions as we find that the
elements of the rotation matrices of the left-handed and right-handed up-type
and down-type fermions do not individually appear in them but only in
combinations that form CKM and PMNS elements. We then show how these relations
can be written in terms of LEFT WCs by integrating out the $Z,W^{\pm}$ and $h$
bosons. We refer the reader to Fig. 1 where this scenario has been pictorially
represented.

The $U(1)_{em}$ invariant lagrangian we have considered is in fact equivalent,
in the unitary gauge, to the HEFT lagrangian which is generally written in an
$SU(2)_{L}\times U(1)_{Y}$ invariant form but with the gauge symmetry being
realized non-linearly. We show this explicitly in Appendix A where we present
a one-to-one mapping between the invariant HEFT operators and the list of
$U(1)_{em}$ invariant operators in Table 1 and Table 4. In the process, we
find some HEFT operators that were missed in earlier literature and others
that were considered but are actually redundant.

The SMEFT predictions we have derived have powerful phenomenological
consequences as they connect observables in different sectors, such as rare
decays in the kaon, B-Meson and charm sectors; decays of the top, $Z,W^{\pm}$
and $\tau$; lepton flavor violating observables and even neutrino NSI. On the
one hand, they can be used to express poorly constrained WCs in terms of
strongly constrained ones, thus allowing us to put new stronger indirect
bounds on the former. On the other hand, if evidence for new physics is seen,
they in general imply that BSM effects cannot appear in a single isolated
channel because these linear relations imply that if one WC is non-zero,
multiple others also must be non-vanishing.

To illustrate the usefulness of these relations in phenomenology, we focus on
the well-motivated UV4f scenario, where the UV physics only involves four-
fermion operators, and HEFT WCs corresponding to BSM couplings of the $Z$,
$W^{\pm}$ and Higgs to fermions are absent. We further restrict ourselves to
the operators with only left-handed fermions, i.e the $LLLL$ class of
operators. In this scenario, there are three sets of relations among the LEFT
WCs. The first set relates the WCs of the neutral-current operators
$(\bar{\nu}_{L}\gamma_{\mu}\nu_{L})\,(\bar{d}_{L}\gamma^{\mu}d_{L})$ and
$(\bar{e}_{L}\gamma_{\mu}e_{L})\,(\bar{u}_{L}\gamma^{\mu}u_{L})$. The second
set consists of relations among the WCs of the neutral-current operators
$(\bar{e}_{L}\gamma_{\mu}e_{L})\,(\bar{d}_{L}\gamma^{\mu}d_{L})$ and
$(\bar{\nu}_{L}\gamma_{\mu}\nu_{L})\,(\bar{u}_{L}\gamma^{\mu}u_{L})$. In the
third set, the charged-current WCs are related to the above neutral-current
coefficients.

The main phenomenological results of this work are as follows:

  1. 1.

Indirect bounds from SMEFT predictions:  In Sec. 3.1\- 3.3, we consider $LLLL$
operators in UV4f models. Using bounds from meson decays and high-$p_{T}$
Drell-Yan searches and applying the SMEFT predictions, we obtain novel bounds
on WCs related to $d\bar{d}\to\nu\bar{\nu}$, $u_{i}\to u_{j}\nu\bar{\nu}$ and
top decays, that are much stronger than the direct bounds. Our main results
are summarised in Fig. 3, 5 and 7 .

  2. 2.

Connecting quark and lepton flavor violation: In Sec. 3.4, we show how the
SMEFT predictions derived by us connect flavor violation in the quark and
lepton sectors. In Table 5, we present a list of processes spanning diverse
observation channels (e.g. LFV tau decays, LFV ${\ell}N\to{\ell}^{\prime}N$
transitions, rare semileptonic $B$, $D$ and $K$ decays, top production and
decays, etc.) that are connected via our analytic relations among the WCs.

  3. 3.

Evidence for new physics from SMEFT predictions: In Sec. 4, we show that the
relations among the WCs of the type $LLLL$ imply that a single nonzero WC
requires that there are at least 9 other nonzero WCs. We then discuss the
specific cases of the observed excess in the $B\to K\nu\nu$ branching fraction
and $R(D^{(*)})$ anomalies, and list other search channels that should see a
correlated signal if these anomalies survive in the future.

In future studies, we aim to extend the approach developed here and apply it
to other flavor physics observables. In this work, we have considered only a
subset of operators appearing in LEFT, HEFT and SMEFT, namely the set of
semileptonic operators. In future work, we will extend our analysis by
including all operators up to dimension-6 in order to find SMEFT-predicted
relations among the corresponding LEFT and HEFT WCs. These predictions will
allow us to interconnect many other important flavor observables. For
instance, predictions can be obtained for dipole operators connected to
observables such as the $b\to s\gamma$ process, for four quark operators that
are associated to the $\Delta F=2$ meson-mixing processes and nonleptonic
meson decays, for four-lepton operators associated to LFV processes such as
$\mu\to 3e$, etc. We, thus, hope that this work will initiate a rich program
in quark and lepton flavor phenomenology that uncovers many more SMEFT-
predicted links between observables.

###### Acknowledgements.

We would like to thank Tuhin S. Roy, Ketan M. Patel, Abhishek M. Iyer, Arnab
Roy, Samadrita Mukherjee, Dibya S. Chattopadhyay and Radhika Vinze for useful
discussions. This work is supported by the Department of Atomic Energy,
Government of India, under Project Identification Number RTI 4002. We
acknowledge the use of computational facilities of the Department of
Theoretical Physics at Tata Institute of Fundamental Research, Mumbai. We
would also like to thank Ajay Salve and Kapil Ghadiali for technical
assistance.

##  Appendix A Semileptonic HEFT operators in $SU(2)_{L}\times U(1)_{Y}$
invariant form

In Table 1 and Table 4, we have presented all possible $U(1)_{em}$-invariant
semileptonic operators relevant to this work. In this Appendix, we show that
these operators can be rewritten as $SU(2)_{L}\times U(1)_{Y}$ invariant
operators of HEFT with the symmetry realized non-linearly. Following the
notation and approach used in Ref. Buchalla:2012qq , we introduce the
Goldstone matrix $U=\exp(2i\varphi^{a}T^{a}/v)$, where $\varphi_{a}$ are the
Goldstones of the breaking of $SU(2)_{L}\times SU(2)_{R}\to SU(2)_{V}$. Under
$SU(2)_{L}\times SU(2)_{R}$, the matrix $U$ transforms as $U\to
g_{L}Ug_{R}^{\dagger}$, where $g_{L}$ and $g_{R}$ are the respective group
elements. We also introduce the $SU(2)_{R}$ quark and lepton doublets denoted
by $r\equiv(u_{R},d_{R})^{T}$ and $\eta\equiv(0,e_{R})^{T}$, respectively.

As the correct symmetry-breaking pattern in SM is $SU(2)_{L}\times U(1)_{Y}\to
U(1)_{em}$, and not $SU(2)_{L}\times SU(2)_{R}\to SU(2)_{V}$, one must include
explicit sources of $SU(2)_{R}$ breaking (see for eg. Ref. murayama ). For
bosonic operators, this is usually done by introducing the two spurions
$L_{\mu}=UD_{\mu}U^{\dagger}$ and $\tau_{L}=UT_{3}U^{\dagger}$. For fermionic
operators, we need other sources of $SU(2)_{R}$ breaking. As shown in Ref.
Buchalla:2012qq , this can be achieved by including factors of $UP_{i}$ in the
operators where the projection matrices $P_{i}$ are defined as

$\displaystyle P_{\pm}\equiv\frac{1}{2}\pm T_{3},\quad P_{12}\equiv
T_{1}+iT_{2},\quad P_{21}\equiv T_{1}-iT_{2}~{}.$ (57)

In the above equation, $T_{i}$ are the $SU(2)_{L}$ generators. One can keep
track of the hypercharge invariance of the operators by keeping in mind that,
while $UP_{+}$ and $UP_{12}$ extract the $Y=-1$ components of $U$, the
projections $UP_{-}$ and $UP_{21}$ extract the $Y=1$ components of $U$.

$LLLL$ | $LLRR$
---|---
${{\mathbf{o}}}_{LL3}=({\bar{l}\gamma_{\mu}l})\,({\bar{q}\gamma^{\mu}q})$ | ${{\mathbf{o}}}_{LR5}=({\bar{l}\gamma_{\mu}l})\,({\bar{u}\gamma^{\mu}u})$
${{\mathbf{o}}}_{LL4}=({\bar{l}\gamma_{\mu}\tau^{a}l})\,({\bar{q}\gamma^{\mu}\tau^{a}q})$ | ${{\mathbf{o}}}_{LR6}=({\bar{l}\gamma_{\mu}l})\,({\bar{d}\gamma^{\mu}d})$
${{\mathbf{o}}}_{LL10}=({\bar{l}\gamma_{\mu}U\tau^{3}U^{\dagger}l})\,({\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}q})$ | ${{\mathbf{o}}}_{FY11}=({\bar{\ell}UP_{-}r})\,({\bar{r}P_{+}U^{\dagger}l})$
${{\mathbf{o}}}_{LL11}=({\bar{l}\gamma_{\mu}l})\,({\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}q})$ | ${{\mathbf{o}}}_{LR14}=({\bar{l}\gamma_{\mu}U\tau^{3}U^{\dagger}l})\,({\bar{u}\gamma^{\mu}u})$
${{\mathbf{o}}}_{LL12}=({\bar{l}\gamma_{\mu}U\tau^{3}U^{\dagger}l})\,({\bar{q}\gamma^{\mu}q})$ | ${{\mathbf{o}}}_{LR15}=({\bar{l}\gamma_{\mu}U\tau^{3}U^{\dagger}l})\,({\bar{d}\gamma^{\mu}d})$
${{\mathbf{o}}}_{LL14}=({\bar{l}\gamma_{\mu}q})\,({\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}l})$ |
$RRLL$ | $RRRR$
${{\mathbf{o}}}_{LR7}=({\bar{e}\gamma_{\mu}e})\,({\bar{q}\gamma^{\mu}q})$ | ${{\mathbf{o}}}_{RR5}=({\bar{e}\gamma_{\mu}e})\,({\bar{u}\gamma^{\mu}u})$
${{\mathbf{o}}}_{LR16}=({\bar{e}\gamma_{\mu}e})\,({\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}q})$ | ${{\mathbf{o}}}_{RR6}=({\bar{e}\gamma_{\mu}e})\,({\bar{d}\gamma^{\mu}d})$
Scalar with $d_{R}$ | Tensor with $d_{R}$
${{\mathbf{o}}}_{FY7}=({\bar{q}UP_{-}r})\,({\bar{\ell}UP_{-}\eta})$ | ${{\mathbf{o}}}_{FY8}=({\bar{q}\sigma^{\mu\nu}UP_{-}r})\,({\bar{l}\sigma_{\mu\nu}UP_{-}\eta})$
${{\mathbf{o}}}_{LR9}=({\bar{q}\gamma^{\mu}l})\,({\bar{e}\gamma_{\mu}d})$ | $\bullet$ ${{\mathbf{o}}}_{ST13}=({\bar{r}P_{-}\sigma^{\mu\nu}Uq})\,({\bar{l}\sigma_{\mu\nu}UP_{-}\eta})$
${{\mathbf{o}}}_{LR18}=({\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}l})\,({\bar{e}\gamma_{\mu}d})$ | $\bullet$ ${{\mathbf{o}}}_{ST14}=({\bar{q}\sigma^{\mu\nu}UP_{12}r})\,({\bar{\eta}\sigma_{\mu\nu}UP_{21}l})$
Scalar with $u_{R}$ | Tensor with $u_{R}$
${{\mathbf{o}}}_{ST9}=({\bar{q}UP_{+}r})\,({\bar{\ell}UP_{-}\eta})$ | ${{\mathbf{o}}}_{ST11}=({\bar{q}\sigma^{\mu\nu}UP_{+}r})\,({\bar{l}\sigma_{\mu\nu}UP_{-}\eta})$
${{\mathbf{o}}}_{FY9}=({\bar{\ell}UP_{-}\eta})\,({\bar{r}P_{+}U^{\dagger}q})$ | $\bullet$ ${{\mathbf{o}}}_{ST16}=({\bar{r}P_{+}\sigma^{\mu\nu}Uq})\,({\bar{l}\sigma_{\mu\nu}UP_{-}\eta})$
${{\mathbf{o}}}_{ST10}=({\bar{q}UP_{21}r})\,({\bar{\ell}UP_{12}\eta})$ | ${{\mathbf{o}}}_{ST12}=({\bar{q}\sigma^{\mu\nu}UP_{21}r})\,({\bar{l}\sigma_{\mu\nu}UP_{12}\eta})$
Table 6:  List of semileptonic $SU(2)_{L}\times U(1)_{Y}$ invariant HEFT
operators. Note that this list is somewhat different from the list presented
in Buchalla:2012qq  (see the text for more details). Some redundant operators
present in Buchalla:2012qq  are omitted from this list. On the other hand,
some operators (preceded by a bullet) which were absent in Buchalla:2012qq
have been added and have been named using a similar nomenclature. Note that
$\tau^{a}=T^{a}/2$ are the Pauli matrices.

We first consider four-fermion operators. In Table 6, we present all possible
$SU(2)_{L}\times U(1)_{Y}$ invariant HEFT operators with two quarks and two
leptons, up to dimension 6. Note that this list has some differences from the
list of operators presented in Ref. Buchalla:2012qq  that we will discuss in
detail in the following. Working in the unitary gauge, i.e. taking $U\to 1$,
we now write each of the operators in Table 6 in terms of the operators in
Table 1. This would confirm that there is a one-to-one mapping between these
two sets of operators in the unitary gauge.

For $LLLL$ vector operators:

$\displaystyle{{\mathbf{o}}}_{LL3}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}+{{\textbf{o}}}_{euLL}^{V}+{{\textbf{o}}}_{\nu
dLL}^{V}+{{\textbf{o}}}_{edLL}^{V}~{},$ (58)
$\displaystyle{{\mathbf{o}}}_{LL4}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}-{{\textbf{o}}}_{euLL}^{V}-{{\textbf{o}}}_{\nu
dLL}^{V}+{{\textbf{o}}}_{edLL}^{V}+2\,{{\textbf{o}}}_{LL}^{V}+2\,{{\textbf{o}}}_{LL}^{\prime
V}~{},$ (59) $\displaystyle{{\mathbf{o}}}_{LL10}$
$\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}-{{\textbf{o}}}_{euLL}^{V}-{{\textbf{o}}}_{\nu
dLL}^{V}+{{\textbf{o}}}_{edLL}^{V}~{},$ (60)
$\displaystyle{{\mathbf{o}}}_{LL11}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}+{{\textbf{o}}}_{euLL}^{V}-{{\textbf{o}}}_{\nu
dLL}^{V}-{{\textbf{o}}}_{edLL}^{V}~{},$ (61)
$\displaystyle{{\mathbf{o}}}_{LL12}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}-{{\textbf{o}}}_{euLL}^{V}+{{\textbf{o}}}_{\nu
dLL}^{V}-{{\textbf{o}}}_{edLL}^{V}~{},$ (62)
$\displaystyle{{\mathbf{o}}}_{LL14}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}+{{\textbf{o}}}_{LL}^{V}-{{\textbf{o}}}_{LL}^{\prime
V}-{{\textbf{o}}}_{edLL}^{V}~{},$ (63)

where we have suppressed the quark and lepton flavor indices. Here
$[{{\textbf{o}}}_{LL}^{\prime V}]^{\alpha\beta
ij}=([{{\textbf{o}}}_{LL}^{V}]^{\beta\alpha ji})^{\dagger}$ and
$[{{\textbf{o}}}_{LL}^{V}]^{\alpha\beta ij}$ are two independent operators.
The 6 operators listed in Table 6, therefore, receive contributions from 6
independent operators of this category in Table 1, providing a one-to-one
mapping between these two lists. In this category, there is one more operator
in Buchalla:2012qq  i.e
${{\mathbf{o}}}_{LL13}=(\bar{q}\gamma^{\mu}U\tau^{3}U^{\dagger}l)(\bar{l}\gamma_{\mu}U\tau^{3}U^{\dagger}q)$.
But this operator is not independent of the 6 operators appearing on the
‘$LLLL$’ block of Table 6. Indeed, it can be written as

$\displaystyle{{\mathbf{o}}}_{LL13}$ $\displaystyle={{\textbf{o}}}_{\nu
uLL}^{V}-{{\textbf{o}}}_{LL}^{V}-({{\textbf{o}}}_{LL}^{\prime
V})+{{\textbf{o}}}_{edLL}^{V}~{},$ (64)

which is equivalent to the relation,

$\displaystyle{{\mathbf{o}}}_{LL13}$
$\displaystyle=\frac{1}{2}({{\mathbf{o}}}_{LL3}+2\,{{\mathbf{o}}}_{LL10}-{{\mathbf{o}}}_{LL4})~{}.$
(65)

This operator has therefore been omitted in our list.

For $LLRR$ vector operators:

$\displaystyle{{\mathbf{o}}}_{LR5}$ $\displaystyle={{\textbf{o}}}_{\nu
uLR}^{V}+{{\textbf{o}}}_{euLR}^{V}~{},$ $\displaystyle{{\mathbf{o}}}_{LR6}$
$\displaystyle={{\textbf{o}}}_{\nu dLR}^{V}+{{\textbf{o}}}_{edLR}^{V}~{},$
(66) $\displaystyle{{\mathbf{o}}}_{FY{11}}$
$\displaystyle=-\frac{1}{2}\,{{\textbf{o}}}_{LR}^{V}~{},$
$\displaystyle{{\mathbf{o}}}_{LR14}$ $\displaystyle={{\textbf{o}}}_{\nu
uLR}^{V}-{{\textbf{o}}}_{euLR}^{V}~{},$ (67)
$\displaystyle{{\mathbf{o}}}_{LR15}$ $\displaystyle={{\textbf{o}}}_{\nu
dLR}^{V}-{{\textbf{o}}}_{edLR}^{V}~{},$ (68)

Note that the operator ${{\mathbf{o}}}_{FY11}$ as defined in Table 6 consists
of two scalar currents. However, this operator maps to a vector operator after
the Fierz transformation and hence it is included in the category $LLRR$.

For $RRLL$ vector operators:

$\displaystyle{{\mathbf{o}}}_{LR7}$
$\displaystyle={{\textbf{o}}}_{euRL}^{V}+{{\textbf{o}}}_{edRL}^{V}~{},\quad{{\mathbf{o}}}_{LR16}={{\textbf{o}}}_{euRL}^{V}-{{\textbf{o}}}_{edRL}^{V}~{}.$
(69)

For $RRRR$ vector operators:

$\displaystyle{{\mathbf{o}}}_{RR5}$
$\displaystyle={{\textbf{o}}}_{euRR}^{V}~{},\quad\quad{{\mathbf{o}}}_{RR6}={{\textbf{o}}}_{edRR}^{V}~{}.$
(70)

For scalar operators:

$\displaystyle{{\mathbf{o}}}_{FY7}$
$\displaystyle={{\textbf{o}}}_{edRLRL}^{\prime S}~{},$
$\displaystyle{{\mathbf{o}}}_{ST9}$
$\displaystyle={{\textbf{o}}}_{euRLRL}^{\prime S}~{},$ (71)
$\displaystyle{{\mathbf{o}}}_{LR9}$
$\displaystyle=-2\,{{\textbf{o}}}_{RLLR}^{S}-2\,{{\textbf{o}}}_{edRLLR}^{S}~{}.$
$\displaystyle{{\mathbf{o}}}_{FY9}$
$\displaystyle={{\textbf{o}}}_{euRLLR}^{\prime S}~{},$ (72)
$\displaystyle{{\mathbf{o}}}_{LR18}$
$\displaystyle=-2\,{{\textbf{o}}}_{RLLR}^{S}+2\,{{\textbf{o}}}_{edRLLR}^{S}~{},$
$\displaystyle{{\mathbf{o}}}_{ST10}$
$\displaystyle={{\textbf{o}}}_{RLRL}^{\prime S}~{}.$ (73)

Here $[{{\textbf{o}}}_{ed(u)RLLR}^{\prime S}]^{\alpha\beta
ij}=([{{\textbf{o}}}_{ed(u)RLLR}^{S}]^{\beta\alpha ji})^{\dagger}$ and
$[{{\textbf{o}}}_{RLRL}^{\prime S}]^{\alpha\beta
ij}=([{{\textbf{o}}}_{RLRL}^{S}]^{\beta\alpha ji})^{\dagger}$. Note that the
operators ${{\mathbf{o}}}_{LR9}$ and ${{\mathbf{o}}}_{LR18}$ are defined as
products of vector currents in Table 6. However, they map to scalar operators
after Fierz transformations, as can be seen from eqs. (72–73).

In Buchalla:2012qq , there is one more scalar operator,
${{\mathbf{o}}}_{ST3}=\varepsilon_{ij}(\bar{q}^{i}u)(\bar{l}^{j}e)$ . This
operator is not independent from the scalar operators appearing in Table 6 and
can be written as

$\displaystyle{{\mathbf{o}}}_{ST3}$
$\displaystyle={{\mathbf{o}}}_{ST9}-{{\mathbf{o}}}_{ST10}~{}.$ (74)

Hence this operator has been omitted in our list.

For tensor operators:

$\displaystyle{{\mathbf{o}}}_{FY8}$
$\displaystyle={{\textbf{o}}}_{edRLRL}^{\prime T}~{},$
$\displaystyle{{\mathbf{o}}}_{ST11}$
$\displaystyle={{\textbf{o}}}_{euRLRL}^{\prime T}~{},$ (75)
$\displaystyle{{\mathbf{o}}}_{ST13}$
$\displaystyle={{\textbf{o}}}_{edRLLR}^{\prime T}~{},$
$\displaystyle{{\mathbf{o}}}_{ST16}$
$\displaystyle={{\textbf{o}}}_{euRLLR}^{\prime T}~{},$ (76)
$\displaystyle{{\mathbf{o}}}_{ST14}$ $\displaystyle={{\textbf{o}}}_{RLLR}^{T}$
$\displaystyle{{\mathbf{o}}}_{S12}$
$\displaystyle=({{\textbf{o}}}_{RLRL}^{\prime T})~{},$ (77)

where $[{{\mathbf{o}}}^{\prime}]^{\alpha\beta
ij}\equiv([{{\mathbf{o}}}]^{\beta\alpha ji})^{\dagger}$. The three tensor
operators ${{\mathbf{o}}}_{ST13}$, ${{\mathbf{o}}}_{ST14}$ and
${{\mathbf{o}}}_{ST16}$ are absent in the list of HEFT operators presented in
Buchalla:2012qq . On the other hand, the operator
${{\mathbf{o}}}_{ST4}=\varepsilon_{ij}\,(\bar{q}^{i}\sigma_{\mu\nu}\,u)\,(\bar{l}^{j}\sigma^{\mu\nu}e)$
included in Buchalla:2012qq  is not an independent one. It can be written as

$\displaystyle{{\mathbf{o}}}_{ST4}$
$\displaystyle={{\mathbf{o}}}_{ST11}-{{\mathbf{o}}}_{ST12}~{},$ (78)

and has been omitted in our list.

HEFT operators with $Z$, $W^{\pm}$ couplings
---
${{\mathbf{o}}}_{\psi V1}=\left(\bar{q}\gamma^{\mu}q\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$ | ${{\mathbf{o}}}_{\psi V2}=\left(\bar{q}\gamma^{\mu}UT_{3}U^{\dagger}q\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$
${{\mathbf{o}}}_{\psi V3}=\left(\bar{q}\gamma^{\mu}UP_{12}U^{\dagger}q\right)\langle U^{\dagger}iD_{\mu}UP_{21}\rangle~{}+{\rm h.c.}$ | ${{\mathbf{o}}}_{\psi V4}=\left(\bar{u}\gamma^{\mu}u\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$
${{\mathbf{o}}}_{\psi V5}=\left(\bar{d}\gamma^{\mu}d\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$ | ${{\mathbf{o}}}_{\psi V6}=\left(\bar{u}\gamma^{\mu}d\right)\langle U^{\dagger}iD_{\mu}UP_{21}\rangle~{}+{\rm h.c.}$
${{\mathbf{o}}}_{\psi V7}=\left(\bar{l}\gamma^{\mu}l\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$ | ${{\mathbf{o}}}_{\psi V8}=\left(\bar{l}\gamma^{\mu}UT_{3}U^{\dagger}l\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$
${{\mathbf{o}}}_{\psi V9}=\left(\bar{l}\gamma^{\mu}UP_{12}U^{\dagger}l\right)\langle U^{\dagger}iD_{\mu}UP_{21}\rangle~{}+{\rm h.c.}$ | ${{\mathbf{o}}}_{\psi V10}=\left(\bar{e}\gamma^{\mu}e\right)\langle U^{\dagger}iD_{\mu}UT_{3}\rangle$
${{\mathbf{o}}}_{\psi h1}=h\,\left(\bar{q}UP_{-}r\right)$ | ${{\mathbf{o}}}_{\psi h2}=h\,\left(\bar{q}UP_{+}r\right)$
${{\mathbf{o}}}_{\psi h3}=h\,\left(\bar{l}UP_{-}\eta\right)$ |
Table 7: HEFT operators in Buchalla:2012qq  with $Z$, $W^{\pm}$ and $h$
couplings to fermions.

For HEFT operators with BSM coupling of $Z$, $W^{\pm}$ to the fermions, we
reproduce the list provided in Buchalla:2012qq  in Table 7. In addition, we
have also included the HEFT operators that modify the coupling of $h$ to
fermions. Once again there is a one-to-one mapping between the operators in
Table 7 and Table 4:

$\displaystyle{{\mathbf{o}}}_{\psi V1}$
$\displaystyle=-\frac{g}{2\cos{\theta}}\left({{\textbf{o}}}_{u_{L}Z}+{{\textbf{o}}}_{d_{L}Z}\right)~{},\quad{{\mathbf{o}}}_{\psi
V2}=-\frac{g}{2\cos{\theta}}\left({{\textbf{o}}}_{u_{L}Z}-{{\textbf{o}}}_{d_{L}Z}\right)~{},$
(79) $\displaystyle{{\mathbf{o}}}_{\psi V3}$
$\displaystyle=-\frac{g}{\sqrt{2}}{{\textbf{o}}}_{ud_{L}W}~{},\qquad\qquad\qquad~{}{{\mathbf{o}}}_{\psi
V4}=-\frac{g}{2\cos{\theta}}{{\textbf{o}}}_{u_{R}Z}~{},$ (80)
$\displaystyle{{\mathbf{o}}}_{\psi V5}$
$\displaystyle=-\frac{g}{2\cos{\theta}}{{\textbf{o}}}_{d_{R}Z}~{},\qquad\qquad\quad~{}~{}{{\mathbf{o}}}_{\psi
V6}=-\frac{g}{\sqrt{2}}{{\textbf{o}}}_{ud_{R}W}~{},$ (81)
$\displaystyle{{\mathbf{o}}}_{\psi V7}$
$\displaystyle=-\frac{g}{2\cos{\theta}}\left({{\textbf{o}}}_{\nu_{L}Z}+{{\textbf{o}}}_{e_{L}Z}\right)~{},\quad{{\mathbf{o}}}_{\psi
V8}=-\frac{g}{2\cos{\theta}}\left({{\textbf{o}}}_{\nu_{L}Z}-{{\textbf{o}}}_{e_{L}Z}\right)~{},$
(82) $\displaystyle{{\mathbf{o}}}_{\psi V9}$
$\displaystyle=-\frac{g}{\sqrt{2}}({{\textbf{o}}}_{e\nu_{L}W})^{\dagger}~{},\qquad\qquad~{}~{}\,{{\mathbf{o}}}_{\psi
V10}=-\frac{g}{2\cos{\theta}}{{\textbf{o}}}_{e_{R}Z}~{},$ (83)
$\displaystyle{{\mathbf{o}}}_{\psi h1}$
$\displaystyle={{\mathbf{o}}}_{dh}~{},\qquad{{\mathbf{o}}}_{\psi
h2}={{\mathbf{o}}}_{uh}~{},\qquad{{\mathbf{o}}}_{\psi
h3}={{\mathbf{o}}}_{eh}~{}.$ (84)

Thus, we have explicitly demonstrated the one-to-one mapping between the HEFT
operators in the $U(1)_{em}$ invariant language and the HEFT operators in
$SU(2)_{L}\times SU(2)_{R}$ language in the unitary gauge.

##  Appendix B Details of the SMEFT basis used

To obtain the SMEFT predictions, we have used the
$\left(m_{W},m_{Z},\alpha_{EM}\right)$ input parameter scheme and the basis as
proposed in Ref. Masso:2014xra . Note that this basis is different from the
Warsaw basis that is conventionally used for SMEFT. In this appendix, we
discuss the difference and the rationale for the choice of this basis.

In the Warsaw basis, the two operators151515Note that in Ref.
Grzadkowski:2010es  the operator ${\cal O}_{T}$ would actually appear as a
linear combination of the operators,
$(H^{\dagger}\,D_{\mu}\,H)^{*}(H^{\dagger}\,D_{\mu}\,H)$ and
$(H^{\dagger}H)\partial_{\mu}\partial^{\mu}(H^{\dagger}H)$. The combination of
these operators orthogonal to ${\cal O}_{T}$, ${\cal
O}_{H}=\frac{1}{2}(\partial_{\mu}|H|^{2})^{2}$, is part of the basis of Ref.
Masso:2014xra .

$\displaystyle{\cal O}_{WB}$
$\displaystyle=gg^{\prime}H^{\dagger}\tau^{I}\,H\,W_{\mu\nu}^{I}\,B^{\mu\nu}~{},$
(85) $\displaystyle{\cal O}_{T}$
$\displaystyle=(H^{\dagger}\,\overleftrightarrow{D}H)^{2}$ (86)

would contribute to the couplings of gauge bosons to the fermions by affecting
their mass and kinetic terms. One needs to carefully normalize the kinetic
term to bring it to the canonical form and also take into account input
parameter shifts. These subtleties become relevant when we try to write the
$Z$ and $W^{\pm}$ coupling modifications in SMEFT as in eqs.(̇32-34).

Instead, in the basis of Ref. Masso:2014xra , the operators ${\cal O}_{WB}$
and ${\cal O}_{T}$ are traded for the following two operators:

$\displaystyle{\cal O}_{WB^{\prime}}$ $\displaystyle={\cal
O}_{WB}-2ig^{\prime}\left(H^{\dagger}\overleftrightarrow{D}^{\mu}H\right)\partial^{\nu}B_{\mu\nu},$
$\displaystyle{\cal O}_{W^{\prime}}$
$\displaystyle=\frac{ig}{2}\left(H^{\dagger}\tau^{a}\overleftrightarrow{D}^{\mu}H\right)D^{\nu}W_{\mu\nu}^{a}-\frac{ig^{\prime}}{2}\left(H^{\dagger}\overleftrightarrow{D}^{\mu}H\right)\partial^{\nu}B_{\mu\nu}.$
(87)

This way of writing the operators eliminates their contributions to any mass
or kinetic term Masso:2014xra , thus allowing us to obtain the SMEFT
predictions in a straightforward way.

Note that the final predictions we obtain should be independent of the basis
being used and the Warsaw basis should also yield the same predictions, albeit
with more complicated intermediate calculations. We show in the following
that, in the Warsaw basis, the contributions of the two operators ${\cal
O}_{T}$ and ${\cal O}_{WB}$ to the HEFT WCs associated with $Z$, $W^{\pm}$
couplings to the fermions cancel out in the final relations. In the Warsaw
basis, these HEFT WCs receive the following SMEFT contributions Efrati:2015eaa
:

$\displaystyle[{{\mathbf{c}}}_{u_{L}Z}]^{ij}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hq}^{(1)}]^{ij}-[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij})+f(1/2,\,2/3)~{},$
(88) $\displaystyle[{{\mathbf{c}}}_{d_{L}Z}]^{ij}$
$\displaystyle=\eta_{LZ}\,([{{\mathcal{C}}}_{Hq}^{(1)}]^{ij}+[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij})+f(-1/2,\,-1/3)~{},$
(89) $\displaystyle[{{\mathbf{c}}}_{ud_{L}W}]^{ij}$
$\displaystyle=\eta_{LW}\,[{{\mathcal{C}}}_{Hq}^{(3)}]^{ij}+f(1/2,\,2/3)-f(-1/2,\,-1/3)~{},$
(90)

where $\eta_{LZ}\equiv-g/(2\,\cos\theta)$, $\eta_{LW}=g/(\sqrt{2})$ and the
term $f(T^{3},Q)$ is defined as Efrati:2015eaa

$\displaystyle f(T^{3},Q)$
$\displaystyle=\mathcal{I}\left[-{{\mathcal{C}}}_{WB}\,Q\,\frac{g^{2}\,g^{\prime
2}}{g^{2}-g^{\prime 2}}+({{\mathcal{C}}}_{T}-\delta
v)\left(T^{3}+Q\,\frac{g^{\prime 2}}{g^{2}-g^{\prime 2}}\right)\right]~{},$
(91)

with $[\delta
v]^{ij}=([{{\mathcal{C}}}_{Hl}^{(3)}]^{11}+[{{\mathcal{C}}}_{Hl}^{(3)}]^{22})/2+[{{\mathcal{C}}}_{{\ell}{\ell}}^{(1)}]^{1221}/4$.
From eqs. (88-90), we obtain the SMEFT prediction

$\displaystyle{{\mathbf{c}}}_{ud_{L}W}=\frac{1}{\sqrt{2}}\cos\theta_{w}\,({{\mathbf{c}}}_{u_{L}Z}-\,{{\mathbf{c}}}_{d_{L}Z})~{},$
(92)

which is the same as the one in Table 3. We see that in the prediction shown
in eq. (92), the function $f(T^{3},Q)$ and the operators within do not appear.
Similarly for $Z$, $W^{\pm}$ coupling to leptons, we recover the prediction
already presented in Table 3,

$\displaystyle{{\mathbf{c}}}_{e\nu_{L}W}=\frac{1}{\sqrt{2}}\cos\theta_{w}\,({{\mathbf{c}}}_{e_{L}Z}-\,{{\mathbf{c}}}_{\nu_{L}Z})~{}.$
(93)

Thus, even in the Warsaw basis, the contributions to our relations from ${\cal
O}_{WB}$ and ${\cal O}_{T}$ cancel out, confirming that the final relations
among the HEFT WCs are independent of the choice of the basis for SMEFT.

##  Appendix C Linear relations among LEFT and HEFT operators

In Sec. 2.3, we have presented SMEFT predictions for LEFT WCs of the class
$LLLL$. In this appendix, we provide a similar analysis for the other classes
of LEFT operators. We first write the matching of four-fermion semileptonic
WCs between LEFT and HEFT. We then substitute the HEFT WCs with the LEFT WCs
for each of the analytic relations presented in Table 3. As a result, we get
relations among the LEFT WCs which also involve the BSM couplings of
$Z,W^{\pm}$ and Higgs bosons to fermions. These relations for the vector
operators are listed in Table 8. For the scalar and the tensor operators, the
relations are presented in Table 9.

Class | Analytic relations for WCs of vector operators | Count
---|---|---
$LLLL$ |  $\begin{aligned} &V_{ik}\left[[{{C}}_{edLL}^{V}]^{\alpha\beta kl}-\left(k_{e_{L}}\,[\hat{{\mathbf{c}}}_{d_{L}Z}]^{kl}\,\delta_{\alpha\beta}+k_{d_{L}}\,[\hat{{\mathbf{c}}}_{e_{L}Z}]^{\alpha\beta}\,\delta_{kl}\right)\right]V^{\dagger}_{{\ell}j}\\\ &~{}~{}=U^{\dagger}_{\alpha\rho}\left[[{{C}}_{\nu uLL}^{V}]^{\rho\sigma ij}-\chi\,\left(k_{\nu_{L}}\,[\hat{{\mathbf{c}}}_{u_{L}Z}]^{ij}\,\delta_{\rho\sigma}+k_{u_{L}}\,[\hat{{\mathbf{c}}}_{\nu_{L}Z}]^{\rho\sigma}\,\delta_{ij}\right)\right]U_{\sigma\beta}\end{aligned}$ | 81 (45)