# Wrinkles in the Froggatt–Nielsen Mechanism and Flavorful New Physics

Pouya Asadi1 Arindam Bhattacharya2 Katherine Fraser2
Samuel Homiller2 and Aditya Parikh3
pasadi (@uoregon.edu), arindamb, kfraser, shomiller (@g.harvard.edu),
aditya.parikh (@stonybrook.edu)
1Institute for Fundamental Science and Department of Physics,
University of Oregon, Eugene, OR 97403, USA
2Department of Physics, Harvard University, Cambridge, MA 02138, USA
3C.N. Yang Institute for Theoretical Physics,
Stony Brook University, Stony Brook, NY 11794, USA

(August 28, 2024)

###### Abstract

When the Froggatt–Nielsen mechanism is used to explain the Standard Model
flavor hierarchy, new physics couplings are also determined by the horizontal
symmetry. However, additional symmetries or dynamics in the UV can sometimes
lead to a departure from this naïve scaling for the new physics couplings. We
show that an effective way to keep track of these changes is by using the new
spurions of the $\mathrm{U}(3)^{5}$ global flavor symmetry, where we
parameterize extra suppression or enhancement factors, referred to as
wrinkles, using the same power counting parameter as in the original
Froggatt–Nielsen model. As a concrete realization, we consider two flavor
spurions of the $S_{1}$ leptoquark, and demonstrate that wrinkles can be used
to make an enhanced value of $\textrm{BR}(B^{+}\to K^{+}\nu\bar{\nu})$
consistent with other flavor observables. We also present example UV models
that realize wrinkles, and comment on choosing consistent charges in ordinary
Froggatt–Nielsen models without the typical monotonicity condition.

###### Contents

  1. 1 Aperitif
  2. 2 Amuse-bouche: Froggatt–Nielsen and BSM Physics
    1. 2.1 Review of the Froggatt–Nielsen Mechanism
    2. 2.2 Froggatt–Nielsen and Flavorful New Physics
  3. 3 Plat Principal: Wrinkles in Froggatt–Nielsen
    1. 3.1 Wrinkled Froggatt–Nielsen Chains
    2. 3.2 Bounds on Wrinkles from Radiative Corrections
    3. 3.3 UV Completions
      1. 3.3.1 Missing Heavy Fermions
      2. 3.3.2 Extra Abelian Symmetries
  4. 4 Dessert: $B\to K\bar{\nu}\nu$ in a Wrinkled Setup
    1. 4.1 $B\to K\bar{\nu}\nu$ in the SM and Beyond
    2. 4.2 Constraints with Different Flavor Ansätze
    3. 4.3 Predictions for Future Measurements
  5. 5 Digestifs
  6. A Full Set of Consistency Conditions
  7. B Calculation of Other Observables
    1. B.1 Dipole Moments
    2. B.2 Lepton Flavor Violating Observables
    3. B.3 Leptonic Meson Decays
      1. B.3.1 $P\rightarrow\ell\nu$
      2. B.3.2 $P\rightarrow\ell\ell^{\prime}$ and $P\rightarrow\nu\nu^{\prime}$
    4. B.4 Semi-leptonic Meson Decays
      1. B.4.1 $R_{D^{(*)}}$
      2. B.4.2 $K\to\pi\nu\bar{\nu}$
    5. B.5 $Z\rightarrow\ell\ell^{\prime}$
    6. B.6 Meson Mixing

##  1 Aperitif

Flavor physics has been a harbinger of physics beyond the Standard Model (BSM)
at various points in time, from predicting the existence of the charm quark
[1, 2] to estimating the mass of the top quark [3, 4, 5, 6] long before its
discovery at the Tevatron [7, 8]. Precision experiments, in particular, help
establish or find violations of the Standard Model (SM) symmetry structures,
and prove to be noteworthy indirect probes of new physics whose mass scale
lies beyond the reach of direct collider searches; see Refs. [9, 10] for
reviews of many such experiments.

A primary goal of flavor physics is to understand the appearance of large
hierarchies in the masses and mixing angles of the SM fermions. The two most
popular solutions to this puzzle are (i) the Froggatt–Nielsen (FN) mechanism
and its variations [11, 12, 13, 14], and (ii) extra dimensional models where
an $\mathcal{O}(1)$ difference in the bulk masses of fermions gives rise to an
exponential hierarchy between the observed masses in the IR [15, 16, 17, 18,
19]. Other notable possibilities include generating the mass hierarchy via
running to the IR in extensions of the SM with scale invariant sectors in the
UV [20], or radiatively generating the Yukawas with the hierarchy governed by
powers of the loop expansion parameter [21, 22, 23, 24]. A review of these and
other dynamical solutions to the flavor puzzle can be found in Refs. [25, 26,
27]. In what follows, we focus our attention on the FN mechanism.

In the FN mechanism, the hierarchies in the SM fermion sector arise as
different powers of a small expansion parameter. This expansion parameter is
given by the ratio of the vacuum expectation value (vev) of a scalar field,
known as the flavon, over a heavy mass scale. The SM Yukawa couplings are
generated by non-renormalizable operators involving the chiral SM fermions,
the Higgs, and the flavon. The dimensionality of these operators—and the
resulting power of the expansion parameter that appears—is dictated by the
charges of the SM fermions under a new Abelian horizontal symmetry,
${\mathrm{U}}(1)_{H}$, which is broken by the flavon. As we will discuss,
there is additional freedom in the assignment of these charges that was
overlooked in Ref. [11]. In the original FN paper, it was supposed that these
irrelevant operators are generated by “chains” including heavy vector-like
matter, also charged under ${\mathrm{U}}(1)_{H}$. A number of variations to
this model have been proposed, including “inverted” models [28], where the
flavon vev is larger than the heavy mass scale.

One of the drawbacks of invoking the FN mechanism is that the new dynamics
responsible for the SM hierarchies can exist at scales far above the weak
scale, beyond the reach of direct experimental probes. Nevertheless, given the
other shortcomings of the SM—the electroweak hierarchy problem in
particular—there is ample reason to expect new physics at or near the TeV
scale. If the new physics is flavorful (i.e., it involves non-universal
couplings to SM matter fields), its flavor structure may also be dictated by
the FN dynamics. This argument can also be run in reverse: given the stringent
constraints from precision measurements of the SM, for new physics to exist at
the TeV scale it must either be flavor-blind or incorporate some symmetry
arguments to suppress flavor-violation [29, 30]. This reasoning is familiar in
the supersymmetric context, where it is understood that squarks must either be
degenerate or flavor-aligned [31].

In this light, it is clearly worthwhile to study the application of the FN
mechanism to the couplings of new BSM fields. This is particularly true when
flavorful new physics is invoked to explain potential discrepancies between
experimental results and the SM expectations: should one of these
discrepancies become an unambiguous signal of new physics, we might glean
information about the dynamics associated with flavor in the UV. This approach
was advocated in Refs. [32, 33, 34], and we will review it extensively in this
work. An immediate consequence of this framework is that many different
experimental observables become correlated. These correlations challenge some
of the simplest solutions to various flavor anomalies, as the couplings and
masses required to explain the discrepancy violate bounds set by other
observables such as lepton flavor violating (LFV) processes or flavor changing
neutral currents.

The goal of this work is to explore how these considerations can change if the
FN setup is amended with additional symmetries or structure in the UV. We do
this by working in an effective field theory (EFT) framework, including the SM
and new BSM fields, with their couplings to fermions treated as spurions under
the ${\mathrm{U}}(3)^{5}$ flavor symmetry of the SM. In this framework, we can
introduce controlled deviations from the size of these spurions dictated by
the horizontal charges. We refer to these deviations as wrinkles, since they
appear in the UV as changes in the length of the chain diagrams responsible
for the Yukawas in the IR. Wrinkles can exist in SM or BSM spurions, and allow
us to relax the correlations between different observables, permitting sizable
new physics contributions to some observables while satisfying other
experimental bounds.

Importantly, while wrinkles allow for much greater flexibility in the
couplings of BSM fields to SM fermions, this flexibility is not without bound.
If the effective theory is to be faithfully embedded in the FN mechanism,
radiative corrections must not spoil the relationship between the couplings in
the IR and the non-renormalizable operators in the UV. This requirement has
been previously formulated as a consistency condition in the context of
minimal flavor violation EFTs [32] (see also Ref. [33]). While these
conditions are trivially satisfied in ordinary FN models, we show that they
put meaningful bounds on wrinkled FN setups.

Since this wrinkled FN setup can be applied to any new physics, we will
illustrate its application in an example, where the SM is extended by a single
leptoquark, denoted $S_{1}$ in the nomenclature of Ref. [35]. See Refs. [36,
34] for previous discussions of the $S_{1}$ leptoquark model with horizontal
symmetries. We will use this leptoquark to enhance the branching ratio of
$B^{+}\rightarrow K^{+}\bar{\nu}\nu$, which currently shows a small
discrepancy with SM predictions [37] and will be precisely measured at the
Belle II experiment in the coming years. Without wrinkles, the charges and
masses required to generate a large $B^{+}\rightarrow K^{+}\bar{\nu}\nu$
signal also imply the existence of large signals in other correlated
observables, such as LFV decays or leptonic meson decays. We will show a
simple example where a wrinkled FN setup evades these bounds while satisfying
the consistency conditions alluded to above. As we will see, the bound on the
wrinkles implies other correlated signals are generated near detection
thresholds in this example, and could potentially be seen in the near future.

In the coming years, troves of new data from colliders and small-scale
experiments searching for signs of flavorful new physics will begin stress-
testing the delicate flavor structure of the SM. Given the substantial
motivation for BSM physics, this structure could break and potentially start
showing signs of deviations from the SM expectation. In preparation for such
deviations, it is timely to develop new model-building tools which enable
embedding their solutions in UV complete frameworks. Wrinkles in an FN Ansatz
are a flexible, bottom-up tool that allow for a broader exploration of the
complementarity of different flavor probes, while reliably parameterizing more
sophisticated UV models of flavor. As such, they present a natural setup to
search for a consistent IR picture of new physics with flavor, should any
deviations from the SM come to light.

This paper is organised as follows: in §2, we review the FN mechanism, its
solution to the flavor hierarchy problem in SM and how it furnishes suitable
Ansätze for couplings arising from new BSM physics. Next, in §3, we introduce
the concept of wrinkles for the FN mechanism, discuss constraints on them, and
provide examples for how they can arise from UV complete models. In §4, we
provide a concrete example of applying wrinkles to the $S_{1}$ scalar
leptoquark embedded in a FN model. We demonstrate that wrinkles allow one to
simultaneously explain bounds on BSM physics from current precision flavor
observables, while also retaining predictive power for potential future
measurements. We conclude in §5. Appendix A provides details about bounds on
wrinkles arising from consistency conditions. Appendix B provides details on
flavor observable computations in the $S_{1}$ leptoquark model.

##  2 Amuse-bouche: Froggatt–Nielsen and BSM Physics

The lepton and quark Yukawas and mixing angles present a clear generational
hierarchy, with the charged particle masses ranging over five orders of
magnitude. This hierarchy implores an explanation in the UV. Searches for
flavorful new physics are carried out in pursuit of such an explanation.
Hence, if any anomaly emerges in these experiments, it is well-motivated to
embed its BSM solutions within UV models that explain the flavor hierarchy as
well.

The FN mechanism [11] provides a four-dimensional, field-theoretic explanation
for this hierarchy, replacing the small dimensionless parameters with a power
counting in powers of an inverse mass scale, fixed by a symmetry. In this
section, we review how this mechanism can explain the parameters in the SM
matter sector, with an emphasis on the EFT point of view. We will then discuss
how this perspective can naturally be extended to BSM physics.

###  2.1 Review of the Froggatt–Nielsen Mechanism

The basic idea of the FN mechanism is to introduce a horizontal symmetry,
${\mathrm{U}}(1)_{H}$, under which different generations of the SM fermions
have different charges. The horizontal symmetry is assumed to be spontaneously
broken by the vacuum expectation value of a SM singlet scalar field,
$\varphi$—the flavon. Assuming our EFT is valid up to some cutoff scale $M$,
we are led to a natural expansion parameter $\lambda=\langle\varphi\rangle/M$,
which appears in non-renormalizable operators involving the SM fermions. Later
on, we will associate this scale $M$ with the mass of new heavy fermions.
Without loss of generality, we take the SM Higgs to be neutral under
${\mathrm{U}}(1)_{H}$ and take the flavon charge to be $-1$.

At scales just below the cutoff, the lowest dimension operators involving the
SM fermions and the Higgs take the form

$\mathcal{L}\,\supset\,r^{u}_{ij}\frac{\varphi^{(\dagger)m_{ij}}}{M^{m_{ij}}}Q_{i}H\bar{u}_{j}+r^{d}_{ij}\frac{\varphi^{(\dagger)n_{ij}}}{M^{n_{ij}}}Q_{i}H^{c}\bar{d}_{j}+r^{e}_{ij}\frac{\varphi^{(\dagger)l_{ij}}}{M^{l_{ij}}}L_{i}H^{c}\bar{e}_{j}+\textrm{h.c.}$
(1)

where $(Q_{i},\leavevmode\nobreak\ \bar{u}_{i},\leavevmode\nobreak\
\bar{d}_{i},\leavevmode\nobreak\ L_{i},\leavevmode\nobreak\ \bar{e}_{i})$ are
different SM fermions, subscripts on fermion fields refer to different
generations, $r_{ij}$ are $\mathcal{O}(1)$ couplings,

$m_{ij}=\big{|}[Q_{i}]+[\bar{u}_{j}]\big{|},\qquad
n_{ij}=\big{|}[Q_{i}]+[\bar{d}_{j}]\big{|},\qquad
l_{ij}=\big{|}[L_{i}]+[\bar{e}_{j}]\big{|},$ (2)

and the square brackets indicate the ${\mathrm{U}}(1)_{H}$ charge. The
hermitian conjugate on $\varphi$ appears if the sum of charges inside the
absolute value is negative. At energies below $\langle\varphi\rangle$, these
operators appear as the Yukawa couplings of the SM Higgs, with the coupling
matrices given by

$Y_{Q\bar{u}}^{ij}=r^{u}_{ij}\frac{\langle\varphi^{(\dagger)}\rangle^{m_{ij}}}{M^{m_{ij}}}\sim\lambda^{m_{ij}},\qquad
Y_{Q\bar{d}}^{ij}=r^{d}_{ij}\frac{\langle\varphi^{(\dagger)}\rangle^{n_{ij}}}{M^{n_{ij}}}\sim\lambda^{n_{ij}},\qquad
Y_{L\bar{e}}^{ij}=r^{e}_{ij}\frac{\langle\varphi^{(\dagger)}\rangle^{l_{ij}}}{M^{l_{ij}}}\sim\lambda^{l_{ij}}.$
(3)

This scaling implies that even modest differences in horizontal charges give
rise to exponential hierarchies in Yukawa couplings. To connect with the
observed flavor structure of the SM, we identify $\lambda$ with the Cabbibo
angle, $\sim 0.2$, so that the CKM matrix hierarchies follow naturally from
the Wolfenstein parameterization [38]. We refer to this setup as vanilla FN.

At the $\mathcal{O}(1)$ level, the masses and mixing angles are

$V_{ij}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[Q_{i}]-[Q_{j}]$}{\scalebox{1.0}[1.05]{$\mid$}}},\quad
U_{ij}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[L_{i}]-[L_{j}]$}{\scalebox{1.0}[1.05]{$\mid$}}},$
(4)
$m^{u}_{i}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[Q_{i}]+[\bar{u}_{i}]$}{\scalebox{1.0}[1.05]{$\mid$}}},\quad
m^{d}_{i}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[Q_{i}]+[\bar{d}_{i}]$}{\scalebox{1.0}[1.05]{$\mid$}}},\quad
m^{l}_{i}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[L_{i}]+[\bar{e}_{i}]$}{\scalebox{1.0}[1.05]{$\mid$}}},$
(5)

where $V$ ($U$) is the CKM [39, 40] (PMNS [41, 42]) matrix.

| Gen. 1 | Gen. 2 | Gen. 3
---|---|---|---
$Q$ | $-q_{0}-3X$ | $-q_{0}-2X$ | $-q_{0}$
$\bar{u}$ | $q_{0}+3X\pm 7$ | $q_{0}-X$ | $q_{0}$
$\bar{d}$ | $q_{0}+3X\pm 6$ | $q_{0}-3X$ | $q_{0}-2X$
$L$ | $l_{0}+Y$ | $l_{0}$ | $l_{0}$
$\bar{e}$ | $-l_{0}-Y\pm 8$ | $-l_{0}+5Y$ | $-l_{0}+3Y$
Table 1: The most general horizontal charge assignment that explains the SM
masses and mixings in FN with $\lambda\sim 0.2$. $q_{0}$ and $l_{0}$ denote
general shifts in quark and lepton charges, respectively, that leave the IR
masses and mixings unchanged. $X,Y=\pm 1$ denote the correlations between
different charges that are required by the CKM and PMNS matrices. For every
value of $(q_{0},l_{0})$, we have $2^{5}$ choices for the charge assignments.
In supersymmetric theories, holomorphy sets $X=-Y=-1$ and picks the positive
sign for first generation RH fermions.

The most general horizontal charge-assignment that gives rise to the observed
structure of the CKM and PMNS matrices and SM fermion masses is given in Table
1.111In general, shifts of $\pm 1$ in most of these charges can be tolerated
when random $\mathcal{O}(1)$ Yukawa couplings in the UV model are taken into
account and the fact that the expansion parameter $\lambda$ is not
particularly small is considered. The anarchic structure of the PMNS matrix,
in particular, leaves room for such small changes in the charges; see Refs.
[43, 44] for further exploration of these shifts. We have the overall freedom
to shift the charges of all quarks (leptons) by the same amounts $q_{0}$
($l_{0}$), respectively. Once these shifts are chosen, the CKM and PMNS
structure constrain the other LH quarks’ and leptons’ charges. As indicated in
Eq. (4), these mixing matrices only fix the absolute value of the difference
between charges, hence the freedom in choosing $X,Y=\pm 1$ in the table. The
appearance of $X,Y$ in multiple entries captures the correlation between those
charges. To find the RH fermion charges we use the measured values of masses
in the SM. As in the case of mixing, Eq. (5) only fixes the absolute value of
the charge difference between LH and RH fermions, leaving the sign
undetermined. We choose the signs so that the eigenvector associated with the
heaviest (lightest) mass eigenstate has the biggest overlap with the third
(first) generation for each type of fermion. To check this, we generated 10000
mass matrices for each charge assignment, drawing new random numbers
$r^{u,d,e}_{ij}\in(0.2,1)$ for each test. For every charge assignment, we
confirmed that a substantial fraction of trials yield the correct mixing
patterns and mass eigenvalues that are within a factor of two of the
experimentally-measured values.

In the original FN proposal, it was assumed that the charges of all five types
of fermions ($Q$, $\bar{u}$, $\bar{d}$, $L$, and $\bar{e}$) are ordered
monotonically between different generations. Table 1 indicates that, while
some correlations between LH and RH fermions of the second and third
generation (captured by $X,Y$) are needed to generate the correct mass
eigenstates, the monotonicity condition can be removed for first generation RH
fermions without distorting the model’s prediction for SM masses. This
manifests itself as a binary choice in the charge of each first generation RH
fermion ($\bar{u},\leavevmode\nobreak\ \bar{d},\leavevmode\nobreak\ \bar{e}$).

It is also popular to consider supersymmetric variations of FN models. In the
supersymmetric case, holomorphy of the superpotential forbids terms with
$\varphi^{\dagger}$ instead of $\varphi$ [12, 13]. This eliminates a great
deal of the freedom in charge assignments tabulated in Table 1. Specifically,
it fixes $X=-Y=-1$ and picks the positive sign for first generation RH
fermions, leaving only the separate overall shifts in the quark and lepton
charges, $q_{0}$ and $l_{0}$. It also enforces the monotonicity of the
horizontal charges across different generations. However, since we do not
explore the supersymmetric case in detail in the rest of this paper, we do not
need to enforce these constraints.

The simplest UV completion of this effective theory (and the one imagined by
Froggatt and Nielsen [11]) is to introduce a set of vector-like fermions $F$
with mass $M$ that live in an SM representation permitting Yukawa couplings
between the Higgs and SM fermions. We assume the existence of heavy fermions
with all horizontal charges necessary to complete the SM Yukawas with Yukawa
couplings to the flavon $\sim\varphi F\bar{F}^{\prime}$. The flavon Yukawa
couplings are assumed to be $\mathcal{O}(1)$, leading to the effective theory
in Eq. (1) with $\mathcal{O}(1)$ Wilson coefficients denoted by $r_{ij}$.

Figure 1: Example diagrams leading to the effective operators for the up-type
Yukawa couplings with vector-like heavy fermions $U$ and $\bar{U}$, where
$\bar{U}$ has the same SM quantum numbers as $\bar{u}$. The subscripts on the
$U$ fields refer to the horizontal charge of ${U}$, and we have taken charges
from Table 1 with $X=-1$ and $q_{0}=0$.

As an example, the up-type Yukawa couplings can be generated by “chain”
diagrams such as those shown in Figure 1. The top Yukawa arises at the
renormalizable level, but the suppressed couplings arise by introducing the
vector-like pair $U$ and $\bar{U}$, where $\bar{U}$ has the same quantum
numbers under the SM gauge groups as $\bar{u}$. The subscripts indicate the
${\mathrm{U}}(1)_{H}$ charge of $U$. For instance, the chain shown on the
right side of Figure 1 gives rise to a $\lambda^{2}$ suppression in the
coupling of $Q_{2}\bar{u}_{3}$. If there exist heavy fermions with SM charges
similar to $Q$ and the correct horizontal charges, chain diagrams with the
Higgs and flavon insertions interchanged will contribute as well. Similar
chains give rise to the Yukawa couplings for other SM fermions.

Models of FN constructions with additional symmetries, multiple expansion
parameters, or expansion parameters that are allowed to freely vary have also
been developed in the literature, e.g. see [12, 13, 45, 43, 46, 47]. For
simplicity, however, in this work we focus on FN setups with only one
expansion parameter, which we identify with the Cabbibo angle, and develop a
systematic way for small deviations from them. We can straightforwardly
generalize our discussions below to more baroque FN setups.

As a final note, in a UV complete model, quantum gravity considerations
require that the horizontal symmetry be embedded in a gauge symmetry [48],
which in turn demands the cancellation of all its anomalies.222The lack of
evidence for the (pseudo-)Nambu–Goldstone boson associated with the
spontaneous breaking of the horizontal symmetry is also often used as
motivation for gauging it. However, models with a potentially viable Goldstone
exist. See Refs. [49, 50, 51, 52] for examples where the Goldstone is
identified with the QCD axion. We have checked that the general charge
assignment of Table 1 can not cancel all gauge anomalies in the typical FN UV
completion, see also Ref. [52] for a similar conclusion. This conclusion is
also corroborated by Refs. [53, 54], which deduce that the general charge
assignments that can explain the SM Yukawa hierarchy can not be anomaly-free
by studying general extensions of the SM with a new anomaly-free
${\mathrm{U}}(1)$ gauge group. As a result, in such a construction one should
resort to either introducing new heavy chiral fermions (and subsequently
extending the scalar sector so as to generate a mass for these fermions) or
the Green-Schwarz mechanism to cancel anomalies [55]. We will leave further
investigations of anomaly cancellation for future work.

###  2.2 Froggatt–Nielsen and Flavorful New Physics

When introducing new physics, some assumptions must be made about the
couplings of SM fields to new particles. These couplings are generically non-
universal unless governed by additional structure such as new gauge
symmetries. Given the hierarchies that exist in the SM fermion couplings, it
is a priori unclear what a “natural” size for such non-universal couplings
should be. However, if one assumes a UV explanation of the flavor hierarchy
such as the FN mechanism, there is a natural Ansatz for the new physics
couplings as well.

The phenomenological significance of such an Ansatz lies in the fact that it
correlates predictions of a BSM model for various flavorful observables in the
IR. Thus, depending on the Ansatz, a model built for explaining a discrepancy
in the data will give rise to correlated signals in other constraining
observables. For instance, any solutions of the $(g-2)_{\mu}$ anomaly with
non-minimal flavor Ansatz gives rise to unacceptably large contributions to
various LFV decays, especially $\tau\rightarrow\mu\gamma$.

To better understand such Ansätze, it is useful to organize our thinking in
terms of the global flavor symmetry of the SM:

$G_{\textrm{flavor}}={\mathrm{SU}}(3)_{Q}\times{\mathrm{SU}}(3)_{u}\times{\mathrm{SU}}(3)_{d}\times{\mathrm{SU}}(3)_{L}\times{\mathrm{SU}}(3)_{e}\times{\mathrm{U}}(1)^{5},$
(6)

where three of the ${\mathrm{U}}(1)$ factors can be identified with
hypercharge, baryon number and lepton number. This symmetry acts on the
generation indices of the chiral matter in the SM, with the unbarred (barred)
fields transforming as triplets (anti-triplets), respectively. The symmetry is
broken explicitly by the Yukawa matrices, but formal invariance under
$G_{\textrm{flavor}}$ can be restored if we promote the Yukawas to transform
as spurions:

$Y_{Q\bar{u}}\sim(\mathbf{\bar{3}}_{Q},\mathbf{3}_{u}),\qquad
Y_{Q\bar{d}}\sim(\mathbf{\bar{3}}_{Q},\mathbf{3}_{d}),\qquad
Y_{L\bar{e}}\sim(\mathbf{\bar{3}}_{L},\mathbf{3}_{e}).$ (7)

This formalism can be extended in a straightforward way to new physics with
any new spurions of $G_{\textrm{flavor}}$ [32, 33, 34]. New fields are taken
to be singlets of the ${\mathrm{SU}}(3)^{5}$ part of the SM flavor group, and
their couplings to SM fermions then have definite transformation properties
under $G_{\textrm{flavor}}$.

As an example, consider the scalar leptoquark $S_{1}$, a color anti-
fundamental with hypercharge $Y=1/3$. This allows for the renormalizable
couplings to SM fields,333The SM gauge symmetries also permit the couplings
$S_{1}\bar{u}\bar{d}$ and $S_{1}Q^{\dagger}Q^{\dagger}$, which lead to proton
decay. We can forbid these couplings by enforcing conservation of baryon
number and endowing the leptoquark with a baryon number of $-1/3$, or by
potentially gauging some discrete subgroup. Therefore, in the rest of this
work, we ignore these couplings. We note that the “wrinkles” introduced in §3
cannot entirely alleviate the proton decay constraint, necessitating a
symmetry-based explanation.

$\mathcal{L}\supset-\Delta_{QL}^{ij}\epsilon^{ab}S_{1}Q_{bi}L_{aj}-\Delta_{\bar{u}\bar{e}}^{ij}S_{1}^{\dagger}\bar{u}_{i}\bar{e}_{j}+\textrm{h.c.},$
(8)

where the spinor indices are implicit, $a,b$ are ${\mathrm{SU}}(2)_{L}$
fundamental indices, $\epsilon^{12}=+1$, and $(i,j)$ are flavor indices. The
$\Delta_{QL}$ coupling also appears in R-parity violating supersymmetric
models, where $S_{1}$ is identified with a down squark; these models have
$\Delta_{\bar{u}\bar{e}}=0$ [56]. The new Yukawa couplings $\Delta_{QL}$ and
$\Delta_{\bar{u}\bar{e}}$ transform as

$\Delta_{QL}\sim(\mathbf{\bar{3}}_{Q},\mathbf{\bar{3}}_{L}),\qquad\Delta_{\bar{u}\bar{e}}\sim(\mathbf{3}_{u},\mathbf{3}_{e}).$
(9)

In the absence of any flavor Ansatz, the matrices $\Delta_{QL}$ and
$\Delta_{\bar{u}\bar{e}}$ are arbitrary $3\times 3$ complex matrices. However,
when embedded in a vanilla FN setup, and assuming the $S_{1}$ leptoquark is
neutral under ${\mathrm{U}}(1)_{H}$, we find an Ansatz for the hierarchies
present in the spurions $\Delta_{QL}$ and $\Delta_{\bar{u}\bar{e}}$. In
analogy with Eq. (3), we find:

$\Delta_{QL}^{ij}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[Q_{i}]+[L_{j}]$}{\scalebox{1.0}[1.05]{$\mid$}}},\qquad\Delta_{\bar{u}\bar{e}}^{ij}\sim\lambda^{{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[\bar{e}_{j}]+[\bar{u}_{i}]$}{\scalebox{1.0}[1.05]{$\mid$}}}.$
(10)

Put differently, the SM charges and flavor symmetries are enough to determine
how the new $S_{1}$ field should be embedded in the effective theory below
$M$. The power counting of the effective theory then dictates that the
expected FN scaling above holds, up to the $\mathcal{O}(1)$ Wilson
coefficients of the effective theory (analogous to the $r_{ij}$ in Eq. (1)).
This Ansatz generalizes to arbitrary new spurions of $G_{\mathrm{flavor}}$
that can arise in other leptoquark models. A complete list of these spurions
is given in Ref. [33].

Once the effective theory is known, we can make predictions for the
contributions of new physics to various observables. Because the same spurion
contributes to multiple observables, these predictions are correlated by a FN
Ansatz. These correlations can lead to inconsistencies with experimental
results. Consequently, it is useful to have a systematic way of deviating from
this scaling while still maintaining the predictivity of FN models. We discuss
a systematic way of doing this in the next section. Specifically, we show how
modifications of the UV spectrum of a FN construction can allow a controlled
deviation from correlations between various observables in the IR, alleviating
violations of experimental bounds.

##  3 Plat Principal: Wrinkles in Froggatt–Nielsen

As described in the previous section, the FN mechanism provides a natural
Ansatz for the hierarchies of new flavor spurions coupled to the SM quarks and
leptons. However, given our lack of knowledge about the dynamics underlying
the flavor structure of the SM, it is worth exploring how this Ansatz could
change within the general framework of horizontal symmetry explanations for
the SM flavor pattern.

In this spirit, we introduce the notion of “wrinkles”, as a way of
parametrically changing the FN Ansatz for the flavor spurions that is
described above without introducing additional scales. In §3.1, we will define
them precisely, and argue that they allow for more flexibility in correlations
between different flavor observables. While this flexibility inherently makes
our Ansatz less predictive, the freedom to introduce wrinkles is not absolute:
there is a bound on the number of wrinkles imparted by radiative corrections,
which we will discuss in §3.2. In §3.3, we give several explicit examples of
realizations of wrinkles in UV models.

###  3.1 Wrinkled Froggatt–Nielsen Chains

In §2.2, we described how the FN Ansatz leads to a natural power counting for
new flavor spurions in powers of $\lambda\equiv\langle\varphi\rangle/M$, which
we identify with the Cabbibo angle. Here, we generalize this power counting by
considering modifications to the power of $\lambda$ that appears in the
spurion.

Consider a flavor spurion $Y_{\psi\bar{\chi}}$, where $\psi$, $\bar{\chi}$ are
given SM matter fields. We introduce what we call “wrinkles” to modify the
scaling of a given element of $Y_{\psi\bar{\chi}}$:

$Y_{\psi\bar{\chi}}^{ij}\sim
W_{\psi\bar{\chi}}^{ij}\lambda^{|[\psi_{i}]+[\bar{\chi}_{j}]|}\equiv\lambda^{\omega_{ij}+|[\psi_{i}]+[\bar{\chi}_{j}]|}.$
(11)

Here we denote the power of $\lambda$ that appears in
$W_{\psi\bar{\chi}}^{ij}$ by $\omega_{\psi\bar{\chi}}^{ij}$ which, for
simplicity, is assumed to be an integer. This additional scaling is motivated
by allowing for additional structure in the UV, such as symmetries inducing
obstructions in the heavy fermion chains which generate the non-renormalizable
operators, and is illustrated schematically in Figure 2. In general, any
modification of the UV theory that gives rise to deviations from predictions
of the vanilla FN setup without changing the number of power counting
parameters can be considered a wrinkle. Different UV completions can lead to
different correlated patterns of matrix entries $\omega_{\psi\bar{\chi}}^{ij}$
as we will discuss in §3.3, but from the IR perspective, these correlations
are not apparent.

To be concrete, consider the example of the spurions $\Delta_{QL}$ and
$\Delta_{\bar{u}\bar{e}}$ for the $S_{1}$ leptoquark, as in Eqs. (8) and (9).
With additional wrinkles, the couplings in Eq. (10) are modified to

$\Delta_{QL}^{ij}\sim\lambda^{\omega_{QL}^{ij}+|[Q_{i}]+[L_{j}]|},\qquad\Delta_{\bar{u}\bar{e}}^{ij}\sim\lambda^{\omega_{\bar{u}\bar{e}}^{ij}+|[\bar{u}_{i}]|+[\bar{e}_{j}]}$
(12)

where $\omega_{QL}$ and $\omega_{\bar{u}\bar{e}}$ are matrices of integers,
whose elements $\omega_{QL}^{ij}$ and $\omega_{\bar{u}\bar{e}}^{ij}$ can vary
across generations independently for both fermions. The idea of wrinkles can
also be extended to models with additional scales by allowing wrinkles for
each power counting parameter. Here we will focus on the case with a single
expansion parameter and not discuss the case of multiple parameters further.

Figure 2: A cartoon illustrating a “wrinkle” in the Yukawa coupling
$\Phi\psi_{i}\bar{\chi}_{j}$, which leads to a change in the predicted scaling
from the FN Ansatz.

Note that there are two distinct possibilities allowed by introducing
wrinkles. The most straightforward one is that the number of factors of
$\lambda$ in some couplings of a new flavor spurion are modified, suppressing
or enhancing their contributions to some flavor observables. For instance,
wrinkles could suppress BSM contributions to observables such as electric and
magnetic dipole moments (EDMs and MDMs) or light meson decays, which are
generally strongly constrained, and allow for spurions with smaller mass
scales to contribute to other observables. We will discuss this possibility
thoroughly, again in the case of the $S_{1}$ leptoquark, in §4.

The second possibility is that wrinkles could exist in SM chains—i.e.,
$Y_{Q\bar{u}}$, $Y_{Q\bar{d}}$, or $Y_{L\bar{e}}$ could have fewer or
additional factors of $\lambda$. In the IR, the SM Yukawa matrices must still
match the measured masses and mixing angles of the quarks and leptons.
Wrinkles in SM chains therefore necessitate different horizontal charges than
the ones shown in Table 1. This changes the expected scaling for BSM spurions,
leading to different couplings than expected in a naïve FN Ansatz between the
SM fermions and new particles. We will not comment in detail on particular
phenomenological applications of this scenario, but highlight that this is an
interesting direction for further exploration.

###  3.2 Bounds on Wrinkles from Radiative Corrections

Allowing for wrinkles would appear to entirely eliminate the predictivity of
the FN Ansatz. However, there is a natural bound on the size of the wrinkles
that arises from demanding that the observed flavor structure in the IR arises
predominantly from tree-level contributions to the effective operators below
the scale $M$. Requiring that the tree-level contribution (including wrinkles)
to the Yukawa coupling is larger than any subleading corrections from loops
leads to a number of consistency conditions on the Yukawas, which in turn set
a bound on the wrinkles. Provided these conditions are satisfied, the flavor
structure in the IR is still determined by the FN mechanism in a predictive
way, with departures from the minimal implementation parameterized by the
wrinkles.

Figure 3: Left: the tree level $S_{1}^{\dagger}\bar{u}_{i}\bar{e}_{j}$
coupling. Right: A loop contribution to the same spurion, leading to the
spurion contribution described by Eq. (13). In both diagrams the dots indicate
a chain of flavon and heavy fermion vertices, whose length is determined by
the horizontal charges of the particles, which we suppress for clarity.

To illustrate these constraints, consider the Yukawa coupling matrix between
the right-handed up-type quarks and the right-handed charged leptons for the
$S_{1}$ leptoquark model in Eq. (8), $\Delta_{\bar{u}\bar{e}}^{ij}$. In a FN
setup, this coupling arises from a non-renormalizable operator with a minimal
number of flavons. It can be UV completed with a tree-level chain of heavy
fermions and flavons with a single leptoquark vertex, as illustrated on the
left in Figure 3. However, the same operator can also be generated at higher
order by including SM fermions in the FN chain $\bar{u}^{i}\to Q^{k}\to
L^{l}\to\bar{e}^{j}$, as shown on the right in Figure 3. The first and last
connections include additional Higgs insertions that are tied together to form
a loop, and the $Q^{k}\to L^{l}$ connection involves a leptoquark interaction.
Thus, the higher-order contribution to $\Delta_{\bar{u}\bar{e}}^{ij}$ is:

$\Delta_{\bar{u}\bar{e}}^{ij}\Big{|}_{\mathrm{loop}}\sim\frac{1}{16\pi^{2}}\left(Y_{Q\bar{u}}^{T}\cdot\Delta_{QL}^{*}\cdot
Y_{L\bar{e}}\right)^{ij}.$ (13)

Demanding this contribution to be smaller than the tree-level contribution,
and assuming the absence of any artificial cancellations, leads to a lower
bound on the Yukawa coupling $\Delta_{\bar{u}\bar{e}}^{ij}$ and an upper bound
on the entries of $\Delta^{*}_{QL}$. This bound begets a set of consistency
conditions on the wrinkles:

$\displaystyle\big{|}\Delta_{\bar{u}\bar{e}}^{ij}\big{|}$
$\displaystyle\gtrsim\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}^{T}\cdot\Delta_{QL}^{*}\cdot
Y_{L\bar{e}}\big{)}^{ij}\Big{|},$ (14)
$\displaystyle\qquad\qquad\implies\lambda^{\omega_{\bar{u}\bar{e}}^{ij}+{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[\bar{u}_{i}]+[\bar{e}_{j}]$}{\scalebox{1.0}[1.05]{$\mid$}}}$
$\displaystyle\gtrsim\frac{1}{16\pi^{2}}\big{|}Y_{Q\bar{u}}^{T}\big{|}^{ik}\lambda^{\omega_{QL}^{kl}+{\scalebox{1.0}[1.05]{$\mid$}}\raisebox{0.70004pt}{\scriptsize$[Q_{k}]+[L_{l}]$}{\scalebox{1.0}[1.05]{$\mid$}}}\big{|}Y_{L\bar{e}}\big{|}^{lj}\
,$

where there is an implicit summation over the indices $k$ and $l$ above. While
the SM Yukawas on the right hand side of this relation may also contain
wrinkles, it is the IR value of the coupling that appears, which is fit to the
SM masses and mixing angles.

Similar consistency conditions were proposed in Refs. [32, 33, 34], neglecting
the loop factor. Other similar constraints (including the loop factor) have
been considered as naturalness constraints on models of flavorful new physics
[57]. We settle for the weaker constraint, including the loop factor, as a
concrete, irreducible bound.444RG evolution of the leptoquark couplings also
does not change the above set of bounds, as long as one imposes the
consistency conditions at the matching scale of order $M$. The structure of
the one loop Yukawa RGEs involves the same higher order operators as appearing
in our consistency condition. Thus, imposing the consistency condition at the
matching scale ensures that running is a small effect and can be neglected.
Consequently, RG evolution to scales below the matching scale ensures that the
consistency condition (inequality) holds at all such scales.  Note that there
are also other higher order contributions to the spurions, such as those from
higher-dimensional operators with the Higgs replaced by its vacuum expectation
value, but they will be smaller than the one in Eq. (13), since
$v^{2}/M^{2}<1/16\pi^{2}$.

More generally, a complete set of consistency conditions can be derived by
again considering the Yukawas as spurions under $G_{\textrm{flavor}}$. In the
absence of any additional symmetries, contributions similar to Eq. (13) arise
from any combination of Yukawa couplings that transform in the same
representation of $G_{\textrm{flavor}}$. The complete list of leading
consistency conditions for all of the Yukawa couplings in the SM extended with
the $S_{1}$ leptoquark are listed in Appendix A.

These inequalities must be satisfied for any wrinkled FN setup involving
additional flavor spurions, and they impose non-trivial constraints on the
size of the wrinkles introduced in Eq. (11).555The consistency conditions, as
written, hold neglecting $\mathcal{O}(1)$ couplings; there may be small
deviations from including them. The details of these constraints depend on the
particular charge assignment of the SM fermions, but once these are fixed, a
degree of predictiveness is returned to the FN Ansatz, even in the presence of
wrinkles. As pointed out in Refs. [32, 33, 34], these consistency conditions
are trivially satisfied in a vanilla FN setup without wrinkles, as a result of
the triangle inequality.

As an example of how this bound works with nonzero wrinkles, consider the
charge assignment in Table 1 with $q_{0}=0$, $l_{0}=-1$, $X=+1$, $Y=-1$ and
all other sign choices being positive. Assuming no wrinkles in the SM Yukawas,
the bound on $\omega_{\bar{u}\bar{e}}^{33}$ from Eq. (14) becomes

$\displaystyle\omega^{33}_{\bar{u}\bar{e}}$ $\displaystyle\leavevmode\nobreak\
\lesssim\leavevmode\nobreak\
\sum_{k,l}\Big{(}\big{|}[Q_{k}]+[\bar{u}_{3}]\big{|}+\big{|}[Q_{k}]+[L_{l}]\big{|}+\omega_{QL}^{kl}+\big{|}[L_{l}]+[\bar{e}_{3}]\big{|}\Big{)}$
(15)
$\displaystyle\qquad\qquad+\log_{\lambda}\\!\frac{1}{16\pi^{2}}-\big{|}[\bar{e}_{3}]+[\bar{u}_{3}]\big{|}$
$\displaystyle\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\
2+\omega_{QL}^{33}+\log_{\lambda}\\!\frac{1}{16\pi^{2}},$

where in the last line we have assumed that $k=l=3$ is the largest entry in
$\omega_{QL}^{kl}$, which is typically the case. We see that, at least for
this consistency condition, up to five wrinkles on
$\Delta_{\bar{u}\bar{e}}^{33}$ are allowed, even without extra wrinkles on
$\Delta_{QL}^{33}$.

A similar argument for general couplings, again using the triangle inequality,
makes it clear that if all $\omega_{\psi\bar{\chi}}^{ij}\geq 0$, a sufficient
condition on the wrinkles is that they are all greater than a loop factor:

$\left(W_{\psi\bar{\chi}}\right)^{ij}\gtrsim\frac{1}{16\pi^{2}}.$ (16)

Note that in this equation, we have assumed a mild separation of scales so
that the logarithms in the loop contribution can be neglected along with other
$\mathcal{O}(1)$ factors in the loop calculation. In this work, we focus on
the bound in Eq. (16) and leave further studies of more accurate lower bounds
on wrinkles for future work. As shown in Eq. (15), this bound may be overly
restrictive, but it provides a useful shortcut for employing wrinkles in an
EFT without having to manually check all the consistency conditions.

###  3.3 UV Completions

We now turn to UV completions of the wrinkles introduced in Eq. (11). Our goal
is not to provide an exhaustive or detailed list of examples, but demonstrate
a proof of principle of potential ways these wrinkles can arise from more
complicated UV completions.

####  3.3.1 Missing Heavy Fermions

As a first concrete realization of the idea sketched in Figure 2, we consider
a situation where one of the heavy fermions with a particular horizontal
charge does not exist in the spectrum. Instead, the chain leading to the
effective operator can only be completed by including additional fermions and
scalars, causing additional suppression.

To illustrate this mechanism, we consider the example in Figure 1 and replace
a single heavy vector-like pair of fermions $U_{1}$, $\bar{U}_{1}$ with two
sets of vector-like pairs, which we will denote by $U_{1}^{(1)}$,
$\bar{U}_{1}^{(1)}$ and $U_{1}^{(2)}$, $\bar{U}_{1}^{(2)}$. These are assumed
to have the same SM and horizontal charges as $U_{1}$, $\bar{U}_{1}$, but also
transform as conjugate pairs under new symmetry groups, $G_{1}$ and $G_{2}$,
respectively. To be explicit, we will take $G_{1}={\mathrm{SU}}(N_{1})$ and
$G_{2}={\mathrm{SU}}(N_{2})$ to be two different continuous, non-Abelian
groups, but the following construction works for arbitrary (continuous or
discrete) groups as well, with straightforward modifications. To complete the
chain diagram, we must also introduce new flavons, which we take to be in the
representations,

$\varphi^{(1)}:(\mathbf{N_{1}},\mathbf{1})_{-1},\qquad\varphi^{(2)}:(\mathbf{1},\mathbf{\overline{N}_{2}})_{-1},\qquad\Phi^{(1,2)}:(\mathbf{\overline{N}_{1}},\mathbf{N_{2}})_{0},$
(17)

where the parentheses indicate the
${\mathrm{SU}}(N_{1})\times{\mathrm{SU}}(N_{2})$ representation, and the
subscript is the horizontal charge. These allow us to construct the diagram
shown in Figure 4, where both of the extra heavy fermion pairs are traversed
between $Q_{2}$ and $\bar{u}_{3}$. The charge assignments forbid the couplings
$\varphi^{(1)}U^{\phantom{(}}_{2}\bar{U}_{1}^{(2)}$ and
$\varphi^{(2)}U_{1}^{(1)}\bar{u}_{3}$, so that this diagram is the leading
effective operator containing $HQ_{2}\bar{u}_{3}$.

Figure 4:  An explicit realization of a “wrinkled” FN chain, where the heavy
quark with horizontal charge $+1$ is replaced by two heavy quarks, along with
additional flavons, transforming under additional symmetries.

Assuming all the scalars acquire vevs $\sim\langle\varphi\rangle$ and that the
new fermions have vector-like masses $\sim M$, this replaces the $\lambda^{2}$
suppression inferred from the horizontal charges with a $\lambda^{3}$
suppression. In other words, this leads to a “wrinkle”,
$W^{23}_{Q\bar{u}}\sim\lambda$.

This construction can be extended to include arbitrarily many wrinkles in
place of a single heavy fermion. For example,
$W^{23}_{Q\bar{u}}\sim\lambda^{2}$ is obtained by introducing additional
mirror quarks, $U_{1}^{(3)}$, $\bar{U}_{1}^{(3)}$, replacing $\varphi^{(2)}$
with a bi-fundamental $\Phi^{(2,3)}$ transforming as
$(\mathbf{1},\mathbf{\overline{N}_{2}},\mathbf{N_{3}})_{0}$, closing the chain
with $\varphi^{(3)}$, which transforms as a
$(\mathbf{1},\mathbf{1},\mathbf{\overline{N}_{3}})_{-1}$; further wrinkles are
obtained for additional mirror quarks. In these types of examples, the Higgs
and chiral fermions of the SM are neutral under the new symmetries, so as to
be compatible with the general arguments in Ref. [12].

Note that with this mechanism, we see an example of the correlation between
wrinkles in different chains. We constructed this wrinkle in the context of
the $Q_{2}$ and $\bar{u}_{3}$ chain, but since we have removed $U_{1}$,
$\bar{U}_{1}$ from the spectrum, the wrinkle necessarily appears in any chain
involving them. For instance, assuming heavy up-like quarks are responsible
for all of the up-type Yukawa couplings, it would also appear in the
$Q_{1}H\bar{u}_{3}$ operator.

####  3.3.2 Extra Abelian Symmetries

Another concrete example in which wrinkles can appear in an effective theory
with the FN Ansatz is realized by considering additional Abelian symmetries in
the UV, under which the SM fermions are charged. In particular, we can
consider gauging the non-anomalous combinations of baryon number, $B$, and the
individual lepton numbers, $L_{e}$, $L_{\mu}$, and $L_{\tau}$, as is
frequently done in model-building for various flavor anomalies [58]. These
symmetries are preserved by the SM Yukawa couplings, but generically violated
by neutrino masses and additional Yukawa couplings between SM fermions and new
BSM fields, such as leptoquarks. For concreteness, we again work with the
$S_{1}$ leptoquark and assume it is neutral under the new symmetry; therefore
the flavor spurion must absorb the remaining ${\mathrm{U}}(1)$ charge. This
means that additional flavons charged under the extra symmetries also must be
included in order to complete the leptoquark Yukawa couplings. The usual
flavon, with ${\mathrm{U}}(1)_{H}$ charge $-1$, is still present, since it is
required to complete the SM Yukawa couplings.666In the presence of neutrino
masses, the extra flavons may also be required to generate the PMNS matrix
structure, depending on the additional symmetries we impose.

In contrast to the UV completions discussed in §3.3.1, where the wrinkles are
always additional suppression factors, the wrinkles that result from these
extra symmetries can naturally either suppress or enhance the size of the
flavor spurions. Another distinction is that we have not removed any fermions
of particular charges from the UV spectrum in this case: we allow fermions
with all required quantum numbers to exist.

Just like other UV models, additional symmetries and the flavons charged under
them can generate a correlated pattern of wrinkles for the different chains.
The details of those correlations depend on whether the new symmetries are
flavor universal or flavor specific; we will discuss examples of both cases.
In order to maintain the predictivity of our example, we also assume
additional symmetries are spontaneously broken at similar scales to the
${\mathrm{U}}(1)_{H}$ symmetry.

The flavor universal case is simpler, but also less flexible because of
interdependence between different chains. For example, assuming
${\mathrm{U}}(1)_{B-L}$ is a symmetry of the theory, we can construct the
leptoquark Yukawa spurions by introducing an additional flavon,
$\tilde{\varphi}$, which we take to have $B-L$ charge $1/3$. The new flavon
will not change any of the SM chains, since they respect the $B-L$ symmetry,
but the leptoquark chains can be different from the usual FN scenario. For
instance, if $\tilde{\varphi}$ has no ${\mathrm{U}}(1)_{H}$ charge, all the
leptoquark Yukawas will become smaller by $\lambda^{2}$, since the external
fermions all have $B-L$ charge difference $\pm 2/3$ without the new flavon. If
$\tilde{\varphi}$ also has ${\mathrm{U}}(1)_{H}$ charge $\geq 1$, then the
pattern of leptoquark chains becomes more intricate. Since each leptoquark
chain must contain exactly two copies of the new $B-L$ flavon and the
remaining difference in horizontal charge requires insertions of the original
flavon, whether a given chain becomes shorter or longer depends on the details
of the assigned horizontal charges.

We have somewhat more freedom in the flavor specific case. As an example,
consider introducing a new ${\mathrm{U}}(1)_{B-3L_{e}}$ symmetry. Now both the
PMNS matrix and chains for the leptoquark Yukawa couplings require new flavons
charged under both ${\mathrm{U}}(1)_{H}$ and ${\mathrm{U}}(1)_{B-3L_{e}}$. We
introduce two additional flavons: $\varphi_{2}$ is necessary to generate PMNS
matrix entries of the correct size, and $\varphi_{3}$ is necessary to complete
the leptoquark chains while respecting the additional symmetries. These
flavons have $B-3L_{e}$ charges

$[\varphi]=0\qquad[\varphi_{2}]=3\qquad[\varphi_{3}]=-1/3.$ (18)

Each also carries ${\mathrm{U}}(1)_{H}$ charge $-1$. Including these extra
symmetries and flavons charged under them creates wrinkles by changing the
required number of vev insertions for the leptoquark couplings compared to the
spurion size we would naively expect with only these ${\mathrm{U}}(1)_{H}$
charges. For example, if we consider only the couplings to the third
generation leptons, we make the right-handed $\mu$ and $\tau$ couplings
smaller while leaving the right-handed $e$ coupling and the left-handed
couplings unaffected. This is shown in Figure 5. Nonetheless, despite the
additional freedom in the flavor specific case, it is still challenging to
obtain certain patterns of wrinkles, such as those constrained by the triangle
inequality.

Figure 5:  Chains before and after adding flavons charged under a
${\mathrm{U}}(1)_{B-3L_{e}}$ symmetry to generate wrinkles. The horizontal
charges correspond to Table 1 with $q_{0}=0$, $l_{0}=2$, and $Y=1$. We observe
that the new ${\mathrm{U}}(1)_{B-3L_{e}}$ symmetry and its flavons modify the
prediction of the model for some of the leptoquark couplings in the IR.

Finally, we comment on a few modifications to the examples above. First, we
note that it is possible to modify this approach by charging the leptoquark
under ${\mathrm{U}}(1)_{H}$ instead of/in addition to additional flavon(s).
Similar to the $B-L$ charged flavons, this is another mechanism to add
wrinkles to the leptoquark couplings without affecting the SM couplings. In
principle, we can also charge the leptoquark under the additional symmetries
we discussed in this section, but note that we are not always guaranteed a
charge assignment which makes all of the couplings invariant. Second, we note
that like the previous case, other modifications such as using discrete
Abelian symmetries also behave similarly. However, we can not replace these
Abelian symmetries with non-Abelian ones [12], because we are charging the SM
fermions under the new symmetry. This is in contrast to the previous case,
where only internal fermions are charged under new non-Abelian symmetries.

While we have provided two different ways in which wrinkles could be
generated, we have not exhausted the possibilities. These are only examples,
and there are undoubtedly many more options for generating wrinkles, which
would be interesting for future work. Since the details of a particular model
are not the central point of this paper, we now move to discussing a full
example in the IR.

##  4 Dessert: $B\to K\bar{\nu}\nu$ in a Wrinkled Setup

To demonstrate the ideas of the previous sections with a specific example, in
this section we study the phenomenology of the $S_{1}$ leptoquark introduced
in Eq. (8) with particular flavor Ansätze in detail. Such Ansätze correlate
the contribution of $S_{1}$ to different observables. As mentioned in the
previous section, the inclusion of wrinkles in a FN Ansatz can change the
relative sizes of predictions for different flavor observables. This could
allow a model to accommodate a significant excess over the SM in one
observable, while suppressing other observables that would otherwise be too
constraining.777Signals of leptoquarks in all flavor experiments can also be
suppressed by choosing $q_{0}$ and $l_{0}$ (defined in Table 1) such that the
quarks’ and leptons’ charges are very far apart, but this requires an
unnaturally large separation of charges. This choice also does not permit the
explanation of any discrepancies in flavor experiments because it suppresses
leptoquark contribution to all observables.

As an illustration, we will focus on constructing a model that can give rise
to a large signal in the semi-leptonic decay $B^{+}\rightarrow
K^{+}\bar{\nu}\nu$. $\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)$ is an interesting test case for several reasons.
Assuming the vanilla FN Ansatz, the mass range preferred for new physics near
the current experimental sensitivity is in the few TeV range, and small hints
of flavorful new physics may have already been detected [37]. Like all flavor-
changing neutral currents, the $b\to s\bar{\nu}\nu$ transition is greatly
suppressed in the SM. It is also relatively clean theoretically, with the
uncertainties in the hadronic form factors and from perturbative effects well
under control [59, 60, 61, 62, 63, 64, 65, 66, 67]. This situation, along with
the prospect of observing the decay at the Belle II experiment in the near
future, make it an intriguing probe of BSM physics [68, 69, 70]. We use this
specific observable as a testbed of various ideas introduced in the previous
section; similar studies can be carried out for any other flavorful anomalies
that may emerge in experimental data.

###  4.1 $B\to K\bar{\nu}\nu$ in the SM and Beyond

In order to understand how various FN Ansätze contribute to
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$, we first need to
discuss the SM and leptoquark contributions, as well as experimental bounds.
Typically, $\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ is
parameterized in terms of the Wilson coefficients $C_{R}^{ij}$ and
$C_{L}^{ij}$, which are defined implicitly in the effective Hamiltonian
governing $b\to s\bar{\nu}\nu$ transitions

$\mathcal{H}_{\textrm{eff}}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}^{\phantom{*}}V_{ts}^{*}\big{(}C_{L}^{ij}\mathcal{O}_{L}^{ij}+C_{R}^{ij}\mathcal{O}_{R}^{ij}\big{)}+\textrm{h.c.}$
(19)

where

$\mathcal{O}^{ij}_{L}=\frac{\alpha_{\mathrm{em}}}{2\pi}\big{(}s_{L}^{\dagger}\bar{\sigma}^{\mu}b_{L}\big{)}\big{(}\nu_{j}^{\dagger}\bar{\sigma}_{\mu}\nu_{i}\big{)},\qquad\mathcal{O}^{ij}_{R}=\frac{\alpha_{\mathrm{em}}}{2\pi}\big{(}s_{R}^{\dagger}\sigma^{\mu}b_{R}\big{)}\big{(}\nu_{j}^{\dagger}\bar{\sigma}_{\mu}\nu_{i}\big{)},$
(20)

and $i,j=e,\mu,\tau$ are neutrino flavor indices.

Figure 6: Example Feynman diagrams leading to $b\to s\bar{\nu}\nu$ transitions
in the SM extended with an $S_{1}$ leptoquark. The left (center) diagram show
the leading one loop SM contributions with the penguin (box) topology, while
the right diagram illustrates the tree-level leptoquark contribution. The $Z$
in the left diagram could also connect to the top line instead.

In the SM (and in the $S_{1}$ leptoquark model we consider below), only
$C_{L}$ is non-zero. The leading contribution to the SM value of the Wilson
coefficient arises from diagrams such as those in Figure 6. Also including NLO
QCD corrections [59, 60, 61] and two-loop electroweak contributions [65], the
SM Wilson coefficient is

$C_{L}^{\,ij,\,\textrm{SM}}=(-6.353\pm 0.074)\,\delta_{ij},$ (21)

where $\delta_{ij}$ captures the fact that the SM contributions are lepton
flavor conserving. This leads to a prediction for the branching ratio [69,
70],

$\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)\Big{|}_{\mathrm{SM}}=(0.46\pm 0.05)\times 10^{-5},$
(22)

where we are inclusive to neutrino flavor.

This process has been searched for at Belle and BaBar by tagging the second
$B$ meson in either a hadronic or semileptonic decay [71, 72, 73]. Similar
searches exist for $\mathrm{BR}(B\rightarrow K^{*}\bar{\nu}\nu)$, e.g. see
Refs. [71, 72]. Each of these channels leads to the same qualitative
conclusions; thus, for the rest of this work we will focus on
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurements, for
simplicity. A combination of these results yields a 90% C.L. upper limit on
the branching ratio of

$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)<1.6\times
10^{-5}.$ (23)

Recently, Belle II has searched for the same decay using an inclusive tagging
technique, which allows them to partially compensate for their smaller dataset
and larger backgrounds [37]. Though not yet statistically significant, a
combination of these results (assuming their uncertainties are uncorrelated)
leads to a best fit value of

$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)=(1.1\pm 0.4)\times
10^{-5},$ (24)

which leaves room for a BSM contribution on top of the SM prediction in Eq.
(22). The uncertainties in all of these estimates—both the tagged and
inclusive searches—are predominantly statistical, and are expected to improve
and become comparable to the theoretical uncertainty in Eq. (22) with the
forthcoming full Belle II dataset [74]. Therefore, while it remains to be seen
if any signals of new physics exist in this channel, it provides an
interesting application of our wrinkled FN setup.

The $S_{1}$ leptoquark contributes to $b\to s\bar{\nu}\nu$ transitions via the
tree-level diagram shown on the right in Figure 6. It generates a Wilson
coefficient

$C_{L}^{ij}\propto\frac{v^{2}}{m_{S_{1}}^{2}}\Delta_{QL}^{3i}\Delta_{QL}^{2j\,*}$
(25)

for the effective theory of Eq. (19). Since this is the same operator as
generated in the SM, it is convenient to capture these effects by considering
the ratio:

$R_{K}^{\nu\nu}\equiv\frac{\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)}{\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)\big{|}}_{\mathrm{SM}}.$ (26)

The contribution from $S_{1}$ is given by [75, 76] (see also Refs. [69, 35])

$R_{K}^{\nu\nu}=1-y\real\left[\frac{(\Delta_{QL}^{3i}\Delta_{QL}^{2i\,*})}{V_{tb}^{\phantom{*}}V_{ts}^{*}}\right]+\frac{3y^{2}}{4}\frac{(\Delta_{QL}^{3i}\Delta_{QL}^{3i\,*})(\Delta_{QL}^{2j}\Delta_{QL}^{2j\,*})}{\big{|}V_{tb}^{\phantom{*}}V_{ts}^{*}\big{|}^{2}},$
(27)

with a sum over repeated lepton indices in each term, and

$y\equiv-\frac{2\pi
v^{2}}{6C_{L}^{\mathrm{SM}}\alpha_{\mathrm{em}}m_{S_{1}}^{2}}\simeq\bigg{(}\frac{1.2\,\mathrm{TeV}}{m_{S_{1}}}\bigg{)}^{2}.$
(28)

In terms of $R_{K}^{\nu\nu}$, the 90% C.L. limit and 68% C.L. preferred values
of the branching ratio in Eqs. (23) and (24) translate to

$R_{K}^{\nu\nu}<3.4,\qquad R_{K}^{\nu\nu}\in[1.5,3.3],$ (29)

respectively. The interpretation of these bounds in the context of the
leptoquark depends on the assumptions made about the hierarchies in
$\Delta_{QL}^{ij}$, to which we now turn.

Observable |  $S_{1}$ Yukawa Couplings | Experimental Result | Future Bounds
---|---|---|---
$\textrm{BR}(B^{+}\to K^{+}\bar{\nu}\nu)$ | $\Delta_{QL}^{3i}\times(\Delta_{QL}^{2j})^{*}$ |  $(1.1\pm 0.4)\times 10^{-5}$ [37] | -
electron EDM | $(V^{*}\Delta_{QL})^{31}\times(\Delta_{\bar{u}\bar{e}}^{31})^{*}$ |  $<4.1\times 10^{-30}\leavevmode\nobreak\ e\,$cm [77] |  $<10^{-31}\leavevmode\nobreak\ e\leavevmode\nobreak\ $cm [78, 79]
$\textrm{BR}(\mu\to e\gamma)$ | $\begin{array}[]{c}(V^{*}\Delta_{QL})^{32}\times\Delta_{\bar{u}\bar{e}}^{31}{}^{\phantom{*}}\\\ \Delta_{\bar{u}\bar{e}}^{32\,*}\times(V^{*}\Delta_{QL})^{31\,*}\end{array}$ |  $<4.2\times 10^{-13}$ [80] |  $<6\times 10^{-14}$ [81]
$\textrm{CR}(\mu\to e)_{N}$ | $(V^{*}\Delta_{QL})^{11\,*}\times(V^{*}\Delta_{QL})^{12}$ |  $<7.0\times 10^{-13}$ [82] |  $<2.5\times 10^{-18}$ [83, 84]
$\textrm{BR}(\tau\to\mu\gamma)$ | $\begin{array}[]{c}(V^{*}\Delta_{QL})^{33}\times\Delta_{\bar{u}\bar{e}}^{32}{}^{\phantom{*}}\\\ \Delta_{\bar{u}\bar{e}}^{33\,*}\times(V^{*}\Delta_{QL})^{32\,*}\end{array}$ |  $<4.2\times 10^{-8}$ [85] |  $<6.9\times 10^{-9}$ [86, 87]
$\textrm{BR}(K^{+}\to\pi^{+}\bar{\nu}\nu)$ | $\Delta_{QL}^{2k}\times(\Delta_{QL}^{1k})^{*}$ |  $<1.88\times 10^{-10}$ [88] |  $(8.4\pm 0.4)\times 10^{-11}$ [89]
$\Delta m_{B_{s}}$ | $(\Delta_{QL}\,\Delta_{QL}^{\dagger})^{32}$ |  $\Delta C_{B_{s}}\leq 0.09$ [90] |  $\Delta C_{B_{s}}\leq 0.026$ [90]

Table 2: Here we show the experimental results for
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ and a few other
constraining observables; we also show the predominant $S_{1}$ Yukawa
couplings contributing to each. Note that for $B$-mixing, we use the
experimental uncertainty on the quantity $C_{B_{s}}$ as defined in Eq. (33).
For $K^{+}\to\pi^{+}\nu\bar{\nu}$, the future bound corresponds to reaching a
$5\%$ experimental uncertainty on the SM branching ratio [91]. The muon to
electron conversion rate in nuclei, $\textrm{CR}(\mu\to e)_{N}$, gets
contributions from both dipole and four-fermion operators; we show the Yukawas
entering the four-fermion operator that is dominant in the FN Ansatz
(associated with a left-handed vector current) here, while the complete set is
given in Appendix B. The current (future) bound listed for it is on the
conversion rate in a gold (aluminum) nucleus.

###  4.2 Constraints with Different Flavor Ansätze

In addition to $b\to s\bar{\nu}\nu$ transitions discussed above, the $S_{1}$
leptoquark can contribute to a number of flavor-changing processes or
precision observables that are constrained by experiments. These include
electric and magnetic dipole moments of SM particles, LFV decays, leptonic and
semi-leptonic meson decays, flavor-violating decays of gauge bosons, and
neutral meson mixing. Some of the most powerful observables, and their
dependence on the leptoquark Yukawa couplings are summarized in Table 2.888For
simplicity, we work with flavor basis neutrinos, so no dependence on the PMNS
matrix appears. As is apparent from the table, the observables depend on
numerous different combinations of the leptoquark couplings. More details
about the observables, including the dependence on the leptoquark couplings
and references to more complete treatments in the literature, are given in
Appendix B.

Because the contributions to various observables are correlated, we need to
pick a particular Ansatz and study it in order to understand these
constraints. In the rest of this section, we study these constraints in the
context of three different flavor Ansätze: flavor anarchy, vanilla FN, and FN
with wrinkles. In particular, we explore how adding wrinkles can alleviate
constraints while maintaining consistency with
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurements.

Without any assumptions about the underlying structure, a minimal assumption
is that all elements of $\Delta_{QL}$ and $\Delta_{\bar{u}\bar{e}}$ are
$\mathcal{O}(1)$. This assumption is commonly referred to as “flavor anarchy”.
Under this assumption, the mass of the leptoquark consistent with the
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurements is
$m_{S_{1}}\in(9,\leavevmode\nobreak\ 18)$ TeV. On the other hand, measurements
of the electron EDM and other flavor-changing processes constrain the mass of
the leptoquark to be above $\sim 10^{5}\leavevmode\nobreak\ \textrm{TeV}$. The
resulting limits for some of the observables considered are shown as yellow
bars in Figure 7. To calculate these ranges for observables that are already
measured experimentally, we demand the leptoquark contribution to be within
one standard deviation of the measured value, while for others we use the
reported upper bounds from Ref. [92].999The exceptions to this are $R_{D}$ and
$a_{\mu}$, where we take the maximum leptoquark mass consistent to within
$3\sigma$ and $4\sigma$, respectively, of the experimental measurement for the
anarchic coupling case, and use the $2\sigma$ ellipse for the preferred mass
range in the wrinkled case.  For the electron EDM, a CP-odd observable, we
assume a purely imaginary coupling to show the maximum reach of the
experimental results.

It is clear that without any flavor texture on the leptoquark Yukawas,
observables such as the electron EDM, LFV decays, or meson-mixing parameters
rule out the leptoquark mass range relevant for
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$.101010Our model
can also contribute to $a_{\mu}$ at one loop to explain the observed anomaly
[93, 94], although recent lattice calculations [95, 96, 97, 98, 99, 100, 101]
and measurements [102] hint toward a smaller discrepancy with the experimental
data. However, other observables already rule out the leptoquark mass range
that has a large enough contribution to $a_{\mu}$. See Refs. [103, 104, 105,
106, 107, 108, 109, 75, 35, 110, 111, 112, 113, 114, 46, 115] for other
solutions to this anomaly, including attempts at embedding the solution in a
FN construction. We have also checked the contribution of our setup to many
other similar observables (electron and tau MDM, $\tau\rightarrow e\gamma$,
$K\rightarrow e\nu$, various other $D$ meson decays, $D_{s}\rightarrow e\nu$,
$B\rightarrow e\nu$, $\pi\rightarrow ee$, $\pi\rightarrow\mu e$), but find
that the constraints they place are not as competitive for our model.

Figure 7:  The leptoquark mass range probed by various observables if the
Yukawa couplings of the leptoquark are either $\mathcal{O}(1)$ (yellow),
follow the vanilla FN setup in Eq. (31) (green), or the same FN setup plus the
wrinkles from Eq. (32) (blue). The preferred range for explaining some
existing anomalies are shown in red, assuming the wrinkled setup. The
undetermined $\mathcal{O}(1)$ factors in the Yukawas (folded in $r_{ij}$ in
Eq. (1)) can further affect the leptoquark contribution and slightly change
the mass range probed by each observable. We see that in our wrinkled setup,
the mass range that explains the current discrepancy in
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurement
(between the horizontal dashed lines) can also be probed by the LFV processes
$\mu\rightarrow e\gamma$ and $\mathrm{CR}(\mu\to e)$, and the electron EDM in
near future measurements.

Thus we are led to consider embedding the $S_{1}$ leptoquark in a FN model of
flavor. This has the benefit of not only alleviating some of the experimental
constraints discussed above, but also relating it to the SM flavor puzzle.

As discussed in §2, aside from the general shifts in the lepton and quark
horizontal charges, there are only a handful of possible charge assignments
that give rise to the correct pattern of SM masses and mixing angles. For
concreteness, we choose horizontal charges from Table 1 with $q_{0}=0$,
$l_{0}=-1$, and $X=-Y=-1$. This yields

$\begin{gathered}([Q_{1}],\,[Q_{2}],\,[Q_{3}])=(3,2,0),\qquad([\bar{u}_{1}],\,[\bar{u}_{2}],\,[\bar{u}_{3}])=(4,1,0),\qquad([\bar{d}_{1}],\,[\bar{d}_{2}],\,[\bar{d}_{3}])=(3,3,2),\\\\[5.0pt]
([L_{1}],\,[L_{2}],\,[L_{3}])=(0,-1,-1),\qquad([\bar{e}_{1}],\,[\bar{e}_{2}],\,[\bar{e}_{3}])=(8,6,4).\end{gathered}$
(30)

With these charge assignments, the FN Ansatz for the leptoquark couplings is:

$\Delta_{QL}\leavevmode\nobreak\ \sim\leavevmode\nobreak\
\begin{pmatrix}\lambda^{3}&\lambda^{2}&\lambda^{2}\\\
\lambda^{2}&\lambda&\lambda\\\
1&\lambda&\lambda\end{pmatrix},\qquad\Delta_{\bar{u}\bar{e}}\leavevmode\nobreak\
\sim\leavevmode\nobreak\
\begin{pmatrix}\lambda^{12}&\lambda^{10}&\lambda^{8}\\\
\lambda^{9}&\lambda^{7}&\lambda^{5}\\\
\lambda^{8}&\lambda^{6}&\lambda^{4}\end{pmatrix}.$ (31)

The resulting bounds, neglecting $\mathcal{O}(1)$ Yukawa factors, are shown as
the green bars in Figure 7. Compared to the anarchic Ansatz, the bounds on the
leptoquark mass are significantly relaxed.

Nevertheless, it is clear that the mass range consistent with the
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurements at
Belle II is still excluded by other observables under the FN Ansatz. We have
checked that—while the exact bounds for different observables can change
significantly—this conclusion remains unchanged for the other possible charge
assignments enumerated in Table 1. If any deviation from SM is observed in
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$, the $S_{1}$
leptoquark embedded in a vanilla FN model cannot explain the anomaly while
respecting bounds from other measurements.

Adding wrinkles to the FN Ansatz as discussed in §3 can ameliorate the tension
with these observables. Using the scaling of the observables with the
leptoquark Yukawas shown in Table 2 as a guide, we add the following wrinkles
(as defined in Eq. (11)) to the leptoquark Yukawa matrices:

$W_{\bar{u}\bar{e}}^{ij}=\lambda^{3},\qquad
W_{QL}=\left(\begin{matrix}\lambda^{3}&\lambda^{3}&\lambda^{3}\\\
\lambda^{3}&1&1\\\ \lambda^{3}&1&1\end{matrix}\right).$ (32)

This is the largest number of wrinkles we can add to suppress the leptoquark
contribution to the most constraining observables (especially electron EDM,
$\mu\rightarrow e\gamma$, $\tau\rightarrow\mu\gamma$, and meson mixing
observables), while retaining consistency with the naïve constraint
$\omega\gtrsim\lambda^{3}\sim 1/16\pi^{2}$ from §3.2 and leaving the
contribution to $\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$
mostly intact. Further suppression with additional powers of $\lambda$ may be
possible, but must be carefully checked with all of the consistency conditions
in Appendix A.

It is worth emphasizing that it is not obvious how to get the pattern of
wrinkles in Eq. (32) from the example UV completions discussed in §3.3.
Nevertheless, we can treat them consistently in an effective field theory
approach, and leave the model-building to future work. Note also that with the
additional suppression of the right-handed Yukawa couplings, the phenomenology
of this model resembles that of the RPV down squark as discussed in §2.2.

The contribution of this wrinkled FN setup to various observables is shown by
blue bars in Figure 7. We find that the set of wrinkles from Eq. (32)
sufficiently suppresses the contribution to other observables, so that they
are all compatible with the mass range of interest for
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$. In particular,
bounds from meson mixing observables and leptonic meson decays are
circumvented. Within this wrinkled setup, the viable leptoquark mass range
that can account for a signal in $\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)$ is slightly above the current direct search bounds
at the LHC (see Refs. [116, 117]) and could be detected in future searches at
the LHC or future hadron [118, 119, 120, 121] or lepton [122, 123, 124, 125,
126, 127] colliders.

There are several observables which probe a similar mass range to
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ which will see
significant improvement in experimental measurements soon. In particular,
these observables include $\mu\to e\gamma$, $\mathrm{CR}(\mu\to e)$, and
electron EDM, though the precise mass range depends on $\mathcal{O}(1)$ Yukawa
couplings in the UV completion. As a result, they could be the smoking gun
signal of an FN-like $S_{1}$ leptoquark solution to any future excess observed
in $\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$. Since the
experimental precision on both of these (and several other) observables is
expected to improve significantly in the near future, we will dedicate the
next subsection to discussing potential discovery prospects for this wrinkled
FN scenario.

###  4.3 Predictions for Future Measurements

We have already seen that adding wrinkles to a FN Ansatz allows for greater
flexibility in simultaneously accommodating experimental deviations from the
SM while satisfying constraints from other observables and explaining the
observed pattern of SM masses and mixing angles. As we will now emphasize,
despite this added flexibility, these choices still make concrete predictions
for other observables, which can be tested in future experiments. The
importance of these tests lies in being able to probe indirect information
about the underlying UV model which is hidden in the charge assignments and
wrinkles in the IR.

Several upcoming experiments will provide concrete tests of our wrinkled
Ansatz. When assuming the wrinkled FN Ansatz from Eq. (32) for the leptoquark
Yukawa couplings, several classes of observables— including LFV processes, the
electron EDM, meson-mixing measurements, and the decay
$K\rightarrow\mu\nu$—have a present sensitivity to roughly the same mass scale
as $\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$. Moreover, the
mass reach of many of these observables is expected to improve significantly
with forthcoming experimental data. Since we have suppressed our model
contribution to these observables as far as possible while satisfying the
bound from the consistency condition in Eq. (16), these correlated signals
allow for a definitive test of these types of wrinkled models within the FN
mechanism.

Figure 8:  Predictions for the $S_{1}$ leptoquark contributions to precision
observables with the wrinkled (blue, solid) and vanilla (green, dashed) FN
Ansätze described in §4.2. We show the $\mu\to e$ conversion rate in an
aluminum nucleus (top left), the electron EDM (bottom left), the relative new
physics contribution to $\Delta m_{B_{s}}$ (top right), and
$\textrm{BR}(\tau\to\mu\gamma)$ (bottom right), using solid (dashed) lines for
current (future) experimental bounds or sensitivity. We do not show the best
current bounds on $\mu\to e$ conversion rate, $<7\times 10^{-13}$, from
SINDRUM II [82] since it was made with a different nucleus (gold). The red
band indicates the mass range of interest for
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$, as in Figure 7.

At the moment, the strongest bound on LFV processes involving muons is the 90%
C.L. limit, $\textrm{BR}(\mu\to e\gamma)<4.2\times 10^{-13}$ set by the MEG
experiment [80]. In the future, however, the most powerful probes of this
model will come from searches for $\mu\to e$ conversion in atomic nuclei. As
discussed in more detail in Appendix B, the conversion rate depends not only
on the dipole operator relevant for $\mu\to e\gamma$ and $\mu\to 3e$ decays,
but also on four-fermion operators including the first generation quarks
generated by integrating out the leptoquark. Future prospects for detecting
$\mu\rightarrow e$ conversion include the COMET experiment, which will set a
limit on the conversion rate of $7\times 10^{-15}$ ($2.6\times 10^{-17}$) in
Phase-I (Phase-II) [128, 129], and at Mu2e, which aims at a final sensitivity
of $2.5\times 10^{-18}$ [83, 84]111111This sensitivity might be achievable at
Mu2e-II, a proposed upgrade of Mu2e using the PIP-II accelerator at Fermilab,
potentially with a target material other than aluminum [130, 131]., both in
aluminum nuclei. For more discussion on current and forthcoming searches for
LFV, see Refs. [132, 133, 134, 10, 130, 131].

In the top left panel of Figure 8, we show the predicted $\mu\to e$ conversion
rate in aluminum nuclei as a function of the leptoquark mass, with the
wrinkled FN Ansatz taken for the Yukawa couplings. The
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$-preferred region
discussed in §4.2 is highlighted in red, while the dashed horizontal lines
show the future sensitivities for the conversion rate. We see that even
Phase-I of the COMET experiment will be sensitive to the mass range preferred
by $B^{+}\rightarrow K^{+}\bar{\nu}\nu$ measurements, while Mu2e will
decisively test all of the relevant parameter space predicted by this model of
flavor.

For the electron EDM, the bounds from the ACME II and JILA experiments [135,
77] are at the level $d_{e}<1.1\times 10^{-29}$ and $4.1\times
10^{-30}\,e\textrm{ cm}$, respectively. For anarchic flavor couplings, this
excludes masses up to $\sim 10^{5}\,\textrm{TeV}$. A vanilla FN Ansatz relaxes
this constraint to $\sim 10^{2}\,\textrm{TeV}$, and with the additional
wrinkles invoked in Eq. (32), this bound weakens to $m_{S_{1}}\gtrsim
2.7\,\textrm{TeV}$. Random factors of $\mathcal{O}(1)$, neglected throughout
our calculations, can slightly affect the reach on $m_{S_{1}}$. The fact that
this is the same mass range as favored by $\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)$ measurements, and that the reach in $m_{S_{1}}$
scales faster with improvements to electron EDM measurements compared to other
observables, underscores the importance of future electron EDM experiments in
probing our model. In the coming years, experimental advances and new
technologies promise to increase the sensitivity of EDM experiments by an
order of magnitude or more [78, 79, 136]. In the lower-left panel of Figure 8,
we show the predicted value of the electron EDM as a function of the
leptoquark mass, alongside current bounds and a projected constraint of
$10^{-31}\,e\textrm{ cm}$, assuming an $\mathcal{O}(1)$ CP-violating phase. As
is clear from the figure, future EDM experiments will decisively test this
model, up to scales $m_{S_{1}}\sim\mathcal{O}(10)\,\textrm{TeV}$.

For the meson mixing observables, we focus in particular on the neutral
$B_{s}$ meson mass difference, $\Delta m_{B_{s}}$, whose matrix element is
directly related to the $B^{+}\rightarrow K^{+}\bar{\nu}\nu$ process for the
$S_{1}$ leptoquark. To understand the current sensitivity to new physics of
$B_{s}-\bar{B}_{s}$ mixing, we follow the UTFit analysis [137, 90, 138] and
compute the quantity $C_{B_{s}}$, defined as

$C_{B_{s}}e^{2i\phi_{B_{s}}}\equiv\genfrac{}{}{}{}{\raisebox{2.0pt}{$\langle
B_{s}|\mathcal{H}_{\textrm{mix}}^{\textsc{SM}+\textsc{NP}}|\bar{B}_{s}\rangle$}}{\raisebox{-2.0pt}{$\langle
B_{s}|\mathcal{H}_{\textrm{mix}}^{\textsc{SM}}|\bar{B}_{s}\rangle$}},$ (33)

where $\mathcal{H}_{\textrm{mix}}$ includes the four-fermion operators
responsible for $\Delta F=2$ transitions, as defined in Appendix B.6. The SM
is defined as the point $C_{B_{s}}=1$, $\phi_{B_{s}}=0$, and the allowed size
of the new physics contribution is determined by a global fit to the flavor
sector, with the range determined primarily by the uncertainties on the input
parameters, such as the CKM matrix elements. To be conservative, we consider
only the absolute value of the matrix elements above, and avoid making any
assumptions about the relative phase between the SM and leptoquark
contributions, which is constrained by $\phi_{B_{s}}$.

The resulting current and future sensitivities (where we assume the current
central value is at the SM, for consistency with future projections) are shown
on the top right in Figure 8. The projected future sensitivity of $\Delta
C_{B_{s}}=0.026$ is taken from Ref. [90], based on projections of HL-LHC
results and Belle II results with $50\,\textrm{ab}^{-1}$ integrated
luminosity. We see that the improved sensitivity will start to probe the
leptoquark mass range preferred by the $\mathrm{BR}\left(B^{+}\rightarrow
K^{+}\bar{\nu}\nu\right)$ measurements. It is also worth emphasizing that
these projections do not account for potential improvements in lattice inputs,
and thus could be quite conservative. A statistically significant signal in
any of the aforementioned channels would also warrant a much more careful
analysis of these $B_{s}$-mixing constraints and projections, including phase
information that depends in more detail on the flavor Ansatz, which could
improve sensitivity even further.

A number of additional flavor-changing or flavor-violating decays will be
probed with increasing sensitivity at Belle II. A notable example is the LFV
decay $\tau\to\mu\gamma$, for which the current bound set by Belle is
$\textrm{BR}(\tau\to\mu\gamma)<4.2\times 10^{-8}$ [85]. Belle II is projected
to improve this bound to $6.9\times 10^{-9}$ [86, 87]. In the lower-right
panel of Figure 8, we show the predicted branching ratio of $\tau\to\mu\gamma$
as a function of mass. We see that, for the mass range preferred by the
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ measurements, the
addition of wrinkles in our flavor Ansatz suppresses what would otherwise be a
predicted signal from assuming the FN mechanism.

Finally, the $K\to\pi\nu\bar{\nu}$ decays, which would rule out the preferred
mass range for $B^{+}\rightarrow K^{+}\bar{\nu}\nu$ without wrinkles, have a
sensitivity $\sim 1\,\mathrm{TeV}$ in the wrinkled FN Ansatz. The
$K^{+}\to\pi^{+}\nu\bar{\nu}$ decay was only recently measured (with a
significance of $3.4\sigma$) at the NA62 experiment [88]. A $10$ – $20\%$
precision on this branching ratio is necessary to start excluding
$m_{S_{1}}\sim 2$ – $3\,\textrm{TeV}$, and the requirement for
$K_{L}\to\pi^{0}\nu\bar{\nu}$ is similar. Both of these may be achievable with
future runs at NA62, or at future experiments planned at the NA62 hall at CERN
[89, 139] and at J-PARC [140], and would be an interesting complementary probe
of the same physics considered here.

The preceding discussion demonstrates that all of these powerful, forthcoming
measurements could have a similar sensitivity to new mass scales for an
appropriate choice of wrinkles. Exactly which search channel is ideal depends
on the precise pattern of charges and wrinkles in the IR. However, the
expectation that we will probe these other correlated signals is relatively
robust since the wrinkles in Eq. (32) were chosen to saturate the bound in Eq.
(16) without diminishing the $B^{+}\rightarrow K^{+}\bar{\nu}\nu$ signal.
While this enhancement to $B^{+}\rightarrow K^{+}\bar{\nu}\nu$ was only for
illustration, and not a fit to a true, significant deviation from the SM, it
reveals that for some motivated UV models of flavor, upcoming experiments can
simultaneously test explanations for the SM flavor puzzle.

##  5 Digestifs

When new physics is embedded in the FN mechanism, the FN Ansatz determines the
size of both the SM and new physics couplings. In this paper, we have put
forward a systematic extension of this Ansatz which can change the expected
scaling of the new physics and SM couplings. These changes, referred to as
wrinkles, deviate from the FN pattern that is dictated by the horizontal
symmetry charges. Wrinkles allow us to demand consistency with other
experimental measurements and searches: modifying the relative size of
couplings restores some theories that would otherwise be unfeasible due to the
correlations between different observables from the FN Ansatz. Therefore, they
vastly increase the FN mechanism’s versatility in accommodating solutions to
flavor anomalies. However, owing to radiative corrections, we have also argued
that wrinkles can not give rise to arbitrarily large deviations from vanilla
FN predictions. There are consistency conditions which must be obeyed by the
size of the new wrinkled Yukawas.

While the primary purpose of wrinkles is to give a consistent IR description
for various flavor observables, we have also explored how they can be UV
completed by various different models. Specifically, in this paper we have
given some simple schematic examples of possible UV realizations. In future
work, it would also be interesting to understand more about what patterns of
wrinkles can be realistically realized in the UV and the various models that
can be used to realize them.

Throughout this work, we focused on the phenomenological example of the
$S_{1}$ leptoquark. We discussed the implementation in the IR when the
leptoquark is embedded in a FN model. We also provided a detailed example of
how an enhancement of the leptoquark contribution to
$\mathrm{BR}\left(B^{+}\rightarrow K^{+}\bar{\nu}\nu\right)$ can consistently
respect other experimental bounds, but only if wrinkles are invoked. This
wrinkled setup also motivates future measurements, since several signals would
be on the verge of discovery in this model, even when the number of wrinkles
is enlarged to saturate the simplest consistency condition. In particular, we
showed predictions for the most sensitive upcoming probes, namely $\mu\to e$
conversion and the electron EDM.

While we limited our exploration to a specific example with the $S_{1}$
leptoquark in this paper, it would be interesting to explore how wrinkles can
be applied more broadly. For instance, in our example we fixed the horizontal
charges of the SM particles, but there are many other possible choices that
reliably yield the SM masses and mixing angles. One could explore how changing
the charges affects the correlations and hence the allowed wrinkled Ansatz,
and see which observables remain correlated to the same mass scale more
generally. It would also be intriguing to include other flavor spurions or to
add wrinkles to the SM couplings in addition to the new physics couplings.
Moreover, it would be useful to do a broad methodical study on the effect of
$\mathcal{O}(1)$ numbers in different spurions to explore naturalness in these
types of models; see Refs. [47, 43, 44] for previous studies of naturalness in
such models.

Aside from the flexibility permitted by wrinkles, it is worthwhile to
emphasize a separate point about FN models in general: there is more than one
charge assignment that can naturally generate the observed SM masses and
mixings, beyond just the overall shift in the quark and lepton charges. In
particular, we find that the charges of first generation fermions can be
either larger than or smaller than other two generations. This is in contrast
to a criterion in Ref. [11], where it was demanded that charges increase
monotonically between generations. However, this general FN charge assignment
is still not anomaly free and requires some cancellation mechanism, such as
Green-Schwarz.

With a number of precision flavor experiments gathering data in the near
future that could probe the underlying mechanisms for the flavor structure of
the SM, it is the right moment to think about sophisticated UV flavor
structures beyond the vanilla FN setup. Wrinkles—a systematic deviation from
the vanilla FN prediction for the relationship between different couplings—are
one such example that significantly increase the versatility of FN
constructions in confronting potential signs of flavorful new physics. We
encourage their use in embedding solutions to anomalous signals in UV complete
models of flavor.

### Acknowledgments

We thank Daniel Aloni, Wolfgang Altmannshofer, Avital Dery, Darius Faroughy,
Seth Koren, Graham Kribs, Clara Murgui, Matthew Reece, Matthew Strassler, and
Lian-Tao Wang for helpful discussions. The work of PA is supported in part by
the U.S. Department of Energy under Grant Number DE-SC0011640. AB, KF and SH
are supported in part by the DOE grant DE-SC0013607. KF and SH are also
supported in part by the Alfred P. Sloan Foundation Grant No. G-2019-12504,
and KF is also supported in part by the NASA ATP Grant NNX16AI12G. The work of
AP is supported in part by the US National Science Foundation Grant PHY2210533
and the Simons Foundation Grant No. 623940. PA thanks Mainz Institute for
Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID
39083149) and KF thanks the Aspen Center for Physics (which is supported by
NSF grant PHY-2210452) for their hospitality during the completion of this
work.

##  Appendix A Full Set of Consistency Conditions

Here we list the full set of consistency conditions that arise for the Yukawa
couplings of the $S_{1}$ leptoquark model embedded in an FN setup. They arise
from considering the representation of the Yukawas under the SM flavor
symmetry group, $G_{\textrm{flavor}}$ (see Eq. (6)), and constructing the
other combinations of Yukawas that transform in the same way. Each combination
produces a one-loop contribution via a diagram analogous to Figure 3. The
representations of $Y_{Q\bar{u}}$, $Y_{Q\bar{d}}$, $Y_{L\bar{e}}$,
$\Delta_{QL}$, and $\Delta_{\bar{u}\bar{e}}$ are listed in Eqs. (7) and (9).
We find

$\begin{split}\Big{|}\Delta_{\bar{u}\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{\bar{u}\bar{e}}\cdot\Delta_{\bar{u}\bar{e}}^{\dagger}\cdot\Delta_{\bar{u}\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{\bar{u}\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{\bar{u}\bar{e}}\cdot
Y_{L\bar{e}}^{\dagger}\cdot Y_{L\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{\bar{u}\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}^{T}\cdot
Y_{Q\bar{u}}^{*}\cdot\Delta_{\bar{u}\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{\bar{u}\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}^{T}\cdot\Delta_{QL}^{*}\cdot
Y_{L\bar{e}}\big{)}^{ij}\Big{|},\\\\[10.00002pt]
\Big{|}Y_{Q\bar{d}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{d}}\cdot
Y_{Q\bar{d}}^{\dagger}\cdot Y_{Q\bar{d}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{d}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}\cdot
Y_{Q\bar{u}}^{\dagger}\cdot Y_{Q\bar{d}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{d}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}\cdot\Delta_{QL}^{\dagger}\cdot
Y_{Q\bar{d}}\big{)}^{ij}\Big{|}.\\\\[10.00002pt]
\Big{|}Y_{L\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{L\bar{e}}\cdot
Y_{L\bar{e}}^{\dagger}\cdot Y_{L\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{L\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}^{T}\cdot\Delta_{QL}^{*}\cdot
Y_{L\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{L\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{L\bar{e}}\cdot\Delta_{\bar{u}\bar{e}}^{\dagger}\cdot\Delta_{\bar{u}\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{L\bar{e}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}^{T}\cdot
Y_{Q\bar{u}}^{*}\cdot\Delta_{\bar{u}\bar{e}}\big{)}^{ij}\Big{|},\end{split}\qquad\begin{split}\Big{|}\Delta_{QL}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}\cdot\Delta_{QL}^{\dagger}\cdot\Delta_{QL}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{QL}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{d}}\cdot
Y^{\dagger}_{Q\bar{d}}\cdot\Delta_{QL}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{QL}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}\cdot
Y^{\dagger}_{Q\bar{u}}\cdot\Delta_{QL}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{QL}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}\cdot
Y_{L\bar{e}}^{*}\cdot Y^{T}_{L\bar{e}}\big{)}^{ij}\Big{|},\\\
\Big{|}\Delta_{QL}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}\cdot\Delta_{\bar{u}\bar{e}}^{*}\cdot
Y^{T}_{L\bar{e}}\big{)}^{ij}\Big{|},\\\\[10.00002pt]
\Big{|}Y_{Q\bar{u}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}\cdot
Y_{Q\bar{u}}^{\dagger}\cdot Y_{Q\bar{u}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{u}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}\cdot\Delta_{QL}^{\dagger}\cdot
Y_{Q\bar{u}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{u}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{u}}\cdot\Delta_{\bar{u}\bar{e}}^{*}\cdot\Delta_{\bar{u}\bar{e}}^{T}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{u}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}Y_{Q\bar{d}}\cdot
Y^{\dagger}_{Q\bar{d}}\cdot Y_{Q\bar{u}}\big{)}^{ij}\Big{|},\\\
\Big{|}Y_{Q\bar{u}}^{ij}\Big{|}&\geq\frac{1}{16\pi^{2}}\Big{|}\big{(}\Delta_{QL}\cdot
Y^{*}_{L\bar{e}}\cdot\Delta_{\bar{u}\bar{e}}^{T}\big{)}^{ij}\Big{|},\end{split}$
(34)

We could also consider additional consistency conditions with more Yukawa
couplings on the right-hand side, but those will be sub-dominant to those
listed above. The consistency conditions listed above are specific to the
spurions we have considered, but this procedure generalizes to arbitrary new
spurions under $G_{\textrm{flavor}}$.

##  Appendix B Calculation of Other Observables

In this appendix we review the contributions of the $S_{1}$ leptoquark to
various flavor observables. The emphasis is on the dependence on the flavor
spurions, $\Delta_{QL}$ and $\Delta_{\bar{u}\bar{e}}$, with many details left
to the references. In what follows, $V$ is the CKM matrix and $v$ is the SM
Higgs vev. We use the CKM parameters as determined in Ref. [138] while the
remainder of our inputs are taken from the PDG [92]. Furthermore, we work with
a set of operators where the neutrinos are left in the flavor basis as the
processes we consider have either a final state neutrino of a specific flavor,
or a sum over all possible final state neutrinos, which can be done in any
basis. Therefore, we do not include explicit factors of the PMNS matrix in the
expressions for the Wilson coefficients. We assume the leptoquark Yukawas are
given in the IR and neglect the running effects. These calculations are used
in §4 to identify the most relevant constraints and the wrinkles which are
useful for evading them. We also employ mostly four-component spinor notation
in this appendix for consistency with the majority of the references.

###  B.1 Dipole Moments

First we calculate the contribution of $S_{1}$ to the electric and magnetic
dipole moments of SM particles. After integrating out the leptoquark, the one
loop diagrams of Figure 9 can give rise to the effective operators

$\mathcal{L}\supset
c^{R}_{ij}\bar{f}_{i}\sigma^{\mu\nu}P_{R}f_{j}F_{\mu\nu}+\mathrm{h.c.,}$ (35)

where $f_{i,j}$ are SM fermions, $F_{\mu\nu}$ is the electromagnetic field
strength, and $c_{ij}^{R}$ is the corresponding Wilson coefficient. By
matching the diagrams in Figure 9 to this operator, we can calculate
$c_{ij}^{R}$ values in our setup. See Refs. [141, 142, 132, 143, 144, 43, 145]
for details of the calculation. Following the notation of Ref. [43], we have

$c^{R}_{ij}=\sum_{\bar{q}}\frac{e}{64\pi^{2}m_{S_{1}}^{2}}\bigg{[}m_{\bar{q}}(V^{*}\Delta_{QL})^{\bar{q}i\,*}\Delta^{\bar{q}j\,*}_{\bar{u}\bar{e}}\Big{(}Q_{S_{1}}A(r)-Q_{\bar{q}}B(r)\Big{)}\\\
+\left(m_{i}\Delta^{\bar{q}i}_{\bar{u}\bar{e}}\Delta^{\bar{q}j\,*}_{\bar{u}\bar{e}}+m_{j}(V^{*}\Delta_{QL})^{\bar{q}i\,*}(V^{*}\Delta_{QL})^{\bar{q}j}\right)\Big{(}Q_{S_{1}}\bar{A}(r)-Q_{\bar{q}}\bar{B}(r)\Big{)}\bigg{]},$
(36)

where the sum is over all possible up-type anti-quarks $\bar{q}$ that can go
in the loop, $Q$ is the electric charge, $m_{i,j}$ are the masses of the
external leptons, $r=m_{\bar{q}}^{2}/m_{S_{1}}^{2}$, and the loop functions
are defined in the appendix of Ref. [43].

Figure 9: Feynman diagram (in two-component notation) for the $S_{1}$
leptoquark contribution to the dipole operators of charged fermions, including
$(g-2)_{\mu}$. The largest contribution arises from the top quark in the loop.
The photon can attach to either internal line in the loop.

In terms of these Wilson coefficients, the electric and magnetic dipole
moments can be written as

$d_{f}=2\leavevmode\nobreak\ \mathrm{Im}\,c_{ff}^{R}\qquad
a_{f}=\frac{4m_{f}}{e}\leavevmode\nobreak\ \mathrm{Re}\,c_{ff}^{R}.$ (37)

Note that because the two fermions in the operator Eq. (35) have opposite
chirality, all the contributions in Eq. (36) are proportional to the external
fermion or internal quark mass. As a result, unless the Yukawas are very
suppressed, the $S_{1}$ contribution to EDMs and MDMs are dominated by
diagrams with the top quark in the loop, which are proportional to $m_{t}$.

###  B.2 Lepton Flavor Violating Observables

The Lagrangian from Eq. (35) also contributes to LFV decays as [141, 132, 43]

$\mathrm{BR}\left(\ell\rightarrow\ell^{\prime}\gamma\right)=\frac{48\pi^{2}}{G_{F}^{2}m_{\ell}^{2}}\left(|c_{\ell\ell^{\prime}}^{R}|^{2}+|c_{\ell^{\prime}\ell}^{R}|^{2}\right).$
(38)

Similar to the previous section, dominant contributions to
$c_{\ell^{\prime}\ell}^{R}$ come from diagrams with the heaviest quarks in the
loop. More concretely, we find that in the limit
$m_{\ell},m_{\ell^{\prime}}\ll m_{S_{1}}$

$\displaystyle
c^{R}_{\ell\ell^{\prime}}\approx\frac{em_{q}}{16\pi^{2}m_{S_{1}}^{2}}\left[\ln(\frac{m_{S_{1}}^{2}}{m_{q}^{2}})-\frac{7}{4}\right](V^{*}\Delta_{QL})^{q\ell\,*}\Delta^{q\ell^{\prime}\,*}_{\bar{u}\bar{e}}.$
(39)

These dipole operators also contribute to the well-constrained LFV processes
$\mu\to 3e$ and $\mu\to e$ conversion in nuclei. In our leptoquark model, the
dipole operator is the only contribution to $\mu\to 3e$, so these branching
ratios are directly correlated:

$\textrm{BR}(\mu\to
3e)=\frac{\alpha}{3\pi}\Big{(}\log\frac{m_{\mu}^{2}}{m_{e}^{2}}-\frac{11}{4}\Big{)}\,\textrm{BR}(\mu\to
e\gamma)\simeq\frac{1}{162}\textrm{BR}(\mu\to e\gamma).$ (40)

The $\mu-e$ conversion process in nuclei, however, also receives contributions
from four-fermion operators coupling the muon and electron to quarks. The
effective Hamiltonian for this process can be written [146, 147]:

$\displaystyle\mathcal{H}$
$\displaystyle\supset\frac{G_{F}}{\sqrt{2}}\sum_{q=u,d,s}\bigg{[}\big{(}c_{LS}^{(q)}\bar{e}P_{R}\mu+c_{RS}^{(q)}\bar{e}P_{L}\mu\big{)}\bar{q}q+\big{(}c_{LP}^{(q)}\bar{e}P_{R}\mu+c_{RP}^{(q)}\bar{e}P_{L}\mu\big{)}\bar{q}\gamma_{5}q$
(41)
$\displaystyle\quad\qquad+\big{(}c_{LV}^{(q)}\bar{e}\gamma^{\mu}P_{L}\mu+c_{RV}^{(q)}\bar{e}\gamma^{\mu}P_{R}\mu\big{)}\bar{q}\gamma_{\mu}q+\big{(}c_{LA}^{(q)}\bar{e}\gamma^{\mu}P_{L}\mu+c_{RA}^{(q)}\bar{e}\gamma^{\mu}P_{R}\mu\big{)}\bar{q}\gamma_{\mu}\gamma_{5}q$
$\displaystyle\quad\qquad+\frac{1}{2}\big{(}c_{LT}^{(q)}\bar{e}\sigma^{\mu\nu}P_{R}\mu+c_{RT}^{(q)}\bar{e}\sigma^{\mu\nu}P_{L}\mu\big{)}\bar{q}\sigma_{\mu\nu}q+\textrm{h.c.}\bigg{]}$

where for the $S_{1}$ leptoquark,

$\displaystyle c_{LS}^{(u)}=+c_{LP}^{(u)}=-c_{LT}^{(u)}$
$\displaystyle=-\frac{1}{2}\frac{v^{2}}{m_{S_{1}}^{2}}(V^{*}\Delta_{QL})^{11\,*}\,\Delta_{\bar{u}\bar{e}}^{12\,*}$
(42) $\displaystyle c_{RS}^{(u)}=-c_{RP}^{(u)}=-c_{RT}^{(u)}$
$\displaystyle=-\frac{1}{2}\frac{v^{2}}{m_{S_{1}}^{2}}(V^{*}\Delta_{QL})^{12}\,\Delta_{\bar{u}\bar{e}}^{11}$
$\displaystyle c_{LV}^{(u)}=-c_{LA}^{(u)}$
$\displaystyle=-\frac{1}{2}\frac{v^{2}}{m_{S_{1}}^{2}}(V^{*}\Delta_{QL})^{12}\,(V^{*}\Delta_{QL})^{11\,*}$
$\displaystyle c_{RV}^{(u)}=c_{RA}^{(u)}$
$\displaystyle=-\frac{1}{2}\frac{v^{2}}{m_{S_{1}}^{2}}\Delta_{\bar{u}\bar{e}}^{11}\,\Delta_{\bar{u}\bar{e}}^{12\,*}$

The conversion rate is then computed by evaluating the overlap integrals of
the fermion wave-function and nucleon densities. This has been performed in
Ref. [148], assuming the coherent conversion process (where the initial and
final state nucleus are the same) dominates. We use the average values of
their overlap integrals for the different nuclei (Al and Au).

###  B.3 Leptonic Meson Decays

####  B.3.1 $P\rightarrow\ell\nu$

The EFT for a generic meson decaying to a neutrino and a charged lepton is
[149, 34, 75]

$\displaystyle\mathcal{H}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}V_{ud}}{\sqrt{2}}\left[C^{V}_{L,ud\ell\nu}\left(\bar{u}_{L}\gamma^{\mu}d_{L}\right)\left(\bar{\ell}_{L}\gamma_{\mu}\nu_{L}\right)+C^{V}_{R,ud\ell\nu}\left(\bar{u}_{R}\gamma^{\mu}d_{R}\right)\left(\bar{\ell}_{L}\gamma_{\mu}\nu_{L}\right)\right.$
$\displaystyle+$
$\displaystyle\left.C^{S}_{L,ud\ell\nu}\left(\bar{u}_{R}d_{L}\right)\left(\bar{\ell}_{R}\nu_{L}\right)+C^{S}_{R,ud\ell\nu}\left(\bar{u}_{L}d_{R}\right)\left(\bar{\ell}_{R}\nu_{L}\right)\right]+\mathrm{h.c.},$

where $u$ ($d$) labels the involved up-type (down-type) quark. In the SM,
these decays are mediated by a $W$ exchange and the overall normalization is
chosen such that $C^{V}_{L}=1$, with other Wilson coefficients set to zero.

For the $S_{1}$ leptoquark, we can show that at the leptoquark mass scale

$\displaystyle C^{V}_{L,ud\ell\nu}$
$\displaystyle=\frac{\Delta_{QL}^{d\nu}(V^{*}\Delta_{QL})^{u\ell\,*}}{V_{ud}}\frac{v^{2}}{4m_{S_{1}}^{2}},$
(44) $\displaystyle C^{S}_{L,ud\ell\nu}$
$\displaystyle=\frac{\Delta_{QL}^{d\nu}\Delta_{\bar{u}\bar{e}}^{u\ell\,}}{V_{ud}}\frac{v^{2}}{4m_{S_{1}}^{2}},$

In our model there are no couplings to RH down-type quarks, so
$C^{S}_{R,ud\ell\nu}=C^{V}_{R,ud\ell\nu}=0$.

The meson branching ratio to $\ell\nu$ is given by

$\mathrm{BR}\left(P_{ud}^{-}\rightarrow\ell\nu\right)=\tau_{P}\frac{m_{P}f_{P}^{2}G_{F}^{2}|V_{ud}|^{2}}{8\pi}m_{\ell}^{2}\left(1-\frac{m_{\ell}^{2}}{m_{P}^{2}}\right)^{2}\\\
\times\Big{|}(C^{V}_{L,ud\ell\nu}-C^{V}_{R,ud\ell\nu})+\frac{m_{P}^{2}}{m_{\ell}(m_{u}+m_{d})}(C^{S}_{R,ud\ell\nu}-C^{S}_{L,ud\ell\nu})\Big{|}^{2},$
(45)

where $\tau_{P}$ is the meson lifetime, $m_{P}$ is the meson mass, $f_{P}$ is
the meson decay constant, $m_{\ell}$ is the final state lepton’s mass, and
$m_{u}$ ($m_{d}$) is the mass of the up-type (down-type) valence quark of the
meson. This equation has been used to calculate the contribution of our model
to various leptonic meson decays in the main text.

####  B.3.2 $P\rightarrow\ell\ell^{\prime}$ and $P\rightarrow\nu\nu^{\prime}$

The Hamiltonian describing a meson $P$ decaying to charged leptons $l$ and
$l^{\prime}$ is [150, 35]

$\displaystyle\mathcal{H}_{\mathrm{eff}}\supset\frac{4G_{F}}{\sqrt{2}}\lambda_{\textsc{CKM}}\left[\sum_{X=S,P,9,10}C_{X}^{qq^{\prime};\ell\ell^{\prime}}\mathcal{O}_{X}^{qq^{\prime};\ell\ell^{\prime}}+C_{X^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}\mathcal{O}_{X^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}+\textrm{h.c.}\right].$
(46)

Here, $\lambda_{\textsc{CKM}}$ is a combination of two CKM entries involving
the valence quarks of the meson, $C_{X}$ are Wilson coefficients, and their
associated operators are

$\displaystyle\mathcal{O}_{S}^{qq^{\prime};\ell\ell^{\prime}}=\frac{\alpha_{\mathrm{em}}}{4\pi}(\bar{q}P_{R}q^{\prime})(\bar{\ell}\ell^{\prime})$
$\displaystyle\mathcal{O}_{P}^{qq^{\prime};\ell\ell^{\prime}}=\frac{\alpha_{\mathrm{em}}}{4\pi}(\bar{q}P_{R}q^{\prime})(\bar{\ell}\gamma^{5}\ell^{\prime})$
(47)
$\displaystyle\mathcal{O}_{9}^{qq^{\prime};\ell\ell^{\prime}}=\frac{\alpha_{\mathrm{em}}}{4\pi}(\bar{q}\gamma^{\mu}P_{L}q^{\prime})(\bar{\ell}\gamma_{\mu}\ell^{\prime})$
$\displaystyle\mathcal{O}_{10}^{qq^{\prime};\ell\ell^{\prime}}=\frac{\alpha_{\mathrm{em}}}{4\pi}(\bar{q}\gamma^{\mu}P_{L}q^{\prime})(\bar{\ell}\gamma_{\mu}\gamma^{5}\ell^{\prime}),$

The operators with a prime on the subscript are obtained by the replacement
$P_{L/R}\rightarrow P_{R/L}$.

At tree level, our leptoquark only gives rise to decays of $D$ and $\pi$ via
$t$-channel diagrams, while decays of $K$, $\leavevmode\nobreak\ B$, and
$B_{s}$ take place at one-loop level and are suppressed. For the tree-level
decays, the Wilson coefficients above can be calculated as a function of the
leptoquark Yukawa couplings [35]

$\displaystyle
C_{9}^{qq^{\prime};\ell\ell^{\prime}}=-C_{10}^{qq^{\prime};\ell\ell^{\prime}}$
$\displaystyle=-\frac{v^{2}\pi}{2\alpha_{\mathrm{em}}\lambda_{\textsc{CKM}}m_{S_{1}}^{2}}(V^{*}\Delta_{QL})_{q^{\prime}\ell^{\prime}}(V\Delta_{QL})_{q\ell}^{*}$
(48) $\displaystyle
C_{9^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}=C_{10^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}$
$\displaystyle=-\frac{v^{2}\pi}{2\alpha_{\mathrm{em}}\lambda_{\textsc{CKM}}m_{S_{1}}^{2}}(\Delta_{\bar{u}\bar{e}})_{q^{\prime}\ell^{\prime}}^{*}(\Delta_{\bar{u}\bar{e}})_{q\ell}$
$\displaystyle
C_{S}^{qq^{\prime};\ell\ell^{\prime}}=C_{P}^{qq^{\prime};\ell\ell^{\prime}}$
$\displaystyle=-\frac{v^{2}\pi}{2\alpha_{\mathrm{em}}\lambda_{\textsc{CKM}}m_{S_{1}}^{2}}(\Delta_{\bar{u}\bar{e}})_{q^{\prime}\ell^{\prime}}^{*}(V^{*}\Delta_{QL})_{q\ell}^{*}$
$\displaystyle
C_{S^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}=-C_{P^{\prime}}^{qq^{\prime};\ell\ell^{\prime}}$
$\displaystyle=-\frac{v^{2}\pi}{2\alpha_{\mathrm{em}}\lambda_{\textsc{CKM}}m_{S_{1}}^{2}}(V^{*}\Delta_{QL})_{q^{\prime}\ell^{\prime}}(\Delta_{\bar{u}\bar{e}})_{q\ell}.$

For $D$ and $\pi$ mesons decays we set
$\lambda_{\textsc{CKM}}=V^{*}_{q^{\prime}b}V_{qb}$ with $q$, $q^{\prime}$
referring to the valence quarks of the meson.

In terms of the Wilson coefficients above, the BR of the meson to $\ell^{-}$
and $\ell^{\prime+}$ is given by [150, 35]

$\displaystyle BR(P\rightarrow\ell^{-}\ell^{\prime+})$
$\displaystyle=\tau_{P}f_{P}^{2}m_{P}^{3}\frac{\alpha_{\mathrm{em}}^{2}G_{F}^{2}}{64\pi^{3}}\lambda_{\textsc{CKM}}^{2}\sqrt{\left(1-\frac{(m_{1}-m_{2})^{2}}{m_{P}^{2}}\right)\left(1-\frac{(m_{1}+m_{2})^{2}}{m_{P}^{2}}\right)}$
(49)
$\displaystyle\mkern-36.0mu\times\left[\left(1-\frac{(m_{1}+m_{2})^{2}}{m_{P}^{2}}\right)\Big{|}(C_{9}-C_{9^{\prime}})\frac{m_{1}-m_{2}}{m_{P}}+\frac{m_{P}}{m_{q^{\prime}}+m_{q}}(C_{S}-C_{S^{\prime}})\Big{|}^{2}\right.$
$\displaystyle\mkern-36.0mu+\left.\left(1-\frac{(m_{1}-m_{2})^{2}}{m_{P}^{2}}\right)\Big{|}(C_{10}-C_{10^{\prime}})\frac{m_{1}+m_{2}}{m_{P}}+\frac{m_{P}}{m_{q^{\prime}}+m_{q}}(C_{P}-C_{P^{\prime}})\Big{|}^{2}\right],$

where $\tau_{P}$ is the meson lifetime, $m_{P}$ is the meson mass, and $m_{1}$
($m_{2}$) is the mass of the $\ell$ ($\ell^{\prime}$) lepton.

We can use Eq. (49) to calculate meson decay to a pair of neutrinos too. For
that, we should set $m_{1}=m_{2}=0$ and only keep couplings to LH fermions in
the SM. Doing that, we find zero contribution for the $S_{1}$ leptoquark.

###  B.4 Semi-leptonic Meson Decays

Next we compute the leptoquark contribution to semi-leptonic meson decays. We
ignore constraints from $B\rightarrow K^{(*)}\ell\ell$, since the $S_{1}$
leptoquark only contributes at loop-level, which is subdominant for leptoquark
masses above a few TeV [69]. Instead, we study the more sensitive observables
$B\rightarrow D^{(*)}l\nu$ and $K\to\pi\nu\bar{\nu}$, which receive
contributions at tree-level.

####  B.4.1 $R_{D^{(*)}}$

$B\rightarrow D^{(*)}l\nu$ proceeds at tree-level via the exchange of the $W$
and the leptoquark [149, 151, 152, 75, 35, 153]. This and other leptoquark
models have generated significant interest in the context of $B\rightarrow
D^{(*)}l\nu$ because some evidence of a lepton flavor non-universal BSM
contribution in this channel, captured by the ratio

$R_{D^{(*)}}\equiv\genfrac{}{}{}{}{\raisebox{3.0pt}{$\mathrm{BR}\left(B\rightarrow
D^{(*)}\tau\nu\right)$}}{\raisebox{-3.0pt}{$\mathrm{BR}\left(B\rightarrow
D^{(*)}\ell\nu\right)$}},$ (50)

has been detected in various experiments [154, 155, 156, 157, 158, 159, 160]
($\ell=e,\mu$).

When computing the decay rate, integrating the heavy mediators out allows us
to work with a set of dimension-6 operators given by

$\mathcal{H}_{\rm
eff}=\frac{4G_{F}V_{cb}}{\sqrt{2}}\bigg{(}\mathcal{O}^{V}_{LL}+\sum_{\begin{subarray}{c}X=S,V,T\\\
M=L,R\end{subarray}}C^{X}_{ML}\mathcal{O}^{X}_{ML}\bigg{)}$ (51)

where

$\mathcal{O}^{S}_{ML}\equiv(\bar{c}P_{M}b)(\bar{\tau}P_{L}\nu)\quad\mathcal{O}^{V}_{ML}\equiv(\bar{c}\gamma^{\mu}P_{M}b)(\bar{\tau}\gamma_{\mu}P_{L}\nu)\quad\mathcal{O}^{T}_{ML}\equiv(\bar{c}\sigma^{\mu\nu}P_{M}b)(\bar{\tau}\sigma_{\mu\nu}P_{L}\nu)$
(52)

Note that we have split apart the contributions to the vector operator such
that the Wilson coefficients only capture leptoquark contributions.

For the process of interest, the helicity amplitude we wish to compute is

$-i\mathcal{M}=\langle\ell(p_{\ell},\lambda_{\ell}),\bar{\nu}_{\ell}(p_{\nu}),D^{(*)}(p_{\mu},\epsilon(\lambda_{M}))|\mathcal{H}_{eff}|B(p_{B})\rangle.$
(53)

Each of these operators can be split apart into the constituent quark and
lepton bilinears, which allows us to split apart the total amplitude into a
product of hadronic and leptonic amplitudes. Details of the calculation can be
found in [161, 162]. The leptonic amplitudes, which are generically functions
of various angles, are identical for both $D$ and $D^{*}$, while the hadronic
amplitudes, which are functions of $q^{2}$, vary and are determined by the
specific helicity of the $D^{(*)}$ meson. The leptonic amplitudes can be found
in multiple references, including [161]; the expressions for the relevant
hadronic functions are taken from [151, 163].121212The correct sign of
$h_{T_{3}}(w)$ is in Ref. [151].

To compute the differential decay rate, we use

$\frac{\differential\Gamma}{\differential
q^{2}\,\differential\cos\theta}=\frac{1}{2m_{B}}\sum_{\ell}\Big{|}\mathcal{M}(q^{2},\cos\theta)\Big{|}^{2}\frac{\sqrt{(m_{B}+m_{D})^{2}-q^{2}}\sqrt{(m_{B}-m_{D})^{2}-q^{2}}}{256\pi^{3}m_{B}^{2}}\left(1-\frac{m_{\tau}^{2}}{q^{2}}\right)$
(54)

where we sum over neutrinos in the final state. Performing the angular
integral over the leptonic functions first, we recover Eqs. (B.6) and (B.8) in
Ref. [162] for the differential decay rates of $B\to D\tau\nu$ and $B\to
D^{*}\tau\nu$ respectively. This is the result for a $\tau$ in the final
state, but making the replacement $m_{\tau}\to m_{\ell}$ gives us the
expression for decays involving any of the SM leptons. The total decay rate
can then be obtained by performing the $q^{2}$ integral over the interval
$[m_{\ell}^{2},(m_{B}-m_{D})^{2}]$.

The expressions from (B.6) and (B.8) in Ref. [162] are given in terms of the
Wilson coefficients defined in Eq. (51), therefore, the last ingredient
required to complete this computation is the set of pertinent Wilson
coefficients for the leptoquark model. They are given by

$\displaystyle C^{S}_{LL}$
$\displaystyle=-\frac{v^{2}}{4m_{LQ}^{2}}\frac{\Delta_{QL}^{3j}\,\Delta_{\bar{u}\bar{e}}^{23}}{V_{cb}}$
(55) $\displaystyle C^{V}_{LL}$
$\displaystyle=\frac{v^{2}}{4m_{LQ}^{2}}\frac{\Delta_{QL}^{3j}(V^{*}\Delta_{QL})^{23\,*}}{V_{cb}}$
$\displaystyle C^{T}_{LL}$
$\displaystyle=\frac{v^{2}}{16m_{LQ}^{2}}\frac{\Delta_{QL}^{3j}\,\Delta_{\bar{u}\bar{e}}^{23}}{V_{cb}}$

####  B.4.2 $K\to\pi\nu\bar{\nu}$

The decays $K^{+}\to\pi^{+}\nu\bar{\nu}$ and $K_{L}\to\pi^{0}\nu\bar{\nu}$ can
be described with an effective Hamiltonian very similar to Eq. (19) [164, 68]:

$\mathcal{H}_{\textrm{eff}}=-\frac{4G_{F}}{\sqrt{2}}\bigg{[}\mathcal{H}_{\textrm{eff}}^{(c)}+V_{td}^{*}V_{ts}^{\phantom{*}}(C_{L}^{K\nu}\mathcal{O}_{L}^{K\nu}+C_{R}^{K\nu}\mathcal{O}_{R}^{K\nu})+\textrm{h.c.}\bigg{]}$
(56)

where

$\mathcal{O}_{L(R)}^{K\nu}=\frac{\alpha_{\mathrm{em}}}{4\pi}(\bar{d}\gamma^{\mu}P_{L(R)}s)(\bar{\nu}\gamma_{\mu}(1-\gamma_{5})\nu),$
(57)

and $\mathcal{H}^{(c)}_{\textrm{eff}}$ includes operators that encode physics
below the weak scale. The branching ratios for $K^{+}\to\pi^{+}\nu\bar{\nu}$
and $K_{L}\to\pi^{0}\nu\bar{\nu}$ are then written as

$\displaystyle\textrm{BR}(K^{+}\to\pi^{+}\nu\bar{\nu})$
$\displaystyle=\kappa_{+}\,\bigg{[}\Big{(}\frac{\textrm{Im}(\lambda_{t}X^{K\nu})}{\lambda^{5}}\Big{)}^{2}+\Big{(}-P_{(u,c)}+\frac{\textrm{Re}(\lambda_{t}X^{K\nu})}{\lambda^{5}}\Big{)}^{2}\bigg{]}$
(58) $\displaystyle\textrm{BR}(K_{L}\to\pi^{0}\nu\bar{\nu})$
$\displaystyle=\kappa_{L}\,\Big{(}\frac{\textrm{Im}(\lambda_{t}X^{K})}{\lambda^{5}}\Big{)}^{2}$

where $X^{K\nu}=-\sin^{2}\theta_{W}(C_{L}^{K\nu}+C_{R}^{K\nu})$,
$\lambda_{t}=V_{td}^{*}V_{ts}^{\phantom{*}}$ and $\lambda=0.2255$ is the
Wolfenstein parameter of the CKM matrix. The $\kappa$-factors encode input
from hadronic matrix elements. Following Ref. [68], we take
$\kappa_{+}=(5.27\pm 0.03)\times 10^{-11}$ and $\kappa_{L}=(2.27\pm
0.01)\times 10^{-10}$. The quantity $P_{(u,c)}=0.41\pm 0.05$ encodes
contributions from charm and light-quark loops. These two decays are related
via the Grossman-Nir bound [165].

The SM Wilson coefficient $C_{L}^{K\nu\,\textsc{SM}}$ is the same as Eq. (21),
while the leptoquark contribution is

$C_{L}^{K\nu}=\frac{v^{2}}{m_{S_{1}}^{2}}\frac{\pi}{2\alpha_{\textrm{em}}}\frac{\Delta_{QL}^{2k}\Delta_{QL}^{1k\,*}}{\lambda_{t}}$
(59)

We set a constraint on the leptoquark mass by demanding that the total
predicted branching ratio (including the SM contribution) be less than the
$2\sigma$ upper limit of the measured branching ratio in Ref. [88]:
$\textrm{BR}(K^{+}\to\pi^{+}\nu\bar{\nu})<1.88\times 10^{-10}$. The analogous
limit for $K_{L}$ decays set by the KOTO experiment,
$\textrm{BR}(K_{L}\to\pi^{0}\nu\bar{\nu})<4.9\times 10^{-9}$ [166] is not yet
competitive in the context of this model.

###  B.5 $Z\rightarrow\ell\ell^{\prime}$

Virtual corrections involving SM fermions and the $S_{1}$ leptoquark can also
contribute to lepton flavor universality violating decays of the SM gauge
bosons. The strongest bound on the leptoquark comes from measurements of the
$Z\to\ell\ell^{\prime}$ decays, which are constrained by ATLAS [167].
Constraints on $Z$ decays can be cast as bounds on anomalous couplings of the
$Z$ boson, $\delta g$, where

$\mathcal{L}\supset\frac{g}{\cos\theta_{W}}\sum_{f,i,j}\bar{f}_{i}\gamma^{\mu}\Big{[}(\delta_{ij}g^{f_{L}}_{\textsc{SM}}+\delta
g^{f_{L}}_{ij})P_{L}+(\delta_{ij}g^{f_{R}}_{\textsc{SM}}+\delta
g^{f_{R}}_{ij})P_{R}\Big{]}f_{j}Z_{\mu},$ (60)

with $g^{f_{L}}_{\textsc{SM}}=T_{3}^{f}-Q_{f}\sin^{2}\theta_{W}$ and
$g^{f_{R}}_{\textsc{SM}}=-Q_{f}\sin^{2}\theta_{W}$ being the left- and right-
handed fermion couplings to the $Z$ boson in the SM.

The $S_{1}$ leptoquark contributions to these anomalous couplings have been
worked out in Refs. [168, 169]. In particular Ref. [169] includes additional
finite terms that are numerically important. The $S_{1}$ leptoquark
contributions to the charged lepton couplings of the $Z$ is

$\displaystyle\delta g^{\ell\,L(R)}_{ij}$
$\displaystyle=\frac{N_{c}}{16\pi^{2}}w_{L(R)}^{tj}(w_{L(R)}^{ti})^{*}\bigg{[}\big{(}g^{u_{L(R)}}_{\textsc{SM}}-g^{u_{R(L)}}_{\textsc{SM}}\big{)}\frac{x_{t}(x_{t}-1-\log
x_{t})}{(x_{t}-1)^{2}}+\frac{x_{Z}}{12}F_{L(R)}(x_{t})\bigg{]}$ (61)
$\displaystyle\qquad+\frac{N_{c}}{48\pi^{2}}x_{Z}\sum_{k=u,c}w_{L(R)}^{kj}(w_{L(R)}^{ki})^{*}\bigg{[}g^{u_{L}(R)}_{\textsc{SM}}\big{(}\log
x_{Z}-i\pi-\frac{1}{6}\big{)}+\frac{1}{6}g^{\ell_{L(R)}}_{\textsc{SM}}\bigg{]}$

where $x_{Z}=m_{Z}^{2}/m_{S_{1}}^{2}$, $x_{t}=m_{t}^{2}/m_{S_{1}}^{2}$,
$w_{L}^{ij}=(V^{*}\Delta_{QL})^{ij}$,
$w_{R}^{ij}=\Delta_{\bar{u}\bar{e}}^{ij}$, and $F_{L(R)}(x)$ are loop
functions, which can be found in Ref. [169].

Ref. [170] sets bounds on combinations of these anomalous couplings with a
variety of flavor Ansätze, by combining the LFV decay bounds with LEP data at
the $Z$-pole [171]. To extract a constraint on the $S_{1}$ leptoquark, we
simply demand that the anomalous couplings computed above satisfy their bounds
assuming generic LFV coupling, which limits

$\sqrt{|\delta g^{\ell_{L}}_{12}|^{2}+|\delta
g^{\ell_{R}}_{12}|^{2}}<1.2\times 10^{-3},\qquad\sqrt{|\delta
g^{\ell_{L}}_{23}|^{2}+|\delta g^{\ell_{R}}_{23}|^{2}}<4.8\times 10^{-3}.$
(62)

The $e\mu$ bound is most constraining for the anarchic and vanilla FN flavor
Ansätze, while the $\mu\tau$ bound is strongest with the additional wrinkles
from Eq. (32).

###  B.6 Meson Mixing

The leptoquark $S_{1}$ also contributes at the one-loop level to operators in
the SM that are responsible for meson mixing. In particular for the down type
quarks, the important operator for meson mixing is the dimension-six, four-
quark bilinear

$\displaystyle\mathcal{H}_{\mathrm{mix}}\supset
C^{ij}_{\mathrm{mix}}\,(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L})\,(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L}).$
(63)

The associated Wilson coefficient for this operator generated by the $S_{1}$
leptoquark is [153]

$\displaystyle
C^{ij}_{\mathrm{mix}}=\frac{1}{128\pi^{2}m_{S_{1}}^{2}}\sum_{k=1}^{3}\left[(\Delta_{QL}^{ik\
*})\Delta_{QL}^{jk}\right]^{2},$ (64)

where the sum above is over all neutrino flavors. Several experimental
quantities of interest can then be derived from this; for instance (in the
limit of negligible CP violating phases) the mass difference $\Delta m$
between the mass eigenstates of the oscillating meson is given by

$\displaystyle\Delta m$
$\displaystyle=\frac{\matrixelement{P}{\mathcal{H}_{\text{mix}}}{\bar{P}}}{m_{P}}\
=\frac{C^{ij}_{\mathrm{mix}}}{m_{P}}\matrixelement{P}{(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L})\
(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L})}{\bar{P}}\ .$ (65)

Here, $P$ denotes the meson whose constituent down-type quarks are in the
$i,j$ generation. The non-perturbative hadronic matrix element above is

$\displaystyle\bra{P}(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L})\
(\bar{d}^{i}_{L}\gamma^{\mu}d^{j}_{L})\ket{\bar{P}}=\frac{2}{3}f_{P}^{2}m_{P}^{2}B_{P},$
(66)

where $f_{P}$ is the meson decay constant and $B_{P}$ is the meson bag factor,
which can be extracted from lattice computations [172, 173, 174].

In order to reduce uncertainties from the hadronic matrix elements, we find it
advantageous to compare ratios of the matrix elements of the mixing operator
(as given in Eq. (33)). We define

$\displaystyle
C_{B_{q}}e^{2i\phi_{B_{q}}}=\genfrac{}{}{}{}{\raisebox{2.0pt}{$\matrixelement{B_{q}}{\mathcal{H}_{\mathrm{mix}}^{\textsc{SM}+\textsc{NP}}}{\bar{B}_{q}}$}}{\raisebox{-2.0pt}{$\matrixelement{B_{q}}{\mathcal{H}_{\mathrm{mix}}^{\textsc{SM}}}{\bar{B}_{q}}$}}\
,$ (67)

where $q=d,s$ and by definition in the SM, $C_{B_{q}}=1$ and $\phi_{B_{q}}=0$.
By definition, the $C_{B_{q}}$ are free from the non-perturbative matrix
elements and depend only on perturbative, short-distance Wilson coefficients.
The aforementioned ratio is experimentally determined by the UTFit
collaboration [137, 90, 138], and can be understood as a short-distance proxy
for the mass difference $\Delta m$. In principle, there can be intricate
interplay between the phases of leptoquark couplings, leading to interference
with the SM contributions in this ratio. In this work, we avoid making any
assumptions on the underlying complex phases of the leptoquark couplings in
$C_{B_{q}}$, and simply compute the absolute value of $C_{B_{q}}$.

Additional CP violation from BSM physics is also strongly constrained by other
meson mixing measurements, especially in the Kaon system. The quantity of
interest is $\epsilon_{K}$, which, following standard assumptions (see e.g.
[175]), is given by

$\displaystyle\epsilon_{K}=\frac{1}{4}\genfrac{}{}{}{}{\raisebox{2.0pt}{$\matrixelement{K_{0}}{\mathcal{H}_{\text{mix}}}{\bar{K}_{0}}$}}{\raisebox{-2.0pt}{$\matrixelement{\bar{K}_{0}}{\mathcal{H}_{\text{mix}}}{K_{0}}$}}-\frac{1}{4}\
.$ (68)

To account for $\epsilon_{K}$, which is much more constraining than the Kaon
mass difference, we define

$C_{\epsilon_{K}}=\genfrac{}{}{}{}{\raisebox{2.0pt}{$\imaginary\langle
K^{0}|\mathcal{H}_{\mathrm{mix}}^{\textsc{SM}+\textsc{NP}}|\bar{K}^{0}\rangle$}}{\raisebox{-2.0pt}{$\imaginary\langle
K^{0}|\mathcal{H}_{\mathrm{mix}}^{\textsc{SM}}|\bar{K}^{0}\rangle$}},$ (69)

where again $C_{\epsilon_{K}}=1$ in the SM.

For all of these quantities, we compute the leptoquark contributions using Eq.
(64). We compare to the SM matrix elements, which are computed following Refs.
[176, 177, 175], including the scale-independent, short-distance QCD
corrections. Then we set constraints using the latest results from UTFit
[138].

We do not consider effects of the $S_{1}$ leptoquark on mixing in mesons with
up-type quarks such as the $D^{0}$, primarily due large hadronic undertainties
[178, 179] in current SM predictions that make it difficult to glean any
information from new physics contributions.

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