# Reflecting on Chirality: CP-violating extensions of the single scalar-
leptoquark solutions for the $(g-2)_{e,\mu}$ puzzles and their implications
for lepton EDMs

Innes Bigaran<EMAIL_ADDRESS>School of Physics, The University
of Melbourne, Victoria 3010, Australia  Raymond R. Volkas
<EMAIL_ADDRESS>School of Physics, The University of Melbourne,
Victoria 3010, Australia

###### Abstract

We study the two scalar leptoquarks capable of generating chirally-enhanced,
sign-dependent contributions to lepton magnetic dipole moments (MDMs) and
electric dipole moments (EDMs), $R_{2}\sim(\mathbf{3},\mathbf{2},7/6)$ and
$S_{1}\sim(\mathbf{3},\mathbf{1},-1/3)$. We consider the case in which the
electron and muon sectors are decoupled, and leptoquark couplings are assigned
complex values. Adopting the coupling ansatz that the electron dipole operator
is generated by charm-containing loops, and muon dipole operator by top-
containing loops, we find that both minimal leptoquark models remain viable
solutions for reconciling anomalies in the muon and electron MDMs, accounting
for either of the two current (disparate) electron MDM results from Cs and Rb
interferometry experiments. We also examine the correlated corrections to the
muon and electron masses generated by these models, and argue that to minimise
fine-tuning this introduces an upper bound on viable leptoquark ($\phi$)
masses, $m_{\phi}<\mathcal{O}(4)$ TeV. Similar arguments allow us to make a
prediction for the upper bound of the muon EDM generated by these models,
$|d_{\mu}|<\mathcal{O}(10^{-22})\;e$ cm, which could be within reach of
upcoming experimental programs, including Muon g$-$2 at Fermilab (FNAL), and
muEDM at Paul Scherrer Institut (PSI).

##  I Introduction

Lepton magnetic (MDM) and electric (EDM) dipole moments are sensitive
precision probes for physics beyond the Standard Model (SM). In terms of the
effective interactions of SM fields, the EDM ($d_{\ell}$) and MDM ($a_{\ell}$)
enter via the following

$\displaystyle\mathcal{L}_{\ell\gamma}$
$\displaystyle=\overline{\ell}\left(a_{\ell}\frac{e}{4m_{\ell}}\sigma_{\mu\nu}-d_{\ell}\frac{i}{2}\sigma_{\mu\nu}\gamma_{5}\right)\ell
F^{\mu\nu},$ (1)

where $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$, and

$\displaystyle a_{\ell}\equiv\frac{1}{2}(g-2)_{\ell},$ (2)

where this represents the deviation of this observable from the tree-level
value, $g=2$.

An accurate determination of the MDMs of charged-leptons has long been an
important test of the Standard Model (SM). Given the precision of the SM
predictions, any deviation from these could provide a ‘smoking gun’ for BSM
physics – particularly where these models could generate loop-level, flavour-
violating interactions.

A persistent deviation between experiment and prediction for the muon MDM has
garnered significant attention in the literature. The most recent measurement
of $a_{\mu}^{\text{exp}}$ from the Fermilab (FNAL) Muon g$-$2 collaboration
[1] yields the following global average

$\Delta a_{\mu}=a_{\mu}^{\text{exp}}-a_{\mu}^{\text{SM}}=(2.51\pm 0.59)\times
10^{-9},$ (3)

when combined with the Brookhaven Muon g$-$2 results [2]. The SM prediction is
taken from the theory white paper [3], the result of a combination of work in
references [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23]. This represents $4.2\,\sigma$ evidence for the existence of BSM
physics.

In contrast to this measurement of the muon, the MDM of the electron can be
indirectly probed using spectroscopic measurements of the fine structure
constant, $\alpha_{em}$, as input for the SM theory calculation. The result
from the most recent measurement in caesium (Cs) suggests an anomaly in the
electron MDM

$\Delta a_{e}^{\text{Cs}}=a_{e}^{\text{exp}}-a_{e}^{\text{SM, Cs}}=-(8.8\pm
3.6)\times 10^{-13},$ (4)

where this corresponds to a $2.5\,\sigma$ deviation from the SM [24]. In
contrast to this, the result from the most recent measurement in rubidium (Rb)
yields

$\Delta a^{\text{Rb}}_{e}=a_{e}^{\text{exp}}-a_{e}^{\text{SM, Rb}}=(4.8\pm
3.0)\times 10^{-13}.$ (5)

which suggests a $1.6\,\sigma$ deviation [25]. These determinations are not
consistent with each other within the stated errors, so one must conclude that
the electron MDM situation requires clarification. For the moment, we thus
treat these two results as independent cases and remain agnostic about where
the resolution will lie.

It is important to note that were the result of equation (4) to be confirmed,
then $\Delta a_{\mu}$ and $\Delta a_{e}$ would be of _opposite sign_ , which
would be an interesting difficulty to be overcome when searching for a common
explanation. Such considerations motivate a search for flavour-violating BSM
effects to achieve a common solution for these two (putative) anomalies.

Leptonic EDMs add to the MDM information by providing a probe of CP-violating
effects beyond the SM. Non-zero permanent EDMs of leptons would break parity
(P) and time-reversal (T) symmetry and, by the CPT theorem, also CP symmetry.
The only verified source of CP violation in the SM is the non-zero phase of
the CKM matrix, which predicts EDMs that are orders of magnitude below current
experimental bounds. Therefore, a measurement of a relatively sizeable
leptonic EDMs at future experiments would necessitate an authentic BSM source
of CP violation.

Given the anomaly or anomalies in the MDMs, we may speculate as to whether
this tension with the SM could also manifest in the EDMs $d_{e}$ and $d_{\mu}$
in anticipation of potential future measurements. The current bounds for the
muon [26] and the electron [27]are given by

$\displaystyle|d_{\mu}|$ $\displaystyle<1.5\times
10^{-19}\;e\;\text{cm}\;\;\;\;90\%\;\text{CL}\;,$ (6) $\displaystyle|d_{e}|$
$\displaystyle<1.1\times 10^{-29}\;e\;\text{cm}\;\;\;\;90\%\;\text{CL}\;.$

In reference [28], we discussed the viability of models of single-scalar
leptoquarks (LQs) to simultaneously ameliorate the anomaly in $\Delta
a_{\mu}$, as it stood prior to the FNAL result quoted in equation (3), and the
potential anomaly $\Delta a_{e}^{\rm Cs}$ of equation (4). Contributions to
$d_{e}$ and $d_{\mu}$ were rendered negligible in the parameter space
explored, by setting the LQ couplings to solely real values. In the present
work, we explore the implications of these established models in an extended
parameter study where contributions to $d_{e}$ and $d_{\mu}$ exist. Lepton
EDMs have been studied in the context of scalar LQ models, for example, in
references [29, 30, 31, 32, 33].

The outline of this paper is as follows; in Section II we introduce scalar
leptoquark models, comment on their manifestation in leptonic MDMs and EDMs,
and outline our LQ coupling flavour ansatz. In Section III, we comment on the
constraints on these models from charged-lepton mass corrections in the
context of fine-tuning, and present the implications of these constraints on
LQ masses and predictions for the generated EDMs. In Section IV, we present an
overview of important additional constraints, and illustrate the viable
parameter space for both LQ models. Finally, in Section V we comment briefly
on the implications of these models on the tau lepton sector, before
concluding in Section VI.

##  II Scalar Leptoquarks and _Getting Chirality Right_

Letoquarks are hypothetical exotic particles that directly Yukawa couple to SM
leptons and quarks. Motivated by the introduction of direct lepton-quark
couplings, rather than separately considering lepton (L) or baryon (B) number
conservation, these are absorbed into the definition of a new conserved
quantity [34] _fermion_ number: $F=3B+L.$ Even though vector LQs have also
been considered in the context of magnetic moment anomalies, we would expect
vector LQ to be associated with gauge-symmetry extensions, complicating the
construction of a full UV-complete model. Therefore, in the interest of
simplicity, we focus only on scalar LQs in this work. Table 1 gives an
overview of the finite set of scalar LQs, including their gauge-group
transformations and fermion numbers, adopting the naming convention of
reference [35].

The fermion number characterises the type of lepton-quark interactions; for
generic charged-leptons ($\ell$) and quarks ($q$), $|F|=2$ LQs couple to
fermion bilinears of the form $\ell q$, and $|F|=0$ couple to
$\overline{\ell}q$. The Yukawa couplings can be expressed as

$\mathcal{L}_{\ell}=\overline{\ell^{(c)}}\;\left[y^{R}P_{R}+y^{L}P_{L}\right]\;q\;\phi^{\dagger}+h.c.$
(7)

for general scalar LQ models. The coupling constants $y^{L}$ and $y^{R}$ are
labelled by the chirality of the quark in the interaction. Leptoquarks which
have both left-handed (LH) and right-handed (RH) couplings, namely $S_{1}$ and
$R_{2}$, are referred to as _mixed-chiral_. We will see that the chirality of
the LQ couplings is paramount for generating sign-dependence in contributions
to $\Delta a_{\ell}$, relevant if equation (4) is correct. Additionally, a
mixed-chiral LQ can yield enhanced contributions to the effective lepton
dipole vertex, via a mass-insertion on an internal fermion leg.

Symbol | $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ | Mixed chiral | $|F|$
---|---|---|---
$\tilde{S}_{1}$ | $(\mathbf{3},\mathbf{1},-4/3)$ | ✗ | 2
$S_{1}$ | $(\mathbf{3},\mathbf{1},-1/3)$ | ✓ | 2
$S_{3}$ | $(\mathbf{3},\mathbf{3},-1/3)$ | ✗ | 2
$\overline{S}_{1}$ | $(\mathbf{3},\mathbf{1},2/3)$ | ✗ | 2
${R}_{2}$ | $(\mathbf{3},\mathbf{2},7/6)$ | ✓ | 0
$\tilde{R}_{2}$ | $(\mathbf{3},\mathbf{2},1/6)$ | ✗ | 0
Table 1: Scalar LQs and their transformation properties, under the hypercharge
convention $Q=I_{3}+Y$. The second-last column indicates whether the LQ has
mixed-chiral couplings.

Note that the $|F|=2$ scalar leptoquarks can also have gauge-invariant di-
quark couplings, leading to a violation of baryon number conservation and
hence strong constraints from proton decay. We thus impose global conservation
of baryon number throughout to forbid diquark couplings of $S_{1}$.

###  II.1 Scalar LQ contributions to MDMs

We comment briefly on this calculation here, but refer the reader to reference
[28] for a more extended discussion.

Following from equation (1), the following Lagrangian parameterises the
contributions to $a_{\ell}$

$\displaystyle\mathcal{L}_{\ell\gamma}$ $\displaystyle\supset
e\overline{\ell}\frac{a_{\ell}}{4m_{\ell}}\sigma_{\mu\nu}F^{\mu\nu}\ell,$ (8)
$\displaystyle\supset\frac{1}{2}e\overline{\ell}\sigma_{\mu\nu}F^{\mu\nu}\left(\sigma_{L}^{\ell}P_{L}+\sigma_{R}^{\ell}P_{R}\right)\ell.$

where $\sigma_{L/R}^{\ell}$ parameterise the effective left- and right-chiral
interactions, as $P_{L(R)}=\frac{1}{2}(1\mp\gamma_{5})$. We also note that
$\sigma_{L/R}^{\ell}$ are not independent: in fact,
$\sigma_{L}^{\ell}=[\sigma_{R}^{\ell}]^{*}$. Equation (8) reveals, via
coefficient matching111Here have defined
$\sigma_{L/R}^{\ell}=i[\sigma_{L/R}^{\ell}]_{2002.12544}$, _i.e._ in contrast
with the definitions used in reference [28]. This is to make the discussion of
EDMs clearer, in that imaginary-valued LQ couplings correspond directly to
imaginary-valued effective interactions. , that

$\Delta
a_{\ell}=m_{\ell}(\sigma_{L}^{\ell}+\sigma_{R}^{\ell})=2m_{\ell}\text{Re}(\sigma_{L}^{\ell}),$
(9)

Figure 1:  Dominant contributions to the $\ell\to\ell^{\prime}\gamma$
processes from scalar LQs – including $d_{\ell}$ and $a_{\ell}$ corrections,
where $\ell=\ell^{\prime}$. The photon line could be connected to either of
the internal legs.

The leading-order topologies for these corrections are illustrated in Figure
1. Their contributions to $\sigma_{L/R}^{\ell}$ are well-established in the
literature [35]. For $F=0$ scalar LQs (_e.g._ $R_{2}$), via the Lagrangian in
equation (7), we have that

$\displaystyle\sigma^{\ell}_{L}=-\frac{N_{c}}{16\pi^{2}m_{\phi}^{2}}\sum_{q}\large[m_{\ell}(|y^{R}_{\ell
q}|^{2}$ $\displaystyle+|y^{L}_{\ell q}|^{2})\kappa$ (10)
$\displaystyle+y^{R}_{\ell q}y^{L*}_{\ell q}m_{q}\kappa^{\prime}\large],$

where

$\displaystyle\kappa(x_{q})=Q_{\phi}f_{S}(x_{q})-f_{F}(x_{q}),\;$ (11)
$\displaystyle\kappa^{\prime}(x_{q})=Q_{\phi}g_{S}(x_{q})-g_{F}(x_{q}),\;$

with $x_{q}=m_{q}^{2}/m_{\phi}^{2}$. These contributions are proportional to
the number of colours, $N_{c}=3$, and are summed over quark flavours $q$
running in the loop. The electric charge of the field $\phi$ is given by
$Q_{\phi}$, and the loop functions in equation (11) are given by [35]

$\displaystyle f_{S}(x)$
$\displaystyle=\frac{x+1}{4(x-1)^{2}}-\frac{x\log(x)}{2(x-1)^{3}},$ (12)
$\displaystyle f_{F}(x)$
$\displaystyle=\frac{x^{2}-5x-2}{12(x-1)^{3}}+\frac{x\log(x)}{2(x-1)^{4}},$
$\displaystyle g_{S}(x)$
$\displaystyle=\frac{1}{x-1}-\frac{\log(x)}{(x-1)^{2}},$ $\displaystyle
g_{F}(x)$ $\displaystyle=\frac{x-3}{2(x-1)^{2}}+\frac{\log(x)}{(x-1)^{3}}.$

For $|F|=2$ LQs (_e.g._ $S_{1}$), the above effective contributions to the MDM
or EDM are calculated similarly, but with $Q_{\phi}\mapsto-Q_{\phi}$. For a
scalar LQ without mixed-chiral Yukawa couplings, the $y^{R}\,y^{L*}$ term in
equation (10) is not present and contributions from each propagator are of
definite relative sign. However, for mixed-chiral scalar LQs, there are terms
proportional to $\kappa^{\prime}$, allowing us to vary the sign of the BSM
contribution.

As summarised in Table 1, by virtue of their chiral nature the $S_{1}$ and
$R_{2}$ leptoquarks are able to induce sign-dependent $\Delta a_{e,\mu}$
contributions enhanced by heavy internal quark masses. These two LQs have also
garnered recent attention in other flavour anomaly studies (see, for example,
those of references [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56]). The relevant LQ couplings for each extension,
represented here as $3\times 3$ Yukawa coupling matrices, are given by222We
implement the interaction basis with neutrinos in their flavour eigenstates
(_i.e._ PMNS rotation matrix is set to the identity throughout) and consider
only terms in which these fields couple as leptoquarks.

$\displaystyle\mathcal{L}_{\text{int}}^{S_{1}}=$
$\displaystyle\left(\overline{L_{L}^{c}}\lambda_{LQ}Q_{L}+\overline{e_{R}^{c}}\lambda_{eu}u_{R}\right)S_{1}^{\dagger}+h.c.,$
(13) $\displaystyle\mathcal{L}_{\text{int}}^{R_{2}}$
$\displaystyle=\left(\overline{L_{L}}\lambda_{Lu}u_{R}+\overline{e_{R}}\lambda_{eQ}Q_{L}\right)R_{2}^{\dagger}+h.c.$
(14)

The doublet, $R_{2}$, can be expressed in terms of its electric charge-
definite components $R_{2}\sim(R_{2}^{5/3},R_{2}^{2/3})$, with charges as
indicated by the superscripts. We assume negligible mass-splitting between the
components of the multiplet, _i.e_ , $m_{R_{2}}\approx m_{R_{2}^{5/3}}\approx
m_{R_{2}^{2/3}},$ so as to avoid constraints from electroweak oblique
corrections [35].

Rotating into the flavour eigenbasis, we adopt the following convention

$\displaystyle\mathfrak{R}_{e}\lambda_{eu}\mathfrak{R}_{u}\mapsto
y^{Seu},\;\;\;\mathfrak{L}_{e}\lambda_{LQ}\mathfrak{L}_{u}\mapsto y^{SLQ},$
(15)
$\displaystyle\mathfrak{L}_{e}^{\dagger}\lambda_{Lu}\mathfrak{R}_{u}\mapsto
y^{RLu},\;\;\;{\mathfrak{R}_{e}^{\dagger}}\lambda_{eQ}\mathfrak{L}_{u}\mapsto
y^{ReQ}.$

Here, $\mathfrak{L}$ and $\mathfrak{R}$ represent the basis mapping between
the gauge and flavour eigenstates, and
$V=\mathfrak{L}_{u}^{\dagger}\mathfrak{L}_{d}$ is the standard CKM matrix.
This is the so-called ‘up-type’ mass-diagonal basis333 The alternative basis
choice would be to select the down-type quark couplings to fix, and allow
associated up-type quark couplings to be generated via CKM mixing. This basis
encourages ansätze that lead to scenarios for $\Delta a_{\mu,e}$ that are
ruled-out by significant contribution to $\mu\to e\gamma$ [29, 57], so we
prefer the up-type basis. – a basis choice that we discussed in detail in
reference [28]. This leaves us with the following

$\displaystyle\mathcal{L}^{S_{1}}\supset
y^{SLQ}_{ij}\left[\overline{e_{L,i}^{c}}u_{L,j}-V_{jk}\;\overline{\nu_{L,i}^{c}}d_{L,k}\right]S_{1}^{\dagger}$
(16)
$\displaystyle+y^{Seu}_{ij}\overline{e_{R,i}^{c}}u_{R,j}S_{1}^{\dagger}+h.c.,$
$\displaystyle\mathcal{L}^{R_{2}}\supset$ $\displaystyle
y^{RLu}_{ij}\left[\overline{\nu_{L,i}}u_{R,j}R_{2}^{2/3,\dagger}-\;\overline{e_{L,i}}u_{R,j}R_{2}^{5/3,\dagger}\right]$
(17)
$\displaystyle+y^{ReQ}_{ij}\overline{e_{R,i}}\left[u_{L,j}R_{2}^{5/3,\dagger}+V_{jk}d_{L,k}R_{2}^{2/3,\dagger}\right]$
$\displaystyle+h.c.$

Matching these Lagrangians onto equation (7) gives the equivalences

$\displaystyle\text{for
$S_{1}$:}\;\;\mathbf{y}^{R}=\mathbf{y}^{Seu}\;\;\;\;\;\text{and}\;\mathbf{y}^{L}=\mathbf{y}^{SLQ},$
(18) $\displaystyle\text{for
$R_{2}$:}\;\;\mathbf{y}^{R}=-\mathbf{y}^{RLu}\;\text{and}\;\mathbf{y}^{L}=\mathbf{y}^{ReQ}.$
(19)

We note that only $R_{2}^{5/3}$ and $S_{1}$ have couplings of both chiralities
with charged-leptons, and that these couplings are to up-type SM quarks.
Therefore, dominant contributions to the processes described by Figure 1 are
from up-type quark loops, and for MDMs via the quark-mass enhanced mixed
chiral term in equation (10).

###  II.2 $S_{1}$ and $R_{2}$ coupling restrictions and the implications for
lepton EDMs

In reference [28], we explored $S_{1}$ and $R_{2}$ models to tackle the
anomalies in the muon and electron MDMs. There we restricted the LQ coupling
texture to represent a dominant single charm-loop contribution to the electron
dipole moments, and top-loop to the muon dipole moments

$\displaystyle\mathbf{y}^{L}\sim\begin{pmatrix}0&\hbox{\pagecolor{Gray}\color[rgb]{0.75,0.75,0.75}'}&0\\\
0&0&\hbox{\pagecolor{Gray}\color[rgb]{0.75,0.75,0.75}'}\\\
0&0&0\end{pmatrix},\;\;\;\;\mathbf{y}^{R}\sim\begin{pmatrix}0&\hbox{\pagecolor{Gray}\color[rgb]{0.75,0.75,0.75}'}&0\\\
0&0&\hbox{\pagecolor{Gray}\color[rgb]{0.75,0.75,0.75}'}\\\ 0&0&0\end{pmatrix}$
(20)

where only grey-shaded entries are nonzero, and by convention rows are
labelled as charged-lepton, and columns as up-type quark, generations. First-
generation quark couplings are avoided as these are known to be highly
constrained. Allowing both anomalies to be generated by the same flavoured up-
type quark leads to dangerous contributions to $\mu\to e\gamma$ [29, 57], and
the alternative coupling texture associated with a _charmphilic_ solution to
$\Delta a_{\mu}$ was ruled-out at one-sigma in reference [58].444We have
performed some estimates of the contributions to $\mu\to e\gamma$ at two-loop
level in these models, and find these not to be very constraining on the
parameter space given the current upper-bound. Nevertheless, this warrants
further careful investigation to determine whether future measurements could
provide a sensitive probe for these effects. For an EFT analysis of
renormalisation group running effects on lepton dipole moments, see ref. [59].

We emphasise that in reference [28] we restricted the coupling entries to be
real-valued, in order to avoid constraints from CP-violating processes. Upon
conducting a phenomenological study, we found that both models could provide
comprehensive solution to the $\Delta a_{e,\mu}$ puzzles so long as the mass
of either leptoquark is $\lesssim 65$ TeV.555In reference [28], we did not
consider the effects of fine-tuning and leptonic mass corrections which would
tighten the upper-bound on LQ mass, as discussed in Section III. Note also
that this analysis was performed before the latest FNAL result for $\Delta
a_{\mu}$, and before the publication of the Rb result for $\Delta a_{e}$.
Considering these updated results, we arrive at the constraints on the
combinations of couplings shown in Table 2666We acknowledge that there is a
negative sign error in the approximate coupling constraints derived in
equations (30)-(33) of reference [28] which has been corrected here. It did
not impact any other derived results in that paper. .

####  II.2.1 Extending the parameter space to complex couplings

Leptoquark | Couplings | Approx. range $\times\left({\hat{m}_{\phi}^{2}}\right)$
---|---|---
$S_{1}$ | Re$[y^{L,*}_{23}y^{R}_{23}]$ | $(3.1\pm 0.7)\times 10^{-3}$
| Re$[y^{L,*}_{12}y^{R}_{12}]_{\text{Cs}}$ | $-(4.6\pm 1.9)\times 10^{-3}$
| Re$[y^{L,*}_{12}y^{R}_{12}]_{\text{Rb}}$ | $(2.5\pm 1.0)\times 10^{-3}$
$R_{2}$ | Re$[y^{L,*}_{23}y^{R}_{23}]$ | $-(1.8\pm 0.4)\times 10^{-3}$
| Re$[y^{L,*}_{12}y^{R}_{12}]_{\text{Cs}}$ | $(4.1\pm 1.7)\times 10^{-3}$
| Re$[y^{L,*}_{12}y^{R}_{12}]_{\text{Rb}}$ | $-(2.2\pm 0.9)\times 10^{-3}$
Leptoquark | Couplings | Upper bound $\times\left(\hat{m}_{\phi}^{2}\right)$
$S_{1}$ |  $|$Im$[y^{L,*}_{23}y^{R}_{23}]|$ | $<1.7$
|  $|$Im$[y^{L,*}_{12}y^{R}_{12}]|$ | $<6.2\times 10^{-9}$
$R_{2}$ |  $|$Im$[y^{L,*}_{23}y^{R}_{23}]|$ | $<1.1$
|  $|$Im$[y^{L,*}_{12}y^{R}_{12}]|$ | $<5.6\times 10^{-9}$
Table 2: Approximate one-sigma allowed ranges for the combination of couplings
in the chirally-enhanced contribution to leptonic MDMs, and upper bounds for
combinations of LQ couplings for contributions to leptonic EDMs. The Cs and Rb
fine-structure constant implications for $a_{e}$ are treated separately.

Naturally, the next step for exploring the implications of these models is to
loosen this real-valued assumption on the LQ Yukawa couplings. This opens up a
rich parameter space, most notably leading to non-zero contributions to lepton
EDMs. The authors of references [29, 60] utilise an EFT framework to assess
the impact of chirally-enhanced models for the $\Delta a_{e,\mu}$ on the size
of lepton EDMs. In this work, we will present a concrete realisation of this
link in the context of single scalar LQ models.

Following from equation (1), the contribution to the lepton EDMs can be
parameterised by

$\displaystyle\mathcal{L}_{\ell\gamma}$
$\displaystyle\supset-\frac{1}{2}id_{\ell}\overline{\ell}\sigma_{\mu\nu}\gamma_{5}\ell
F^{\mu\nu},$ (21)
$\displaystyle\supset\frac{1}{2}e\overline{\ell}\sigma_{\mu\nu}F^{\mu\nu}\left(\sigma_{L}^{\ell}P_{L}+\sigma_{R}^{\ell}P_{R}\right)\ell.$

Matching coefficients in equation (21), we find that

$\displaystyle
d_{\ell}=\frac{e}{2i}(\sigma_{L}^{\ell}-\sigma_{L}^{\ell*})=\text{Im}(\sigma_{L}^{\ell})\;e\,[\text{GeV}^{-1}].$
(22)

Note that a consequence of equation (10) is that for pure-real Yukawa
couplings, $\text{Im}(\sigma_{L}^{\ell})$ is exactly zero. The parameter space
explored in reference [28] is therefore a subspace of that explored in this
work.

In equation (22), the squared brackets indicate the units of the result if
masses are input in GeV.777 We also recognise that $\text{GeV}^{-1}=1.98\times
10^{-14}$ cm, for conversion to appropriate units for EDM measurements. In
Table 2 we provide an overview of the approximate upper-bounds on LQ couplings
for satisfying the constraints from $d_{\ell}$ measurements. Here, and later
in this work, we use the following notation for brevity

$\displaystyle\hat{m}_{\phi}\equiv\frac{{m}_{\phi}}{\text{TeV}}.$ (23)

The results in Table 2 imply a bound on the phase difference between the input
couplings relevant for both MDM and EDM of each lepton. We may write

$\displaystyle
y_{ij}^{L*}y^{R}_{ij}=\left|y_{ij}^{L*}y^{R}_{ij}\right|\exp(i\xi_{ij}),$ (24)

where $\xi_{ij}\in[0,2\pi)$ is given by

$\displaystyle\xi_{ij}={\rm Arg}(y_{ij}^{R})-{\rm Arg}(y_{ij}^{L})$ (25)

For electrons, assuming that the central value of either MDM result is
satisfied, the bound implies that

$\displaystyle|\tan(\xi_{12})|\lesssim 10^{-6},$ (26)

This is satisfied for angles very close to $\xi_{12}=0$ or $\pi$. In other
words, the relative phase between the RH and LH couplings is extremely small
because satisfying the EDM constraints severely restricts the size of the
imaginary component. The current bound of the muon EDM considered together
with the MDM anomaly does not restrict the muon relative phase to the anywhere
near the same extent.

####  II.2.2 Counting the free parameters

The CP-violating phases introduced by extending the parameter space to include
complex-valued couplings _a priori_ yield four additional parameters for each
LQ model. However, by rephasing fields it is straightforward to show that only
two of these phases are physical when neutrino masses are neglected. They can
be taken to be the relative phases between the exotic RH and LH couplings for
the electron-charm and muon-top subsectors. In the limit that each such phase
approaches zero, an approximate CP-symmetry is restored to its respective
subsector. By this argument, we can say that setting $\xi_{12}=0$ or $\pi$ is
technically natural for that subsector considered in isolation, with the
overall technical naturalness of the full theory being an open question.

For the remainder of this work we proceed in such a way that we set
$\xi_{12}=0$ or $\pi$, and therefore do not generate a contribution to the
electron EDM. This is consistent with the preference of current experimental
bounds demonstrated by equation (26). The choice between $0$ or $\pi$
corresponds to scanning over both positive and negative values for the
relevant couplings between the electron and charm quark.

##  III Fine-tuning and corrections to the muon and electron masses

A sizable, chirally-enhanced, contribution to lepton dipole moments also,
consequentially, generates a lepton mass correction888See reference [61] for a
discussion of how these class of models could be responsible for lepton
(specifically, muon) mass generation.. Theoretically, this can be absorbed by
adjusting the pole mass, $m_{\ell}^{0}$. However, a critique of these simple
models is that having this correction greater than the physical lepton mass
introduces a fine-tuning problem [31, 62]. In this section, we outline the
constraint that this requirement introduces for our models. Note that for
simplicity we have switched-off quartic couplings between the Higgs and LQs,
but acknowledge that these additional free parameters could extend the viable
parameter space for these models – although, they may also introduce a new
suite of fine-tuning problems. The Feynman diagram for this radiative
correction is shown in Figure 2,

Figure 2:  One-loop correction from scalar LQ to the lepton ($\ell$)
propagator. Vertex chirality is labelled $X$ and $Y$ = $\\{R,L\\}$ for
reference in Section III Figure 3: Implications of the constraint on the
electron mass correction on $S_{1}$ LQ solutions to the electron MDM, for (a)
$m_{S_{1}}=2$ TeV , (b) $m_{S_{1}}=4$ TeV and (c) $m_{S_{1}}=6$ TeV. These
illustrations of parameter space assume that the imaginary components of the
electron LQ couplings are negligible. The shaded grey region, common in all
plots, shows the allowed points by restriction on the electron mass correction
in equation (34). For the Cs and Rb MDM results, solid lines show the central
values, shaded regions are the one sigma region, and the two sigma region is
denoted by a dashed line.

Referring again to Figure 2, the contribution of this correction to a charged-
lepton mass, $m_{\ell}$, is defined via a one-loop self energy,
$\Sigma_{\ell}$. For the full calculation of these corrections, we refer to
reference [45], but quote the relevant results here. Up to linear order in the
self-energy expansion, the lepton mass correction is

$\displaystyle\delta m_{\ell}$
$\displaystyle=m_{\ell}^{0}\left(\frac{1}{2}\Sigma_{\ell}^{LL}+\frac{1}{2}\Sigma_{\ell}^{RR}\right)+\Sigma_{\ell}^{LR},$
(27)

where $\Sigma_{\ell}^{XY}$= $\Sigma_{\ell}^{XY}(p^{2}=0)$, _i.e_ we take the
momentum independent contributions of the self-energies to generate a physical
mass correction. Here $X$ and $Y$ refer to the chirality of the labelled
vertices in Figure 2.

The general expressions for these self-energies evaluated at energy scale
$\mu$ are given by

$\displaystyle\Sigma_{\ell}^{LR}$
$\displaystyle=-\sum_{q}\frac{m_{q}N_{c}}{16\pi^{2}}\mathcal{I}_{0}\left(\frac{\mu^{2}}{m_{\phi}^{2}},\frac{m_{q}^{2}}{m_{\phi}^{2}}\right)y^{R*}_{\ell
q}y^{L}_{\ell q},$ (28) $\displaystyle\Sigma_{\ell}^{LL}+\Sigma_{\ell}^{RR}$
$\displaystyle=-\frac{N_{c}}{32\pi^{2}}\sum_{q}\mathcal{I}_{1}\left(\frac{\mu^{2}}{m_{\phi}^{2}},\frac{m_{q}^{2}}{m_{\phi}^{2}}\right)\left[|y^{L}_{\ell
q}|^{2}+|y^{R}_{\ell q}|^{2}\right].$ (29)

where the loop functions are given by

$\displaystyle\mathcal{I}_{0}(x,y)$
$\displaystyle=\frac{1}{\epsilon}+1+\log(x)+y\log(y),$ (30)
$\displaystyle\mathcal{I}_{1}(x,y)$
$\displaystyle=\frac{1}{\epsilon}+\frac{1}{2}+\log(x)-y,$ (31)

and the couplings are as defined in equations (18) and (19). Evaluating the
masses at the LQ scale $\mu=m_{\phi}$, and taking only the finite
contributions to $m_{\ell}$, we presume to cancel the self-energy divergences
via $\overline{MS}$ as per reference [45]. This yields the following:

$\displaystyle\mathcal{I}_{0}(1,m_{q}^{2}/m_{\phi}^{2})$ $\displaystyle\mapsto
1+m_{q}^{2}/m_{\phi}^{2}\log(m_{q}^{2}/m_{\phi}^{2})$ (32)
$\displaystyle\mathcal{I}_{0}(1,m_{q}^{2}/m_{\phi}^{2})$ $\displaystyle\mapsto
1-m_{q}^{2}/m_{\phi}^{2}$ (33)

Assuming dominance of the mixed-chiral contribution $\Sigma_{\ell}^{LR}$ and
we require for both leptons that

$\displaystyle\left|\frac{\delta m_{\ell}}{m_{\ell}^{0}}\right|$
$\displaystyle=\left|\frac{\Sigma_{\ell}^{LR}}{m_{\ell}^{0}}\right|\lesssim
1,$ (34)

which is to say, it can account for the lepton mass but should not require
significant, tree-level, fine-tuning. Therefore, now noting that in our model
only a single quark species runs in the loop at a time, we can drop the
summation over $q$ and we arrive at

$\displaystyle|\delta m_{\ell}|$
$\displaystyle\simeq\frac{m_{q}N_{c}}{16\pi^{2}}|y^{R*}_{\ell q}y^{L}_{\ell
q}|\left[1+\frac{2m_{q}^{2}}{m_{\phi}^{2}}\log\left(\frac{m_{q}}{m_{\phi}}\right)\right].$
(35)

We take the pole masses of the two leptons at the TeV scale to be
$m_{\mu}^{0}=0.10468806$ GeV and $m_{e}^{0}=4.959016\times 10^{-4}$ GeV [63].
We will explore the implications of the correction on the electron and muon
mass, and their interplay with dipole moments, below.

###  III.1 Preliminary comment on the considered LQ masses

Presently the scalar LQ mass, $m_{\phi}$, is most strongly constrained using
LHC searches for the decay of pairs of scalar LQs with couplings predominantly
to first-generation leptons [64],

${\hat{m}_{\phi}}>1.8\;\;\text{at}\;95\%\text{CL}.$ (36)

Given that we allow such couplings in the models detailed in this work, and
these couplings can take order-one values, we adopt equation (36) as a
conservative lower-bound on the LQ mass.

For the remainder of section, we selected three benchmark LQ masses for
specific study

$\displaystyle{\hat{m}_{\phi}}\in\\{2,4,6\\}.$ (37)

which we will see span the allowed regions of parameter space, and extend to
those forbidden by the mass corrections and not preferred by the present
dipole measurements. This is discussed further in forthcoming sections.

###  III.2 Electron MDM implications for the electron mass

We elaborate further here on the constraints that the electron mass, in
particular, places on the viability of these models for explaining the
anomalies in the electron AMM. For this study, we assume that the imaginary
component for both the electron RH and LH couplings are negligible, _i.e._
$\xi_{12}=$ 0 or $\pi$.

Combining the electron result with the central values for MDM results, we can
derive a conservative bound

$\displaystyle\hat{m}_{\phi}\lesssim 3.12\;.$ (38)

Figure 3 shows a cross section of $S_{1}$ parameter space. We find that the
upper bound on the couplings from the correction to $m_{e}$ is largely LQ mass
independent for the considered masses. It is evident from Figure 3(a), that
for smaller masses this electron mass correction is small for solutions to
both MDM results. For a mass of 4 TeV, this model can no longer reach the
central value for the Cs anomaly, whereas there is sizeable parameter space
remaining to explain the Rb result. At a mass of 6 TeV, it appears that for
the region of parameter space illustrated, the $S_{1}$ leptoquark cannot
accommodate the Rb or Cs MDM result, within one sigma.

Note that the $R_{2}$ leptoquark was found to demonstrate similar behaviour in
the $y^{L}_{12}-y^{R}_{12}$ plane, but with reflection over the
Re$(y^{L}_{12})$ and Re$(y^{L}_{12})$ axes, illustrating the preference for
opposite-signed coupling combinations.

Figure 4: Implications of the constraint on the muon mass correction on
$S_{1}$ and $R_{2}$ LQ solutions to the muon dipole moments. Subfigure (a)
($S_{1}$) and subfigure (b) ($R_{2}$) demonstrate the interplay of constraints
for benchmark masses $\hat{m}_{\phi}=2,4,6$. For each mass, solid lines show
the central values, shaded regions are the one sigma region, and the two sigma
region is denoted by a dashed line. Horizontally bound regions show the
projected reach of PSI and FNAL, such that the coloured regions represent
parameter space that would generate an EDM beyond their reach. In subfigure
(c) we depict benchmark phases $\xi_{23}$ for a mass of $\hat{m}_{\phi}=2$,
demonstrate the overlap with viable muon MDM solutions, for each LQ. Contours
representing the trivial phases $\xi_{23}=0(\pi)$ are not explicitly shown,
but would lie along the positive (negative) x axis.

###  III.3 Interplay of corrections to the muon mass and muon dipole moments

We first consider the muon EDM and the relationship between dipole
contributions and the muon mass correction, without considering the muon AMM.
Following from equation (35), we can derive a bound on the muon coupling
magnitudes,

$\displaystyle|y_{23}^{L}y_{23}^{R}|=\sqrt{{\rm
Im}\left(y_{23}^{L,*}y_{23}^{R}\right)^{2}+{\rm
Re}\left(y_{23}^{L,*}y_{23}^{R}\right)^{2}}\lesssim 0.04,$ (39)

which is roughly independent of mass for the considered mass-range. Now, we
can arrive at a conservative upper bound on the imaginary component by letting
${\rm Re}\left(y_{23}^{L}y_{23}^{R}\right)\to 0$,

$\displaystyle|{\rm Im}\left(y_{23}^{L,*}y_{23}^{R}\right)|\lesssim 0.04$ (40)

Following from equation (22), we can derive a prediction for the upper bound
for the $S_{1}$ and $R_{2}$ muon EDM, and therefore a measure of at what point
future experiments will be sensitive to these models. This gives the following

$\displaystyle|d_{\mu}|_{S_{1}}\lesssim\left\\{\begin{array}[]{*{2}{c}}2.77,&\hat{m}_{\phi}=2\\\
0.96,&\hat{m}_{\phi}=4\\\ 0.50,&\hat{m}_{\phi}=6\\\ \end{array}\right\\}\times
10^{-22}\;e\;\text{cm};$ (44)
$\displaystyle|d_{\mu}|_{R_{2}}\lesssim\left\\{\begin{array}[]{*{2}{c}}3.91,&\hat{m}_{\phi}=2\\\
1.25,&\hat{m}_{\phi}=4\\\ 0.62,&\hat{m}_{\phi}=6\\\ \end{array}\right\\}\times
10^{-22}\;e\;\text{cm}.$ (48)

The Paul Scherrer Institut (PSI) muEDM experiment is projected to have a
sensitivity within the ballpark of $5\times 10^{-23}$ $e$ cm [65, 66].
Similarly, the FNAL Muon g$-$2 experiment aims to probe muon EDMs of order
$10^{-21}e$ cm [67]. Therefore, muon EDMs generated by these models are well
within the reach of the PSI experiment, and maybe even by FNAL depending on
the value of $\xi_{23}$. Projected reach of both experiments can be seen in
Figure 4 as horizontally bound regions, such that the coloured regions
represent the remaining parameter space that would generate an EDM beyond
their future sensitivity.

We now analyse the implications for the muon MDM from the fine-tuning issue
raised by muon mass corrections. This interplay can be seen explicitly in
Figure 4. We cannot immediately assume the relative suppression of the
imaginary component of the coupling component to the real, as we did for the
electron as a result of current experimental bounds. We can, however,
recognise that a similar logic to that which goes into deriving equation (40)
can be applied here, to achieve the bound (agnostic to masses within the
considered range),

$\displaystyle|{\rm Re}\left(y_{23}^{L,*}y_{23}^{R}\right)|\lesssim 0.04,$
(49)

which is required to ensure that the muon mass correction is not too large.
Combining this with the bounds quoted in Table 2 we can arrive at the
following mass constraint for these models for explaining the muon AMM: for
the $S_{1}$ LQ,

$\displaystyle\hat{m}_{S_{1}}\lesssim\left\\{\begin{array}[]{*{2}{c}}4.08,&\text{at}\;\text{CV}\\\
4.71,&\text{at}\;1\;\sigma\\\ 4.85,&\text{at}\;2\;\sigma\\\
\end{array}\right\\},$ (53)

and for the $R_{2}$ leptoquark,

$\displaystyle\hat{m}_{R_{2}}\lesssim\left\\{\begin{array}[]{*{2}{c}}3.59,&\text{at}\;\text{CV}\\\
5.36,&\text{at}\;1\;\sigma\\\ 6.32,&\text{at}\;2\;\sigma\\\
\end{array}\right\\}.$ (57)

where CV denotes the present central value for the muon MDM anomaly, followed
by the one and two sigma limits. These upper mass limits are generally
applicable to any $S_{1}$ or $R_{2}$ LQ model whose contribution to the muon
MDM is generated by a top-containing loop. Note that for either of our models,
these mass constraints are less stringent than that derived from the electron
mass correction in equation (38).

The interplay of above constraints can be seen in Figure 4. In Figure 4(c) we
focus on a LQ mass of $\hat{m}_{\phi}=2$ and show contours representing
benchmark values of the phase parameter

$\displaystyle\xi_{23}=\\{0,\frac{\pi}{4},\frac{\pi}{3}\\}\;\;\text{for}\;S_{1}$
(58)
$\displaystyle\xi_{23}=\\{\pi,\frac{2\pi}{3},\frac{3\pi}{4}\\}\;\;\text{for}\;R_{2},$
(59)

consistent with the sign preference for each model, as detailed in Table 2.
Contours representing the trivial phases $\xi_{23}=0(\pi)$ are not explicitly
shown, but would lie along the positive (negative) x axis. For the non-trivial
phases shown, an EDM generated by either of the LQ models satisfying the muon
MDM central value, could feasibly be probed by either PSI or the FNAL
experiment.

##  IV  Additional constraints

Many of the additional constraints considered in this work overlap with those
considered in reference [28], particularly in the kaon sector, for high
$p_{T}$ tails, and for leptonic $Z$ decays. Here we provide an overview of
these and additional constraints, as well as indicating the importance of some
future probes of our models. A summary of present experimental values for the
pertinent constraints can be found in Table 3.

###  IV.1 Contributions to composite EDM measurements

The contributions of the LQ couplings to QCD effective interactions at
$\mu=m_{\phi}$ are very small, as $\alpha(m_{\phi})$ is small. Nevertheless,
the RGE evolution of the generated four-fermion interactions permit mixing
into these contributions as they evolve to the low-scale. This makes the
contributions from the effective quark and gluon (chromo) EDMs important to
consider, as these could lead to nonzero predictions for nucleon EDMs, and
those of paramagnetic, polar molecular systems [31].

Thus far, EDM searches in general have null results, yet they yield stringent
upper bounds. For muon-top couplings of LQs, Dekens _et al_ [30] find that the
muon EDM provides stronger constraints than the composite measurements.
Similarly, for the electron-charm couplings, the $d_{e}$ constraint is most
constraining.

With improved modelling of composite structures these constraints may become
tighter, but for now and in the foreseeable future our model couplings are
most strongly constrained by the charged-lepton EDMs. For this reason, we do
not discuss these additional EDM contributions further. A detailed discussion
on how the $R_{2}$ and $S_{1}$ leptoquarks can generate fundamental and
composite EDMs is provided in reference [30].

Process | Observable | Constraint
---|---|---
$Z\to\ell_{i}\ell_{j}$ | $g_{A}^{e}/g_{A}^{e,\text{SM}}$ | $0.999681\pm 0.000698227$
| $g_{A}^{\mu}/g_{A}^{\mu,\text{SM}}$ | $0.99986\pm 0.00107726$
$Z\to\nu\nu$ | $N_{\nu}^{\text{Eff.}}$ | $2.9840(82)$
$K^{+}\to\pi^{+}\nu\nu$ | Br | $(1.7\pm 1.1)\times 10^{-10}$
$K^{0}_{L}\to\pi^{0}\nu\nu$ | Br | $<2.6\times 10^{-8}$
$K^{0}_{L}\to e^{+}e^{-}$ | Br | $(9^{+6}_{-4})\times 10^{-12}$
$K^{0}_{L}\to\mu^{+}\mu^{-}$ | Br | $(6.84\pm 0.11)\times 10^{-9}$
$K^{0}_{L}\to\mu^{+}e^{-}$ | Br | $<4.7\times 10^{-12}$
$K_{0}$-$\overline{K_{0}}$ mixing | $|\epsilon_{K}|$ | $(2.228\pm 0.011)\times 10^{-3}$
| $\Delta M_{K}$ | $(3.484\pm 0.006)\times 10^{-12}$
$pp\to\ell\ell$ | $|y^{Seu}_{12}|$ | $<0.648~{}\hat{m}_{\phi}$
[68] | $|y^{RLu}_{12}|$ | $<0.524~{}\hat{m}_{\phi}$
Table 3: Processes most constraining on this model. Values quoted without
citation are from PDG [69]. Constraints from $pp\to\ell\ell$ are derived from
Table 1 of reference [68].

###  IV.2 Leptonic $Z$ and Higgs decays

Given the chirally-enhanced nature of contributions to the lepton effective
dipole vertex, correlations with other observables sensitive to electroweak
symmetry breaking are to be expected [70]. In particular, $Z$-boson decays to
dilepton final states are expected to constrain the same set of LQ couplings
that are sensitive to lepton dipole moments. Also, given the sizable
corrections to the lepton masses established above, we would expect BSM
effects to also be present in the dilepton decays of the higgs, in particular
$h\to\mu\mu$. We discuss these two sets of constraints below.

####  IV.2.1 Z effective couplings

The couplings $g_{f_{L(R)}}$ are the effective LH and RH couplings of the Z
boson to fermions, $f$. The effective Lagrangian for describing the BSM
contributions to these interactions is given by:

$\mathcal{L}^{Z}_{\text{eff}}=\frac{g}{\cos(\theta_{W})}\sum_{i,j}\overline{f_{i}}\gamma^{\mu}\left[g^{ij}_{f_{L}}P_{L}+g^{ij}_{f_{R}}P_{R}\right]f_{j}Z_{\mu},$
(60)

where $\theta_{W}$ is the weak-mixing angle. For this study, we use
constraints from corrections to effective vector and axial-vector couplings
about their central values, where

$\displaystyle g_{f_{V(A)}}^{ij}=g_{f_{L}}^{ij}\pm g_{f_{R}}^{ij}.$ (61)

For $i=j$, we relabel the coupling $g^{ii}_{f_{A(V)}}\equiv g^{f_{i}}_{A(V)}$
We follow the calculation of effective $Z$ couplings to charged-leptons
$\ell$, and associated observables, from reference [71]. For TeV scale LQs,
the top-mass enhancement of the contribution to the effective coupling to the
muons gives

$\displaystyle\delta
g^{\mu}_{A(V)}\simeq-\frac{N_{c}}{32\pi^{2}}\frac{m_{t}^{2}}{m_{\phi}^{2}}\left(1+\log\frac{m_{t}^{2}}{m_{\phi}^{2}}\right)\left(|y^{R}_{23}|^{2}\pm|y^{L}_{23}|^{2}\right)$
(62)

Moreover, the axial vector couplings are the most tightly constrained, so we
focus on these for constraining these models. These constraints are quoted in
Table 3.

Complementary constraints on the LQ interaction with neutrinos can be inferred
from the effective number of light neutrino species, $N_{\nu}^{\text{Eff}}$,
which parameterises the decay rate of the $Z$ to invisible final states
($Z\to\nu\bar{\nu}$). These are particularly important for the $R_{2}$
leptoquark, where the neutrinos also couple to up-type quarks and these
couplings are not subject to CKM suppression. Again, the top-quark
contribution is dominant and following reference [71], the $R_{2}$
contribution is calculated as

$\displaystyle N_{\nu}^{\text{Eff}}$ $\displaystyle\simeq
3-N_{c}\frac{|y^{RLu}_{23}|^{2}}{8\pi^{2}}\frac{m_{t}^{2}}{m_{\phi}^{2}}\left(1+\log\frac{m_{t}^{2}}{m_{\phi}^{2}}\right).$
(63)

For $S_{1}$, neutrinos only couple to down-type quarks, therefore, no top-mass
enhancement, and also these couplings are generated via the CKM. Therefore,
for this LQ, $Z\to\nu\nu$ does not provide a competitive constraint.

####  IV.2.2 Higgs dilepton decays

The LQ contribution to Higgs decay to the dilepton final state is strongly
correlated to the aforementioned lepton mass corrections, and to the electric
and magnetic dipole moments.999Correlations in the context of the EFT between
$h\to\mu\mu$ and $\Delta a_{\mu}$ were considered in references [45, 72], and
$h\to\tau\tau$ and $\Delta a_{\tau}$ in reference [73]. For years LHC
measurements of Higgs decays to two charged-leptons were limited to the those
of the third generation. Recently, the first evidence for decays
$h\to\mu^{+}\mu^{-}$ from the ATLAS and CMS collaborations[74, 75] indicate
that there is an enhancement of the signal strength over the SM prediction,
although there is considerable uncertainty.

Following the derivation in reference [76], we find the precise approximation
for the contribution of the $S_{1}$ leptoquark from a loop containing the
quark $q$

$\displaystyle\frac{\mathrm{Br}(h\to\ell^{+}\ell^{-})}{\mathrm{Br}(h\to\ell^{+}\ell^{-})_{\mathrm{SM}}}\approx\left|1+\frac{m_{q}^{3}}{8m_{\ell}}\frac{N_{c}}{8\pi^{2}}\frac{y^{R*}_{\ell
q}y^{L}_{\ell
q}}{m_{\phi}^{2}}\mathcal{J}\left(\frac{m_{h}^{2}}{m_{\phi}^{2}},\frac{m_{q}^{2}}{m_{\phi}^{2}}\right)\right|,$
$\displaystyle\;\;\approx
1+\frac{m_{q}^{3}}{8m_{\ell}}\frac{N_{c}}{8\pi^{2}}\frac{\text{Re}(y^{L*}_{\ell
q}y^{R}_{\ell
q})}{m_{\phi}^{2}}\mathcal{J}\left(\frac{m_{h}^{2}}{m_{\phi}^{2}},\frac{m_{q}^{2}}{m_{\phi}^{2}}\right),$
(64)

where the second line follows from assuming that the magnitude of the LQ
correction is small. The form of equation (64) is identical for $R_{2}$ except
that we switch $y^{L}_{\ell q}\leftrightarrow y^{R}_{\ell q}$.101010Reference
[76] adopts the labelling of the chirality of the interaction by that of the
lepton in the LQ interaction, hence the relabelling required. The loop
function here is given by:

$\displaystyle\mathcal{J}(x,y)=2(x-4)\log(y)-8+\frac{13}{3}x.$ (65)

For the dimuon final state, LQ couplings in equation (64),
Re($y^{R*}_{23}y^{L}_{23}$), also appears in, and is constrained by, the
expression for $\Delta a_{\mu}$. Moreover, the sign of this coupling
combination required for the $R_{2}$ and $S_{1}$ LQ to generated the measured
result for the muon $\Delta a_{\mu}$ is opposite. This means that the
observable $\mathrm{Br}(h\to\mu^{+}\mu^{-})$ provides a probe for
distinguishing the effects of these two scalar LQs. Following Table 2,
observing an enhancement over the SM would be consistent with an $S_{1}$ model
for $\Delta a_{\mu}$, and a suppression would be consistent with that of
$R_{2}$.

We will not discuss this notion further here as the current measurements are
not yet sensitive enough to provide a definitive test, but note that this is
discussed in detail in references [76, 72].

We may ponder whether that same probing capacity would be true of
$\mathrm{Br}(h\to e^{+}e^{-})$. This is such an extraordinarily rare process
in the SM that any measurement of this decay would signal BSM physics [77]. If
this process were to be measured at high precision in future experiments,
these two scalar LQ models of $\Delta a_{e}$ _could_ be distinguished, but
only once the sign conflict of this anomaly is resolved.

###  IV.3 Constraints from the kaon sector

Interactions involving strange-mesons are generated by these LQ couplings via
$SU(2)_{L}$ invariance of the associated interaction, explicitly manifest in
equations (16) and (17). This also means that the constraints here are most
directly relevant for refining the parameter space for LH LQ couplings. The
implications for scalar LQ models from constraints in the kaon sector have
been studied in references [78, 79, 80].

Following the derivations in reference [80], we re-derive the relevant
constraints for each LQ model. As $S_{1}$ is $|F|=2$ and $R_{2}$ is $|F|=0$,
this makes identifying the strongest constraints heavily model-dependent. We
find that the $R_{2}$ parameter space is much more tightly constrained by the
kaon sector than that of $S_{1}$.

Additionally, because of the CKM dependence of couplings between the LQs and
down-type quarks, we find that these constraints do not tightly constrain
parameter phases. This follows from the discussion in Section II.2.2, where we
have argued that the only physical phases can be phrased as the difference
between the LH and RH couplings’ phase. Physical observables should be
independent of the basis choice, and we can choose a basis in which the phases
can all lie in the RH LQ couplings. Thus, we do not expect phase constraints
to arise from considering the kaon sector, as a consequence of our coupling
ansatz.

Figure 5: An overview of the constraints on the magnitude of the $S_{1}$ LQ
couplings for a benchmark mass of 2 TeV. The plots show (a) the $y^{L}$
coupling plane, (b) electron couplings and (c) muon couplings. The hashed
regions indicate that the region is ruled out in the direction of the hashing.
The $y^{L}$ couplings are subject to stronger bounds, so their axes are
logarithmic while the RH couplings are shown on linear axes. For the electron
couplings, the shaded region around the indicated MDM lines show the one-sigma
margin of error on the experimental values, from equations (4) and (5). For
the muon couplings, the central value is indicated for a number of benchmark
angles $\xi_{23}$.

####  IV.3.1 $K^{+}\to\pi^{+}\nu\overline{\nu}$

Contributions to the theoretically clean rare process
Br$(K^{+}\to\pi^{+}\nu\overline{\nu})$ are found to give competitive
constraints for the $S_{1}$ model. As the effective interaction here is purely
vector in nature, it does not run in QCD. Taking the leading-order
contribution and following the derivation in reference [80], we obtain the
following bound from the lepton-flavour conserving contribution:

$\displaystyle|y_{12}^{SLQ}|<0.041\;{\hat{m}_{\phi}},$ (66)

and for the flavour-violating contribution:

$\displaystyle|y_{12}^{SLQ}y_{23}^{SLQ}|<0.0467\;{\hat{m}_{\phi}}^{2}.$ (67)

We find constraints on these couplings are stronger for this process in
contrast with the alternative decay $K^{0}_{L}\to\pi^{0}\nu\overline{\nu}$. We
also note that given that we permit lepton flavour violating processes, the
Grossman-Nir (GN) bound does not directly apply to this model [81, 82].The
final state contributions that violate lepton flavour can generate CP-
conserving contributions to $K^{0}_{L}\to\pi^{0}\nu\overline{\nu}$, otherwise
negligible in the SM. This makes future measurements of this process very
likely to generate competitive constraints for the relevant LQ couplings.
Nevertheless, given the relatively low statistics for the current KOTO [83]
and NA62 [84], we refrain from discussing the GN bound further here.

####  IV.3.2 $K^{0}_{L}\to\ell\ell^{\prime}$

These decays are forbidden at tree-level in the SM, and therefore reference
[80] conservatively assumes that the LQ contribution saturates the upper
limit. Conversely, for the $R_{2}$ model the strongest constraint from the
kaon sector comes from tree-level leptoquark exchange to the helicity
suppressed $K^{0}_{L}\to e^{+}e^{-}$ transition. Noting the vector nature of
the effective contribution to this process, and following again from reference
[80], we derive an upper bound at $90\%$ confidence on the associated
coupling:

$\displaystyle|y^{ReQ}_{12}|\lesssim 0.0956\;{\hat{m}_{\phi}}.$ (68)

Similarly for the muon coupling, the significantly less-suppressed process
$K^{0}_{L}\to\mu^{+}\mu^{-}$ yields:

$\displaystyle|y^{ReQ}_{23}|\lesssim 0.367\;{\hat{m}_{\phi}},$ (69)

and for the lepton-flavour violating decay $K^{0}_{L}\to e^{\pm}\mu^{\mp}$
yields:

$\displaystyle|y_{23}^{ReQ}y_{12}^{ReQ}|<3.78\;{\hat{m}_{\phi}}^{2}\times
10^{-4}.$ (70)

For each of these contributions, the couplings involved are between down-type
and charged-leptons, generated via CKM mixing. This means that, due to the
refined coupling texture, the constraints enter for the magnitudes of these
couplings rather than having phase-dependence. These combinations of
constraints carve out viable parameter-space for these models in the magnitude
$|y_{23}^{ReQ}|-|y_{12}^{ReQ}|$ plane.

####  IV.3.3 Kaon mixing: $|\epsilon_{K}|$

An additional constraint on these models arises from considering neutral kaon
mixing. Because of the restricted coupling structure, dependence on the phases
of the input couplings drop out, however the relative sizes are constrained by
the parameter $\epsilon_{K}$, such that

$\displaystyle|y_{23}^{L}|^{2}(|y_{12}^{L}|^{2}$
$\displaystyle-0.0016|y_{23}^{L}|^{2})$
$\displaystyle\in[-0.077,0.232]\times\;{\hat{m}_{\phi}}^{2}.$ (71)

The above relation holds for both $S_{1}$ and $R_{2}$, derived following
reference [80]. In both instances, this constraint is found to be stronger for
this model than the complementary constraint from neutral kaon mixing $\Delta
M_{K}$. For the considered mass range, the effects on kaon mixing are a less
stringent constraint than those from other kaon sector observables discussed
above. This is based on the current constraints listed in Table 3.

###  IV.4 High $p_{T}$ leptonic constraints

We briefly comment here on the constraints from contributions to
$pp\to\ell\ell$ (dilepton production) and $pp\to\ell\nu$ (monolepton
production) via tree-level t-channel processes. These can be probed directly
in the high-$p_{T}$ tails of Drell-Yan processes at the LHC [85]. For these
processes, numerical analyses find minimal interference between NP effective
operator contributions and _a priori_ several NP operators can be
simultaneously constrained [68].

In reference [28] we presented an outline of the impact of these constraints
on the magnitudes of couplings in both the $R_{2}$ and $S_{1}$ model. As these
constraints also apply to the parameter space explored in this paper, we
refrain from discussing the details of these calculations here but direct the
reader there for further detail. Only RH coupling constraints as consequence
of the dilepton high-$p_{T}$ study are listed in Table 3. The LH couplings are
more tightly constrained by the meson decays discussed earlier. Of the
constraints from charm monolepton decays derived from reference [86], none
were found to be more constraining than the requirement for the LQ couplings
to generate the MDM experimental values.

Note that with increased LHC luminosity, the allowed regions for the effective
interactions manifest in high-$p_{T}$ leptonic tails are forecast to shrink
significantly [86, 68], further restricting parameter space for both models.
However, we limit our consideration here to present high-$p_{T}$ constraints.

###  IV.5 Summary of additional constraints

We select a benchmark mass of $\hat{m}_{\phi}=2$ to provide demonstrative
cross sections of these models’ parameter space. Our aim here is to show the
interplay between different additional constraints discussed in this section,
together with the constraints from lepton mass correction and satisfying the
MDM experimental values.

Firstly, Figures 5(a) and 6(a) show the plane of $y^{L}$ coupling magnitudes,
$|y^{L}_{23}|$ and $|y^{L}_{12}|$. The constraints appearing here originate
from the kaon sector. We see that the $R_{2}$ leptoquark parameter space is
much more strongly constrained than that of $S_{1}$, as discussed in Section
IV.3.

Figure 6: An overview of the constraints on the magnitude of the $R_{2}$ LQ
couplings for a benchmark mass of 2 TeV. The plots show (a) the $y^{L}$
coupling plane, (b) electron couplings and (c) muon couplings. The hashed
regions indicate that the region is ruled out in the direction of the hashing.
The $y^{L}$ couplings are subject to stronger bounds, so their axes are
logarithmic while the RH couplings are shown on a linear axis. For the
electron couplings, the shaded region around the indicated MDM lines show the
one-sigma region about the central value on the experimental values, from
equations (4) and (5). For the muon couplings, the central value is indicated
for a number of benchmark angles $\xi_{23}$.

In Figures 5 and 6 we show the $|y^{L}_{ij}|$ and $|y^{R}_{ij}|$ parameter
space separately for electron and muon sectors, in subfigures (b) and (c)
respectively. The hashed contours show parameter space that is ruled-out by an
array of constraints detailed in this section. In the central plots, we
illustrate regions that satisfy the electron MDM constraints for Cs and Rb, to
within one-sigma. For the muon sector in Figures 5(c) and 6(c), we show
contours representing the benchmark values of $\xi_{23}$, consistent with the
sign preference for each model – as per equations (58) and (59).

One way to interpret the results in Figures 5 and 6 is as follows:

  1. 1.

Identify a point on subfigure (c) that is consistent with the muon MDM, and
within experimental constraints. Note the LH coupling at this point;

  2. 2.

Move to subfigure (a) and use the LH muon coupling to find a LH electron
coupling consistent with constraints;

  3. 3.

Use the electron coupling to check subfigure (b) for compatibility with either
electron MDM result.

Following this procedure, it is easy to verify that there is viable parameter
space for both $S_{1}$ and $R_{2}$ models under our prescribed coupling
ansatz, capable of explaining the muon and electron MDM results – regardless
of whether we take the Cs or Rb result to be true. Taking these results
together with Figure 4 (c), we can infer the predicted muon EDM from the
determined viable parameters.

##  V A brief comment on the third lepton flavour

One may notice at this point that there has been no mention of the couplings
to the tau lepton. This is because the precision with which the tau MDM can be
measured is not yet sensitive enough to be directly contrasted with the SM
prediction [87].

Nevertheless, consider the case that a LQ coupling exists between the tau
lepton and either the charm or the top quark. If it were to couple directly to
the up quark, we would expect it would rapidly approach bounds from couplings
to the first generation of quarks, however the below bounds would not be
applicable.

Considering first the case that the tau lepton couples solely to the top
quark. We note that LQ corrections to the branching ratio
Br$(\tau\to\mu\gamma)$ are also generated in the case of nonzero $\Delta
a_{\tau}$ [29] such that

$\text{Br}[\tau\to\mu\gamma]=\frac{\alpha}{16\Gamma_{\tau}}\frac{m_{\tau}^{2}}{m_{\mu}}|\Delta
a_{\mu}\Delta a_{\tau}|<4.2\times 10^{-8},$ (72)

where we have sourced the upper-bound on $\text{Br}[\tau\to\mu\gamma]$ from
the Belle collaboration [46]. This implies the following bound on $\Delta
a_{\tau}$, if the experimental value of $\Delta a_{\mu}$ remains at its
current value:

$|\Delta a_{\tau}|_{\rm{top}}<2.5\times 10^{-9}$ (73)

Alternatively, if the LQ model couples the tau solely to the charm quark, we
can relate the branching ratio Br$(\tau\to e\gamma)$ to $\Delta a_{e}$ and
$\Delta a_{\tau}$, such that

$\text{Br}[\tau\to
e\gamma]=\frac{\alpha}{16\Gamma_{\tau}}\frac{m_{\tau}^{2}}{m_{e}}|\Delta
a_{e}\Delta a_{\tau}|<3.3\times 10^{-8},$ (74)

again sourcing the upper-bound from the Belle collaboration [46]. Assuming
that $|\Delta a_{e}|$ remains of order $10^{-13}$, this implies that

$|\Delta a_{\tau}|_{\rm{charm}}\lesssim 2.3\times 10^{-7}.$ (75)

The results above show that if an anomaly in the tau MDM is established, then
these single LQ solutions could be extended to the third family, and be probed
using charged-lepton flavour violating decays. These results can be used to
guide model building with scalar LQs in the context of lepton flavour
violation. Tau leptons also have strong prospects for the measurement of CP-
violating observables on the horizon [88], which could lead to an interesting
study of parameter space considering the implications of complex LQ couplings
in this sector – see, for example, references [32, 43].

##  VI Conclusions

In this paper, we have studied the two mixed-chiral scalar leptoquarks $S_{1}$
and $R_{2}$, and the interplay between their contributions to MDMs, EDMs and
radiative charged-lepton mass corrections. For each LQ, we adopted a flavour
ansatz such that the LQ’s contribution to the muon dipole operator was
generated (at leading-order) by a top-containing loop, and the electron by a
charm-containing loop. We allowed LQ couplings to take complex values, and
thus built upon the parameter space studied under a similar framework in
reference [42]. This extension of parameter space was represented via the
difference in phase between LH and RH electron-charm ($\xi_{12}$) and muon-top
($\xi_{23}$) couplings – a quantity directly probed by the measurement of
lepton electric dipole moments. For the electron-charm sector, current
experimental constraints prefer a phase difference very close to $\xi_{12}=0$
or $\pi$. For this reason, we restricted electron-charm couplings to real
values.

We also addressed a common critique of these chirally-enhanced models for
lepton dipole moments, specifically the fine-tuning required to suppress
radiative corrections to the associated lepton masses. To quantify what we
assessed not to be fine-tuned, we imposed the requirement that the magnitude
of the LQ correction not exceed the lepton pole mass. Using this, we argued
that the corrections to the electron mass considered together with satisfying
either Cs or Rb MDM results, led to an upper bound for either LQ mass of

$\displaystyle m_{\phi}\lesssim 3.12\;\text{TeV}.$ (76)

The combination of muon mass constraint and the muon MDM generated weaker
bounds on the LQ mass, but these were provided as they are applicable for any
$R_{2}$ or $S_{1}$ models with top-mass enhanced contributions to $\Delta
a_{\mu}$, such as have become common in the literature.

Utilising the the fine-tuning restriction on the muon mass, we derived
indicative upper-bounds on the magnitude of the muon EDM generated by our
models,

$\displaystyle|d_{\mu}|<\mathcal{O}(10^{-22})\;e\;{\rm cm},$ (77)

which are within reach of future experiments, such as muEDM at PSI, and even
FNAL Muon g$-$2 (depending on the value of $\xi_{23}$). The extension of these
simple scalar LQ models to address MDM anomalies with CP-violating couplings
can thus be probed directly in the near future.

We also performed a comprehensive study of parameter constraints, including
those from electroweak processes, the kaon sector, and high $p_{T}$ dilepton
tails. Our results establish that both $S_{1}$ and $R_{2}$ remain viable BSM
candidates for explaining both $\Delta a_{\mu}$ and $\Delta a_{e}$ anomalies,
regardless of whether the results for Cs or Rb are confirmed. The future of
muon EDM experiments provide exciting prospects for probing such models in the
case of nonzero CP violation in the LQ coupling matrices.

Note that the structure of preferred nonzero LQ couplings in the textures
proposed in this work could be motivated by flavour symmetry models.
Approaches can be explored to generate such coupling structures in UV-complete
BSM models; for example, Froggatt-Nielson mechanisms [89] . Embedding these
textures in a UV-complete model is beyond the scope of this work, although we
identify it as an avenue for future exploration.

Considering our analysis in the context of wider flavour studies, we motivated
future exploration of these LQs’ novel parameter spaces. We addressed the
viability of extensions to our outlined models for explaining potential
anomalies in the $\tau$ MDM, if these were to be measured with precision in
future experimental programs. Our work here can be straightforwardly
incorporated into broader studies of lepton-flavour violation in scalar LQ
models.

###### Acknowledgements.

IB would like to thank Daniel Flynn and Harry M. Quiney for useful discussion
about molecular EDMs, Peter Athron and Michael A. Schmidt for helpful dialogue
about the lepton mass corrections, Martin Hoferichter and Olcyr Sumensari for
their advice on the study of $Z$ effective couplings. This work is supported
in part by the Australian Research Council and the Australian Government
Research Training Program Scholarship initiative.

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