aainstitutetext: AHEP Group, Instituto de Física Corpuscular –
CSIC/Universitat de València, Apartado 22085, E–46071 València,
Spainbbinstitutetext: Bethe Center for Theoretical Physics & Physikalisches
Institut der Universität Bonn,
Nußallee 12, 53115 Bonn, Germanyccinstitutetext: Dipartimento di Fisica
“Enrico Fermi”, Università di Pisa and INFN, Sezione di Pisa,
Largo Bruno Pontecorvo 3, I–56127 Pisa, Italyddinstitutetext: Department of
Physics, National Tsing Hua University, Hsinchu 300, Taiwaneeinstitutetext:
Center for Theory and Computation, National Tsing Hua University, Hsinchu 300,
Taiwan

# Heavy neutral leptons from kaons in effective field theory

Rebeca Beltrán<EMAIL_ADDRESS>b Julian Günther
<EMAIL_ADDRESS>a Martin Hirsch<EMAIL_ADDRESS>c Arsenii
Titov<EMAIL_ADDRESS>d,e Zeren Simon Wang<EMAIL_ADDRESS>
###### Abstract

In the framework of the low-energy effective theory containing in addition to
the Standard Model fields heavy neutral leptons (HNLs), we compute the decay
rates of neutral and charged kaons into HNLs. We consider both lepton-number-
conserving and lepton-number-violating four-fermion operators, taking into
account also the contribution of active-heavy neutrino mixing. Assuming that
the produced HNLs are long-lived, we perform simulations and calculate the
sensitivities of future long-lived-particle (LLP) detectors at the high-
luminosity LHC to the considered scenario. When applicable, we also recast the
existing bounds on the minimal mixing case obtained by NA62, T2K, and PS191.
Our findings show that for some of the effective operators considered the
future LLP detectors, in particular, MATHUSLA and ANUBIS, will probe (i) new
parts of the parameter space spanned by HNL mass and active-heavy neutrino
mixing, and (ii) new-physics scales in excess of 2000 TeV.

††preprint:  IFIC/23-43

##  1 Introduction

A rich program of searches for long-lived particles (LLPs) is planned at the
LHC in the next decades Curtin:2018mvb ; Alimena:2019zri ; Feng:2022inv . It
envisions the construction of several dedicated experiments sensitive to decay
lengths of $\mathcal{O}(1-100)$ m. Some LLP detectors, namely, FASER
Feng:2017uoz ; FASER:2018eoc  and MoEDAL-MAPP1 Pinfold:2019nqj ;
Pinfold:2019zwp , are already operational, whereas several more experiments
are planned for the high-luminosity (HL) phase of the LHC. These include
ANUBIS Bauer:2019vqk , CODEX-b Gligorov:2017nwh , FACET Cerci:2021nlb , FASER2
Feng:2022inv , MoEDAL-MAPP2 Pinfold:2019nqj ; Pinfold:2019zwp , and MATHUSLA
Curtin:2018mvb ; Chou:2016lxi ; MATHUSLA:2020uve .

LLPs are present in many models that can account for unresolved problems in
particle physics and cosmology, such as the mechanism of neutrino mass
generation and the nature of dark matter. The main focus of phenomenological
studies so far has been on various renormalizable models, including the Higgs
portal, the neutrino portal, the dark photon portal, as well as on non-
renormalizable models featuring long-lived axion-like particles (ALPs), see
e.g. Refs. Feng:2022inv ; Antel:2023hkf  for reviews and references.

However, it is plausible that in addition to the renormalizable couplings,
LLPs may interact with the Standard Model (SM) via effective, non-
renormalizable interactions. In this case, the SM viewed as an effective field
theory (EFT) has to be extended to include LLPs and their effective
interactions. One such example is the $N_{R}$SMEFT delAguila:2008ir ;
Aparici:2009fh ; Bhattacharya:2015vja ; Liao:2016qyd , which assumes the
existence of heavy neutral leptons (HNLs), $N$, with masses below or around
the electroweak scale, $v$. Recently, a dictionary of tree-level UV
completions of the dimension-six and dimension-seven operators with $N_{R}$
has been provided in Ref. Beltran:2023ymm .

There are two basic ways of HNL production at the LHC: (i) directly from
partonic collisions, and (ii) for $m_{N}\leq$ (few) GeV, in decays of mesons,
which in turn are copiously produced in $pp$ collisions. The long-lived HNLs
produced in the first way via the four-fermion operators with two quarks and
either two or one $N_{R}$ have been studied in Ref. Cottin:2021lzz  and Ref.
Beltran:2021hpq , respectively. These two sets of effective operators lead to
distinct phenomenology. The pair-$N_{R}$ operators may enhance the production
cross section while not contributing to the decay of the lightest HNL. This
makes the far LLP detectors introduced above an ideal place to probe such
effective interactions Cottin:2021lzz . On the contrary, single-$N_{R}$
operators, if large enough to enhance the HNL production, will also induce HNL
decays. Still, displaced-vertex searches at the local LHC detectors (ATLAS and
CMS) are very efficient to probe this set of effective operators
Beltran:2021hpq . The reach of the LLP detectors to the neutrino dipole
operator involving an active neutrino and $N_{R}$ has been recently estimated
in Ref. Liu:2023nxi .

Meson decays into long-lived HNLs triggered by non-renormalizable interactions
have been considered in Refs. DeVries:2020jbs ; Beltran:2022ast ;
Barducci:2022gdv ; Barducci:2023hzo . The EFT appropriate for the description
of meson decays is $N_{R}$LEFT Bischer:2019ttk ; Chala:2020vqp ; Li:2020lba ;
Li:2020wxi , the low-energy theory where the top quark, the Higgs boson and
the heavy SU$(2)_{L}$ gauge bosons are not present as dynamical degrees of
freedom. In Ref. DeVries:2020jbs , the authors have investigated the reach of
the proposed LHC far detectors to HNLs produced in the decays of $D$\- and
$B$-mesons via single-$N_{R}$ operators with a charged lepton, demonstrating
that the new physics scale, $\Lambda$, as high as $100$ TeV could be probed by
these detectors. The same set of single-$N_{R}$ operators, but with the $\tau$
lepton, would lead to HNL production in $\tau$ decays at Belle II Zhou:2021ylt
(see also Ref. Han:2022uho ). Pair-$N_{R}$ operators triggering $D$\- and
$B$-meson decays have been thoroughly examined in Ref. Beltran:2022ast . It
has been shown that for certain operators $\Lambda$ as large as 300 TeV and
active-heavy neutrino mixing squared, $|U_{eN}|^{2}$, as small as $10^{-15}$
could be tested by MATHUSLA, with the reach of ANUBIS being only a factor of a
few smaller. In Ref. Barducci:2022gdv , the sensitivities of FASER2, FACET,
ANUBIS, CODEX-b, and MAPP, to the dimension-five sterile neutrino dipole
operator (existing for at least two generations of HNLs, $N_{1}$ and $N_{2}$)
have been estimated. This operator leads to two-body decays of vector mesons
into $N_{1}$ and $N_{2}$ mediated by a photon, $\gamma$, and a subsequent
decay of the heavier HNL $N_{2}\to N_{1}\gamma$. Similarly, the sensitivity
reach of FASER2 and FACET to the dipole operator coupling an HNL to the photon
has recently been investigated in Ref. Barducci:2023hzo  for the HNLs produced
from meson decays.

In the present work, we perform state-of-the-art numerical simulations and
derive the sensitivities of the future LLP detectors at the LHC to HNLs
produced from neutral and charged kaon decays in the $N_{R}$LEFT. First, we
study the scenario in which the HNL production is induced by pair-$N_{R}$
operators, while the HNL decay proceeds via active-heavy neutrino mixing
$U_{eN}$. Second, we investigate the case in which both HNL production and
decay are induced by the same single-$N_{R}$ operator structure, but with
different quark flavor indices. Finally, we consider the situation where the
production proceeds via a single-$N_{R}$ operator as well as via $U_{eN}$,
while the $N$ decays via mixing only. We consider both lepton-number-
conserving (LNC) and lepton-number-violating (LNV) operators and discuss the
differences.

HNLs can also mediate meson decays. In particular, LNV decays
$K^{\mp}\to\pi^{\pm}\ell^{\mp}\ell^{\mp}$ mediated by light sterile neutrinos
have been studied in Ref. Zhou:2021lnl  adopting the EFT approach. We also
mention Ref. Fernandez-Martinez:2023phj . Here, the authors have derived
limits on EFT operators from HNL searches in kaon decays (among others).

The remainder of the paper is organized as follows. In Sec. 2, we briefly
recap neutral kaon mixing, summarize the $N_{R}$LEFT operators of interest,
and then discuss the HNL production from kaon decays. In Sec. 3, we describe
various experiments we consider and provide the details of numerical
simulations. Sec. 4 contains the derived sensitivities and the relevant
discussion. Finally, we provide summary and conclusions of our work in Sec. 5.
Additionally, we add two appendices at the end of the paper which explain in
detail the computation of the decay widths of the kaons into the HNLs and
those of the HNLs into a charged lepton and a pion, in the EFT framework.

##  2 HNLs from kaons in effective field theory

###  2.1 The neutral kaon system

The neutral kaons $K^{0}$ ($d\bar{s}$) and $\overline{K^{0}}$ ($s\bar{d}$) are
flavor eigenstates that can be produced in strong interactions. Weak
interactions cause the mixing between these two neutral states. If $CP$ were a
symmetry of the total hamiltonian $\mathcal{H}$ (including strong,
electromagnetic, and weak interactions), $CP$ eigenstates would also be
eigenstates of $\mathcal{H}$. With the convention (see e.g. Ref. Buras:1998raa
)

$\widehat{CP}|K^{0}\rangle=-|\overline{K^{0}}\rangle\,,\qquad\widehat{CP}|\overline{K^{0}}\rangle=-|K^{0}\rangle\,,$
(1)

we can define the $CP$ eigenstates as:

$|K_{1}\rangle=\frac{1}{\sqrt{2}}\left(|K^{0}\rangle-|\overline{K^{0}}\rangle\right)\,,\qquad|K_{2}\rangle=\frac{1}{\sqrt{2}}\left(|K^{0}\rangle+|\overline{K^{0}}\rangle\right)\,,$
(2)

where the former has $CP=+1$ and the latter $CP=-1$. However, $CP$ is mildly
violated by weak interactions and therefore the mass eigenstates
$|K_{S}\rangle$ and $|K_{L}\rangle$, characterized by definite lifetimes, are
different from $|K_{1}\rangle$ and $|K_{2}\rangle$:

$|K_{S}\rangle=\frac{|K_{1}\rangle+\varepsilon|K_{2}\rangle}{\sqrt{1+|\varepsilon|^{2}}}\,,\qquad|K_{L}\rangle=\frac{|K_{2}\rangle+\varepsilon|K_{1}\rangle}{\sqrt{1+|\varepsilon|^{2}}}\,,$
(3)

where $\varepsilon$ is the parameter accounting for indirect $CP$ violation in
neutral kaon decays. $K_{S}$ ($K_{L}$) denotes the neutral kaon with the
shorter (longer) lifetime. Experimentally, $|\varepsilon|\approx 2.23\times
10^{-3}$ ParticleDataGroup:2022pth , and for the purposes of this work we can
safely neglect it. Thus, in what follows we assume
$|K_{S}\rangle\approx|K_{1}\rangle$ and $|K_{L}\rangle\approx|K_{2}\rangle$.

###  2.2 Effective field theory with right-handed neutrinos

We will work in the framework of the low-energy effective field theory
extended with right-handed neutrinos, $N_{R}$, dubbed as $N_{R}$LEFT, see e.g.
Bischer:2019ttk ; Chala:2020vqp ; Li:2020lba ; Li:2020wxi . We assume $N_{R}$
to be a Majorana particle and allow for both lepton-number-conserving (LNC)
and lepton-number-violating (LNV) operators. Charm and bottom meson decays
triggered by four-fermion effective operators with $N_{R}$ have been studied
in detail in Refs. DeVries:2020jbs ; Beltran:2022ast . Here, we are interested
in the decays of kaons induced by four-fermion interactions. These
interactions can be grouped into pair-$N_{R}$ operators given in table 1 and
single-$N_{R}$ operators provided in table 2.111In what follows, we will not
consider single-$N_{R}$ operators with an active neutrino $\nu_{L}$.

LNC operators
---
Name | Structure | $N_{\mathrm{pars}}$
${\cal O}_{dN}^{V,RR}$ | $\left(\overline{d_{R}}\gamma_{\mu}d_{R}\right)\left(\overline{N_{R}}\gamma^{\mu}N_{R}\right)$ | 9
${\cal O}_{uN}^{V,RR}$ | $\left(\overline{u_{R}}\gamma_{\mu}u_{R}\right)\left(\overline{N_{R}}\gamma^{\mu}N_{R}\right)$ | 4
${\cal O}_{dN}^{V,LR}$ | $\left(\overline{d_{L}}\gamma_{\mu}d_{L}\right)\left(\overline{N_{R}}\gamma^{\mu}N_{R}\right)$ | 9
${\cal O}_{uN}^{V,LR}$ | $\left(\overline{u_{L}}\gamma_{\mu}u_{L}\right)\left(\overline{N_{R}}\gamma^{\mu}N_{R}\right)$ | 4

LNV operators
---
Name | Structure | $N_{\mathrm{pars}}$
${\cal O}_{dN}^{S,RR}$ | $\left(\overline{d_{L}}d_{R}\right)\left(\overline{N_{R}^{c}}N_{R}\right)$ | 18
${\cal O}_{uN}^{S,RR}$ | $\left(\overline{u_{L}}u_{R}\right)\left(\overline{N_{R}^{c}}N_{R}\right)$ | 8
${\cal O}_{dN}^{S,LR}$ | $\left(\overline{d_{R}}d_{L}\right)\left(\overline{N_{R}^{c}}N_{R}\right)$ | 18
${\cal O}_{uN}^{S,LR}$ | $\left(\overline{u_{R}}u_{L}\right)\left(\overline{N_{R}^{c}}N_{R}\right)$ | 8

Table 1: Four-fermion operators in $N_{R}$LEFT, involving two quarks and two
$N_{R}$’s, assuming one generation of $N_{R}$. LNV operator structures require
“+h.c.”. In the third column, we provide the number of independent real
parameters, $N_{\mathrm{pars}}$, associated with each operator structure.

LNC operators
---
Name | Structure
${\cal O}_{udeN}^{V,RR}$ | $\left(\overline{u_{R}}\gamma_{\mu}d_{R}\right)\left(\overline{e_{R}}\gamma^{\mu}N_{R}\right)$
${\cal O}_{udeN}^{V,LR}$ | $\left(\overline{u_{L}}\gamma_{\mu}d_{L}\right)\left(\overline{e_{R}}\gamma^{\mu}N_{R}\right)$
${\cal O}_{udeN}^{S,RR}$ | $\left(\overline{u_{L}}d_{R}\right)\left(\overline{e_{L}}N_{R}\right)$
${\cal O}_{udeN}^{T,RR}$ | $\left(\overline{u_{L}}\sigma_{\mu\nu}d_{R}\right)\left(\overline{e_{L}}\sigma^{\mu\nu}N_{R}\right)$
${\cal O}_{udeN}^{S,LR}$ | $\left(\overline{u_{R}}d_{L}\right)\left(\overline{e_{L}}N_{R}\right)$

LNV operators
---
Name | Structure
${\cal O}_{udeN}^{V,LL}$ | $\left(\overline{u_{L}}\gamma_{\mu}d_{L}\right)\left(\overline{e_{L}}\gamma^{\mu}N_{R}^{c}\right)$
${\cal O}_{udeN}^{V,RL}$ | $\left(\overline{u_{R}}\gamma_{\mu}d_{R}\right)\left(\overline{e_{L}}\gamma^{\mu}N_{R}^{c}\right)$
${\cal O}_{udeN}^{S,LL}$ | $\left(\overline{u_{R}}d_{L}\right)\left(\overline{e_{R}}N_{R}^{c}\right)$
${\cal O}_{udeN}^{T,LL}$ | $\left(\overline{u_{R}}\sigma_{\mu\nu}d_{L}\right)\left(\overline{e_{R}}\sigma^{\mu\nu}N_{R}^{c}\right)$
${\cal O}_{udeN}^{S,RL}$ | $\left(\overline{u_{L}}d_{R}\right)\left(\overline{e_{R}}N_{R}^{c}\right)$

Table 2: Four-fermion operators in $N_{R}$LEFT, involving two quarks, one
charged lepton, and one $N_{R}$. Both LNC and LNV operator structures require
“+h.c.”. For one generation of $N_{R}$, there are 36 independent real
parameters associated with each operator structure.

Since the top quark is not in the spectrum of the $N_{R}$LEFT, we have
$n_{u}=2$ and $n_{d}=n_{e}=n_{\nu}=3$, with $n_{f}$ denoting the number of
generations of a fermion $f$. In addition, we assume one generation of
$N_{R}$.

Generically, $N_{R}$ mixes with the active neutrinos at the renormalizable
level. Integrating out the $W$ boson leads to the following contribution to
the effective Lagrangian:

$\mathcal{L}_{\mathrm{mix}}=-\frac{4G_{F}}{\sqrt{2}}V_{ij}U_{\ell
N}\left(\overline{u_{iL}}\gamma_{\mu}d_{jL}\right)\left(\overline{e_{\ell
L}}\gamma^{\mu}N_{R}^{c}\right)+\text{h.c.}\,,$ (4)

where $G_{F}$ is the Fermi constant, $V$ is the CKM quark mixing matrix, and
$U_{\ell N}$ is active-heavy neutrino mixing. For charged kaon decays, the
relevant CKM matrix element is $V_{us}$. In what follows, we will separate
this contribution from the corresponding operator in table 2 by denoting its
Wilson coefficient (WC) as

$c_{\mathrm{mix}}=-\frac{4G_{F}}{\sqrt{2}}V_{us}U_{\ell
N}\,,\qquad\text{with}\qquad|c_{\mathrm{mix}}|\approx\frac{0.45\,|U_{\ell
N}|}{v^{2}}\,.$ (5)

For a more detailed discussion of $N_{R}$LEFT, including the running of the
considered operators and their matching to the $N_{R}$SMEFT operators, see
Ref. Beltran:2022ast .

###  2.3 HNL production in kaon decays

The pair-$N_{R}$ operators $\mathcal{O}_{dN}$ from table 1 involving $d$ and
$s$ quarks trigger the following two- and three-body decays: $K_{S/L}\to NN$,
$K_{S/L}\to\pi^{0}NN$ and $K^{\pm}\to\pi^{\pm}NN$. The single-$N_{R}$
operators $\mathcal{O}_{udeN}$ from table 2 containing $u$ and $s$ quarks
induce $K^{\pm}\to e^{\pm}N$, $K^{\pm}\to\pi^{0}e^{\pm}N$ and
$K_{S/L}\to\pi^{\pm}e^{\mp}N$. In appendix A, we provide the formulae for the
two-body decay widths and the three-body decay amplitudes. The three-body
decay widths are then computed according to the procedure explained in Refs.
DeVries:2020jbs ; Beltran:2022ast .

If the products of HNL decays are not detected, the kaon decay modes we are
interested in contribute to $K_{S/L}\to\text{inv.}$,
$K_{S/L}\to\pi^{0}\nu\overline{\nu}$, $K^{\pm}\to\pi^{\pm}\nu\overline{\nu}$,
and $K^{+}\to e^{+}\nu_{e}$, $K^{+}\to\pi^{0}e^{+}\nu_{e}$,
$K_{S/L}\to\pi^{\pm}e^{\mp}\nu_{e}$. The branching ratios of the decays into
final states with at least one charged particle have been measured, whereas
for $K_{L}\to\pi^{0}\nu\overline{\nu}$ a stringent upper limit on the
branching ratio has been obtained. We summarize the current experimental
results in table 3.

Decay | Branching ratio
---|---
$K_{L}\to\pi^{0}\nu\overline{\nu}$ |  $<3.0\times 10^{-9}$ at 90% C.L.
$K^{+}\to\pi^{+}\nu\overline{\nu}$ | $\left(1.14^{+0.40}_{-0.33}\right)\times 10^{-10}$

Decay | Branching ratio
---|---
$K^{+}\to e^{+}\nu_{e}$ | $\left(1.582\pm 0.007\right)\times 10^{-5}$
$K^{+}\to\pi^{0}e^{+}\nu_{e}$ | $\left(5.07\pm 0.04\right)\times 10^{-2}$
$K_{S}\to\pi^{\pm}e^{\mp}\nu_{e}$ | $\left(7.04\pm 0.08\right)\times 10^{-4}$
$K_{L}\to\pi^{\pm}e^{\mp}\nu_{e}$ | $\left(40.55\pm 0.11\right)\times 10^{-2}$

Table 3: Branching ratios of semi-invisible kaon decays
ParticleDataGroup:2022pth . For $K_{S/L}\to\pi^{\pm}e^{\mp}\nu_{e}$, the
values are for the sum of particle and antiparticle states indicated.

We will take them into account when deriving the limits on the WCs of the
pair-$N_{R}$ and single-$N_{R}$ operators. In the case of a measured branching
ratio, we will require the new contribution not to exceed twice the
experimental error, whereas for $K_{L}\to\pi^{0}NN$, we will demand that its
branching ratio is smaller than $3.0\times 10^{-9}$ in accordance with the
current upper bound on the branching ratio of $K_{L}\to\pi^{0}\nu\bar{\nu}$
ParticleDataGroup:2022pth .

In figure 1, we display the branching ratios of kaon decays triggered by the
LNC operator $\mathcal{O}_{dN,21}^{V,RR}$ and by the LNV operator
$\mathcal{O}_{dN,21}^{S,RR}$. In each case, we assume that the corresponding
WC is either real (left panel) or purely imaginary (right panel). For the LNC
operator, a real (purely imaginary) WC does not allow for $K_{S}\to NN$ and
$K_{L}\to\pi^{0}NN$ ($K_{L}\to NN$ and $K_{S}\to\pi^{0}NN$), as can be
inferred from eqs. (21) and (28) (eqs. (22) and (27)).222We note that this is
strictly true only in the limit $\varepsilon\to 0$. For small $m_{N}$, the
allowed two-body decay is suppressed, since the corresponding decay width
scales with $m_{N}^{2}$, see eqs. (21) and (22). In this figure, we fix the
absolute value of the operator coefficient to $10^{-6}v^{-2}$ in order to
comply with the measurements reported in the left panel of table 3, in
particular, with that of $K^{+}\to\pi^{+}\nu\overline{\nu}$. Switching on
$\mathcal{O}_{dN,21}^{V,LR}$ and $\mathcal{O}_{dN,21}^{S,LR}$ (one at a time)
would lead to the same results as for $\mathcal{O}_{dN,21}^{V,RR}$ and
$\mathcal{O}_{dN,21}^{S,RR}$, respectively.

Figure 1: Branching ratios of kaon decays triggered by the LNC (top) and LNV
(bottom) pair-$N_{R}$ operators. In the left (right) panel, the corresponding
WC is purely real (imaginary).

In figure 2, we plot the branching ratios of kaon decays induced by the LNC
operators $\mathcal{O}_{udeN,12}^{V,RR}$, $\mathcal{O}_{udeN,12}^{S,RR}$, and
$\mathcal{O}_{udeN,12}^{T,RR}$, with respective WC $c_{\mathcal{O}}$, as well
as by active-heavy neutrino mixing $U_{eN}$. We show three representative
cases: (i) $U_{eN}=10^{-5}$ and $c_{\mathcal{O}}=0$, (i) $U_{eN}=0$ and
$c_{\mathcal{O}}=10^{-5}v^{-2}$, and (iii) $U_{eN}=10^{-5}$ and
$c_{\mathcal{O}}=10^{-5}v^{-2}$. The value of the operator coefficient ensures
that the branching ratios are compatible with the errors in the measurements
presented in the right panel of table 3, most importantly, with that of
$K^{+}\to e^{+}\nu_{e}$. The value $U_{eN}=10^{-5}$ ($|U_{eN}|^{2}=10^{-10}$)
is such that it is below the existing constraints. For HNLs lighter than the
kaons, the leading constraints on the mixing parameter come from NA62
NA62:2020mcv , PS191 Bernardi:1987ek , and T2K T2K:2019jwa , which set upper
limits at the level of $|U_{eN}|^{2}\sim 5\times 10^{-10}$. For even lighter
HNLs, below the pion mass, PIENU PIENU:2017wbj  has ruled out parameter space
corresponding to $|U_{eN}|^{2}>10^{-7}\sim 10^{-8}$. Moreover, with the chosen
value for the mixing, its contribution is comparable to the operator one for
$c_{\mathcal{O}}\sim 10^{-5}v^{-2}$, cf. eq. (5).

Figure 2: Branching ratios of kaon decays triggered by the LNC single-$N_{R}$
operators with electron, as well as by active-heavy mixing $U_{eN}$.

In figure 3, we show the branching ratios for the same processes as in figure
2, but for the LNV operators $\mathcal{O}_{udeN,12}^{V,RL}$,
$\mathcal{O}_{udeN,12}^{S,RL}$, and $\mathcal{O}_{udeN,12}^{T,LL}$. In the
absence of mixing, the results are identical to those for the corresponding
LNC operators (switched on one at a time). In the presence of mixing, there is
an interference between the effective operator generated by new physics and
the four-fermion interaction (see eq. (4)) arising from integrating out the
$W$ boson. Its effect is more pronounced for the vector-type operators, and it
is stronger for the LNV operator, as can be understood from eq. (23).

Figure 3: Branching ratios of kaon decays triggered by the LNV single-$N_{R}$
operators with electron, as well as by active-heavy mixing $U_{eN}$.

##  3 Experiments and simulation

A whole list of far detectors has been proposed for operation in the vicinity
of various interaction points of the LHC, either during Run 3 or the HL-LHC
phase; some of them have even been approved and are running, including FASER
and MoEDAL-MAPP1. Since they are all sensitive to signatures of tracks
stemming from displaced decays of LLPs taking place inside their fiducial
volumes, we will perform the numerical analysis taking into account these
proposals comprehensively.

We classify these detectors according to their associated IP at the LHC. For
the ATLAS IP, ANUBIS Bauer:2019vqk , FASER Feng:2017uoz ; FASER:2018eoc , and
FASER2 Feng:2022inv  are relevant. ANUBIS333Very recently, a new design of
ANUBIS has been discussed ANUBIS_talk_slides ; Satterthwaite:2839063 . Instead
of being placed inside one of the service shafts, ANUBIS is now considered to
be installed at the ATLAS cavern ceiling or shaft bottom. Given the changing
status of the proposal, we stick to the original design for sensitivity study
as a good reference for interested readers. is a detector proposed to be
installed inside one of the service shaft above the ATLAS IP. It has a
cylindrical shape of 56 m height and 18 m diameter. Being close to the ATLAS
IP, it is expected to suffer from certain background sources such as neutral
kaons. Nevertheless, exclusion bounds at 95% confidence level (C.L.) are
usually shown with 3 signal events, assuming zero background, and we will
follow the practice in this paper. FASER is currently collecting data at the
LHC; first results can be found in Refs. FASER:2023zcr ; Petersen:2023hgm . It
is a small cylindrical detector installed in the very forward position along
the beam direction, with a distance of 480 m from the interaction point.
Further, an upgraded program of FASER, FASER2, has been suggested to be built
at the site of the proposed Forward Physics Facility Feng:2022inv . It is
larger than FASER and has hence better acceptances to LLPs. ANUBIS and FASER2
should collect in total 3 ab-1 integrated luminosity data, while FASER will
have order of 150 fb-1 integrated luminosity.

Near the CMS IP, MATHUSLA Curtin:2018mvb ; Chou:2016lxi ; MATHUSLA:2020uve
and FACET Cerci:2021nlb  have been proposed. MATHUSLA would be an experiment
on the ground, with about 100 m distance from the CMS IP. It would have a huge
effective volume of $100~{}\rm{m}\times 100~{}\rm{m}\times 25~{}\rm{m}$. FACET
is suggested to be a sub-system of the CMS experiment; with a cylindrical
shape, it has a distance of 101 m from the CMS IP, enclosing the beam pipe.
Moreover, with a radius of 0.5 m and a length of 18 m, FACET is relatively
large compared to FASER. Both MATHUSLA and FACET would be running during the
HL-LHC and will hence have an integrated luminosity of 3 ab-1.

Finally, for the LHCb IP, we have CODEX-b Gligorov:2017nwh , and MoEDAL-MAPP1
and MoEDAL-MAPP2 Pinfold:2019nqj ; Pinfold:2019zwp . CODEX-b has been proposed
as a cubic box with a dimension of 10 m $\times$ 10 m $\times$ 10 m covering
the pseudorapidity range $[0.2,0.6]$, roughly 10 m away from IP8. MoEDAL-MAPP1
is a small detector of about 140 m3 in a gallery of negative pseudorapidity
with respect to the LHCb IP, under operation during Run 3, and MoEDAL-MAPP2 is
an enlarged version of MAPP1, to be running during the HL-LHC period. The
integrated luminosity associated with the LHCb interaction point is lower
compared to that of ATLAS or CMS. CODEX-b and MoEDAL-MAPP2 will have an
integrated luminosity of 300 fb-1 while MoEDAL-MAPP1 has only 30 fb-1.

For a more detailed summary of these proposals, we refer the reader to e.g.
Refs. Lee:2018pag ; Alimena:2019zri ; DeVries:2020jbs , as well as to the
respective references proposing the detectors.

In order to obtain the kinematics of the HNLs produced from kaons, we make use
of the tool Pythia8 with the module SoftQCD:all. The kaons are generated in
$pp$ collisions with a center-of-mass-energy 14 TeV, and are set to decay
exclusively in the signal-event channels. Pythia8 provides the boost factor
and the polar angle of each simulated HNL. We note that since kaons are
themselves long-lived, we let Pythia8 decide the decay positions of the kaons
and thus take into account the production position of the HNL ($=$ the decay
position of the kaon) as well as the polar angle and boost factor of the HNL,
in the computation of the HNLs’ average decay probability inside the far
detectors. The total signal-event rates at each detector can be computed with
the following formula

$\displaystyle N_{S}=\sum_{K^{\prime}s}N_{K}\cdot n\cdot\text{BR}\left(K\to
n\,N+\text{anything}\right)\cdot\epsilon\cdot\text{BR}\left(N\to\text{visible}\right)\,,$
(6)

where the summation goes over different kaons, $N_{K}$ is the total number of
kaons, $n=1,2$ is the number of HNLs produced in the considered kaon decays,
BR$(N\to\text{visible})$ is the visible decay branching ratio of the HNL, and
$\epsilon$ is the average decay probability in a far detector. We estimate the
number of kaons $N_{K}$, with the help of the tool EPOS LHC Pierog:2013ria
provided in the CRMC simulation package CRMC , to be $N_{K^{\pm}}=2.38\times
10^{18}$, $N_{K_{S}}=1.31\times 10^{18}$, and $N_{K_{L}}=1.30\times 10^{18}$,
over the whole $4\pi$ solid angle. The computation procedure of $\epsilon$ is
based on the exact formulae given in Refs. Dercks:2018eua ; Dercks:2018wum ;
DeVries:2020jbs ; Hirsch:2020klk ; Beltran:2022ast  and the further
development presented in Ref. DUNE_upcoming  for LLPs from kaons. As the
kaons, especially $K^{\pm}$ and $K_{L}$, travel macroscopic distances, the
infrastructure surrounding the IPs may affect the kinematics of the kaons. To
simplify the analysis, we neglect the influence of any magnetic fields present
at the IP. Additionally, we introduce cut-offs for the production positions of
the HNL that are included in the signal-event rates $N_{S}$, resulting in
conservative estimates. Here, we provide a brief summary of the cut-offs we
employ in our simulation; for more detail, see Ref. DUNE_upcoming . For
example, a lead shield covering the total fiducial volume in order to veto
neutral background events is placed approximately $5\text{\,}\mathrm{m}$ in
front of CODEX-b. Hence, we require the kaons to decay before reaching the
shield. For detectors in the far forward region (FASER, FASER2, and FACET), we
employ the beamline geometry of the ATLAS and CMS IP, which involve absorbers
for charged and neutral particles in order to protect beamline infrastructure.
The hadron calorimetry of ATLAS and CMS restricts the decay region of the
kaons for ANUBIS and MATHUSLA, respectively. Lastly, for MoEDAL-MAPP1 and
MAPP2, the natural rock between the IP and the detectors is the limiting
factor. Hence, the kaons are required to decay within the
$3.8\text{\,}\mathrm{m}$ wide beam cavern.

Finally, we discuss the procedure we apply for recasting the bounds on the
HNLs in the minimal scenario, obtained in some past searches, into those on
the HNLs in the EFT scenarios considered here. In general, we follow the
approaches laid out in Refs. Beltran:2023nli ; Fernandez-Martinez:2023phj ;
Dreiner:2023gir  (see also Ref. Barouki:2022bkt ). We consider three searches
at NA62 NA62:2020mcv , PS191 Bernardi:1987ek , and T2K T2K:2019jwa . Since
these searches all require a prompt charged lepton, they are applicable only
to the single-$N_{R}$ scenarios. We first consider the NA62 search, which
looked for HNL production in $K^{+}$ decays to positrons and missing energy,
assuming the proper lifetime of $N$ is larger than 50 ns. The search obtained
bounds on BR$(K^{+}\to e^{+}N)$ and hence those on the active-sterile neutrino
mixing, as functions of the sterile neutrino mass. We simply convert
BR$(K^{+}\to e^{+}N)$ to the production Wilson coefficient of the
single-$N_{R}$ scenario in question, for each mass value, and obtain the
corresponding recast bounds. Both the PS191 and T2K searches are for both a
prompt charged lepton and a displaced vertex at the detached detector. For
PS191, we extract the sensitivity curve presented in the plane $|U_{eN}|^{2}$
vs. $m_{N}$ for the signal process $K^{+}\to e^{+}N$, $N\to e^{-}\pi^{+}$ and
its charge-conjugate channel, and for the T2K near detector ND280, the signal
process $K^{\pm}\to e^{\pm}N$, $N\to e^{\pm}\pi^{\mp}$ is considered. By
rescaling the production and visible decay rates, we obtain the recast bounds
in the EFT parameter space.

##  4 Results

For presenting the numerical results, we choose several benchmarks
characterized by the different couplings responsible for the HNL production
and decay. We summarize them in table 4.

Benchmark | Production | Decay
---|---|---
B1.1 | $c_{dN,21}^{V,RR}\in\mathbb{R}$ | $U_{eN}$
B1.2 | $c_{dN,21}^{V,RR}\in i\,\mathbb{R}$ | $U_{eN}$
B2.1 | $c_{dN,21}^{S,RR}\in\mathbb{R}$ | $U_{eN}$
B2.2 | $c_{dN,21}^{S,RR}\in i\,\mathbb{R}$ | $U_{eN}$

Benchmark | Production | Decay
---|---|---
B3 | $c_{udeN,12}^{V,RR}$ | $c_{udeN,11}^{V,RR}$
B4 | $c_{udeN,12}^{S,RR}$ | $c_{udeN,11}^{S,RR}$
B5 |  $c_{udeN,12}^{V,RR}$ and $U_{eN}$ | $U_{eN}$
B6 |  $c_{udeN,12}^{S,RR}$ and $U_{eN}$ | $U_{eN}$
B7 |  $c_{udeN,12}^{V,RL}$ and $U_{eN}$ | $U_{eN}$
B8 |  $c_{udeN,12}^{S,RL}$ and $U_{eN}$ | $U_{eN}$

Table 4: Benchmarks for the scenarios with pair-$N_{R}$ (left) and
single-$N_{R}$ (right) operators.

In total, we consider four scenarios with pair-$N_{R}$ operators (benchmarks
B1 and B2, each having two sub-cases) and six scenarios with single-$N_{R}$
operators (benchmarks B3–B8).

For benchmark B1 (B2), HNL production is governed by the LNC (LNV)
pair-$N_{R}$ operator $\mathcal{O}_{dN,21}^{V,RR}$
($\mathcal{O}_{dN,21}^{S,RR}$), while HNL decay proceeds via active-heavy
neutrino mixing $U_{eN}$. For each of these two operators, we consider two
cases: (i) real WC and (ii) purely imaginary WC, cf. figure 1 and the related
discussion. First, in figure 4, we fix the absolute value of the WC to
$10^{-6}v^{-2}$ (to respect the constraint coming from
$K^{+}\to\pi^{+}\nu\overline{\nu}$) and display the projected exclusion limits
in the plane $|U_{eN}|^{2}$ vs. $m_{N}$ for three signal events.

Figure 4: Projected exclusion limits in the plane $|U_{eN}|^{2}$ vs. $m_{N}$
for pair-$N_{R}$ operator benchmarks B1.1 and B1.2 (top), and B2.1 and B2.2
(bottom). The absolute value of the corresponding WC has been fixed to
$10^{-6}v^{-2}$. The current bounds from NA62, T2K, and PS191, as well as the
type-I seesaw target region where $m_{\nu}=0.05$–$0.12$ eV, are also shown.

The curves correspond to 95% C.L. limits (under the assumption of zero
background). For the LNC operator, the sensitivities are approximately two
orders of magnitude weaker in the case of purely imaginary WC (B1.2, top-right
plot) than in the case of real WC of the same size (B1.1, top-left plot). This
can be understood from figure 1, which shows that for the same size of the WC,
BR$(K_{S}\to NN)$ is about two orders of magnitude smaller than BR$(K_{L}\to
NN)$, whereas the production numbers of $K_{S}$ and $K_{L}$ are nearly the
same. The difference in the sensitivities is much milder for the LNV operator,
for which the branching ratios of the relevant kaon decays differ only
slightly between the cases of real and purely imaginary WC.

Among the considered LLP detectors, the best limit comes from MATHUSLA, which
can probe $|U_{eN}|^{2}$ down to $4\times 10^{-8}$ ($1.4\times 10^{-9}$) at
$m_{N}\approx 0.23$ GeV for B1.1 (B2.1). It is followed by ANUBIS, which has a
factor of a few weaker sensitivity. MAPP2, FACET and CODEX-b provide very
similar exclusion limits, which are approximately two orders of magnitude
weaker than the expected limits from MATHUSLA. Finally, FASER2 and MAPP1 have
the weakest depicted sensitivities.444For FASER, there is no isocurve shown in
figure 4 and figure 5. Because of the small geometric acceptance and
comparatively lower integrated luminosity, the simulated number of signal
events is less than three in the shown parameter region. We also show the
existing limits on active-heavy neutrino mixing obtained by the NA62
NA62:2020mcv , T2K T2K:2019jwa , and PS191 Bernardi:1987ek  experiments.
Though derived under the hypothesis of the minimal mixing case, these limits
apply to the considered EFT scenarios as well. As can be seen, NA62
outperforms all the far detectors and already touches the (naive) type-I
seesaw band, where the values of $m_{N}$ and $U_{eN}$ yield the light neutrino
mass $m_{\nu}=0.05$–$0.12$ eV.

In figure 5, we fix $|U_{eN}|^{2}=10^{-10}$ and show the exclusion limits in
the plane WC vs. $m_{N}$.

Figure 5: Exclusion limits in the plane WC vs. $m_{N}$ for pair-$N_{R}$
operator benchmarks B1.1 and B1.2 (top), and B2.1 and B2.2 (bottom). The
active-heavy neutrino mixing parameter has been fixed as
$|U_{eN}|^{2}=10^{-10}$. The dotted line represents the constraint originating
from the measured branching ratio of $K^{+}\to\pi^{+}\nu\overline{\nu}$.

For benchmark B1.1 (B1.2), MATHUSLA will be able to probe the WC as small as
$2\times 10^{-5}v^{-2}$ ($2\times 10^{-4}v^{-2}$) for $m_{N}$ in the range
0.22–0.24 GeV. These numbers translate to the new physics scale $\Lambda$ of
55 (17) TeV.555We associate the new physics scale $\Lambda$ with the effective
operators in the $N_{R}$SMEFT, assuming their WCs are $\Lambda^{4-d}$, with
$d$ denoting the operator mass dimension. The LNC four-fermion operators in
the $N_{R}$LEFT arise from $d=6$ four-fermion operators in the $N_{R}$SMEFT,
whereas the LNV four-fermion operators in the $N_{R}$LEFT originate from $d=7$
operators in the $N_{R}$SMEFT. For details of the matching between the two
effective theories, see e.g. Refs. Beltran:2022ast ; DeVries:2020jbs . In the
case of the LNV operator, we find that for benchmark B2.1 (B2.2), MATHUSLA can
exlude the effective couplings down to $4\times 10^{-6}v^{-2}$ for
$m_{N}\approx 0.23$ GeV (8$\times 10^{-6}$ for $m_{N}\approx 0.21$ GeV).
However, being an LNV operator, it originates at dimension 7 in the
$N_{R}$SMEFT, and hence, these exclusion limits translate to $\Lambda$ of
around 11 (9) TeV. We also show the constraint originating from the
measurement of the branching ratio of $K^{+}\to\pi^{+}\nu\overline{\nu}$. It
is complementary to the projected limits for $m_{N}\lesssim 0.16$–0.17 GeV.

Next, we consider the single-$N_{R}$ operator benchmarks summarized in the
right part of table 4. We start with benchmarks B3 and B4, for which we assume
that both HNL production and decay proceed via the same effective operator,
but carrying different quark flavor indices. The indices 12 lead to the HNL
production in kaon decays, while the indices 11 realize the decay $N\to
e^{\mp}\pi^{\pm}$. For these benchmarks, we also assume that there is no
active-heavy neutrino mixing. We consider two single-$N_{R}$ operators
$\mathcal{O}_{udeN}^{V,RR}$ and $\mathcal{O}_{udeN}^{S,RR}$, both conserving
the lepton number. (The results for the corresponding LNV operators are the
same under the assumption of zero active-heavy neutrino mixing.) For graphical
presentation, we choose to set the production and decay couplings equal, i.e.
$c_{udeN,12}^{V,RR}=c_{udeN,11}^{V,RR}$ for B3, and
$c_{udeN,12}^{S,RR}=c_{udeN,11}^{S,RR}$ for B4, and show the derived exclusion
limits in the plane WC vs. $m_{N}$, see figure 6.

Figure 6: Exclusion limits in the plane WC vs. $m_{N}$ for single-$N_{R}$
operator benchmarks B3 (left) and B4 (right). The production and decay
couplings have been assumed to be equal. The recast bounds from NA62, T2K, and
PS191, as well as the constraint originating from the measured branching ratio
of $K^{+}\to e^{+}\nu_{e}$, are also shown.

In addition to the sensitivities of the proposed LLP detectors at the LHC, we
show the recast bounds from NA62 NA62:2020mcv , T2K T2K:2019jwa , and PS191
Bernardi:1987ek  obtained according to the procedure explained at the end of
Sec. 3. All the far detectors at the LHC will have better reach to these
scenarios than NA62, which excludes WC values larger than approximately
(2–3)$\times 10^{-5}v^{-2}$ ((2–3)$\times 10^{-6}v^{-2}$) for benchmark B3
(B4). For the vector-type operator and $m_{N}\approx 0.35$–$0.40$ GeV,
MATHUSLA will probe the effective couplings as small as $9.5\times
10^{-8}v^{-2}$, and FASER down to $1.4\times 10^{-5}v^{-2}$, with the
sensitivities of the other experiments lying between these two extremes.
Translating these numbers to the new-physics scale $\Lambda$, we find that
MATHUSLA (FASER) will be sensitive to $\Lambda$ as high as 798 (65) TeV. For
the scalar-type operator, the reach in the $N_{R}$LEFT WC is around one order
of magnitude better, owing to the larger branching ratio of $K^{\pm}\to
e^{\pm}N$ in this case, see figure 2. The new-physics scales, which could be
probed by MATHUSLA (FASER) for $m_{N}\approx 0.25$–0.35 GeV, are in excess of
2000 (200) TeV.

Finally, we turn to benchmarks B5–B8, where both an effective operator and
active-heavy neutrino mixing contribute to HNL production, while the HNL decay
proceeds via mixing only, see table 4. Here, we consider both LNC vector (B5)
and scalar (B6) operators, and LNV vector (B7) and scalar (B8) operators,
since the interference between the effective interaction and the mixing term
is slightly different in the LNC and LNV cases, cf. figures 2 and 3. In figure
7, we fix the corresponding WC to $10^{-5}v^{-2}$ (to satisfy the constraint
coming from $K^{+}\to e^{+}\nu_{e}$) and present the sensitivities in the
plane $|U_{eN}|^{2}$ vs. $m_{N}$.

Figure 7: Projected exclusion limits in the plane $|U_{eN}|^{2}$ vs. $m_{N}$
for single-$N_{R}$ operator benchmarks B5 and B6 (top), and B7 and B8
(bottom). The absolute value of the corresponding WC has been fixed to
$10^{-5}v^{-2}$. The recast bounds from NA62, T2K, and PS191 are also shown.

For the vector-type operators and $m_{N}\approx 0.35$–$0.40$ GeV, MATHUSLA
could probe $|U_{eN}|^{2}$ down to $3.0~{}(2.0)\times 10^{-10}$ for the LNC
(LNV) operator. For the scalar-type operators, the MATHUSLA reach is the same
for the LNC and LNV operators: $|U_{eN}|^{2}\approx 1.0\times 10^{-11}$ for
$m_{N}\approx 0.35$ GeV.

The sensitivity curves in the case of the LNC scalar-type operator (top-right
plot) has a different shape compared to the other operators. The difference is
caused by the interference terms of the single-$N_{R}$ operator and
$c_{\mathrm{mix}}$ in eq. (23). This branching ratio, corresponding to the
decay $K^{-}\rightarrow e^{-}+N$, dominates the production of the HNLs and
thus can influence the shape significantly. For the LNC scalar-type operator
(B6), under the assumption that $m_{e}\ll m_{N}$, there is a destructive
interference term proportional to $m_{N}(m_{K}^{2}-m_{N}^{2})$ in the decay
rate. In contrast, for the LNV scalar-type operator (B8), the interference
term is proportional to $m_{e}(m_{K}^{2}+m_{N}^{2})$, and, consequently, it is
suppressed by the small value of $m_{e}$ across the entire range of the
$m_{N}$ values considered in this work. If the corresponding coefficient of
the scalar-type operator is fixed to $10^{-5}v^{-2}$, this stronger
interference in B6 causes a local minimum in the leading branching ratio
BR$(K^{-}\to e^{-}+N)$ for every HNL mass $m_{N}$ in the $U_{eN}$-axis. Hence,
the shape of the isocurves might be influenced, as can be seen in the
isocurves of ANUBIS and MATHUSLA. This effect can even lead to isolated
isocurves for detectors such as CODEX-b and MAPP2.

We also display the recast bounds from NA62, T2K, and PS191. As can be seen,
the NA62 limit is more stringent than the expected limits from the future LLP
detectors for $m_{N}\lesssim 0.35$ GeV. For larger $m_{N}$, MATHUSLA and the
other far detectors take over, excluding new parts of the parameter space. For
the scalar-type operators (benchmarks B6 and B8), the recast NA62 bound covers
the naive seesaw band for $m_{N}\lesssim 0.43$ GeV, and the projected MATHUSLA
(ANUBIS) exclusion limits go beyond the seesaw target region for
$0.17~{}\text{GeV}\lesssim m_{N}\lesssim 0.47$ GeV ($0.20~{}\text{GeV}\lesssim
m_{N}\lesssim 0.44$ GeV).

In figure 8, we fix $|U_{eN}|^{2}=10^{-10}$ and show the exclusion limits in
the plane WC vs. $m_{N}$.

Figure 8: Exclusion limits in the plane WC vs. $m_{N}$ for single-$N_{R}$
operator benchmarks B5 and B6 (top), and B7 and B8 (bottom). The active-heavy
neutrino mixing parameter has been fixed as $|U_{eN}|^{2}=10^{-10}$. The
recast bounds from NA62, T2K, and PS191, as well as the constraint originating
from the measured branching ratio of $K^{+}\to e^{+}\nu_{e}$, are also shown.

The recast NA62 bound covers the ranges which will be accessible to the future
LLP detectors for $m_{N}\lesssim 0.35$ GeV. For larger HNL masses, MATHUSLA
will probe an unexplored region of the parameter space, ruling out the WC
$\gtrsim 2\times 10^{-5}v^{-2}$ ($\gtrsim 3\times 10^{-6}v^{-2}$) for the
vector-type (scalar-type) operators. Overall, for benchmarks B5–B8, the future
far detectors will access new parameter space for larger HNL masses.

##  5 Summary

In this work, we have studied the potential of present and future far-detector
experiments at the LHC for probing long-lived heavy neutral leptons (HNLs) of
Majorana nature produced from rare kaon decays, in the theoretical framework
of low-energy effective field theory extended with sterile neutrinos
($N_{R}$LEFT). We have focused on dimension-6 effective operators consisting
of a pair of quarks together with a charged lepton and an HNL, or a pair of
HNLs. Besides the effective operators, we also take into account the minimal
mixing parameter between the HNLs and the standard-model active neutrinos. For
simplicity, we assume in this work that there is only one kinematically
relevant HNL, that the HNL mixes with the electron neutrino only, and that for
the effective operators also only the first-generation leptons are considered.

There are both lepton-number-conserving and lepton-number-violating operators;
we have investigated both of them and elaborated on their differences. We have
computed the kaons’ decay branching ratios into the HNLs with the considered
effective operators, as a function of the HNL mass, the effective couplings,
and the mixing parameter $U_{eN}$. In addition, the decay rates of the HNLs in
the $N_{R}$LEFT are calculated with care, including the interference between
the EFT operators and the minimal mixing contributions. We have further
performed detailed Monte-Carlo simulations, in order to determine the
acceptance of the far detectors for the long-lived HNLs. These experiments
include ANUBIS, CODEX-b, FASER and FASER2, FACET, MATHUSLA, and MoeDAL-MAPP1
and MAPP2. In particular, because of the long-lifetime nature of the kaons
($K^{\pm}$, $K_{S}$, and $K_{L}$), we cannot assume they decay essentially at
the interaction points; we have thus taken into account their decay positions
in the simulation when we compute the decay probability of the HNLs in the
detector fiducial volumes. Moreover, for the various experiments, we have
properly placed a cut-off position beyond which the kaons are vetoed.

For a series of benchmark scenarios classified by the number of HNLs in the
operators as well as the Lorentz structure of the operators, we have obtained
numerical results. Besides the projection for the LHC far detectors, we have
recast existing bounds on the HNLs in the minimal mixing scenario into those
on the HNLs in the considered EFT benchmarks. We find that for the
pair-$N_{R}$ scenarios, the existing bounds from NA62 are already so strong
that it has excluded all the parameter space that could be probed by future
far-detector experiments. On the other hand, for the single-$N_{R}$
benchmarks, the studied future experiments are sensitive to regions of
parameter space currently unexcluded. Particularly, for benchmarks B3 and B4,
the projected limits on the effective couplings can be orders of magnitude
stronger than the existing bounds.

To summarize, our findings in this work show the potential of the proposed far
detectors at the LHC for studying long-lived HNLs in the EFT framework, and
highly motivate the construction and operation of the proposed far detectors
at the LHC.

## Acknowledgements

We thank Juan Carlos Helo and Felix Kling for useful discussions, and thank
Giovanna Cottin for contributions in the early stage of the project. This work
is supported by the Spanish grants PID2020-113775GB-I00 (AEI/10.13039/
501100011033) and CIPROM/2021/054 (Generalitat Valenciana). R.B. acknowledges
financial support from the Generalitat Valenciana (grant ACIF/2021/052).

##  Appendix A Kaon decays into HNLs

Throughout this work, we assume $N$ to be a Majorana particle. We neglect
$|\varepsilon|\ll 1$ responsible for indirect $CP$ violation in the neutral
kaon system, such that $K_{S}$ and $K_{L}$ coincide with $K_{1}$ and $K_{2}$
given by eq. (2). Since the latter are superpositions of flavor eigenstates
and the effective operators we consider are written in the flavor basis, it
proves convenient to define the following combinations of the Wilson
coefficients:

$\displaystyle a^{V}_{ij}$ $\displaystyle\equiv
c_{dN,ij}^{V,RR}-c_{dN,ij}^{V,LR}\qquad\text{and}\qquad a^{S}_{ij}\equiv
c_{dN,ij}^{S,RR}-c_{dN,ij}^{S,LR}\,,$ (7) $\displaystyle b^{V}_{ij}$
$\displaystyle\equiv c_{dN,ij}^{V,RR}+c_{dN,ij}^{V,LR}\qquad\text{and}\qquad
b^{S}_{ij}\equiv c_{dN,ij}^{S,RR}+c_{dN,ij}^{S,LR}\,.$ (8)

In addition, for the single-$N_{R}$ operators with a charged lepton, we define

$\displaystyle c^{V}_{ij}$ $\displaystyle\equiv
c_{udeN,ij}^{V,RR}-c_{udeN,ij}^{V,LR}\qquad\text{and}\qquad c^{S}_{ij}\equiv
c_{udeN,ij}^{S,RR}-c_{udeN,ij}^{S,LR}\,,$ (9) $\displaystyle d^{V}_{ij}$
$\displaystyle\equiv
c_{udeN,ij}^{V,RL}-c_{udeN,ij}^{V,LL}\qquad\text{and}\qquad d^{S}_{ij}\equiv
c_{udeN,ij}^{S,RL}-c_{udeN,ij}^{S,LL}\,,$ (10) $\displaystyle g^{V}_{ij}$
$\displaystyle\equiv
c_{udeN,ij}^{V,RR}+c_{udeN,ij}^{V,LR}\qquad\text{and}\qquad g^{S}_{ij}\equiv
c_{udeN,ij}^{S,RR}+c_{udeN,ij}^{S,LR}\,,$ (11) $\displaystyle h^{V}_{ij}$
$\displaystyle\equiv
c_{udeN,ij}^{V,RL}+c_{udeN,ij}^{V,LL}\qquad\text{and}\qquad h^{S}_{ij}\equiv
c_{udeN,ij}^{S,RL}+c_{udeN,ij}^{S,LL}\,.$ (12)

###  A.1 Form factors

The non-zero hadronic matrix elements entering the computations of kaon decay
amplitudes are given by

$\displaystyle\langle 0|\bar{s}\gamma^{\mu}\gamma_{5}d|K^{0}\rangle$
$\displaystyle=if_{K}p_{K}^{\mu}\,,$ (13) $\displaystyle\langle
0|\bar{s}\gamma_{5}d|K^{0}\rangle$
$\displaystyle=i\frac{m_{K}^{2}}{m_{d}+m_{s}}f_{K}\equiv if_{K}^{S}\,,$ (14)

for leptonic kaon decays. Here $m_{K}=493.6$ MeV is the neutral kaon mass, and
$f_{K}=155.7$ MeV FLAG:2021npn , and $f_{K}^{S}$ are the kaon decay constants.
For $K^{0}\rightarrow\pi^{-}$ transitions, the relevant matrix elements read

$\displaystyle\langle\pi^{-}|\bar{s}\gamma^{\mu}u|K^{0}\rangle$
$\displaystyle=f_{+}(q^{2})P^{\mu}+\left(f_{0}(q^{2})-f_{+}(q^{2})\right)\frac{m_{K}^{2}-m_{\pi}^{2}}{q^{2}}q^{\mu}\,,$
(15) $\displaystyle\langle\pi^{-}|\bar{s}u|K^{0}\rangle$
$\displaystyle=\frac{m_{K}^{2}-m_{\pi}^{2}}{m_{u}-m_{s}}f_{0}(q^{2})\equiv
f_{S}(q^{2})\,,$ (16)
$\displaystyle\langle\pi^{-}|\bar{s}\sigma^{\mu\nu}u|K^{0}\rangle$
$\displaystyle=\frac{i}{m_{K}}\left(p_{K}^{\mu}p_{\pi}^{\nu}-p_{K}^{\nu}p_{\pi}^{\mu}\right)B_{T}(q^{2})\,,$
(17)

where $P=p_{K}+p_{\pi}$ is the sum of the kaon and pion 4-momenta, and
$q=p_{K}-p_{\pi}$ is their difference. The three form factors ($f_{+}$,
$f_{0}$, $B_{T}$) can be parameterized in terms of $q^{2}$ as follows (see
Ref. Falkowski:2021bkq  for more detail)

$\displaystyle f_{+}(q^{2})$
$\displaystyle=f_{+}(0)+\Lambda_{+}\frac{q^{2}}{m_{\pi}^{2}}\,,$ (18)
$\displaystyle f_{0}(q^{2})$ $\displaystyle=f_{+}(0)+\left(\log
C-G(0)\right)\frac{m_{\pi}^{2}}{m_{K}^{2}-m_{\pi}^{2}}\frac{q^{2}}{m_{\pi}^{2}}\,,$
(19) $\displaystyle B_{T}(q^{2})$
$\displaystyle=B_{T}(0)\left(1-s_{T}^{K\pi}q^{2}\right)\,,$ (20)

and the numerical values of the parameters entering these expressions are
reported in table 5.

Parameter | $f_{+}(0)$ | $\Lambda_{+}$ | $\log C$ | $G(0)$ | $B_{T}(0)/f_{+}(0)$ | $s_{T}^{K\pi}~{}[\text{GeV}^{-2}]$
---|---|---|---|---|---|---
Central value | 0.9706 | 0.02422 | 0.1198 | 0.0398 | 0.68 | 1.10
Table 5: Parameters entering the form factors $f_{+}$, $f_{0}$ and $B_{T}$ in
eqs. (18)–(20).

###  A.2 Two-body decays

The partial decay widths of $K_{S/L}\to NN$ mediated by the pair-$N_{R}$
operators $\mathcal{O}_{dN}$ given in table 1 read:

$\displaystyle\Gamma(K_{S}\to NN)$
$\displaystyle=\frac{m_{K}}{32\pi}\sqrt{1-\frac{4m_{N}^{2}}{m_{K}^{2}}}\,\bigg{[}4f_{K}^{2}m_{N}^{2}\left(\operatorname{Im}a^{V}_{21}\right)^{2}$
$\displaystyle+\left(f_{K}^{S}\right)^{2}\bigg{\\{}\left[\operatorname{Im}\left(a^{S}_{21}-a^{S}_{12}\right)\right]^{2}+\left[\operatorname{Re}\left(a^{S}_{21}-a^{S}_{12}\right)\right]^{2}\left(1-\frac{4m_{N}^{2}}{m_{K}^{2}}\right)\bigg{\\}}$
$\displaystyle+4f_{K}f_{K}^{S}m_{N}\operatorname{Im}a^{V}_{21}\operatorname{Im}\left(a^{S}_{21}-a^{S}_{12}\right)\bigg{]}\,,$
(21) $\displaystyle\Gamma(K_{L}\to NN)$
$\displaystyle=\frac{m_{K}}{32\pi}\sqrt{1-\frac{4m_{N}^{2}}{m_{K}^{2}}}\,\bigg{[}4f_{K}^{2}m_{N}^{2}\left(\operatorname{Re}a^{V}_{21}\right)^{2}$
$\displaystyle+\left(f_{K}^{S}\right)^{2}\bigg{\\{}\left[\operatorname{Re}\left(a^{S}_{21}+a^{S}_{12}\right)\right]^{2}+\left[\operatorname{Im}\left(a^{S}_{21}+a^{S}_{12}\right)\right]^{2}\left(1-\frac{4m_{N}^{2}}{m_{K}^{2}}\right)\bigg{\\}}$
$\displaystyle+4f_{K}f_{K}^{S}m_{N}\operatorname{Re}a^{V}_{21}\operatorname{Re}\left(a^{S}_{21}+a^{S}_{12}\right)\bigg{]}\,.$
(22)

In this computation, we have neglected the contribution from active-heavy
mixing for two reasons: (i) the amplitude for $K_{S/L}\to NN$ is proportional
to $U_{\ell N}^{2}$ and $|U_{\ell N}|^{2}\ll 1$, and (ii) flavor-changing
neutral currents are further suppressed, since in the SM they do not occur at
tree level.

The single-$N_{R}$ operators with a charged lepton summarized in table 2 as
well as the standard active-heavy neutrino mixing trigger $K^{-}\to\ell^{-}N$.
For the corresponding decay width, we find

$\displaystyle\Gamma(K^{-}\to\ell^{-}N)$
$\displaystyle=\frac{\lambda^{1/2}\left(m_{K}^{2},m_{\ell}^{2},m_{N}^{2}\right)}{64\pi
m_{K}^{3}}\bigg{\\{}f_{K}^{2}\left(\left|c^{V}_{12}\right|^{2}+\left|d^{V}_{12}\right|^{2}+|c_{\text{mix}}|^{2}-2\operatorname{Re}\left(d^{V}_{12}c_{\text{mix}}^{\ast}\right)\right)$
$\displaystyle\times\left[m_{K}^{2}\left(m_{\ell}^{2}+m_{N}^{2}\right)-\left(m_{\ell}^{2}-m_{N}^{2}\right)^{2}\right]$
$\displaystyle+\left(f_{K}^{S}\right)^{2}\left(\left|c^{S}_{12}\right|^{2}+\left|d^{S}_{12}\right|^{2}\right)\left[m_{K}^{2}-m_{\ell}^{2}-m_{N}^{2}\right]$
$\displaystyle+2f_{K}f_{K}^{S}\operatorname{Re}\left(c^{V}_{12}c^{S\ast}_{12}+d^{V}_{12}d^{S\ast}_{12}-d^{S\ast}_{12}c_{\text{mix}}\right)m_{\ell}\left[m_{K}^{2}-m_{\ell}^{2}+m_{N}^{2}\right]$
$\displaystyle-4f_{K}^{2}\operatorname{Re}\left(c^{V}_{12}d^{V\ast}_{12}-c^{V}_{12}c_{\text{mix}}^{\ast}\right)m_{K}^{2}m_{\ell}\,m_{N}-4\left(f_{K}^{S}\right)^{2}\operatorname{Re}\left(c^{S}_{12}d^{S\ast}_{12}\right)m_{\ell}\,m_{N}$
$\displaystyle-2f_{K}f_{K}^{S}\operatorname{Re}\left(c^{V}_{12}d^{S\ast}_{12}+c^{S}_{12}d^{V\ast}_{12}-c^{S}_{12}c_{\text{mix}}^{\ast}\right)m_{N}\left[m_{K}^{2}+m_{\ell}^{2}-m_{N}^{2}\right]\bigg{\\}}\,,$
(23)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$. This result complements
eq. (52) of Ref. DeVries:2020jbs , where only one LNV operator
($C_{\mathrm{VLL}}^{(6)}$ in their notation) arising from the $N_{R}$SMEFT at
$d=6$ has been included. In our notation, it means
$d^{V}_{12}=-c_{udeN,12}^{V,LL}$ and $d^{S}_{12}=0$. We find agreement with
eq. (52) of Ref. DeVries:2020jbs  in this limit.

###  A.3 Three-body decays

We rely on the isospin symmetry and express the results in terms of the form
factors $f_{+}$, $f_{0}$, $f_{S}$ and $B_{T}$ for
$\langle\pi^{-}|\overline{s}\,\Gamma\,u|K^{0}\rangle$ given in eqs. (18)–(20).
By the isospin symmetry, we have

$\displaystyle\langle\pi^{0}|\overline{s}\,\Gamma\,d|K^{0}\rangle$
$\displaystyle=-\frac{1}{\sqrt{2}}\langle\pi^{-}|\overline{s}\,\Gamma\,u|K^{0}\rangle\,,$
(24) $\displaystyle\langle\pi^{+}|\overline{s}\,\Gamma\,d|K^{+}\rangle$
$\displaystyle=\langle\pi^{-}|\overline{s}\,\Gamma\,u|K^{0}\rangle\,.$ (25)

The amplitude for $K_{S/L}\to\pi^{0}NN$ is given by

$\displaystyle\mathcal{M}(K_{S/L}\to\pi^{0}NN)$
$\displaystyle=\frac{1}{4}\left(b^{V}_{21}\pm
b^{V\ast}_{21}\right)\left[f_{+}P_{\mu}+\left(f_{0}-f_{+}\right)\frac{m_{K}^{2}-m_{\pi}^{2}}{q^{2}}q_{\mu}\right]\left[\overline{u_{N}}\gamma^{\mu}\gamma_{5}v_{N^{\prime}}\right]$
$\displaystyle+\frac{f_{S}}{2}\left(b^{S}_{21}\mp
b^{S}_{12}\right)\left[\overline{u_{N}}P_{R}v_{N^{\prime}}\right]+\frac{f_{S}}{2}\left(b^{S}_{12}\mp
b^{S}_{21}\right)^{\ast}\left[\overline{u_{N}}P_{L}v_{N^{\prime}}\right]\,.$
(26)

The squared amplitudes summed over spins then read:

$\displaystyle\sum_{\mathrm{spins}}\left|\mathcal{M}(K_{S}\to\pi^{0}NN)\right|^{2}$
$\displaystyle=2\left(\operatorname{Re}b^{V}_{21}\right)^{2}\bigg{\\{}-4f_{+}^{2}(p\cdot
p_{N})^{2}+2f_{+}^{2}(a+m_{K}^{2}-m_{\pi}^{2})(p\cdot p_{N})$ $\displaystyle-
f_{+}^{2}\left[a(m_{K}^{2}+m_{N}^{2})-2m_{N}^{2}(m_{K}^{2}+m_{\pi}^{2})\right]+\frac{(f_{0}^{2}-f_{+}^{2})}{a}m_{N}^{2}(m_{K}^{2}-m_{\pi}^{2})^{2}\bigg{\\}}$
$\displaystyle+\frac{f_{S}^{2}}{2}\,\bigg{\\{}\left[\operatorname{Re}\left(b^{S}_{21}-b^{S}_{12}\right)\right]^{2}a+\left[\operatorname{Im}\left(b^{S}_{21}-b^{S}_{12}\right)\right]^{2}\left(a-4m_{N}^{2}\right)\bigg{\\}}$
$\displaystyle+2f_{0}f_{S}\,m_{N}\left(m_{K}^{2}-m_{\pi}^{2}\right)\operatorname{Re}b^{V}_{21}\operatorname{Re}\left(b^{S}_{21}-b^{S}_{12}\right)\,,$
(27)
$\displaystyle\sum_{\mathrm{spins}}\left|\mathcal{M}(K_{L}\to\pi^{0}NN)\right|^{2}$
$\displaystyle=2\left(\operatorname{Im}b^{V}_{21}\right)^{2}\bigg{\\{}-4f_{+}^{2}(p\cdot
p_{N})^{2}+2f_{+}^{2}(a+m_{K}^{2}-m_{\pi}^{2})(p\cdot p_{N})$ $\displaystyle-
f_{+}^{2}\left[a(m_{K}^{2}+m_{N}^{2})-2m_{N}^{2}(m_{K}^{2}+m_{\pi}^{2})\right]+\frac{(f_{0}^{2}-f_{+}^{2})}{a}m_{N}^{2}(m_{K}^{2}-m_{\pi}^{2})^{2}\bigg{\\}}$
$\displaystyle+\frac{f_{S}^{2}}{2}\,\bigg{\\{}\left[\operatorname{Im}\left(b^{S}_{21}+b^{S}_{12}\right)\right]^{2}a+\left[\operatorname{Re}\left(b^{S}_{21}+b^{S}_{12}\right)\right]^{2}\left(a-4m_{N}^{2}\right)\bigg{\\}}$
$\displaystyle+2f_{0}f_{S}\,m_{N}\left(m_{K}^{2}-m_{\pi}^{2}\right)\operatorname{Im}b^{V}_{21}\operatorname{Im}\left(b^{S}_{21}+b^{S}_{12}\right)\,.$
(28)

For $K^{+}\to\pi^{+}NN$, we find:

$\displaystyle\mathcal{M}(K^{+}\to\pi^{+}NN)$
$\displaystyle=\frac{b^{V}_{21}}{2}\left[f_{+}P_{\mu}+\left(f_{0}-f_{+}\right)\frac{M^{2}-m^{2}}{q^{2}}q_{\mu}\right]\left[\overline{u_{N}}\gamma^{\mu}\gamma_{5}v_{N^{\prime}}\right]$
$\displaystyle+f_{S}\,b^{S}_{21}\left[\overline{u_{N}}P_{R}v_{N^{\prime}}\right]+f_{S}\,b^{S\ast}_{12}\left[\overline{u_{N}}P_{L}v_{N^{\prime}}\right]\,,$
(29)

and

$\displaystyle\sum_{\mathrm{spins}}\left|\mathcal{M}(K^{+}\to\pi^{+}NN)\right|^{2}$
$\displaystyle=2\left|b^{V}_{21}\right|^{2}\bigg{\\{}-4f_{+}^{2}(p\cdot
p_{N})^{2}+2f_{+}^{2}(a+m_{K}^{2}-m_{\pi}^{2})(p\cdot p_{N})$ $\displaystyle-
f_{+}^{2}\left[a(m_{K}^{2}+m_{N}^{2})-2m_{N}^{2}(m_{K}^{2}+m_{\pi}^{2})\right]+\frac{(f_{0}^{2}-f_{+}^{2})}{a}m_{N}^{2}(m_{K}^{2}-m_{\pi}^{2})^{2}\bigg{\\}}$
$\displaystyle+f_{S}^{2}\,\bigg{\\{}\left(\left|b^{S}_{21}\right|^{2}+\left|b^{S}_{12}\right|^{2}\right)\left(a-2m_{N}^{2}\right)-4m_{N}^{2}\operatorname{Re}\left(b^{S}_{21}b^{S}_{12}\right)\bigg{\\}}$
$\displaystyle+2f_{0}f_{S}\,m_{N}\left(m_{K}^{2}-m_{\pi}^{2}\right)\operatorname{Re}\left[b^{V}_{21}\left(b^{S\ast}_{21}-b^{S}_{12}\right)\right]\,.$
(30)

Turning to the single-$N_{R}$ operators with a charged lepton, we have

$\displaystyle\mathcal{M}(K^{-}\to\pi^{0}\ell^{-}N)$
$\displaystyle=\frac{1}{2\sqrt{2}}\bigg{\\{}g^{V}_{12}\left[\overline{u_{\ell}}\gamma^{\mu}P_{R}v_{N}\right]+(h^{V}_{12}+c_{\text{mix}})\left[\overline{u_{\ell}}\gamma^{\mu}P_{L}v_{N}\right]\bigg{\\}}$
$\displaystyle\times\left[f_{+}P_{\mu}+\left(f_{0}-f_{+}\right)\frac{m_{K}^{2}-m_{\pi}^{2}}{q^{2}}q_{\mu}\right]$
$\displaystyle+\frac{f_{S}}{2\sqrt{2}}\bigg{\\{}g^{S}_{12}\left[\overline{u_{\ell}}P_{R}v_{N}\right]+h^{S}_{12}\left[\overline{u_{\ell}}P_{L}v_{N}\right]\bigg{\\}}$
$\displaystyle+\frac{i}{2\sqrt{2}}\frac{B_{T}}{m_{K}}\bigg{\\{}\left[p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu}+i\,\epsilon_{\mu\nu\alpha\beta}\,p^{\alpha}p^{\prime\beta}\right]c_{udeN,12}^{T,RR}\left[\overline{u_{\ell}}\sigma^{\mu\nu}P_{R}v_{N}\right]$
$\displaystyle+\left[p^{\mu}p^{\prime\nu}-p^{\nu}p^{\prime\mu}-i\,\epsilon_{\mu\nu\alpha\beta}\,p^{\alpha}p^{\prime\beta}\right]c_{udeN,12}^{T,LL}\left[\overline{u_{\ell}}\sigma^{\mu\nu}P_{L}v_{N}\right]\bigg{\\}}\,.$
(31)

Finally, up to an unphysical sign,

$\mathcal{M}(K_{S/L}\to\pi^{+}\ell^{-}N)=\mathcal{M}(K^{-}\to\pi^{0}\ell^{-}N)\,.$
(32)

We do not provide the full expression for the modulus square of amplitude (31)
summed over spins, since it is rather cumbersome. However, it is
straightforward to obtain it, especially, assuming only one operator (either
V, S, or T) at a time. In the limit of zero mixing ($c_{\text{mix}}=0$), as in
the case of two-body decay in eq. (23), the LNV single-$N_{R}$ operators
(switched on one at a time) lead to the same results as their LNC
counterparts. Finally, the three-body decay widths are computed following the
procedure explained in Refs. DeVries:2020jbs ; Beltran:2022ast .

##  Appendix B HNL decays

Effective operators in $N_{R}$LEFT not only enhance the HNL production but
also trigger their decay. Assuming only one generation of $N_{R}$, the
pair-$N_{R}$ operators in table 1 cannot make the HNL decay, whereas the
single-$N_{R}$ operators in table 2 do contribute to it. In this appendix we
provide the partial decay width of $N\rightarrow\ell^{-}\pi^{+}$, which is the
only kinematically allowed channel if HNLs are produced in kaon decays and if
$m_{N}>m_{\pi}+m_{\ell}$. In the computation we take into account the
contribution from (i) the $\mathcal{O}_{udeN}$ operators in table 2, (ii) eq.
(4), i.e. the standard mixing to active neutrinos, and (iii) the interference
terms between (i) and (ii). The final expression reads

$\displaystyle\Gamma(N\to\ell^{-}\pi^{+})$
$\displaystyle=\frac{\lambda^{1/2}\left(m_{N}^{2},m_{\ell}^{2},m_{\pi}^{2}\right)}{128\pi
m_{N}^{3}}\bigg{\\{}f_{\pi}^{2}\left(\left|c^{V}_{11}\right|^{2}+\left|d^{V}_{11}\right|^{2}+|c_{\text{mix}}|^{2}-2\operatorname{Re}\left(d^{V}_{11}c_{\text{mix}}^{\ast}\right)\right)$
$\displaystyle\times\left[\left(m_{\ell}^{2}-m_{N}^{2}\right)^{2}-m_{\pi}^{2}\left(m_{\ell}^{2}+m_{N}^{2}\right)\right]$
$\displaystyle+\left(f_{\pi}^{S}\right)^{2}\left(\left|c^{S}_{11}\right|^{2}+\left|d^{S}_{11}\right|^{2}\right)\left[m_{\ell}^{2}+m_{N}^{2}-m_{\pi}^{2}\right]$
$\displaystyle-2f_{\pi}f_{\pi}^{S}\operatorname{Re}\left(c^{V}_{11}c^{S\ast}_{11}+d^{V}_{11}d^{S\ast}_{11}-c_{\text{mix}}d^{S\ast}_{11}\right)m_{\ell}\left[m_{\ell}^{2}-m_{N}^{2}-m_{\pi}^{2}\right]$
$\displaystyle+4f_{\pi}^{2}\operatorname{Re}\left(c^{V}_{11}d^{V\ast}_{11}-c^{V}_{11}c^{\ast}_{\text{mix}}\right)m_{\pi}^{2}m_{\ell}\,m_{N}+4\left(f_{\pi}^{S}\right)^{2}\operatorname{Re}\left(c^{S}_{11}d^{S\ast}_{11}\right)m_{\ell}\,m_{N}$
$\displaystyle-2f_{\pi}f_{\pi}^{S}\operatorname{Re}\left(c^{V}_{11}d^{S\ast}_{11}+c^{S}_{11}d^{V\ast}_{11}-c^{S}_{11}c^{\ast}_{\text{mix}}\right)m_{N}\left[m_{\ell}^{2}-m_{N}^{2}+m_{\pi}^{2}\right]\bigg{\\}}\,.$
(33)

The previous result can be derived from the amplitude leading to eq. (23) by
substituting $K$ with $\pi$ and perfomring the interchange
$p_{\pi}\leftrightarrow p_{N}$ along the computation. The decay constants are
given by $f_{\pi}=130.2$ MeV FLAG:2021npn  and
$f_{\pi}^{S}=\frac{m_{\pi}^{2}}{m_{u}+m_{d}}f_{\pi}$. We have also used the
notation introduced in appendix A for the WC, and $c_{\rm mix}$ is the
coefficient defined in eq. (5).

In the scenarios outlined in table 4, the decay of the HNL is governed by
either the neutrino mixing parameter (scenarios B1–B2 and B5–B8) or one of the
WC of the effective operators (scenarios B3 and B4). In cases where the former
applies, eq. (33) reduces to the one in the minimal scenario
Bondarenko:2018ptm . Conversely, in the latter cases, eq. (33) can be
simplified to

$\displaystyle\Gamma(N\to\ell^{-}\pi^{+})$
$\displaystyle=\frac{\lambda^{1/2}\left(m_{N}^{2},m_{\ell}^{2},m_{\pi}^{2}\right)}{128\pi
m_{N}^{3}}\bigg{\\{}f_{\pi}^{2}\left(\left|c^{V}_{11}\right|^{2}+\left|d^{V}_{11}\right|^{2}\right)\left[\left(m_{\ell}^{2}-m_{N}^{2}\right)^{2}-m_{\pi}^{2}\left(m_{\ell}^{2}+m_{N}^{2}\right)\right]$
$\displaystyle+\left(f_{\pi}^{S}\right)^{2}\left(\left|c^{S}_{11}\right|^{2}+\left|d^{S}_{11}\right|^{2}\right)\left[m_{\ell}^{2}+m_{N}^{2}-m_{\pi}^{2}\right]$
$\displaystyle+4f_{\pi}^{2}\operatorname{Re}\left(c^{V}_{11}d^{V\ast}_{11}\right)m_{\pi}^{2}m_{\ell}\,m_{N}+4\left(f_{\pi}^{S}\right)^{2}\operatorname{Re}\left(c^{S}_{11}d^{S\ast}_{11}\right)m_{\ell}\,m_{N}$
$\displaystyle-2f_{\pi}f_{\pi}^{S}\operatorname{Re}\left(c^{V}_{11}c^{S\ast}_{11}+d^{V}_{11}d^{S\ast}_{11}\right)m_{\ell}\left[m_{\ell}^{2}-m_{N}^{2}-m_{\pi}^{2}\right]$
$\displaystyle-2f_{\pi}f_{\pi}^{S}\operatorname{Re}\left(c^{V}_{11}d^{S\ast}_{11}+c^{S}_{11}d^{V\ast}_{11}\right)m_{N}\left[m_{\ell}^{2}-m_{N}^{2}+m_{\pi}^{2}\right]\bigg{\\}}\,.$
(34)

It is worth mentioning that in scenarios B1–B2 and B5–B8, there are additional
decay channels for $N$, such as the purely leptonic $\nu\nu\nu$ and
$\nu\ell\ell$ channels, which enhance the HNL total decay width. The possible
open channels depend on the HNL mass Bondarenko:2018ptm . Meanwhile, in
scenarios B3–B4, the total decay width becomes twice the result in eq. (34),
since the charge conjugated channel is also open for Majorana $N$.

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