Resonant Majorana neutrino effects in $\Delta$L=2 four-body hyperon decays

Gerardo Hernández-Tomé†, Diego Portillo-Sánchez⋆, and Genaro Toledo†

$\dagger$ Instituto de Física, Universidad Nacional Autónoma de México, AP
20-364, Ciudad de México 01000, México
$\star$Departamento de Física, Centro de Investigación y de Estudios Avanzados
del Instituto Politécnico Nacional
Apartado Postal 14-740, 07000 México D.F., México

###### Abstract

We computed the $\Sigma^{-}\to n\pi^{+}e^{-}\ell^{-}$ ($\ell=e,\mu$),
$\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$, and $\Lambda\to p\pi^{+}e^{-}e^{-}$
lepton number violating (LNV) hyperon decays mediated by a resonant Majorana
neutrino. The expected hyperon production rate of experiments like BES-III of
around $10^{6}-10^{8}$ may allow searching for these rare hyperon decays at
enough sensitivities. We illustrate the limits on the new heavy mixing
parameters derived from these hyperon channels and compare them with other LNV
meson decays in similar mass regions of the resonant neutrino state.

##  1 Introduction

The study of hyperon decay properties had a golden era some sixty years ago
when Cabibbo proposed the universality of charged weak interactions in
semileptonic decays [1]. Hyperon semileptonic decays were used to measure the
weak charges in strangeness-changing transitions and to extract the Cabibbo
angle $\sin\theta_{c}$. On the other hand, non-leptonic decays allowed to
measure the hyperon polarizations and to determine the final state
interactions phases [2, 3, 4]. The field of hyperon physics was somehow
abandoned with the advent of high intensity kaon beams which allowed to
extract the Cabibbo angle with reduced strong interactions uncertainties.
Until the late nineties, only a few searches of rare and forbidden hyperon
decays were reported [5]. In the last twenty-five years, a few more data on
allowed, rare and forbidden hyperon decays were reported by the HyperCP [6],
NA48 [7], LHCb [8], KTeV [9], BESIII [11, 12] collaborations.

The BESIII hyperon physics program has brought a renewed interest in this
field thanks to the large dataset of baryon-antibaryon pairs produced in
$J/\psi$ and $\psi(2S)$ decays [11]. Owing to the non-negligible branching
fractions for these decays, the large production rate of these charmonium
states would allow the production of $10^{6}-10^{8}$ hyperon pairs of
different species. This opens the possibility of improving measurements of
allowed and rare hyperon decays that will set strong limits, for example, on
the rare FCNC hyperon decays with charged lepton or neutrinos pairs [11].
Similarly, searches for forbidden (lepton number or baryon number) decays can
be pursued, allowing to test models that include the violation of these
accidental symmetries [11].

At present, the observation of neutrino oscillation represents one of the most
thrilling discoveries in particle physics, setting new questions about the
nature and origin of their tiny masses. The most promissing approach to
establish if neutrinos are their own antiparticles is to search for lepton
number violating processes ($\Delta$L=2) which would be only possible if that
is the case. Despite the neutrinoless double beta decays in nuclei
($0\nu\beta\beta$) are the most extensively studied and promising laboratory
to give an answer on this matter, alternative and complementary searches for
other LNV processes can play also a crucial role in current and future
experiments, since they provide information on specific energy windows.

In this paper we focus on the $B_{A}(p_{A})\to
B_{B}(p_{B})\ell_{1}^{-}(p_{1})\ell_{2}^{-}(p_{2})\pi^{+}(p_{\pi})$ lepton
number violating (LNV) decays ($B_{A,B}$ denote hyperon states, see Fig. 1).
Specifically, we will consider the following channels: $\Sigma^{-}\to
n\pi^{+}e^{-}\ell^{-}$ ($\ell=e,\mu$), $\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$,
and $\Lambda\to p\pi^{+}e^{-}e^{-}$. This kind of decays have not been studied
before, and they can be induced by the resonant enhancement of intermediate
mass Majorana neutrinos***These novel channels extend the search of similar
LNV effects performed in semileptonic baryon, meson and tau decays [17, 23,
19, 18, 20, 30, 31, 32, 21, 22, 24, 25, 26, 27, 28, 29, 33, 34].. LNV hyperon
decays of the form $B_{A}^{-}\to B_{B}^{+}\ell^{-}\ell^{\prime-}$ have been
studied before in Refs. [13, 14, 15, 16]. These processes are mediated by a
virtual Majorana neutrino and are similar to neutrinoless double beta decays.
On the other hand, resonant production of Majorana neutrinos are possible for
a limited range of their masses in such a way that they can be produced on
their mass-shell. Contrary to production of virtual Majorana neutrinos
processes with rates of $O(G_{F}^{4})$, the rates for production of resonant
Majorana neutrinos becomes of $O(G_{F}^{2})$ [17, 23], which allows to place
better constraints of their parameter space even with upper limits given by
current experimental sensitivities.

In the following we present the formalism to describe these processes and
introduce the integration method for four-body decays, which extend the one
followed in the 3-body case [17, 23] and allows to properly account for the
different charged leptons flavor case. Given the clean experimental signature,
one may expect that very strong limits can be set on the branching fractions
of these decays, similar to existing limits on other $\Delta$L=2 meson decays.
Therefore, it would be interesting to explore if similar limits on the
parameter space of resonant Majorana neutrinos can be obtained from the
proposed four body hyperon decays.

##  2 Computation

(a) | (b)
---|---
Figure 1: Feynman diagrams for the four-body $\Delta L$=2 hyperon decays
mediated by a resonant heavy Majorana neutrino $N$. We consider the following
channels: $\Sigma^{-}\to n\pi^{+}e^{-}\ell^{-}$ ($\ell=e,\mu$),
$\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$, and $\Lambda\to p\pi^{+}e^{-}e^{-}$.
Note that diagram (a) is the dominant one when the neutrino is on-shell
because its contribution is enhanced due to a resonance effect, opposite to
diagram (b) where the neutrino can not become into a resonant state.

Adopting the convention for the neutrino states on Ref. [17], let us consider
an scenario where the leptonic sector incorporates a number $n$ of singlet
right-handed neutrinos $N_{R_{j}}$ ($j=1,2,\ldots n$) in addition to the usual
three left-handed $SU(2)$ lepton doublets $L^{T}_{iL}=(\nu_{i},\ell_{i})_{L}$.
In such scenario, after the proper mass matrix diagonalization, the charged
lepton current relevant for our computation can be written as follows

$\displaystyle\mathcal{L}_{W}=-\frac{g}{\sqrt{2}}W^{+}\bigg{(}\sum_{\ell=e,\mu,\tau}\sum_{i=1}^{3}U_{\ell
i}^{*}\bar{\nu_{i}}\gamma_{\mu}P_{L}\ell+\sum_{\ell=e,\mu,\tau}\sum_{j=4}^{3+n}V_{\ell
j}^{*}\bar{N}_{j}^{c}\gamma_{\mu}P_{L}\ell\bigg{)}+\textrm{h.c.},$ (2.1)

where $P_{L}=(1-\gamma_{5})/2$ is the left-handed chirality projector,
$N^{c}=C\bar{N}^{T}$ is the charge conjugate spinor, and $U_{\ell j}$
($V_{\ell j}$) describes the lepton mixing matrix elements for the light
(heavy) neutrino states.

Similar to previous works, we base our analysis considering the case of a
simply minimal scenario with only one heavy Majorana neutrino $N$, with the
corresponding mass $m_{N}$ and mixing with the charged lepton flavor $V_{\ell
N}$ ($\ell=e,\mu,\tau$) †††This minimal scenario is not able to explain the
current data coming from neutrino oscillations experiments but represents a
simple approach to encode the effects of a larger number of heavy states
present in well-justified massive neutrino models. Recently, the interference
effects in extensions with at least two heavy neutrino states for three-body
meson LNV decays have been reported in [39, 41, 40].. The relevant diagram for
the $B_{A}(p_{A})\to
B_{B}(p_{B})\ell_{1}^{-}(p_{1})\ell_{2}^{-}(p_{2})\pi^{+}(p_{\pi})$ LNV
hyperon decays is depicted in Fig. 1(a), and its amplitude can be written as
follows

$\displaystyle\mathcal{M}_{1}=\left(\frac{G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N}}{a_{1}+i\Gamma_{N}m_{N}}\right)\ell_{\mu\nu}(p_{1},p_{2})\,H^{\mu}(p_{B},p_{A})\,p_{\pi}^{\nu},$
(2.2)

where $a_{1}\equiv(p_{A}-p_{B}-p_{1})^{2}-m_{N}^{2}$, and
$p_{A}-p_{B}-p_{1}=p_{\pi}+p_{2}$ is the momentum carried out by the heavy
neutrino $N$, and we have defined $G\equiv G_{F}^{2}V_{us}V_{ud}$. The
leptonic and hadronic parts are given by

$\displaystyle\ell_{\mu\nu}(p_{1},p_{2})$
$\displaystyle\equiv\bar{u}(p_{1})\gamma_{\mu}\gamma_{\nu}(1+\gamma_{5})v(p_{2}),$
(2.3) $\displaystyle H^{\mu}(p_{B},p_{A})$ $\displaystyle\equiv\langle
B_{B}(p_{B})|J_{\mu}|B_{A}(p_{A})\rangle.$ (2.4)

The hadronic current $J_{\mu}$ is parametrized in terms of six form factors
which are determined from the well-known lepton number conserving hyperon
decays $B_{A}\to B_{B}\ell^{-}\bar{\nu}_{\ell}$ ($\ell=e,\,\mu$) [35, 37, 36,
38]:

$\displaystyle\langle B_{B}(p_{B})|J_{\mu}|B_{A}(p_{A})\rangle$
$\displaystyle=\bar{u}(p_{B})\bigg{[}f_{1}(q^{2})\gamma_{\mu}+if_{2}(q^{2})\frac{\sigma_{\mu\nu}q^{\nu}}{M_{A}}+\frac{q_{\mu}f_{3}(q^{2})}{M_{A}}$
(2.5)
$\displaystyle\quad\quad\quad\,\,\,+g_{1}(q^{2})\gamma_{\mu}\gamma_{5}+ig_{2}(q^{2})\frac{\sigma_{\mu\nu}q^{\nu}\gamma_{5}}{M_{A}}+\frac{q_{\mu}g_{3}(q^{2})\gamma_{5}}{M_{A}}\bigg{]}u(p_{A}),$

where $q^{2}=(p_{A}-p_{B})^{2}$ is the squared momentum transferred in the
hadronic transition, $u(p_{A})$ and $M_{A}$ ($\bar{u}(p_{B})$, and $M_{B}$)
are the spinor and mass of the initial (final) baryon, respectively.
Nevertheless, the contributions of $f_{3}$, and $g_{3}$ form factors in Eq.
(2.5) are negligible in comparison with the other form factors since they pick
up a factor proportional to the mass of the charged-lepton $m_{\ell}$ involved
in the transition [35, 37, 36]. Furthermore, $f_{2}$ and $g_{2}$ are in
principle not negligible, but they become subleading in the SU(3)-flavor
symmetry of QCD [42, 43]. Therefore, in the following we will consider that
the hadronic current describing the hadronic transition in Eq. (2.4) is
dominated by the vector and axial form factors as follows:

$\displaystyle\langle
B_{B}(p_{B})|J_{\mu}|B_{A}(p_{A})\rangle=\bar{u}(p_{B})\gamma_{\mu}\big{[}f_{1}(q^{2})+g_{1}(q^{2})\gamma_{5}\big{]}u(p_{A}).$
(2.6)

Now, from neutrino and electron scattering off nucleons it has been found that
the observed distributions can be described by a dipole parametrization. In
such a way that an extrapolation to the time-like region leads to

$\displaystyle f_{1}(q^{2})$
$\displaystyle=f_{1}(0)\left(1-\frac{q^{2}}{m_{df}^{2}}\right)^{-2},$ (2.7)
$\displaystyle g_{1}(q^{2})$
$\displaystyle=g_{1}(0)\left(1-\frac{q^{2}}{m_{dg}^{2}}\right)^{-2},$ (2.8)

with $m_{df}=0.84$ GeV and $m_{dg}=1.08$ GeV. Since these pole masses
corresponds to strangeness-conserving form factors, a rescaling using the
values of vector and axial mesons masses allows to assume that $m_{df}=0.97$
GeV and $m_{dg}=1.25$ GeV would be a good guess for the dipole masses in the
strangeness-changing case [35, 36]. The values of the form factors at zero
momentum transfer, $f_{1}(0)$ and $g_{1}(0)$ are given in Table 1 and in the
case of the vector form factors they incorporate the effects of SU(3) flavor
symmetry breaking [35, 37, 36, 38].

Transition | $f_{1}(0)$ | $g_{1}(0)$
---|---|---
$\Sigma^{-}\to n$ | -1 | 0.341
$\Xi^{-}\to\Lambda$ | $\sqrt{3/2}$ | 0.239
$\Lambda\to p$ | $-\sqrt{3/2}$ | -0.895
Table 1: Vector and axial transition form factors for weak hyperon decays at
zero momentum transfer ($q^{2}=0$) [35].

It is important to note that the amplitude $\mathcal{M}_{1}$ in Eq. (2.2) has
a resonant effect when $(p_{\pi}+p_{2})^{2}\approx m_{N}^{2}$ ‡‡‡ For the
$B_{A}(p_{A})\to
B_{B}(p_{B})\ell_{1}^{-}(p_{1})\ell_{2}^{-}(p_{2})\pi^{+}(p_{\pi})$ decays
mediated by an intermediate neutrino state produced on-shell its mass must
satisfies that $m_{\ell_{2}^{-}}+m_{\pi^{+}}\leq m_{N}\leq
m_{A}-m_{B}-m_{\ell_{1}^{-}}$.. Besides, if the experiment is unable to
distinguish which lepton was emitted at each stage for non-identical charged
leptons or, for the antisymmetrization of identical leptons we also need to
consider the diagram contribution with the final charged leptons interchanged
$\ell_{1}(p_{1})\leftrightarrow\ell_{2}(p_{2})$ in Fig. 1. This second diagram
has a resonant effect when $(p_{\pi}+p_{1})^{2}\approx m_{N}^{2}$. Since in
general $(p_{\pi}+p_{2})^{2}\neq(p_{\pi}+p_{1})^{2}$, it turns out convenient
to apply the Single-Diagram-Enhanced multi-channel integration method [44].
This method has been implemented for three-body channels. Here we generalize
it to four-body decays, along the same lines by defining the functions

$\displaystyle
f_{{PS}_{1}}=\frac{\overline{|\mathcal{M}_{1}|}^{2}}{\overline{|\mathcal{M}_{1}|}^{2}+\overline{|\mathcal{M}_{2}|}^{2}}\overline{|\mathcal{M}|}^{2},\quad
f_{{PS}_{2}}=\frac{\overline{|\mathcal{M}_{2}|}^{2}}{\overline{|\mathcal{M}_{1}|}^{2}+\overline{|\mathcal{M}_{2}|}^{2}}\overline{|\mathcal{M}|}^{2},$
(2.9)

with $\mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}$. In this way, Eq. (A.3) can
be rewritten as $\overline{|\mathcal{M}|}^{2}=f_{{PS}_{1}}+f_{{PS}_{2}}$, and
consequently the decay width is given by

$\displaystyle\Gamma_{B_{A}\to B_{B}\ell_{1}^{-}\ell_{2}^{-}\pi^{+}}=$
$\displaystyle\frac{N}{4(4\pi)^{6}m_{A}^{3}}\left[\int
f_{{PS}_{1}}dPS_{1}+\int f_{{PS}_{2}}dPS_{2}\right],$ (2.10)

with $N=1/2,(1)$ for the case where the two charged final leptons are same
(different) particles. In our case, the functions $f_{{PS}_{1}}$ and
$f_{{PS}_{2}}$ can be written as follows (see the Appendix A for details)

$\displaystyle f_{{PS}_{1}}$
$\displaystyle=\frac{(G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N})^{2}A}{a_{1}^{2}+\Gamma_{N}^{2}m_{N}^{2}}\bigg{[}1+2\frac{(a_{1}a_{2}+\Gamma_{N}^{2}m_{N}^{2})C_{1}+(a_{2}-a_{1})\Gamma_{N}m_{N}C_{2}}{(a_{2}^{2}+\Gamma_{N}^{2}m_{N}^{2})A+(a_{1}^{2}+\Gamma_{N}^{2}m_{N}^{2})B}\bigg{]},$
(2.11) $\displaystyle f_{{PS}_{2}}$
$\displaystyle=f_{{PS}_{1}}(p_{1}\leftrightarrow p_{2}),$ (2.12)

where the $A$, $B$, $C_{1}$, and $C_{2}$ functions are reported for the first
time in the Appendix A. Now, the phase space integration can be done for
$f_{{PS}_{1}}$ and $f_{{PS}_{2}}$ separately, and added up after the proper
phase space integration. Regarding the first integral in Eq. (2.10) §§§The
phase space variables for the second integral in Eq. (2.10) are chosen
conveniently as ($s_{B2},\,s_{1\pi},\,\theta_{B},\,\theta_{1},\,\phi$) with
the mass invariants $s_{B2}=(p_{B}+p_{2})^{2}$, and
$s_{1\pi}=(p_{1}+p_{\pi})^{2}$., this is described conveniently in terms of
the five independent variables
($s_{B1},\,s_{2\pi},\,\theta_{B},\,\theta_{2},\,\phi$) (see Fig. 1 in
reference [45]):

  * •

$s_{B1}=(p_{B}+p_{1})^{2}$ and $s_{2\pi}=(p_{2}+p_{\pi})^{2}$ stand for the
invariant masses of the $B_{B}\ell_{1}^{-}$ and $\ell_{2}^{-}\pi^{+}$ systems,
respectively.

  * •

$\theta_{B}$ ($\theta_{2}$) is the angle between the three-momentum of $B_{B}$
($\pi^{+}$) in the rest frame of the pair $B_{B}\ell_{1}^{-}$
($\pi^{+}\ell_{2}^{-}$) with respect to the line of flight of the
$B_{B}\ell_{1}^{-}$ ($\pi^{+}\ell_{2}^{-}$) in the rest frame of the particle
$B_{A}$.

  * •

$\phi$ is the angle between the planes defined by the $B_{B}\ell_{1}^{-}$ and
$\pi^{+}\ell_{2}^{-}$ pairs systems in the rest frame of the particle $B_{A}$.

In order to evaluate Eq. (2.10) we need to consider the total decay width for
the new heavy neutrino states. This can be obtained by adding up the
contributions of all its partial decay widths $(\Gamma_{i}^{\textrm{p.w.}})$
that can be opened at the mass $m_{N}$ [17]

$\displaystyle\Gamma_{N}=\sum_{i}\Gamma_{i}^{\textrm{p.w.}}\cdot\theta(m_{N}-\sum_{j}m_{j}),$
(2.13)

where $\theta$ is the Heaviside function and $m_{j}$ stand for the masses of
all the final states particles involved in $\Gamma_{i}^{\textrm{p.w.}}$. Let
us illustrate this point by considering the $\Sigma^{-}\to n\pi^{+}e^{-}e^{-}$
channel, here, the mass of the resonant state must be inside the range
$m_{e^{-}}+m_{\pi^{+}}\leq m_{N}\leq m_{\Sigma^{-}}-m_{n}-m_{e^{-}}$, then the
possible decay channels of the heavy $N$ state (induced by charged and neutral
currents) that contribute to its total decay width $\Gamma_{N}$ are the
following $N\to\ell^{\pm}\pi^{\mp}$, $N\to\pi^{0}\nu_{\ell}$,
$N\to\ell_{1}^{\mp}\ell_{2}^{\pm}\nu_{\ell_{2}}$,
$N\to\ell_{2}^{-}\ell_{2}^{+}\nu_{\ell_{1}}$, and $\nu_{\ell_{1}}\nu\bar{\nu}$
(with $\ell,\,\ell_{1},\,\ell_{2}=e,\,\mu.$). The analytical expressions for
these partial widths can be found in Ref. [17], they depend on each particular
channel considered, and they are given as a function of both the neutrino mass
and the norm of the squared mixings involved, that is
$\Gamma_{i}^{\textrm{p.w}}=\Gamma_{i}^{\textrm{p.w}}(m_{N},|V_{\ell N}|^{2})$.
Then we have considered the indirect limits on the mixing elements of the
heavy neutrino with the three charged leptons [46] in order to estimate the
total neutrino width, namely

$\displaystyle|V_{eN}|\leq 0.050,\quad|V_{\mu N}|\leq 0.021,\quad|V_{\tau
N}|\leq 0.075.$ (2.14)

Using the above values in Eq. (2.13) the total decay width $\Gamma_{N}$ varies
from 0.07 neV to 4.4 neV into the resonant mass region for the $\Sigma^{-}\to
n\pi^{+}e^{-}e^{-}$ decay. The decay width is very small compare with mass of
the new neutral state $\Gamma_{N}\ll m_{N}$, and since
$(p_{2}+p_{\pi})^{2}=s_{2\pi}\approx m_{N}^{2}$ in Eq. (2.11), the narrow
width approximation can be applied. That means, that we can replace

$\displaystyle\frac{1}{(s_{2\pi}-m_{N}^{2})^{2}+m_{N}^{2}\Gamma_{N}^{2}}\rightarrow\frac{\pi}{m_{N}\Gamma_{N}}\delta(s_{2\pi}-m_{N}^{2})$
(2.15)

transforming the five-variable integral in Eq. (2.11) into a four-variable
one:

$\displaystyle\int f_{{PS}_{1}}dPS_{1}$
$\displaystyle=\frac{\pi(G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N})^{2}}{\Gamma_{N}m_{N}}\int
X\beta_{B1}\beta_{2\pi}$ $\displaystyle\times$
$\displaystyle\left[A\left(1+2\frac{\Gamma_{N}^{2}m_{N}^{2}C_{1}+\Gamma_{N}m_{N}a_{2}C_{2}}{(a_{2}^{2}+\Gamma_{N}^{2}m_{N}^{2})A+\Gamma_{N}^{2}m_{N}^{2}B}\right)\right]d{s_{B1}}\,d{\cos\theta_{B}}\,d{\cos\theta_{2}}\,d\phi,$
(2.16)

with the following integration limits:

$\displaystyle(m_{B}+m_{1})^{2}$ $\displaystyle\leq
s_{B1}\leq(m_{A}-m_{2}-m_{\pi})^{2},\quad-1\leq\cos{\theta_{B}}\leq 1,$
$\displaystyle-1$ $\displaystyle\leq\cos{\theta_{2}}\leq
1,\quad\quad\quad\quad\quad\quad\quad\,\,-\pi\leq\phi\leq\pi.$ (2.17)

This provides all the formalism we need to compute the decay width and set the
region of the parameters, given on the expected experimental branching ratio,
as we show below.

##  3 Numerical Analysis

Figure 2: Exclusion regions on the ($m_{N},|V_{eN}|^{2}$) plane by assuming a
$\textrm{BR}(B_{A}\to B_{B}\pi^{+}e^{-}e^{-})<10^{-8}$ limit. The purple line
stand for the $\Sigma^{-}\to\pi^{+}ne^{-}e^{-}$ channel, the blue one for the
$\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$, and the cyan color for the
$\textrm{BR}(\Lambda\to p\pi^{+}e^{-}e^{-})$ decay (see main text for further
details). The solid (dashed) lines correspond to estimates considering a
parameters dependent neutrino width $\Gamma_{N}$ (fixed). Figure 3: Exclusion
regions on the ($m_{N},|V_{eN}V_{\mu N}|$) plane by assuming the
$\textrm{BR}(\Sigma^{-}\to n\pi^{+}e^{-}\mu^{-})<10^{-8}$ limit in the search
of lepton flavor violating hyperon decays.

The projected sensitivity of BES-III for the search of rare and forbidden
hyperon three-body hyperon decays at BES-III is of the order of
$10^{-6}-10^{-8}$ [11] with clean backgrounds ¶¶¶It is also worthy to mention
that these kinds of transitions can be also searched by the LHCb collaboration
with higher sensitivities because of the huge production cross-section there.
. However, there is not an estimation for similar four-body hyperon decays. In
this work, we will assume an optimistic scenario considering similar
sensitivities for three and four-body processes.

In Fig. 2 we show the exclusion region on the plane ($m_{N}$, $|V_{eN}|^{2}$)
for the neutrino resonant state obtained by assuming a rate of BR$(B_{A}\to
B_{B}\pi^{+}e^{-}e^{-})<10^{-8}$ for the channels involving a pair of
electrons in the final state. We have considered here two benchmarks to
evaluate the total neutrino width. On one side, the solid lines represent the
universal coupling assumption, that is, we consider that $V_{eN}=V_{\mu
N}=V_{\tau N}$ in Eq. (2.13); therefore, the total neutrino width (and
consequently the branching ratio of the $B_{A}\to B_{B}\pi^{+}e^{-}e^{-}$
hyperon decays) can be expressed only as a function of
$\Gamma_{N}=\Gamma_{N}(|V_{eN}|^{2},m_{N})$. On the other hand, the dashed
lines represent a scenario where the total neutrino width is fixed to the
razonable value $\Gamma_{N}=10^{-15}$ MeV (consistent with the estimation of
the total neutrino width using the indirect limits reported in Eq. (2.14)).
From this plot, we can observe that, in general, the exclusion region will
depend on which assumption we considered, although in general, they are of the
same order for all the allowed mass of the resonant neutrino in the different
channels. In any case, the most restrictive limits will come from the
$\Sigma^{-}\to n\pi^{+}e^{-}e^{-}$ channel, follow by $\Lambda\to
p\pi^{+}e^{-}e^{-}$, and finally the much less restrictive
$\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$ channel. Additionally to the processes
with a pair of electrons in the final state, the $\Sigma^{-}\to
n\pi^{+}e\mu^{-}$ channel is the only possible kinematically allowed four-body
LNV hyperon decay. As we can see in Fig. 3, if the search for this transition
can achieve a rate of BR$(\Sigma^{-}\to n\pi^{+}e^{-}\mu^{-})<10^{-8}$ then
the limits set on the plane ($m_{N},|V_{eN}V_{\mu N}|$) are much less
restrictive than the di-electronic case because phase space restrictions are
more stringent. For the case with two different flavors notice that the limits
are split into two disconnected regions. The left (right) region on Fig 3, is
associated with the case where the muon (electron) was created along with the
resonant neutrino state, and the electron (muon) comes after the subsequent
neutrino decay. Overlap of these regions can be achieved in other scenarios,
provided the kinematical conditions allow them to do so. The formalism here
developed allow to address both cases regardless of invoking the direct narrow
width approximation or not (see appendix B).

##  4 Conclusions

The search for $\Delta$L=2 processes is crucial for unraveling the Dirac or
Majorana nature of neutrinos. Except possibly for neutrinoless double-beta
decay in nuclei, diverse neutrino mass models predict that LNV effects can lie
beyond the reach of current experiments. However, if new hypothetical heavy
Majorana neutrinos with masses from $\sim 100$ MeV to few GeV can be produced
on-shell as an intermediate state in LNV decays of mesons, baryons, or the tau
lepton, then their branching ratios can be amplified due to a resonant effect.
The no observation of such processes sets limits on the parameter space of
these new heavy neutrinos states. In this regard, most of the studies have
focused on three-body LNV meson or tau decays, however, recently the study of
similar four-body LNV channels has also drawn attention as complementary means
because they can provide information about different kinematical phase-space
regions. In this work, we studied the four-body LNV decays of hyperons
mediated by a resonant Majorana neutrino.

Our results suggest that the direct limits derived on $|V_{eN}|^{2}$ from the
$\Sigma^{-}\to n\pi^{+}e^{-}e^{-}$ channel can be of the same order ($\sim
10^{-3}$) than those obtained from the meson decays, such as
$D^{+}\to\pi^{-}e^{+}e^{+}$ and $D_{s}^{+}\to\pi^{-}e^{+}e^{+}$, but far away
from the current most stringent ones from the semileptonic kaon decay
$K^{+}\to\pi^{-}e^{+}e^{+}$ which is around $\mathcal{O}(10^{-9})$ [23]
∥∥∥Current bounds for $K^{+}\to\pi^{-}e^{+}e^{+}\leq 5.3\times 10^{-11}$ and
$K^{+}\to\pi^{-}e^{+}\mu^{+}\leq 4.2\times 10^{-11}$ are reported by the NA62
experiment at CERN in [48, 47].. Moreover, less restrictive limits on
$|V_{eN}|^{2}$ ($\sim 10^{-1}$) can be obtained from the $\Lambda\to
p\pi^{+}e^{-}e^{-}$ and $\Xi^{-}\to\Lambda\pi^{+}e^{-}e^{-}$ which are
comparable with the limits from the $B^{+}\to\pi^{-}e^{+}e^{+}$ meson channel.
On the other hand, the $\Sigma^{-}\to n\pi^{+}e^{-}\mu^{-}$ channel is the
only possible four-body LNV hyperon decay mediated by a Majorana neutrino
involving a muon as final state, but places a very weak contraints on
$|V_{eN}V_{\mu N}|$ for the small mass on-shell neutrino regions allowed
assuming the expected sensitivity for rare hyperon decays of BES III.

## Acknowledgements

We would like to thank G. López Castro for many helpful discussions. G.H.T.
and G.T. acknowledge the support of DGAPA-PAPIIT UNAM, grant no. IN110622 for
financial support. G.H.T. is funded by PROGRAMA DE BECAS POSDOCTORALES DGAPA-
UNAM. The work of D.P.S. was supported by Ciencia de Frontera Conacyt project
No. 428218 and the program “BECAS CONACYT NACIONALES”.

##  Appendix A Appendix

Following the Feynman rules for fermion number-violating interactions reported
in Ref. [49], the contribution of the second diagram with the charged leptons
interchanged in Fig. 1(a) is given by

$\displaystyle\mathcal{M}_{2}=\left(\frac{G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N}}{a_{2}+i\Gamma_{N}m_{N}}\right)\ell_{\nu\mu}(p_{1},p_{2})\,H^{\mu}(p_{B},p_{A})\,p_{\pi}^{\nu},$
(A.1)

with $a_{2}\equiv(p_{A}-p_{B}-p_{2})^{2}-m_{N}^{2}$. Therefore,
$\mathcal{M}=\mathcal{M}_{1}+\mathcal{M}_{2}$ can be written as follows

$\displaystyle\mathcal{M}=G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N}\bar{u}(p_{1})\left(\frac{\gamma_{\mu}\gamma_{\nu}}{{a_{1}+i\Gamma_{N}m_{N}}}+\frac{\gamma_{\nu}\gamma_{\mu}}{{a_{2}+i\Gamma_{N}m_{N}}}\right)(1+\gamma_{5})v(p_{2})\,H^{\mu}(p_{B},p_{A})\,p_{\pi}^{\nu}.$
(A.2)

The total amplitude squared is given by

$\displaystyle\overline{|\mathcal{M}|}^{2}$
$\displaystyle=\frac{1}{2}\sum_{\textrm{spins}}|\mathcal{M}|^{2}=\frac{1}{2}\sum_{\textrm{spins}}\left(|\mathcal{M}_{1}|^{2}+|\mathcal{M}_{2}|^{2}+2\textrm{Re}[\mathcal{M}_{1}\mathcal{M}_{2}^{\dagger}]\right),$
(A.3)

where the individual contributions can be written as follows

$\displaystyle\overline{|\mathcal{M}_{1}|}^{2}$
$\displaystyle=\frac{1}{2}\sum_{\textrm{spins}}|\mathcal{M}_{1}|^{2}=\frac{1}{2}\frac{(G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N})^{2}A}{\left(a_{1}^{2}+\Gamma_{N}^{2}m_{N}^{2}\right)}$
(A.4) $\displaystyle\overline{|\mathcal{M}_{2}|}^{2}$
$\displaystyle=\frac{1}{2}\sum_{\textrm{spins}}|\mathcal{M}_{2}|^{2}=\frac{1}{2}\frac{(G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N})^{2}B}{\left(a_{2}^{2}+\Gamma_{N}^{2}m_{N}^{2}\right)},$

while the interference term

$\displaystyle\mathcal{M}_{1}\mathcal{M}_{2}^{\dagger}$
$\displaystyle=(G\,V_{\ell_{1}N}V_{\ell_{2}N}f_{\pi}m_{N})^{2}(C_{1}+iC_{2})\frac{\left[a_{1}a_{2}+\Gamma_{N}^{2}m_{N}^{2}+i(a_{1}-a_{2})\right]\Gamma_{N}m_{N}}{\left(a_{1}^{2}+\Gamma_{N}^{2}m_{N}^{2}\right)\left(a_{2}^{2}+\Gamma_{N}^{2}m_{N}^{2}\right)},$
(A.5)

with the $A,$ $B,$ $C_{1}$, and $C_{2}$ functions given by:

$\displaystyle A$
$\displaystyle=64[f_{1}^{2}(q^{2})\xi_{1}+g_{1}^{2}(q^{2})\xi_{2}+f_{1}(q^{2})g_{1}(q^{2})\xi_{3}],$
(A.6) $\displaystyle B$ $\displaystyle=A(p_{1}\leftrightarrow p_{2}),$ (A.7)
$\displaystyle C_{1}$
$\displaystyle=64[f_{1}^{2}(q^{2})\xi_{4}+g_{1}^{2}(q^{2})\xi_{5}+f_{1}(q^{2})g_{1}(q^{2})\xi_{6}]$
$\displaystyle C_{2}$
$\displaystyle=64\epsilon_{\mu\nu\lambda\rho}p_{B}^{\mu}p_{1}^{\nu}p_{2}^{\lambda}p_{\pi}^{\rho}[-(f_{1}^{2}+g_{2}^{2})(r_{A\pi}+r_{B\pi})+f_{1}(q^{2})g_{1}(q^{2})2(r_{1\pi}+r_{2\pi})],$
(A.8)

and the following definitions

$\displaystyle\xi_{1}$
$\displaystyle=m_{A}m_{B}(m_{\pi}^{2}r_{12}-2r_{1\pi}r_{2\pi})-m_{\pi}^{2}(r_{A1}r_{B2}+r_{A2}r_{B1})+2r_{2\pi}(r_{A1}r_{B\pi}+2r_{A\pi}r_{B1}),$
(A.9) $\displaystyle\xi_{2}$
$\displaystyle=\xi_{1}-2m_{A}m_{B}(m_{\pi}^{2}r_{12}-2r_{12}r_{2\pi}),$ (A.10)
$\displaystyle\xi_{3}$
$\displaystyle=2[m_{\pi}^{2}(r_{A2}r_{B1}-r_{A1}r_{B2})+2r_{2\pi}(r_{A1}r_{B\pi}-r_{A\pi}r_{B1})],$
(A.11) $\displaystyle\xi_{4}$
$\displaystyle=-2m_{A}m_{B}r_{1\pi}r_{2\pi}+m_{\pi}^{2}(r_{12}r_{AB}-r_{A1}r_{B2}-r_{A2}r_{B1})-2r_{12}r_{A\pi}r_{B\pi}$
$\displaystyle+r_{1\pi}(r_{A2}r_{B\pi}+r_{A\pi}r_{B2})+r_{2\pi}(r_{A1}r_{B\pi}+r_{A\pi}r_{B1})$
(A.12) $\displaystyle\xi_{5}$
$\displaystyle=\xi_{4}-2m_{A}m_{B}(m_{\pi}^{2}r_{12}-2r_{1\pi}r_{2\pi}),$
(A.13) $\displaystyle\xi_{6}$
$\displaystyle=2[r_{1\pi}(r_{A2}r_{B\pi}-r_{A\pi}r_{B2})+r_{2\pi}(r_{A1}r_{B\pi}-r_{A\pi}r_{B1})].$
(A.14)

In the above expression, we have defined $r_{ij}\equiv p_{i}\cdot p_{j}$ with
$p_{i}$ and $p_{j}$ denoting any of the momenta of the external particles
(that is $p_{i,j}=p_{A},p_{B},p_{1},p_{2},p_{\pi}$). Now, for the set of
variables chosen in Section 2, the scalar products $r_{ij}$ involved in Eqs.
(A.9-A.14) are given as follows

$\displaystyle r_{B1}$
$\displaystyle=\frac{1}{2}(s_{B1}-m_{B}^{2}-m_{1}^{2}),\quad\quad
r_{2\pi}=\frac{1}{2}(s_{2\pi}-m_{2}^{2}-m_{\pi}^{2}),$ (A.15) $\displaystyle
r_{B2}$
$\displaystyle=\frac{1}{4}(\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}),\quad
r_{B\pi}=\frac{1}{4}(\alpha_{1}-\alpha_{2}+\alpha_{3}-\alpha_{4}),$ (A.16)
$\displaystyle r_{12}$
$\displaystyle=\frac{1}{4}(\alpha_{1}+\alpha_{2}-\alpha_{3}-\alpha_{4}),\quad
r_{1\pi}=\frac{1}{4}(\alpha_{1}-\alpha_{2}-\alpha_{3}+\alpha_{4}),$ (A.17)
$\displaystyle r_{AB}$
$\displaystyle=\frac{1}{2}(\alpha_{5}+\alpha_{6}),\quad\quad\quad\quad\quad
r_{A1}=\frac{1}{2}(\alpha_{5}-\alpha_{6}),$ (A.18) $\displaystyle r_{A2}$
$\displaystyle=\frac{1}{2}(\alpha_{7}+\alpha_{8}),\quad\quad\quad\quad\quad
r_{A\pi}=\frac{1}{2}(\alpha_{7}-\alpha_{8}),$ (A.19)
$\displaystyle\epsilon_{\mu\nu\lambda\rho}p_{B}^{\mu}p_{1}^{\nu}p_{2}^{\lambda}p_{\pi}^{\rho}$
$\displaystyle=-\sqrt{s_{B1}s_{2\pi}}\beta_{B1}\beta_{2\pi}X\sin{\theta_{B}}\sin{\theta_{2}}\sin{\phi},$
(A.20)

with the definitions

$\displaystyle\alpha_{1}$
$\displaystyle=\frac{1}{2}(m_{A}^{2}-s_{B1}-s_{2\pi}),$ (A.21)
$\displaystyle\alpha_{2}$
$\displaystyle=X\beta_{2\pi}\cos{\theta_{2}}+\left(\frac{m_{2}^{2}-m_{\pi}^{2}}{s_{2\pi}}\right)\alpha_{1},$
(A.22) $\displaystyle\alpha_{3}$
$\displaystyle=X\beta_{B1}\cos{\theta_{B}}+\left(\frac{m_{B}^{2}-m_{1}^{2}}{s_{B1}}\right)\alpha_{1},$
(A.23) $\displaystyle\alpha_{4}$
$\displaystyle=\left(\frac{m_{B}^{2}-m_{1}^{2}}{s_{B1}}\right)\left(\frac{m_{2}^{2}-m_{\pi}^{2}}{s_{2\pi}}\right)\alpha_{1}+\left(\frac{m_{B}^{2}-m_{1}^{2}}{s_{B1}}\right)X\beta_{2\pi}\cos{\theta_{2}}$
(A.24)
$\displaystyle+\left(\frac{m_{2}^{2}-m_{\pi}^{2}}{s_{2\pi}}\right)X\beta_{B1}\cos{\theta_{B}}+\beta_{B1}\beta_{2\pi}\left(\alpha_{1}\cos{\theta_{B}}\cos_{\theta_{2}}-\sqrt{s_{B1}s_{2\pi}}\sin{\theta_{B}}\sin{\theta_{2}}\cos{\phi}\right)$
$\displaystyle\alpha_{5}$
$\displaystyle=\frac{1}{2}(m_{A}^{2}+s_{B1}-s_{2\pi}),$ (A.25)
$\displaystyle\alpha_{6}$
$\displaystyle=(m_{B}^{2}-m_{1}^{2})\left(1+\frac{\alpha_{1}}{s_{B1}}\right)+X\beta_{B1}\cos{\theta_{B}},$
(A.26) $\displaystyle\alpha_{7}$
$\displaystyle=\frac{1}{2}(m_{A}^{2}-s_{B1}+s_{2\pi}),$ (A.27)
$\displaystyle\alpha_{8}$
$\displaystyle=(m_{2}^{2}-m_{\pi}^{2})\left(1+\frac{\alpha_{1}}{s_{2\pi}}\right)+X\beta_{2\pi}\cos{\theta_{2}},$
(A.28)

and

$\displaystyle\lambda(a,b,c)$ $\displaystyle=a^{2}+b^{2}+c^{2}-2(ab+bc+ac),$
(A.29) $\displaystyle X$
$\displaystyle=\frac{\lambda(m_{A}^{2},s_{B1},s_{2\pi})^{1/2}}{2},\quad\beta_{B1}=\frac{\lambda(s_{B1},m_{B}^{2},m_{1}^{2})^{1/2}}{s_{B1}},\quad\beta_{2\pi}=\frac{\lambda(s_{2\pi},m_{2}^{2},m_{\pi}^{2})^{1/2}}{s_{2\pi}}.$
(A.30)

##  Appendix B Comparison with direct narrow width approximation computation

For completeness and as a crosscheck of our computation we have verified that
for the cases where we can distinguish the flavour of the charged lepton
created as a product of the decay of the resonant state or the channels with
two identical external charged leptons, the results using the Single-Diagram-
Enhanced multi-channel integration method can be reproduced by applying
directly the narrow width approximation

$\displaystyle\textrm{BR}(B_{A}\to
B_{B}\ell_{1}^{-}\ell_{2}^{-}\pi^{+})=\textrm{BR}(B_{A}\to
B_{B}\ell_{1}^{-}N)\times\Gamma(N\to\ell_{2}^{-}\pi^{+})\tau_{N}/\hbar,$ (B.1)

where $\tau_{N}$ is the lifetime of the intermediate neutrino state. In this
case, the partial decay width $\Gamma(N\to\ell_{2}^{-}\pi^{+})$ can be
computed straightforwardly by [17]:

$\displaystyle\Gamma(N\to\ell_{2}^{-}\pi^{+})=\frac{G_{F}^{2}}{16\pi}|V_{ud}|^{2}|V_{\ell_{2}N}|^{2}f_{\pi}^{2}m_{N}\,\lambda^{\frac{1}{2}}\left(m_{N}^{2},m_{\ell_{2}}^{2},m_{\pi}^{2}\right)\big{[}\left(1-x_{\ell_{2}}\right)^{2}-x_{\pi}\left(1+x_{\ell_{2}}\right)\big{]},$
(B.2)

with $x_{y}\equiv m_{y}^{2}/m_{N}^{2}$, $\lambda$ is the Källen function
defined previously, and $f_{\pi}$ is the pion decay constant. Regarding the
subprocess $B_{A}(p_{A})\to B_{B}(p_{B})\ell_{1}^{-}(p_{1})N(p_{N})$ in Eq.
(B.1), the amplitude is given by

$\displaystyle\mathcal{M}(B_{A}\to
B_{B}\ell_{1}^{-}N)=-\frac{G_{F}}{\sqrt{2}}V_{us}V_{\ell_{1}N}\langle
B_{B}(p_{B})|J_{\mu}|B_{A}(p_{A})\rangle L^{\mu},$ (B.3)

with $J_{\mu}$ hadronic previously defined in Eq. (2.6), and the leptonic
current defined as follows

$\displaystyle L^{\mu}\equiv\bar{u}(p_{1})\gamma^{\mu}(1-\gamma_{5})v(p_{N}).$
(B.4)

The squared amplitude of Eq. (B.3) is given by

$\displaystyle|\mathcal{M}|^{2}=\mathcal{M}\mathcal{M}^{\dagger}=$
$\displaystyle
32G_{F}^{2}|V_{us}|^{2}|V_{\ell_{1}N}|^{2}\bigg{[}m_{A}m_{B}\big{(}g_{1}^{2}(q^{2})-f_{1}^{2}(q^{2})\big{)}$
(B.5) $\displaystyle+\big{(}f_{1}(q^{2})-g_{1}(q^{2})\big{)}^{2}(p_{1}\cdot
p_{B})(p_{A}\cdot p_{N})$
$\displaystyle+\big{(}f_{1}(q^{2})+g_{1}(q^{2})\big{)}^{2}(p_{1}\cdot
p_{A})(p_{B}\cdot p_{N})\bigg{]}.$

Now, by defining $s_{1N}\equiv(p_{1}+p_{N})^{2}$ and
$s_{1B}\equiv(p_{1}+p_{B})^{2}$, the branching ratio can be expressed in terms
of these two Lorentz invariants as follows

$\displaystyle\textrm{BR}(B_{A}\to
B_{B}\ell_{1}^{-}N)=\frac{G_{F}^{2}|V_{us}|^{2}|V_{\ell_{1}N}|^{2}}{64\pi^{3}m_{A}^{3}\Gamma_{B_{A}}}\int_{s_{{1B}_{\textrm{min}}}}^{s_{{1B}_{\textrm{max}}}}\int_{s_{{1N}_{\textrm{min}}}}^{s_{{1N}_{\textrm{max}}}}\mathcal{F}(s_{1B},s_{1N})ds_{1N}ds_{1B},$
(B.6)

where

$\displaystyle\mathcal{F}(s_{1B},s_{1N})$
$\displaystyle=2m_{A}m_{B}\big{(}s_{1N}-m_{N}^{2}-m_{1}^{2}\big{)}\bigg{[}\frac{g_{1}^{2}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{g}}^{2}}\big{)}^{4}}-\frac{f_{1}^{2}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{f}}^{2}}\big{)}^{4}}\bigg{]}$
(B.7)
$\displaystyle+\big{(}s_{1B}-m_{B}^{2}-m_{1}^{2}\big{)}\big{(}m_{A}^{2}+m_{N}^{2}-s_{1B}\big{)}\bigg{[}\frac{f_{1}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{f}}^{2}}\big{)}^{2}}-\frac{g_{1}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{g}}^{2}}\big{)}^{2}}\bigg{]}^{2}$
$\displaystyle+\big{(}m_{1}^{2}+m_{A}^{2}-s_{1B}-s_{1N}\big{)}\big{(}s_{1B}+s_{1N}-m_{N}^{2}-m_{B}^{2}\big{)}\bigg{[}\frac{f_{1}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{f}}^{2}}\big{)}^{2}}+\frac{g_{1}(0)}{\big{(}1-\frac{s_{1N}}{m_{d_{g}}^{2}}\big{)}^{2}}\bigg{]}^{2},$

and the phase-space integration limits are

$\displaystyle s_{1B}^{\pm}=$ $\displaystyle
m_{A}^{2}+m_{1}^{2}-s_{1N}-\frac{1}{2s_{1N}}\bigg{[}(m_{A}^{2}-m_{B}^{2}-s_{1N})(s_{1N}-m_{1}^{2}+m_{2}^{2})$
(B.8)
$\displaystyle\pm\sqrt{\lambda(m_{A}^{2},m_{B}^{2},s_{1N})\lambda(s_{1N},m_{1}^{2},m_{2}^{2})}\bigg{]},$

and

$\displaystyle(m_{1}+m_{N})^{2}\leq s_{1N}\leq(m_{A}-m_{B})^{2}.$ (B.9)

Finally, the numerical values (central values) for the masses, lifetimes, and
CKM elements used in our numerical analysis are reported in [50].

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