DESY 13-128 ISSN 0418-9833

LC-PHSM-2013-024

December 2013

Bilinear $R$ Parity Violation at the ILC – Neutrino Physics at Colliders

Benedikt Vormwald1,2 and Jenny List1

1- Deutsches Elektronen-Synchrotron DESY

Notkestr. 85, 22607 Hamburg, Germany

2- Universität Hamburg, Institut für Experimentalphysik

Luruper Chaussee 149, 22761 Hamburg, Germany

Supersymmetry (SUSY) with bilinearly broken $R$ parity (bRPV) offers an
attractive possibility to explain the origin of neutrino masses and mixings.
In such scenarios, the study of neutralino decays at colliders gives access to
neutrino sector parameters.

The ILC offers a very clean environment to study the neutralino properties as
well as its subsequent decays, which typically involve a $W$ or $Z$ boson and
a lepton. This study is based on ILC beam parameters according to the
Technical Design Report for a center of mass energy of $500\,\mathrm{GeV}$. A
full detector simulation of the International Large Detector (ILD) has been
performed for all Standard Model backgrounds and for neutralino pair
production within a simplified model. The bRPV parameters are fixed according
to current neutrino data.

In this scenario, the $\tilde{\chi}^{0}_{1}$ mass can be reconstructed with an
uncertainty of $\delta
m^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}=(40\text{(stat.)}\oplus
50\text{(syst.)})\,\mathrm{\,MeV}$ for an integrated luminosity of
$500\,\mathrm{fb}^{-1}$ from direct $\tilde{\chi}^{0}_{1}$ pair production,
thus, to a large extent independently of the rest of the SUSY spectrum. The
achievable precision on the atmospheric neutrino mixing angle
$\sin^{2}\theta_{23}$ from measuring the neutralino branching fractions
BR($\tilde{\chi}_{1}^{0}\rightarrow W\mu$) and
BR($\tilde{\chi}_{1}^{0}\rightarrow W\tau$) at the ILC is in the same range
than current uncertainties from neutrino experiments. Thus, the ILC could have
the opportunity to unveil the mechanism of neutrino mass generation.

##  1 Introduction

Supersymmetry (SUSY) [1, 2] is a very appealing extension of the Standard
Model (SM). It provides an elegant solution for the Higgs hierarchy problem,
makes gauge unification possible and apart from that, SUSY is the only non-
trivial extension of the Lorentz algebra [3]. In the most general
renormalizable Lagrangian of the Minimal Supersymmetric Standard Model (MSSM)
trilinear and bilinear terms appear, which violate the conservation of baryon
number $B$ and lepton number $L$.

The presence of all these terms would lead to proton decay, which is
experimentally not observed. A common way to circumvent this problem is to
introduce a discrete $\mathbb{Z}_{2}$ symmetry assigned to each field in order
to suppress these terms. This quantum number, called $R$ parity, has the form

$\displaystyle R=(-1)^{3B+L+2S},$ (1)

where $B$ is the baryon number, $L$ the lepton number and $S$ the spin of the
field. Hence, SM particles always carry $R=+1$ and SUSY particles $R=-1$. The
conservation of this quantum number has the consequence, that all $B$ and $L$
breaking terms in the SUSY Lagrangian are forbidden and the proton remains
stable.

However, proton decay only appears if $B$ and $L$ violation is present at the
same time. So, breaking either $B$ or $L$ is well consistent with proton
stability. Thus, $R$ parity violating (RPV) SUSY scenarios are also viable
alternatives to the widely studied $R$ parity conserving (RPC) scenarios.

We will focus in the following on bilinear $R$ parity violation (bRPV), which
has the interesting feature to be able to introduce neutrino masses and
mixings. The phenomenology of this mechanism has already been discussed in
detail in the literature [4, 5, 6, 7, 8, 9]. Our aims is to investigate the
performance of the International Linear Collider and one of its proposed
detector concepts for measuring the atmospheric neutrino mixing angle in such
a bRPV SUSY scenario, based on full detector simulation and current beam
parameters [10, 11].

In this section, we briefly summarise the basic concept of bilinear $R$ parity
violation and its connection to collider physics. The superpotential and the
corresponding soft SUSY breaking terms in bRPV SUSY have the form

$\displaystyle W^{\mathrm{MSSM}}=$ $\displaystyle
W^{\mathrm{MSSM}}_{\mathrm{RPC}}+\epsilon_{i}L_{i}H_{u},$ (2)
$\displaystyle\mathcal{L}^{\mathrm{MSSM}}_{\mathrm{soft}}=$
$\displaystyle\mathcal{L}^{\mathrm{MSSM}}_{\mathrm{soft},{RPC}}+\epsilon_{i}B_{i}L_{i}H_{u},$
(3)

where $i=\\{e,\mu,\tau\\}$ is the generation index. $H_{u}$ indicates the
${SU(2)}$ doublet of the Higgs superfield and $L_{i}$ the ${SU(2)}$ doublet of
the lepton superfield. $\epsilon_{i}$ and $B_{i}$ are bRPV parameters. In
addition to that, the three sneutrinos acquire a vacuum expectation value
(VEV) $\langle\widetilde{\nu}_{i}\rangle=v_{i}$. Because of three additional
tadpole equations one ends up with six free parameters for bilinear $R$ parity
violation. These parameters can be fixed by fitting them to neutrino
observables, like neutrino mass differences and mixing angles.

The introduction of lepton number violation allows the neutrinos to mix with
the other neutral fermions of the model, i.e. the gauginos and higgsinos.
Thus, in the basis of neutral fermions
$\Psi^{0T}=\left(\widetilde{B},\widetilde{W}^{0},\widetilde{H}^{0}_{d},\widetilde{H}^{0}_{u},\nu_{e},\nu_{\mu},\nu_{\tau}\right)$
the corresponding mass term in the Lagrangian looks like

$\displaystyle\mathcal{L}=-\frac{1}{2}\left(\Psi^{0}\right)^{T}\mathbf{M}_{N}\Psi^{0}+c.c.,$
(4)

where the mass matrix $\mathbf{M}_{N}$ has additional off–diagonal entries due
to bilinear $R$ parity breaking.

Diagonalising $\mathbf{M}_{N}$ generates one neutrino mass at tree level as
well as two neutrino mixing angles. The atmospheric neutrino mixing angle, for
instance, writes as

$\displaystyle\tan(\theta_{23})$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{\mu}}{\Lambda_{\tau}},$ (5)

where $\Lambda_{i}=\mu v_{i}+v_{d}\epsilon_{i}$ are so called alignment
parameters. Herein, $\mu$ is the MSSM higgsino mass parameter and $v_{d}$
represents the VEV of the down-type Higgs. It has been show in [7] that the
remaining neutrino mixing angle and neutrino masses can be derived on 1-loop
level.

A very interesting feature of this model is that the left-handed part of the
$\widetilde{\chi}^{0}_{1}-W-l_{i}$–coupling is approximately proportional to
the alignment parameters

$\displaystyle O_{\widetilde{\chi}^{0}_{1}Wl_{i}}\simeq\Lambda_{i}\cdot
f(M_{1},M_{2},\mu,v_{d},v_{u})\propto\Lambda_{i},$ (6)

where $f$ is a function of the soft SUSY breaking parameters. The full
expression of $O_{\widetilde{\chi}^{0}_{1}Wl_{i}}$ can be found in [8].
Combining eq. (5) with eq. (6) makes clear that neutrino mixing can be
determined from measuring branching rations of the neutralino decays:

$\displaystyle\tan^{2}(\theta_{23})\simeq\frac{O_{\widetilde{\chi}^{0}_{1}W\mu}^{2}}{O_{\widetilde{\chi}^{0}_{1}W\tau}^{2}}=\frac{\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow
W\mu)}{\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow W\tau)}$ (7)

The exact relation only holds at tree level. Ref. [8] shows that via loop
contributions additional SUSY parameters enter into the mass matrix and thus
into the relation between branching ratios and neutrino mixing angles. For
this study, we use SPheno3.2.4beta [12] to fit the alignment parameters to
current neutrino data and to extract the physical observables, in particular
the neutralino branching ratios, at loop level. This procedure is repeated for
every considered point in SUSY parameter space.

It is worth mentioning that for bRPV SUSY there is always a connection between
LSP decays and neutrino physics independently of the type of the LSP, which is
shown in [9].

The remainder of the paper is organised as follows: In section 2 we summarise
the status of RPV searches at the LHC. In the following section 3 we focus on
the study at the ILC. Here we introduce the studied simplified scenario, give
some details on the used Monte-Carlo samples, define the event selection and
comment on sources for systematic uncertainties. The results are presented in
section 4, where we discuss the LSP mass measurement, the expected signal
significance in the parameter space of the simplified model and the precision
in measuring the atmospheric mixing angle from the ratio of two LSP branching
ratios. We finally conclude by summarising our obtained results in section 5.

##  2 Status at the LHC

In pp collisions the dominant SUSY production mode is via squark and gluino
production. Those coloured particles then decay via cascades down to the
lightest SUSY particle (LSP), which in the RPV case then decays into Standard
Model particles. The main difference to studies assuming $R$ parity
conservation is that the cut on missing energy is relaxed significantly, since
the LSP does not escape undetected anymore.

The ATLAS collaboration performed a dedicated bRPV SUSY search in the
framework of the CMSSM, where the RPV parameters have been fitted to neutrino
data and have not been taken as free [13]. In this study contributions from
all possible production modes have been taken into account. This study
excludes a wide range of the CMSSM parameter plane reaching up to
$m_{1/2}\approx 600\,\mathrm{\,GeV}$ or $m_{0}\approx 1.2\,\mathrm{\,TeV}$.
However, most of the exclusions of the parameter space result from limits on
coloured particles for the specific parameter points.

Except for this, various RPV SUSY searches have been performed in the
simplified model framework [14, 15, 16, 17, 18, 19, 20]. Many of these studies
assume strong production, which is dominant for not too high squark or gluino
masses. So, the derived limits are again predominantly limits on the coloured
sector of the model and the electroweak sector remains untested.

The LHC now starts to become sensitive to direct electroweakino production and
is able to set limits on electroweakino masses in the RPC [21, 22, 23] as well
as in the RPV case [24]. However, the cross section for direct
$\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1}$ production at the LHC is
extremely small, unless a specific Higgsino-Wino mixing is assumed [24]. So,
usually a production of heavier electroweakinos is considered, which decay via
the LSP to Standard Model particles. Therefore, the limits can only be given
as a combination of $\widetilde{\chi}^{0}_{1}$ mass and the mass of a heavier
electroweakino. In the RPV case, currently one study is present that assumes
direct chargino production and one additional non-vanishing trilinear RPV
coupling [24]. The resulting limits in the
$m_{\widetilde{\chi}^{0}_{1}}$-$m_{\widetilde{\chi}^{\pm}_{1}}$ plane strongly
depend on the assumed RPV coupling. Under the most optimistic assumptions,
chargino masses up to $750\,\mathrm{\,GeV}$ have been probed. Due to the
strong dependency on the assumed type and strength of the RPV couplings, it is
not possible to directly re-interpret these limits in a bRPV scenario which
accounts for neutrino data.

##  3 Bilinear RPV at the ILC

###  3.1 Model definition

At the ILC the situation is complementary to the LHC: Here, direct
electroweakino production is dominant and the electroweak sector can be probed
directly, including LSP pair production.

In our scenario we assume the lightest neutralino to be a bino, which leaves
only the $t/u$-channel production for direct $\widetilde{\chi}^{0}_{1}$ pair
production (see fig. 1). In presence of bRPV couplings, not only selectrons
are possible as exchange particle, but due to the additional terms in
principle all other charged scalars could contribute. However, these
contributions are strongly suppressed by the small RPV couplings. We define a
simplified model in which we set all masses of the SUSY particles to the
multi-TeV scale, except for $m_{\widetilde{\chi}^{0}_{1}}$ and
$m_{\widetilde{e}_{R}}$. Thus, those two parameters fix the production cross
section.

Figure 1: Left: Main production channel for an bino-like LSP at the ILC:
t-channel exchange of an selectron. Right: LSP decay to an on-shell $W$ boson
and a lepton offers direct access to bRPV alignment parameters $\Lambda_{i}$
that account for neutrino mixing.

In the case of a wino LSP, the cross section would drop due to the missing
coupling to the right selectron. For a light left selectron, however, the
situation is comparable to the bino case. A higgsino-like LSP would allow
$s$-channel associate production of $\widetilde{\chi}^{0}_{1}$ and
$\widetilde{\chi}^{0}_{2}$, predominantly via a $Z$ boson. The cross section
for this production process is about $100\,\mathrm{\,fb}-200\,\mathrm{\,fb}$
[25]. In a light higgsino scenario $\widetilde{\chi}^{0}_{1}$ and
$\widetilde{\chi}^{0}_{2}$ are usually close in mass and the decay products of
$\widetilde{\chi}^{0}_{2}$ to $\widetilde{\chi}^{0}_{1}$ are rather soft.
Thus, experimentally the situation is comparable to direct
$\widetilde{\chi}^{0}_{1}$ pair production.

Figure 2: Unpolarised and polarised production cross section at the ILC500 in
the described simplified model. Beam polarisation can significantly enhance
the production cross section. The shaded area shows the region of the
parameter space, where the selectron becomes the LSP.

At the ILC, the electron beam is polarised to $80\%$ and the positron beam to
up to $60\%$ [11]. The advantages of beam polarisation have been discussed in
many studies. A comprehensive overview can be found, for instance, in [26]. In
the case of $t/u$-channel production with selectron exchange different
combinations of beam polarisation influence the production cross section
significantly. Figure 2 shows the cross section in the
$\tilde{e}_{R}$-$\tilde{\chi}^{0}_{1}$ mass plane for unpolarised beams (left)
and for the baseline polarisation of $P(e^{+},e^{-})=(-30\%,+80\%)$ (right),
which enhances the cross section considerably.

Since the RPV couplings are very small if they are used to describe neutrino
data correctly, light LSPs can become rather long-lived with a decay length of
meters up to kilometers and escape the detector. Therefore, scenarios with
very light LSPs would behave very similar to RPC SUSY scenarios, where the
LSPs are stable. However, those escaping pair-produced LSPs could still be
detected via radiative LSP production for cross sections of
$\mathcal{O}(10\,\mathrm{\,fb})$ as demonstrated in [27]. In this study, we
focus on scenarios where the on-shell $W$-decay channel is available, i.e. on
LSP masses larger than the $W$ mass. Thus, the decay length of the LSP lies
between $10\,\mathrm{\,cm}$ and $100\,\mathrm{\,\mu m}$ as depicted in fig. 3
(left). The right-hand panel fo figure 3 shows the impact parameter resolution
of the ILD detector concept as determined from full detector simulation [10].
It indicates that the arising displaced vertices in this scenarios are well
detectable for ILC detectors.

Figure 3: Left: Decay length of the LSP in dependence of its mass in the
simplified model. For a light spectrum of the remaining sparticles, the decay
length can be smaller. Right: Impact parameter resolution of the ILD detector
concept (from [10]).

For the ILC study the following example point has been used:

$\displaystyle m_{\widetilde{\chi}^{0}_{1}}$ $\displaystyle=$ $\displaystyle
98.48\,\mathrm{\,GeV}$ (8) $\displaystyle m_{\widetilde{e}_{R}}$
$\displaystyle=$ $\displaystyle 280.72\,\mathrm{\,GeV}$ (9)

The mass of the neutralino has been selected as a worst case scenario, where
$m_{\widetilde{\chi}^{0}_{1}}\simeq m_{W/Z}$. This is most challenging, since
in this case the LSP signal is expected to overlay significantly with SM
background and the involved leptons from the LSP decay to $W$ become
relatively soft. For higher LSP masses the study is almost SM background free.
The production cross section for the polarisation
$\mathcal{P}(e^{+},e^{-})=(-30\%,+80\%)$ and a center of mass energy of
$\sqrt{s}=500\,\mathrm{\,GeV}$ amounts to $344\,\mathrm{\,fb}$, driven by the
choice of $m_{\widetilde{e}_{R}}$. As can be seen from figure 2, the
production cross section is still $\sim 100\,$fb when $m_{\widetilde{e}_{R}}$
is twice as large. The branching ratios for the decay modes that are relevant
for measuring the neutrino atmospheric mixing angle (see eq. (7)) read111These
branching ratios correspond to bRPV couplings
$O_{\widetilde{\chi}^{0}_{1}Wl_{2,3}}$ in the order of $10^{-7}$.
$BR(\widetilde{\chi}^{0}_{1}\rightarrow\mu^{\pm}W^{\mp})=0.43$ and
$BR(\widetilde{\chi}^{0}_{1}\rightarrow\tau^{\pm}W^{\mp})=0.47$. The remaining
fraction comes mainly from on-shell $Z$ decay modes
$\widetilde{\chi}^{0}_{1}\rightarrow\nu_{i}Z$, where the relative fraction of
$\nu_{i}Z$ vs. $l_{i}W$ decays depends only on the $\widetilde{\chi}^{0}_{1}$
mass. Decays to electrons have a negligibly small branching fraction due to
the smallness of the reactor neutrino mixing angle $\theta_{13}$, since
$\mathrm{BR}(\tilde{\chi}_{1}^{0}\rightarrow
We)\propto\Lambda_{e}\propto\tan^{2}(\theta_{13})$ [8]. Three-body decays are
negligible in our case. They become sizable only for small
$\widetilde{\tau}$-$\widetilde{\chi}^{0}_{1}$ mass differences, which would
allow direct $\widetilde{\tau}$ production at the ILC and thus offer a wealth
of additional observables which we don’t discuss in this paper.

For an integrated luminosity of $100\,\mathrm{\,fb}^{-1}$ one expects $34400$
produced neutralino pairs and among them $6361$ events ending up in the
$\mu\mu$-channel, $7599$ events in the $\tau\tau$-channel, and $13904$ events
in the mixed $\mu\tau$-channel. As the ILC is expected to deliver an
integrated luminosity of $250\,\mathrm{\,fb}^{-1}/\mathrm{year}$ at
$500\,\mathrm{\,GeV}$, this amount of data is going to be collected within
approximately five months of operation at design luminosity.

###  3.2 Data samples

For the given example point a full detector simulation of the International
Large Detector (ILD) based on the recently published detector descripton [10]
has been performed. The ILD concept is one of two proposed detector concepts
at the ILC. Its design is optimised for Particle Flow reconstruction, which
aims at reconstructing each individual particle with the most precise detector
component. To this end, properties of charged particles are only measured by
the tracking system and the highly segmented calorimeters are only used for
the measurement of neutral particles. The main part of the proposed tracking
system at the ILD is a time-projection-chamber, which is complemented by
silicon strip and pixel detectors. This system is expected to obtain a
tracking resolution of up to $\sigma_{1/p_{t}}=2\cdot
10^{-5}\,\mathrm{\,GeV}^{-1}$ for high momenta. Due to the more benign
radiation environment at the ILC, the inner detectors can be built with very
little material, amounting to only $10\%$ of a radiation length in front of
the electromagnetic calorimeter in the barrel region. The electromagnetic
calorimeter is foreseen as a 30-layer silicon-tungsten sampling calorimeter
with cell sizes of $5\times 5\,\mathrm{\,mm}^{2}$. In testbeam operation of a
prototype [28], an energy resolution of $\Delta E/E=(16.6\pm
0.1)\%/\sqrt{E}\oplus(1.1\pm 0.1)\%$ has been achieved. The highly segmented
hadronic calorimeter with cell sizes of $3\times 3\,\mathrm{\,cm}^{2}$ in
connection with the Particle Flow Concept allows for a jet energy resolution
of $\Delta E/E=(3-4)\%$ [29].

The full detector simulation has been performed for an integrated luminosity
of $100\,\mathrm{\,fb}^{-1}$ and at a center of mass energy of
$\sqrt{s}=500\,\mathrm{\,GeV}$. For the SM background, samples produced for
the benchmarking of the ILD detector for the Technical Design Report have been
used [10].

In the case of bRPV events the program Sarah [30] has been used to generate
model files for the event generator. As for the SM samples, Whizard [31] has
been used as event generator of the hard process and Pythia [32] for
fragmentation and hadronization. These events have been passed through Mokka
[33], the full Geant4-based [34] simulation of the ILD detector and finally
reconstructed with MarlinReco [35, 36]. For the event generation, realistic
beam parameters have been taken into account, in particular the ILC specific
beam energy spectrum at $500\,\mathrm{\,GeV}$ [11].

With the instantaneous luminosity forseen at the ILC, on average $\langle
N\rangle=1.2$ interactions of photons leading to the production of low $p_{t}$
hadrons are expected per bunch-crossing [10]. This takes into account
contributions from real photons accompanying the electron beam due to
bremsstrahlung and synchrotron radiation as well as from virtual photons
radiated off the primary beam electrons. Therefore, each hard-interaction
event (from SUSY or SM background) has been overlaid with a Poissonian
distributed number of such $\gamma\gamma\rightarrow\mathrm{hadrons}$ events
before the reconstruction step.

At the time of the MC production a too large number of overlay events per
hard-interaction had been assumed ($\langle N\rangle=1.7$), which results in a
conservative estimate of the $\gamma\gamma\rightarrow\mathrm{hadrons}$
background in this study. However, as it will be demonstrated in the next
section, even this larger background could be removed very efficiently.

###  3.3 Event selection

The signal events have a rather clear signature: The produced LSPs decay into
either a $\mu$ or $\tau$ plus a $W$ boson. In the following we restrict
ourselves to the hadronic $W$ decay mode. Thereby, the event is – except for
some missing energy from a potential $\tau$ decay – fully reconstructable with
six visible objects in the final state.

#### Event preparation and preselection

Before the actual event selection is performed, the
$\gamma\gamma\rightarrow\mathrm{hadrons}$ background is removed from the event
by the following procedure: Since we expect to have six final state objects,
an exclusive $k_{T}$ jet clustering algorithm ($R=1.3$) [37, 38] which is
forced to find six jets is applied to the reconstructed objects in the events.
This algorithm builds up six jets which are assigned to the hard interaction,
as well as two very forward directed beam jets which are treated as beam
background. Removing those beam jets and using only the objects which end up
clustered in the main jets for the further analysis, recovers very well the
bare event without background overlay. Fig. 4 shows the impact of overlaid
$\gamma\gamma$ events on the visible energy (left) and the ability to remove
this background with the described method (right).

Figure 4: Left: Effect of $\gamma\gamma\rightarrow\mathrm{hadrons}$ background
overlay on the visible energy in the bRPV SUSY sample. Right: Recovered
visible energy in the event after $\gamma\gamma\rightarrow\mathrm{hadrons}$
background removal procedure described in the text.

Two hadronically decaying $W$ bosons imply a relatively high particle
multiplicity $N_{\mathrm{objects}}$ in the event. Due to the fact that there
is no major source for missing energy, the visible energy in the event is
close to the center of mass energy of $500\,\mathrm{\,GeV}$. So, the following
preselection cuts have been used:

$\displaystyle E_{\mathrm{visible}}\geq 350\,\mathrm{\,GeV}$ (10)
$\displaystyle 50\geq N_{\mathrm{objects}}\geq 150,$ (11)

This preselection on the one hand reduces Standard Model background and on the
other hand cuts away some of the leptonic $W$ contribution in the LSP decay,
which is considered as background in this analysis. $95\%$ of the signal
events pass this preselection.

In order to be able to measure the ratio of different branching ratios of the
LSP decay (see eq. (7)) we have to define a selection to distinguish between
the different event classes:

####  $\mathbf{\mu\mu}$ class

Here we look for an event with at least two reconstructed muons. The muon
identification is provided by the Pandora Particle Flow Algorithms [29],
seeded by a minimum-ionizing signature in the calorimeters and the
instrumented flux return yoke. The two most energetic muons are removed from
the event. No further lepton isolation criteria is required. The remaining
objects of the event are clustered by the Durham jets clustering algorithm
into four jets. All pair-permutations of the four jets are used to find the
best $W$ candidates, where the objects with the smallest

$\displaystyle\chi^{2}_{W_{i}}=\left(\frac{m_{\mathrm{reco,}i}-m_{W}}{\sigma_{\mathrm{res}}}\right)^{2}$
(12)

are used for the proceeding analysis. Herein, $\sigma_{\mathrm{res}}$ is an
estimated resolution factor of $5\,\mathrm{\,GeV}$. For all combinations of
$W$ candidates and muons the invariant mass is determined and the
reconstructed objets from the pair with smallest

$\displaystyle\chi^{2}_{\mathrm{eqm}}=\left(\frac{m_{\mathrm{reco,}1}-m_{\mathrm{reco,}2}}{\sigma_{\mathrm{res}}}\right)^{2}$
(13)

are considered as LSP candidates. If the event fulfills the condition

$\displaystyle\chi^{2}_{W_{1}}<2\quad\mathrm{and}\quad\chi^{2}_{W_{2}}<2\quad\mathrm{and}\quad\chi^{2}_{\mathrm{eqm}}<2,$
(14)

the event is counted as $\mu\mu$ event, else it is tested against the
$\mu\tau$ class.

####  $\mathbf{\mu\tau}$ class

In this class at least one reconstructed muon in the event is required. The
most energetic muon is removed and the rest of the event is forced into five
jets by an inclusive Durham jet clustering algorithm. Since the muon
requirement is relaxed, this class is more prone to background. Therefore we
test in addition whether the chosen jet configuration describes the event
well. We employ a cut on the Durham jet algorithm parameters $y_{i-1,i}$ and
$y_{i,i+1}$ [37]. $y_{i-1,i}$ is a measure of the distance in energy-momentum
space between the two of the $i$ jets which would be merged if a $i-1$
configuration was required. $y_{i,i+1}$ gives the analoguos distance measure
for the two jets which have been merged in the last step when clustering from
the $i+1$ to the final $i$ configuration. For events well fitting to the 5-jet
configuration, $y_{4,5}-y_{5,6}$ gets maximal, so we use the following cut:

$\displaystyle y_{4,5}-y_{5,6}>5\cdot 10^{-5}.$ (15)

The jet with the smallest number of constituents is considered as $\tau$
candidate if $N_{\mathrm{const.}}<10$. Additionally it is required that the
jet does not contain a muon. The further approach to find $W$ candidates and
finally LSP candidates is identical to the $\mu\mu$ class. If condition (14)
is satisfied this event is counted as $\mu\tau$ event, otherwise the
$\tau\tau$ class is tested.

####  $\mathbf{\tau\tau}$ class

The event is forced into six jets by an inclusive Durham jet clustering
algorithm. In analogy to the $\mu\tau$ class, it is required that
$y_{5,6}-y_{6,7}>5\cdot 10^{-5}$. The two jets with the smallest numbers of
constituents are considered as $\tau$ candidates if each fulfills
$N_{\mathrm{const.}}<10$ and the muon-veto. The further approach to find $W$
candidates and finally LSP candidates is again identical to the $\mu\mu$ and
$\mu\tau$ class. If condition (14) is satisfied this event is counted as
$\tau\tau$ event. Otherwise, the event is rejected.

Figure 5: Reconstruction mass of the LSP for the different event classes.
Yellow indicates the signal events, orange depicts background originating from
non-signal LSP decays and red shows the remaining Standard Model background.
For the $\mu\mu$ and $\mu\tau$ channel the SM background peaks at
$M_{W}/M_{Z}$ and the signal at $m_{\widetilde{\chi}^{0}_{1}}$. In the
$\tau\tau$ channel the purity is lowest.

After this selection fig. 5 shows the reconstructed neutralino mass in each of
the event classes for $100\,\mathrm{\,fb}^{-1}$ of fully simulated events. The
remaining background is dominated by SM $W$ pair production, whereas the
contribution from other LSP decays is very small.

###  3.4 Systematic uncertainties

#### Mass reconstruction

For the reconstruction of the LSP mass the two most important sources of
uncertainties arise from the reconstruction of the $\mu$ momentum as well as
on the reconstruction of the momentum and energy of the $W$ candidates. In our
scenario, the muons originating from an LSP decay carry a momentum of up to
$80\,\mathrm{\,GeV}$, while the jets originating from the $W$ bosons from the
signal decay have an energy of up to $200\,\mathrm{\,GeV}$.

The muon momentum scale can be calibrated using $Z$ boson decays. The
unpolarised cross section of the process $e^{+}e^{-}\rightarrow
Z\rightarrow\mu^{+}\mu^{-}$ at the ILC500 amounts to $2.5\,\mathrm{\,pb}$. For
an assumed integrated luminosity of
$\int\mathcal{L}\mathrm{d}t=500\,\mathrm{\,fb}^{-1}$ this yields in total
$N_{\mu\mu}=12.5\cdot 10^{5}$ muon pairs. Assuming that ultimatively the
precision of the scale calibration is limited by the available statistics for
the calibration process, we estimate that a precision of
$1/\sqrt{N_{\mu\mu}}=0.09\%$ can be reached.

The jet energy calibration of hadronically decaying $W$ bosons can be derived
analoguously from hadronic $Z$ decays. Since the cross section for
$e^{+}e^{-}\rightarrow Z\rightarrow q\bar{q}$ is ten times larger compared to
the muonic decay channel, which means that the from the point of view of
available control sample statistics, the jet energy scale uncertainty could
reach $0.03\%$, assuming that the calibration is sufficiently stable over
time. The resulting impact on the neutralino mass determination is
$11\,\mathrm{\,MeV}$. Thus it could be a factor four larger before it becomes
comparable to the contribution from the momentum scale. Alternatively,
exploiting kinematic fits [39] in conjunction with the well-known beam energy,
at the ILC forseen to be controlled to $10^{-4}$ [10], could significantly
reduce the dependence of the reconstructed neutralino mass on the jet energy
scale, leading to similar final precision estimates.

Systematic errors on luminosity, beam energy and beam polarisation do not
enter into the mass measurement, which depends solely on the reconstructed
detector signals. The selection efficiency does not show any dependency on the
reconstructed LSP mass within the available Monte-Carlo statistics. Thus we
conclude that any potential bias due to the selection efficiency plays an
insignificant role in the mass measurement.

For LSP masses below $\sim 105\,\mathrm{\,GeV}$, the SM background has a
steeply falling invariant mass distribution. Therefore any uncertainty related
to the modelling of this slope, e.g. assumptions on hadronisation, colour-
reconnection etc., could enter into the LSP mass determination. However the
ILC itself will offer numerous opportunities for SM precision measurements
beyond today’s knowledge. In particular the cross section for the dominating
background process, $W$ pair production, is in the several pb range, thus
providing ample possibilities to tune the modelling of this process. We thus
estimate that the residual effect on the LSP mass determination via
subtraction of the SM background is not larger than $15\,\mathrm{\,MeV}$.

source | calibration process | cross section |  $1/\sqrt{N}$@$500\,\mathrm{\,fb}^{-1}$ | effect on $m_{\widetilde{\chi}^{0}_{1}}$
---|---|---|---|---
$|\vec{p}_{\mu}|$ scale | $Z\rightarrow\mu^{+}\mu^{-}$ | $2.5\,\mathrm{\,pb}$ | $0.09\%$ | $46\,\mathrm{\,MeV}$
$E_{\mathrm{jet}}$ scale | $Z\rightarrow q\bar{q}$ | $25\,\mathrm{\,pb}$ | $0.03\%$ | $11\,\mathrm{\,MeV}$
background modelling |  $WW\rightarrow$ hadrons | $7\,\mathrm{\,pb}$ | $0.05\%$ | $15\,\mathrm{\,MeV}$
total |  |  |  | $\sim 50\,\mathrm{\,MeV}$
Table 1: Main sources for systematic uncertainties on the LSP mass
reconstruction. The largest contribution in the current estimate stems from
the muon momentum scale calibration.

Table 1 summarises the main sources for the systematic uncertainty on the LSP
mass reconstruction and its propagation to the LSP mass reconstruction. The
total systematic error is estimated to be about $50\,\mathrm{\,MeV}$.

#### Measurement of ratio of branching ratios

The measurement of the ratio of branching ratios is a measurement of the ratio
of the number of events reconstructed in the different event classes. For this
reason, all systematic uncertainties which factorise with the number of events
cancel. The same is true for reconstruction effects which affect all event
classes simultanousely, like systematic uncertainties on jet energy scales,
for instance.

The main source of systematic error is expected to arise from the
determination of the selection efficiencies and purities of the different
event classes from Monte-Carlo. Therefore, validation of the Monte Carlo
simulation with data is very important.

The process $e^{+}e^{-}\rightarrow ZZ\rightarrow l^{+}l^{-}q\bar{q}$ with
$l=\\{\mu,\tau\\}$ offers the possibility to study the Monte-Carlo description
of $\tau$ \+ jets and $\mu$ \+ jets events under comparable experimental
conditions as for the signal decay. The unpolarised cross section per process
at a center of mass energy of $500\,\mathrm{\,GeV}$ is $225\,\mathrm{\,fb}$.
Thus, for $\int\mathcal{L}\mathrm{d}t=500\,\mathrm{\,fb}^{-1}$ the expected
precision on Monte-Carlo and data comparison amounts to $0.3\%$. As a second
process, $e^{+}e^{-}\rightarrow W^{+}W^{-}\rightarrow l\nu_{l}q\bar{q}$ with
$l=\\{\mu,\tau\\}$ can also be studied in order to validate Monte-Carlo. This
process has a significantly larger cross section of $2.5\,\mathrm{\,pb}$ per
process, which leads in the end to a statistical uncertainty of $0.09\%$ for
$\int\mathcal{L}\mathrm{d}t=500\,\mathrm{\,fb}^{-1}$.

As for the mass measurement, we allow some modelling uncertainty for the main
SM background, $WW\rightarrow$ hadrons, of $0.05\%$.

Taking these considerations into account, a conservative estimate on the
systematic uncertainties on Monte-Carlo is $\mathcal{O}(0.5\%)$ (c.f. table
2).

| control process | cross section |  $1/\sqrt{N}$@$500\,\mathrm{\,fb}^{-1}$
---|---|---|---
$\mu$ / $\tau$ ID with $Z/W\rightarrow$ jets | $ZZ\rightarrow llq\bar{q}$ | $225\,\mathrm{\,fb}$ | $0.3\%$
$WW\rightarrow l\nu_{l}q\bar{q}$ | $2.5\,\mathrm{\,pb}$ | $0.09\%$
SM background modelling |  $WW\rightarrow$ hadrons | $7\,\mathrm{\,pb}$ | $0.05\%$
total |  |  | $\sim 0.5\%$
Table 2: Possible control samples to verify efficiencies and purities obtained
from Monte-Carlo simulation; $l$ denotes either a $\mu$ or a $\tau$. The
largest contribution originates from the limited statistics of the SM control
sample for verifying the migrations between the $\mu\mu$, $\mu\tau$ and
$\tau\tau$ classes.

##  4 Results

###  4.1 Mass measurement and resolution

The $\mu\mu$ channel shows a clear signal peak, which can be used for
measuring the LSP mass accurately. Figure 6 shows the reconstructed LSP mass
spectrum for an integrated luminosity of
$\int\mathcal{L}\mathrm{d}t=100\,\mathrm{\,fb}^{-1}$ after subtracting the SM
background. Since the natural width of the LSP is negligibly small
($\Gamma_{\widetilde{\chi}^{0}_{1}}=\mathcal{O}(10^{-14}\,\mathrm{\,GeV}$)),
the width of the distribution is dominanted by the detector resolution. From a
gaussian fit, the LSP mass can be determined to

$\displaystyle m^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}$
$\displaystyle=(98.401\pm 0.092\text{(stat.)})\,\mathrm{\,GeV}.$ (16)

This value is within the error in very good agreement with the input mass of
the example point of $98.48\,\mathrm{\,GeV}$ (c.f. eq. (8)). The obtained
width of the gaussian is about
$\sigma^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}=3\,\mathrm{\,GeV}$, which is
in very good agreement with the ILD design goals [40].

Scaling the statistical uncertainty to an integrated luminosity of
$\int\mathcal{L}\mathrm{d}t=500\,\mathrm{\,fb}^{-1}$ and combining the
systematic uncertainties as discussed in section 3.4, the total uncertainty on
the LSP mass measurement becomes

$\displaystyle\delta m^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}$
$\displaystyle=(40\text{(stat.)}\oplus 50\text{(syst.)})\,\mathrm{\,MeV}.$
(17)

Figure 6: Mass reconstruction of the LSP in the $\mu\mu$ channel. The Standard
Model background is subtracted, but fluctuations are taken into account in the
bin errors. The distribution is slightly washed out towards lower masses,
which originates from misreconstructions. For this reason, this part of the
distribution has not been taken into account for the fit.

###  4.2 Signal significance

The precision measurement of the LSP mass in the $\mu\mu$ channel can be used
to define a signal region $m^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}\pm
3\sigma^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}$. This further reduces the
background fraction in the selected event classes. The decomposition of number
of measured events in the different event classes $N^{\mathrm{reco}}$ into the
number of events in the different truth classes $N^{\mathrm{true}}$ is shown
in the the following matrix $\mathbf{N}$:

$\displaystyle\mathbf{N}=\bordermatrix{&N^{\mathrm{true}}_{\mu\mu}&N^{\mathrm{true}}_{\mu\tau}&N^{\mathrm{true}}_{\tau\tau}&N^{\mathrm{true}}_{\mathrm{LSPBG}}&N^{\mathrm{true}}_{\mathrm{SMBG}}\cr
N^{\mathrm{sel}}_{\mu\mu}&858&173&11&40&69\cr
N^{\mathrm{sel}}_{\mu\tau}&16&410&45&17&67\cr
N^{\mathrm{sel}}_{\tau\tau}&0&2&107&4&60\cr}$ (18)

Thereby, $N^{\mathrm{true}}_{\mathrm{LSPBG}}$ counts events in which at least
one LSP decays differently than the targeted two-body decay
$\widetilde{\chi}^{0}_{1}\rightarrow Wl$. As expected, the $\tau\tau$ channel
is by far the one with the lowest purity.

The signal over Standard Model background ratio for the different classes can
be derived. Herein, every selected event originating from an LSP decay is
counted as signal.

| $S/\sqrt{B}$
---|---
$\mu\mu$ class | 130
$\mu\tau$ class | 60
$\tau\tau$ class | 15

The described event selection is still significant on a $5\sigma$ level for
very large selectron masses well above $1.5\,\mathrm{\,TeV}$ and a large range
of $m_{\widetilde{\chi}^{0}_{1}}$, as depicted in fig. 7. Positron beam
polarisation further enhances the production cross section and, thus,
increases the sensitivity of the analysis to selectron masses of almost
$2\,\mathrm{\,TeV}$ for $P(e^{+})=-60\%$.

Figure 7: 5$\sigma$ contour of signal significance in the $\mu\mu$ channel for
different beam polarisations and an integrated luminosity of
$\int\mathcal{L}\mathrm{d}t=500\,\mathrm{\,fb}^{-1}$ based on a log likelihood
ratio. The di-muon selection efficiency obtained from full simulation for
$m_{\widetilde{\chi}^{0}_{1}}=98.48\,\mathrm{\,GeV}$ has been assumed for all
values of the neutralino mass. For larger $m_{\widetilde{\chi}^{0}_{1}}$, this
is a conservative estimate. Higher positron beam polarisation helps to further
increase the significance in this model.

It has already been pointed out that this studied parameter point is a worst
case scenario with respect to the neutralino mass. It is clearly visible in
fig. 5 that for another parameter point with a higher LSP mass the signal peak
would shift into an almost completely background-free region. Though, the LSPs
in this model become rather long-lived at lower masses (compare fig. 3), where
the Standard Model background is large. Adding this information to the
analysis and requiring from the reconstructed objects not to point to the
primary vertex, would reduce the Standard Model background drastically.
However, this would on the other hand introduce a strong model dependency to
the analysis. Therefore the exploitation of the lifetime information is left
as a future option for improvements.

Likewise, the requirement for same sign leptons in the event classes is an
option for further improving the analysis. This restriction would reduce the
number of signal events by a factor of $2$, but could heavily suppress the
remaining SM background.

###  4.3 Branching ratio measurement

We assume in the following that the average number of Standard Model
background events can be estimated from Monte Carlo with a precision of
$0.05\%$ (cf. section 3.4). The LSP non-signal background consists mainly of
events in which one of the two LSPs decayed non-signal like into $Z\nu$. As
soon as the LSP mass is known, the relative fraction
$\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow
Z\nu)/\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow Wl)$ is determined
and, thus, also the number of LSP background events can be predicted. Under
this assumption we can subtract the backgrounds and build a $3\times 3$
efficiency matrix $\mathbf{E}$, which is defined like

$\displaystyle(\mathbf{E})_{ij}=\frac{(\mathbf{N})_{ij}}{N^{\mathrm{true}}_{j}}=\left(\begin{array}[]{ccc}0.2981&0.0277&0.0032\\\
0.0056&0.0658&0.0129\\\ 0.0000&0.0004&0.0306\end{array}\right)_{ij},$ (22)

where $i,j={\mu\mu,\mu\tau,\tau\tau}$. The error on the entries of the
efficiency matrix is dominated by the assumed systematic uncertainty of
$0.5\%$ on the Monte-Carlo prediction on the migrations between the signal
classes (cf. section 3.4).

The vector of selected events becomes

$\displaystyle\vec{N}_{\mathrm{sig}}^{\mathrm{sel}}$
$\displaystyle=\left(\begin{array}[]{c}N^{\mathrm{sel}}_{\mu\mu}-\langle
N^{\mathrm{MC}}_{BG,\mu\mu}\rangle\\\ N^{\mathrm{sel}}_{\mu\tau}-\langle
N^{\mathrm{MC}}_{BG,\mu\tau}\rangle\\\ N^{\mathrm{sel}}_{\tau\tau}-\langle
N^{\mathrm{MC}}_{BG,\tau\tau}\rangle\end{array}\right)$ (26)
$\displaystyle\Delta\vec{N}_{\mathrm{sig}}^{\mathrm{sel}}$
$\displaystyle=\left(\begin{array}[]{c}\sqrt{N^{\mathrm{sel}}_{\mu\mu}+\delta\langle
N^{\mathrm{MC}}_{BG,\mu\mu}\rangle^{2}}\\\
\sqrt{N^{\mathrm{sel}}_{\mu\tau}+\delta\langle
N^{\mathrm{MC}}_{BG,\mu\tau}\rangle^{2}}\\\
\sqrt{N^{\mathrm{sel}}_{\tau\tau}+\delta\langle
N^{\mathrm{MC}}_{BG,\tau\tau}\rangle^{2}}\end{array}\right).$ (30)

Herein, $\delta\langle N^{\mathrm{MC}}_{BG,ll}\rangle$ with
$ll=\\{\mu\mu,\mu\tau,\tau\tau\\}$ are the systematic errors on the SM
background estimation in the event classes. They are negligible compared to
the statistical fluctuations of the reconstructed events per event class.

The efficiency matrix can then be inverted and used to unfold the different
event classes obtaining the number of reconstructed events in the event
classes

$\displaystyle\vec{N}^{\mathrm{reco}}=\mathbf{E}^{-1}\vec{N}^{\mathrm{sel}}_{\mathrm{sig}}.$
(31)

The ratio of the two branching rations can be extracted in different ways

$\displaystyle\frac{\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow
W\mu)}{\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow W\tau)}$
$\displaystyle=\frac{2N^{\mathrm{reco}}_{\mu\mu}}{N^{\mathrm{reco}}_{\mu\tau}}=\frac{N^{\mathrm{reco}}_{\mu\tau}}{2N^{\mathrm{reco}}_{\tau\tau}}=\sqrt{\frac{N^{\mathrm{reco}}_{\mu\mu}}{N^{\mathrm{reco}}_{\tau\tau}}},$
(32)

since for the expected number of events the following relations hold:

$\displaystyle N^{\mathrm{reco}}_{\mu\mu}$ $\displaystyle=$ $\displaystyle
N_{\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1}}\cdot\operatorname{BR}^{2}(\widetilde{\chi}^{0}_{1}\rightarrow
W\mu)$ (33) $\displaystyle N^{\mathrm{reco}}_{\mu\tau}$ $\displaystyle=$
$\displaystyle N_{\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1}}\cdot
2\cdot\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow
W\mu)\cdot\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow W\tau)$ (34)
$\displaystyle N^{\mathrm{reco}}_{\tau\tau}$ $\displaystyle=$ $\displaystyle
N_{\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1}}\cdot\operatorname{BR}^{2}(\widetilde{\chi}^{0}_{1}\rightarrow
W\tau),$ (35)

where
$N_{\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1}}=\sigma(e^{+}e^{-}\rightarrow\widetilde{\chi}^{0}_{1}\widetilde{\chi}^{0}_{1})\cdot\int\mathcal{L}\mathrm{d}t$
is the number of produced LSP pairs. Because of the low selection purity in
the $\tau\tau$ channel, we have chosen the relation involving only the
$\mu\mu$ and $\mu\tau$ channel for the further analysis. Using all three
relations in eq. (32) as input for a constrained fit can improve the precision
by a factor of $2$, but this is not persued in the following.

In order to estimate the uncertainty on the resulting ratio of event numbers,
an error propagation has been performed. The uncertainty is depicted in fig. 8
in dependence of the integrated luminosity at the ILC500. For the studied
parameter point the achievable precision for
$\int\mathcal{L}\mathrm{d}t=100\,\mathrm{\,fb}^{-1}$ is about $9\%$. This
scales down to roughly $4\%$ for $500\,\mathrm{\,fb}^{-1}$, which is the
desired integrated luminosity at ILC500 in a first stage. The uncertainty
contains a systematic error of $0.85\%$ arising from the propagation of the
assumed systematic uncertainties on the efficiencies 2 through the unfolding
procedure.

Figure 8: Achievable measurement precision of the ratio of the muonic and
tauonic two-body decay modes of the LSP at the ILC.

###  4.4 Neutrino interpretation

The measured ratio of branching ratios can now be translated into the
atmospheric mixing angle following eq. (7). As mentioned earlier, the given
relation is only valid on tree-level and there are additional parametric
uncertainties coming from residual SUSY parameter dependencies [8]. For this
reason we define two scenarios: In the first scenario we assume that the LSP
is the only accessible SUSY particle at ILC500. All other supersymmetric
particles are randomly chosen to be heavier than $300\,\mathrm{\,GeV}$. In a
random parameter scan consisting of $~{}6000$ scan points [41] we find that
for $95\%$ of all found viable SUSY parameter points the deviation of the
correlation between atmospheric mixing angle and the ratio of the branching
ratios is below $17\%$. Assuming that the remaining three electroweakinos are
measureable at ILC1000, which is an optional upgrade of the ILC to a center of
mass energy of $1\,\mathrm{\,TeV}$, the correlation uncertainty reduces to
$7\%$.

Figure 9: Random scan of the SUSY parameter space [41] in order to estimate
the parametric uncertainty on the correlation between the atmospheric mixing
angle and the ratio of branching ratios
$\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow
W\mu)/\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow W\tau)$.

The derived precision of the measurement of the atmospheric neutrino mixing
angle for different assumed parametric uncertainties is shown in fig. 10
(left). One can now compare this precision with the uncertainty of current
neutrino experiments [42], which is done in fig. 10 (right). The middle red
line indicates the best fit value of the atmospheric neutrino mixing angle and
the upper and lower dashed red lines indicate the $1\sigma$ uncertainty. An
agreement between the collider and neutrino experiment data would clearly
establish bRPV as origin of neutrino masses. Improvements from future neutrino
experiments or a reduction of the parametric uncertainty by observation of
additional SUSY particles at the ILC or the LHC would strengthen this
conclusion even further.

Figure 10: Precision of the measurement of the atmospheric mixing angle at the
ILC. Left: Relative uncertainty assuming different parametric uncertainties on
the relation between ratio of branching rations and atmospheric neutrino
mixing angle. Right: Comparison between achievable precision at the ILC and
the precision at current neutrino experiments assuming present best fit value
[42] as central value.

##  5 Conclusions

We have presented a full ILD detector simulation of a bRPV SUSY model, which
is an attractive possibility to explain neutrino mass generation and mixing. A
highly detailed ILD model as well as realistic ILC beam parameters have been
taken into account for the simulation. As studied parameter point a worst case
scenario has been used, where $m_{\widetilde{\chi}^{0}_{1}}\simeq m_{W/Z}$
and, thus, the signal significantly overlaps with SM background.

We have developed a model-independent selection strategy to disentangle the
different event classes involving the two decay modes of the LSP
$\widetilde{\chi}^{0}_{1}\rightarrow\mu^{\pm}W^{\mp}$ and
$\widetilde{\chi}^{0}_{1}\rightarrow\tau^{\pm}W^{\mp}$. It has been
demonstrated that in the $\mu\mu$ event class a very accurate mass measurement
with an uncertainty of $\delta
m^{\mathrm{fit}}_{\widetilde{\chi}^{0}_{1}}=(40\text{(stat.)}\oplus
50\text{(syst.)})\,\mathrm{\,MeV}$ is possible for an integrated luminosity of
$500\,\mathrm{\,fb}^{-1}$. With the described selection, a signal to
background ratio of $130$ in the $\mu\mu$ event class and $60$ in the
$\mu\tau$ event class has been achieved. Even for very large selectron masses
of up to $1.5\,\mathrm{\,TeV}$ a $5\sigma$ discovery is possible for a large
range of $m_{\widetilde{\chi}^{0}_{1}}$.

The $\mu\mu$ and $\mu\tau$ event class have been used to determine the ratio
of the two branching ratios
$\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow\mu^{\pm}W^{\mp})/\operatorname{BR}(\widetilde{\chi}^{0}_{1}\rightarrow\tau^{\pm}W^{\mp})$,
which is related to the atmospheric neutrino mixing angle
$\sin^{2}\theta_{23}$. For an integrated luminosity of
$500\,\mathrm{\,fb}^{-1}$ the total uncertainty on this ratio, including
statistical and systematic uncertainties, has been determined to $4\%$.

Finally, we have shown that the precision in measuring the atmospheric
neutrino mixing angle is in the same range than measurements from neutrino
oscillation experiments, even when taking parametric uncertainties due to the
unknown parts of the SUSY spectrum into account. Therefore, the International
Linear Collider is highly capable to test bRPV SUSY as origin of neutrino
masses and mixings.

## Acknowledgement

We would like to thank the ILC Generators group and the ILD MC production team
for providing the SM background samples, as well as Frank Gäde and the iLCSoft
team for the great support. We are particularly grateful to Werner Porod for
helpful discussions and for providing the neutrino data fit for the simplified
model approach and the input for fig. 9. We thankfully acknowledge the support
by the DFG through the SFB 676 “Particles, Strings and the Early Universe”.

## References

  * [1] J. Wess and B. Zumino, Phys. Lett. B 49 (1974) 52.
  * [2] J. Wess and B. Zumino, Nucl. Phys. B 70 (1974) 39.
  * [3] R. Haag, J. T. Lopuszanski and M. Sohnius, Nucl. Phys. B 88 (1975) 257.
  * [4] J. C. Romao and J. W. F. Valle, Nucl. Phys. B 381 (1992) 87.
  * [5] B. Mukhopadhyaya, S. Roy and F. Vissani, Phys. Lett. B 443 (1998) 191 [hep-ph/9808265].
  * [6] S. Y. Choi, E. J. Chun, S. K. Kang and J. S. Lee, Phys. Rev. D 60 (1999) 075002 [hep-ph/9903465].
  * [7] M. Hirsch, M. A. Diaz, W. Porod, J. C. Romao and J. W. F. Valle, Phys. Rev. D 62 (2000) 113008 [Erratum-ibid. D 65 (2002) 119901] [hep-ph/0004115].
  * [8] W. Porod, M. Hirsch, J. Romao and J. W. F. Valle, Phys. Rev. D 63 (2001) 115004 [hep-ph/0011248].
  * [9] M. Hirsch and W. Porod, Phys. Rev. D 68 (2003) 115007 [hep-ph/0307364].
  * [10] T. Behnke, J. E. Brau, P. N. Burrows, J. Fuster, M. Peskin, M. Stanitzki, Y. Sugimoto and S. Yamada et al., “The International Linear Collider Technical Design Report - Volume 4: Detectors,” arXiv:1306.6329 [physics.ins-det].
  * [11] C. Adolphsen, M. Barone, B. Barish, K. Buesser, P. Burrows, J. Carwardine, J. Clark and Hélèn. M. Durand et al., “The International Linear Collider Technical Design Report - Volume 3.II: Accelerator Baseline Design,” arXiv:1306.6328 [physics.acc-ph].
  * [12] W. Porod, Comput. Phys. Commun. 153 (2003) 275 [hep-ph/0301101], W. Porod and F. Staub, Comput. Phys. Commun. 183 (2012) 2458 [arXiv:1104.1573 [hep-ph]], and W. Porod, private communication.
  * [13] [ATLAS Collaboration], Search for supersymmetry at $sqrt{s}=7$ TeV in final states with large jet multiplicity, missing transverse momentum and one isolated lepton with the ATLAS detector, ATLAS-CONF-2012-140.
  * [14] [ATLAS Collaboration], Search for strongly produced superpartners in final states with two same sign leptons with the ATLAS detector using 21 fb-1 of proton-proton collisions at sqrt(s)=8 TeV., ATLAS-CONF-2013-007.
  * [15] G. Aad et al. [ATLAS Collaboration], Search for a heavy narrow resonance decaying to e mu, e tau, or mu tau with the ATLAS detector in sqrt(s) = 7 TeV pp collisions at the LHC, arXiv:1212.1272 [hep-ex].
  * [16] G. Aad et al. [ATLAS Collaboration], JHEP 1212 (2012) 086 [arXiv:1210.4813 [hep-ex]].
  * [17] [CMS Collaboration], Search for RPV SUSY resonant second generation slepton production in same-sign dimuon events at sqrt(s) = 7 TeV, CMS-PAS-SUS-13-005.
  * [18] [CMS Collaboration], Search for RPV supersymmetry with three or more leptons and b-tags, CMS-PAS-SUS-12-027.
  * [19] [CMS Collaboration], Search for light stop RPV supersymmetry with three or more leptons and b-tags, CMS-PAS-SUS-13-003.
  * [20] [CMS Collaboration], Search for RPV SUSY in the four-lepton final state, CMS-PAS-SUS-13-010.
  * [21] [CMS Collaboration], Search for direct EWK production of SUSY particles in multilepton modes with 8TeV data, CMS-PAS-SUS-12-022.
  * [22] [ATLAS Collaboration], Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in 21 fb-1 of pp collisions at $\sqrt{s}=8\,$TeV with the ATLAS detector, ATLAS-CONF-2013-035.
  * [23] [ATLAS Collaboration], Search for electroweak production of supersymmetric particles in final states with at least two hadronically decaying taus and missing transverse momentum with the ATLAS detector in proton-proton collisions at $\sqrt{s}=8$ TeV, ATLAS-CONF-2013-028.
  * [24] [ATLAS Collaboration], Search for supersymmetry in events with four or more leptons in 21 fb-1 of pp collisions at $\sqrt{s}=8\,$TeV with the ATLAS detector, ATLAS-CONF-2013-036.
  * [25] H. Baer, V. Barger and P. Huang, JHEP 1111 (2011) 031 [arXiv:1107.5581 [hep-ph]].
  * [26] G. Moortgat-Pick, T. Abe, G. Alexander, B. Ananthanarayan, A. A. Babich, V. Bharadwaj, D. Barber and A. Bartl et al., Phys. Rept. 460 (2008) 131 [hep-ph/0507011].
  * [27] C. Bartels, M. Berggren and J. List, Eur. Phys. J. C 72 (2012) 2213 [arXiv:1206.6639 [hep-ex]].
  * [28] C. Adloff et al. [CALICE Collaboration], J. Phys. Conf. Ser. 160 (2009) 012065 [arXiv:0811.2354 [physics.ins-det]].
  * [29] M. A. Thomson, Nucl. Instrum. Meth. A 611 (2009) 25 [arXiv:0907.3577 [physics.ins-det]].
  * [30] F. Staub, Comput. Phys. Commun. 181 (2010) 1077 [arXiv:0909.2863 [hep-ph]].
  * [31] W. Kilian, T. Ohl and J. Reuter, Eur. Phys. J. C 71 (2011) 1742 [arXiv:0708.4233 [hep-ph]].
M. Moretti, T. Ohl and J. Reuter, O’Mega: An Optimizing matrix element
generator, In *2nd ECFA/DESY Study 1998-2001* 1981-2009 [hep-ph/0102195].

  * [32] T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 0605 (2006) 026 [hep-ph/0603175].
  * [33] P. Mora de Freitas and H. Videau, “Detector simulation with MOKKA / GEANT4: Present and future,” LC-TOOL-2003-010.
  * [34] S. Agostinelli et al. [GEANT4 Collaboration], Nucl. Instrum. Meth. A 506 (2003) 250.
J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. A. Dubois, M. Asai, G.
Barrand and R. Capra et al., IEEE Trans. Nucl. Sci. 53 (2006) 270.

  * [35] F. Gäde, Nucl. Instrum. Meth. A 559 (2006) 177.
  * [36] O. Wendt & al., Pramana 69, (2007) 1109, [physics/0702171].
  * [37] M. Cacciari, G. P. Salam and G. Soyez, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097 [hep-ph]].
  * [38] M. Cacciari and G. P. Salam, Phys. Lett. B 641 (2006) 57 [hep-ph/0512210].
  * [39] M. Beckmann, B. List and J. List, Nucl. Instrum. Meth. A 624 (2010) 184 [arXiv:1006.0436 [hep-ex]].
  * [40] T. Abe et al. [ILD Concept Group - Linear Collider Collaboration], The International Large Detector: Letter of Intent, [arXiv:1006.3396 [hep-ex]].
  * [41] Random scans provided by W. Porod, private communication.
  * [42] G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D 86 (2012) 013012 [arXiv:1205.5254 [hep-ph]].


components of fields that contribute under the total differential hence not
affecting the final integrated results. Since the extension to other HS models
is straightforward we will not discuss it in more detail here. Analogously,
the differential homotopy approach can be applied to the multi-particle
Coxeter HS theory [14] anticipated to shed light on the relation between HS
theory and String Theory, one of the favorite ideas of Lars Brink.

## Acknowledgments

I am grateful to Olga Gelfond for careful reading the manuscript and many
useful discussions and comments. I acknowledge with gratitude useful comments
by Vyatcheslav Didenko, Philipp Kirakosiants, Aleksandr Tarusov, Kirill
Ushakov, Danila Valeriev, and, especially, Vyatcheslav Vereitin and the
referee. Also I wish to thank Ofer Aharony, Theoretical High Energy Physics
Group of Weizmann Institute of Science for hospitality at the final stage of
this work.

##

## References

  * [1] C. Fronsdal, Phys. Rev. D 18 (1978), 3624.
  * [2] E. S. Fradkin and M. A. Vasiliev, Phys. Lett. B 189 (1987) 89.
  * [3] E. S. Fradkin and M. A. Vasiliev, Annals Phys. 177 (1987), 63.
  * [4] F. A. Berends, G. J. H. Burgers and H. van Dam, Nucl. Phys. B 260 (1985), 295-322.
  * [5] M. A. Vasiliev, JETP Lett. 50 (1989), 374-377.
  * [6] E. Bergshoeff, M. P. Blencowe and K. S. Stelle, Commun. Math. Phys. 128 (1990), 213.
  * [7] M. A. Vasiliev, Int. J. Mod. Phys. A 6 (1991), 1115-1135.
  * [8] S. F. Prokushkin and M. A. Vasiliev, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236].
  * [9] A. K. H. Bengtsson, I. Bengtsson and L. Brink, Nucl.Phys. B227 (1983) 31.
  * [10] O. A. Gelfond and M. A. Vasiliev, Phys. Lett. B 786 (2018), 180-188 [arXiv:1805.11941 [hep-th]].
  * [11] M. A. Vasiliev, in *Sellin 1987, Proceedings, Theory of elementary particles* 234-252.
  * [12] D. J. Gross and P. F. Mende, Nucl. Phys. B 303 (1988) 407.
  * [13] D. J. Gross, Phys. Rev. Lett. 60 (1988) 1229.
  * [14] M. A. Vasiliev, JHEP 08 (2018), 051 [arXiv:1804.06520 [hep-th]].
  * [15] L. Brink, T. H. Hansson and M. A. Vasiliev, Phys. Lett. B 286 (1992), 109-111 [arXiv:hep-th/9206049 [hep-th]].
  * [16] L. Brink, T. H. Hansson, S. Konstein and M. A. Vasiliev, Nucl. Phys. B 401 (1993), 591-612 [arXiv:hep-th/9302023 [hep-th]].
  * [17] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].
  * [18] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109].
  * [19] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
  * [20] I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 550 (2002) 213 [hep-th/0210114].
  * [21] E. Sezgin and P. Sundell, Nucl. Phys. B 644 (2002) 303 [Erratum-ibid. B 660 (2003) 403] [arXiv:hep-th/0205131].
  * [22] R. G. Leigh and A. C. Petkou, JHEP 0306 (2003) 011 [hep-th/0304217].
  * [23] E. Sezgin and P. Sundell, JHEP 07 (2005), 044 [arXiv:hep-th/0305040 [hep-th]].
  * [24] O. Aharony, G. Gur-Ari and R. Yacoby, JHEP 1203 (2012) 037 [arXiv:1110.4382 [hep-th]].
  * [25] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia and X. Yin, Eur. Phys. J. C 72 (2012), 2112 [arXiv:1110.4386 [hep-th]].
  * [26] M. Flato and C. Fronsdal, Lett. Math. Phys. 2, 421 (1978); Phys. Lett. B 97, 236 (1980).
  * [27] X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, JHEP 1511 (2015) 149 [arXiv:1508.04292 [hep-th]].
  * [28] C. Sleight and M. Taronna, Phys. Rev. Lett. 121 (2018) no.17, 171604 [arXiv:1704.07859 [hep-th]].
  * [29] D. Ponomarev, Universe 4 (2018) no.1, 2 [arXiv:1710.00403 [hep-th]].
  * [30] Y. Neiman, [arXiv:2302.00852 [hep-th]].
  * [31] X. Bekaert, N. Boulanger and S. Cnockaert, JHEP 01 (2006), 052 [arXiv:hep-th/0508048 [hep-th]].
  * [32] A. K. H. Bengtsson, J. Math. Phys. 48 (2007), 072302 [arXiv:hep-th/0611067 [hep-th]].
  * [33] R. Manvelyan, K. Mkrtchyan and W. Ruhl, Nucl. Phys. B 836 (2010) 204 [arXiv:1003.2877].
  * [34] A. Sagnotti and M. Taronna, Nucl. Phys. B 842 (2011), 299-361 [arXiv:1006.5242 [hep-th]].
  * [35] A. Fotopoulos and M. Tsulaia, JHEP 1011 (2010) 086 [arXiv:1009.0727 [hep-th]].
  * [36] M. A. Vasiliev, Nucl. Phys. B 862 (2012) 341 [arXiv:1108.5921 [hep-th]].
  * [37] E. Joung, L. Lopez and M. Taronna, JHEP 1207 (2012) 041 [arXiv:1203.6578].
  * [38] I. L. Buchbinder, T. V. Snegirev and Y. M. Zinoviev, J. Phys. A 46 (2013), 214015 [arXiv:1208.0183 [hep-th]].
  * [39] D. Francia, G. L. Monaco and K. Mkrtchyan, JHEP 04 (2017), 068 [arXiv:1611.00292 [hep-th]].
  * [40] I. L. Buchbinder, S. J. Gates and K. Koutrolikos, Universe 4 (2018) no.1, 6 [arXiv:1708.06262 [hep-th]].
  * [41] I. L. Buchbinder and A. A. Reshetnyak, [arXiv:2304.10358 [hep-th]].
  * [42] A. K. H. Bengtsson, “Systematics of Higher-spin Light-front Interactions,” [arXiv:1205.6117 [hep-th]].
  * [43] R. R. Metsaev, Mod. Phys. Lett. A 6 (1991), 359-367.
  * [44] R. R. Metsaev, Mod.Phys.Lett. A8 (1993) 2413–-2426.
  * [45] R. R. Metsaev, Nucl. Phys. B 563 (1999), 295-348 [arXiv:hep-th/9906217 [hep-th]].
  * [46] R. R. Metsaev, Nucl.Phys. B759 (2006) 147–-201, [hep-th/0512342].
  * [47] R. R. Metsaev, Nucl.Phys. B859 (2012) 13 [arXiv:0712.3526].
  * [48] V. Ivanovskiy and D. Ponomarev, JHEP 09 (2023), 014 [arXiv:2306.13441 [hep-th]].
  * [49] S. Giombi and X. Yin, J. Phys. A 46 (2013) 214003 [arXiv:1208.4036 [hep-th]].
  * [50] M. A. Vasiliev, Phys. Lett. B 285 (1992) 225.
  * [51] V. Didenko, O. Gelfond, A. Korybut and M. Vasiliev, J. Phys. A 51 (2018) no.46, 465202 [arXiv:1807.00001 [hep-th]].
  * [52] V. Didenko, O. Gelfond, A. Korybut and M. Vasiliev, JHEP 12 (2019), 086 [arXiv:1909.04876 [hep-th]].
  * [53] O. Gelfond and M. Vasiliev, JHEP 03 (2020), 002 [arXiv:1910.00487 [hep-th]].
  * [54] V. E. Didenko, O. A. Gelfond, A. V. Korybut and M. A. Vasiliev, JHEP 12 (2020), 184 [arXiv:2009.02811 [hep-th]].
  * [55] O. A. Gelfond and A. V. Korybut, Eur. Phys. J. C 81 (2021) no.7, 605 [arXiv:2101.01683 [hep-th]].
  * [56] V. E. Didenko, JHEP 10 (2022), 191 [arXiv:2209.01966 [hep-th]].
  * [57] A. Sharapov, E. Skvortsov, A. Sukhanov and R. Van Dongen, Nucl. Phys. B 990 (2023), 116152 [arXiv:2209.15441 [hep-th]].
  * [58] M. A. Vasiliev, Phys. Lett. B 834 (2022), 137401 [arXiv:2208.02004 [hep-th]].
  * [59] M. A. Vasiliev, JHEP 10, 111 (2017) [arXiv:1605.02662 [hep-th]].
  * [60] O. A. Gelfond and M. A. Vasiliev, Nucl. Phys. B 931 (2018), 383-417 [arXiv:1706.03718 [hep-th]].
  * [61] O. A. Gelfond and M. A. Vasiliev, unpublished.
  * [62] M. A. Vasiliev, Annals Phys. 190 (1989), 59-106.
  * [63] A. A. Tarusov, K. A. Ushakov and M. A. Vasiliev, JHEP 03 (2023), 128 [arXiv:2212.01908 [hep-th]].
  * [64] D. De Filippi, C. Iazeolla and P. Sundell, JHEP 10 (2019), 215 [arXiv:1905.06325 [hep-th]].
  * [65] D. De Filippi, C. Iazeolla and P. Sundell, JHEP 07 (2022), 003 [arXiv:2111.09288 [hep-th]].
  * [66] M. A. Vasiliev, arXiv:hep-th/9910096.
  * [67] V. E. Didenko and A. V. Korybut, JHEP 05 (2023), 133 [arXiv:2212.05006 [hep-th]].
  * [68] O. A. Gelfond, [arXiv:2308.16281 [hep-th]].
  * [69] M. A. Vasiliev, Phys. Lett. B 567 (2003), 139-151 [arXiv:hep-th/0304049 [hep-th]].
  * [70] A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky and E. Sokatchev, Class. Quant. Grav. 1 (1984), 469-498 [erratum: Class. Quant. Grav. 2 (1985), 127].
  * [71] I. Buchbinder, E. Ivanov and N. Zaigraev, Phys. Part. Nucl. Lett. 20 (2023) no.3, 300-305 [arXiv:2211.09501 [hep-th]].
  * [72] I. Buchbinder, E. Ivanov and N. Zaigraev, JHEP 03 (2023), 036 [arXiv:2212.14114 [hep-th]].
  * [73] E. Ivanov, [arXiv:2306.10401 [hep-th]].