IFJPAN-IV-2021-6

LA-UR-21-24456

Leptonic anomalous magnetic moments in $\nu$SMEFT

Vincenzo Ciriglianoa, Wouter Dekensb, Jordy de Vriesc,d, Kaori Fuyutoa,
Emanuele Mereghettia, Richard Ruize

a Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA

b Department of Physics, University of California at San Diego, La Jolla, CA
92093, USA

c Institute for Theoretical Physics Amsterdam and Delta Institute for
Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH
Amsterdam, The Netherlands

d Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

e Institute of Nuclear Physics, Polish Academy of Sciences, ul.
Radzikowskiego, Cracow 31-342, Poland

We investigate contributions to the anomalous magnetic moments of charged
leptons in the neutrino-extended Standard Model Effective Field Theory
($\nu$SMEFT). We discuss how $\nu$SMEFT operators can contribute to a lepton’s
magnetic moment at one- and two-loop order. We show that only one operator can
account for existing electronic and muonic discrepancies, assuming new physics
appears above $1$ TeV. In particular, we find that a right-handed charged
current in combination with minimal sterile-active mixing can explain the
discrepancy for sterile neutrino masses of $\mathcal{O}(100)$ GeV while
avoiding direct and indirect constraints. We discuss how searches for sterile
neutrino production at the (HL-)LHC, measurements of
$h\rightarrow\mu^{+}\mu^{-}$ and searches for $h\rightarrow e^{+}e^{-}$,
neutrinoless double beta decay experiments, and improved unitarity tests of
the CKM matrix can further probe the relevant parameter space.

##  1 Introduction

The E989 experiment at Fermilab [1], which has recently measured the anomalous
magnetic moment of the muon $a_{\mu}$ to the highest precision [2], confirms
the larger-than-predicted measurement of the E821 experiment at Brookhaven [3]
and could at last indicate the existence of beyond-the-Standard-Model (BSM)
physics outside the neutrino and dark sectors. A combination of the two
measurements gives the average

$a^{\rm average}_{\mu}=(116\,592\,061\pm 41)\cdot 10^{-11}.$ (1)

This should be compared to the Standard Model (SM) prediction of

$a^{\rm SM\,\,\,\,\,\,\,}_{\mu}=(116\,591\,810\pm 43)\cdot 10^{-11}\,,$ (2)

as reported in the white paper by the global theory initiative [4], implying a
discrepancy of

$\Delta a_{\mu}=(251\pm 59)\cdot 10^{-11}.$ (3)

Of course, further experimental and theoretical investigations must carefully
examine possible sources of uncertainty. For example: the recent lattice-QCD
calculation of the hadronic vacuum polarization contribution to $\Delta
a_{\mu}$ [5] can shift the SM prediction much closer to the experimental
result, essentially eliminating the discrepancy. However, this would
simultaneously introduce tensions elsewhere [6, 7, 8]. Here we will use Eq.
(2) as our SM prediction for $a^{\rm SM}_{\mu}$.

The anomalous magnetic moment of the electron is measured with even higher
accuracy [9]:

$a_{e}^{\rm exp}=\left(1\,159\,652\,180.73\pm 0.28\right)\cdot 10^{-12}.$ (4)

In this case, the error of the SM prediction is dominated by the extraction of
the fine structure constant $\alpha_{\rm em}$ from experiments that measure
the recoil of alkali atoms. The two most recent extractions of $a_{e}$ from
Cesium [10] and Rubidium [11] differ by more than 5$\sigma$, and lead to

$\displaystyle\Delta a_{e}^{\rm Cs}$ $\displaystyle=$ $\displaystyle
a_{e}^{\rm exp}-a_{e}^{\rm SM,\,Cs}=-\left(0.88\pm 0.36\right)\cdot
10^{-12}\,,$ $\displaystyle\Delta a_{e}^{\rm Rb}$ $\displaystyle=$
$\displaystyle a_{e}^{\rm exp}-a_{e}^{\rm SM,\,Rb}=\left(0.48\pm
0.30\right)\cdot 10^{-12}\,.$ (5)

This indicates a $-2.4\sigma$ ($+1.6\sigma$) tension with the SM that is not
too significant in comparison to the muon case. For the $\tau$ lepton, the
existing bounds on the anomalous magnetic moment [12],

$\displaystyle-0.052<$ $\displaystyle a_{\tau}$ $\displaystyle<0.013\,,\qquad
e^{+}e^{-}\to e^{+}e^{-}\tau^{+}\tau^{-}\,\,\text{
\cite[cite]{[\@@bibref{}{Abdallah:2003xd}{}{}]}}\,,$ $\displaystyle-0.007<$
$\displaystyle a_{\tau}$ $\displaystyle<0.005\,,\qquad
e^{+}e^{-}\to\tau^{+}\tau^{-}\hskip 28.45274pt\text{
\cite[cite]{[\@@bibref{}{GonzalezSprinberg:2000mk}{}{}]}}\,,$ (6)

are still one order of magnitude away from the SM prediction of
$a_{\tau}=\left(117\,721\pm 5\right)\cdot 10^{-8}$ [15]. While at the moment
$a_{\tau}$ is not very sensitive to BSM physics, the limit in Eq. (1) will be
improved at Belle-II and at the Large Hadron Collider (LHC) [16, 17, 18, 19].

The muonic discrepancy in particular has lead to an overwhelming amount of
theoretical work looking for suitable explanations. For a recent overview, see
Ref. [20]. Most models involve adding new fields and interactions to the SM
that go on to generate novel loop contributions to $a_{\mu}$ and $a_{e}$.
Under the assumption that new fields are heavy111We note that exceptions
exist, for instance in the form of light dark photons [21]., motivated by the
absence of evidence for BSM physics in direct searches at the LHC and
elsewhere, these models can be efficiently described by effective field theory
(EFT) techniques. A recent investigation in the framework of the Standard
Model EFT (SMEFT) concluded that only a handful of dimension-six SMEFT
operators could account for $\Delta a_{\mu}$ while not being excluded by other
measurements and still obey the EFT demand that $v/\Lambda\ll 1$ [22]. Here,
$v=\sqrt{2}\langle H^{0}\rangle\simeq 246$ GeV is the Higgs field’s vacuum
expectation value (vev) and $\Lambda$ is the scale of BSM physics. In the EFT
language, any BSM scenario that explains $\Delta a_{\mu}$ will most likely
need to generate either the muon dipole operators themselves,
$\mathcal{O}_{eB},~{}\mathcal{O}_{eW}$, which induce corrections to $a_{\mu}$
at tree level, or semi-leptonic tensor operators, $\mathcal{O}_{lequ}^{(3)}$,
which contribute to the dipole moments through one-loop diagrams [22]. The
latter can be significant due to top (or charm) loops, which give rise to a
chirally enhanced contributions and scale as $\sim m_{t(c)}/m_{\mu}$. Similar
conclusions hold in case when trying to explain corrections to $a_{e}$ at the
level of Eq. (1).

In this work, we investigate whether slightly relaxing the SMEFT assumptions
could give rise to additional mechanisms. We do so by considering the
neutrino-extended SMEFT ($\nu$SMEFT) [23, 24]. This framework is motivated by
the need to account for neutrino masses, whose explanation involves right-
handed (RH) neutrinos in many (but not all) BSM scenarios. A minimal SM
extension would be the addition of a set of $\nu_{R}$ fields that, in
combination with the SM’s left-handed (LH) neutrino $\nu_{L}$ and Higgs
fields, can generate neutrino Dirac masses through the usual Higgs mechanism.
However, being a gauge singlet, nothing forbids the presence of a RH Majorana
mass term $m_{\nu_{R}}$ for $\nu_{R}$. The simultaneous presence of a Dirac
and Majorana masses leads to three light active Majorana neutrinos and $n$
sterile Majorana states, where $n$ is the number of $\nu_{R}$ fields that are
added to the theory. A priori nothing can be said about the RH Majorana masses
of sterile neutrinos and they might as well be light, with $m_{\nu_{R}}\leq
v$. While the neutrinos look sterile at low energies, in a large class of BSM
models they interact with SM and BSM fields at high energies. Examples are
leptoquark models [25], left-right-symmetric models [26, 27], gauged baryon
and lepton number models [28, 29, 30], and Grand Unified Theories [31]. At
lower energies, such interactions can be described in terms of local operators
in the $\nu$SMEFT Lagrangian. Here we consider the effect of $\nu_{R}$
operators on $a_{\mu}$ and $a_{e}$.

This paper is organized as follows: In Sect. 2 we introduce the $\nu$SMEFT
Lagrangian and its rotation into the mass basis. In Sect. 3, we investigate
which $\nu$SMEFT operators can generate corrections to $a_{\mu}$ and $a_{e}$
at the one- and two-loop levels and which operators are phenomenologically
viable. Sect. 3.2 focuses on the most promising $\nu$SMEFT operator and
discusses its contributions to $a_{e,\mu}$. We investigate in detail what
parameter space can account for $\Delta a_{e,\mu}$ and how this parameter
space can be tested in the future in Sect. 4. Our main findings are then
discussed in Sect. 5. Finally, we conclude in Sect. 6.

##  2 The $\nu$SMEFT Lagrangian

We consider an EFT that consists of the SM supplemented by $n$ SM gauge
singlets $\nu_{R}$ (although we mainly consider $n=1$), and include higher-
dimensional operators up to dimension six. The Lagrangian of the resulting
EFT, known as $\nu$SMEFT, can then be written as [24]

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{SM}+\bar{\nu}_{R}\,i\not{\partial}\nu_{R}-\left[\frac{1}{2}\bar{\nu}^{c}_{R}\,\bar{M}_{R}\nu_{R}+\bar{L}\tilde{H}Y_{\nu}\nu_{R}+\rm{H.c.}\right]$
(7)
$\displaystyle+\mathcal{L}^{(5)}_{\nu_{L}}+\mathcal{L}^{(5)}_{\nu_{R}}+\mathcal{L}^{(6)}_{\nu_{L}}+\mathcal{L}^{(6)}_{\nu_{R}}\,,$

where $\mathcal{L}_{SM}$ contains renormalizable operators of the SM and do
not involve $\nu_{R}$. We use $\Psi^{c}=C\bar{\Psi}^{T}$ for a spinor field
$\Psi$ in terms of $C=-C^{-1}=-C^{T}=-C^{\dagger}$, the charge conjugation
matrix. $L=(\nu_{L},\,e_{L})^{T}$ is the LH lepton doublet with generation
indices suppressed for clarity. $\tilde{H}=i\tau_{2}H^{*}$ with $H$ denoting
the Higgs doublet

$H=\frac{v}{\sqrt{2}}U(x)\left(\begin{array}[]{c}0\\\
1+\frac{h(x)}{v}\end{array}\right)\,.$ (8)

Here, $h(x)$ is the Higgs boson field and $U(x)$ is an $SU(2)$ matrix encoding
the electroweak (EW) Goldstone modes. $\nu_{R}$ is a column vector of $n$ RH
neutrinos such that $Y_{\nu}$ is a $3\times n$ Yukawa matrix. $\bar{M}_{R}$ is
a complex, symmetric $n\times n$ mass matrix containing the Majorana masses of
$\nu_{R}$. The terms $\mathcal{L}^{(5)}_{\nu_{L}}$ and
$\mathcal{L}^{(6)}_{\nu_{L}}$ were constructed in Refs. [32] and [33],
respectively, and describe the dimension-five and -six operators containing
just SM fields (no $\nu_{R}$). Similarly, $\mathcal{L}^{(5)}_{\nu_{R}}$ and
$\mathcal{L}^{(6)}_{\nu_{R}}$ contain the complete set of dimension-five and
-six operators involving at least one $\nu_{R}$ field and were derived in
Refs. [34] and [23, 24], respectively.

The dimension-five terms generate Majorana masses for $\nu_{L}$ and $\nu_{R}$
and can be written as

$\mathcal{L}^{(5)}_{\nu_{L}}=\epsilon_{kl}\epsilon_{mn}(L_{k}^{T}\,C^{(5)}\,CL_{m})H_{l}H_{n}\,,\qquad\mathcal{L}^{(5)}_{\nu_{R}}=-\bar{\nu}^{c}_{R}\,\bar{M}_{R}^{(5)}\nu_{R}H^{\dagger}H\,,$
(9)

where $C^{(5)}$ and $M_{R}^{(5)}$ are dimensionful Wilson coefficients.
Throughout this work we employ a notation where Wilson coefficients of gauge-
invariant $\nu$SMEFT operators have absorbed all factors of the EFT cutoff
scale $\Lambda$. The coefficient of an operator of dimension $d$ then has mass
dimension $4-d$ and scales as $C^{(d)}\sim\Lambda^{4-d}$.

Transition dipole moments appear at the same mass dimension as the terms in
Eq. (9), but require at least two sterile states. The first interesting
contributions to $\Delta a_{\ell}$ for $\ell=e,\mu$, involving only one
$\nu_{R}$, then arise from dimension-six operators. The effects of the
operators in (normal) SMEFT on the leptonic dipole moments were recently
discussed in Ref. [22] and here we focus on the dimension-six operators
involving right-handed neutrinos, which are collected in Table 1.

$\psi^{2}H^{3}$ | $\psi^{2}H^{2}D$ | $\psi^{2}HX(+\mbox{H.c.})$
---|---|---
$\mathcal{O}_{L\nu H}(+\mbox{H.c.})$ | $(\bar{L}\nu_{R})\tilde{H}(H^{\dagger}H)$ | $\mathcal{O}_{H\nu}$ | $(\bar{\nu}_{R}\gamma^{\mu}\nu_{R})(H^{\dagger}i\overleftrightarrow{D_{\mu}}H)$ | $\mathcal{O}_{\nu B}$ | $(\bar{L}\sigma_{\mu\nu}\nu_{R})\tilde{H}B^{\mu\nu}$
|  | $\mathcal{O}_{H\nu e}(+\mbox{H.c.})$ | $(\bar{\nu}_{R}\gamma^{\mu}e)({\tilde{H}}^{\dagger}iD_{\mu}H)$ | $\mathcal{O}_{\nu W}$ | $(\bar{L}\sigma_{\mu\nu}\nu_{R})\tau^{I}\tilde{H}W^{I\mu\nu}$
$(\bar{R}R)(\bar{R}R)$ | $(\bar{L}L)(\bar{R}R)$ | $(\bar{L}R)(\bar{L}R)(+\mbox{H.c.})$
$\mathcal{O}_{\nu\nu}$ | $(\bar{\nu}_{R}\gamma^{\mu}\nu_{R})(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ | $\mathcal{O}_{L\nu}$ | $(\bar{L}\gamma^{\mu}L)(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ | $\mathcal{O}_{L\nu Le}$ | $(\bar{L}\nu_{R})\epsilon(\bar{L}e)$
$\mathcal{O}_{e\nu}$ | $(\bar{e}\gamma^{\mu}e)(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ | $\mathcal{O}_{Q\nu}$ | $(\bar{Q}\gamma^{\mu}Q)(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ | $\mathcal{O}_{L\nu Qd}$ | $(\bar{L}\nu_{R})\epsilon(\bar{Q}d)$
$\mathcal{O}_{u\nu}$ | $(\bar{u}\gamma^{\mu}u)(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ |  |  | $\mathcal{O}_{LdQ\nu}$ | $(\bar{L}d)\epsilon(\bar{Q}\nu_{R})$
$\mathcal{O}_{d\nu}$ | $(\bar{d}\gamma^{\mu}d)(\bar{\nu}_{R}\gamma_{\mu}\nu_{R})$ |  |  |  |
$\mathcal{O}_{du\nu e}(+\mbox{H.c.})$ | $(\bar{d}\gamma^{\mu}u)(\bar{\nu}_{R}\gamma_{\mu}e)$ |  |  |  |
$(\bar{L}R)(\bar{R}L)$ | $(\not{L}\cap B)(+\mbox{H.c.})$ | $(\not{L}\cap\not{B})(+\mbox{H.c.})$
$\mathcal{O}_{Qu\nu L}(+\mbox{H.c.})$ | $(\bar{Q}u)(\bar{\nu}_{R}L)$ | $\mathcal{O}_{\nu\nu\nu\nu}$ | $(\bar{\nu}_{R}^{c}\nu_{R})(\bar{\nu}_{R}^{c}\nu_{R})$ | $\mathcal{O}_{QQd\nu}$ | $\epsilon_{ij}\epsilon_{\alpha\beta\sigma}(Q^{i}_{\alpha}CQ^{j}_{\beta})(d_{\sigma}C\nu_{R})$
|  |  |  | $\mathcal{O}_{udd\nu}$ | $\epsilon_{\alpha\beta\sigma}(u_{\alpha}Cd_{\beta})(d_{\sigma}C\nu_{R})$

Table 1: The complete basis of dimension-six operators involving $\nu_{R}$
taken from Ref. [24]. The operators are expressed in terms of a column vector
of $n$ gauge singlet fields, $\nu_{R}$, and of SM fields, the lepton and Higgs
doublets, $L$ and $H$, the quark left-handed doublet $Q=(u_{L},d_{L})^{T}$,
and the right-handed fields $e$, $u$, and $d$.

We work in the basis where charged leptons $e^{i}_{L,R}$ and quarks
$u^{i}_{L,R}$, $d^{i}_{R}$ are mass eigenstates ($i=1,2,3$). After EW symmetry
breaking this means that $d^{i,\,\rm gauge}_{L}=V^{ij}d_{L}^{j,\,\rm mass}$,
where $V$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. At dimension six, the
mass terms of the neutrinos after EW symmetry breaking but before mass-
diagonalization take the form,

$\displaystyle\mathcal{L}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bar{\mathcal{N}}^{c}M_{\nu}\mathcal{N}+{\rm
H.c.}\,,\qquad M_{\nu}=\begin{pmatrix}M_{L}&M_{D}^{*}\\\
M_{D}^{\dagger}&M_{R}^{\dagger}\end{pmatrix}\,,$ $\displaystyle M_{L}$
$\displaystyle=$ $\displaystyle-v^{2}C^{(5)}\,,\qquad
M_{R}=\bar{M}_{R}+v^{2}\bar{M}_{R}^{(5)}\,,\qquad
M_{D}=\frac{v}{\sqrt{2}}\left[Y_{\nu}-\frac{v^{2}}{2}C_{L\nu
H}^{(6)}\right]\,.$ (10)

Here $\mathcal{N}=(\nu_{L},\,\nu_{R}^{c})^{T}$ making $M_{\nu}$ a
$(3+n)\times(3+n)$ symmetric matrix. This matrix can be diagonalized by a
transformation involving the unitary matrix $U$, given by

$\displaystyle U^{T}M_{\nu}U=m_{\nu}={\rm
diag}(m_{1},\dots,m_{3+n})\,,\quad\text{with}\quad\mathcal{N}=U\mathcal{N}_{m}\,.$
(11)

The mass eigenstates can be written as
$\nu=\mathcal{N}_{m}+\mathcal{N}_{m}^{c}=\nu^{c}$. The flavor eigenstates are
then related by $\nu_{L}=P_{L}(PU)\nu$ and $\nu_{R}=P_{R}(P_{s}U^{*})\nu$,
where $P=\begin{pmatrix}\mathcal{I}_{3\times 3}&0_{3\times n}\end{pmatrix}$
and $P_{s}=\begin{pmatrix}0_{n\times 3}&\mathcal{I}_{n\times n}\end{pmatrix}$
are projection matrices. For simplicity we will often consider the case $n=1$,
implying four massive neutrinos with masses $m_{i}\ll\Lambda$. The first three
states $\nu_{1,2,3}$ approximately correspond to active neutrinos with sub-eV
masses [35], while $\nu_{4}$ is a nearly sterile neutrino with mass $m_{4}$
that could be significantly larger. For $n=1$, the mass terms involving
$\nu_{R}$, $M_{D}$, and $M_{R}$, are not sufficient to generate the masses of
the light neutrinos. Two sterile states are in principle sufficient, but would
require significant tuning to obtain the sizable mixing matrix elements that
are needed to resolve the discrepancy in the lepton’s magnetic moments. We
will assume a non-zero value of $M_{L}$ that accounts for the measured
neutrino masses and mixing. Such an $M_{L}$ can for instance be induced at
tree level by integrating out even heavier neutrino mass eigenstates and
$SU(2)$ triplets, or at loop level via more complicated field content [36, 37,
38].

Throughout this work we generically denote the mixing between active, lepton-
flavor states $\alpha\in\\{e,\mu,\tau\\}$ with (light and heavy) neutrino mass
eigenstates $j\in\\{1,\dots,3+n\\}$ by $U_{\alpha j}$. From oscillation data,
$|U_{\alpha j}|\sim\mathcal{O}(1)$ for $j=1,2,3$, while from various
constraints (as described below) $|U_{\alpha j}|\ll 1$ for $j\geq 4$. In this
notation, the $3\times 3$ Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix that
parametrizes active-light neutrino mixing is non-unitary and is embedded in
the $(3+n)\times(3+n)$ mixing matrix. For the case of $n=1$, we label the
active-sterile mixing matrix element associated with $\nu_{4}$ by $U_{\alpha
4}$. In addition, we adopt the mixing formalism of Ref. [39] and denote the
mixing between “sterile” flavors $S=\\{1,\dots n\\}$ with mass eigenstates
$j=\\{1,\dots,3+n\\}$ by $U_{Sj}$. By unitarity, $|U_{Sj}|\ll 1$ for
$j=1,2,3$, while $|U_{Sj}|$ can be $\mathcal{O}(1)$ for $j\geq 4$.

##  3 Contributions to $a_{\ell}$ from dimension-six $\nu_{R}$ operators

In principle, several of the dimension-six operators involving sterile
neutrinos will give rise to loop-level corrections to $\Delta a_{e}$ and
$\Delta a_{\mu}$, which we henceforth denote as $\Delta a_{\ell}$. In the EFT
below the EW scale, $\Delta a_{\ell}$ is generated at tree-level for lepton
$e$ by the leptonic dipole moment

$\mathcal{L}_{\rm dipole}=L_{\begin{subarray}{c}e\gamma\\\
pr\end{subarray}}\,\bar{e}_{L\,p}\sigma^{\mu\nu}e_{R\,r}\,F_{\mu\nu}+{\rm
H.c.}\,,$ (12)

where $p,r$ are flavor indices, and the Wilson coefficient
$L_{\begin{subarray}{c}e\gamma\\\ \ell\ell\end{subarray}}$ gives a
contribution to $a_{\ell}$ of

$\displaystyle\Delta a_{\ell}=\frac{4m_{\ell}}{e}{\rm
Re}\left(L_{\begin{subarray}{c}e\gamma\\\ \ell\ell\end{subarray}}\right)\,.$
(13)

In the denominator is the quantity $e=|e|=\sqrt{4\pi\alpha_{\rm em}}$.

We now write the loop-level contributions of $\nu$SMEFT interactions to
$L_{\begin{subarray}{c}e\gamma\\\ \mu\mu\end{subarray}}$ and
$L_{\begin{subarray}{c}e\gamma\\\ ee\end{subarray}}$ as

$L_{\begin{subarray}{c}e\gamma\\\
\ell\ell\end{subarray}}=\left[e\frac{b_{i}}{v}(v^{2}C_{i})\right]_{\ell\ell}\,.$
(14)

The dimensionless factors $b_{i}$ parametrize the contributions of gauge and
Yukawa couplings as well as the loop order at which the corresponding
$\nu$SMEFT operator $\mathcal{O}_{i}$ generates the dipole interactions.

We do not pursue a comprehensive calculation of $b_{i}$, and hence
$L_{\begin{subarray}{c}e\gamma\\\ \ell\ell\end{subarray}}$, for all operators
in Table 1, as simple order-of-magnitude estimates show that most interactions
cannot induce sufficiently large anomalous magnetic moments to account for
measured $\Delta a_{\mu}$. We provide such estimates for all $B$\- and
$L$-conserving dimension-six $\nu$SMEFT operators in Table 2, where we show a
naïve expression for each generated $L_{e\gamma}$, as well as its effect on
$\Delta a_{\mu}$ and $\Delta a_{e}$, which can be written as

$\displaystyle\Delta a_{\ell}=\frac{4m_{\ell}}{e}{\rm
Re}\left(L_{\begin{subarray}{c}e\gamma\\\
\ell\ell\end{subarray}}\right)\,=\frac{4m_{\ell}}{v}{\rm
Re}\left({b_{i}}v^{2}C_{i}\right)\,.$ (15)

For the numerical estimates we use $L=(4\pi)^{-2}$ for loop factors and
$Y_{e,d,u}=\sqrt{2}m_{\mu(e),b,t}/v$ for $\Delta a_{\mu}~{}(\Delta a_{e})$. As
we assume $|\mathcal{C}_{i}|=v^{2}|C_{i}|$ and $|Y_{\nu}|$ to be
$\ll\mathcal{O}(1)$, contributions of the form $\Delta a_{\ell}/\Delta
a_{\ell}^{\rm expt.}=c_{i}\mathcal{C}_{i}$ or $\Delta a_{\ell}/\Delta
a_{\ell}^{\rm expt.}=c_{i}^{\prime}Y_{\nu}\mathcal{C}_{i}$, with
$c_{i},c^{\prime}_{i}\leq\mathcal{O}(1)$, are unlikely to explain $\Delta
a_{\mu}^{\rm expt.}$ or $\Delta a_{e}^{\rm expt.}$. In only five cases do we
estimate that ${c}_{i},~{}{c}_{i}^{\prime}\geq 1$ and denote these with
asterisks.

class | $\psi^{2}H^{3}$ | $\psi^{2}H^{2}D$ | $(\bar{R}R)(\bar{R}R)$
---|---|---|---
$\mathcal{C}_{i}=v^{2}C_{i}$ | $\mathcal{C}_{L\nu H}$ | $\mathcal{C}_{H\nu}$ | ${}^{*}\mathcal{C}_{H\nu e}$ | $\mathcal{C}_{\nu\nu}$ | $\mathcal{C}_{u\nu,d\nu}$ | $\mathcal{C}_{e\nu}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{{}^{*}\mathcal{C}_{du\nu e}}$
$b_{i}$ | $L^{2}Y_{\nu}^{\dagger}Y_{e}$ | $L^{2}Y_{e}$ | $LY_{\nu}$ | $L^{3}Y_{e}\,Y_{\nu}^{\dagger}Y_{\nu}$ | $L^{2}Y_{e}\\{Y_{\nu}Y_{\nu}^{\dagger},L\\}$ | $L^{2}Y_{e}$ | $L^{2}Y_{u}^{\dagger}Y_{d}Y_{\nu}$
$\Delta a_{\mu}/(\mathcal{C}_{i}\Delta a_{\mu}^{{\rm expt.}})$ | $10^{-2}Y_{\nu}$ | $10^{-2}$ | $3\cdot 10^{3}\,Y_{\nu}$ | $10^{-4}Y_{\nu}^{\dagger}Y_{\nu}$ | $10^{-2}\\{Y_{\nu}Y_{\nu}^{\dagger},10^{-2}\\}$ | $10^{-2}$ | $0.4Y_{\nu}^{\dagger}$
$\Delta a_{e}/(\mathcal{C}_{i}\Delta a_{e}^{{\rm expt.}})$ | $10^{-3}Y_{\nu}$ | $10^{-3}$ | $4\cdot 10^{4}\,Y_{\nu}$ | $10^{-5}Y_{\nu}^{\dagger}Y_{\nu}$ | $10^{-3}\\{Y_{\nu}Y_{\nu}^{\dagger},10^{-2}\\}$ | $10^{-3}$ | $\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{6Y_{\nu}^{\dagger}}$
class | $\psi^{2}HX(+\mbox{H.c.})$ | $(\bar{L}L)(\bar{R}R)$ | $(\bar{L}R)(\bar{L}R)$ | $(\bar{L}R)(\bar{R}L)$
$\mathcal{C}_{i}=v^{2}C_{i}$ | ${}^{*}\mathcal{C}_{\nu B,\nu W}$ | $\mathcal{C}_{Q\nu}$ | $\mathcal{C}_{L\nu}$ | ${}^{*}\mathcal{C}_{L\nu Le}$ | $\mathcal{C}_{L\nu Qd}$ | $\mathcal{C}_{LdQ\nu}$ | $\mathcal{C}_{Qu\nu L}$
$b_{i}$ | $LY_{\nu}^{\dagger}Y_{e}$ | $L^{2}Y_{e}\\{Y_{\nu}Y_{\nu}^{\dagger},L\\}$ | $L^{2}Y_{e}$ | $L^{2}Y_{\nu}^{\dagger}$ | $L^{2}Y_{d}^{\dagger}\,Y_{\nu}^{\dagger}Y_{e}$ | $L^{2}Y_{d}^{\dagger}\,Y_{\nu}^{\dagger}Y_{e}$ | $L^{2}Y_{u}^{\dagger}\,Y_{e}^{\dagger}Y_{\nu}$
$\Delta a_{\mu}/(\mathcal{C}_{i}\Delta a_{\mu}^{{\rm expt.}})$ | $2Y_{\nu}^{\dagger}$ | $10^{-2}\\{Y_{\nu}Y_{\nu}^{\dagger},10^{-2}\\}$ | $10^{-2}$ | $20Y_{\nu}^{\dagger}$ | $10^{-4}Y_{\nu}$ | $10^{-4}Y_{\nu}^{\dagger}$ | $10^{-2}Y_{\nu}$
$\Delta a_{e}/(\mathcal{C}_{i}\Delta a_{e}^{{\rm expt.}})$ | $0.1Y_{\nu}^{\dagger}$ | $10^{-3}\\{Y_{\nu}Y_{\nu}^{\dagger},10^{-2}\\}$ | $10^{-3}$ | $300Y_{\nu}^{\dagger}$ | $10^{-5}Y_{\nu}$ | $10^{-5}Y_{\nu}^{\dagger}$ | $10^{-3}Y_{\nu}$

Table 2: Estimates of the contributions of the dimension-six Wilson
coefficients to the coefficients $b_{i}$ (defined in Eq. (14)), where $\Delta
a_{\ell}=(4m_{\ell}/v)b_{i}{\cal C}_{i}$. The third (fourth) row is the ratio
$\Delta a_{\ell}/\Delta a_{\ell}^{\rm expt.}$ for $\ell=\mu~{}(e)$ in units of
$\mathcal{C}_{i}=v^{2}C_{i}$. $L$ stands for a loop factor and $Y_{i}$ are the
different Yukawa couplings. The cases with ${c}_{i},~{}{c}_{i}^{\prime}\geq 1$
are highlighted (in green) with an asterisk.

In many cases the impact on $\Delta a_{\ell}$ is too small because the first
possible diagrammatic insertions appear beyond one-loop order, or because of
the appearance of small Yukawa couplings. For example: in the operator class
$(\overline{L}R)(\overline{R}L)$, the impact of $\mathcal{O}_{Qu\nu L}$ on
$\Delta a_{\mu}$ is suppressed by $b_{Qu\nu L}\sim
L^{2}Y_{u}^{\dagger}Y_{e}^{\dagger}Y_{\nu}\sim 10^{-8}Y_{\nu}$, which leads to
the ratio $\Delta a_{\mu}/\Delta a_{\mu}^{\rm expt.}\sim 10^{-2}\times
Y_{\nu}v^{2}C_{Qu\nu L}$. Only in the extreme limit where $Y_{\nu}\sim
v^{2}C_{Qu\nu L}\sim\mathcal{O}(10)$ and the EFT breaks down can
$\mathcal{O}_{Qu\nu L}$ account for experimental measurements. In summary,
while the contributions from most operators are significantly smaller than the
current discrepancies, five operators generate potentially interesting
contributions. Indicated (in green) with asterisks in Table 2, these operators
are:

$\displaystyle\mathcal{O}_{H\nu e},\quad\mathcal{O}_{\nu
B},\quad\mathcal{O}_{\nu W},\quad\mathcal{O}_{L\nu
Le}\quad\text{and}\quad\mathcal{O}_{du\nu e}.$ (16)

The corresponding one- and two-loop contributions to $\Delta a_{\ell}$ are
depicted in Fig. 1. The left diagram is a threshold contribution induced by
the $\mathcal{O}_{H\nu e}$ operator. The two diagrams in the middle depict the
dipoles, $\mathcal{O}_{\nu B}$ and $\mathcal{O}_{\nu W}$, and four-lepton
operators, $\mathcal{O}_{L\nu Le}$, which generate magnetic moments through
renormalization group running. In the electron case, the semileptonic operator
$\mathcal{O}_{du\nu e}$ also induces a somewhat sizable corrections to $\Delta
a_{e}$ as in the right diagram.

Figure 1: Examples of the contribution from the dim-6 operators to $\Delta
a_{l}$. The gray box represents an insertion of a $\nu$SMEFT operator. The
solid line corresponds to fermions, while dashed and wavy lines denote the
$SU(2)$ Higgs doublet and a SM gauge boson, respectively.

###  3.1 Dimension-six dipole and four-fermion interactions

We first discuss the dipole operators, $\mathcal{O}_{\nu B}$,
$\mathcal{O}_{\nu W}$, which could, in principle, give significant
contributions to $a_{\mu}$. These couplings also generate magnetic moments
$\mu_{\nu}$ for the active neutrinos that are described at tree level by the
interaction
$\mathcal{L}\supset-\frac{1}{4}[\mu_{\nu}]_{ij}{\nu}^{T}_{i}C\sigma^{\mu\nu}P_{R}\nu_{j}F_{\mu\nu}+{\rm
h.c.}$ [40, 41], where $C$ is the charge conjugation matrix and $i,j$ run over
light neutrino mass eigenstates222The coefficient $\mu_{\nu}$ used here
corresponds to $\lambda^{\dagger}$ defined in Ref. [41].. Barring
cancellations, current experiment limits require $v^{2}\big{|}U^{*}_{\ell
i}\left[C_{\nu B,\nu W}\right]_{\ell S}U^{*}_{Sj}\big{|}\lesssim 10^{-6}$ [42,
12]. Since $|U_{\ell i}|\sim\mathcal{O}(1)$, the Wilson coefficients $C_{\nu
B}$ and $C_{\nu W}$ can only be sizable if the sterile-light mixing element
$U_{Sj}$ saturates the constraint with $|U^{*}_{Sj}|\sim\mathcal{O}(10^{-6})$.
(Smaller values of $U_{Sj}$ require that $v^{2}C_{\nu B,\nu
W}\gtrsim\mathcal{O}(1)$, implying a breakdown of the EFT.) Notably, this
value of $|U_{Sj}|$ is consistent with the high-scale Type I seesaw featuring
$\nu_{4}$ with mass $m_{4}\sim\mathcal{O}(100)$ GeV, i.e.,
$|U_{Sj}|\sim|U_{\ell 4}|\sim\sqrt{m_{j}/m_{4}}\sim 10^{-6}$. Nevertheless,
this limit in conjunction with Table 2 implies that there is no room for
significant contributions to either lepton’s magnetic moment.

The third promising operator that could be relevant for both $a_{\mu}$ and
$a_{e}$ is $\mathcal{O}_{L\nu Le}$. While this four-fermion interaction
contributes to active neutrinos’ magnetic moments, the contributions are
suppressed by a small lepton mass in addition to a loop factor. Hence, no
strong constraint can be set. We therefore go beyond the naïve estimate of
Table 2 and first consider its leading-log contributions to the coefficient
$L_{\begin{subarray}{c}e\gamma\\\ \mu\mu\end{subarray}}$. The required RGEs
are those of the mixing of $C_{L\nu Le}$ into the leptonic dipole moments
$C_{eB}$ and $C_{eW}$, as well as the running and mixing among $C_{eB}$ and
$C_{eW}$. The former are given by

$\displaystyle\frac{d}{d\ln\mu}C_{\begin{subarray}{c}eB\\\
pr\end{subarray}}=\frac{1}{2}\frac{g_{1}{\rm
y}_{\ell}}{16\pi^{2}}C_{\begin{subarray}{c}L\nu Le\\\
pStr\end{subarray}}\left[Y^{*}_{\nu}\right]_{tS}\,,\qquad\frac{d}{d\ln\mu}C_{\begin{subarray}{c}eW\\\
pr\end{subarray}}=-\frac{1}{4}\frac{g}{16\pi^{2}}C_{\begin{subarray}{c}L\nu
Le\\\ pStr\end{subarray}}\left[Y^{*}_{\nu}\right]_{tS}\,,$ (17)

where ${\rm y}_{\ell}=-1/2$ is the hypercharge of charged lepton $\ell$, the
indices $p,r,t=\\{e,\,\mu,\,\tau\\}$ indicate the charged lepton flavor, and
$S=\\{1,\dots n\\}$ label sterile neutrino flavors. This induces the
combination of dipole moments that couples to the $Z$ boson, but not the
photon, below the EW scale. Thus, to generate the electromagnetic dipole
moment one requires the mixing between $C_{eB}$ and $C_{eW}$ as well. Using
the RGEs of Refs. [43, 44] we obtain

$\displaystyle\Delta a_{\mu}/\Delta a_{\mu}^{{\rm expt.}}$
$\displaystyle\simeq$ $\displaystyle 0.2\,v^{2}C_{\begin{subarray}{c}L\nu
Le\\\ \mu
St\mu\end{subarray}}\left[Y^{*}_{\nu}\right]_{tS}\ln^{2}\left(\frac{m_{4}}{\Lambda}\right)\,,$
$\displaystyle\Delta a_{e}/\Delta a_{e}^{{\rm Cs}}$ $\displaystyle\simeq$
$\displaystyle-2.8\,v^{2}C_{\begin{subarray}{c}L\nu Le\\\
eSte\end{subarray}}\left[Y^{*}_{\nu}\right]_{tS}\ln^{2}\left(\frac{m_{4}}{\Lambda}\right)\,.$
(18)

As estimate, setting $\nu_{4}$’s mass to $m_{4}\sim 100$ GeV and $\Lambda=1$
TeV gives coefficients of $\mathcal{O}(1)\times
v^{2}C_{\begin{subarray}{c}L\nu Le\\\ \mu St\mu\end{subarray}}$ and
$\mathcal{O}(10)\times v^{2}C_{\begin{subarray}{c}L\nu Le\\\
eSte\end{subarray}}$, which are both a factor $20$ smaller than the naïve
estimates in Table 2. The quantities $v^{2}C_{i}$ and the relevant entry of
the Yukawa coupling $\left[Y_{\nu}\right]_{\ell S}\sim U_{\ell
i}m_{i}U_{Si}/v$ are constrained to be well below $\mathcal{O}(1)$, either due
to the EFT assumptions requiring $m_{4}<\Lambda$ or experimental constraints.
In particular, the mixing elements are bounded by $|U_{e4}|\lesssim 6\cdot
10^{-2}$ and $|U_{\mu 4}|\lesssim 10^{-2}$ (see discussions below), making
significant contributions unlikely in the range of validity of the EFT,
$\Lambda\gg v$. For the same reasons we exclude the, in principle, promising
contributions to $a_{e}$ from $\mathcal{O}_{du\nu e}$. We conclude that only
$\mathcal{O}_{H\nu e}$ provides a contribution that can be large enough to
explain the discrepancies surrounding $a_{e}$ and $a_{\mu}$.

###  3.2 A dimension-six right-handed current

We now focus on the remaining dim-6 operator $O_{H\nu e}$, which in the gauge
basis is given by

$\displaystyle\mathcal{L}_{H\nu e}$ $\displaystyle=$
$\displaystyle\left(i\tilde{\varphi}^{\dagger}D_{\mu}\varphi\right)\,\bar{\nu}_{R}\gamma^{\mu}\,C_{H\nu
e}e_{R}+\mathrm{H.c.}\,,$ (19)

where $D_{\mu}=\partial_{\mu}-ig_{1}{\rm
y}B_{\mu}-i\frac{g}{2}\tau^{I}W_{\mu}^{I}-ig_{s}t^{a}G^{a}_{\mu}$ is the usual
SM covariant derivative, and $C_{H\nu e}$ is generally an $n\times 3$ matrix
in flavor space with mass dimension GeV-2. In the unitary gauge after EW
symmetry breaking, this operator can be written in the neutrino mass basis as

$\displaystyle\mathcal{L}_{H\nu e}$ $\displaystyle=$
$\displaystyle\frac{gv^{2}}{2\sqrt{2}}\sum_{j=1}^{n+3}\,\bar{\nu}_{j}\gamma^{\mu}\left[U^{T}P_{s}^{T}C_{H\nu
e}\right]_{j\alpha}e_{R\alpha}\,W_{\mu}^{+}\left(1+\frac{h}{v}\right)^{2}+\text{H.c.}$
(20)

Focusing on couplings between charged leptons and heavy neutrino states, we
obtain

$\displaystyle\mathcal{L}_{H\nu e}$ $\displaystyle\approx$
$\displaystyle\frac{gv^{2}}{2\sqrt{2}}\sum_{j=4}^{n+3}\left[\bar{C}_{H\nu
e}\right]_{j\alpha}\,(\bar{\nu}_{j}\gamma^{\mu}\,e_{R\alpha})\,W_{\mu}^{+}\left(1+\frac{h}{v}\right)^{2}+\text{H.c.},$
(21)

where $\alpha\in\\{e,\mu,\tau\\}$, $S$ again denotes the sterile flavor
states, and we have defined the effective coupling $\left[\bar{C}_{H\nu
e}\right]_{j\alpha}\equiv U_{Sj}(C_{H\nu e})_{S\alpha}$. For $n$ heavy mass
eigenstates, this interaction between sterile neutrinos and RH charged leptons
leads to the following one-loop correction to $\Delta a_{\ell}$:

$\displaystyle\Delta a_{\ell}$ $\displaystyle=$
$\displaystyle-\frac{2m_{\ell}}{(4\pi)^{2}}\sum_{j=4}^{n+3}m_{j}\,w(x_{j})\,{\rm
Re}\left(U_{\ell j}\left[\bar{C}_{H\nu e}\right]_{j\ell}\right)\,.$ (22)

Here $x_{j}=m_{j}^{2}/m_{W}^{2}$, and the loop function $w(x)$ is given by

$\displaystyle w(x)=\frac{4+x(x-11)}{2(x-1)^{2}}+3\frac{x^{2}\ln
x}{(x-1)^{3}}\,.$ (23)

As expected from the earlier estimate, this contribution to a lepton’s
anomalous magnetic moment is chirally enhanced by the sterile neutrinos’
masses. The same loops induce contributions to leptonic electric dipole
moments (EDMs) proportional to ${\rm Im}[U_{\ell j}\left[\bar{C}_{H\nu
e}\right]_{j\ell}]$. These contributions to $d_{\ell}$ can be obtained by
multiplying Eq. (22) by the factor $\left[-e/(2m_{\ell})\right]$ and replacing
Re$\left[\dots\right]\to$Im$\left[\dots\right]$. Although the experimental
limit on the electron EDM [45] leads to stringent constraints on the CP-
violating phase of the couplings with $\ell=e$, it probes a different
combination of couplings than $\Delta a_{\ell}$. In what follows we consider
the most important experimental probes of the sterile couplings in Eq. (22),
namely $\bar{C}_{H\nu e}$ and $U_{\ell 4}$.

##  4 Experimental probes of the leptonic right-handed current

We now discuss experimental constrains and projected sensitivity at near-
future facilities for the RH current operator $\mathcal{O}_{H\nu e}$ and the
active-sterile mixing element $U_{\ell 4}$. Discussions on the parameter space
preferred by $\Delta a_{\mu}$ and $\Delta a_{e}$ are deferred to Section 5.

###  4.1 Contributions to $h\to\ell\ell$

Following Ref. [46], we consider the contributions of $\mathcal{O}_{H\nu e}$
to the Yukawa couplings of the charged leptons. In particular, a nonzero
$C_{H\nu e}$ can induce large corrections to the lepton’s Yukawa coupling
$y_{\ell}$ that are proportional to the sterile neutrino mass $m_{j}$. We
start from

$\displaystyle\mathcal{L}\supset-
m_{\ell}\,\bar{\ell}_{L}\ell_{R}-y_{\ell}\,\bar{\ell}_{L}\ell_{R}\,h+{\rm
H.c.}\,\,,$ (24)

where $y_{\ell}$ here is related to the $Y_{e}$ in Table 2 by
$y_{\ell}=\sqrt{1/2}\left[Y_{e}^{\dagger}\right]_{\ell\ell}$. The running of
the lepton’s Yukawa and mass are modified at a renormalization scale $\mu_{r}$
by the sterile neutrino as follows:

$\displaystyle\frac{dm_{\ell}}{d\ln\mu_{r}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{g^{2}v^{2}}{(4\pi)^{2}}U_{\ell
j}\,\left[\bar{C}_{H\nu e}\right]_{j\ell}m_{j}\,(x_{j}-3)\,,$ (25)
$\displaystyle\frac{dy_{\ell}}{d\ln\mu_{r}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{g^{2}v}{(4\pi)^{2}}U_{\ell
j}\,\left[\bar{C}_{H\nu e}\right]_{j\ell}m_{j}\,(x_{h}+3x_{j}-9)\,,$ (26)

where $x_{h(j)}=m_{h(j)}^{2}/m_{W}^{2}$ and the equations holds for
$m_{j},m_{W}<\mu_{r}$. These RGEs can be obtained from the contributions of
the SMEFT interaction $C_{Hud}$ to the SM down-type Yukawa couplings $Y_{d}$
and dimension-six operator $C_{dH}$ determined in Refs. [43, 44, 47].
Explicitly, one replaces the couplings of Refs. [43, 44, 47] by

$C_{Hud}\to C_{H\nu e},\quad Y_{u}\to Y_{\nu}^{\dagger},\quad C_{dH}\to
C_{eH},\quad C_{uH}\to C_{L\nu H},\quad\text{and}\quad Y_{d}\to Y_{e}\,.$ (27)

With this we can make the identification

$\displaystyle y_{\ell}=\sqrt{1/2}$
$\displaystyle\left[Y_{e}^{\dagger}-\frac{3}{2}v^{2}C_{eH}\right]_{\ell\ell},\quad
m_{\ell}=\frac{v}{\sqrt{2}}\left[Y_{e}^{\dagger}-\frac{1}{2}v^{2}C_{eH}\right]_{\ell\ell}\,,$
(28)
$\displaystyle\text{and}\quad\left[Y_{\nu}\right]_{pS}=\frac{\sqrt{2}}{v}\left[M_{D}\right]_{pS}=\frac{\sqrt{2}}{v}U_{pi}m_{i}U_{Si}\,.$
(29)

In addition, threshold corrections to the muon mass and Yukawa appear after
integrating out a sterile neutrino or the $W$ boson.

For the lepton mass below the EW scale, we obtain the $\mu_{r}$-independent
result

$m_{\ell}=m_{\ell}(\mu_{r})-\frac{m_{W}^{2}}{(4\pi)^{2}}\left[x_{j}-1+(x_{j}-3)\log\frac{\mu_{r}^{2}}{m_{W}^{2}}-\frac{x_{j}-4}{x_{j}-1}x_{j}\log
x_{j}\right]U_{\ell j}\,\left[\bar{C}_{H\nu e}\right]_{j\ell}m_{j}\,.$ (30)

The effective Yukawa coupling that is probed in Higgs decays is then given by

$\displaystyle y_{\ell}^{\rm(eff)}$ $\displaystyle=$ $\displaystyle
y_{\ell}(\mu_{r})-\frac{1}{4}\frac{g^{2}v}{(4\pi)^{2}}U_{\ell
j}\,\left[\bar{C}_{H\nu
e}\right]_{j\ell}m_{j}\bigg{\\{}(x_{h}+3x_{j}-9)\log\frac{\mu_{r}^{2}}{m_{W}^{2}}-3\frac{x_{j}(x_{j}-3)}{x_{j}-1}\log
x_{j}$ (31)
$\displaystyle+x_{j}\beta_{j}\log\left(\frac{\beta_{j}-1}{\beta_{j}+1}\right)+(x_{h}-2)\beta_{W}\log\left(\frac{\beta_{W}-1}{\beta_{W}+1}\right)+[2x_{h}+x_{j}(x_{j}-7)]f_{1}(x_{h},x_{j})$
$\displaystyle-[4+(2-x_{h})x_{j}]f_{2}(x_{h},x_{j})+(-5+2x_{h}+4x_{j})\bigg{\\}}\,.$

In the above, the function $y_{\ell}(\mu_{r})$ is given by the solution of Eq.
(25), $\beta_{W(j)}=\sqrt{1-4m_{W(j)}^{2}/m_{h}^{2}}$ is a kinematic factor,
and the loop functions $f_{1,2}$ are

$\displaystyle
f_{1}(x_{h},x_{j})=\int_{0}^{1}dz\frac{1}{1-x_{j}+x_{h}z}\log\left(\frac{x_{j}-x_{h}(1-z)z}{1+(x_{j}-1)z}\right)\,,$
$\displaystyle
f_{2}(x_{h},x_{j})=\int_{0}^{1}dz\frac{1}{-1+x_{j}+x_{h}z}\log\left(\frac{1-x_{h}(1-z)z}{x_{j}+z(1-x_{j})}\right)\,.$
(32)

We stress that $y_{\ell}^{\rm(eff)}$ is independent of the renormalization
scale $\mu_{r}$.

To obtain constraints from measurements and searches for Higgs decays to
leptons, we assume only $C_{H\nu e}$ is generated at a scale
$\mu_{r}=\Lambda$. We thus use the measured value of the lepton mass as input
for $m_{\ell}$ in Eq. (30). This determines $m_{\ell}(\Lambda)$ from which we
obtain $y_{\ell}(\Lambda)=m_{\ell}(\Lambda)/v$ and subsequently
$y_{\ell}(\mu_{r})$. This procedure sets the BSM Yukawa interactions to zero
at $\mu_{r}=\Lambda$, i.e. $C_{eH}(\Lambda)=0$. One might expect a general BSM
scenarios to generate multiple operators in the EFT, which could give
additional contributions to $h\to\ell\ell$. Our assumptions lead to
conservative constraints as long as there are no significant cancellations
between the different contributions.

For $\ell=\mu$, we compare the obtained value for
$|y_{\mu}^{\rm(eff)}/y_{\mu}^{\rm(SM)}|^{2}$ to recent CMS measurements, which
using $\mathcal{L}\approx 137{\rm~{}fb^{-1}}$ of data at
$\sqrt{s}=13{\rm~{}TeV}$ reports a signal strength of [48]

$\mathcal{\mu}_{h\to\mu\mu}=\frac{{\rm Br}(h\to\mu^{+}\mu^{-})_{\phantom{\rm
SM}}}{{\rm Br}(h\to\mu^{+}\mu^{-})_{\rm SM}}=1.19^{+0.44}_{-0.42}\,,$ (33)

and the projected HL-LHC sensitivity of $\mu_{h\to\mu\mu}=1.0\pm 0.1$ [49].
ATLAS has also performed a measurement and reports a comparable result but
with a slightly larger uncertainty [50].

We use an analogous procedure to obtain the effective Yukawa coupling of the
electron, $y_{e}^{\rm(eff)}$, which we constrain by using the recent limit
Br$(h\to e^{+}e^{-})<3.6\cdot 10^{-4}$ [51]. This translates to

$\displaystyle\left|\frac{y_{e}^{\rm(eff)}}{y_{e}^{\rm(SM)}}\right|\leq\bigg{\\{}\begin{matrix}[l]268&\quad{\rm
LHC\,(current)}&\mathcal{L}=139\,{\rm fb}^{-1}\\\ 58&\quad{\rm HL-
LHC}&\mathcal{L}=3\,{\rm ab}^{-1}\end{matrix}\,,$ (34)

where we used a factor of $\sqrt{139\,{\rm fb}^{-1}/3\,{\rm ab}^{-1}}$ to
estimate the sensitivity of the HL-LHC from the current limit. For the reach
of a future Higgs factory to BR$(H\to\ell^{+}\ell^{-})$, see Ref. [52].

###  4.2 Constraints on electromagnetic interactions of light neutrinos

Here, we briefly consider the impact of dimension-six $\nu$SMEFT operators on
the magnetic moment of light, active neutrinos. Specifically, a diagram with
the same topology as the first diagram in Fig. 1, but with neutrino mass
eigenstates $\nu_{i}$ and $\nu_{j}$ on the external legs and charged leptons
$\ell_{\alpha}$ running inside the loop, generates a photon dipole operator
with coefficient

$\left[\mu_{\nu}\right]_{ij}=-2\sqrt{2}U_{\alpha i}^{*}\left[c_{w}C_{\nu
B}+s_{w}C_{\nu W}\right]_{\alpha
S}U_{Sj}^{*}=-\frac{e}{(4\pi)^{2}}\frac{\sqrt{2}m_{\ell_{\alpha}}}{v}U^{*}_{Sj}\left[C^{*}_{H\nu
e}\right]_{S\alpha}U^{*}_{\alpha i}.$ (35)

For $\nu_{j}\to\nu_{i}\gamma$, this corresponds to a neutrino magnetic moment
of

$\left[\mu_{\nu}\right]_{ij}=\frac{8}{(4\pi)^{2}}{y_{\ell_{\alpha}}}y_{e}\mu_{B}\,U^{*}_{Sj}\left[v^{2}C^{*}_{H\nu
e}\right]_{S\alpha}U^{*}_{\alpha i}\overset{\tiny\alpha=\mu}{\approx}4.4\cdot
10^{-11}\mu_{B}\,U^{*}_{Sj}\left[v^{2}C^{*}_{H\nu e}\right]_{S\mu}U^{*}_{\mu
i},$ (36)

where for the numerical estimate we specified $\alpha=\mu$. Presently, the
Borexino experiment sets an upper bound of about $\mu_{\nu}^{(\rm
eff)}<2.8\cdot 10^{-11}\mu_{B}$ at 90% CL on the effective magnetic moment of
light neutrinos [53]. This corresponds to limits at the same order of
magnitude on the individual elements $\left[\mu_{\nu}\right]_{ij}$. Thus,
current limits only probe large coefficients with $v^{2}|C_{H\nu
e}|\sim\mathcal{O}(1)$ and mixing angles $|U_{Sj}|\sim\mathcal{O}(1)$, which
are weak in comparison to the experiments discussed below.

###  4.3 Lepton-flavor violation

The same diagrams that induce $\Delta a_{\ell}$ also contribute to lepton-
flavor-changing dipoles, and hence $\ell_{\beta}\to\ell_{\alpha}\gamma$
transitions, through the coupling

$\displaystyle L_{\begin{subarray}{c}e\gamma\\\ \alpha\beta\end{subarray}}$
$\displaystyle=$ $\displaystyle-\frac{e}{32\pi^{2}}\sum_{j=4}^{n+3}U_{\alpha
j}m_{j}\left[\bar{C}_{H\nu e}\right]_{j\beta}\,w(x_{j})\,.$ (37)

The loop function $w(x)$ is defined in Eq. (23). The branching rate for
$\mu\rightarrow e\gamma$ is then given by

$\displaystyle\textrm{BR}(\mu\rightarrow
e\gamma)=\tau_{\mu}\frac{m_{\mu}^{3}}{4\pi}\left(\left|L_{\begin{subarray}{c}e\gamma\\\
\mu e\end{subarray}}\right|^{2}+\left|L_{\begin{subarray}{c}e\gamma\\\
e\mu\end{subarray}}\right|^{2}\right)$ $\displaystyle=\frac{3\alpha_{\rm
em}}{8\pi\hat{\Gamma}_{\mu}}\left(\left|\sum_{j=4}^{n}U_{\mu
j}\frac{m_{j}}{m_{\mu}}\left[v^{2}\bar{C}_{H\nu
e}\right]_{je}\,w(x_{j})\right|^{2}+\left|\sum_{j=4}^{n}U_{ej}\frac{m_{j}}{m_{\mu}}\left[v^{2}\bar{C}_{H\nu
e}\right]_{j\mu}\,w(x_{j})\right|^{2}\right)\,,$ (38)

where we expressed the muon lifetime $\tau_{\mu}$ as

$\tau^{-1}_{\mu}=\frac{G_{F}^{2}m_{\mu}^{5}}{192\pi^{3}}\hat{\Gamma}_{\mu}.$
(39)

The dimensionless number $\hat{\Gamma}_{\mu}$ accounts for radiative
corrections and finite $m_{e}$ and is adjusted to reproduce the measured muon
lifetime. Using Ref. [12], we obtain $\hat{\Gamma}_{\mu}=0.996$. Eq. (4.3)
should then be compared with the stringent limit on the $\mu\rightarrow
e\gamma$ branching ratio, $\textrm{BR}(\mu\rightarrow e\gamma)<4.2\cdot
10^{-13}$ [54].

###  4.4 Constraints on $U_{\mu 4}$ and $U_{e4}$ from CKM unitarity

From the expression for $\Delta a_{\ell}$ in Eq. (22), we observe that the
correction to a lepton’s anomalous magnetic moment by a heavy neutrino
$\nu_{j}$ $(4\leq j\leq n+3)$ through $\mathcal{O}_{H\nu e}$ is proportional
to the combination $(U_{\ell j}\,\left[\bar{C}_{H\nu e}\right]_{j\ell})$.
Constraints on the elements of the mixing matrix $U_{\ell j}$ between active
and sterile states have been studied extensively in the literature. These
papers mainly focus on minimal scenarios in the absence of higher-dimensional
operators but many of the results can be applied here as well. For recent
reviews, see Refs. [55, 56, 57, 58, 59].

For simplicity we consider $n=1$ and begin with discussing the $U_{\mu 4}$
mixing element. Laboratory constraints on $U_{\mu 4}$ are very dependent on
the mass of the sterile neutrino. At small masses
$m_{4}\lesssim\mathcal{O}(\rm eV)$, the bounds mainly come from oscillation
experiments and require $|U_{\mu 4}|\lesssim 0.1$. Strong bounds exists for
$\mathcal{O}(1\,\mathrm{MeV})<m_{4}<m_{W}$ from a large class of accelerator
experiments, where sterile neutrinos are produced directly in meson and weak
boson decays; for details see Sect. 4.6. These bounds weaken significantly for
$m_{4}>m_{W}$. To account for the measured value of $\Delta a_{\mu}$, a larger
$m_{4}$ is preferred to maximize the chiral enhancement. This automatically
selects the region $m_{W}<m_{4}<\Lambda$. In this mass range the constraint on
$U_{\mu 4}$ is driven by muonic, mesonic, and nuclear $\beta$ decays. In
particular, the interplay of these limits manifests as stringent constraint on
$U_{\mu 4}$ from CKM unitarity.

Most of the discussion for $U_{e4}$ is similar to that of the muonic mixing
element, with the exception that for light sterile masses, i.e., $m_{4}<1$
MeV. There, a strong bound arise from looking for kinks in the electron
spectrum in nuclear $\beta$-decays [60, 61]. For instance: $|U_{e4}|\lesssim
0.03$ for $m_{4}\sim 0.1$ MeV from the $\beta$-decay spectrum of 35S [62].
Because of these constraints and the chiral enhancement, again sterile
neutrino masses are preferred above the $W$ boson mass.

We proceed by following Refs. [55, 63] and update some of the theoretical
input for the extraction of $V_{ud}$ from superallowed $0^{+}\rightarrow
0^{+}$ decays, and of $V_{us}$ from leptonic and semileptonic kaon decays. The
existence of a sterile neutrino that is not kinematically accessible in muon
decay causes a reduction of the muon decay width (relative to SM expectations)
due to the non-unitarity of the PMNS matrix. This affects the extraction of
the Fermi constant from muon decay through the relationship

$G^{(\mu)}_{F}=G^{(0)}_{F}\left(1-\frac{1}{2}|U_{\mu
4}|^{2}-\frac{1}{2}|U_{e4}|^{2}\right)\,.$ (40)

Here, $G_{F}^{(0)}=1/\sqrt{2}v^{2}$ is the Fermi constant in the SM at tree
level, and we assume $|U_{\mu 4}|,|U_{e4}|\ll 1$. The modification of $G_{F}$
propagates to the semileptonic decays that are used to extract the $V_{ud}$
and $V_{us}$ elements of the CKM matrix and test first-row unitarity. In
particular,

$\displaystyle V^{0^{+}\rightarrow 0^{+}}_{ud}$ $\displaystyle=$
$\displaystyle V_{ud}\left(1+\frac{1}{2}|U_{\mu 4}|^{2}\right)\,,$ (41)
$\displaystyle|V_{us}|^{K\to\pi e\nu}$ $\displaystyle=$
$\displaystyle|V_{us}|\left(1+\frac{1}{2}|U_{\mu 4}|^{2}\right)\,,$ (42)
$\displaystyle|V_{us}|^{K\to\pi\mu\nu}$ $\displaystyle=$
$\displaystyle|V_{us}|\left(1+\frac{1}{2}|U_{e4}|^{2}\right)\,.$ (43)

The branching ratio $\Gamma(K\rightarrow\mu\nu)/\Gamma(\pi\rightarrow\mu\nu)$,
which determines $V_{us}/V_{ud}$, is not affected by $U_{\mu 4}$ and $U_{e4}$.
For $V^{0^{+}\rightarrow 0^{+}}_{ud}$ we use the recent extraction reported in
the Particle Data Group (PDG) [12] , which adopts the improved evaluation of
radiative corrections from Refs. [64, 65, 66]. We also consider the more
conservative analysis of Towner and Hardy (TH) [67], which quotes a larger
nuclear theory error. Respectively, the extracted values of $V_{ud}^{0^{+}\to
0^{+}}$ are

$|V^{0^{+}\to 0^{+}}_{ud}|\big{|}_{\rm PDG}=0.97370(14),\qquad|V^{0^{+}\to
0^{+}}_{ud}|\big{|}_{\rm TH}=0.97373(31)\,.$ (44)

For the extraction of $V_{us}$, we use the experimental values of
$|V_{us}f^{K\pi}(0)|^{K\rightarrow\pi\ell\nu}$ given in Ref. [68], which lists
the electron and muon channels separately. The theoretical input consists of
the ratio of decay constants $f_{K}/f_{\pi}$ and of the form factor
$f^{K\pi}(0)$, for which we take the FLAG‘19 average with $N_{f}=2+1+1$
flavors[69]. Assuming unitarity of the CKM matrix, we then perform a two-
parameter fit to $|U_{\mu 4}|$ and
$|V_{us}|=\lambda\simeq\sqrt{1-V_{ud}^{2}}$. Since the current theoretical and
experimental input indicate $|V_{ud}|^{2}+|V_{us}|^{2}<1$ at the
$3\sigma$-$4\sigma$ level and the presence of sterile neutrinos would pull in
the opposite direction, i.e., drive $|V_{ud}|^{2}+|V_{us}|^{2}>1$, a scenario
with $\nu_{4}$ and $|U_{\mu 4}|\neq 0$ is disfavored with respect to the SM.
This tension could be accounted for by including dimension-six SMEFT
operators, see e.g. [70, 71, 72]. Using $V^{0^{+}\to 0^{+}}_{ud}\big{|}_{\rm
PDG}$, we find

$\displaystyle|U_{\mu 4}|<0.01\,\quad\text{at}\quad 95\%\,{\rm C.L.}\,,$ (45)

with the bound weakening to $|U_{\mu 4}|<0.02$ if we use $V^{0^{+}\to
0^{+}}_{ud}\big{|}_{\rm TH}$. The electron case is slightly less constrained
and is less affected by the extraction of $V_{ud}$. Using $V^{0^{+}\to
0^{+}}_{ud}\big{|}_{\rm PDG}$ we find

$\displaystyle|U_{e4}|<0.06\,,\quad\text{at}\quad 95\%\,{\rm C.L.}\,,$ (46)

which becomes $|U_{e4}|<0.07$ with $V^{0^{+}\to 0^{+}}_{ud}\big{|}_{\rm TH}$.

So far we have neglected $\mathcal{O}_{H\nu e}$. The contributions of this
operator to muonic and mesonic $\beta$ decays scale as $\sim
U_{Sj}^{2}\bar{C}^{2}_{H\nu e}$, with $j=1,2,3$. Since $|U_{Sj}|\sim|U_{\mu
4}|\ll 1$, these corrections can be safely neglected.

Throughout this discussion, we worked in the context of $n=1$ sterile
neutrinos. Eqs. (41)-(43) can be generalized to the $n>1$ case by the
replacement $|U_{\alpha 4}|^{2}\rightarrow\sum_{j=4}^{3+n}|U_{\alpha j}|^{2}$.
The limits in Eqs. (45) and (46) then still apply to each $|U_{\alpha j}|$,
after marginalizing over other elements.

###  4.5 Neutrinoless double beta decay

The impact of sterile neutrinos with non-minimal interactions on neutrinoless
double beta decays ($0\nu\beta\beta$) was studied in Refs. [73, 74]. These
constraints are only relevant for interactions with electrons as there is not
enough energy available in $0\nu\beta\beta$ to produce final-state muons.
Assuming $m_{j}\gtrsim 1$ GeV for $j\geq 4$, the heavier neutrinos can be
integrated out as $0\nu\beta\beta$ takes place at energies of
$\mathcal{O}({\rm MeV})$. This generates two dimension-nine scalar operators
of the form

$\mathcal{L}^{(9)}=\frac{1}{v^{5}}C^{(9)}_{1R}\,\bar{q}_{L}\gamma^{\mu}\tau^{+}q_{L}\,\bar{q}_{L}\gamma_{\mu}\tau^{+}q_{L}\,\bar{e}_{R}e_{R}^{c}+\frac{1}{v^{5}}C^{(9)}_{1L}\,\bar{q}_{L}\gamma^{\mu}\tau^{+}q_{L}\,\bar{q}_{L}\gamma_{\mu}\tau^{+}q_{L}\,\bar{e}_{L}e_{L}^{c}\,,$
(47)

where $q=(u,\,d)^{T}$, $\tau^{+}=(\tau_{1}+i\tau_{2})/2$, and $C^{(9)}_{i}$
are dimensionful coefficients given by

$\displaystyle C^{(9)}_{1\,L}$ $\displaystyle=$
$\displaystyle-2vV_{ud}^{2}\sum^{n+3}_{j=4}\frac{U^{2}_{ej}}{m_{j}}\,,\qquad
C^{(9)}_{1\,R}=-\frac{v^{5}}{2}V_{ud}^{2}\sum^{n+3}_{j=4}\left[\bar{C}_{H\nu
e}\right]^{*}_{je}\frac{1}{m_{j}}\left[\bar{C}_{H\nu e}\right]^{*}_{je}.$ (48)

The second effect of heavy sterile neutrinos is that in the definition of the
effective electron neutrino Majorana mass $m_{\beta\beta}$, the sum is
restricted to the three lightest neutrinos

$m_{\beta\beta}=\sum_{j=1}^{3}m_{j}U^{2}_{ej}\,.$ (49)

This is the usual expression in SMEFT at dimension five but with the slight
difference being that the $3\times 3$ PMNS matrix is not by itself unitary.

The $0\nu\beta\beta$ half-life $T^{0\nu}_{1/2}$ can then be expressed as

$\left(T^{0\nu}_{1/2}\right)^{-1}=g_{A}^{4}\Big{\\{}G_{01}\left(|\mathcal{A}_{L}|^{2}+|\mathcal{A}_{R}|^{2}\right)-2(G_{01}-G_{04})\textrm{Re}(\mathcal{A}_{L}^{*}\mathcal{A}_{R})\Big{\\}}\,,$
(50)

where $G_{01}$ and $G_{04}$ are phase space factors, and $\mathcal{A}_{L}$ and
$\mathcal{A}_{R}$ are the product of nuclear matrix elements (NMEs), hadronic
couplings and Wilson coefficients of $\nu$SMEFT operators. In our case

$\displaystyle\mathcal{A}_{L}$ $\displaystyle=$
$\displaystyle-\frac{m_{\beta\beta}}{m_{e}}{V_{ud}^{2}}\left(-\frac{g_{V}^{2}}{g_{A}^{2}}M_{F}+M_{GT}+M_{T}+\frac{2}{g_{A}^{2}}g_{\nu}^{NN}m_{\pi}^{2}M_{F\,sd}\right)+\frac{m_{\pi}^{2}}{m_{e}v}C^{(9)}_{1\,L}\mathcal{A}_{sd}\,,$
$\displaystyle\mathcal{A}_{R}$ $\displaystyle=$
$\displaystyle\frac{m_{\pi}^{2}}{m_{e}v}C^{(9)}_{1\,R}\mathcal{A}_{sd}\,,$
$\displaystyle\mathcal{A}_{sd}$ $\displaystyle=$
$\displaystyle\left(\frac{5}{6}g_{1}^{\pi\pi}\mathcal{M}_{PS,sd}+\frac{1}{2}\left(g_{1}^{\pi
N}-\frac{5}{6}g_{1}^{\pi\pi}\right)(M^{AP}_{GT,\,sd}+M^{AP}_{T,\,sd})-\frac{2}{g_{A}^{2}}g_{1}^{NN}M_{F,sd}\right)\,.$
(51)

Here $M_{F}$, $M_{GT}$ and $M_{T}$ are the NMEs induced by the exchange of
light Majorana neutrinos, while the ‘short distance’ ($sd$) nuclear matrix
elements $\mathcal{M}_{PS,sd}$, $M^{AP}_{GT,\,sd}$, $M^{AP}_{T,\,sd}$ and
$M_{F,\,sd}$ are the pion- and short-range NMEs induced by the dimension-nine
operators $O^{(9)}_{1\,L}$ and $O^{(9)}_{1\,R}$ in Eq. (47), and from the
exchange of light neutrinos with hard momenta [75, 76]. For both space phase
factors and NMEs, we use the definitions of Ref. [74].

The low-energy constants (LECs) $g_{1}^{\pi\pi}$, $g_{1}^{\pi N}$ and
$g_{1}^{NN}$ are all expected to be $\mathcal{O}(1)$. $g_{1}^{\pi\pi}$ has
been computed on the lattice, yielding $g_{1}^{\pi\pi}=0.36\pm 0.02$ [77],
while $g_{1}^{\pi N}$ and $g_{1}^{NN}$ are at the moment unknown. We set them
to the factorization-inspired values $g_{1}^{\pi N}=1$ and
$g_{1}^{NN}(2~{}{\rm GeV})=(1+3g^{2}_{A})/4+1$ [74]. The short-distance LEC
$g_{\nu}^{NN}$ arises from the exchange of hard Majorana neutrinos and was
recently estimated in Refs. [78, 79]. Following Ref. [74], for simplicity we
set it to $g_{\nu}^{NN}=-1/(2F_{\pi})^{2}$ with $F_{\pi}=92.2~{}$MeV, and note
that we will not use the amplitude proportional to $m_{\beta\beta}$ in
obtaining constraints (see discussion below).

To keep the expressions compact we now focus on the case of $n=1$. Using the
NMEs of Ref. [80] and the phase space factors extracted from Ref. [81], the
half-life of 136Xe becomes

$\displaystyle\left[T^{0\nu}_{1/2}\left({}^{136}{\rm
Xe}\right)\right]^{-1}=\Bigg{\\{}2.4\cdot
10^{-24}\left|\frac{m_{\beta\beta}}{1\,{\rm eV}}\right|^{2}$ (52)
$\displaystyle+1.5\cdot
10^{-7}\frac{\textrm{Re}(m_{\beta\beta}U^{*2}_{e4})}{m_{4}}-7.4\cdot
10^{-9}\frac{\textrm{Re}(m_{\beta\beta}\left[v^{2}\bar{C}_{H\nu
e}\right]^{2}_{4e})}{m_{4}}$ $\displaystyle+10^{-10}\left(\frac{1\,\rm
GeV}{m_{4}}\right)^{2}\left(23\,|U_{e4}|^{4}-2.3\,\textrm{Re}\left(\left[v^{2}\bar{C}_{H\nu
e}\right]^{2}_{4e}U^{\,2}_{e4}\right)+1.4\left|\left[v^{2}\bar{C}_{H\nu
e}\right]_{4e}\right|^{4}\right)\Bigg{\\}}\,{\rm yr}^{-1}.\qquad$

The strongest constraint on the $0\nu\beta\beta$ decay half-life comes from
the 2016 KamLAND-Zen measurement [82], which implies the lower limit
$T^{0\nu}_{1/2}({}^{136}\mathrm{Xe})>1.07\cdot 10^{26}$ yr at 90% C.L. Next-
generation experiments such as nEXO [83] will provide sensitivity at the level
of $10^{28}$ yr.

Some care is needed when using these expressions to set limits, as the
contributions from $m_{\beta\beta}$ and from $C^{(9)}_{1\,L}$ can depend
strongly on the mechanism that accounts for the active neutrino masses. In our
present set-up, assuming $M_{R}\gg M_{D}\gg M_{L}$, the light neutrino masses
consists of Weinberg operator and Type I seesaw contributions, i.e.,
$M_{\nu_{\rm light}}\simeq
M_{L}-\left[M_{D}M_{R}^{-1}M_{D}^{T}\right]^{\dagger}$, which can be generated
in a high-scale Type I+II seesaw scenario. Here,
$\left[M_{L}\right]_{\alpha\beta}=U_{\alpha j}^{*}m_{j}U_{\beta j}^{*}$ and we
would expect a similar scaling, $\sim U_{\alpha j}^{*}m_{j}U_{\beta j}^{*}$,
for the Type I seesaw contribution. As we consider $m_{j}>m_{W}$ and
$|U_{\alpha j}|\sim 0.01$ for $j\geq 4$, a naïve estimate of these
contributions to the light neutrino masses turns out to be too large
$\gtrsim\mathcal{O}(10)$ MeV. This implies that a realistic scenario of
neutrino masses will require non-trivial cancellations, which can also affect
the $C_{1\,L}^{(9)}$ contributions. One scenario with such cancellations is
that of a low-scale Type I seesaw, e.g., the so-called Inverse Type I seesaw,
which features pseudo-Dirac neutrinos [84, 85, 86]. In such a scenario, the
case of $n=2$ corresponds to nearly degenerate $m_{4}$ and $m_{5}$. In this
limit, we can reproduce the masses and mixing of the light neutrinos with
$M_{L}=0$, and obtain $m_{4,5}>m_{W}$ simultaneously with $|U_{\alpha j}|\sim
0.01$. Apart from giving masses to light neutrinos that are smaller than
expected from the above estimate, one also finds that $C_{1\,L}^{(9)}$ scales
as $m_{\nu_{\rm light}}/m_{4}^{2}$, significantly below the naïve estimate
$\sim U_{e4}^{2}/m_{4}$. As we do not wish to go into the details of the mass
mechanism of neutrinos, we will take a conservative approach and only consider
the contributions from $C^{(9)}_{1\,R}$, given by the last term in Eq. (52)
when setting limits. This implies that $0\nu\beta\beta$ experiments constrain
$\left[\bar{C}_{H\nu e}\right]_{4e}$ independently of the $U_{4e}$ mixing
angle.

The same matching procedure can be followed in the muonic sector, where
operators analogous to $C^{(9)}_{1\,L}$ and $C^{(9)}_{1\,R}$ contribute to
lepton-number-violating decays such as
$K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+}$, studied in Ref. [87]. The limit on
the branching ratio, BR$(K^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})<4.2\cdot
10^{-11}$ [12] is, however, too weak to provide competitive constraints on
$[\bar{C}_{H\nu e}]_{4\mu}$.

###  4.6 Accelerator Constraints and LHC Prospects

If the $\nu$SMEFT operator can indeed account for the measured anomalous
magnetic moments of the muon and/or electron, then one anticipates an impact
on the production of $\nu_{4}$ directly through (anomalous) EW currents at
accelerator facilities. Consequentially, such production is strongly
constrained by direct searches. For instance, short baseline and beam dump
experiments require $|U_{\mu 4}|\lesssim 10^{-5}-10^{-3}$ for 150 MeV$\lesssim
m_{4}\lesssim 450$ MeV [88, 89]. For larger masses but still below $m_{W}$,
ATLAS [90], CMS [91, 92], and LHCb [93] measurements imply that $|U_{\mu
4}|\lesssim 3\cdot 10^{-3}$ [91, 90]. As discussed below, these bounds weaken
significantly for $m_{4}>m_{W}$. In this region the strongest indirect limits
arise from EW precision data; the strongest direct limits are set by CMS [91,
92].

In addition to these experimental limits are theoretical constraints. For
ultra heavy $\nu_{4}$ with $m_{4}\gg m_{W}$, the total width of $\nu_{4}$
$(\Gamma_{4})$ grows according to the Goldstone Equivalence Theorem. The
perturbativity bound of $\Gamma_{4}<m_{4}$ then requires [94]:

$|U_{\mu 4}|^{2}<\frac{16\pi}{g^{2}}\frac{m_{W}^{2}}{m_{4}^{2}}\approx
76\cdot\left(\frac{100{\rm~{}GeV}}{m_{4}}\right)^{2}.$ (53)

As these limits, much like those from $s$-wave unitarity in
$W^{\pm}W^{\pm}\to\ell^{\pm}_{\alpha}\ell^{\pm}_{\beta}$ scattering [95, 96],
are much weaker than experimental bounds, we neglect them for the remainder of
this study.

Figure 2:  Lowest order, Born-level graph depicting $d\overline{u}\to
W^{-*}\to\mu^{-}\nu_{4}$ scattering with leading helicity configurations
(subscripts) in $\nu$SMEFT at dimension $d=4$ (L) and $d=6$ (R). Drawn using
JaxoDraw [97].

We now consider the sensitivity of ATLAS and CMS to direct production of
$\ell\nu_{4}$ pairs through dimension-four and -six interactions. We mainly
focus here on the muonic case and discuss couplings to electrons at the end of
this section. For the sterile neutrino masses under consideration, i.e.,
$m_{4}\sim 100-1000{\rm~{}GeV}$, the production of $\mu^{\pm}\nu_{4}$ pairs
due to minimal mixing ($d=4$ operators) is dominated by two processes. The
first is the charged-current Drell-Yan (DY) process [98]

$q\overline{q}\to W^{\pm}\to\mu^{\pm}\nu_{4}\,,$ (54)

as shown in the left diagram of Fig. 2 for the
$d_{L}\overline{u}_{R}\to\mu^{-}_{L}\nu_{4}$ partonic configuration. An
additional contribution (not shown) comes from $W\gamma$ fusion [99, 100,
101], and proceeds through

$q\gamma\to\mu^{\pm}\nu_{4}q^{\prime}.$ (55)

Both processes have matrix elements that are linear in active-sterile mixing
$U_{\mu 4}$ and therefore exhibit production-level cross sections
$\sigma(pp\to\mu^{\pm}\nu_{4}+X)$ that scale as $|U_{\mu 4}|^{2}$.

At the dimension-six level ($d=6$) both channels receive corrections from
$\mathcal{O}_{H\nu e}$. This is shown diagrammatically in the right panel of
Fig. 2 for the $d_{L}\overline{u}_{R}\to\mu^{-}_{R}\nu_{4}$ partonic channel.
As stressed above, the $d=4$ contribution occurs through a LH chiral coupling
while the $d=6$ contribution occurs through a RH chiral coupling. Since the
muon can be treated as massless for typical LHC collision scales, the DY and
$W\gamma$ mechanisms at $d=4~{}(6)$ are driven almost exclusively by LH (RH)
$\mu^{-}$. (In the same manner, the RH (LH) $\mu^{+}$ drives the charge-
conjugated processes.)

The helicity amplitudes for RH (LH) $\mu^{-}$ at $d=4~{}(6)$ are helicity-
inverting and hence result in squared amplitudes that are suppressed by a
factor of $|\mathcal{M}_{\rm helicity-inverting}|^{2}\propto
m_{\mu}^{2}/Q^{2}\sim m_{\mu}^{2}/m_{4}^{2}$. Here $Q$ is the invariant mass
of the $(\mu\nu_{4})$-system and scales as $Q\sim m_{4}$. For the range of
$m_{4}$ that is of interest, this translates to a helicity suppression of
about $m_{\mu}^{2}/m_{4}^{2}\lesssim 10^{-6}$. Similarly, interference between
$d=4$ and $d=6$ requires projecting RH (LH) chiral operators on LH (RH)
helicity states of $\mu^{-}$, and again is suppressed by a factor of ${{\rm
Re\,}[}\mathcal{M}^{*}_{d=4}\mathcal{M}_{d=6}]\propto m_{\mu}^{2}/Q^{2}\sim
m_{\mu}^{2}/m_{4}^{2}$.

Figure 3:  (L) Upper: As a function of $\nu_{4}$’s mass, the total
$pp\to\mu^{\pm}\nu_{4}+X$ cross section at NLO in QCD with scale uncertainty
(band thickness) at the $\sqrt{s}=13$ TeV LHC for the Drell-Yan (DY) and
$W\gamma$ fusion channels purely at dimension $d=4$ assuming $|U_{\mu
4}|^{2}=1$ (upper curves), and purely at $d=6$ assuming
$\left[\overline{C}_{H\nu e}\right]_{4\mu}=(1~{}{\rm TeV})^{-2}$ (lower
curves). Lower: the QCD $K$-factor. (R) Same as (L) but for the combined
$d=4+6$ rate assuming (upper curves) $|U_{\mu 4}|^{2}=10^{-6}$ with
$\left[\overline{C}_{H\nu e}\right]_{4\mu}=(1~{}{\rm TeV})^{-2}$ and (lower
curves) $|U_{\mu 4}|^{2}=10^{-4}$ with $\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(10~{}{\rm TeV})^{-2}$.

Neglecting contributions of $\mathcal{O}(m_{\mu}^{2}/Q^{2})$, the total,
lowest order cross sections for the DY channels at the partonic level, along
with their dependence on mixing and EFT inputs, are:

$\displaystyle\hat{\sigma}^{d=4}(\overline{u}_{R}d_{L}\to\mu^{-}_{L}\nu_{4})$
$\displaystyle=\frac{g^{4}|U_{\mu 4}|^{2}}{2^{7}\pi
N_{c}^{2}}\frac{(Q^{2}-m_{4}^{2})^{2}(2Q^{2}+m_{4}^{2})}{Q^{4}\left[(Q^{2}-m_{W}^{2})^{2}+(m_{W}\Gamma_{W})^{2}\right]}\,,\quad$
(56)
$\displaystyle\hat{\sigma}^{d=6}(\overline{u}_{R}d_{L}\to\mu^{-}_{R}\nu_{4})$
$\displaystyle=\left(\frac{\left|\mathcal{C}_{H\nu e}\right|^{2}}{4|U_{\mu
4}|^{2}}\right)\times\hat{\sigma}^{d=4}(\overline{u}_{R}d_{L}\to\mu^{-}_{L}\nu_{4})\,,\quad\text{where}$
(57) $\displaystyle\mathcal{C}_{H\nu e}$
$\displaystyle=v^{2}\left[\overline{C}_{H\nu e}\right]_{4\mu}\,.$ (58)

For the $W\gamma$ channel, analogous expressions exist as does the scaling
relationship between $d=4$ and $d=6$ contributions. The fact that phase-space-
integrated expressions at $d=4$ and $d=6$ differ by only prefactors is a
consequence that both interactions are maximally parity violating. Upon
integration over the $\mu^{-}$’s polar angle $\theta_{\mu}$, relative
differences of the form $(1\mp\cos\theta_{\mu})$ vanish. Due to the strong
kinematic suppression that scales as $\hat{\sigma}\sim 1/Q^{2}$, heavy
neutrinos with masses above $m_{W}$ are produced very close to the kinematic
threshold [94]. Even at the LHC, this ensures the validity of our EFT for
$\nu_{4}$ masses satisfying $m_{W}\lesssim m_{4}\ll\Lambda$.

The (effective) non-interference of $d=4$ and $d=6$ contributions subsequently
allows us to express the total of the two as their linear combination, with
coefficients set to the mixing and EFT inputs respectively. For the DY and
$W\gamma$ channels, this is given by

$\displaystyle\hat{\sigma}^{d=4+d=6}_{{\rm DY}~{}(W\gamma)}$
$\displaystyle~{}=~{}|U_{\mu 4}|^{2}\times\hat{\sigma}^{d=4}_{{\rm
DY}~{}(W\gamma)}\left[|U_{\mu 4}|^{2}=1\right]$
$\displaystyle\qquad~{}+~{}\left|\left[\overline{C}_{H\nu
e}\right]\right|^{2}\times(1{\rm~{}TeV})^{4}\times\hat{\sigma}^{d=6}_{{\rm
DY}~{}(W\gamma)}\left[\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(1{\rm~{}TeV})^{-2}\right]$ (59)
$\displaystyle~{}=~{}\left(|U_{\mu 4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu
e}|^{2}\right)\times\hat{\sigma}^{d=4}_{{\rm DY}~{}(W\gamma)}\left[|U_{\mu
4}|^{2}=1\right]$ (60)

In the first line, the total partonic cross section $\hat{\sigma}^{d=4+6}$ is
expressed in terms of $d=4$ and $d=6$ cross sections evaluated that are
respectively evaluated assuming

$|U_{\mu 4}|^{2}=1\quad\text{and}\quad\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(1{\rm~{}TeV})^{-2}\,,$ (61)

and appropriately re-scaled for arbitrary active-sterile mixing and EFT
parameters. In Eq. (60), we alternatively cast the total cross section solely
in terms of the $d=4$ contribution. Regardless of the parametrization, the
dependence on mixing and EFT parameters are not altered by convolutions with
QCD parton distribution functions (PDFs) nor QCD corrections [102]. The
dependence therefore also hold for cross sections at the hadron level.

With this guidance, we model the production of $\mu^{\pm}\nu_{4}$ pairs in LHC
collisions by expanding333 The SM_HeavyN_vSMEFTdim6_NLO and
SM_HeavyN_vSMEFTdim6_XLO UFOs are available from the URL
feynrules.irmp.ucl.ac.be/wiki/HeavyN. In the model files, the relevant Wilson
coefficient CmuN1 is dimensionless and the EFT scale is given by Lambda. These
are related to $\overline{C}_{H\nu e}$ by $\left[\overline{C}_{H\nu
e}\right]_{4\mu}=\texttt{CmuN1}/\texttt{Lambda}^{2}$. the HeavyN UFO libraries
[101, 103] by the operator $\mathcal{O}_{H\nu e}$. Using the procedure of
Refs. [104, 103] we can compute the scattering rates for the DY and $W\gamma$
channels up to NLO in QCD at both $d=4$ and $d=6$. For cross section
computations, we use the Monte Carlo suite MadGraph5_aMC@NLO [105, 106] and
tune our simulation inputs to match the heavy neutrino analysis of Ref. [94].

$m_{4}$ | 95 GeV | 100 GeV | 130 GeV | 150 GeV | 200 GeV
---|---|---|---|---|---
$|U_{\mu 4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu e}|^{2}$ | $3.46\times 10^{-3}$ | $4.32\times 10^{-3}$ | $7.17\times 10^{-3}$ | $7.97\times 10^{-3}$ | $8.88\times 10^{-3}$
$m_{4}$ | 400 GeV | 600 GeV | 800 GeV | 1000 GeV | 1200 GeV
$|U_{\mu 4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu e}|^{2}$ | $3.01\times 10^{-2}$ | $8.35\times 10^{-2}$ | $2.06\times 10^{-1}$ | $4.41\times 10^{-1}$ | $8.48\times 10^{-1}$
Table 3: Limits on $d=6$ coupling $\mathcal{C}_{H\nu
e}=v^{2}\left[\overline{C}_{H\nu e}\right]_{4\mu}$ and active-sterile neutrino
mixing $|U_{\mu 4}|$ as derived from CMS searches for heavy neutrinos with
$\mathcal{L}\approx 36{\rm~{}fb^{-1}}$ at $\sqrt{s}=13{\rm~{}TeV}$ [91].

In the top-left panel of Fig. 3, we plot as a function of $m_{4}$ the total,
hadronic $pp\to\mu^{\pm}\nu_{4}+X$ cross section for the DY and $W\gamma$
channels, at $d=4$ (upper curves) assuming a benchmark active-sterile mixing
of $|U_{\mu 4}|^{2}=1$, and at $d=6$ (lower curves) assuming a benchmark
Wilson coefficient $\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(1{\rm~{}TeV})^{-2}$. The band thickness corresponds to the
residual QCD scale uncertainty at NLO and reaches the few percent level for
all channels for the range of $m_{4}$ under consideration. While not shown,
PDF uncertainties reach a few percent for all curves. For both sets of
scattering processes and for the inputs assumed, we see that the $d=6$ rates
are suppressed compared to the $d=4$ rates by a factor $4|U_{\mu
4}|^{2}/|\mathcal{C}_{H\nu e}|^{2}\sim 10^{3}$ as seen in Eq. (57).
Consequentially, the $W\gamma$ rate at $d=6$ overtakes its DY counterpart at
$m_{4}\sim 900$ GeV, just as it does at $d=4$ [101, 103]. In the lower panel
of the same plot we show the QCD factor,

$K=\sigma^{\rm NLO}/\sigma^{\rm LO}.$ (62)

As anticipated, the impact of QCD corrections to the $d=6$ processes are
comparable to the $d=4$ cases. For our choices of SM inputs and QCD evolution
scales, we find that QCD corrections at NLO increase the DY $(W\gamma)$
channel at $d=6$ by about $10$-$20\%$ $(0$-$20\%)$.

For the representative combinations of neutrino mixing and EFT parameters:

$\displaystyle\text{Scenario~{}I~{}(small mixing/small cutoff)}:$
$\displaystyle\quad|U_{\mu 4}|^{2}=10^{-6},\quad\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(1{\rm~{}TeV})^{-2}\,,$ (63)
$\displaystyle\text{Scenario~{}II~{}(large mixing/large cutoff)}:$
$\displaystyle\quad|U_{\mu 4}|^{2}=10^{-4},\quad\left[\overline{C}_{H\nu
e}\right]_{4\mu}=(10{\rm~{}TeV})^{-2}\,,$ (64)

and using the linear relationship of Eq. (59), we plot in the upper panel of
Fig. 3(R) the total $d=4+6$ contribution to the $pp\to\mu^{\pm}\nu_{4}+X$
production cross section at the $\sqrt{s}=13$ TeV LHC, at NLO in QCD and for
the DY and $W\gamma$ channels. For the “small” mixing / “small” cutoff (upper
curves) scenario, which is dominated by physics at $d=6$, we find that
hadronic cross section rates at NLO can reach about $\sigma^{d=4+6}\sim
10{\rm~{}fb}~{}(5{\rm~{}ab})$ for $m_{4}\sim 150~{}(1000)$ GeV. Conversely,
for the “large” mixing / “large” cutoff (lower curves), which is dominated by
physics at $d=4$, we find rates are about $10\times$ smaller than the first
scenario.

At $\sqrt{s}=13$ TeV and with $\mathcal{L}\approx 36{\rm~{}fb^{-1}}$ of data,
searches by the CMS experiment are presently the most constraining for
$m_{4}\approx 100-1000{\rm~{}GeV}$ due to their inclusion of the $W\gamma$
channel [91, 92], though the sensitivity of ATLAS is projected to be
comparable [94, 96]. In its trilepton analysis [91], which is sensitive to
(pseudo)Dirac and Majorana neutrinos, the CMS experiment reports upper limits
on $|U_{\mu 4}|^{2}$ at the 95% CL. Using the scaling behavior for the DY and
$W\gamma$ channels at $d=4$, limits on active-sterile mixing can be
interpreted back as limits on the production cross section of
$\mu^{\pm}\nu_{4}$ pairs. For a given mass $m_{4}$, we define the 95% upper
limit on the cross section set by CMS to be

$\sigma_{\rm CMS~{}36{\rm~{}fb^{-1}}}^{\rm
95\%~{}CL}(pp\to\mu^{\pm}\nu_{4})=|U_{\mu 4}^{\rm
95\%~{}CL}|^{2}\times\left(\sigma^{d=4}_{\rm DY}\left[|U_{\mu
4}|^{2}=1\right]+\sigma^{d=4}_{W\gamma}\left[|U_{\mu 4}|^{2}=1\right]\right).$
(65)

Assuming that the production of $\mu^{\pm}\nu_{4}$ pairs is governed by both
$d=4$ and $d=6$ operators, then by Eq. (60), CMS data constrain neutrino
mixing and EFT parameters at 95% CL to satisfy:

$\displaystyle\left(|U_{\mu 4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu
e}|^{2}\right)<\frac{\sigma_{\rm CMS~{}36{\rm~{}fb^{-1}}}^{\rm
95\%~{}CL}(pp\to\mu^{\pm}\nu_{4})}{\sigma^{d=4}_{\rm DY}\left[|U_{\mu
4}|^{2}=1\right]+\sigma^{d=4}_{W\gamma}\left[|U_{\mu 4}|^{2}=1\right]}.$ (66)

Constraints on mixings and cutoff scales are summarized in Table 3 for
representative $\nu_{4}$ masses. For $m_{4}\sim 100-1000{\rm~{}GeV}$, direct
searches by CMS constrain the quantity $(|U_{\mu 4}|^{2}+\mathcal{C}_{H\nu
e}|^{2}/4)$ to be below the $3\cdot 10^{-3}-4\cdot 10^{-1}$ level at 95% CL.
In the limit that $|U_{\mu 4}|^{2}$ vanishes, which can be the case if
$|U_{\mu 4}|^{2}\sim m_{1}/m_{4}\lesssim 10^{-10}$ as in the high-scale Type I
Seesaw, then CMS data only weakly constrain the $d=6$ coupling to be
$(v^{2}\left[\overline{C}_{H\nu e}\right]_{4\mu})\lesssim 0.13~{}(1.3)$ for
$m_{4}=100~{}(1000)$ GeV.

$m_{4}$ | 90 GeV | 100 GeV | 130 GeV | 150 GeV | 200 GeV
---|---|---|---|---|---
$|U_{e4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu e}|^{2}$ | $6.22\times 10^{-3}$ | $6.52\times 10^{-3}$ | $1.10\times 10^{-2}$ | $1.43\times 10^{-2}$ | $1.36\times 10^{-2}$
$m_{4}$ | 400 GeV | 600 GeV | 800 GeV | 1000 GeV | 1200 GeV
$|U_{e4}|^{2}+\frac{1}{4}|\mathcal{C}_{H\nu e}|^{2}$ | $5.24\times 10^{-2}$ | $1.67\times 10^{-1}$ | $4.28\times 10^{-1}$ | $9.49\times 10^{-1}$ | $1.84$
Table 4: Same as Table 3 but for the $e$-flavor channel and based on search of
Ref. [91].

As a brief remark, we note that the above limits assume that the branching
fractions (BRs) of $\nu_{4}$ at dimension $d=4+6$ can be well approximated by
those at $d=4$. Due to an interplay between kinematic thresholds and the
Goldstone Equivalence Theorem, BR$(\nu_{4}\to W\ell)$ at $d=4$ spans about
$\sim 100\%-50\%$, over the range $m_{4}\sim 90{\rm~{}GeV}-1{\rm~{}TeV}$. A
large $d=6$ contribution could at most increase BR$(\nu_{4}\to W\mu)$ to 100%,
thereby doubling (at most) the signal yield in the CMS’s search for $\nu_{4}$.
Consequentially, the limits reported in Table 3 would be tightened by (at
most) a factor of $\sqrt{2}\approx 1.4$. An ambiguity exists, however, because
it is possible that other $d=6$ operators comparably enhance the $\nu_{4}\to
Z\nu_{\alpha}/H\nu_{\alpha}$ channels, which then reduce BR$(\nu_{4}\to
W\mu)$. We avoid this complication by assuming the $d=4$ branching fractions
for $\nu_{4}$.

With the full Run II data set, we anticipate that the sensitivity limit set in
Eq. (66) can be improved by a roughly factor of
$\sqrt{140{\rm~{}fb^{-1}}/36{\rm~{}fb^{-1}}}\sim 2$, assuming no improvements
to the analysis are made. Likewise, at the high luminosity LHC (HL-LHC) and
again assuming a comparable acceptance and selection rates, one can further
improve upon Eq. (66) roughly by a factor of
$\sqrt{3{\rm~{}ab^{-1}}/36{\rm~{}fb^{-1}}}\sim 10$.

For the case of electrons (and taus), the entire discussion above holds. In
particular, searches for the $pp\to e^{\pm}\nu_{4}+X$ process by the CMS
experiment constrain $|U_{e4}|$ and $|\overline{C}_{H\nu e}|$ to a comparable
level as reported for the muon flavor channel [91, 92]. These constraints are
summarized in Table 4 for representative $m_{4}$. Categorically, limits on
effective electron couplings are about $1.5-2\times$ weaker than the limits on
effective muon couplings. This is due to the larger background rate in the
$e^{\pm}\nu_{4}$ channel, and is driven by the mis-identification of jets as
electrons [91]. Likewise, limits on effective tau lepton couplings are
expected to be even weaker due to the smaller $\tau$-tagging efficiencies
[107, 94]. Finally, we estimate that the full Run II and HL-LHC data sets will
improve sensitivity to the $e^{\pm}\nu_{4}$ channel to the same degree as the
muon case.

##  5 The preferred parameter space

We now discuss the parameter space that can account for the discrepancies in
the muon’s and electron’s anomalous magnetic moment. We begin with the muonic
case in Sect. 5.1, where the deviation from the SM is more significant, and
address the electronic case in Sect. 5.2.

Figure 4: The green area depicts parameter space in the $m_{4}$-$\Lambda$
plane that can account for the $\Delta a_{\mu}$ discrepancy for a mixing angle
of $U_{\mu 4}=0.01$ (left panel) and $U_{\mu 4}=0.02$ (right panel). The red
region is excluded by CMS and the blue regions are excluded by measurements of
the $h\to\mu^{+}\mu^{-}$ branching ratio. The red and blue dashed lines show
prospects for the high-luminosity LHC. The gray area shows the region with
$m_{4}\gtrsim\Lambda$, where the $\nu$SMEFT approach breaks down.

###  5.1 $\Delta a_{\mu}$

We present our findings for the $n=1$ case in Fig. 4 in the $m_{4}$-$\Lambda$
plane. In the left (right) panel, motivated by the CKM unitarity bounds we set
$|U_{\mu 4}|=10^{-2}$ ($|U_{\mu 4}|=2\cdot 10^{-2}$), and chose
$\left[\bar{C}_{H\nu e}\right]_{4\mu}=-1/\Lambda^{2}$, where the sign is
needed to get a positive $\Delta a_{\mu}$. The green band corresponds to
parameter space that would reconcile the recent E989 measurement of $\Delta
a_{\mu}$ with the SM prediction of Ref. [4]. In the same plot, we depict
constraints from several experiments: The red band is the constraint from the
CMS experiment on direct $\nu_{4}$ production as described in Sect. 4.6. At
present, this does not reach the preferred parameter space yet. The large
increase in data from the HL-LHC however is expected to cover the
small-$m_{4}$ part of the green band. The blue area denotes parameter space
that is excluded by the observed $h\to\mu^{+}\mu^{-}$ signal strength (the
small gap in this area arises from parameter space where dimension-six
corrections vanish). The prospects for measurements at the HL-LHC of the
$h\to\mu^{+}\mu^{-}$ signal strength are at the $15\%$ level for ATLAS and
$13\%$ for CMS [49]. A combination should give a sensitivity at the $10\%$
level, which is indicated by the dashed blue line in Fig. 4, and covers much
of the preferred parameter space. Finally, the dark gray area indicates
parameter space where $m_{4}>\Lambda$, which is inconsistent with the
$\nu$SMEFT framework. The difference between the left and right panel
illustrates the sensitivity to $U_{\mu 4}$. Larger mixing angles are
consistent with larger values of $\Lambda$, but experimental constraints
become tighter as well.

Figure 5:  The upper (lower) green area depicts parameter space in the
$m_{4}$-$\Lambda$ plane that can account for the $\Delta a_{e}^{\rm Rb}$
($\Delta a_{e}^{\rm Cs}$) discrepancy for a mixing angle $U_{e4}=0.01$ (left
panel) and $U_{e4}=0.02$ (right panel). The red region depicts the excluded
area due to $0\nu\beta\beta$ and the blue regions are excluded by measurements
of the $h\to ee$ branching ratio. The red and blue dashed lines show prospects
for future $0\nu\beta\beta$ experiments and the high-luminosity LHC. The gray
area shows the region with $m_{4}\gtrsim\Lambda$, where the $\nu$SMEFT
approach breaks down.

For the representative values of $|U_{\mu 4}|$ that we use, the parameter
space that is not ruled out corresponds to a mass range for the sterile
neutrino of $m_{W}<m_{4}<700\,\mathrm{GeV}$, where $0.1<m_{4}/\Lambda<0.5$.
Larger masses are ruled out by $h\rightarrow\mu\mu$ measurements, while for
lighter masses the constraints on the mixing angle $U_{\mu 4}$ are
significantly more stringent such that CMS and ATLAS rule out this part of the
parameter space. The green band can be shifted (slightly) upwards by choosing
a larger mixing angle closer to the current upper bound $|U_{\mu 4}|=0.02$, or
downwards by choosing a smaller mixing angle. We find that assuming a smaller
mixing angle puts more of the preferred $\Delta a_{\mu}$ window in reach of
searches for $\mu\nu_{4}$ pairs by ATLAS and CMS.

The plot and discussions in Sect. 4 make it clear that the $\nu$SMEFT
parameter space is very constrained and only a small window is left to account
for the muon $g-2$ anomaly. The existing window will be largely covered at Run
III of the LHC and the HL-LHC by searches for direct $\nu_{4}$ production as
well as indirect constraints from $h\to\mu^{+}\mu^{-}$ precision measurements.
Additional constraints can arise from low-energy improvements of unitarity
tests, which would further shrink allowed values of $U_{\mu 4}$. Finally,
following the argument of recent no-lose theorems [108, 109], a confirmation
of the $\nu$SMEFT interpretation of $g-2$ results by any of the indirect
probes we have discussed suggests that the underlying physics can be directly
probed by muon collider proposals that are actively discussed in the
literature [110, 111, 112].

###  5.2 $\Delta a_{e}$

We now turn to $\Delta a_{e}$. Again considering the case of $n=1$, we show
the $m_{4}-\Lambda$ plane with $|U_{e4}|=0.01$ ($|U_{e4}|=0.02$) in the left
(right) panel of Fig. 5. The upper and lower green bands correspond to the
regions preferred by $\Delta a_{e}^{\rm Rb}$ and $\Delta a_{e}^{\rm Cs}$,
which correspond to a different choice of sign, $\left[\bar{C}_{H\nu
e}\right]_{4e}=\pm\frac{1}{\Lambda^{2}}$, respectively444This choice of sign
hardly impacts the remaining constraints depicted in the figure as they
approximately probe $|\bar{C}_{H\nu e}|$, given the experimental
sensitivities.. The blue regions are excluded by the current LHC (solid) and
prospective HL-LHC (dashed) measurements of Br$(h\to ee)$ discussed in Sect.
4.1. The solid blue line falls almost completely falls in the gray area where
$m_{4}\geq\Lambda$, indicating that present $h\rightarrow e^{+}e^{-}$ searches
are not accurate enough to set limits. The searches for $e\nu_{4}$ production,
discussed in Sect. 4.6, lead to similar limits as in the $\mu$ case. As
$\Delta a_{e}$ can be explained with larger values of $\Lambda$ than $\Delta
a_{\mu}$, these direct limits do not significantly constrain the $\Delta
a_{e}$-preferred regions and are not shown. More stringent constraints come
from $0\nu\beta\beta$, see Sect. 4.5, where the KamLAND-Zen measurement [82]
currently rules out the solid red region, while the reach of next-generation
experiments is shown by the dashed line.

From the left panel, we see that explanations of $\Delta a_{e}^{\rm Rb}$
require $m_{4}\gtrsim 200$ GeV and $\Lambda\gtrsim 6.5$ TeV, while larger
$m_{4}\gtrsim 650$ GeV and smaller $\Lambda>5.5$ TeV are preferred when using
$\Delta a_{e}^{\rm Cs}$. Unlike the case of the muon’s $g-2$, the sensitivity
of the HL-LHC to $h\to e^{+}e^{-}$ is not expected to rule out a significant
part of the parameters space. On the other hand, future $0\nu\beta\beta$
experiments can be seen to push up the lower limits to $m_{4}\gtrsim 500$ GeV
and $\Lambda\gtrsim 9$ TeV for explanations of $\Delta a_{e}^{\rm Rb}$, while
they would probe all of the parameter space up to $m_{4}=1$ TeV in the case of
$\Delta a_{e}^{\rm Cs}$. Similar conclusions hold for the scenario with
$|U_{e4}|=0.02$.

Finally, we note that the experimental limit on the electron EDM,
$|d_{e}|<1.1\cdot 10^{-29}e$ cm [45], provides a stringent constraint on the
imaginary part of the combination of couplings,

$\left|\sum_{j=4}^{n+3}\frac{m_{j}}{v}\,w(x_{j})\,{\rm
Im}\left(U_{ej}\left[v^{2}\bar{C}_{H\nu e}\right]_{je}\right)\right|\leq
2.2\cdot 10^{-11}\,.$ (67)

The real part of the same combination induces $\Delta a_{e}$ and would need to
be roughly six orders of magnitude larger to contribute at the level of
$\Delta a_{e}^{\rm Cs,Rb}$, implying explanations of these discrepancies will
need be CP-conserving to good precision. In contrast, the constraint due to
$d_{\mu}$ [113] allows the corresponding combination of muon couplings to have
an imaginary part that is larger than the real part needed to explain $\Delta
a_{\mu}$.

###  5.3 $\Delta a_{\mu}$ and $\Delta a_{e}$

As the sterile neutrino operator $\mathcal{O}_{H\nu e}$ seems to be able to
explain the discrepancies in both $a_{e}$ and $a_{\mu}$, a natural question is
whether it could explain both types of anomalies simultaneously. For the case
of a single sterile neutrino that we have focused on so far, the severe limits
from searches for $\mu\rightarrow e\gamma$ exclude this possibility. In this
scenario one can express the branching ratio as $\textrm{BR}(\mu\rightarrow
e\gamma)\sim\big{|}\frac{\Delta
a_{e}}{m_{e}\xi}\big{|}^{2}+\big{|}\xi\frac{\Delta
a_{\mu}}{m_{\mu}}\big{|}^{2}$, with $\xi=U_{e4}/U_{\mu 4}$. Using $\Delta
a_{e}^{\rm Cs}$, this leads to a lower bound on the prediction for the
branching fraction of

$\textrm{BR}(\mu\rightarrow e\gamma)_{\rm
min}=\tau_{\mu}\frac{\alpha_{em}m_{\mu}^{3}}{8}\Bigg{|}\frac{\Delta
a_{e}\Delta a_{e}}{m_{e}m_{\mu}}\Bigg{|}\simeq 1.4\cdot 10^{-4}\,.$ (68)

Such a value is clearly well above the current limit of
$\textrm{BR}(\mu\rightarrow e\gamma)<4.2\cdot 10^{-13}$ [54]. These limits
can, in principle, be avoided by introducing two sterile states, i.e., $n=2$,
with each mass eigenstate coupling almost exclusively to the electron and
muon, indicating a hierarchy within the active-sterile mixing sector. In such
scenarios, the $\mu\to e\gamma$ limits would require certain mixing matrix
elements, e.g. $U_{e5}$ and $U_{\mu 4}$ for $n=2$, to be tuned to zero to very
high precision compared to the nonzero elements, $U_{e4}$ and $U_{\mu 5}$.

##  6 Conclusions

In this work we investigated contributions to the anomalous magnetic moments
of leptons in the framework of the neutrino-extended Standard Model Effective
Field Theory. We investigated which $\nu$SMEFT operators can give sizable
contributions to $\Delta a_{e}$ and $\Delta a_{\mu}$ and concluded that
remarkably only one dimension-six $\nu$SMEFT operator is relevant for
$\Lambda\gg v$. This operator effectively induces a right-handed, leptonic
charged current, which modifies leptonic anomalous magnetic moments at one
loop. Due to a chiral enhancement and experimental constraints the
contribution can be sizable if the associated sterile neutrino has a mass
above $m_{W}$.

These results extend the contributions of SMEFT operators obtained in Ref.
[22] to the $\nu$SMEFT. Following the notation of Ref. [22] and assuming BSM
physics arises at a scale $\Lambda=1$ TeV, we can write the complete
$\nu$SMEFT contribution to $\Delta a_{\mu}$ and $\Delta a_{e}$ as

$\displaystyle\Delta a_{\ell}$ $\displaystyle=$
$\displaystyle\frac{m_{\ell}}{m_{\mu}}{\rm Re}\Bigg{[}{1.8_{\mu}\atop
1.7_{e}}\cdot 10^{-4}\widetilde{C}_{\begin{subarray}{c}eB\\\
\ell\ell\end{subarray}}-{9.6_{\mu}\atop 9.3_{e}}\cdot
10^{-5}\widetilde{C}_{\begin{subarray}{c}eW\\\
\ell\ell\end{subarray}}-{10_{\mu}\atop 9.7_{e}}\cdot
10^{-6}\widetilde{C}_{\begin{subarray}{c}lequ\\\ \ell\ell
33\end{subarray}}^{(3)}$ (69) $\displaystyle-(1.9+0.2c_{T}^{(c)})\cdot
10^{-7}\widetilde{C}^{(3)}_{\begin{subarray}{c}lequ\\\ \ell\ell
22\end{subarray}}+1.4\cdot 10^{-8}\widetilde{C}_{\begin{subarray}{c}lequ\\\
\ell\ell 33\end{subarray}}^{(1)}-4.9\cdot
10^{-9}\widetilde{C}_{\begin{subarray}{c}le\\\ \ell 33\ell\end{subarray}}$
$\displaystyle+\left(3.4c_{T}-0.4\right)\cdot
10^{-9}\widetilde{C}_{\begin{subarray}{c}lequ\\\ \ell\ell
11\end{subarray}}^{(3)}+\left(2.1c_{T}^{(c)}+{10_{\mu}\atop
9.7_{e}}\right)\cdot 10^{-10}\widetilde{C}_{\begin{subarray}{c}lequ\\\
\ell\ell 22\end{subarray}}^{(1)}$ $\displaystyle-2.9\cdot
10^{-10}\widetilde{C}_{\begin{subarray}{c}le\\\ \ell
22\ell\end{subarray}}+\frac{m_{\ell}}{m_{\mu}}\left(2.9\cdot
10^{-9}\widetilde{C}_{HWB}-2.3\cdot 10^{-9}\widetilde{C}_{HB}\right)$
$\displaystyle-1.1\cdot
10^{-7}\sum_{j=4}^{3+n}\frac{m_{j}}{m_{W}}w(x_{j})U_{\ell
j}\left[\widetilde{C}_{H\nu e}\right]_{j\ell}\Bigg{]}\,.$

Here the first four lines are due to SMEFT interactions, where the most
significant contributions come from the leptonic dipole moments, $C_{eB}$ and
$C_{eW}$, and semi-leptonic four-fermion interactions, $C_{lequ}^{(3)}$. The
dimensionless couplings $\widetilde{C}$ are related to the usual SMEFT Wilson
coefficients by $\widetilde{C}=\Lambda^{2}C(\Lambda)$ and we dropped terms
whose coefficients for $\ell=\mu$ ($\ell=e$) are smaller than $10^{-9}$
($10^{-12}$). $c_{T}$ and $c_{T}^{(c)}$ are related to non-perturbative
contributions which are proportional to the matrix elements of light quarks
and charm quarks, respectively, see Ref. [22] for details. Finally, the last
line gives the $\nu$SMEFT contribution where we used
$\left[\widetilde{C}_{H\nu e}\right]_{j\ell}=\Lambda^{2}\left[\bar{C}_{H\nu
e}(\Lambda)\right]_{j\ell}$ with $\Lambda=1$ TeV, where the loop function
$w(x)\to 1/2$ in the limit of $x\to\infty$.

Evaluating the contribution due to $\bar{C}_{H\nu e}$, we find a small window
of preferred masses, $100<m_{4}/(1\,\mathrm{GeV})<700$, for $|U_{\mu
4}|=\mathcal{O}(10^{-2})$, and $v/\Lambda=\mathcal{O}(0.2)$, where $\Delta
a_{\mu}$ can be accounted for. This parameter space will be largely tested by
future measurements at the high-luminosity LHC. Similarly, for $|U_{e4}|=0.01$
and $m_{4}>200$ GeV a values of $\Lambda>6.5$ TeV exists where $\Delta
a_{e}^{\rm Rb}$ can be accommodated. In this case, future $0\nu\beta\beta$
experiments will be able to probe a significant part of the parameters space,
up to $m_{4}<1$ TeV.

The main focus of this work is to identify whether new $\nu$SMEFT
contributions could be relevant and not to speculate about potential UV
completions. Nevertheless, it is worthwhile to consider what kind of models
could reproduce the low-energy EFT. The dimension-six $\mathcal{O}_{H\nu e}$
operator could hint towards a RH $W_{R}^{\pm}$ boson that couples mainly to
muons or electrons and mixes with the SM $W^{\pm}$ fields. Similar
$W_{R}^{\pm}$ bosons appear for instance in left-right symmetric models.
However, these models typically couple the $W_{R}$ to first-generation quarks
as well, in which case $m_{W_{R}}\sim 1$ TeV, as required in this work to
account for $\Delta a_{\mu}$, is already ruled out by searches for $W_{R}$
production at the LHC [114, 115, 116]. Models can be envisioned where
$W_{R}^{\pm}$ does not couple to quarks at all or at least not to first-
generation quarks to avoid such constraints.

Other possible scenarios are models with vector-like leptons which have been
studied in the context of explanations of other anomalies such as the first-
row CKM unitarity and lepton-flavor universality violation in $B$ meson decays
[117, 118, 71]. These hypothetical leptons have been searched for at the LHC
[119, 120, 121]. In such models the vector-like leptons can mix with the RH
neutrinos and muons, giving rise to the $\mathcal{O}_{H\nu e}$ operator at
tree level. For example, in the case of the $SU(2)$ doublet vector-like lepton
$\Delta_{1}\sim(1,2,-1/2)$, we obtain $C_{H\nu
e}\sim\lambda_{\Delta_{1}\nu_{R}}\lambda_{\Delta_{1}\mu_{R}}/m^{2}_{\Delta_{1}}$
with $\lambda_{ij}$ being the Yukawa coupling involving $i$ and $j$ particles.
The parameter space for the successful $\Delta a_{\mu}$ explanation can be
achieved if
$\lambda_{\Delta_{1}\nu_{R}}\lambda_{\Delta_{1}\mu_{R}}\sim\mathcal{O}(1)$ and
$m_{\Delta_{1}}\sim\mathcal{O}(1)~{}$TeV. Current LHC limits are below 1 TeV
[121], while future colliders are expected to reach the TeV range [122]. We
leave a more detailed analysis of possible scenarios to future work.

## Acknowledgements

The work of VC, KF and EM is supported by the US Department of Energy through
the Los Alamos National Laboratory. KF is also supported by the LANL LDRD
Program. Los Alamos National Laboratory is operated by Triad National
Security, LLC, for the National Nuclear Security Administration of U.S.
Department of Energy (Contract No. 89233218CNA000001). WD is supported by U.S.
Department of Energy Office of Science, under contract DE-SC0009919. RR
acknowledges the support of Narodowe Centrum Nauki under Grant No.
2019/34/E/ST2/00186.

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