aainstitutetext: Instituto de Física Corpuscular (IFIC), Universitat de
Valencia-CSIC, E-46980 Valencia, Spainbbinstitutetext: SISSA, International
School for Advanced Studies, INFN, Sezione di Trieste, Via Bonomea 265,
I-34136 Trieste, Italyccinstitutetext: Jožef Stefan Institute, Jamova 39, 1000
Ljubljana, Sloveniaddinstitutetext: Department of Physics and Astronomy,
University College London, London WC1E 6BT, United Kingdom

#  Probing Heavy Neutrino Magnetic Moments at the LHC using Long-Lived
Particle Searches

Rebeca Beltrán b,c Patrick D. Bolton d Frank F. Deppisch a Chandan Hati a
Martin Hirsch<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore long-lived particle (LLP) searches using non-pointing photons at
the LHC as a probe for sterile-to-sterile and active-to-sterile transition
magnetic dipole moments of sterile neutrinos. We consider heavy sterile
neutrinos with masses ranging from a few GeV to several hundreds of GeV. We
discuss transition magnetic dipole moments using the Standard Model effective
field theory and low-energy effective field theory extended by sterile
neutrinos ($N_{R}$SMEFT and $N_{R}$LEFT) and also provide a simplified UV
complete model example. LLP searches at the LHC using non-pointing photons
will probe sterile-to-sterile dipole moments two orders of magnitude below the
current best constraints from LEP, while an unprecedented sensitivity to
sterile neutrino mass of about 700 GeV is expected for active-to-sterile
dipole moments. For the UV-model example with one-loop transition magnetic
moments, the searches for charged lepton flavour violating processes in
synergy with LLP searches at the LHC can probe new physics at several TeV mass
scales and provide valuable insights into the lepton flavour structure of new
physics couplings.

##  1 Introduction

Neutrino oscillations currently provide the only concrete laboratory-based
evidence of physics beyond the Standard Model (SM). In the absence of direct
signs of physics beyond the SM from resonance searches at the Large Hadron
Collider (LHC), it is therefore interesting to consider the possibility that
the existence of new physics (NP) could manifest itself as non-trivial
neutrino properties. One such intriguing attribute is the neutrino magnetic
dipole moment. Minimally extending the SM to include non-zero active neutrino
masses, the induced neutrino magnetic moments are tiny, $\mu_{\nu}\simeq
3\times 10^{-20}\,\mu_{B}~{}(m_{\nu}/0.1~{}\text{eV})$, where $m_{\nu}$ is the
neutrino mass scale and $\mu_{B}$ is the commonly employed unit of the Bohr
magneton Marciano:1977wx ; Lee:1977tib ; Fujikawa:1980yx . Thus, a positive
experimental signal of neutrino magnetic moments could provide a smoking gun
for physics beyond the SM.

Sizeable active-to-active neutrino magnetic moments with naturally small
neutrino masses have been a lively avenue of research in recent times
Voloshin:1987qy ; Barbieri:1988fh ; Babu:2020ivd ; Babu:2021jnu . With the
addition of right-handed (RH) or sterile neutrinos to the SM field content,
often invoked as a means to dynamically generate the light neutrino masses,
the prospect of active-to-sterile neutrino transition magnetic moments has
also been of much interest in the literature Magill:2018jla ; Brdar:2020quo ;
Schwetz:2020xra ; Bolton:2021pey ; Fernandez-Martinez:2023phj . A wide variety
of flavour-dependent signatures of transition magnetic moments have been
examined, such as the Primakoff upscattering of an active neutrino to a heavy
state (with its subsequent decay to a light neutrino and a photon) in coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) experiments Miranda:2019wdy ;
Li:2020lba ; Miranda:2021kre ; Bolton:2021pey , NOMAD Gninenko:1998nn ;
Magill:2018jla , CHARM-II Coloma:2017ppo , DONUT DONUT:2001zvi , LSND
Magill:2018jla , Borexino Brdar:2020quo ; Plestid:2020vqf , XENON1T
Shoemaker:2018vii ; Brdar:2020quo ; Li:2022bqr , IceCube Huang:2022pce ;
Coloma:2017ppo , MiniBooNE Gninenko:2009ks ; Magill:2018jla , DUNE
Schwetz:2020xra ; Atkinson:2021rnp ; Ovchynnikov:2022rqj , Super-Kamiokande
Plestid:2020vqf ; Atkinson:2021rnp , and FASER$\nu$ Jodlowski:2020vhr ;
Ismail:2021dyp ; the production of heavy states via meson decays, leading to
invisible final states Li:2020lba  and displaced vertex signatures
Barducci:2023hzo ; monophoton plus missing energy at LEP, LHC, and future
colliders Magill:2018jla ; Zhang:2023nxy ; Ovchynnikov:2023wgg ;
Barducci:2024kig ; modifications to Big Bang nucleosynthesis (BBN) affecting
the abundance of 4He Magill:2018jla ; and excess energy loss in SN1987A and
low-energy supernovae Magill:2018jla ; Chauhan:2024nfa . If active-to-sterile
neutrino transition magnetic moments are discovered in the future, further
observations may be used to discriminate the Dirac versus Majorana nature of
the heavy states. For example, the photon emitted in the decay of a heavy
state produced via upscattering in a CE$\nu$NS experiment exhibits different
kinematic distributions in the two scenarios Bolton:2021pey . From the model-
building perspective, large active-to-active and active-to-sterile neutrino
transition magnetic moments are intimately connected with the radiative
corrections to neutrino masses, requiring non-trivial symmetries or
cancellations to realise a consistent ultraviolet (UV) completion Bell:2005kz
; Bell:2006wi .

On the other hand, sterile-to-sterile neutrino transition magnetic moments
remain relatively unconstrained for sterile neutrinos with masses above the
heavy meson masses. The transition magnetic moments between GeV scale sterile
neutrinos have been extensively studied in the context of long-lived particle
(LLP) searches at AL3X, ANUBIS, CODEX-b, DUNE, FACET, FASER/FASER$\nu$, MAPP,
MATHUSLA, and SHiP Barducci:2022gdv ; Barducci:2023hzo ; Gunther:2023vmz . For
sterile neutrinos heavier than the energies accessible in meson decays, LLP
searches at the LHC can provide an alternative window to sterile-to-sterile
neutrino magnetic moments, as recently explored in Liu:2023nxi ; Chun:2024mus
; Duarte:2023tdw . The leading constraint in this mass regime (up to the $Z$
mass) comes from the monophoton plus missing energy signature at LEP
Magill:2018jla .

In this work, we start with the model-independent effective field theory (EFT)
approach, examining the operators relevant for GeV to TeV scale neutrino
magnetic moments within the frameworks of $N_{R}$SMEFT (SM effective field
theory extended with RH neutrinos) and its low-energy counterpart, $N_{R}$LEFT
(low-energy effective field theory extended with RH neutrinos). To complement
this, we consider a UV-complete model example that generates these operators
at the one-loop level. Our model is similar to the one proposed in
Aparici:2009oua , but we generalise it further by removing the $Z_{2}$
symmetry imposed in the original construction. The broadening of the model
parameter space leads to the interesting scenario where large active-to-
sterile magnetic moments are also generated at one-loop, without relying on
the highly suppressed active-sterile neutrino mixing. This opens up a wider
range of phenomenology for LLP searches at the LHC. We also consider
constraints from other probes on the UV model, such as electroweak precision
observables and charged lepton flavour violation (cLFV).

We focus on the heavy sterile neutrino mass ranging from the GeV to several
$100$ GeV to explore the constraints on the transition magnetic moments from
LLP searches at the LHC using non-pointing photons. Our results for the
simplified model example can be straightforwardly generalised to many
extensions of the SM involving sterile neutrinos, an extended scalar sector,
and vector-like fermions Singer:1980sw ; Babu:1989rb ; Pilaftsis:1991ug ;
Deppisch:2017vne ; Hati:2018tge ; Hati:2018fzc ; Deppisch:2016jzl ;
Fonseca:2016tbn ; Hati:2020fzp . We find that to probe the heavy neutrino
dipole moments with a non-pointing photon final state (without any prompt
charged particles) at the LHC, the transverse impact parameter can be used to
select events with a displaced vertex. Our numerical simulations show that,
depending on the dominant production and decay mechanisms for the sterile
neutrinos, searches for LLPs with non-pointing photons will be sensitive to
unexplored parts of the parameter space in sterile-to-sterile and active-to-
sterile neutrino transition dipole moments for sterile neutrino mass ranging
from tens of GeV to well beyond the $Z$ mass.

The plan for the rest of the paper is as follows. In Section 2, we present an
EFT description of the operators relevant to the sterile neutrino magnetic
moment in the context of different EFTs. We then present a simple UV-complete
model example for realising the sterile-to-sterile and active-to-sterile
transition magnetic dipole moments in Section 3. In this Section, we also
present the derivation of these magnetic moments and their correlation with
neutrino masses. In Section 4, we review the various phenomenological
constraints on the model from direct collider searches and searches for cLFV
processes. In Section 5, we present different interesting final state
signatures for probing the sterile-to-sterile and active-to-sterile transition
magnetic dipole moments using LLP searches with non-pointing photons. We
further discuss the implementation and simulation details. In Section 6, we
present the numerical results in terms of the EFT coefficients and the
parameters of the explicit UV model. Finally, we conclude in Section 7.

##  2 Sterile Neutrino Transition Magnetic Moments

In the presence of RH or sterile neutrinos $N_{R}$ with masses below the
electroweak (EW) scale, transition magnetic moments are described by the
effective Lagrangian

$\displaystyle\mathcal{L}_{N_{R}\text{LEFT}}$ $\displaystyle\supset
d_{\scriptscriptstyle NN\gamma}^{\prime ij}\,\mathcal{O}_{\scriptscriptstyle
NN\gamma}^{ij}+d_{\scriptscriptstyle\nu N\gamma}^{\prime\alpha
i}\,\mathcal{O}_{\scriptscriptstyle\nu N\gamma}^{\alpha i}+\text{h.c.}\,,$ (1)

with the dimension-five ($d=5$) operators111While we omit it here, it is
common to include a factor of $1/2$ in the definition of
$\mathcal{O}_{\scriptscriptstyle NN\gamma}$ so that the normalisation is
analogous to the Majorana and Dirac mass terms, as seen in Section 3.1.

$\displaystyle\mathcal{O}_{\scriptscriptstyle
NN\gamma}^{ij}=(\bar{N}_{Ri}^{c}\sigma_{\mu\nu}N_{Rj})F^{\mu\nu}\,,\quad\mathcal{O}_{\scriptscriptstyle\nu
N\gamma}^{\alpha i}=(\bar{\nu}_{L\alpha}\sigma_{\mu\nu}N_{Ri})F^{\mu\nu}\,,$
(2)

where $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$, $F_{\mu\nu}$
is the photon field strength and $N_{R}^{c}=\mathcal{C}\bar{N}_{R}^{T}$, with
$\mathcal{C}$ the charge conjugation matrix. Here, we use
$\alpha\in\\{1,2,3\\}$ and $i,j\in\\{1,2,\ldots\\}$ to denote the generation
of SM lepton and sterile neutrino, respectively. The operators
$\mathcal{O}_{\scriptscriptstyle NN\gamma}^{ij}$ and
$\mathcal{O}_{\scriptscriptstyle\nu N\gamma}^{\alpha i}$ correspond to
sterile-to-sterile and active-to-sterile neutrino transition magnetic moments,
respectively. The former is antisymmetric in the indices $i,j$ and vanishes
for a single sterile state. We note that these effective interactions are only
applicable well below the EW symmetry breaking scale and are often referred to
as $N_{R}$LEFT operators. We stress that the prime superscript is used to
distinguish the $N_{R}$LEFT Wilson coefficients from the basis of operators
obtained after rotating $N_{R}$SMEFT operators at the EW symmetry breaking
scale, but not integrating out degrees of freedom such as $W$ and $Z$, which
will be defined shortly.

The $N_{R}$LEFT operators in Eq. (2) arise from operators above the EW scale
in the $N_{R}$SMEFT, in particular, the operators (up to $d=6$)

$\displaystyle\mathcal{L}_{N_{R}\text{SMEFT}}$ $\displaystyle\supset
C^{(5)ij}_{\scriptscriptstyle NNB}\,\mathcal{O}_{\scriptscriptstyle
NNB}^{(5)ij}+C_{\scriptscriptstyle NB}^{(6)\alpha
i}\,\mathcal{O}_{\scriptscriptstyle NB}^{(6)\alpha i}+{C}_{\scriptscriptstyle
NW}^{(6)\alpha i}\,\mathcal{O}_{\scriptscriptstyle NW}^{(6)\alpha
i}+\text{h.c.}\,,$ (3)

with

$\displaystyle\mathcal{O}_{\scriptscriptstyle
NNB}^{(5)ij}=(\bar{N}_{Ri}^{c}\sigma_{\mu\nu}N_{Rj})B^{\mu\nu}\,$ , (4)
$\displaystyle\mathcal{O}_{\scriptscriptstyle NB}^{(6)\alpha
i}=(\bar{L}_{\alpha}\sigma_{\mu\nu}N_{Ri})\tilde{H}B^{\mu\nu}\,$
$\displaystyle,\quad\mathcal{O}_{\scriptscriptstyle NW}^{(6)\alpha
i}=(\bar{L}_{\alpha}\sigma_{\mu\nu}N_{Ri})\tau^{I}\tilde{H}W^{I\mu\nu}\,,$ (5)

where $L=(\nu_{L}~{}\ell_{L})^{T}$ and $H=(\varphi^{+},\varphi^{0})^{T}$ are
the SM lepton and Higgs doublets, $B_{\mu\nu}$ and $W^{I}_{\mu\nu}$ are the
field strengths of $U(1)_{Y}$ and $SU(2)_{L}$, respectively, and
$\tau^{I}=\sigma_{I}/2$ are the generators of $SU(2)_{L}$, with $\sigma_{I}$
being the Pauli matrices. In Eq. (5), we additionally have
$\tilde{H}=i\sigma_{2}H^{*}$. The operator $\mathcal{O}_{\scriptscriptstyle
NN\gamma}^{ij}$ arises from the $N_{R}$SMEFT operator at $d=5$, while the
operator $\mathcal{O}_{\scriptscriptstyle\nu N\gamma}^{\alpha i}$ arises from
those at $d=6$ after expanding around the Higgs vacuum expectation value as
$H=(\varphi^{+},(v+h+i\varphi_{Z})/\sqrt{2})^{T}$ and transforming to the
weak-rotated basis of gauge fields. At the next lowest dimension in the
($N_{R}$)SMEFT are the $d=7$ operators Fridell:2023rtr

$\displaystyle\mathcal{O}_{\scriptscriptstyle
LHB}^{(7)\alpha\beta}=(\bar{L}_{\alpha}\tilde{H})\sigma_{\mu\nu}(\tilde{H}^{T}L_{\beta}^{c})B^{\mu\nu}\,$
$\displaystyle,\quad\mathcal{O}_{\scriptscriptstyle
LHW}^{(7)\alpha\beta}=(\bar{L}_{\alpha}\tilde{H})\sigma_{\mu\nu}(\tilde{H}^{T}\tau^{I}L_{\beta}^{c})W^{I\mu\nu}\,,$
(6) $\displaystyle\mathcal{O}_{\scriptscriptstyle
NHB}^{(7)ij}=(\bar{N}_{Ri}^{c}\sigma_{\mu\nu}N_{Rj})(H^{\dagger}H)B^{\mu\nu}\,$
$\displaystyle,\quad\mathcal{O}_{\scriptscriptstyle
NHW}^{(7)ij}=(\bar{N}_{Ri}^{c}\sigma_{\mu\nu}N_{Rj})(H^{\dagger}\tau^{I}H)W^{I\mu\nu}\,,$
(7)

which generate active-to-active neutrino magnetic moments
$\mathcal{O}_{\scriptscriptstyle\nu\nu\gamma}^{\alpha\beta}$ and further
modify $\mathcal{O}_{\scriptscriptstyle NN\gamma}^{ij}$.

In the context of high-energy collider processes, it is convenient to define a
basis of rotated $N_{R}$SMEFT operators at the EW symmetry breaking scale. In
this basis, there are two additional $d=5$ operators that involve the $Z$
boson field strength,

$\displaystyle\mathcal{O}_{\scriptscriptstyle
NNZ}^{ij}=(\bar{N}_{Ri}^{c}\sigma_{\mu\nu}N_{Rj})Z^{\mu\nu}\,,\quad\mathcal{O}_{\scriptscriptstyle\nu
NZ}^{\alpha i}=(\bar{\nu}_{L\alpha}\sigma_{\mu\nu}N_{Ri})Z^{\mu\nu}\;.$ (8)

These operators are not present in the usual $N_{R}$LEFT because $Z$ is
integrated out Liao:2016qyd ; Li:2020lba ; Li:2021tsq ; however, they are
convenient to use near the $Z$ pole, where they describe the production and
subsequent decay of the sterile neutrinos via a dipole-like coupling to the
$Z$ boson. We will denote the weak-rotated interactions near the $Z$ pole as

$\displaystyle\mathcal{L}^{\text{$Z$-pole}}_{N_{R}\text{LEFT}}$
$\displaystyle\supset d_{\scriptscriptstyle
NN\gamma}^{ij}\,\mathcal{O}_{\scriptscriptstyle
NN\gamma}^{ij}+d_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}\,\mathcal{O}_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}+d_{\scriptscriptstyle NNZ}^{ij}\,\mathcal{O}_{\scriptscriptstyle
NNZ}^{ij}+d_{\scriptscriptstyle\nu NZ}^{\alpha
i}\,\mathcal{O}_{\scriptscriptstyle\nu NZ}^{\alpha i}+\text{h.c.}\,.$ (9)

Now, considering the $N_{R}$SMEFT operators up to $d=6$, the dipole moments in
Eq. (9) are related to the $N_{R}$SMEFT Wilson coefficients in the unbroken
phase as

$\displaystyle
d_{\scriptscriptstyle{NN\gamma}}^{ij}=c_{w}C_{\scriptscriptstyle
NNB}^{(5)ij}\,$ $\displaystyle,\quad d_{\scriptscriptstyle
NNZ}^{ij}=-s_{w}C_{\scriptscriptstyle NNB}^{(5)ij}\,,$ (10) $\displaystyle
d_{\scriptscriptstyle{\nu N\gamma}}^{\alpha
i}=\frac{v}{\sqrt{2}}\left(c_{w}C_{\scriptscriptstyle{NB}}^{(6)\alpha
i}+\frac{s_{w}}{2}C_{\scriptscriptstyle{NW}}^{(6)\alpha i}\right)\,$
$\displaystyle,\quad d_{\scriptscriptstyle\nu NZ}^{\alpha
i}=\frac{v}{\sqrt{2}}\left(-s_{w}C_{\scriptscriptstyle NB}^{(6)\alpha
i}+\frac{c_{w}}{2}C_{\scriptscriptstyle NW}^{(6)\alpha i}\right)\,.$ (11)

where $s_{w}=\sin\theta_{w}$ and $c_{w}=\cos\theta_{w}$, with $\theta_{w}$ the
weak mixing angle, and $v$ is the Higgs vacuum expectation value. From Eq.
(10), we observe the ratio $d_{\scriptscriptstyle NNZ}^{\alpha
i}/d_{\scriptscriptstyle NN\gamma}^{\alpha i}=-t_{w}$ between the sterile-to-
sterile couplings, with $t_{w}=s_{w}/c_{w}$. In general, a UV-complete model
will also predict some relation between the Wilson coefficients
$C_{\scriptscriptstyle NB}^{(6)\alpha i}$ and $C_{\scriptscriptstyle
NW}^{(6)\alpha i}$; if we take $C_{\scriptscriptstyle NW}^{(6)\alpha
i}/C_{\scriptscriptstyle NB}^{(6)\alpha i}=a(g/g^{\prime})=a/t_{w}$, where $a$
is an arbitrary parameter and $g$ and $g^{\prime}$ are the $SU(2)_{L}$ and
$U(1)_{Y}$ couplings, respectively, we obtain the following ratio between the
active-to-sterile couplings

$\displaystyle\frac{d_{\scriptscriptstyle\nu NZ}^{\alpha
i}}{d_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}}=\frac{a-2t_{w}^{2}}{(2+a)t_{w}}\,.$ (12)

There are therefore two flat directions in the parameter space of couplings;
$d_{\scriptscriptstyle\nu N\gamma}^{\alpha i}=0$ for $a=-2$ and
$d_{\scriptscriptstyle\nu NZ}^{\alpha i}=0$ for $a=2t_{w}^{2}\approx 0.6$.

In addition to the operators in Eq. (8), there is also the $d=5$ charged-
current dipole operator,

$\displaystyle{}\mathcal{L}^{\text{$Z$-pole}}_{N_{R}\text{LEFT}}$
$\displaystyle\supset d_{\scriptscriptstyle\ell NW}^{\alpha
i}\left(\bar{\ell}_{L\alpha}\sigma_{\mu\nu}N_{Ri}\right)W^{\mu\nu}+\text{h.c.}\,,$
(13)

where $W^{\mu\nu}$ is the field strength of the $W$ boson. The associated
dipole moment is related to the $N_{R}$SMEFT Wilson coefficient
$C_{\scriptscriptstyle NW}^{(6)}$ in the unbroken phase as

$\displaystyle d_{\scriptscriptstyle\ell NW}^{\alpha
i}=\frac{v}{2}C_{\scriptscriptstyle NW}^{(6)\alpha
i}\quad\Rightarrow\quad\frac{d_{\scriptscriptstyle\ell NW}^{\alpha
i}}{d_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}}=\frac{\sqrt{2}a}{(2+a)s_{w}}\,.$ (14)

Clearly, we have $d_{\scriptscriptstyle\ell NW}^{\alpha i}=0$ for $a=0$. This
operator is important for the single production of a heavy $N$ with a charged
lepton or in the decay of $N$ to a charged lepton plus di-jet final state.

We note that, from a phenomenological point of view, the free parameters of
the model can be taken to be $\\{m_{N},C_{\scriptscriptstyle
NNB}^{(5)ij},C_{\scriptscriptstyle NB}^{(6)\alpha i},C_{\scriptscriptstyle
NW}^{(6)\alpha i}\\}$ or $\\{m_{N},d_{\scriptscriptstyle
NN\gamma}^{ij},d_{\scriptscriptstyle\nu N\gamma}^{\alpha i},a\\}$, with
general flavour structure. In any UV-complete realisation of these couplings,
however, the flavour structure will be fixed. We take both approaches in this
paper. First, we consider model-independent constraints on the couplings from
LLP searches at the HL-LHC. We then express $d_{\scriptscriptstyle
NN\gamma}^{ij}$ and $d_{\scriptscriptstyle\nu N\gamma}^{\alpha i}$ in terms of
the parameters of a specific UV-complete scenario, which fixes both the
flavour structure and the parameter $a$. We will review this UV scenario in
the next section.

##  3 A UV Model for Sterile Neutrino Magnetic Moments

Field(s) | Irrep | Couplings | $Z_{2}$ | Tree-level, $d>4$ | One-loop, $d>4$
---|---|---|---|---|---
$N_{R}$ | $(\mathbf{1},\mathbf{1})_{0}$ | $Y_{\nu}$ | $\checkmark$ | – | –
$E$ | $(\mathbf{1},\mathbf{1})_{-1}$ | $Y_{E}$ | $\times$ |  $\mathcal{O}_{Hl}^{(1)}$, $\mathcal{O}_{Hl}^{(3)}$ |  $\mathcal{O}_{eB}$, $\mathcal{O}_{eW}$, $\ldots$
$\phi$ | $(\mathbf{1},\mathbf{1})_{-1}$ |  $f$, $\lambda_{\phi H}$ | $\times$ | $\mathcal{O}_{ll}$ |  $\mathcal{O}_{eB}$, $\mathcal{O}_{eW}$, $\ldots$
$N_{R},\phi$ | – | $f^{\prime}$ | $\times$ |  $\mathcal{O}_{lNle}$, $\mathcal{O}_{eN}$ |  $\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$, $\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$, $\dots$
$N_{R},E$ | – | – | – | – |  $\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$, $\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$, $\dots$
$N_{R},E,\phi$ | – |  $h$, $h^{\prime}$ | $\checkmark$ | – |  $\mathcal{O}_{\scriptscriptstyle NNB}^{(5)}$, $\ldots$
Table 1: Fields in the UV model that generate sterile-to-sterile and active-
to-sterile neutrino magnetic moments. Shown are the representations of the
fields under the SM gauge group and the renormalisable couplings when one, two
or all of the fields are present; check marks indicate the allowed couplings
when $E$ and $\phi$ are odd under $Z_{2}$. Also shown are a selection of
($N_{R}$)SMEFT operators generated at tree-level and one-loop after
integrating out $E$ and $\phi$, giving interesting observable effects at low
energies.

In this work, we consider a simple UV extension of the SM with RH gauge-
singlet fermions $N_{R}$. To generate magnetic moments for these states, we
add the same minimal ingredients as Aparici:2009oua ; Chala:2020vqp ; a single
vector-like lepton $E$ and a scalar $\phi$.222Comparing to other notation used
in the literature, $E\Leftrightarrow X_{N}^{c}$ Chala:2020vqp  and
$\phi\Leftrightarrow\varphi$ Chala:2020vqp , $h$ Felkl:2021qdn  and
$\mathcal{S}_{1}^{*}$ deBlas:2013gla ; Beltran:2023ymm . Both are singlets
under $SU(3)_{c}$ and $SU(2)_{L}$ and have the hypercharges333We employ the
convention $D_{\mu}=\partial_{\mu}+ig\tau^{I}W^{I\mu}+ig^{\prime}YB_{\mu}$ for
the covariant derivative, such that SM fields have the hypercharges
$Y(L)=-\frac{1}{2}$, $Y(\ell_{R})=-1$, $Y(H)=\frac{1}{2}$. $Y(E)=Y(\phi)=-1$.
However, we do not impose an additional $Z_{2}$ symmetry under which the new
fields transform as $E\to-E$ and $\phi\to-\phi$ Aparici:2009oua . Without this
extra discrete symmetry, the following renormalisable terms can be written,
which are not present in the SM,

$\displaystyle\mathcal{L}$
$\displaystyle\supset\bar{E}\left(i\not{D}-m_{E}\right)E+(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m_{\phi}^{2}|\phi|^{2}-\lambda_{\phi\phi}|\phi|^{4}-\lambda_{\phi
H}|\phi|^{2}(H^{\dagger}H)$
$\displaystyle\quad-\Big{[}\bar{N}_{R}hE_{L}\phi^{*}+\bar{N}_{R}^{c}h^{\prime}E_{R}\phi^{*}$
$\displaystyle\quad\quad~{}~{}+\bar{L}Y_{E}HE_{R}+\bar{E}_{L}m_{eE}\ell_{R}+\bar{L}f\tilde{L}\phi+\bar{N}_{R}^{c}f^{\prime}\ell_{R}\phi^{*}+\text{h.c.}\Big{]}\,,$
(15)

where $\tilde{L}=i\sigma_{2}L^{c}$. The RH neutrinos $N_{R}$ can also have the
Yukawa-type and Majorana mass terms
$-\bar{L}Y_{\nu}\tilde{H}N_{R}-\frac{1}{2}\bar{N}^{c}_{R}M_{R}N_{R}+\text{h.c.}$,
which will be discussed in Section 3.1. Note that, because of the removal of
the $Z_{2}$ symmetry, we obtain four additional terms in the last line of Eq.
(3) with respect to Aparici:2009oua . The field content of the model is also
summarised in Tab. 1, where we show the representations of the fields under
the SM gauge group and the renormalisable couplings which are present when
one, two or all of the fields are considered. With check marks (crosses), we
indicate which couplings are allowed (forbidden) when the $Z_{2}$ symmetry,
under which $E$ and $\phi$ are odd, is enforced. We also show the
($N_{R}$)SMEFT operators which are generated after integrating out $E$ and
$\phi$. Among these are the operators $\mathcal{O}_{\scriptscriptstyle
NNB}^{(5)}$, $\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$ and
$\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$ which induce sterile-to-sterile
and active-to-sterile neutrino magnetic moments, but also operators which are
relevant for other low-energy probes, such as cLFV processes.

After EW symmetry breaking, mixing is induced between the SM charged leptons
$\ell$ and the vector-like lepton $E$. In the broken phase, the Lagrangian
contains the following extended $4\times 4$ mass matrix

$\displaystyle\mathcal{L}\supset-\begin{pmatrix}\bar{\ell}_{L}&\bar{E}_{L}\end{pmatrix}\mathcal{M}_{E}\begin{pmatrix}\ell_{R}\\\
E_{R}\end{pmatrix}+\text{h.c.}\,;\quad\mathcal{M}_{E}=\begin{pmatrix}\frac{vY_{e}}{\sqrt{2}}&\frac{vY_{E}}{\sqrt{2}}\\\
m_{eE}&m_{E}\end{pmatrix}\,,$ (16)

where the upper-left entry arises from the SM lepton Yukawa term
$-\bar{L}Y_{e}H\ell_{R}+\text{h.c.}$. Without loss of generality, the weak
eigenbasis RH fields $\ell_{R}$ and $E_{R}$, which have identical gauge
interactions, can be chosen such that all of the entries of $m_{eE}$, which is
a $3\times 1$ matrix, are zero Alves:2023ufm . This is equivalent to rotating
away non-zero $m_{eE}$ via redefinitions of $Y_{e}$, $Y_{E}$ and $m_{E}$. Now,
the mass matrix $\mathcal{M}_{E}$ can be diagonalised with the following
rotation of fields,

$\displaystyle\begin{pmatrix}\ell_{L\alpha}\\\
E_{L}\end{pmatrix}=\begin{pmatrix}V_{\alpha\beta}^{L}&V_{\alpha E}^{L}\\\
V_{E\beta}^{L}&V_{EE}^{L}\end{pmatrix}P_{L}\begin{pmatrix}\ell_{\beta}^{\prime}\\\
E^{\prime}\end{pmatrix}\,,\quad\begin{pmatrix}\ell_{R\alpha}\\\
E_{R}\end{pmatrix}=\begin{pmatrix}V_{\alpha\beta}^{R}&V_{\alpha E}^{R}\\\
V_{E\beta}^{R}&V_{EE}^{R}\end{pmatrix}P_{R}\begin{pmatrix}\ell_{\beta}^{\prime}\\\
E^{\prime}\end{pmatrix}\,,$ (17)

such that
$V^{L\dagger}\mathcal{M}_{E}V^{R}=\mathcal{M}_{E}^{\text{diag}}=\text{diag}(m_{e},m_{\mu},m_{\tau},m_{E^{\prime}})$.
Here, $\ell_{\beta}^{\prime}$ and $E^{\prime}$ represent the physical mass
eigenstate fields, with the former corresponding to the known charged leptons
$\ell_{\beta}^{\prime}\in\\{e,\mu,\tau\\}$ and the latter a single heavy
charged lepton. With abuse of notation, we relabel
$\ell_{\alpha}^{\prime}\to\ell_{\alpha}$ and $E^{\prime}\to E$ in the
following. In the limit $Y_{e}\sim Y_{E}\ll\sqrt{2}m_{E}/v$, it becomes
possible to block-diagonalise $\mathcal{M}_{E}$ with the approximate seesaw-
like mixing

$\displaystyle V^{L}_{\alpha
E}=-V_{E\alpha}^{L*}\approx\frac{vY_{E}^{\alpha}}{\sqrt{2}m_{E}}\,,\quad
V^{R}_{\alpha
E}=-V_{E\alpha}^{R*}\approx\frac{v^{2}[Y_{e}]_{\alpha\gamma}^{*}Y_{E}^{\gamma}}{2m_{E}^{2}}\,,$
(18)

where we see that the mixing between the RH fields is further suppressed by
$v/m_{E}$ with respect to the mixing between LH fields. Given that the LH
mixing is less suppressed, we can also consider the following non-unitarity in
the $V^{L}_{\alpha i}$ entry,

$\displaystyle
V^{L}_{\alpha\beta}\approx\delta_{\alpha\beta}-\frac{v^{2}Y_{E}^{\alpha}Y_{E}^{\beta*}}{4m_{E}^{2}}\,.$
(19)

Similarly, non-unitarity is seen in the $V^{L}_{EE}$ entry, but is irrelevant
phenomenologically.

Expanding the covariant derivatives in the SM Lagrangian and Eq. (3), and
transforming to the weak-rotated gauge fields, the gauge interactions of
$\nu_{L}$, $\ell_{L/R}$, $E_{L/R}$ and $\phi$ in the broken phase are found to
be

$\displaystyle\mathcal{L}$ $\displaystyle\supset-
eQ\Big{[}\bar{\ell}\gamma_{\mu}\ell+\bar{E}\gamma_{\mu}E+i\phi^{*}\overset{\leftrightarrow}{\partial}_{\mu}\phi\Big{]}A^{\mu}-\bigg{[}\frac{g}{\sqrt{2}}\bar{\ell}_{L}\gamma_{\mu}\nu_{L}W^{\mu}+\text{h.c.}\bigg{]}$
$\displaystyle\quad~{}-\frac{g}{c_{w}}\bigg{[}g_{L}^{\nu}\bar{\nu}_{L}\gamma_{\mu}\nu_{L}+g_{L}^{\ell}\bar{\ell}_{L}\gamma_{\mu}\ell_{L}+g_{R}^{\ell}\bigg{(}\bar{E}_{L}\gamma_{\mu}E_{L}+\begin{pmatrix}\bar{\ell}_{R}&\bar{E}_{R}\end{pmatrix}\gamma_{\mu}\begin{pmatrix}\ell_{R}\\\
E_{R}\end{pmatrix}+i\phi^{*}\overset{\leftrightarrow}{\partial}_{\mu}\phi\bigg{)}\bigg{]}Z^{\mu}\,,$
(20)

where, as usual, the electric charge is $Q=\tau^{3}+Y$ and the LH and RH $Z$
couplings are $g_{L}^{f}=\tau^{3}-Qs_{w}^{2}$ and $g_{R}^{f}=-Qs_{w}^{2}$,
respectively. Additionally, the charged lepton and vector-like lepton fields
must be rotated to the mass basis according to Eq. (17); the structure of Eq.
(3) results in the photon couplings and $Z$ couplings to $\ell_{R}$ and
$E_{R}$ being diagonal in the mass basis, as insertions of the LH and RH
mixing cancel due to unitarity. However, the LH mixing $V^{L}$ remains in the
charged-current term and the neutral-current terms involving $\ell_{L}$ and
$E_{L}$. In the seesaw limit, the Lagrangian then becomes

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\bigg{[}\frac{g}{\sqrt{2}}\bar{\ell}_{\alpha}\bigg{(}\delta_{\alpha\beta}-\frac{v^{2}Y_{E}^{\alpha}Y_{E}^{\beta*}}{4m_{E}^{2}}\bigg{)}\gamma_{\mu}P_{L}\nu_{\beta}W^{\mu}+\text{h.c.}\bigg{]}$
$\displaystyle\quad\quad-\frac{g}{c_{w}}\bar{\ell}_{\alpha}\bigg{(}g_{L}^{\ell}\delta_{\alpha\beta}+\frac{v^{2}Y_{E}^{\alpha}Y_{E}^{\beta*}}{4m_{E}^{2}}\bigg{)}\gamma_{\mu}P_{L}\ell_{\beta}Z^{\mu}\,.$
(21)

The former induces lepton flavour universality (LFU) violating charged-current
processes. The latter describes off-diagonal $Z$ couplings or flavour-changing
neutral-currents (FCNCs), which are subject to a plethora of stringent bounds
from EW precision observables and cLFV. We note that Eq. (3) is equivalent to
integrating out $E$ at low energies, which gives non-zero tree-level matching
conditions for the coefficients of the $d=6$ SMEFT operators
$\mathcal{O}_{Hl}^{(1)}=(H^{\dagger}i\overset{\leftrightarrow}{D}_{\mu}H)(\bar{L}\gamma^{\mu}L)$
and
$\mathcal{O}_{Hl}^{(3)}=(H^{\dagger}i\overset{\leftrightarrow}{D}\phantom{D}^{I}_{\mu}H)(\bar{L}\tau^{I}\gamma^{\mu}L)$,
i.e.,

$\displaystyle
C_{Hl}^{(1)\alpha\beta}=C_{Hl}^{(3)\alpha\beta}=-\frac{Y_{E}^{\alpha}Y_{E}^{\beta*}}{4m_{E}^{2}}\,,$
(22)

which has been used previously in the literature delAguila:2008pw ;
deBlas:2017xtg ; Crivellin:2020ebi . Constraints on the Yukawa coupling
$Y_{E}$ from these probes are detailed in Section 4.

Next, the Lagrangian in Eq. (3) results in the Higgs interactions

$\displaystyle\mathcal{L}\supset-\frac{1}{\sqrt{2}}\,\bar{\ell}_{L}\begin{pmatrix}Y_{e}&Y_{E}\end{pmatrix}\begin{pmatrix}\ell_{R}\\\
E_{R}\end{pmatrix}h+\text{h.c.}\,.$ (23)

This term can also be rotated to the mass basis, where the RH mixing $V^{R}$
is no longer cancelled due to unitarity. The resulting couplings of the Higgs
to charged leptons are

$\displaystyle\mathcal{L}\supset-\frac{1}{\sqrt{2}}\bar{\ell}_{\alpha}\bigg{(}\delta_{\alpha\gamma}-\frac{3v^{2}Y_{E}^{\alpha}Y_{E}^{\gamma*}}{4m_{E}^{2}}\bigg{)}[Y_{e}]_{\gamma\beta}P_{R}\ell_{\beta}h+\text{h.c.}\,,$
(24)

which is nothing other than the tree-level matching of the UV theory onto the
$d=6$ SMEFT operator $\mathcal{O}_{eH}=(H^{\dagger}H)\bar{L}H\ell_{R}$, with
the coefficient

$\displaystyle
C_{eH}^{\alpha\beta}=\frac{Y_{E}^{\alpha}Y_{E}^{\gamma*}[Y_{e}]_{\gamma\beta}}{2m_{E}^{2}}\,.$
(25)

Eq. (24) modifies SM Higgs decays and induces cLFV Higgs decays via the off-
diagonal couplings.

Moving now to the scalar sector of the UV theory, the term
$-\bar{L}f\tilde{L}\phi$ in Eq. (3) can be written explicitly as

$\displaystyle\mathcal{L}\supset-\begin{pmatrix}\bar{\nu}_{L}&\bar{\ell}_{L}\end{pmatrix}f\begin{pmatrix}\ell_{L}^{c}\\\
-\nu_{L}^{c}\end{pmatrix}\phi+\text{h.c.}=-2\bar{\nu}_{L}f\ell_{L}^{c}\phi+\text{h.c.}\,,$
(26)

where, in the second equality, we have used that $f$ is antisymmetric in
flavour space, $f_{\alpha\beta}=-f_{\beta\alpha}$. This interaction also
induces LFU violating and cLFV processes at tree-level and one-loop. If the
scalar $\phi$ is heavy and integrated out at low energies, Eq. (26) results in
the tree-level matching to the SMEFT operator
$\mathcal{O}_{ll}=(\bar{L}\gamma_{\mu}L)(\bar{L}\gamma^{\mu}L)$, with the
coefficient deBlas:2014mba ; Crivellin:2020klg ; Felkl:2021qdn ,

$\displaystyle
C_{ll}^{\alpha\beta\gamma\delta}=\frac{f_{\alpha\gamma}f_{\delta\beta}^{*}}{m_{\phi}^{2}}\,.$
(27)

Including also the term $\bar{N}_{R}^{c}f^{\prime}\ell_{R}\phi^{*}$,
integrating out $\phi$ also gives tree-level matching to the $N_{R}$SMEFT
operators $\mathcal{O}_{lNle}=(\bar{L}N_{R})\epsilon(\bar{L}\ell_{R})$ and
$\mathcal{O}_{eN}=(\bar{\ell}_{R}\gamma_{\mu}\ell_{R})(\bar{N}_{R}\gamma^{\mu}N_{R})$
Beltran:2023ymm , with

$\displaystyle C_{lNle}^{\alpha
i\beta\gamma}=\frac{2f_{\alpha\beta}f_{i\gamma}^{\prime}}{m_{\phi}^{2}}\,,\quad
C_{eN}^{\alpha\beta
ij}=\frac{f_{i\alpha}^{\prime*}f_{j\beta}^{\prime}}{2m_{\phi}^{2}}\,.$ (28)

This exhausts the tree-level phenomenology of the UV model at low energies,
which would have been absent if the $Z_{2}$ symmetry had been imposed. At the
one-loop level, many more operators are induced in the ($N_{R}$)SMEFT, with
partial matching results, predominantly in the SMEFT, available in the
literature Chala:2020vqp ; Zhang:2021jdf ; Carmona:2021xtq ; Fuentes-
Martin:2022jrf . The focus of the remainder of this section is the one-loop
matching of the operators $\mathcal{O}_{\scriptscriptstyle NNB}^{(5)}$,
$\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$ and
$\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$, which induce LLP signatures at
the LHC. First, in Section 3.1, we review the generation of neutrino masses.

###  3.1 Neutrino Masses

For sterile-to-sterile neutrino magnetic moments to be present, at least two
RH neutrinos $N_{R}$ are required in the model. As mentioned below Eq. (3), it
is possible to write the renormalisable terms
$-\bar{L}Y_{\nu}\tilde{H}N_{R}-\frac{1}{2}\bar{N}_{R}^{c}M_{R}N_{R}+\text{h.c.}$
in the Lagrangian. Below the EW scale, these terms lead to mixing between the
active neutrinos $\nu_{L}$ and $N_{R}$. The Lagrangian contains

$\displaystyle\mathcal{L}\supset-\frac{1}{2}\begin{pmatrix}\bar{\nu}_{L}&\bar{N}_{R}^{c}\end{pmatrix}\mathcal{M}_{\nu}\begin{pmatrix}\nu^{c}_{L}\\\
N_{R}\end{pmatrix}+\text{h.c.}\,;\quad\mathcal{\mathcal{M}}_{\nu}=\begin{pmatrix}0&\frac{vY_{\nu}}{\sqrt{2}}\\\
\frac{vY_{\nu}^{T}}{\sqrt{2}}&M_{R}\end{pmatrix}\,,$ (29)

where the extended neutrino mass matrix $\mathcal{M}_{\nu}$ is symmetric.
Without loss of generality, $M_{R}$ can be made to be diagonal via a rotation
among the RH neutrinos, such that
$M_{R}=\text{diag}(m_{N_{1}},m_{N_{2}},\ldots)$ and the resulting massive
states are Majorana fermions, with $N_{i}=N_{i}^{c}$. However, we will keep
$M_{R}$ general in the following. The mass matrix can be diagonalised with the
following unitary rotation Schechter:1980gr

$\displaystyle\begin{pmatrix}\nu_{L\alpha}^{c}\\\
N_{Rj}\end{pmatrix}=\begin{pmatrix}U_{\alpha i}^{*}&V_{\alpha N_{i}}^{*}\\\
\tilde{V}_{ji}&\tilde{U}_{jN_{i}}\end{pmatrix}P_{R}\begin{pmatrix}\nu_{i}^{\prime}\\\
N_{i}^{\prime}\end{pmatrix}\,,$ (30)

such that
$V^{T}\mathcal{M}_{\nu}V=\mathcal{M}_{\nu}^{\text{diag}}=\text{diag}(m_{1},m_{2},m_{3},m_{N_{1}^{\prime}},\ldots)$.
Here, $\nu_{i}^{\prime}=\nu_{i}^{\prime c}$ and $N^{\prime}_{i}=N_{i}^{\prime
c}$ are the physical Majorana mass eigenstate fields in the broken phase, with
the former corresponding to the observed light neutrinos and the latter to
heavy sterile neutrinos. Similar to the charged leptons, we will relabel
$\nu_{i}^{\prime}\to\nu_{i}$ and $N_{i}^{\prime}\to N_{i}$. In the limit
$Y_{\nu}\ll\sqrt{2}M_{R}/v$, $\mathcal{M}_{\nu}$ can be diagonalised with

$\displaystyle V_{\alpha N_{i}}$
$\displaystyle\approx\frac{v}{\sqrt{2}}\big{(}Y_{\nu}M_{R}^{-1}\big{)}_{\alpha
j}\tilde{U}_{jN_{i}}^{*}\,,\quad\tilde{V}_{ji}\approx-\frac{v}{\sqrt{2}}\big{(}M_{R}^{-1}Y_{\nu}^{T}\big{)}_{j\alpha}U_{\alpha
i}^{*}\,,$ (31)

to obtain the sub-blocks

$\displaystyle[M_{\nu}]_{\alpha\beta}=U_{\alpha i}U_{\beta i}m_{i}$
$\displaystyle\approx-\frac{v^{2}}{2}\big{(}Y_{\nu}M_{R}^{-1}Y_{\nu}^{T}\big{)}_{\alpha\beta}\,,\quad[M_{N}]_{ij}=\tilde{U}^{*}_{iN_{i}}\tilde{U}^{*}_{jN_{i}}m_{N_{i}}\approx[M_{R}]_{ij}\,,$
(32)

where $U$ and $\tilde{U}$ diagonalise $M_{\nu}$ and $M_{N}$, respectively.
Note that $\tilde{U}_{jN_{i}}=\delta_{ij}$ if we had taken $M_{R}$ to be
diagonal. Combining Eqs. (31) and (32), it is possible to find the well-known
relation for the active-sterile neutrino mixing Casas:2001sr

$\displaystyle V_{\alpha N_{i}}=iU_{\alpha
j}\mathcal{R}_{ji}\sqrt{\frac{m_{j}}{m_{N_{i}}}}\,,$ (33)

where $\mathcal{R}$ is an arbitrary orthogonal matrix, i.e.
$\mathcal{R}_{ji}\mathcal{R}_{ki}=\delta_{jk}$.

In this work, we consider heavy sterile states $N_{i}$ within the kinematic
reach of the HL-LHC, i.e., $m_{N_{i}}$ from 1 GeV up to 1 TeV. For this range
of masses, there are two possible ways to produce the observed light neutrino
masses via the tree-level relations Eq. (32). The first scenario is that the
heavy states $N_{i}$ are dominantly Majorana (or, in other words, the masses
$m_{N_{i}}$ are well separated). Then, the light neutrino masses are given by
$[M_{\nu}]_{\alpha\beta}=-\frac{v^{2}}{2}[Y_{\nu}]_{\alpha i}[Y_{\nu}]_{\beta
i}/m_{N_{i}}$, which is the standard Type-I seesaw relation Minkowski:1977sc ;
Gell-Mann:1979vob ; Yanagida:1979as ; Mohapatra:1979ia ; Schechter:1980gr .
For $m_{N}\sim 100$ GeV, neutrino masses $m_{\nu}\sim 0.1$ eV then require
Yukawa couplings of size $[Y_{\nu}]_{\alpha i}\sim 10^{-6}$. The active-
sterile mixing is governed by Eq. (33) with $|\mathcal{R}_{ji}|\leq 1$, or
$V_{\ell N}\sim\sqrt{m_{\nu}/m_{N}}$. However, for heavy sterile states in the
mass range relevant for the HL-LHC, this active-sterile mixing $V_{\ell N}$ is
too suppressed to give observable effects unless some additional NP is
present.

The second scenario, which can be more relevant for LLP searches, is when
pairs of heavy sterile states $N_{i}$ form pseudo-Dirac states with small mass
splittings. In this limit, cancellations in the matrix product
$Y_{\nu}M_{R}^{-1}Y_{\nu}^{T}$ ensure small light neutrino masses instead of
suppressed Yukawa couplings. Now that $Y_{\nu}$ can in principle be large, so
can the active-sterile mixing in Eq. (31); equivalently, this corresponds to
the limit $|\mathcal{R}_{ji}|\gg 1$ in Eq. (33). This interesting scenario is
minimally obtained in the inverse seesaw mechanism Mohapatra:1986aw ;
Nandi:1985uh ; Mohapatra:1986bd ; Gonzalez-Garcia:1988okv ; Deppisch:2004fa ;
Ma:2009du ; Dev:2012sg ; BhupalDev:2012jvh ; CentellesChulia:2020dfh , which
does not require any additional Higgs doublets or symmetries. To demonstrate
this mechanism, we can consider the simplified scenario where two heavy
sterile states, $N_{R1}\equiv N_{R}$ and $N_{R2}\equiv S_{L}^{c}$, only
interact with one generation of active neutrino. The relevant mass terms can
be written as

$\displaystyle\mathcal{L}\supset-m_{\nu N}\bar{\nu}_{L}N_{R}-m_{\nu
S}\bar{\nu}_{L}S_{L}^{c}-{1\over
2}\left(\mu_{N}\bar{N}_{R}^{c}N_{R}+\mu_{S}\bar{S}_{L}S_{L}^{c}\right)-m_{N_{D}}\bar{S}_{L}N_{R}+{\rm
h.c.}\,,$ (34)

where $(m_{\nu N},m_{\nu S})\equiv vY_{\nu}/\sqrt{2}$. As such, the extended
neutrino mass matrix (assuming one generation of active neutrino) in the basis
$(\nu_{L}^{c},N_{R},S^{c}_{L})$ takes the form

$\displaystyle\mathcal{M}_{\nu}=\left(\begin{matrix}0&m_{\nu N}&m_{\nu S}\cr
m_{\nu N}&\mu_{N}&m_{N_{D}}\cr m_{\nu
S}&m_{N_{D}}&\mu_{S}\end{matrix}\right)\,,$ (35)

where $m_{\nu S}$ can be rotated away without any loss of generality Ma:2009du
; Abada:2014vea  and the lepton number violating masses $\mu_{N,S}\ll
m_{N_{D}}$ are naturally small in the sense of ’t Hooft tHooft:1979rat , i.e.
total lepton number symmetry is restored in the limit $\mu_{N,S}\to 0$. The
symmetric mass matrix $\mathcal{M}_{\nu}$ can be diagonalised as before; the
inclusion of $N_{R}$ and $S_{L}^{c}$ leads to two Majorana eigenstates $N_{1}$
and $N_{2}$ with a small mass splitting compared to the Dirac mass
$m_{N_{D}}$. The unitary rotation matrix $V$ can be solved perturbatively by
using the Hermitian combination $\mathcal{M}_{\nu}^{\dagger}\mathcal{M}_{\nu}$
(or $\mathcal{M}_{\nu}\mathcal{M}_{\nu}^{\dagger}$),

$\displaystyle
V^{\dagger}\mathcal{M}^{\dagger}_{\nu}\mathcal{M}_{\nu}V=\left(V^{T}\mathcal{M}_{\nu}V\right)^{\dagger}\left(V^{T}\mathcal{M}_{\nu}V\right)=\text{diag}(m_{\nu}^{2},m_{N_{1}}^{2},m_{N_{2}}^{2})\,,$
(36)

where the $V$ diagonalising $\mathcal{M}_{\nu}\mathcal{M}_{\nu}^{\dagger}$ is
the same as that in Eq. (36). The most straightforward case is obtained in the
limit where $\mu_{N},\mu_{S}\ll m_{\nu N}\ll m_{N_{D}}$ and, without loss of
generality, $m_{\nu S}$ is set to zero via a rotation among the sterile
states. In this limit, the mass eigenvalues can be solved perturbatively to
obtain

$\displaystyle m_{\nu}$ $\displaystyle\simeq\frac{|m_{\nu
N}|^{2}|\mu_{S}|}{|m_{\nu N}|^{2}+|m_{N_{D}}|^{2}}\,,$ (37) $\displaystyle
m_{N_{1}}^{2}$ $\displaystyle\simeq|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}-\frac{\left|\mu_{S}^{*}m_{N_{D}}^{2}+\mu_{N}\left(|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}\right)\right|}{\sqrt{|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}}}\,,$ (38) $\displaystyle m_{N_{2}}^{2}$
$\displaystyle\simeq|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}+\frac{\left|\mu_{S}^{*}m_{N_{D}}^{2}+\mu_{N}\left(|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}\right)\right|}{\sqrt{|m_{\nu
N}|^{2}+|m_{N_{D}}|^{2}}}\,.$ (39)

Another interesting case occurs when $\mu_{N}\sim\mu_{S}$ and $m_{\nu N}\sim
m_{\nu S}$, requiring some additional symmetry preventing the freedom to
rotate away $m_{\nu S}$. The mass matrix can be exactly solved in this
scenario, leading to the masses

$\displaystyle m_{\nu}$
$\displaystyle=\frac{1}{2}(m_{N_{D}}+\mu_{S})\left[1-\sqrt{1+\frac{8m_{\nu
N}^{2}}{(m_{N_{D}}+\mu_{S})^{2}}}\right]\simeq-\frac{2m_{\nu
N}^{2}}{m_{N_{D}}+\mu_{S}}\,,$ (40) $\displaystyle m_{N_{1}}$
$\displaystyle=\frac{1}{2}(m_{N_{D}}+\mu_{S})\left[1+\sqrt{1+\frac{8m_{\nu
N}^{2}}{(m_{N_{D}}+\mu_{S})^{2}}}\right]\simeq(m_{N_{D}}+\mu_{S})-\frac{2m_{\nu
N}^{2}}{m_{N_{D}}+\mu_{S}}\,,$ (41) $\displaystyle m_{N_{2}}$
$\displaystyle=m_{N_{1}}-\mu_{S}\,,$ (42)

where the approximate relations are valid in the limit $m_{\nu N}\ll
m_{N_{D}}$. This case corresponds to the scenario where $N_{2}$ is decoupled,
which can be protected in the presence of an explicit symmetry, and only
$N_{1}$ mixes with the active state. We note that any new interactions
involving $N_{1}$ and $N_{2}$, e.g., a transition magnetic dipole moment, is
in general not diagonal in this basis and may lead to corrections to the mass
matrix $\mathcal{M}_{\nu}$. However, to restrict ourselves to these two
limiting cases of the inverse seesaw mechanism, we will assume any such
correction to be small.

Figure 1: Diagrams at one-loop in the UV model which renormalise the RH
neutrino mass matrix $M_{R}$ (left) and Yukawa coupling $Y_{\nu}$ (right). For
the Yukawa coupling, there are two other one-loop diagrams; one similar to
that above with the replacements $E_{R}^{c}\to\ell^{c}_{R\rho}$,
$h_{i}^{\prime}\to f_{i\rho}^{\prime}$ and
$Y_{E}^{\beta*}\to[Y_{e}]_{\beta\rho}^{*}$ and another with an internal Higgs
line and an insertion of $[Y_{\nu}]_{\alpha i}$.

In the presence of the fields $E$ and $\phi$ and their interactions in Eq.
(3), it is possible to draw the one-loop diagrams in Fig. 1 which renormalise
the Yukawa coupling $Y_{\nu}$ and the RH neutrino mass $M_{R}$. The threshold
corrections from these diagrams are

$\displaystyle[Y_{\nu}]_{\alpha i}$ $\displaystyle=[Y_{\nu}^{(0)}]_{\alpha
i}-\frac{1}{8\pi^{2}}\bigg{[}f_{\alpha\beta}Y_{E}^{\beta*}h_{i}^{\prime}\bigg{(}1+\frac{r\log
r}{1-r}+\log\frac{\mu^{2}}{m_{E}^{2}}\bigg{)}+f_{\alpha\beta}[Y_{e}]_{\beta\rho}^{*}f_{i\rho}^{\prime}\bigg{(}1+\log\frac{\mu^{2}}{m_{\phi}^{2}}\bigg{)}$
$\displaystyle\hskip
80.00012pt-\frac{1}{2}Y_{E}^{\alpha}Y_{E}^{\beta*}[Y_{\nu}^{(0)}]_{\beta
i}\bigg{(}1-\frac{\mu_{H}^{(0)2}}{m_{E}^{2}}\bigg{)}\bigg{(}1+\log\frac{\mu^{2}}{m_{E}^{2}}\bigg{)}\bigg{]}\,,$
(43) $\displaystyle[M_{R}]_{ij}$
$\displaystyle=[M_{R}^{(0)}]_{ij}-\frac{(h_{i}^{\prime}h_{j}^{*}+h_{i}^{*}h_{j}^{\prime})m_{E}}{16\pi^{2}}\left(1+\frac{r\log
r}{1-r}+\log\frac{\mu^{2}}{m_{E}^{2}}\right)\,,$ (44)

where $r=m_{\phi}^{2}/m_{E}^{2}$ and $\mu_{H}$ is the usual parameter in the
Higgs potential. While it is always possible to absorb the finite corrections
to $Y_{\nu}$ and $M_{R}$ in their renormalised values, these expressions are
nevertheless useful to verify if fine-tuning between the tree-level and one-
loop contributions is required to obtain the desired heavy masses $m_{N_{i}}$
and transition magnetic moments, calculated in the following sections.

###  3.2 Sterile-to-Sterile Neutrino Magnetic Moments

In our model, the $d=5$ $N_{R}$SMEFT dipole operator
$\mathcal{O}_{\scriptscriptstyle NNB}^{(5)}$ is generated at one-loop, as
shown by the diagrams in Fig. 2, which then induces sterile-to-sterile
transition magnetic moments $\mathcal{O}_{\scriptscriptstyle NN\gamma}$ and
$\mathcal{O}_{\scriptscriptstyle NNZ}$ in the broken phase. We perform the
matching of the UV model to the effective operator coefficient
$C_{\scriptscriptstyle NNB}^{(5)}$ using the diagrammatic approach, which
involves computing the one-light-particle-irreducible (1LPI) amplitudes in the
effective and UV theories and requiring that they coincide at the matching
scale. All one-loop amplitudes are performed using dimensional regularisation
with $d=4-2\epsilon$ space-time dimensions and introducing the renormalisation
scale $\mu$. We then subtract divergent pieces in the $\overline{\text{MS}}$
scheme.

The amplitude $\braket{N_{i}N_{j}B}$ in the $N_{R}$SMEFT via the operator
$\mathcal{O}_{\scriptscriptstyle NNB}^{(5)ij}$ is

$\displaystyle i\mathcal{M}_{ij}^{\text{EFT}}$
$\displaystyle=4\bar{u}(p_{N_{i}})\sigma_{\mu\nu}p_{B}^{\nu}\Big{[}C_{\scriptscriptstyle
NNB}^{(5)ij}P_{R}-C_{\scriptscriptstyle
NNB}^{(5)ij*}P_{L}\Big{]}u(p_{N_{j}})\epsilon^{\mu*}(p_{B})\,,$ (45)

where $p_{B}=p_{N_{j}}-p_{N_{i}}$. The amplitude $\braket{N_{i}N_{j}B}$ in the
UV model is instead given by

$\displaystyle i\mathcal{M}_{ij}^{\text{UV}}$
$\displaystyle=g^{\prime}Y\bar{u}(p_{N_{i}})\Big{[}(h_{i}^{\prime}P_{R}+h_{i}P_{L})\Gamma_{\mu}(h_{j}^{*}P_{R}+h_{j}^{\prime*}P_{L})$
$\displaystyle\hskip
60.00009pt-(h_{i}^{*}P_{R}+h_{i}^{\prime*}P_{L})\Gamma_{\mu}(h_{j}^{\prime}P_{R}+h_{j}P_{L})\Big{]}u(p_{N_{j}})\epsilon^{\mu*}(p_{B})\,,$
(46)

where $Y(E)=Y(\phi)=-1$ and $\Gamma_{\mu}$ denotes the loop integral

$\displaystyle\Gamma_{\mu}$
$\displaystyle=\mu^{2\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\,\bigg{[}\frac{(\not{p}\phantom{p}_{N_{i}}-\not{k}+m_{E})\gamma_{\mu}(\not{p}\phantom{p}_{N_{j}}-\not{k}+m_{E})}{\big{[}k^{2}-m_{\phi}^{2}\big{]}\big{[}(p_{N_{i}}-k)^{2}-m_{E}^{2}\big{]}\big{[}(p_{N_{j}}-k)^{2}-m_{E}^{2}\big{]}}$
$\displaystyle\hskip
75.0001pt-\frac{(\not{k}+m_{E})(p_{N_{i}}+p_{N_{j}}-2k)_{\mu}}{\big{[}k^{2}-m_{E}^{2}\big{]}\big{[}(p_{N_{i}}-k)^{2}-m_{\phi}^{2}\big{]}\big{[}(p_{N_{j}}-k)^{2}-m_{\phi}^{2}\big{]}}\bigg{]}\,,$
(47)

where $k$ is the loop momentum. The first and second terms inside the square
brackets of Eq. (3.2) originate from the coupling of $B_{\mu}$ to $E$ and
$\phi$, respectively. We now expand the loop integral in powers of the
external momenta and, assuming $m_{N_{i}}\ll m_{E},m_{\phi}$, obtain the
amplitude

$\displaystyle i\mathcal{M}_{ij}^{\text{UV}}$
$\displaystyle=\frac{g^{\prime}}{16\pi^{2}m_{E}}f(r)\bar{u}(p_{N_{i}})\sigma_{\mu\nu}p_{B}^{\nu}\Big{[}(h^{\prime}_{i}h^{*}_{j}-h^{*}_{i}h^{\prime}_{j})P_{R}-(h^{\prime*}_{i}h_{j}-h_{i}h^{\prime*}_{j})P_{L}\Big{]}u(p_{N_{j}})\epsilon^{\mu*}(p_{B})\,,$
(48)

where we define the loop function

$\displaystyle f(r)=\frac{1}{1-r}+\frac{r\log r}{(1-r)^{2}}\,,$ (49)

again with $r=m_{\phi}^{2}/m_{E}^{2}$. For $m_{E}=m_{\phi}$, the limit of this
function is $f(r)\big{|}_{r\to 1}=\frac{1}{2}$.

Figure 2: Diagrams at one-loop in the UV model which generate the $d=5$ dipole
operator $\mathcal{O}_{\scriptscriptstyle NNB}^{(5)}$. The black square
indicates an insertion of the vector-like lepton mass $m_{E}$. At low
energies, these induce electromagnetic sterile-to-sterile dipole moments
$\mathcal{O}_{\scriptscriptstyle NN\gamma}$. Near the $Z$ pole, sterile-to-
sterile dipole couplings to $Z$, $\mathcal{O}_{\scriptscriptstyle NNZ}$, are
also induced.

The amplitudes in Eqs. (45) and (48) can now be equated to find

$\displaystyle C_{\scriptscriptstyle
NNB}^{(5)ij}=\frac{g^{\prime}(h_{i}^{\prime}h_{j}^{*}-h_{i}^{*}h_{j}^{\prime})}{64\pi^{2}m_{E}}f(r)\,,$
(50)

which is in agreement with the result of Aparici:2009oua . We see that the
flavour structure is determined entirely by the combination
$h_{i}^{\prime}h_{j}^{*}-h_{i}^{*}h_{j}^{\prime}$, with the diagonal elements
vanishing, as expected. In the broken phase of the SM, but at energies where
the $Z$ boson is not yet integrated out, ${C}_{\scriptscriptstyle NNB}^{(5)}$
results in the $\gamma$ and $Z$ couplings ($d_{\scriptscriptstyle{NN\gamma}}$
and $d_{\scriptscriptstyle{NNZ}}$) according to Eq. (10). For
$m_{E}=m_{\phi}$, Eqs. (50) and (10) give dipole couplings of size,

$\displaystyle d_{\scriptscriptstyle NN\gamma}^{ij}=c_{w}C_{\scriptscriptstyle
NNB}^{(5)ij}\approx 2.4\times
10^{-6}~{}\text{GeV}^{-1}\bigg{(}\frac{h_{i}^{\prime}h_{j}^{*}-h_{i}^{*}h_{j}^{\prime}}{10}\bigg{)}\bigg{(}\frac{1~{}\text{TeV}}{m_{E}}\bigg{)}\,.$
(51)

Thus, even with values of the couplings at the perturbative limit,
$h,h^{\prime}\lesssim\sqrt{4\pi}$, values of the dipole coupling can only be
obtained up to $d_{\scriptscriptstyle NN\gamma}^{ij}\sim
10^{-4}~{}\text{GeV}^{-1}$ with $m_{E}$ and $m_{\phi}$ just above the current
lower limits from collider searches, $m_{E},m_{\phi}\gtrsim 200$ GeV (see
Section 4 for further details).

Examining Eq. (44), we observe that the one-loop correction to $[M_{R}]_{ij}$
depends on the same couplings entering $C_{\scriptscriptstyle NNB}^{(5)ij}$ in
Eq. (50). Therefore, to obtain large sterile-to-sterile neutrino magnetic
moments for sterile states in the 10 GeV to 1 TeV range, some fine-tuning
between the tree-level and one-loop contributions to $[M_{R}]_{ij}$ may be
required.

###  3.3 Active-to-Sterile Neutrino Magnetic Moments

Figure 3: Diagrams at one-loop in the UV model which generate the $d=6$ dipole
operators $\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$ and
$\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$. At low energies, these induce the
electromagnetic active-to-sterile dipole moment
$\mathcal{O}_{\scriptscriptstyle\nu N\gamma}$. Additionally, at energies near
the EW scale, the dipole moments $\mathcal{O}_{\scriptscriptstyle\ell NW}$ and
$\mathcal{O}_{\scriptscriptstyle\nu NZ}$ are generated. Note that there are
additional diagrams with $E_{R}^{c}\to\ell^{c}_{R\rho}$, $h_{i}^{\prime}\to
f_{i\rho}^{\prime}$ and $Y_{E}^{\beta*}\to[Y_{e}]_{\beta\rho}^{*}$ and others
with an internal Higgs line and an insertion of $[Y_{\nu}]_{\alpha i}$, which
also contribute to Eq. (53).

The $d=6$ operators $\mathcal{O}_{\scriptscriptstyle NB}^{(6)}$ and
$\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$, which induce active-to-sterile
neutrino transition magnetic moments $\mathcal{O}_{\scriptscriptstyle\nu
N\gamma}$ and $\mathcal{O}_{\scriptscriptstyle\nu NZ}$, as well as the
charged-current dipole $\mathcal{O}_{\scriptscriptstyle\ell NW}$, are also
generated in the UV model at one-loop, as depicted in Fig. 3. These diagrams
do not require mixing between the active and sterile neutrinos and thus avoid
mixing suppression. However, they do rely on the Yuwawa coupling $Y_{E}$ of
the lepton doublet $L$ with vector-like lepton $E_{R}$ and Higgs doublet $H$.
In the following, we use the same procedure as the previous section to find
the matching between $C_{\scriptscriptstyle NB}^{(6)}$ and
$C_{\scriptscriptstyle NW}^{(6)}$ and the parameters of the UV model.

In the $N_{R}$SMEFT, the amplitude $\braket{\nu_{\alpha}N_{i}Bh}$ is given by

$\displaystyle i\mathcal{M}_{\alpha
i}^{\text{EFT}}=\sqrt{2}C_{\scriptscriptstyle NB}^{(6)\alpha
i}\bar{u}(p_{\nu_{\alpha}})\sigma_{\mu\nu}p_{B}^{\nu}P_{R}u(p_{N_{i}})\epsilon^{\mu*}(p_{B})\,,$
(52)

where $p_{B}=p_{N_{i}}-p_{\nu_{\alpha}}$. In the UV model, the amplitude
$\braket{\nu_{\alpha}N_{i}Bh}$ is determined in part by the three one-loop
diagrams shown in Fig. 3. It is also possible to draw one-loop diagrams
analogous to those in Fig. 3, but with $E_{R}^{c}\to\ell^{c}_{R\rho}$,
$h_{i}^{\prime}\to f_{i\rho}^{\prime}$ and
$Y_{E}^{\beta*}\to[Y_{e}]_{\beta\rho}^{*}$, and another with an internal Higgs
line proportional to $Y_{E}^{\alpha}Y_{E}^{\beta*}[Y_{\nu}]_{\beta i}$.
Including all of these contributions, the UV amplitude
$\braket{\nu_{\alpha}N_{i}Bh}$ can be calculated and compared with the EFT
amplitude, as enacted in the previous section. This gives the matching
condition,

$\displaystyle C_{\scriptscriptstyle NB}^{(6)\alpha
i}=\frac{g^{\prime}}{64\pi^{2}m_{E}^{2}}\left[3f_{\alpha\beta}Y_{E}^{\beta*}h_{i}^{\prime}f(r)-\frac{f_{\alpha\beta}[Y_{e}]_{\beta\rho}^{*}f_{i\rho}^{\prime}}{r}\bigg{(}\frac{5}{2}+3\log\frac{\mu^{2}}{m_{\phi}^{2}}\bigg{)}+Y_{E}^{\alpha}Y_{E}^{\beta*}[Y_{\nu}]_{\beta
i}\right]\,,$ (53)

where the flavour indices $\beta$ and $\rho$ are summed over and the loop
function $f(r)$ is the same as in Eq. (49), with $r=m_{\phi}^{2}/m_{E}^{2}$.

As seen in the diagram to the right of Fig. 3, the intermediate charged lepton
can also couple to $W_{\mu}^{3}$, which generates the operator
$\mathcal{O}_{\scriptscriptstyle NW}^{(6)}$. The amplitude for
$\braket{\nu_{\alpha}N_{i}W^{3}h}$ from this operator is

$\displaystyle i\mathcal{M}_{\alpha
i}^{\text{EFT}}=\frac{C_{\scriptscriptstyle NW}^{(6)\alpha
i}}{\sqrt{2}}\bar{u}(p_{\nu_{\alpha}})\sigma_{\mu\nu}p_{W}^{\nu}P_{R}u(p_{N_{i}})\epsilon^{\mu*}(p_{W})\,,$
(54)

for $p_{W}=p_{N_{i}}-p_{\nu_{\alpha}}$. The UV amplitude
$\braket{\nu_{\alpha}N_{i}W^{3}h}$ can be calculated from the diagrams
entering $\braket{\nu_{\alpha}N_{i}Bh}$ where it is possible to replace $B\to
W^{3}$. The resulting matching condition is

$\displaystyle C_{\scriptscriptstyle NW}^{(6)\alpha
i}=\frac{g}{32\pi^{2}m_{E}^{2}}\left[f_{\alpha\beta}Y_{E}^{\beta*}h_{i}^{\prime}f(r)-\frac{f_{\alpha\beta}[Y_{e}]_{\beta\rho}^{*}f_{i\rho}^{\prime}}{r}\bigg{(}\frac{3}{2}+\log\frac{\mu^{2}}{m_{\phi}^{2}}\bigg{)}+Y_{E}^{\alpha}Y_{E}^{\beta*}[Y_{\nu}]_{\beta
i}\right]\,.$ (55)

From Eqs. (53) and (55) we see that, in principle, there are multiple
contributions to these operators. To simplify this dependence on the UV model
parameters, we note that we are only interested in the scenario where the
coefficient $C_{\scriptscriptstyle NNB}^{(5)ij}$ is sizeable, requiring large
$h^{\prime}$. Then, if we assume that the first terms in Eqs. (53) and (55)
dominate over the others, we obtain the particularly simple relation between
the coefficients

$\displaystyle C_{\scriptscriptstyle NW}^{(6)\alpha
i}=\frac{2}{3t_{w}}C_{\scriptscriptstyle NB}^{(6)\alpha i}\approx
1.22\,C_{\scriptscriptstyle NB}^{(6)\alpha i}\,.$ (56)

Thus, as discussed in Section 2, the UV model sets the parameter $a=2/3$. Now,
using Eqs. (10) and Eq. (55), we find the corresponding active-to-sterile
dipole couplings in the broken phase to be

$\displaystyle d_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}=\frac{4vc_{w}}{3\sqrt{2}}C_{\scriptscriptstyle NB}^{(6)\alpha i}\approx
1.7\times
10^{-9}~{}\text{GeV}^{-1}\bigg{(}\frac{f_{\alpha\beta}Y_{E}^{\beta*}h_{i}^{\prime}}{10^{-2}}\bigg{)}\bigg{(}\frac{1~{}\text{TeV}}{m_{E}}\bigg{)}^{2}\,.$
(57)

Using Eqs. (12) and $a=2/3$, we also find the ratio $d_{\scriptscriptstyle\nu
NZ}^{\alpha i}/d_{\scriptscriptstyle\nu N\gamma}^{\alpha
i}=(1-3t_{w}^{2})/(4t_{w})\approx 4.5\times 10^{-2}$. The active-to-sterile
dipole coupling with $Z$ is therefore relatively suppressed with respect to
$d_{\scriptscriptstyle\nu N\gamma}^{\alpha i}$ in this scenario. Finally, we
note that the flavour structure of the couplings $d^{\alpha
i}_{\scriptscriptstyle\nu N\gamma}$ and $d^{\alpha i}_{\scriptscriptstyle\nu
NZ}$ exhibits an interesting dependence on the parameters of the UV model. In
the case of flavour universal couplings, at least one of the entries of
$d^{\alpha i}_{\scriptscriptstyle\nu N\gamma}$ and $d^{\alpha
i}_{\scriptscriptstyle\nu NZ}$ for $\alpha\in\\{e,\mu,\tau\\}$ must vanish,
with the other two being equal and opposite in sign; for the choice
$Y_{E}^{e}=Y_{E}^{\mu}=Y_{E}^{\tau}$ and $f_{e\mu}=f_{e\tau}=f_{\mu\tau}$, we
obtain $d^{\mu i}_{\scriptscriptstyle\nu N\gamma}=d^{\mu
i}_{\scriptscriptstyle\nu NZ}=0$, for example. In the scenario where only the
$\mu$-$\tau$ couplings are non-zero, for example $Y_{E}^{\mu}=Y_{E}^{\tau}$
and $f_{\mu\tau}$, we would instead have $d^{ei}_{\scriptscriptstyle\nu
N\gamma}=d^{ei}_{\scriptscriptstyle\nu NZ}=0$, $d^{\mu
i}_{\scriptscriptstyle\nu N\gamma}=-d^{\tau i}_{\scriptscriptstyle\nu
N\gamma}$ and $d^{\mu i}_{\scriptscriptstyle\nu NZ}=-d^{\tau
i}_{\scriptscriptstyle\nu NZ}$. Finally, if only couplings involving $\tau$
are present (e.g., $Y_{E}^{\tau}$, $f_{e\tau}=f_{\mu\tau}$), we would obtain
$d^{\tau i}_{\scriptscriptstyle\nu N\gamma}=d^{\tau i}_{\scriptscriptstyle\nu
NZ}=0$, $d^{ei}_{\scriptscriptstyle\nu N\gamma}=d^{\mu
i}_{\scriptscriptstyle\nu N\gamma}$ and $d^{ei}_{\scriptscriptstyle\nu
NZ}=d^{\mu i}_{\scriptscriptstyle\nu NZ}$. These particular limits can be
readily translated to other flavour combinations.

###  3.4 Active-to-Active Neutrino Magnetic Moments

Figure 4: Diagrams at two loops in the UV model which contribute negligibly to
the $d=7$ dipole operators $\mathcal{O}_{\scriptscriptstyle LHB}^{(7)}$ and
$\mathcal{O}_{\scriptscriptstyle LHW}^{(7)}$. The crossed circle in the left
diagram refers to the sterile-to-sterile one-loop diagrams in Fig. 2, while
the circle in the other two diagrams correspond to the active-to-sterile
diagrams in Fig. 3.

Here, we finally comment that active-to-active neutrino magnetic moments are
naturally suppressed in the UV model. With the fields and couplings available
in Eq. (3) and assuming the RH neutrinos satisfy $m_{N_{i}}\ll
m_{E},m_{\phi}$, the lowest-order 1LPI amplitudes that contribute to the $d=7$
SMEFT operators $C_{\scriptscriptstyle LHB}^{(7)\alpha\beta}$ and
$C_{\scriptscriptstyle LHW}^{(7)\alpha\beta}$ are at two loops, as shown in
Fig. 4.

However, in the broken phase, and at energies where the RH neutrinos may also
be integrated out, active-to-active neutrino magnetic moments are induced via
the active-sterile mixing $V_{\ell N}$, i.e.,

$\displaystyle d_{\scriptscriptstyle\nu\nu\gamma}^{\alpha\beta}$
$\displaystyle=V_{\alpha N_{i}}^{*}V_{\beta
N_{j}}^{*}d^{ij}_{\scriptscriptstyle NN\gamma}+\Big{[}V_{\alpha
N_{i}}^{*}d_{\scriptscriptstyle\nu N\gamma}^{\beta i}-V_{\beta
N_{i}}^{*}d_{\scriptscriptstyle\nu N\gamma}^{\alpha i}\Big{]}$
$\displaystyle\approx
10^{-16}~{}\text{GeV}^{-1}\bigg{[}\bigg{(}\frac{V_{\alpha N_{i}}^{*}V_{\beta
N_{j}}^{*}}{10^{-10}}\bigg{)}\bigg{(}\frac{d_{\scriptscriptstyle
NN\gamma}^{ij}}{10^{-6}~{}\text{GeV}^{-1}}\bigg{)}+\bigg{(}\frac{V_{\alpha
N_{i}}^{*}}{10^{-5}}\bigg{)}\bigg{(}\frac{d_{\scriptscriptstyle\nu
N\gamma}^{\beta i}}{10^{-11}~{}\text{GeV}^{-1}}\bigg{)}\bigg{]}\,,$ (58)

where in the second line we have assumed that $\alpha\neq\beta$ and $d^{\alpha
i}_{\scriptscriptstyle\nu N\gamma}=0$. For active-sterile mixing of the size
predicted by the Type-I seesaw mechanism, $V_{\ell
N}\sim\sqrt{m_{\nu}/m_{N}}\sim 10^{-6}$, this contribution is highly
suppressed both by the mixing and the one-loop suppressed
$d_{\scriptscriptstyle NN\gamma}^{\alpha i}$ and $d_{\scriptscriptstyle\nu
N\gamma}^{\alpha i}$. Thus, even with the larger mixing $V_{\ell N}$ in the
inverse seesaw scenario, we can still expect the active-to-active neutrino
magnetic moments to safely satisfy the bounds from TEXONO TEXONO:2006xds ,
GEMMA Beda:2007hf , LSND LSND:2001akn  and Borexino Borexino:2008dzn ,
$d_{\scriptscriptstyle\nu\nu\gamma}^{\alpha\beta}\lesssim 2\times
10^{-8}~{}\text{GeV}^{-1}$.

##  4 Phenomenological Constraints on the UV Model

In this section, we review the constraints on the UV-complete extension of the
SM outlined in Section 3. We begin by examining the direct production of $E$
and $\phi$ in collider experiments. We then consider rare low-energy processes
which are not present or highly-suppressed in the SM, but could be enhanced by
non-diagonal flavour couplings after integrating out $E$ and $\phi$.

###  4.1 Constraints from Direct Collider Searches

A recasting of direct searches for selectrons and smuons at the LHC provides
useful constraints on the singly-charged scalar $\phi$ in our UV scenario. The
dominant limit comes from the Drell-Yan pair production process
$pp\to\gamma/Z\to\phi^{+}\phi^{-}$ followed by the decay
$\phi^{\pm}\rightarrow\ell^{\pm}\nu$. In Cao:2017ffm ; Alcaide:2017dcx ;
Alcaide:2019kdr ; Crivellin:2020klg , the most recent ATLAS search
ATLAS:2019lff  for oppositely-charged electron and muon pairs with 139
$\text{fb}^{-1}$ of collected data was recast for the singly-charged scalar
scenario. The ATLAS search places a lower bound on the slepton masses of
approximately 450 GeV for a 100% branching ratio of the slepton in the
specific channel. The major difference between this analysis and the pair
production of $\phi^{+}\phi^{-}$ is the cross-section in the two scenarios. In
Crivellin:2020klg , a simple scaling factor was used to map the leading-order
Madgraph-simulated production cross-section for $pp\to\phi^{+}\phi^{-}$ onto
the production cross-section given by ATLAS for the RH slepton pair. Allowing
for further uncertainties, a conservative lower limit of $m_{\phi}\gtrsim 200$
GeV was found. In addition, monophoton searches for dark matter at LEP
DELPHI:2003dlq ; Fox:2011fx  can also be recast to obtain the limit
$m_{\phi}/|f_{e\mu}|\gtrsim 350~{}\text{GeV}$ Crivellin:2020klg .

In general, $SU(2)_{L}$ singlet vector-like leptons $E$ are subject to much
weaker bounds compared to their $SU(2)_{L}$ doublet counterparts, due to their
relatively smaller production cross-sections. Initially, searches at LEP
L3:2001xsz  were able to exclude the masses $m_{E}<101.2$ GeV. More recently,
the ATLAS collaboration has performed a search for heavy charged leptons
decaying to a $Z$ boson and an electron or muon, excluding the mass range
129–176 GeV (114–168 GeV) for mixing with only electrons (muons), except for
the interval 144–163 GeV (153–160 GeV) ATLAS:2015qoy . In Kumar:2015tna , the
prospects of probing singlet vector-like leptons with multi-lepton searches at
the LHC were explored, with some reach for exclusion at the HL-LHC. In our
specific model, the pair-produced vector-like leptons can also decay to a RH
neutrino and the singly-charged scalar, $E^{\pm}\to N_{i}\phi^{\pm}$. The
subsequent decay $\phi^{\pm}\rightarrow\ell^{\pm}\nu$ therefore leads to the
signature of two oppositely charged leptons plus missing energy. This is a
signature similar to that from $\phi^{+}\phi^{-}$ pair production and is
expected to yield a similar constraint, $m_{E}\gtrsim 200$ GeV.

Finally, EW precision data from LEP, complemented by the measurement of the
Higgs mass, have been used to perform a global fit for the SM plus heavy NP
effects, parametrised by $d=6$ SMEFT operators delAguila:2008pw ;
deBlas:2013gla ; deBlas:2014mba . From this fit, flavour-dependent upper
bounds can be placed on the mixing between the charged leptons and $E$, which
alternatively can be written as the lower limits $m_{E}/|Y_{E}^{e}|>8.3$ TeV,
$m_{E}/|Y_{E}^{\mu}|>5.8$ TeV and $m_{E}/|Y_{E}^{\tau}|>5.3$ TeV (95% C.L.).
The singly-charged scalar is similarly constrained as
$m_{\phi}/|f_{e\mu}|>12.5$ TeV (95% C.L.).

###  4.2 Constraints from Charged Lepton Flavour Violation

Figure 5: Diagrams at tree-level in the UV extension which result in
violations of LFU and non-vanishing rates for the cLFV processes
$\ell_{\alpha}\to\ell_{\beta}\ell\bar{\ell}$ and $\tau\to\ell_{\beta}X$, where
$X$ is a light meson.

As mentioned in Section 3, charged lepton flavour violating (cLFV), lepton
flavour universality (LFU) violating observables and precision tests of
observables such as $(g-2)_{\mu}$ can be used to constrain the parameter space
of the UV model Falkowski:2013jya ; Crivellin:2017rmk ; Poh:2017tfo ;
Crivellin:2020klg ; Crivellin:2020ebi ; CarcamoHernandez:2021yev ;
Felkl:2021qdn ; Hamaguchi:2022byw ; Altmannshofer:2022fvz . In the presence of
the Yukawa couplings $Y_{E}$, mixing is induced between the SM charged leptons
and the vector-like lepton $E$, modifying the SM charged-current, neutral-
current and Higgs interactions at low energies, as seen in Eqs. (3) and
(24).444The Yukawa coupling $Y_{\nu}$, which induces the active-sterile mixing
$V_{\ell N}$, also modifies the charged- and neutral-current interactions and
therefore contributes to cLFV processes Alonso:2012ji . For simplicity, we
assume that $Y_{\nu}\ll 1$ for the purposes of deriving bounds on the UV
model.

Equivalently, from the EFT perspective, the operators
$\mathcal{O}_{Hl}^{(1)}$, $\mathcal{O}_{Hl}^{(3)}$ and $\mathcal{O}_{eH}$ are
induced at tree-level after integrating out $E$. Likewise, the operator
$\mathcal{O}_{ll}$ is generated after integrating out $\phi$. SM processes
such as $\ell_{\alpha}\to\ell_{\beta}\nu\bar{\nu}$, $\pi\to\ell_{\alpha}\nu$,
$\tau\to\pi\nu$ and $B\to D^{(*)}\ell_{\alpha}\nu$ are modified in the
presence of these operators, while cLFV processes such as
$\ell_{\alpha}\to\ell_{\beta}\gamma$,
$\ell_{\alpha}\to\ell_{\beta}\ell\bar{\ell}$, $\mu-e$ conversion in nuclei,
$Z\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$ and
$h\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$ are induced. The
$\ell_{\alpha}\to\ell_{\beta}\gamma$ process additionally receives
contributions directly from the dipole operators $\mathcal{O}_{eB}$ and
$\mathcal{O}_{eW}$, both of which are generated after integrating out $E$ and
$\phi$ at one-loop. In the following, we assume that $E$ and $\phi$ are
integrated out at a scale well above the low energies of the considered
processes. Thus, one should consider the running of the operators from the
high scale down to the EW scale, match to $SU(3)_{c}\times U(1)_{Y}$ invariant
operators, and then run down to the low scale. However, we neglect the sub-
leading effects of running as we only aim to obtain bounds on the model for
comparison with those from LLP searches in Section 5.

For the bounds from LFU violation, we consider here as an example only the
purely leptonic LFU ratios, which measure deviations from the SM predictions
for $\ell_{\alpha}\to\ell_{\beta}\nu\bar{\nu}$. In the UV model, we obtain the
ratio Crivellin:2020ebi ; Allwicher:2021ndi

$\displaystyle\frac{\Gamma(\ell_{\alpha}\to\ell_{\beta}\nu\bar{\nu})}{\Gamma(\ell_{\alpha}\to\ell_{\beta}\nu\bar{\nu})\big{|}_{\text{SM}}}=1+v^{2}\,\bigg{[}\frac{|Y_{E}^{\alpha}|^{2}+|Y_{E}^{\beta}|^{2}}{2m_{E}^{2}}+\frac{2|f_{\alpha\beta}|^{2}}{m_{\phi}^{2}}\bigg{]}\,,$
(59)

where we only take into account the interference between the SM and the NP
contributions. The first and second terms in the square brackets of Eq. (59)
arise from the exchange of $W$ and $\phi$, respectively, as seen in the left-
and right-most diagrams of Fig. 5. The $Z$ and $h$ exchange diagrams do not
contribute at this level, because there are no FCNCs at tree-level in the SM.
This ratio can be used to determine the ratio of couplings
$|g_{\alpha}/g_{\beta}|$, obtained as

$\displaystyle\left|\frac{g_{\alpha}}{g_{\beta}}\right|\approx
1+v^{2}\,\bigg{[}\frac{|Y_{E}^{\alpha}|^{2}-|Y_{E}^{\beta}|^{2}}{4m_{E}^{2}}+\frac{|f_{\alpha\rho}|^{2}-|f_{\beta\rho}|^{2}}{m_{\phi}^{2}}\bigg{]}\,.$
(60)

From Eq. (60), it is clear that LFU violation vanishes for flavour universal
couplings, e.g., the choice $Y_{E}^{e}=Y_{E}^{\mu}=Y_{E}^{\tau}$ and
$f_{e\mu}=f_{e\tau}=f_{\mu\tau}$. Current experimental values of these
coupling ratios are $|g_{\tau}/g_{\mu}|=1.0009(14)$,
$|g_{\tau}/g_{e}|=1.0027(14)$, $|g_{\mu}/g_{e}|=1.0019(14)$ Pich:2013lsa ;
HFLAV:2022esi . From the first of these results, taking only $Y_{E}^{\tau}$ to
be non-zero gives $m_{E}/|Y_{E}^{\tau}|>4.1~{}\text{TeV}$.

The tree-level $Z$ and Higgs exchange diagrams in Fig. 5 induce the cLFV
processes $\ell_{\alpha}\to\ell_{\beta}\ell\bar{\ell}$, which are subject to
stringent constraints from SINDRUM SINDRUM:1987nra  and Belle Hayasaka:2010np
. The branching ratio for this general process, neglecting final-state lepton
masses, is Kuno:1999jp ; Crivellin:2013hpa

$\displaystyle\mathcal{B}(\ell_{\alpha}\to\ell_{\beta}\ell_{\gamma}\bar{\ell}_{\delta})$
$\displaystyle\approx\frac{m_{\alpha}^{5}}{768\pi^{3}\Gamma_{\alpha}(1+\delta_{\beta\gamma})}\frac{|Y_{E}^{\alpha*}Y_{E}^{\beta}|^{2}}{m_{E}^{4}}\left[(1+\delta_{\beta\gamma})(g_{L}^{\ell})^{2}+(g_{R}^{\ell})^{2}\right]\,.$
(61)

We assume that the dominant contribution to Eq. (61) comes from the $Z$
exchange diagrams, or correspondingly the operators $\mathcal{O}_{Hl}^{(1)}$
and $\mathcal{O}_{Hl}^{(3)}$. The contribution of the Higgs exchange diagram,
or the operator $\mathcal{O}_{eH}$, can be safely neglected as it is further
suppressed by the lepton masses, as seen in Eqs. (23) and (24). In Eq. (61),
we also neglect the contributions from $E$ and $\phi$ at one-loop via penguin
and box diagrams, or equivalently via the operators $\mathcal{O}_{eB}$,
$\mathcal{O}_{eW}$ and $\mathcal{O}_{ll}$ Crivellin:2020klg . The one-loop
contribution of $\phi$ will only be relevant if the Yukawa couplings inducing
$\ell_{\alpha}\to\ell_{\beta}\ell\bar{\ell}$ at tree-level satisfy
$Y_{E}^{\alpha},Y_{E}^{\beta}\ll f_{\alpha\beta}$. If we assume that the tree-
level contribution dominates, the branching ratio in Eq. (61) and the
corresponding experimental upper limits can be used to place lower bounds on
the combination $m_{E}/|Y_{E}^{\alpha*}Y_{E}^{\beta}|^{1/2}$ for
$\alpha\neq\beta$. The SINDRUM experiment provides the upper limit
$\mathcal{B}(\mu^{+}\to e^{+}e^{+}e^{-})<1\times 10^{-12}$ (90% C.L.), giving
$m_{E}/|Y_{E}^{\mu*}Y_{E}^{e}|^{1/2}>120~{}\text{TeV}$. Likewise, the upper
limits $\mathcal{B}(\tau^{-}\to e^{-}e^{-}e^{+})<2.7\times 10^{-8}$ and
$\mathcal{B}(\tau^{-}\to\mu^{-}\mu^{-}\mu^{+})<2.1\times 10^{-8}$ from Belle
translate to $m_{E}/|Y_{E}^{\tau*}Y_{E}^{e}|^{1/2}>5.9~{}\text{TeV}$ and
$m_{E}/|Y_{E}^{\tau*}Y_{E}^{\mu}|^{1/2}>6.3~{}\text{TeV}$, respectively. The
equivalent constraints from $\mathcal{B}(\tau^{-}\to e^{-}\mu^{-}\mu^{+})$ and
$\mathcal{B}(\tau^{-}\to\mu^{-}e^{-}e^{+})$ are comparable. We note that the
cLFV processes $\ell_{\alpha}\to\ell_{\beta}q\bar{q}$, i.e., hadronic $\tau$
decays $\tau\to\ell_{\beta}X$, where $X$ is a light pseudoscalar, scalar or
vector meson, are also generated. Constraints on such decay modes from Belle
and BaBar are comparible to those on $\tau\to\ell_{\beta}\ell\bar{\ell}$
Banerjee:2022xuw .

In the presence of the flavour-changing $Z$ couplings at tree-level, the
exotic process of muon conversion to electrons in nuclei can also occur. The
rate for this process, divided by the total capture rate
$\Gamma_{\text{capt}}$, is Kitano:2002mt ; Davidson:2018kud

$\displaystyle\text{CR}(\mu\to e)$
$\displaystyle=\frac{m_{\mu}^{5}}{\Gamma_{\text{capt}}}\frac{\big{|}Y_{E}^{\mu*}Y_{E}^{e}\big{|}^{2}}{m_{E}^{4}}\bigg{|}(g_{L}^{u}+g_{R}^{u})(2V^{(p)}+V^{(n)})+(g_{L}^{d}+g_{R}^{d})(V^{(p)}+2V^{(n)})\bigg{|}^{2}\,,$
(62)

where $V^{(p)}$ and $V^{(n)}$ are nucleus-dependent overlap integrals over the
proton and neutron densities and the muon and electron wavefunctions
Kitano:2002mt . In Eq. (62), we again assume that the dominant contribution
arises at tree-level via the operators $\mathcal{O}_{Hl}^{(1)}$ and
$\mathcal{O}_{Hl}^{(3)}$, neglecting the contributions of $E$ and $\phi$ at
one-loop via penguin diagrams. The SINDRUM experiment has placed an upper
bound on the $\mu-e$ conversion rate on ${}^{197}_{~{}79}$Au of
$\text{CR}(\mu\to e)<7\times 10^{-13}$ (90% C.L.) SINDRUMII:2006dvw , which
results in the constraint
$m_{E}/|Y_{E}^{\mu*}Y_{E}^{e}|^{1/2}>290~{}\text{TeV}$.

Next, the radiative cLFV process $\ell_{\alpha}\to\ell_{\beta}\gamma$ is
generated at one-loop, as depicted in Fig. 6. The corresponding branching
ratio is given by Crivellin:2013hpa ; Pruna:2014asa ; Davidson:2016edt

$\displaystyle\mathcal{B}(\ell_{\alpha}\to\ell_{\beta}\gamma)$
$\displaystyle\approx\frac{m_{\alpha}^{5}}{512\pi^{3}\Gamma_{\alpha}}\frac{\alpha}{2\pi}\Bigg{[}\bigg{|}\frac{Y_{E}^{\alpha*}Y_{E}^{\beta}}{4m_{E}^{2}}\bigg{(}1+\frac{8}{3}g_{L}^{\ell}\bigg{)}+\frac{f_{\rho\alpha}^{*}f_{\rho\beta}}{3m_{\phi}^{2}}\bigg{|}^{2}+\bigg{|}\frac{f_{i\alpha}^{\prime}f_{i\beta}^{\prime*}}{12m_{\phi}^{2}}\bigg{|}^{2}\Bigg{]}\,,$
(63)

where the indices $\rho$ and $i$ are summed over. The term in Eq. (63)
proportional to $Y_{E}^{\alpha*}Y_{E}^{\beta}$ originates from the operators
$\mathcal{O}_{Hl}^{(1)}$, $\mathcal{O}_{Hl}^{(3)}$, $\mathcal{O}_{eB}$ and
$\mathcal{O}_{eW}$, generated after integrating out $E$, while the terms
proportional to $f_{\rho\alpha}^{*}f_{\rho\beta}$ and
$f_{i\alpha}^{\prime}f_{i\beta}^{\prime*}$ arise from the contributions of
$\phi$ to $\mathcal{O}_{eB}$ and $\mathcal{O}_{eW}$. Contributions of the
operator $\mathcal{O}_{eH}$ via two-loop Barr-Zee type diagrams are neglected,
as $C_{eH}$ is suppressed by the lepton masses. If we assume only non-zero
$Y_{E}^{\alpha}$ values, the current upper limit from the MEG experiment
MEG:2016leq , $\mathcal{B}(\mu\to e\gamma)<4.2\times 10^{-13}$ (90% C.L.),
corresponds to the constraint
$m_{E}/|Y_{E}^{\mu*}Y_{E}^{e}|^{1/2}>75~{}\text{TeV}$. For $\tau$ decays, the
BaBar experiment gives the upper limits $\mathcal{B}(\tau\to
e\gamma)<3.3\times 10^{-8}$ and $\mathcal{B}(\tau\to\mu\gamma)<4.4\times
10^{-8}$ (90% C.L.) BaBar:2009hkt , enforcing
$m_{E}/|Y_{E}^{\tau*}Y_{E}^{e}|^{1/2}>2.9~{}\text{TeV}$ and
$m_{E}/|Y_{E}^{\tau*}Y_{E}^{\mu}|^{1/2}>2.7~{}\text{TeV}$, respectively.

Figure 6: Diagrams at one-loop in the UV model which induce the cLFV process
$\ell_{\alpha}\to\ell_{\beta}\gamma$ and contribute to the anomalous magnetic
moment of charged leptons, $a_{\ell}$. From left to right, the first three
diagrams are the result of the modified $W$, $Z$ and Higgs interactions in
Eqs. (3) and (23), respectively, while the fourth diagram is induced by the
$-\bar{L}f\tilde{L}\phi$ and $-\bar{N}^{c}_{R}f^{\prime}\ell_{R}\phi^{*}$
interactions. The first and last diagrams are further modified in the presence
of the active-sterile mixing $V_{\ell N}$.

Finally, limits can be placed from the cLFV decays
$Z\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$ and
$h\to\ell_{\alpha}^{+}\ell_{\beta}^{-}$. With non-zero values of
$Y_{E}^{\alpha}$, the branching ratios of these are Harnik:2012pb ;
Blankenburg:2012ex ; Falkowski:2013jya ; Altmannshofer:2022fvz

$\displaystyle\mathcal{B}(Z\to\ell_{\alpha}^{+}\ell_{\beta}^{-})$
$\displaystyle=\frac{m_{Z}}{6\pi\Gamma_{Z}}\frac{m_{Z}^{2}}{v^{2}}\left[\left(\big{(}g_{L}^{\ell}\big{)}^{2}+\big{(}g_{R}^{\ell}\big{)}^{2}+g_{L}^{\ell}\frac{v^{2}|Y_{E}^{\alpha}|^{2}}{2m_{E}^{2}}\right)\delta_{\alpha\beta}+\frac{v^{4}|Y_{E}^{\alpha*}Y_{E}^{\beta}|^{2}}{16m_{E}^{4}}\right]\,,$
(64) $\displaystyle\mathcal{B}(h\to\ell_{\alpha}^{+}\ell_{\beta}^{-})$
$\displaystyle=\frac{m_{h}}{8\pi\Gamma_{h}}\frac{m_{\alpha}^{2}}{v^{2}}\left[\left(1-\frac{3v^{2}|Y_{E}^{\alpha}|^{2}}{2m_{E}^{2}}\right)\delta_{\alpha\beta}+\frac{9v^{4}|Y_{E}^{\alpha*}Y_{E}^{\beta}|^{2}}{16m_{E}^{4}}\right]\,.$
(65)

We note that $\mathcal{B}(Z\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp})$ and
$\mathcal{B}(h\to\ell_{\alpha}^{\pm}\ell_{\beta}^{\mp})$ are found by adding
the branching ratios with $\alpha\leftrightarrow\beta$ to those above. Upper
bounds on these decays have been placed by OPAL OPAL:1995grn , ATLAS
ATLAS:2014vur ; ATLAS:2019pmk  and CMS CMS:2021rsq . Taking the most
stringent, $\mathcal{B}(Z\to e^{\pm}\mu^{\mp})<4.2\times 10^{-7}$ at 90% C.L.
ATLAS:2014vur , gives $m_{E}/|Y_{E}^{\mu*}Y_{E}^{e}|^{1/2}>4.1~{}\text{TeV}$.

To conclude this section, we observe that
$\ell_{\alpha}\to\ell_{\beta}\ell\bar{\ell}$ and $\mu-e$ conversion in nuclei
provide the most stringent constraints on the UV model, with the latter
probing values of $m_{E}$ up to 290 TeV for $\mathcal{O}(1)$ Yukawa couplings
$Y_{E}$. The coupling $Y_{E}^{\alpha}$ enters the active-to-sterile dipole
couplings, as seen in Eq. (53) and (55), and therefore these constraints must
be taken into account when considering non-zero $d_{\scriptscriptstyle\nu
N\gamma}^{\alpha i}$. However, we note that these constraints can be easily
evaded if the couplings of $E$ and $\phi$ are not flavour universal. For
example, one can consider only non-zero couplings in the $\mu$-$\tau$
($e$-$\tau$) sector; for $Y_{E}^{e}=0$ ($Y_{E}^{\mu}=0$), the strong bounds
from $\mu\to 3e$ and $\mu-e$ conversion are no longer applicable. The only
bounds that then apply are the much weaker limits from LFU violation,
$\tau\to\mu\gamma$ ($\tau\to e\gamma$), and $\tau\to 3\mu$ ($\tau\to 3e$). One
can also consider the case where only one Yukawa coupling is non-zero, e.g.
$Y_{E}^{\tau}$. Then the bounds strongest bounds come from the one-loop
contributions of $\phi$ to $\mu\to e\gamma$, $\mu\to 3e$ and $\mu-e$
conversion.

##  5 Long-Lived Particle Searches at the LHC using Non-Pointing Photons

In this section, we discuss the potential of searches for non-pointing photons
at the LHC to constrain sterile neutrino magnetic moments. To keep our
analysis as general as possible, here we will follow the same conventions as
the EFT notation introduced in Section 2.

###  5.1 Sterile Neutrino Production and Decay Mechanisms

In what follows, we first discuss the sterile neutrino production and decay
processes triggered by the EFT Wilson coefficients
$C_{\scriptscriptstyle{NNB}}^{(5)ij}$ and
$C_{\scriptscriptstyle{NX}}^{(6)\alpha i}$, where $X=B,W$.555Here, we assume
$C_{\scriptscriptstyle{NW}}^{(6)\alpha
i}=2/(3t_{w})C_{\scriptscriptstyle{NB}}^{(6)\alpha i}$, as discussed at the
end of Section 2. We also comment on the contributions from the active-sterile
neutrino mixing.

Hereafter we simplify notation by omitting flavour indices. First, we take
$C_{\scriptscriptstyle{NNB}}^{(5)12}\equiv C_{\scriptscriptstyle{NNB}}^{(5)}$
and since $\mathcal{O}_{\scriptscriptstyle{NNB}}^{(5)}$ is an antisymmetric
operator, we are implicitly assuming
$C_{\scriptscriptstyle{NNB}}^{(5)21}=-C_{\scriptscriptstyle{NNB}}^{(5)12}$.
Second, we also omit the lepton flavour $\alpha$ in
$C_{\scriptscriptstyle{NX}}^{(6)\alpha i}$,
$(C_{\scriptscriptstyle{NX}}^{(6)\alpha i}\equiv
C_{\scriptscriptstyle{N_{i}X}}^{(6)})$, as processes which involve active
neutrinos do not reveal which neutrino flavour is participating. These
simplifications will also apply to the coefficients in the broken phase.

Figure 7: Example cross-sections at the LHC ($\sqrt{s}=14$ TeV). The left plot
shows the contribution of $C_{\scriptscriptstyle NNB}^{(5)}$ to $\sigma(pp\to
N_{1}N_{2})$ as a function of $m_{N_{1}}+m_{N_{2}}$. Dashed lines represent
the partial contributions from the $Z$ and $\gamma$ channels. The right plot
displays the production cross-section of the processes mediated by
$C_{\scriptscriptstyle{NB}}^{(6)}$ and $C_{\scriptscriptstyle{NW}}^{(6)}$. The
cross-section for the charged-current process given by the chosen values of
these couplings is the same as the cross-section for the production of $N_{2}$
via the mixing $|V_{eN_{2}}|^{2}\simeq 3\times 10^{-6}$.

The coupling $C_{\scriptscriptstyle{NNB}}^{(5)}$ contributes to pair-
production of the sterile neutrinos via $pp\rightarrow\gamma/Z\rightarrow
N_{1}N_{2}$, while $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ leads to the
production of a single $N_{i}$ through $pp\rightarrow\gamma/Z\rightarrow
N_{i}\nu$ and $pp\rightarrow W^{\pm}\rightarrow N_{i}\ell^{\pm}$. Fig. 7 shows
some example cross-sections for fixed values of
$C_{\scriptscriptstyle{NNB}}^{(5)}$ (left) and
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ (right). For
$m_{N_{1}}+m_{N_{2}}<m_{Z}$, pair production through
$C_{\scriptscriptstyle{NNB}}^{(5)}$ is dominated by the $Z$ exchange diagram,
whereas for larger masses photon exchange dominates. Single sterile neutrino
production through $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$, on the other hand,
is dominated by the charged current. The neutral channel is subdominant
compared to the charged one in this case because of the suppression between
$d_{\scriptscriptstyle\nu N_{i}\gamma}$ and $d_{\scriptscriptstyle\nu N_{i}Z}$
discussed below Eq. (57). Note that sterile neutrino production at the LHC
through mixing (non-zero $V_{\ell N_{i}}$) proceeds through charged current
events with the same dependence on the mass $m_{N_{i}}$ as for
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$. We therefore do not show production
via mixing separately in Fig. 7 (right).

For sterile neutrino decays, the interaction
$C_{\scriptscriptstyle{NNB}}^{(5)}$ induces the two-body decay
$N_{2}\rightarrow N_{1}\gamma$ and, if $N_{2}$ is heavy enough,
$N_{2}\rightarrow N_{1}Z$. Note that the lightest $N_{1}$ can not decay
through $C_{\scriptscriptstyle{NNB}}^{(5)}$. The relevant partial decay widths
of $N_{2}$, in terms of the broken phase parameters defined in (10), are given
by

$\Gamma(N_{2}\rightarrow N_{1}\gamma)=\frac{2|d_{\scriptscriptstyle
NN\gamma}|^{2}}{\pi}m_{N_{2}}^{3}\bigg{(}1-\frac{m_{N_{1}}^{2}}{m_{N_{2}}^{2}}\bigg{)}^{3}=\frac{2|d_{\scriptscriptstyle
NN\gamma}|^{2}}{\pi}m_{N_{2}}^{3}\left(2-\delta\right)^{3}\delta^{3}\,,$ (66)

with $\delta\equiv 1-\frac{m_{N_{1}}}{m_{N_{2}}}$ denoting the mass splitting
between the two sterile states, and

$\Gamma(N_{2}\rightarrow N_{1}Z)=\frac{|d_{\scriptscriptstyle
NNZ}|^{2}}{\pi}\,m_{N_{2}}^{3}\,f_{Z}\left(\frac{m_{N_{1}}}{m_{N_{2}}},\frac{m_{Z}}{m_{N_{2}}}\right)\,\lambda^{1/2}\bigg{(}1,\frac{m_{N_{1}}^{2}}{m_{N_{2}}^{2}},\frac{m_{Z}^{2}}{m_{N_{2}}^{2}}\bigg{)}\,,$
(67)

with

$\displaystyle f_{Z}(x,y)$ $\displaystyle=$
$\displaystyle\left((1-x)^{2}-y^{2}\right)\left(2(1+x)^{2}+y^{2}\right)\,,$
(68) $\displaystyle\lambda(x,y,z)$ $\displaystyle=$
$\displaystyle(x-y-z)^{2}-4yz\,.$ (69)

We recall that we use the notation $d_{\scriptscriptstyle NN\gamma}^{12}\equiv
d_{\scriptscriptstyle NN\gamma}$ in this section, and because of the
antisymmetric nature of the coupling, there is an additional factor of 4 in
the decay width of $N_{2}\to N_{1}\gamma$ compared to that of
$N_{i}\to\nu\gamma$ (see Eqs. (66) and (70)).

The couplings $C_{\scriptscriptstyle{N_{i}B}}^{(6)}$ and
$C_{\scriptscriptstyle{N_{i}W}}^{(6)}$ generate the decays
$N_{i}\rightarrow\nu\gamma$, $N_{i}\rightarrow\nu Z$ and
$N_{i}\rightarrow\ell^{\pm}W^{\mp}$. While the $\gamma$ final state is
kinematically allowed for practically all values of $m_{N_{i}}$, $Z$ and $W$
final states contribute only for $m_{N_{i}}$ larger than the gauge boson
masses.666For $m_{N_{i}}$ below the weak gauge boson masses, three-body decays
through off-shell $Z/W$ occur, but we have checked that these are always
subdominant in comparison to the $\gamma$ channel. Hence these partial widths
are neglected in our numerical study. The relevant partial decay widths,
written in terms of the broken phase parameters defined in Eqs. (11) and (14),
are given by

$\displaystyle\Gamma(N_{i}\rightarrow\nu\gamma)$ $\displaystyle=$
$\displaystyle\frac{|d_{\scriptscriptstyle\nu
N_{i}\gamma}|^{2}}{2\pi}m_{N_{i}}^{3},$ (70)
$\displaystyle\Gamma(N_{i}\rightarrow\nu Z)$ $\displaystyle=$
$\displaystyle\frac{|d_{\scriptscriptstyle\nu
N_{i}Z}|^{2}}{8\pi}m_{N_{i}}(2m_{N_{i}}^{2}+m_{Z}^{2})\lambda\bigg{(}1,0,\frac{m_{Z}^{2}}{m_{N_{i}}^{2}}\bigg{)},$
(71) $\displaystyle\Gamma(N_{i}\rightarrow\ell^{-}W^{+})$ $\displaystyle=$
$\displaystyle\frac{|d_{\scriptscriptstyle\ell
N_{i}W}|^{2}}{16\pi}m_{N_{i}}(2m_{N_{i}}^{2}+m_{W}^{2})\lambda\bigg{(}1,0,\frac{m_{W}^{2}}{m_{N_{i}}^{2}}\bigg{)},$
(72)

where, for simplicity, we have neglected the charged lepton masses. In the
computation of the total width of a Majorana $N_{i}$, the latter two partial
decay widths are multiplied by 2 to account for the Majorana nature of $\nu$
and the charged conjugated channel $\ell^{+}W^{-}$. Sterile neutrinos will
also decay via mixing. These decays have been thoroughly studied in the
literature. For our numerical analysis, we use the decay width formulas of
Atre:2009rg ; Bondarenko:2018ptm .

Figure 8: Partial decay widths of the decay modes induced by neutrino mixing,
$C_{\scriptscriptstyle{NNB}}^{(5)}$ and $C_{\scriptscriptstyle{N_{i}B}}^{(6)}$
($C_{\scriptscriptstyle{N_{i}W}}^{(6)}=2/(3t_{w})C_{\scriptscriptstyle{N_{i}B}}^{(6)}$).
We have chosen different benchmark values of the coefficients in each plot,
see the corresponding legend.

In Fig. 8 we show the previous partial decay widths as a function of the
sterile neutrino masses for various coefficient values. Here,
$C_{\scriptscriptstyle{N_{i}B}}^{(6)}$ and
$C_{\scriptscriptstyle{N_{i}W}}^{(6)}$ are fixed as
$C_{\scriptscriptstyle{N_{i}B}}^{(6)}=5\times 10^{-12}$ GeV-2 and
$C_{\scriptscriptstyle{N_{i}W}}^{(6)}=2/(3t_{w})C_{\scriptscriptstyle{N_{i}B}}^{(6)}\approx
6.22\times 10^{-12}$ GeV-2. These control the decay channels to $\nu\gamma$,
$\nu Z$, $e^{-}W^{+}$. For comparison, we also show the neutrino mixing
partial decay width for two different values of $|V_{\ell N_{i}}|^{2}$. In the
left (right) plot, we fix $|V_{\ell N_{i}}|^{2}=10^{-9}$ $(10^{-14})$,
motivated by an inverse seesaw (or naive Type-I seesaw) estimation.
Additionally, we show the partial decay width of $N_{2}\to N_{1}\gamma$ and
$N_{2}\to N_{1}Z$ controlled by $C_{\scriptscriptstyle{NNB}}^{(5)}$ and
$\delta$. In the left plot, these parameters have been fixed to
$C_{\scriptscriptstyle{NNB}}^{(5)}=10^{-6}$ GeV-1 and $\delta=0.01$, and in
the right plot, to $C_{\scriptscriptstyle{NNB}}^{(5)}=10^{-8}$ GeV-1 and
$\delta=0.5$. Notice that only the right plot shows a curve for $N_{2}\to
N_{1}Z$, as a large mass splitting is needed for this channel.

Figure 9: Nine different process classes: This classification distinguishes
scenarios, depending on which couplings dominate $N$ production and decay. The
sterile-to-sterile (active-to-sterile) magnetic moment is controlled by
$C_{\scriptscriptstyle{NNB}}^{(5)}$ ($C_{\scriptscriptstyle{NB}}^{(6)}$,
$C_{\scriptscriptstyle{NW}}^{(6)}$). The active-sterile neutrino mixing is
parameterised by $V_{\ell N}$.

From Figs. 7 and 8 we can compare the relative sizes of
$C_{\scriptscriptstyle{NNB}}^{(5)}$, $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$
and $V_{\ell N_{i}}$, determining the dominant sterile neutrino production and
decay mechanisms. There are nine distinct possibilities, which we define as
shown in Fig. 9. The scenarios in the same row have the same dominant
production mechanism, whereas the scenarios in the same column share the same
dominant decay mode for the sterile neutrino. In practice, not all
combinations will lead to non-pointing photons and we present the features of
all these scenarios below.

Scenarios B1-B3 assume that production is controlled by
$C_{\scriptscriptstyle{NNB}}^{(5)}$, thus $N_{2}$ and $N_{1}$ are pair
produced. Then:

  * •

B1: $N_{2}$ will decay to $N_{1}\gamma$, whereas $N_{1}$ escapes undetected.
The decay length of $N_{2}$ is controlled by
$C_{\scriptscriptstyle{NNB}}^{(5)}$ and the mass difference
$\delta=1-m_{N_{1}}/m_{N_{2}}$. $N_{2}$ is long-lived if $\delta\ll 1$. Too
small $\delta$, however, leads to photons too soft to be tagged. The signal
consists of one non-pointing photon plus missing energy.

  * •

B2: $N_{1}$ will decay to $\nu\gamma$ and for small enough
$C_{\scriptscriptstyle{N_{1}X}}^{(6)}$ this decay is long-lived. $N_{2}$ will
decay to $N_{1}\gamma$ either promptly or, if $\delta\ll 1$, with a finite
decay length. The signature of this scenario contains up to three photons,
either two or three of which are non-pointing. There is also missing energy in
the event, from $\nu$ and the potential undetected photons.

  * •

B3: The mixing parameter $V_{\ell N_{1}}$ controls the decay of $N_{1}$.
Again, $N_{2}$ will decay to $N_{1}\gamma$ promptly (or delayed if $\delta\ll
1$). The potential final state signal is a prompt (or non-pointing) photon
plus one or two displaced vertices (DV) with charged tracks from the $N_{1}$
decays. Unless two displaced vertices and the photon are all found, the event
again contains missing energy.

In scenarios B4-B6, production is controlled by
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$:

  * •

B4: Comparing the production cross-sections in Fig. 7 to the decay widths in
Eqs. (66) and (70) it becomes quickly clear that this scenario can not be
realised, since $C_{\scriptscriptstyle{NNB}}^{(5)}$ large enough to dominate
the decay width over $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ would also
dominate the cross-section.

  * •

B5: In this scenario $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ is assumed to be
also dominant in the decay of $N_{i}$. Comparing the width (70) to the
production cross-section in Fig. 7 (right), it is clear that the decays
$N_{i}\to\nu\gamma$ are too fast to lead to non-pointing photons since there
is no kinematic suppression ($\delta\simeq 1$) if
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ is large enough to give a sizeable
cross-section. The signal in this case is a prompt lepton accompanied by a
prompt photon.

  * •

B6: $V_{\ell N_{i}}$ is assumed to dominate the decay. Given the discussion
above for (B5), it is clear that decays will be prompt in this case. There are
no photons in the final state, thus no difference in signal to the minimal
mixing case.

Finally, for B7-B9 we assume production is dominated by mixing, $V_{\ell
N_{i}}$.

  * •

B7: The conditions defining this scenario can be fulfilled only in a narrow
range of parameters. The cross-section from the dipole and the one from mixing
are roughly of the same order for some particular ratio of
$C_{\scriptscriptstyle{NNB}}^{(5)}$ to $V_{\ell N_{i}}$. For this ratio, the
width from $C_{\scriptscriptstyle{NNB}}^{(5)}$ will be smaller than the one
from mixing for $m_{N_{2}}>15$ GeV, for $\delta=0.01$. For a smaller ratio
(and thus a more dominant production via mixing), the partial width through
the dipole can dominate only for smaller values of $m_{N_{2}}$. In this narrow
parameter range some events with a non-pointing photon would be generated,
however, always together with a certain number of displaced vertex events.

  * •

B8: The decay caused by $C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ can easily
dominate over the decay via mixing for $m_{N_{i}}<m_{Z}$, without
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$ dominating the cross-section. Since a
minimum size of $V_{\ell N_{i}}$ is needed to give a sizeable cross-section,
however, decay lengths will be shorter than in the pure mixing scenario. The
signal is a prompt lepton plus a photon (prompt or non-pointing, depending on
$C_{\scriptscriptstyle{N_{i}X}}^{(6)}$).

  * •

B9: This scenario is phenomenologically the same as the minimal mixing
scenario. It has been discussed extensively in the literature.

Given the above discussion, the most interesting scenarios are B1-B3. We will
therefore study these three scenarios in detail below.

###  5.2 Simulation Details

In order to determine the sensitivity reach of ATLAS for probing sterile
neutrino magnetic moments, we performed a numerical study of the
representative chosen benchmarks, using Monte-Carlo (MC) techniques. We first
implemented our model described by Eq. (3) (only up to $d=6$) plus the
interactions induced by active-sterile neutrino mixing in `FeynRules`
Christensen:2008py ; Alloul:2013bka . The generated UFO files Degrande:2011ua
were embedded in `MadGraph5` Alwall:2014hca , where we generated the collision
process $pp\rightarrow N_{1}N_{2}$ at $\sqrt{s}=14$ TeV, induced by
$C_{\scriptscriptstyle{NNB}}^{(5)}$ in the selected benchmarks. We generated
$10^{5}$ events at each point of the grid covering the parameter plane
$(m_{\text{LLP}},c_{\text{decay}})$, where the LLP is the sterile neutrino
providing the displaced signature at the detector and $c_{\text{decay}}$
denotes the interaction dominating the LLP decay width. These two parameters
define the two-dimensional space we scan over in our simulation for each
benchmark scenario, as shown in the second column of Tab. 2. In the third
column, we display the remaining parameters of the model, which are fixed to
some numerical value. In B1, we keep the mass splitting $\delta$ fixed such
that $m_{N_{1}}$ varies in the scan along with $m_{N_{2}}$. Whereas in B2 and
B3 we fix $m_{N_{2}}$ and $C_{\scriptscriptstyle{NNB}}^{(5)}$, since they do
not affect the long-lived nature of $N_{1}$.

Scenario | Model parameters | Simulated decay
---|---|---
Scan | Fixed
B1 |  $m_{N_{2}}$, $C_{\scriptscriptstyle{NNB}}^{(5)}$ | $\delta$ | $N_{2}\rightarrow N_{1}\gamma$
B2 |  $m_{N_{1}}$, $C_{\scriptscriptstyle{N_{1}X}}^{(6)}$ |  $m_{N_{2}}$, $C_{\scriptscriptstyle{NNB}}^{(5)}$ |  | $N_{2}\rightarrow N_{1}\gamma$
---
$N_{1}\rightarrow\nu\gamma$
B3 |  $m_{N_{1}}$, $|V_{eN_{1}}|^{2}$ |  $m_{N_{2}}$, $C_{\scriptscriptstyle{NNB}}^{(5)}$ |  | $N_{2}\rightarrow N_{1}\gamma$
---
$N_{1}\rightarrow ejj$
Table 2: Parameters entering the numerical simulation in each benchmark. The
scan parameters define the two-dimensional space to be surveyed, while the
other model parameters remain fixed. The last column indicates the decay
channel studied in the simulation.

The sterile neutrino decays are treated separately in `MadSpin`
Artoisenet:2012st . To ensure numerical stability for small decay widths, the
$N_{i}$ was forced to decay into the final state of interest (shown in the
last column of Tab. 2) in all simulated events. The decay LHE event files,
given as an output by `MadSpin`, were then processed by `Pythia8`
Sjostrand:2014zea  to estimate the ATLAS efficiency in detecting the
characteristic LLP signature of each benchmark: non-pointing photons in B1 and
B2, and displaced vertices in B3. Specific details of each search strategy are
outlined below.

Non-pointing photons are emitted in the decay of the long-lived sterile
neutrinos. These decays occur at a secondary vertex, displaced from the
collision point or primary vertex (PV), causing the photon direction to point
away from the PV. The ATLAS (and CMS) Electromagnetic Calorimeter (ECal) can
detect energetic photons and reconstruct their trajectories precisely. The
photon displacement is quantified using the impact parameter (IP), defined as
the minimum distance of the photon trajectory to the PV. The IP can be
decomposed into its transverse and longitudinal components,
$d_{\scriptscriptstyle{XY}}$ and $d_{\scriptscriptstyle{Z}}$, which can be
obtained with

$\displaystyle
d_{\scriptscriptstyle{XY}}=x_{\scriptscriptstyle{LLP}}\frac{p_{\scriptscriptstyle{Y}}}{p_{\scriptscriptstyle{T}}}-y_{\scriptscriptstyle{LLP}}\frac{p_{\scriptscriptstyle{X}}}{p_{\scriptscriptstyle{T}}},$
(73) $\displaystyle
d_{\scriptscriptstyle{Z}}=\frac{z_{\scriptscriptstyle{LLP}}-(\vec{r}\cdot\vec{p})p_{\scriptscriptstyle{Z}}/|\vec{p}|^{2}}{1-p_{\scriptscriptstyle{Z}}^{2}/|\vec{p}|^{2}},$
(74)

where
$\vec{r}=\\{x_{\scriptscriptstyle{LLP}},y_{\scriptscriptstyle{LLP}},z_{\scriptscriptstyle{LLP}}\\}$
is the LLP decay position and $\vec{p}$ is the photon momentum.

In our simulation, we have access to the true decay positions of the sterile
neutrinos as well as to the kinematic properties of the outgoing particles.
Therefore, we can use the previous equations to compute
$d_{\scriptscriptstyle{XY}}$ or $d_{\scriptscriptstyle{Z}}$ for each photon
emitted in the decay of $N_{i}$. In practice, measuring
$d_{\scriptscriptstyle{Z}}$ requires knowing the exact location of the PV
along the beamline ($z$-axis). However, this can be challenging in the high-
luminosity conditions at the LHC, where multiple collisions occur per bunch
crossing, leading to the production of several PVs. Moreover, in our specific
case (scenarios B1-B3), sterile neutrinos are produced at the collision point
without any accompanying charged particle. This limitation extends, in
general, to scenarios where only neutral particles are produced at the origin
and they decay far from the PV. Therefore, we can only use the variable
$d_{\scriptscriptstyle{XY}}$, which can be measured with respect to the
beamline.

It is worth mentioning that recent works have been performed in this
direction. ATLAS, for instance, presented results from a search using non-
pointing and delayed photons in ATLAS:2022vhr , and in Duarte:2023tdw ;
Delgado:2022fea , the authors have recast it for constraining Higgs decays
into sterile neutrinos in the context of dimension-5 operators. However, the
search relies on $d_{Z}$ and the time delay $t_{\gamma}$, which also requires
knowing the PV location. We cannot apply this search to our selected
benchmarks and accordingly propose an alternative search strategy. Our
approach draws inspiration from the CMS search in CMS:2012idp , which looked
for non-pointing photons using instead a trigger on
$d_{\scriptscriptstyle{XY}}$.

In Tab. 3, we summarise the cuts for the non-pointing photon search that we
apply in scenarios B1 and B2. The event selection starts by triggering photons
satisfying $|p_{T}^{\gamma}|>10$ GeV and $|\eta^{\gamma}|<2.47$. Subsequently,
we demand the LLP to decay before it reaches the outer layer of the ECal. This
requirement imposes the following cuts on the transverse plane and
longitudinal axis: $r_{\text{DV}}<1450$ mm and $|z_{\text{DV}}|<3450$ mm,
respectively. Finally, the energetic photons passing the initial selection
criteria and originating from one of the DVs are required to have
$|d_{\scriptscriptstyle{XY}}^{\gamma}|>6$ mm. This last cut is expected to
reduce SM background significantly CMS:2012idp .

On the other hand, in scenario B3, the LLP candidate decays via the active-
sterile neutrino mixing. The dominant decay modes contain charged particles
that leave tracks that originate from a displaced vertex (DV). ATLAS and CMS
have looked for long-lived sterile neutrinos using a DV search strategy and
have placed bounds on the neutrino mixing parameter ATLAS:2022atq ;
CMS:2022fut . Here, we follow Cottin:2021lzz  and use their DV search, which
targets a heavy $N$ decaying into $ejj$. We summarise the main selection cuts
of this search in the lower part of Tab. 3. The event selection starts by
identifying electrons with $|p_{T}^{e}|>120$ GeV, $|\eta^{e}|<2.47$. Then we
select events with a DV lying inside the ATLAS inner tracker detector, which
imposes a cut in the transverse and longitudinal positions: $4$ mm
$<r_{\text{DV}}<300$ mm, $|z_{\text{DV}}|<300$ mm. To reconstruct the DV, at
least four displaced charged particle tracks are needed. We require them to
have a transverse impact parameter of $|d_{0}|>2$ mm.777The approximate
transverse impact parameter in this case is defined as
$d_{0}=r\cdot\Delta\phi$, where $r$ is the transverse distance of the track
from the interaction point, and $\Delta\phi$ is the azimuthal angle between
the track and the direciton of $N_{1}$. Additionally, one of the displaced
tracks must correspond to the energetic electrons passing the initial trigger.
A final cut on the invariant mass of the DV ($m_{\text{DV}}>5$ GeV) is applied
to remove SM background from B-mesons. Further details can be found in Section
III of Cottin:2021lzz .

Scenario | Signature | Selection cuts
---|---|---
B1 | Non-pointing $\gamma$ |  $|p_{T}^{\gamma}|>10$ GeV , $|\eta^{\gamma}|<2.47$
B2 | Non-pointing $\gamma$ ($\times 2$) |  $r_{\text{DV}}<1450$ mm , $|z_{\text{DV}}|<3450$ mm
(+ prompt $\gamma$) |  $|d_{\scriptscriptstyle{XY}}^{\gamma}|>6$ mm
B3 |  |  $|p_{T}^{e}|>120$ GeV, $|\eta^{e}|<2.47$
Displaced Vertex ($\times 2$) |  $4$ mm $<r_{\text{DV}}<300$ mm, $|z_{\text{DV}}|<300$ mm
(+ prompt $\gamma$) | 4 tracks with $|d_{0}|>2$ mm
|  $m_{\text{DV}}>5$ GeV
Table 3: Summary of selection cuts for a non-pointing photon search (B1 and
B2) and a displaced vertex search (B3) at the ATLAS detector.

The total number of signal events for the three selected benchmarks is
obtained with the following expressions:

$N_{\text{sig.}}^{\text{B1}}=\sigma\cdot\mathcal{L}\cdot\mathcal{B}\left(N_{2}\rightarrow
N_{1}\gamma\right)\cdot\epsilon_{\text{sel}}^{\text{B1}},$ (75)
$N_{\text{sig.}}^{\text{B2}}=\sigma\cdot\mathcal{L}\cdot\mathcal{B}\left(N_{2}\rightarrow
N_{1}\gamma\right)\cdot
2\cdot\mathcal{B}\left(N_{1}\rightarrow\nu\gamma\right)\cdot\epsilon_{\text{sel}}^{\text{B2}},$
(76)
$N_{\text{sig.}}^{\text{B3}}=\sigma\cdot\mathcal{L}\cdot\mathcal{B}\left(N_{2}\rightarrow
N_{1}\gamma\right)\cdot 2\cdot\mathcal{B}\left(N_{1}\rightarrow
ejj\right)\cdot\epsilon_{\text{sel}}^{\text{B3}},$ (77)

where $\sigma$ is the cross-section of the process $pp\rightarrow N_{1}N_{2}$
that depends on the sterile neutrino masses and the interaction
$C_{\scriptscriptstyle{NNB}}^{(5)}$. $\mathcal{L}$ corresponds to the total
integrated luminosity during the high-luminosity phase, $3$ ab-1. The
branching ratio into the appropriate final state is also a function of the
masses and the coupling responsible for the decay in each scenario (see second
column in Tab. 2). The factor $\epsilon_{\text{sel}}$ denotes the efficiency
of event selection in ATLAS after applying the cuts in Tab. 3. Notice that
there is an additional factor of two in the formula for B2 and B3 since we
only require one displaced photon/vertex and there are two $N_{1}$ in each
event.

Under the assumption of zero background, we derive the $95$% C.L. sensitivity
prospects of ATLAS by requiring 3 signal events. Exclusion limits can be
placed in the corresponding parameter space if no signal events are found at
the end of the experiment. However, in case of a discovery, a larger number of
events would be necessary to identify the scenario. For instance, looking for
a second photon in the non-pointing photon search could distinguish a
potential signal event from B1 and B2. In scenario B3, on top of the displaced
vertex search, an additional trigger on a prompt photon associated with the
event containing the DV would distinguish this scenario from other models,
such as the minimal scenario.

##  6 Results and Discussion

Following the methodology described above, we obtain the experimental
sensitivity of ATLAS for probing sterile neutrinos interacting via dipole
interactions. We work at the level of $N_{R}$SMEFT operators in the unbroken
phase and then make the conversion to the broken phase parameters using Eqs.
(10) and (11). Besides, we consider the active-sterile neutrino mixing as an
independent parameter and, for simplicity, focus on the mixing of the light
sterile neutrino with the electron sector only, i.e. $V_{\ell
N_{1}}=V_{eN_{1}}$. The scenarios under study are B1-B3 in Fig. 9. In these
scenarios, sterile neutrinos are pair-produced in proton-proton collisions
through the operator $\mathcal{O}_{\scriptscriptstyle{NNB}}^{(5)}$. The
difference stems from the interaction dominating the long-lived sterile
neutrino decays: $C_{\scriptscriptstyle{NNB}}^{(5)}$,
$C_{\scriptscriptstyle{N_{1}X}}^{(6)}$ ($X=B,W$) or $V_{eN_{1}}$.

Figure 10: Sensitivity prospects of the non-pointing photon search at ATLAS
($\sqrt{s}=14$ TeV, 3 ab-1) for scenario B1 $\left(pp\rightarrow
N_{1}N_{2}\rightarrow N_{1}(N_{1}\gamma)^{\text{LLP}}\right)$. The 3 (30)
event contours are shown as solid (dashed) lines in the plane $m_{N_{2}}$ vs.
$d_{\scriptscriptstyle NN\gamma}$ for three values of the mass splitting
$\delta=1-m_{N_{1}}/m_{N_{2}}$. The grey area shows the current constraint
from LEP Magill:2018jla .

Fig. 10 shows the sensitivity reach of the non-pointing photon search applied
to scenario B1, in which $N_{2}$ is the LLP decaying within the detector.
Contours corresponding to 3 (30) events are shown as solid (dashed) lines in
the $m_{N_{2}}$ vs $d_{\scriptscriptstyle NN\gamma}$ plane, for three mass
splittings: $\delta=0.05$, $\delta=0.01$ and $\delta=0.005$. The sensitivity
in mass goes beyond $m_{N_{2}}\simeq 5$ GeV, $m_{N_{2}}\simeq 14$ GeV and
$m_{N_{2}}\simeq 16$ GeV, respectively. Dipole couplings as small as
$d_{\scriptscriptstyle NN\gamma}\simeq 5\times 10^{-7}$ $(8\times 10^{-7})$
GeV-1 can be probed if $\delta=0.05$ ($0.01$), improving the current limits
from LEP Magill:2018jla  (shown as the grey area) by more than one order of
magnitude in the corresponding mass ranges. For $\delta=0.005$, the
sensitivity reaches $d_{\scriptscriptstyle NN\gamma}\simeq 2\times 10^{-6}$
GeV-1. The shape of the curve can be understood in the following way. The
lower part of the curve is the long-lived limit, for a given mass and
$\delta$, a smaller coupling results in $N_{2}$ being too long-lived, escaping
the detector without decaying. The upper part represents the opposite
situation. A larger coupling value would make the $N_{2}$ decay too promptly
and the emitted photon would point back to the primary vertex, losing the LLP
signature. Moreover, since the production cross-section is proportional to
$|d_{\scriptscriptstyle{NN\gamma}}|^{2}$, we quickly lose events as the dipole
coefficient becomes smaller. However, we determined that the results are
highly sensitive to the $p_{T}^{\gamma}$ cut in this scenario. The smaller the
mass splitting, the less energetic the final photons are and for
$\delta\lesssim 0.005$ they become too soft to be detected in the ECal. This
effect can already be observed in the top left corner of the contour for
$\delta=0.005$, where the search is no longer sensitive to low sterile
neutrino masses.

Figure 11:  Sensitivity prospects of the non-pointing photon search at ATLAS
($\sqrt{s}=14$ TeV, 3 ab-1) for scenario B2: $pp\to N_{1}N_{2}(\to
N_{1}\gamma)\to(\nu\gamma+\nu\gamma)^{\text{LLP}}\gamma$. Solid (dashed) lines
corresponding to 3 (30) events are shown in the $m_{N_{1}}$ vs.
$d_{\scriptscriptstyle\nu N\gamma}$ plane for fixed values of $m_{N_{2}}$ and
$C_{\scriptscriptstyle{NNB}}^{(5)}$. The vertical lines in the upper mass
reach come from the kinematic threshold ($m_{N_{1}}<m_{N_{2}}$). Current
exclusion bounds are weaker and are not shown.

Fig. 11 shows the results for scenario B2 in the $m_{N_{1}}$ vs.
$d_{\scriptscriptstyle\nu N_{1}\gamma}$ plane. In this case, production and
decay of the LLP candidate, $N_{1}$, are decoupled since they depend on
different interactions. We fix $C_{\scriptscriptstyle{NNB}}^{(5)}=10^{-5}$
GeV-1, corresponding to $d_{\scriptscriptstyle{NN\gamma}}\simeq 8.8\times
10^{-6}$ GeV-1, to ensure it dominates sterile neutrino production. Two mass
values of $N_{2}$ are considered: $m_{N_{2}}=700$ GeV and $m_{N_{2}}=80$ GeV.
We recall that the scan parameters are $m_{N_{1}}$ and
$C_{\scriptscriptstyle{N_{1}B}}^{(6)}$, which then we convert onto
$d_{\scriptscriptstyle\nu N_{1}\gamma}$ using Eq. (11).888The contribution of
$d_{\scriptscriptstyle\nu N_{1}Z}$ and $d_{\scriptscriptstyle\ell N_{1}W}$ to
the total decay width of $N_{1}$ are also taken into account, since we are
considering the EFT operators in the unbroken phase. The solid (dashed) lines
correspond to 3 (30) events. Equivalently, these dashed lines correspond to 3
events for $C_{\scriptscriptstyle NNB}^{(5)}\approx 3\times 10^{-6}$ GeV-1.
The shape of the curves is similar to that of Fig. 10, but the probed
parameter space is much larger, precisely because the production and decay
mechanisms of $N_{1}$ are decoupled. The sensitivity reach in mass is limited
by the kinematic threshold $m_{N_{1}}\lesssim m_{N_{2}}$, resulting in these
vertical lines at $m_{N_{1}}=700$ GeV and $m_{N_{1}}=80$ GeV. As concerns
active-to-sterile magnetic moments, values as small as
$d_{\scriptscriptstyle\nu N_{1}\gamma}\simeq 5\times 10^{-13}~{}(2\times
10^{-12})$ GeV-1 can be probed for $m_{N_{2}}=700~{}(80)$ GeV. In contrast to
the previous scenario, the non-pointing photons are considerably more
energetic since they arise in the decay $N_{1}\rightarrow\nu\gamma$.
Consequently, almost all simulated photons pass the corresponding cuts of the
non-pointing photon search, giving a large event selection efficiency. Even
with a stronger $p_{T}^{\gamma}$ cut, the area enclosed by the sensitivity
curves comprises unexplored parameter space. Indeed, current exclusion bounds
are much weaker and do not appear in the figure.

Figure 12:  Sensitivity prospects of the displaced vertex search at ATLAS
($\sqrt{s}=14$ TeV, 3 ab-1) for scenario B3: $pp\to N_{1}N_{2}(\to
N_{1}\gamma)\to(ejj+ejj)^{\text{LLP}}\gamma$. Solid (dashed) lines
corresponding to 3 (10) events are shown in the $m_{N_{1}}$ vs. $|V_{eN}|^{2}$
plane for fixed values of $m_{N_{2}}$ and $C_{\scriptscriptstyle{NNB}}^{(5)}$.
The kinematic threshold ($m_{N_{1}}<m_{N_{2}}$) cuts the upper reach in mass
with the vertical lines. The black and grey solid lines represent the CMS and
ATLAS sensitivity prospects of a DV + prompt lepton search Drewes:2019fou ;
Cottin:2018nms ; Beltran:2021hpq .

For scenario B3, the results of the displaced vertex search are shown in Fig.
12. We display the 3 (solid) and 10 (dashed) event contours in the $m_{N_{1}}$
vs. $|V_{eN_{1}}|^{2}$ plane. Analogously to B2, production and decay of
$N_{1}$ are decoupled. We fix the production coupling
($C_{\scriptscriptstyle{NNB}}^{(5)}=10^{-5}$ GeV-1) and choose two values for
$m_{N_{2}}$. For $m_{N_{2}}=600$ GeV, the sensitivity extends across 10 orders
of magnitude in $|V_{eN_{1}}|^{2}$, reaching down to $10^{-17}$, and saturates
the kinematic threshold of $m_{N_{1}}\simeq 600$ GeV. The slope of the curve
changes around $m_{N_{1}}\approx m_{W}$ because the $N_{1}$ can decay to on-
shell weak bosons via the two-body decays $N_{1}\to W^{\pm}e^{\mp}$ and
$N_{1}\to Z\nu$. This increase in the decay width is compensated by smaller
values of the mixing parameter. For the chosen parameter values, 30 events are
not reached and in the 10 event contour, we already observe two isolated
regions in the parameter space. Conversely, for $m_{N_{2}}=80$ GeV, 30 signal
events can be obtained although we do not show the corresponding curve in the
plot. The probed region, in this case, is one order of magnitude larger in
$|V_{eN_{1}}|^{2}$ than for $m_{N_{2}}=600$ GeV, reaching values as small as
$|V_{eN_{1}}|\simeq 10^{-13}$ at $m_{N_{1}}\simeq 80$ GeV. Present exclusion
bounds on the mixing parameter $|V_{eN}|^{2}$ are weaker in this sterile
neutrino mass range and hence are not shown. However, we display the
sensitivity prospects of displaced vertex searches by ATLAS and CMS
Drewes:2019fou ; Cottin:2018nms ; Beltran:2021hpq , represented with the grey
and black lines. Their results are derived in the context of the minimal
scenario (only neutrino mixing) and, to be precise, these searches cannot be
directly recast into our parameter space as they require prompt leptons at the
origin. Nevertheless, we show them for illustrative purposes, revealing that
much smaller squared mixing values can be explored in the presence of sizeable
sterile neutrino dipole moments, even accessing the Type-I seesaw band. This
light grey region in the plot is obtained assuming the naive Type-I seesaw
relation for active neutrino masses of $0.05$ and $0.001$ eV.

Figure 13: Top left panel: Sterile-to-sterile neutrino magnetic moment as a
function of the relevant parameters of the UV model. The coloured areas
correspond to the probed regions in Fig. 10: $\delta=0.005$ in red,
$\delta=0.01$ in orange, $\delta=0.05$ in yellow. The grey shaded area is
excluded by direct collider searches. Bottom panels: Active-to-sterile
neutrino magnetic moment as a function of the relevant parameters of the UV
model. The pink ($m_{N_{2}}=80$ GeV) and blue ($m_{N_{2}}=700$ GeV) areas
cover the probed dipole values in Fig. 11 for $C_{\scriptscriptstyle
NNB}^{(5)}=3\times 10^{-6}$ GeV-1. The parameter space excluded by collider
searches and cLFV processes is shown in grey for two flavour scenarios. The
legend with the different constraints is displayed in the top right panel.

Finally, we can map the values of the dipole couplings to which experiments
are sensitive onto the parameters of the UV-complete model, namely the masses
of the singly-charged scalar and vector-like lepton ($m_{\phi},m_{E}$) and the
Yukawa-type couplings in Eq. (3) ($f,f^{\prime},Y_{E},h,h^{\prime}$). The
sterile-to-sterile and active-to-sterile dipoles are related to these
parameters via

$\displaystyle
d_{\scriptscriptstyle{NN\gamma}}^{ij}=\frac{e}{128\pi^{2}m_{E}}\big{(}h_{i}^{\prime}h_{j}^{\ast}-h_{i}^{\ast}h_{j}^{\prime}\big{)}\,,\quad
d_{\scriptscriptstyle{\nu N\gamma}}^{\alpha
i}=\frac{ev}{32\sqrt{2}\pi^{2}m_{E}^{2}}f_{\alpha\beta}Y_{E}^{\beta\ast}h_{i}^{\prime}\,,$
(78)

where we have assumed that $m_{E}=m_{\phi}$, hence replacing
$f(r)\big{|}_{r\to 1}=\frac{1}{2}$. Since there are products of two or even
three couplings entering these equations, we find it convenient to plot the
generated magnetic moments as functions of the specific coupling combinations
and the vector-like lepton mass. In the top panel of Fig. 13, we show fixed
values of $d_{\scriptscriptstyle{NN\gamma}}$
($d_{\scriptscriptstyle{NN\gamma}}^{12}$) ranging from $10^{-5}$ to $10^{-9}$
GeV-1 in the corresponding plane. Additionally, the coloured areas correspond
to the probed regions in Fig. 10, marginalising the dependence on $m_{N_{2}}$.
Direct collider searches exclude the low mass region of $m_{E}$ (see
discussion in Section 4). The bottom panels in Fig. 13 show active-to-sterile
magnetic moments ranging from $10^{-7}$ to $10^{-13}$ GeV-1. Here, the pink
and blue areas cover the probed regions illustrated in Fig. 11 for the 30
event contours assuming $C_{\scriptscriptstyle NNB}^{(5)}=10^{-5}$ GeV-1
(which correspond to 3 event contours assuming $C_{\scriptscriptstyle
NNB}^{(5)}=3\times 10^{-6}$ GeV-1), again marginalising the dependence on
$m_{N_{1}}$. In addition to the collider constraints, we show bounds from cLFV
processes (see discussion in Section 4) for two different flavour scenarios.
The left plot assumes $\mu$-$\tau$ couplings for the vector-like lepton
($Y_{E}^{\mu}=Y_{E}^{\tau}\gg Y_{E}^{e}\approx 0$ and $f_{\mu\tau}\neq 0$),
while the right plot assumes couplings only to $\tau$ ($Y_{E}^{\tau}\gg
Y_{E}^{e,\mu}\approx 0$ and $f_{e\tau}=f_{\mu\tau}\neq 0$). As we have fixed
$C_{\scriptscriptstyle{NNB}}^{(5)}$, there is an upper limit on $m_{E}$ above
which the couplings of the sterile-to-sterile magnetic moment enter the non-
perturbative regime. Masses above the grey dashed line lead to
$h,h^{\prime}>\sqrt{4\pi}$.

Figure 14: Top panel: Sensitivity projections from Fig. 10 in the plane
$m_{E}$ vs. $m_{N_{2}}$ for fixed coupling values. Direct collider searches
exclude the lower grey band. Bottom panels: Sensitivity projections from Fig.
11 for $C_{\scriptscriptstyle NNB}^{(5)}=3\times 10^{-6}$ GeV-1 in the plane
$m_{E}$ vs. $m_{N_{1}}$. The left (right) plot assumes $\mu$–$\tau$ (only
$\tau$) couplings of the vector-like lepton. Above the grey dashed line we
enter the non-perturbative regime of $h,h^{\prime}>\sqrt{4\pi}$, given the
value of $C_{\scriptscriptstyle{NNB}}^{(5)}$. The grey region in the lower
part of the plots is excluded by different cLFV processes and EWPD, which
depend on the flavour scenario.

In Fig. 14, we display in three different panels the sensitivity projections
from Figs. 10 and 11. These are shown in the planes $m_{E}$ vs. $m_{N_{2}}$
for B1 (top panel) and $m_{E}$ vs. $m_{N_{1}}$ for B2 (bottom panels). We
overlay current constraints from direct collider searches and cLFV processes
as before. In the bottom panels, we also show the non-perturbative regime
$h,h^{\prime}>\sqrt{4\pi}$. Additionally, the region corresponding to
$m_{E}<m_{N_{2}}=700$ GeV compromises the EFT validity. As can be seen from
the top plot of Fig. 14, for scenario B1 (where the decay of sterile neutrinos
is dominated by the sterile-to-sterile transition magnetic moment) LLP
searches can be sensitive to vector-like lepton masses as high as several TeV
for sterile neutrino masses less than 10 GeV. However, the cLFV processes do
not place additional constraints on this scenario because the couplings
entering the cLFV rates are independent of the sterile-to-sterile transition
magnetic moment.

In contrast, for scenario B2 (where the decay of sterile neutrinos is
dominated by the active-to-sterile transition magnetic moment), the
complementarity between LLP searches at the LHC and the constraints from cLFV
becomes evident, as can be seen from the bottom two plots of Figs. 14 and 10.
In this scenario, for a lepton flavour universal coupling of the vector-like
lepton ($Y_{E}^{e}=Y_{E}^{\mu}=Y_{E}^{\tau}$), the stringent constraints from
the current measurements of cLFV processes involving the first two generations
of the charged leptons (combined with the perturbativity of Yukawa couplings)
exclude the whole model parameter space. On the other hand, for the lepton
flavour non-universal case, we have viable parameter space to be explored by
LLP searches with non-pointing photons. If the vector-like lepton couples
mainly to $\mu$ and $\tau$ with similar strength (with negligible coupling to
the first-generation charged lepton, $Y_{E}^{\mu}=Y_{E}^{\tau}\approx
f_{\mu\tau}\gg Y_{E}^{e}\approx 0$) then the viable parameter space of the
model in vector-like lepton mass spans between ($0.30-2.3$) TeV, with the
lower limit coming from the current constraint from the
$\tau\rightarrow\mu\gamma$ and the current measurement of LFY from tau decays.
The upper limit corresponds to the perturbativity of the Yukawa couplings. If
the vector-like lepton couples only to $\tau$ (with negligible coupling to the
first two generations of charged lepton, $Y_{E}^{\tau}\approx
f_{e\tau}=f_{\mu\tau}\gg Y_{E}^{e,\mu}\approx 0$) then the viable parameter
space of the model in vector-like lepton mass spans between ($0.8-2.3$) TeV,
with the lower limit coming from the current constraints from the cLFV process
$\mu\rightarrow e\gamma$ and the upper limit from the perturbativity of the
Yukawa couplings.

##  7 Conclusions

In this work, we have examined long-lived particle (LLP) searches at the LHC,
using non-pointing photons to investigate sterile-to-sterile and active-to-
sterile transition magnetic dipole moments. We considered two Majorana sterile
neutrinos with masses ranging from a few GeV to several hundred GeV with
sizable sterile-to-sterile and active-to-sterile transition magnetic moments,
in addition to the usual active-sterile neutrino mixing. We first discussed
the operators relevant to neutrino magnetic moments within an EFT framework;
specifically, $N_{R}$SMEFT and its low-energy counterpart, $N_{R}$LEFT.
Subsequently, we considered as an example a simplified UV-complete model,
illustrating the emergence of substantial transition magnetic moments between
the heavy sterile neutrinos at the loop level. We discussed relevant
constraints on this model from direct collider searches and charged lepton
flavour violating processes.

For the phenomenological analysis of sterile neutrino magnetic moments, we
began by taking the EFT approach. We considered the impact of two independent
$N_{R}$SMEFT operators, $\mathcal{O}_{\scriptscriptstyle{NNB}}^{(5)}$ and
$\mathcal{O}_{\scriptscriptstyle{NB}}^{(6)}$, together with the active-sterile
neutrino mixing. We identified nine possible scenarios based on the dominant
production and decay modes at the LHC. Of these nine possibilities, we
explored the three physically realisable cases in detail. In these, the
sterile-to-sterile magnetic moment dominates the production cross-section of
sterile neutrinos at the LHC. Since there are no charged tracks from the
primary vertex for the scenarios of interest, we have put forward a search
strategy employing the transverse impact parameter of non-pointing photons,
which does not rely on the location of the primary vertex.

We presented detailed sensitivities for the three physically realisable
scenarios at the high-luminosity LHC. Our numerical simulations indicate that
for sterile neutrinos decaying primarily to photons, either via sterile-to-
sterile (scenario B1) or active-to-sterile magnetic dipole moments (scenario
B2), searches for LLPs with non-pointing photons can probe unexplored regions
of the parameter space for sterile neutrino masses ranging from several GeV to
several hundred GeV. The high-luminosity LHC will improve constraints on the
sterile-to-sterile neutrino transition magnetic moment from LEP searches by
more than an order of magnitude for the sub-ten-GeV sterile neutrino mass
regime. On the other hand, for scenario B2, where much more energetic photons
are produced, the active-to-sterile neutrino transition dipole moment can be
probed to much lower values with respect to existing constraints. In this
case, an unprecedented reach for sterile neutrino masses up to hundreds of GeV
is expected. For the scenario where the lighter sterile neutrino decays
dominantly through mixing, our results reveal that LLP searches using
displaced vertices could probe active-sterile neutrino mixing values much
lower than those in the minimal scenario (where mixing controls both the
production cross-sections and sterile neutrino lifetimes).

Applying the results from the model-independent EFT approach to our realistic
simplified model example, we found that the synergy between LLP searches using
non-pointing photons at the LHC and the constraints from cLFV processes can
potentially probe and distinguish different flavour structures of new physics
couplings. For the scenario where the decay of sterile neutrinos is dominated
by the sterile-to-sterile transition magnetic moment, LLP searches will be
sensitive to vector-like lepton masses as high as several TeV. However, such a
scenario remains largely unconstrained from the cLFV processes owing to the
freedom of choice for some of the couplings entering the cLFV rates, which are
independent of the sterile-to-sterile transition magnetic moment. In contrast,
for the scenario where the decay of sterile neutrinos is dominated by the
active-to-sterile transition magnetic moment, the complementarity of LLP
searches at the LHC and the constraints from cLFV can play a pivotal role in
understanding the flavour structure of the model parameter space. For a lepton
flavour universal coupling of the vector-like lepton, the stringent
constraints from the current measurements of cLFV processes involving the
first two generations of the charged leptons already exclude the whole viable
model parameter space. On the other hand, if the vector-like lepton couples
mainly to $\mu$ and (or) $\tau$, then future cLFV measurements will also
provide excellent synergy to the LLP searches at the LHC using non-pointing
photons.

#### Acknowledgements

R. B. is supported by the grants ACIF/2021/052 and CIBEFP/2022/62 (Generalitat
Valenciana). P. D. B. is supported by the Slovenian Research Agency under the
research core funding No. P1-0035 and in part by the research grants N1-0253
and J1-4389. P. D. B. also acknowledges support from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 860881-HIDDeN. F. F. D. acknowledges support from the
UK Science and Technology Facilities Council (STFC) via the Consolidated
Grants ST/P00072X/1 and ST/T000880/1. C. H. is supported by the Generalitat
Valenciana under Plan Gen-T via CDEIGENT grant No. CIDEIG/2022/16. R. B. also
acknowledges partial support from CDEIGENT grant No. CIDEIG/2022/16. R. B., C.
H., and M. H. acknowledge partial support from the Spanish grants PID2020-
113775GBI00 (AEI/10.13039/501100011033), and Prometeo CIPROM/2021/054
(Generalitat Valenciana).

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