# Probing heavy triplet leptons of the type-III seesaw mechanism at future
muon colliders

Tong Li<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin
300071, China  Han Qin<EMAIL_ADDRESS>PITT PACC, Department of Physics and
Astronomy, University of Pittsburgh, Pittsburgh, PA 15217, USA  Chang-Yuan Yao
<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071,
China  Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg,
Germany  Man Yuan<EMAIL_ADDRESS>School of Physics, Nankai
University, Tianjin 300071, China

###### Abstract

We study the search potential of heavy triplet leptons in Type III Seesaw
mechanism at future high-energy and high-luminosity muon colliders. The impact
of up-to-date neutrino oscillation results is taken into account on neutrino
parameters and the consequent decay patterns of heavy leptons in Type III
Seesaw. The three heavy leptons in the Casas-Ibarra parametrization with
diagonal unity matrix are distinguishable in their decay modes. We consider
the pair production of charged triplet leptons $E^{+}E^{-}$ through both
$\mu^{+}\mu^{-}$ annihilation and vector boson fusion (VBF) processes. The
leading-order framework of electroweak parton distribution functions are
utilized to calculate the VBF cross sections. The charged triplet leptons can
be probed for masses above $\sim 1$ TeV. The $E^{+}E^{-}\to
ZZ\ell^{+}\ell^{-}$ channel can be further utilized to fully reconstruct the
three triplet leptons and distinguish neutrino mass patterns. The pair
production of heavy neutrinos $N$ and the associated production $E^{\pm}N$ are
only induced by VBF processes and lead to lepton-number-violating (LNV)
signature. We also study the search potential of LNV processes at future high-
energy muon collider with $\sqrt{s}=30$ TeV.

##  I Introduction

Neutrino oscillation experiments have provided compelling evidence that
neutrinos have small but finite masses. It is well-known that, in the context
of the Standard Model (SM), the small Majorana neutrino masses can be
generated at leading order through a dimension-5 “Weinberg operator” Weinberg
(1979)

$\displaystyle{\kappa\over\Lambda}LLHH\;,$ (1)

where $L$ and $H$ stand for the SM left-handed lepton doublet and the Higgs
doublet, respectively. The ultraviolet completions of Eq. (1) at tree level Ma
(1998) are only implemented by the Type I Minkowski (1977); Yanagida (1979);
Gell-Mann  _et al._ (1979); Glashow (1980); Mohapatra and Senjanovic (1980);
Shrock (1981); Schechter and Valle (1980), Type II Konetschny and Kummer
(1977); Cheng and Li (1980); Lazarides  _et al._ (1981); Schechter and Valle
(1980); Mohapatra and Senjanovic (1981) and Type III Foot  _et al._ (1989)
Seesaw mechanisms. As proved in Refs. Kersten and Smirnov (2007); Moffat  _et
al._ (2017), requiring all three light neutrinos to be massless is equivalent
to requiring lepton number to be conserved at all orders in perturbation
theory. In other words, in light of non-zero neutrino masses, lepton number is
nearly conserved in low-scale seesaw models with fermionic singlets. Thus, for
low-scale Type I Seesaw, the observable lepton number violation (LNV) is
decoupled in high energy processes. Any observation of lepton number violation
at high-energy colliders implies an extension of the canonical Type I Seesaw.
In particular, the Type III Seesaw introduces at least two SU$(2)_{L}$ triplet
lepton representations $\Sigma_{L}$ with $(1,3,0)$ SM quantum numbers. The
active neutrino mass matrix has the same form as that in Type I Seesaw and the
high energy scale $\Lambda$ is given by integrating out the mass of triplet
leptons which can be as low as a few TeV.

On the other hand, several reactor experiments have provided precise
measurements for the neutrino mixing parameters. For instance, Double Chooz
Abe  _et al._ (2012), RENO Ahn  _et al._ (2012) and in particular Daya Bay An
_et al._ (2012) have reported non-zero measurements of $\theta_{13}$ by
looking for the disappearance of anti-electron neutrino. More recently, two
long-baseline neutrino experiments T2K Abe  _et al._ (2021) and NOvA Acero
_et al._ (2021) reported the indication of a non-zero CP-violating phase.
Despite a slight tension at $\sim 2\sigma$ level between them, these
experiments provide us up-to-date neutrino oscillation results to investigate
the impact on neutrino mass models. As the sign of the mass-squared difference
$\Delta m_{31}^{2}$ has not been determined up to now, the so-called normal
hierarchy (NH) with positive $\Delta m_{31}^{2}$ and the inverted hierarchy
(IH) with negative $\Delta m_{31}^{2}$ are normally considered in the data
analysis. Future neutrino oscillation experiments, such as T2HK Abe  _et al._
(2015), JUNO Djurcic  _et al._ (2015) and DUNE Acciarri  _et al._ (2015), are
very likely to further explain the discrepancy and make a distinction between
these two neutrino mass patterns. Besides the neutrino experiments, high-
energy colliders can also provide complementary strategies to reveal the
neutrino properties by searching for the triplet leptons and leptonic
signatures.

The new triplet leptons in TeV Type III Seesaw mechanism can be experimentally
accessible at the Large Hadron Collider (LHC) (see Refs. Arhrib  _et al._
(2010); Li and He (2009); Biggio and Bonnet (2012); Bandyopadhyay  _et al._
(2012); Aguilar-Saavedra  _et al._ (2013); Cai  _et al._ (2018); Jana  _et
al._ (2020); Das and Mandal (2021); Ashanujjaman and Ghosh (2021); Sen  _et
al._ (2022); Ashanujjaman and Ghosh (2022) and the references therein) and
lepton colliders Goswami and Poulose (2018); Das  _et al._ (2020);
Bandyopadhyay  _et al._ (2020); Bhattacharya  _et al._ (2022). The triplet
leptons are produced in pairs through pure electroweak (EW) gauge couplings
and followed by multi-lepton signatures. After combining two-, three- and
four-lepton channels together, the observed lower limit on the mass of heavy
leptons is above 900 GeV at the 95% confidence level (CL) Sirunyan  _et al._
(2017); Aad  _et al._ (2022). According to the theoretical estimations, it is
difficult to discover the lepton-number-conserving or lepton-number-violating
signal events for triplet leptons heavier than $\sim 1$ TeV at the LHC. One
needs a bigger machine with higher energy and integrated luminosity. Recently,
high energy muon colliders have received much attention in the community due
to the technological developments Bogomilov  _et al._ (2020); Delahaye  _et
al._ (2019); Bartosik  _et al._ (2020). The community considers the high-
energy option of muon collider with multi-TeV center-of-mass (c.m.) energies
and a high integrated luminosity scaling with energy quadratically Delahaye
_et al._ (2019)

$\displaystyle\mathcal{L}=\left(\frac{\sqrt{s}}{10\ {\rm
TeV}}\right)^{2}10~{}\textrm{ab}^{-1}\;.$ (2)

It thus provides excellent opportunities to produce and discover new heavy EW
particles such as the triplet leptons in Type III Seesaw.

In this work, we investigate the search for heavy triplet leptons at high-
energy muon colliders. The pair productions of heavy triplet leptons are
categorized into the topologies of direct $\mu^{+}\mu^{-}$ annihilation and
the vector boson fusion (VBF). The mass eigenstates of charged triplet lepton
(denoted by $E^{\pm}$) can be produced in pairs through both of these two
processes. The large total cross section of $E^{+}E^{-}$ production makes it
an appealing discovery channel of the triplet lepton. By contrast, the
associated production of $E^{\pm}$ and the neutral heavy neutrino $N$ as well
as the pair production of $N$ are only induced by VBF processes. In spite of
their small production cross sections, they can produce distinctively rare LNV
signal and play as crucial channels to determine Majorana neutrino property
and distinguish triplet mass eigenstates at high energies.

This paper is organized as follows. In Sec. II, we first outline the Type III
Seesaw model and discuss the constraints from neutrino oscillation experiments
on the decay patterns of heavy triplet leptons. Then we simulate the pair
production of heavy triplet leptons via $\mu^{+}\mu^{-}$ annihilation and VBF
at high-energy muon colliders in Sec. III. The results of projected
sensitivity for heavy leptons are also given. Finally, in Sec. IV we summarize
our conclusions.

##  II Type III Seesaw and heavy triplet leptons

In addition to the SM fields, the Type III Seesaw Foot  _et al._ (1989)
consists of SU$(2)_{L}$ triplet leptons with zero hypercharge, i.e.,
$\Sigma_{L}\sim(1,3,0)$ under SM gauge group
SU$(3)_{C}\times$SU$(2)_{L}\times$U$(1)_{Y}$:

$\displaystyle\Sigma_{L}=\left(\begin{array}[]{cc}\Sigma_{L}^{0}/\sqrt{2}&\Sigma^{+}_{L}\\\
\Sigma_{L}^{-}&-\Sigma_{L}^{0}/\sqrt{2}\\\ \end{array}\right)\ .$ (5)

The Type III Seesaw Lagrangian is given by the kinetic term, mass term and
Yukawa interactions in terms of $\Sigma_{L}$ and its charge conjugated form
$\Sigma_{L}^{c}$

$\mathcal{L}_{T}={\rm
Tr}\left[\overline{\Sigma_{L}}i\cancel{D}\Sigma_{L}\right]-\left({1\over
2}{\rm
Tr}\left[\overline{\Sigma_{L}^{c}}M_{\Sigma}\Sigma_{L}\right]+Y_{\Sigma}\overline{l_{L}}\tilde{H}\Sigma_{L}^{c}+\text{H.c.}\right)\;,$
(6)

where $\tilde{H}=i\sigma_{2}H^{\ast}\sim(1,2,-1)$. Besides the neutrino mass
mixing for $(\nu_{L},\Sigma_{L}^{0})^{T}$, the Yukawa term in Eq. (6) also
induces a mass mixing between the charged SM leptons and the charged triplet
leptons

$\displaystyle\left(\begin{array}[]{cc}\overline{\nu_{L}^{c}}&\overline{\Sigma_{L}^{0c}}\\\
\end{array}\right)\left(\begin{array}[]{cc}0&Y_{\Sigma}^{T}v_{0}/2\sqrt{2}\\\
Y_{\Sigma}v_{0}/2\sqrt{2}&M_{\Sigma}/2\\\
\end{array}\right)\left(\begin{array}[]{c}\nu_{L}\\\ \Sigma_{L}^{0}\\\
\end{array}\right)+\left(\begin{array}[]{cc}\overline{l_{R}}&\overline{\Sigma_{L}^{+c}}\\\
\end{array}\right)\left(\begin{array}[]{cc}m_{l}&0\\\
Y_{\Sigma}v_{0}&M_{\Sigma}\\\
\end{array}\right)\left(\begin{array}[]{c}l_{L}\\\ \Sigma_{L}^{-}\\\
\end{array}\right)+\text{H.c.}\;,$ (17)

where $v_{0}\approx 246$ GeV is the vacuum expectation value of Higgs. After
introducing unitary matrices to transit light doublet and heavy triplet lepton
fields, one obtains the diagonal mass matrices and mass eigenvalues for
neutrinos and charged leptons. The mass eigenstates of heavy neutral and
charged leptons are denoted by $N$ and $E^{\pm}$, respectively. Expanding the
unitary matrices to leading order in $Y_{\Sigma}v_{0}M_{\Sigma}^{-1}$, the
mixing between the SM charged leptons and triplet lepton mass eigenstates is
governed by $V_{\ell N}=-Y_{\Sigma}^{\dagger}v_{0}M_{\Sigma}^{-1}/\sqrt{2}$.
In the mass basis, to the leading order, we have the following interactions
involving heavy triplet leptons

$\displaystyle\mathcal{L}_{\rm Type~{}III}$ $\displaystyle=$ $\displaystyle
e\overline{E}\gamma^{\mu}EA_{\mu}+g\cos\theta_{W}\overline{E}\gamma^{\mu}EZ_{\mu}$
$\displaystyle+$ $\displaystyle{g\over
2\cos\theta_{W}}\Big{[}\overline{\nu_{\ell}}(V_{PMNS}^{\dagger}V_{\ell
N}\gamma^{\mu}P_{L}-V_{PMNS}^{T}V^{\ast}_{\ell
N}\gamma^{\mu}P_{R})N+\sqrt{2}\overline{\ell}V_{\ell
N}\gamma^{\mu}P_{L}E\Big{]}Z_{\mu}$ $\displaystyle-$ $\displaystyle
g\Big{[}\overline{E}\gamma^{\mu}N+{1\over\sqrt{2}}\overline{\ell}V_{\ell
N}\gamma^{\mu}P_{L}N+\overline{E}V_{\ell
N}^{T}V_{PMNS}^{\ast}\gamma^{\mu}P_{R}\nu_{\ell}\Big{]}W^{-}_{\mu}$
$\displaystyle+$ $\displaystyle{g\over
2M_{W}}\Big{[}\overline{\nu_{\ell}}(V_{PMNS}^{\dagger}V_{\ell
N}M_{N}^{diag}P_{R}+V_{PMNS}^{T}V^{\ast}_{\ell
N}M_{N}^{diag}P_{L})N+\sqrt{2}\overline{\ell}V_{\ell
N}M_{E}^{diag}P_{R}E\Big{]}h+\text{H.c.}\;,$

where all fields are mass eigenstates.

The above gauge interactions between SM leptons and heavy triplet leptons are
all given by $V_{\ell N}$ Aguilar-Saavedra (2010); Li and He (2009) and the
partial widths of both heavy charged lepton and heavy neutrino are
proportional to $|V_{\ell N}|^{2}$. In the limit of $M_{E}\approx M_{N}\gg
M_{W},M_{Z},M_{h}$, the partial widths become del Aguila and Aguilar-Saavedra
(2009); Aguilar-Saavedra (2010)

$\displaystyle{1\over
2}\Gamma(N\to\sum_{\ell}\ell^{+}W^{-}+\ell^{-}W^{+})\approx\Gamma(N\to\sum_{\nu}\nu
Z+\bar{\nu}Z)\approx\Gamma(N\to\sum_{\nu}\nu h+\bar{\nu}h)$ (20)
$\displaystyle\approx$ $\displaystyle{1\over
2}\Gamma(E^{\pm}\to\sum_{\nu}\overset{(-)}{\nu}W^{\pm})\approx\Gamma(E^{\pm}\to\sum_{\ell}\ell^{\pm}Z)\approx\Gamma(E^{\pm}\to\sum_{\ell}\ell^{\pm}h)$
$\displaystyle\approx$ $\displaystyle{G_{F}\over
8\sqrt{2}\pi}\sum_{\ell}|V_{\ell N}|^{2}M_{\Sigma}^{3}\;.$

Thus, the decay branching ratios of heavy leptons exhibit asymptotic behavior
consistent with the Goldstone Equivalence Theorem Chanowitz and Gaillard
(1985); Lee  _et al._ (1977), and are given by the following relations
Franceschini  _et al._ (2008); del Aguila and Aguilar-Saavedra (2009);
Aguilar-Saavedra (2010); Arhrib  _et al._ (2010)

$\displaystyle{1\over 2}{\rm
BR}(N\to\sum_{\ell}\ell^{+}W^{-}+\ell^{-}W^{+})\approx{\rm
BR}(N\to\sum_{\nu}\nu Z+\bar{\nu}Z)\approx{\rm BR}(N\to\sum_{\nu}\nu
h+\bar{\nu}h)$ (21) $\displaystyle\approx$ $\displaystyle{1\over 2}{\rm
BR}(E^{\pm}\to\sum_{\nu}\overset{(-)}{\nu}W^{\pm})\approx{\rm
BR}(E^{\pm}\to\sum_{\ell}\ell^{\pm}Z)\approx{\rm
BR}(E^{\pm}\to\sum_{\ell}\ell^{\pm}h)\approx{1\over 4}\;.$

As the triplet mass grows, this asymptotic behavior can be seen explicitly in
the triplet lepton partial widths.

There is a relation between diagonalized neutrino mass matrices and mixing
matrices

$\displaystyle U^{\ast}_{\rm PMNS}m_{\nu}^{diag}U^{\dagger}_{\rm
PMNS}+V^{\ast}_{\ell N}M_{N}^{diag}V^{\dagger}_{\ell N}=0\ ,$ (22)

with mass eigenvalues being $m_{\nu}^{diag}=(m_{1},m_{2},m_{3})$ and
$M_{N}^{diag}=(M_{N_{1}},M_{N_{2}},M_{N_{3}})$. Here the masses and mixing of
the light neutrinos in the first term are measurable from the neutrino
oscillation experiments. The second term contains the masses and mixing of the
new heavy neutrinos. The general solution of the $V_{\ell N}$ in Eq. (22) can
be parameterized in terms of an arbitrary orthogonal complex matrix $\Omega$
in the Casas-Ibarra parametrization Casas and Ibarra (2001)

$\displaystyle V_{\ell N}=U_{\rm
PMNS}(m_{\nu}^{diag})^{1/2}\Omega(M_{N}^{diag})^{-1/2},$ (23)

with the orthogonality condition $\Omega\Omega^{T}=I$. Using the SM
electroweak current for heavy Majorana neutrinos $N_{i}$, in the mixed mass-
flavor basis, one can obtain the partial width of their decay into charged
lepton Atre  _et al._ (2009)

$\displaystyle\Gamma(N_{i}\to\ell^{\pm}W^{\mp})={G_{F}\over
8\sqrt{2}\pi}|V_{\ell
N_{i}}|^{2}M_{N_{i}}(M_{N_{i}}^{2}+2M_{W}^{2})\left(1-{M_{W}^{2}\over
M_{N_{i}}^{2}}\right)^{2},$ (24)

where $\ell=e,\mu,\tau$. Note that generally the Dirac mass matrix can be
quite arbitrary with three complex angles parameterizing the orthogonal matrix
$\Omega$ Fileviez Perez  _et al._ (2009); Biggio  _et al._ (2020); Das and
Mandal (2021); Fileviez Pérez and Plascencia (2020). The above asymptotic
behavior of decay branching ratios of heavy neutrino indicates
BR$(N_{i}\to\sum_{\ell}\ell^{\pm}W^{\mp})$ is nearly equal to $25\%$ for any
heavy neutrino $N_{i}~{}(i=1,2,3)$. The branching fraction of each heavy
neutrino $N_{i}$ decaying into any charged lepton flavor is thus less than
$25\%$ for any possible $\Omega$ matrix. In one simple case for $\Omega$,
i.e., a diagonal unity matrix $\Omega=I$, the values of $|V_{\ell N}|^{2}$ for
each $N_{i}$ are proportional to one and only one light neutrino mass Fileviez
Perez  _et al._ (2009). The branching ratio of $N_{i}\to\ell^{\pm}W^{\mp}$ for
any charged lepton flavor is thus independent of active neutrino mass. This
representative choice of $\Omega=I$ can also lead to different leading decay
into charged lepton flavor for each heavy neutrino, so one can distinguish
different heavy neutrinos Fileviez Pérez and Plascencia (2020); Han  _et al._
(2021a). Thus, to reduce the parameter dependence and obtain certain
predictions, we take this illustrative case of Casas-Ibarra parametrization
with $\Omega=I$ in the following study. Note that our results below depend on
this specific case and a different choice of $\Omega$ would lead to different
conclusions for individual $N_{i}$.

In order to consider the implication of the neutrino experiments, we then
discuss the neutrino mass and mixing parameters in light of oscillation data.
The neutrino mixing matrix can be parameterized as

$\displaystyle
U_{\text{PMNS}}=\left(\begin{array}[]{lll}c_{12}c_{13}&c_{13}s_{12}&e^{-\text{i$\delta$}}s_{13}\\\
-c_{12}s_{13}s_{23}e^{\text{i$\delta$}}-c_{23}s_{12}&c_{12}c_{23}-e^{\text{i$\delta$}}s_{12}s_{13}s_{23}&c_{13}s_{23}\\\
s_{12}s_{23}-e^{\text{i$\delta$}}c_{12}c_{23}s_{13}&-c_{23}s_{12}s_{13}e^{\text{i$\delta$}}-c_{12}s_{23}&c_{13}c_{23}\end{array}\right)\times\text{diag}(e^{i\Phi_{1}/2},1,e^{i\Phi_{2}/2})\;,$
(28)

where $s_{ij}\equiv\sin{\theta_{ij}}$, $c_{ij}\equiv\cos{\theta_{ij}}$,
$0\leq\theta_{ij}\leq\pi/2$ and $0\leq\delta,\Phi_{i}\leq 2\pi$ with $\delta$
being the Dirac CP phase and $\Phi_{i}$ are the Majorana phases. Taking into
account the atmospheric neutrino data from the Super-Kamiokande collaboration,
the latest best global fit results of the neutrino masses and mixing
parameters are given by Esteban  _et al._ (2020); nuf (2021)

$\displaystyle\Delta m_{21}^{2}$ $\displaystyle=$ $\displaystyle 7.42\times
10^{-5}\,{\rm eV}^{2}\;,$ $\displaystyle\Delta m_{31}^{2}$ $\displaystyle=$
$\displaystyle 2.510\times 10^{-3}\,{\rm eV}^{2}\ (\Delta
m_{32}^{2}=-2.490\times 10^{-3}\,{\rm eV}^{2})\;,$
$\displaystyle\sin^{2}{\theta_{12}}$ $\displaystyle=$ $\displaystyle
0.304\;,~{}~{}\sin^{2}{\theta_{23}}=0.450\ (0.570)\;,$
$\displaystyle\sin^{2}{\theta_{13}}$ $\displaystyle=$ $\displaystyle 0.02246\
(0.02241)\;,~{}~{}\delta_{\rm CP}=230^{\circ}(278^{\circ})\;,$ (30)

for the spectrum of the neutrino masses of NH (IH). In addition, the sum of
neutrino masses is constrained to be $\sum_{i=1}^{3}m_{i}<0.12$ eV by
combining the Planck, lensing and baryon acoustic oscillation data Aghanim
_et al._ (2020) at $95\%$ CL. By inputting the above experimental constraints
to the mixing matrix $V_{\ell N}$ and the above partial decay width formulas,
we can obtain the preferred values of decay branching ratios for heavy
neutrino decay patterns. In our numerical calculations below, we adopt the
benchmark decay branching ratios of heavy neutrinos with $\Omega=I$ for NH and
IH as shown in Table 1. One can also obtain the decay branching fractions of
charged leptons $E_{i}^{\pm}$ as the relations in Eq. (21) apply.

BR$(N_{i})$ | $e^{\pm}W^{\mp}$ | $\mu^{\pm}W^{\mp}$ | $\tau^{\pm}W^{\mp}$
---|---|---|---
$N_{1}$ | 17% (17%) | 3.26% (3.73%) | 4.74% (4.27%)
$N_{2}$ | 7.43% (7.43%) | 10.75% (7.34%) | 6.82% (10.23%)
$N_{3}$ | 0.56% (0.56%) | 11% (13.93%) | 13.44% (10.51%)
Table 1: Benchmark decay branching ratios of $N_{i}$ in $\Omega=I$ case for NH
(IH). Here we assume the smallest light neutrino mass as $10^{-4}$ eV and
$M_{N_{i}}\gg M_{W}$.

##  III Pair production of heavy leptons at muon collider

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Feynman diagrams of heavy triplet lepton pair productions in Type
III Seesaw, including $\mu^{+}\mu^{-}$ annihilation (a) and VBF processes
(b-f).

The pair production channels of heavy triplet leptons at muon colliders
include

  * •

$\mu^{+}\mu^{-}\to\gamma^{\ast}/Z^{\ast}\to E^{+}E^{-}$ ,

  * •

$W^{+}W^{-}\to\gamma^{\ast}/Z^{\ast}\to E^{+}E^{-}$ , $\gamma/Z+\gamma/Z\to
E^{+}E^{-}$ , $W^{+}W^{-}\to NN,E^{+}E^{-}$ ,

  * •

$W^{\pm}+\gamma/Z\to W^{\pm\ast}\to E^{\pm}N$, $W^{\pm}+\gamma/Z\to E^{\pm}N$
.

The Feynman diagrams of heavy triplet lepton pair productions in Type III
Seesaw are collected in Fig. 1. They are given by the pure EW gauge
interactions in Eq. (LABEL:eq:L) and not suppressed by the mixing $V_{\ell
N}$. For c.m. energies $\sqrt{s}\gg M_{W}$, the initial muon beams
substantially radiate EW gauge bosons under an approximately unbroken SM gauge
symmetry. The vector bosons are treated as initial state partons and produce
“inclusive” processes. We use MadGraph5_aMC@NLO Alwall  _et al._ (2014) to
calculate the VBF cross sections, by utilizing the leading-order framework of
electroweak parton distribution functions (EW PDFs) Han  _et al._ (2021b); Al
Ali  _et al._ (2021); Ruiz  _et al._ (2021) with a dynamical scale
$Q=\sqrt{\hat{s}}/2$, where $\sqrt{\hat{s}}$ is the partonic c.m. energy. For
the initial gauge bosons $V_{a}$ and $V_{b}$, the VBF production cross section
can be factorized as the product of the parton luminosity
$d\mathcal{L}_{ab}/d\tau$ and the partonic cross section $\hat{\sigma}$ Han
_et al._ (2021b)

$\displaystyle\sigma(\mu^{+}\mu^{-}\to F+X)$ $\displaystyle=$
$\displaystyle\int^{1}_{\tau_{0}}d\tau\sum_{ab}{d\mathcal{L}_{ab}\over
d\tau}\hat{\sigma}(V_{a}V_{b}\to F)\;,$ $\displaystyle{d\mathcal{L}_{ab}\over
d\tau}$ $\displaystyle=$ $\displaystyle{1\over
1+\delta_{ab}}\int^{1}_{\tau}{d\xi\over\xi}\Big{[}f_{a}(\xi,Q^{2})f_{b}({\tau\over\xi},Q^{2})+(a\leftrightarrow
b)\Big{]}\;,$ (31)

where $F(X)$ denotes an exclusive final state (the underlying remnants),
$f_{a}(\xi,Q^{2})$ is the EW PDF for vector $V_{a}$ with $Q$ being the
factorization scale, $\tau_{0}=m_{F}^{2}/s$ and $\tau=\hat{s}/s$. By summing
over all gauge boson initial states in Fig. 1, one can obtain the total cross
section of the “inclusive” VBF production processes.

We consider three benchmark choices of muon collider energies and the
corresponding integrated luminosities for illustration

$\displaystyle\sqrt{s}=3,\ 10\ {\rm and}\ 30\ {\rm TeV},\quad{\mathcal{L}}=1,\
10\ {\rm and}\ 90\ {\rm ab}^{-1}\;.$ (32)

The heavy triplet leptons are followed by the decays into charged leptons or
active neutrinos as well as EW gauge bosons. We smeared the parton level
events with an energy resolution of $\Delta E/E=2\%$. Note that we consider
the hadronic decay of EW gauge bosons into two jets. The two jets generally
form a fat jet here as the gauge bosons are from heavy triplet decay and thus
are highly boosted. We perform the parton showering and detector simulation to
identify the fat jets and then reconstruct the vector bosons efficiently. The
hadronic decay branching fractions and the efficiencies of the gauge boson
reconstruction are included in our numerical analysis. The details of this
reconstruction will be described below. We also take a conservative way to
estimate the significance as

$\displaystyle\mathcal{S}={N_{S}\over\sqrt{N_{S}+N_{B}}}\;,$ (33)

where $N_{S}$ and $N_{B}$ denote the signal and SM background event numbers,
respectively.

###  III.1 $E^{+}E^{-}$ Pair Production

We first perform the calculation of the pair production of heavy charged
leptons. The $E^{+}E^{-}$ productions are decomposed to $\mu^{+}\mu^{-}$
direct annihilation and the electroweak VBF processes initiated by $\gamma/Z$
or $W^{+}W^{-}$. In Fig. 2, we show the production cross sections of
$E^{+}E^{-}$ as a function of $\sqrt{s}$ (left) and $M_{E}$ (right). The
productions via $\mu^{+}\mu^{-}$ annihilation, VBF and their sum are
represented as dashed lines, dotted lines and solid lines, respectively. The
$\mu\mu$ annihilation cross sections approach $\sigma^{\rm Ann}\sim\beta/s$
with the velocity as $\beta=\sqrt{1-\frac{4M_{E}^{2}}{s}}$. Thus, they
decrease with the increasing c.m. energy. Well above the threshold, the
triplet mass can be ignored at high energies and the heavy charged leptons
tend to be indistinguishable in this channel. On the other hand, the VBF cross
sections $\sigma^{\rm VBF}$ are enhanced by collinear logarithm ${\rm
ln}(\hat{s}/m_{\mu}^{2})$ for photon or ${\rm ln}(\hat{s}/m_{V}^{2})$ for
massive gauge boson $V$ at high beam energies. They are also very sensitive to
the heavy lepton masses and decrease quickly with increasing $M_{E}$. The VBF
process is thus important at low $M_{E}$ and high c.m. energies.

We consider the above productions followed by two decay modes of $E^{\pm}$

$\displaystyle E^{\pm}\to\nu W^{\pm}\;,~{}~{}E^{\pm}\to\ell^{\pm}Z\;.$ (34)

For the first mode, the SM neutrinos are summed over and thus one has ${\rm
BR}(E^{\pm}\to\sum_{\nu}\nu W^{\pm})\approx 50\%$ according to the relation in
Eq. (21). In the second mode, we only study the decay $E^{\pm}\to\mu^{\pm}Z$
for illustration and the decay branching fractions of the heavy neutrino in
Table 1 apply here as ${\rm BR}(N_{i}\to\ell^{\pm}W^{\mp})\approx{\rm
BR}(E_{i}^{\pm}\to\ell^{\pm}Z)$ for degenerate triplet leptons. Note that we
take the channel $E\to\mu Z$ here for illustrative estimation of the heavy
triplet search prospect. The channel $E\to eZ$ can also be analyzed following
the same strategies and their SM background analysis is even simpler as the
electrons cannot come from the fermion line of initial states.

Figure 2: Cross sections of $E^{+}E^{-}$ pair production as a function of
$\sqrt{s}$ (left) and $M_{E}$ (right) at muon colliders by $\mu^{+}\mu^{-}$
annihilation (dashed lines), VBF (dotted lines), and the sum (solid lines).

####  III.1.1 $E^{+}E^{-}\to W^{+}W^{-}\nu\bar{\nu}$

In this case, our signal is composed of $W^{+}W^{-}$ and large missing energy
induced by SM neutrinos. The SM background in this channel thus becomes

$\displaystyle W^{+}W^{-}Z\;,$ (35)

followed by decay $Z\to\nu\bar{\nu}$ and is generated by MadGraph at the
parton level. We also consider both the $\mu^{+}\mu^{-}$ annihilation and VBF
productions for the above background. In the beginning, the size of SM
background is comparable to our signal. We thus employ the following judicious
cuts to reduce the background:

  * •

We take into account the $W$ bosons identified in central region and set the
rapidity cut as follows

$\displaystyle\left|\eta(W)\right|<3.0\;.$ (36)

  * •

For a few TeV $E^{\pm}$, the decay products from heavy triplet lepton are very
energetic as shown in Fig. 3 (left) 111The differential cross sections of the
$\mu\mu$ annihilation and VBF processes are individually weighted by their
total cross sections and summed together here.. We therefore tight up the
kinematical cut with

$\displaystyle p_{T}^{\rm{max}}(W)>\frac{M_{E}}{2}\;,$ (37)

for the leading $W$ boson.

  * •

To remove the $W^{+}W^{-}Z$ background with neutrinos from $Z$ boson decay as
seen in the right panel of Fig. 3, we require the events with large missing
energy from neutrinos

$\displaystyle\cancel{E}_{T}>\frac{M_{E}}{4}\;.$ (38)

As a result, the SM background can be significantly suppressed.

Figure 3: The normalized $p_{T}^{\rm max}(W)$ (left) and $\cancel{E}_{T}$
(right) distributions of the signal $E^{+}E^{-}\to W^{+}W^{-}\nu\bar{\nu}$ and
the SM background for $\sqrt{s}=3$ TeV and $M_{E}=1$ TeV.

We then perform the local significance analysis. The event numbers of signal
($N_{S}$) and background ($N_{B}$) are given by

$\displaystyle N_{S}$ $\displaystyle=$ $\displaystyle(\sigma_{S}^{\rm
Ann}\epsilon_{S}^{\rm
Ann}\epsilon_{S,W}^{\rm{Ann}}+\sigma_{S}^{\rm{VBF}}\epsilon_{S}^{\rm{VBF}}\epsilon_{S,W}^{\rm{VBF}})\times{\rm
BR}(E^{\pm}\to W^{\pm}\nu)^{2}\times\mathcal{L}\times\rm{BR}(W^{\pm}\to
q\bar{q}^{\prime})^{2}\;,$ $\displaystyle N_{B}$ $\displaystyle=$
$\displaystyle(\sigma_{B}^{\rm Ann}\epsilon_{B}^{\rm
Ann}\epsilon_{B,W}^{\rm{Ann}}+\sigma_{B}^{\rm{VBF}}\epsilon_{B}^{\rm{VBF}}\epsilon_{B,W}^{\rm{VBF}})\times{\rm
BR}(Z\to\nu\bar{\nu})\times\mathcal{L}\times\rm{BR}(W^{\pm}\to
q\bar{q}^{\prime})^{2}\;,$ (39)

where $\sigma_{S}$ is the signal cross section, $\sigma_{B}$ is the cross
section of the SM background $W^{+}W^{-}Z$, the superscripts “${\rm Ann}$” and
“${\rm VBF}$” denote $\mu^{+}\mu^{-}$ direct annihilation and VBF process,
respectively, $\epsilon_{S,B}$ represent the efficiencies of the above cuts
and $\mathcal{L}$ denotes the integrated luminosity. In practise, the $W$
bosons subsequently decay into leptonic or hadronic products. Here and below
we consider the hadronic decay of vector bosons into a dijet. As being from
heavy lepton decay, the vector bosons are highly boosted and the two jets are
likely to merge into a single fat jet $J$. To perform the simulation of vector
boson reconstruction, we pass the parton-level events to Pythia 8 Sjöstrand
_et al._ (2015) and Delphes 3 de Favereau  _et al._ (2014) for parton shower
and detector simulation, respectively. We choose the detector card of muon
collider in the latest version of Delphes for detector-level simulation. The
jets from $W$ decay are clustered in FastJet Cacciari  _et al._ (2012) and we
require at least two fat jets. The fat jet is reconstructed via the “Valencia”
algorithm with $R=0.5$ and the one with $65<M_{J}<95$ GeV ($75<M_{J}<105$ GeV
for $Z$ boson used in Sec. III.2) is identified as boosted $W$ candidate.
$\epsilon_{S,W}$ and $\epsilon_{B,W}$ in Eq. (39) are the efficiencies of the
vector boson reconstruction.

In Fig. 4, we show the integrated luminosities required for $3\sigma$ and
$5\sigma$ significance at muon collider with $\sqrt{s}=3$ TeV (left) and
$\sqrt{s}=10$ TeV (right). One can see that the charged triplet lepton as
heavy as one-half of the colliding energy can be probed. A $5\sigma$
significance can be reached through the $E^{+}E^{-}\to W^{+}W^{-}\nu\bar{\nu}$
process with the integrated luminosity below 0.1 ab-1 for $\sqrt{s}=3$ TeV and
2 ab-1 for $\sqrt{s}=10$ TeV. Despite this optimistic result, however, we are
not able to reconstruct the heavy leptons due to the missing neutrinos from
$E^{\pm}$ decay. Moreover, the different neutrino mass patterns and the three
triplet leptons cannot be identified and distinguished through this process as
the missing neutrinos lose lepton flavor information in detector. We need to
discuss other signal channels in the following section to reveal the neutrino
characteristics.

Figure 4: The integrated luminosities for $3\sigma$ and $5\sigma$ significance
versus $M_{E}$ for $E^{+}E^{-}\to W^{+}W^{-}\nu\bar{\nu}$ in muon collider
with $\sqrt{s}=3$ TeV (left) and 10 TeV (right).

####  III.1.2 $E^{+}E^{-}\to ZZ\mu^{+}\mu^{-}$

Next, we discuss the $E^{+}E^{-}$ pair production followed by
$E^{\pm}\to\ell^{\pm}Z$ decay mode. In this case, we only consider the decay
channel $E^{\pm}\to\mu^{\pm}Z$ for illustration. The signal is thus composed
of two $Z$ bosons and $\mu^{+}\mu^{-}$. The SM background is also categorized
into $\mu^{+}\mu^{-}$ annihilation and VBF production. Since there are two
muons in the final states of this channel, we employ the following basic cuts

$\displaystyle p_{T}(\mu)>50~{}{\rm
GeV}\,,~{}~{}\left|\eta(\mu,Z)\right|<3.0\;.$ (40)

We also employ the judicious cuts to reduce the background:

  * •

For the energetic products from a few TeV $E^{\pm}$, we set the following
kinematical cuts

$\displaystyle p_{T}^{\rm max}(Z)>\frac{M_{E}}{3}\,,~{}~{}p_{T}^{\rm
max}(\mu)>\frac{M_{E}}{3}\;.$ (41)

  * •

Since the final particles $\mu^{\pm}$ and $Z$ are both observable in detector,
one can reconstruct the triplet lepton $E^{\pm}$. To reconstruct heavy fermion
$E^{\pm}$, we pair the final particles ($Z,\mu$) by taking advantage of the
feature that they have equal masses $M_{Z_{1}\mu_{1}}=M_{Z_{2}\mu_{2}}$, which
is in order to make sure they come from the correct mother particle. Then we
take the invariant masses close to $M_{E}$ with

$\displaystyle\left|M_{Z\mu}-M_{E}\right|<\frac{M_{E}}{5}\;.$ (42)

The reconstructed masses are shown in Fig. 5.

These cuts help to reduce the SM backgrounds.

Figure 5: The invariant mass $M_{Z\mu}$ distribution of the signal
$E^{+}E^{-}\to ZZ\mu^{+}\mu^{-}$ and the background for $M_{E}=1$ TeV,
$\sqrt{s}=$ 3 TeV (left) and 10 TeV (right).

For the local significance analysis, we apply the benchmark decay branching
fraction of $E^{\pm}\to\mu^{\pm}Z$ for $N_{1,2,3}$ in both NH and IH mass
patterns in Table 1. As a result, in Fig. 6, we show the integrated
luminosities for $5\sigma$ significance in the muon collider with $\sqrt{s}=3$
TeV (left) and $\sqrt{s}=10$ TeV (right). With the optimistic integrated
luminosity of 1 ab-1 for $\sqrt{s}=3$ TeV or 10 ab-1 for $\sqrt{s}=10$ TeV,
both $N_{2}$ and $N_{3}$ can be discovered. One can distinguish $E_{2}$ and
$E_{3}$ for IH and they are not distinguishable for NH. To discover $N_{1}$,
we need much higher luminosity. A 5$\sigma$ significance for $\sqrt{s}=3$ TeV
will be reached with the integrated luminosity above 1 ab-1 for NH and IH. The
reachable branching ratio of $E^{\pm}\to Z\mu^{\pm}$ corresponding to
3$\sigma$ and 5$\sigma$ significance is given in Fig. 7, with $\sqrt{s}=3$
TeV, $\mathcal{L}=1~{}{\rm ab}^{-1}$ (left) and $\sqrt{s}=10$ TeV,
$\mathcal{L}=10~{}{\rm ab}^{-1}$ (right). One can see that BR$(E^{\pm}\to
Z\mu^{\pm})$ below 10.7% (6.4%) can be reached for 5$\sigma$ (3$\sigma$)
significance for $\sqrt{s}=3$ TeV and $\mathcal{L}=1~{}{\rm ab}^{-1}$. For
$\sqrt{s}=10$ TeV and $\mathcal{L}=10~{}{\rm ab}^{-1}$, the branching fraction
can be as large as 13.0% (7.8%) for 5$\sigma$ (3$\sigma$) significance.

Figure 6: The integrated luminosities for 5$\sigma$ significance versus
$M_{E}$ for $E^{+}E^{-}\to ZZ\mu^{+}\mu^{-}$ in muon colliders with
$\sqrt{s}=3$ TeV (left) and 10 TeV (right), for $E_{1,2,3}$ and NH (solid
lines) or IH (dashed lines).

Figure 7: The reachable branching ratio of $E^{\pm}\to Z\mu^{\pm}$
corresponding to 3$\sigma$ and 5$\sigma$ significance versus $M_{E}$ for
$E^{+}E^{-}\to ZZ\mu^{+}\mu^{-}$ in muon colliders with $\sqrt{s}=3$ TeV,
$\mathcal{L}=1~{}{\rm ab}^{-1}$ (left) and $\sqrt{s}=10$ TeV,
$\mathcal{L}=10~{}{\rm ab}^{-1}$ (right).

###  III.2 $E^{\pm}N$ Associated Production

Figure 8: Cross sections of $E^{\pm}N$ associated production as a function of
$\sqrt{s}$ (left) and $M_{E,N}$ (right) at muon colliders by VBF process.

The $E^{\pm}N$ associated production is only from VBF processes initiated by
$\gamma/Z$ and $W^{\pm}$. The cross sections are shown in Fig. 8, as a
function of $\sqrt{s}$ (left) and $M_{E(N)}$ (right), and are enhanced by EW
PDF at higher beam energies. We find that muon collider with energy $3$ TeV or
10 TeV can not generate enough signal events, and we thus take $\sqrt{s}=30$
TeV in the following studies. For $\sqrt{s}=30$ TeV, under the assumption that
the masses of charged and neutral triplet leptons are degenerate, the cross
sections are less than $10^{-3}$ fb for triplet lepton masses greater than 1
TeV. The associated production of charged and neutral heavy leptons followed
by decay modes $E^{\pm}\to\ell^{\pm}Z$ and $N\to\ell^{\pm}W^{\mp}$ can lead to
the rare signature of lepton number violation with same-sign charged leptons

$\displaystyle E^{\pm}N\to ZW^{\mp}\mu^{\pm}\mu^{\pm}\;.$ (43)

We will only consider the sensitivity of muon collider with $\sqrt{s}=30$ TeV
to the distinctive LNV signature.

The leading SM background to this channel becomes

$\displaystyle ZW^{\mp}W^{\pm}W^{\pm}\;,$ (44)

followed by decay $W^{\pm}\to\mu^{\pm}\nu_{\mu}(\bar{\nu}_{\mu})$. We simulate
VBF production of the above background using MadGraph at the parton level.
Firstly, we employ the basic cuts for the gauge bosons in final states

$\displaystyle p_{T}(W,Z)>300~{}{\rm
GeV}\,,~{}~{}\left|\eta(W,Z)\right|<3.0\;.$ (45)

We also employ further judicious cuts to reduce the background

  * •

For a few TeV $E^{\pm}$ and $N$, we set the advanced $p_{T}$ cuts on the gauge
bosons as follows

$\displaystyle p_{T}(W)>\frac{M_{N}}{3}\,,\quad p_{T}(Z)>\frac{M_{E}}{3}\;.$
(46)

  * •

Note that the $\nu_{\mu}$ or $\bar{\nu}_{\mu}$ from $W$ boson decay only
appear in the SM background. For our signal, the $E^{\pm}N$ or $NN$ production
is induced by the VBF process in which there is no missing neutrino.
Nevertheless, in the process of event analysis, we consider the smearing of
the transverse momentum with a resolution of the detector mentioned below Eq.
(32). As a result, there appears a small transverse missing energy in the
signal. As shown in left panel of Fig. 9, we veto the events with large
missing energy from neutrinos

$\displaystyle\cancel{E}_{T}<500~{}{\rm GeV}\;.$ (47)

  * •

To suppress the continuum background, we reconstruct the heavy lepton
$E^{\pm}$ and $N$. Firstly, we pair the final particles $W\mu$ and $Z\mu$
according to the feature that the real combination comes from heavy triplets
with degenerate masses. The reconstructed heavy lepton masses are shown in the
right panel of Fig. 9. Then we take the mass windows of the resonances

$\displaystyle\left|M_{Z\mu_{1}}-M_{E}\right|<\frac{M_{E}}{5}\,,\quad\left|M_{W\mu_{2}}-M_{N}\right|<\frac{M_{N}}{5}\;.$
(48)

Figure 9:  The normalized $\cancel{E}_{T}$ (left) and invariant mass
$M_{Z\mu}$ (right) distributions of the heavy leptons in the signal
$E^{\pm}N\to ZW^{\mp}\mu^{\pm}\mu^{\pm}$ and the background for $\sqrt{s}=30$
TeV and $M_{E(N)}=3$ TeV.

One can efficiently suppress the SM background and the cut efficiency becomes
$\epsilon_{B}\sim 0.1\%$ by using the above cuts, so it is enough to only
consider the signal events. The event number of signal is given by

$\displaystyle N_{S}=$
$\displaystyle\sigma_{S}^{\rm{VBF}}\times\epsilon_{S}^{\rm{VBF}}\times\epsilon^{\rm
VBF}_{S,W/Z}\times{\rm BR}(E^{\pm}\to Z\mu^{\pm})\times{\rm BR}(N\to
W^{\mp}\mu^{\pm})$ (49) $\displaystyle\times\mathcal{L}\times{\rm
BR}(W^{\pm}\to q\bar{q}^{\prime})\times{\rm BR}(Z\to q\bar{q})\;,$

where $\epsilon^{\rm VBF}_{S,W/Z}$ denotes the efficiency of reconstructing
$W$ and $Z$ bosons, ${\rm BR}(E^{\pm}\to Z\mu^{\pm})\approx{\rm BR}(N\to
W^{\mp}\mu^{\pm})$ as shown in Table 1. The number of signal events at muon
collider with $\sqrt{s}=30$ TeV is shown in left panel of Fig. 10, with the
integrated luminosity of 90 ${\rm ab}^{-1}$. One can see that the three heavy
triplet leptons and two neutrino mass patterns exhibit different event
numbers. For $M_{E(N)}$ greater than 6 TeV, it is impossible to receive more
than one signal event. The reachable branching ratio of $E^{\pm}\to
Z\mu^{\pm}$ and $N\to W^{\mp}\mu^{\pm}$ corresponding to $N_{S}=5$ and
$N_{S}=10$ is given in the right panel of Fig. 10. ${\rm BR}(E^{\pm}\to
Z\mu^{\pm})$ below 40.3$\%$ (28.5$\%$) can be reached for $N_{S}=10$
($N_{S}=5$) and $M_{E(N)}<5$ TeV at muon collider with $\sqrt{s}=30$ TeV and
$\mathcal{L}=90$ ${\rm ab}^{-1}$.

Figure 10:  Left: The number of signal events versus $M_{E(N)}$ for
$E^{\pm}N\to W^{\mp}Z\mu^{\pm}\mu^{\pm}$ at muon collider with $\sqrt{s}=30$
TeV and $\mathcal{L}=90~{}{\rm ab}^{-1}$, for $E(N)_{1,2,3}$ and NH (solid
lines) or IH (dashed lines). Right: The reachable branching ratio of
$E^{\pm}\to Z\mu^{\pm}$ or $N\to W^{\mp}\mu^{\pm}$ corresponding to $N_{S}=5$
and $N_{S}=10$ versus $M_{E(N)}$ for $E^{\pm}N\to W^{\mp}Z\mu^{\pm}\mu^{\pm}$
at muon collider with $\sqrt{s}=30$ TeV and $\mathcal{L}=90~{}{\rm ab}^{-1}$.

###  III.3 $NN$ Pair Production

Figure 11: Cross sections of $NN$ pair production as a function of $\sqrt{s}$
(left) and $M_{N}$ (right) at muon colliders by VBF process.

In this subsection, we discuss the $NN$ pair production from VBF process
initiated by $W^{+}W^{-}$. The VBF cross sections as a function of $\sqrt{s}$
and $M_{N}$ are shown in Fig. 11. The cross sections are slightly smaller than
those of $E^{\pm}N$ associated production as there is only $W^{+}W^{-}$
initiated VBF process. We will only consider the distinctive LNV signature at
muon collider with $\sqrt{s}=30$ TeV, i.e.,

$\displaystyle NN\to W^{\mp}W^{\mp}\mu^{\pm}\mu^{\pm}\;.$ (50)

The leading SM background becomes

$\displaystyle W^{+}W^{+}W^{-}W^{-}\;,$ (51)

followed by decay $W^{\pm}\to\mu^{\pm}\nu_{\mu}(\bar{\nu}_{\mu})$ for one pair
of same-sign $W$ bosons. The judicious cuts to reduce the background are

  * •

For a few TeV $N$, we set the advanced $p_{T}$ cuts for the leading $W$ boson
and $\mu^{\pm}$ as follows

$\displaystyle p_{T}^{\rm max}(W)>\frac{M_{N}}{3}\,,\quad p_{T}^{\rm
max}(\mu)>\frac{M_{N}}{3}\;.$ (52)

  * •

Since the $\nu_{\mu}$ or $\bar{\nu}_{\mu}$ only appears in the SM background,
we can set the cuts on missing energy from neutrinos as shown in left panel of
Fig. 12

$\displaystyle\cancel{E}_{T}<500~{}{\rm GeV}\;.$ (53)

  * •

In order to reconstruct the heavy neutrino $N$, we pair the particles $W\mu$
according to the feature that the real combination comes from heavy triplets
with degenerate masses. The invariant mass distribution is shown in the right
panel of Fig. 12. Then we take the mass windows of the resonances

$\displaystyle\left|M_{W\mu}-M_{N}\right|<\frac{M_{N}}{5}\;.$ (54)

Figure 12:  The normalized $\cancel{E}_{T}$ (left) and invariant mass
$M_{W\mu}$ (right) distributions of the heavy leptons in the signal $NN\to
W^{\mp}W^{\mp}\mu^{\pm}\mu^{\pm}$ and the background for $\sqrt{s}=30$ TeV and
$M_{N}=3$ TeV.

Since the SM background is extremely suppressed, we will only consider the
signal events. The event number of signal is given by

$\displaystyle
N_{S}=\sigma_{S}^{\rm{VBF}}\times\epsilon_{S}^{\rm{VBF}}\times\epsilon^{\rm
VBF}_{S,W}\times{\rm BR}^{2}(N\to W^{\mp}\mu^{\pm})\times\mathcal{L}\times{\rm
BR}(W^{\pm}\to q\bar{q}^{\prime})^{2}\times 2\ \;,$ (55)

where the factor 2 accounts for the charge conjugation of decay products, as
shown in the left panel of Fig. 13. For $M_{N}\gtrsim 6$ TeV, it is almost
impossible to obtain more than one signal event. The reachable branching ratio
of $N\to W^{\mp}\mu^{\pm}$ is given in the right panel of Fig. 13. ${\rm
BR}(N\to W^{\mp}\mu^{\pm})$ below $22.2\%~{}(15.8\%)$ can be reached for
$N_{S}=10~{}(N_{S}=5)$ and $M_{N}<5$ TeV at muon collider with $\sqrt{s}=30$
TeV and $\mathcal{L}=90~{}{\rm ab}^{-1}$.

Figure 13:  Left: The number of signal events versus $M_{N}$ for $NN\to
W^{\mp}W^{\mp}\mu^{\pm}\mu^{\pm}$ at muon collider with $\sqrt{s}=30$ TeV and
$\mathcal{L}=90~{}{\rm ab}^{-1}$, for $N_{1,2,3}$ and NH (solid lines) or IH
(dashed lines). Right: The reachable branching ratio of $N\to
W^{\mp}\mu^{\pm}$ corresponding to $N_{S}=5$ and $N_{S}=10$ versus $M_{N}$ for
$NN\to W^{\mp}W^{\mp}\mu^{\pm}\mu^{\pm}$ at muon collider with $\sqrt{s}=30$
TeV and $\mathcal{L}=90~{}{\rm ab}^{-1}$.

Finally, we comment on the decay length and possible displaced vertex of heavy
triplet leptons. The total decay widths of heavy triplet leptons are
proportional to $|V_{\ell N}|^{2}\sim m_{\nu}/M_{\Sigma}$ which may lead to
long decay length. We calculate the decay length of the triplet leptons by the
formula $L=\gamma\beta c\tau$, where $\tau=1/\Gamma$ is the life-time,
$\gamma$ is the boost factor and $\beta$ is the ratio of velocity $v_{\Sigma}$
to the speed of light $c$. The product $\gamma\beta$ can then be given by
$\gamma\beta=\sqrt{E_{\Sigma}^{2}/M_{\Sigma}^{2}-1}$. To evaluate the
displaced vertex, we take into account the light neutrino masses with the
minimal one being $10^{-4}$ eV. In our analysis, $E_{\Sigma}$ is roughly taken
to be $\sqrt{s}/2$ in the c.m. frame. Fig. 14 shows the decay length versus
$M_{N}$ for $N_{i}~{}(i=1,2,3)$ and $\Omega=I$ at different collider energies.
We find that in most cases the decay length is less than $\mathcal{O}(1)$ mm
and the heavy leptons can be viewed as decaying promptly. Only for the cases
of $N_{1}$ in NH and $N_{3}$ in IH, the typical decay length can be longer
than 1 mm in relatively low mass region. In particular, for $M_{N}\sim 1$ TeV
and $\sqrt{s}=30$ TeV, the decay length could be as large as 20 mm, which is
sufficiently long to perform the displaced vertex searches Jana  _et al._
(2020); Sen  _et al._ (2022). If this clear difference is observed in future,
it could serve as an indication to distinguish the heavy leptons and neutrino
patterns.

Figure 14: Decay length versus $M_{N}$ for different heavy neutrinos
$N_{i}~{}(i=1,2,3)$ and different neutrino mass hierarchies NH (left) and IH
(right) at muon collider with energy $3$ TeV (dotted line), $10$ TeV (dashed
line) and $30$ TeV (solid line) respectively.

##  IV Conclusion

We perform a study of search for heavy triplet leptons in Type III Seesaw
mechanism at future high-energy and high-luminosity muon colliders. The impact
of up-to-date neutrino oscillation results is taken into account on neutrino
mass model and the consequent lepton flavor signatures of heavy leptons in
Type III Seesaw. According to the asymptotic behavior, the heavy lepton
branching fractions exhibit ${\rm
BR}(E^{\pm}\to\sum_{\nu}\overset{(-)}{\nu}W^{\pm})\approx 50\%$ and ${\rm
BR}(N\to\ell^{\pm}W^{\mp})\approx{\rm BR}(E^{\pm}\to\ell^{\pm}Z)$ for
$M_{E(N)}\gg M_{W}$. The three heavy leptons in the Casas-Ibarra
parametrization with diagonal unity matrix can be distinguished in terms of
their decay modes.

We consider the pair production of charged triplet leptons $E^{+}E^{-}$
through both $\mu^{+}\mu^{-}$ annihilation and VBF processes. The charged
triplet lepton as heavy as one-half of colliding energy can be probed. A
$5\sigma$ significance can be reached through the $E^{+}E^{-}\to
W^{+}W^{-}\nu\bar{\nu}$ process with the integrated luminosity below 0.1 ab-1
for $\sqrt{s}=3$ TeV and 2 ab-1 for $\sqrt{s}=10$ TeV. The $E^{+}E^{-}\to
ZZ\ell^{+}\ell^{-}$ channel can be further utilized to fully reconstruct the
three triplet leptons and distinguish neutrino mass patterns. BR$(E^{\pm}\to
Z\mu^{\pm})$ below 10.7% (6.4%) can be reached for 5$\sigma$ (3$\sigma$)
significance for $\sqrt{s}=3$ TeV and $\mathcal{L}=1~{}{\rm ab}^{-1}$.

The associated production $E^{\pm}N$ and the pair production of heavy
neutrinos $NN$ are only induced by VBF processes and lead to rare LNV
signature. We study the search potential of LNV processes at future high-
energy muon collider with $\sqrt{s}=30$ TeV. Through $E^{\pm}N$ ($NN$)
production, ${\rm BR}(N\to\mu^{\pm}W^{\mp})$ below 40.3% (22.2%) can be
reached for 10 signal events and $M_{E(N)}<5$ TeV at muon collider with
$\sqrt{s}=30$ TeV and $\mathcal{L}=90$ ${\rm ab}^{-1}$.

###### Acknowledgements.

We would like to thank Tao Han for very useful discussions. T.L. is supported
by the National Natural Science Foundation of China (Grant No. 11975129,
12035008) and “the Fundamental Research Funds for the Central Universities”,
Nankai University (Grants No. 63196013). C.Y.Y. is supported in part by the
Grants No. NSFC-11975130, No. NSFC-12035008, No. NSFC-12047533, the Helmholtz-
OCPC International Postdoctoral Exchange Fellowship Program, the National Key
Research and Development Program of China under Grant No. 2017YFA0402200 and
the China Postdoctoral Science Foundation under Grant No. 2018M641621.

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### Author biography

[./images/liliwang.jpg]Lili Wang is a professor at the School of Computer
Science and Engineering, Beihang University, and a researcher of the State Key
Laboratory of Virtual Reality Technology and Systems. Her research interests
include virtual reality, augmented reality, and rendering.

[./images/XuehuaiShi.jpg]Xuehuai Shi is a Ph.D student at the School of
Computer Science and Engineering, Beihang University, China. His current
research focuses on virtual reality and foveated rendering.

[./images/yiliu.jpg]Yi Liu is a master student at the School of Computer
Science and Engineering, Beihang University, China. His current research
focuses on virutal reailty, augumented reality and rendering.


HYP2022 - 14${}^{\textrm{th}}$ International Conference on Hypernuclear and
Strange Particle Physics

#  $\Xi$-nuclear constraints from $\Xi^{-}$ emulsion capture events

Eliahu Friedman<EMAIL_ADDRESS>Institute of Physics,
The Hebrew University, Jerusalem 9190401, Israel1 Avraham Gal
<EMAIL_ADDRESS>
###### Abstract

All five KEK and J-PARC two-body $\Xi^{-}$+AZ $\to$
${}^{A^{\prime}}_{\leavevmode\nobreak\ \Lambda}$Z’+
${}^{A^{\prime\prime}}_{\leavevmode\nobreak\ \Lambda}$Z” capture events in
light emulsion nuclei, including KISO and IBUKI in 14N, are consistent with
Coulomb-assisted $1p_{\Xi^{-}}$ nuclear states. The underlying $\Xi$-nuclear
potential is strongly attractive, with nuclear-matter depth $V_{\Xi}$ larger
than 20 MeV. The recent 14N capture events KINKA and IRRAWADDY assigned by
J-PARC E07 to $1s_{\Xi^{-}}$ nuclear states, and implying considerably
shallower $V_{\Xi}$, have also another interpretation as $1p_{\Xi^{0}}$
nuclear states.

##  1 Introduction: $\Xi^{-}$ capture events

The nuclear interactions of $\Xi$ hyperons are poorly known GHM16 ; HN18 .
Because of the large momentum transfer in the standard $(K^{-},K^{+})$
production reaction, induced by the two-body $K^{-}p\to K^{+}\Xi^{-}$
strangeness exchange reaction, the $\Xi^{-}$ hyperons are produced dominantly
in the quasi-free continuum region, with less than 1% expected to form
$\Xi^{-}$-nuclear bound states that decay by a $\Xi^{-}p\to\Lambda\Lambda$
strong-interaction capture reaction. In fact, no $\Xi^{-}$ or $\Lambda\Lambda$
hypernuclear bound states have ever been observed unambiguously in such
experiments, e.g., KEK-PS E224 E224  and BNL-AGS E885 E885 , both on 12C, and
BNL-AGS E906 E906  on 9Be. A potential depth of $V_{\Xi}=17\pm 6$ MeV was
deduced recently from the quasi-free $\Xi^{-}$ spectrum taken in the
9Be($K^{-},K^{+}$) reaction HH21 .

Here we focus on $\Xi$ nuclear constraints derived by observing $\Xi^{-}$
capture events in exposures of light-emulsion CNO nuclei to the
$(K^{-},K^{+})$ reaction. A small fraction of the produced high-energy
$\Xi^{-}$ hyperons slows down in the emulsion, undergoing an Auger process to
form high-$n$ atomic states, and cascades down radiatively. Strong-interaction
capture takes over atomic radiative cascade in a 3D atomic orbit bound by 126,
175, 231 keV in C, N, O, respectively, affected to less than 1 keV by the
strong interaction BFG99 . Interestingly, all two-body $\Xi^{-}$ capture
events $\Xi^{-}+{{}^{A}Z}\to$ ${}^{A^{\prime}}_{\leavevmode\nobreak\
\Lambda}$Z’+${}^{A^{\prime\prime}}_{\leavevmode\nobreak\ \Lambda}$Z” to twin
single-$\Lambda$ hypernuclei reported in KEK and J-PARC light-emulsion $K^{-}$
exposure experiments E176 ; E373a ; E373b ; E07a ; E07b  are consistent with
$\Xi^{-}$ capture from a lower orbit: a Coulomb-assisted $1p_{\Xi^{-}}$
nuclear state bound by $\sim$1 MeV. These capture events are listed in Table
1. Expecting the final two $\Lambda$ hyperons in $\Xi^{-}p\to\Lambda\Lambda$
capture to form in a spin $S=0,\,1s_{\Lambda}^{2}$ configuration, the initial
$\Xi^{-}$ hyperon and the proton on which it is captured must satisfy
$l_{\Xi^{-}}=l_{p}$ Zhu91 , which for $p$-shell nuclear targets favors the
choice $l_{\Xi^{-}}=1$. Not listed in the table are multi-body capture events
that require for their interpretation undetected capture products, usually
neutrons, on top of a pair of single-$\Lambda$ hypernuclei. Several of these
new J-PARC E07 events E07b , like KINKA and IRRAWADDY, imply $\Xi^{-}$ capture
from $1s_{\Xi^{-}}$ nuclear states, with estimated capture rates of order 1%
of capture rates from the $1p_{\Xi^{-}}$ nuclear states considered here Zhu91
; Koike17 . We discuss these states below.

Table 1: Two-body $\Xi^{-}$ capture emulsion events from KEK and J-PARC experiments. Experiment | Event | ${}^{A}Z$ |  ${}^{A^{\prime}}_{\Lambda}$Z’+${}^{A^{\prime\prime}}_{\Lambda}$Z” |  $B_{\Xi^{-}}$ (MeV)
---|---|---|---|---
KEK E176 E176  | 10-09-06 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be | 0.82$\pm$0.17
KEK E176 E176  | 13-11-14 |  12C |  ${}^{4}_{\Lambda}$H+${}^{9}_{\Lambda}$Be∗ | 0.82$\pm$0.14
KEK E176 E176  | 14-03-35 |  14N |  ${}^{3}_{\Lambda}$H+${}^{12}_{\leavevmode\nobreak\ \Lambda}$B | 1.18$\pm$0.22
KEK E373 E373b  | KISO |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{\leavevmode\nobreak\ \Lambda}$Be∗ | 1.03$\pm$0.18
J-PARC E07 E07a  | IBUKI |  14N |  ${}^{5}_{\Lambda}$He+${}^{10}_{\leavevmode\nobreak\ \Lambda}$Be | 1.27$\pm$0.21

##  2 $\Xi$ nuclear optical potential

$\Xi^{-}$ atomic and nuclear bound states are calculated using a standard
$t\rho$ optical-potential form FG21

$V_{\rm
opt}(r)=-\frac{2\pi}{\mu}\,(1+\frac{A-1}{A}\frac{\mu}{m_{N}})\,[b_{0}\,\rho(r)+b_{1}\,\rho_{\rm
exc}(r)],$ (1)

where $\mu$ is the $\Xi^{-}$-nucleus reduced mass and the complex strength
parameters $b_{0}$ and $b_{1}$ are effective, generally density dependent $\Xi
N$ isoscalar and isovector c.m. scattering amplitudes respectively. The
density $\rho=\rho_{n}+\rho_{p}$ is a nuclear density distribution normalized
to the number of nucleons $A$ and $\rho_{\rm exc}=\rho_{n}-\rho_{p}$ is a
neutron-excess density with $\rho_{n}=(N/Z)\rho_{p}$, implying that $\rho_{\rm
exc}=0$ for the $N=Z$ emulsion nuclei 12C and 14N considered here. A finite-
size Coulomb potential $V_{c}$, including vacuum-polarization terms is added.
For densities we used mostly harmonic-oscillator (HO) densities Elton61  where
the r.m.s. radius of $\rho_{p}$ was set equal to that of the nuclear charge
density AM13 . Folding reasonably chosen $\Xi N$ interaction ranges other than
corresponding to the proton charge radius, or using Modified Harmonic
Oscillator densities, or replacing HO densities by realistic three-parameter
Fermi density distributions, made little difference: all the calculated
binding energies changed by a small fraction, about 0.03 MeV, of the
uncertainty imposed by the $\pm$0.15 MeV experimental uncertainty of the 0.82
MeV $1p_{\Xi^{-}}$ binding energy in 12C listed in Table 1. This holds also
for adding a $\rho_{\rm exc}\neq 0$ term induced by considering realistic
differences of neutron and proton r.m.s. radii.

Accepting the binding energy interval $B_{\Xi^{-}}^{1p}$=0.82$\pm$0.15 MeV for
the two KEK E176 events listed in Table 1, a $\Xi$-nuclear potential strength
of Re$\,b_{0}=0.32\pm 0.01$ fm follows for a fixed value Im$\,b_{0}=0.01$ fm.
The sensitivity to variations of Im$\,b_{0}$ is minimal: choosing
Im$\,b_{0}=0.04$ fm BFG99  instead of 0.01 fm increases Re$\,b_{0}$ by 0.01 fm
to $0.33\pm 0.01$ fm. The value Re$\,b_{0}=0.32\pm 0.01$ fm implies in the
limit $A\to\infty$ and $\rho(r)\to\rho_{0}$=0.17 fm-3 a depth value
$V_{\Xi}=24.3\pm 0.8$ MeV in nuclear matter, compatible with that derived from
AGS-E906 in Ref. HH21  and in agreement with the range of values 21–24 MeV
extracted from old emulsion events DG83 .

So far we have discussed a density independent $t$-matrix element $b_{0}$ in
$V_{\rm opt}$, Eq. (1), to fit the $\Xi^{-}$ capture events in 12C from Table
1. To explore how robust the deduced $\Xi$ potential-depth value
$V_{\Xi}=24.3\pm 0.8$ MeV is, we introduce the next to leading-order density
dependence of $V_{\rm opt}$, replacing Re$\,b_{0}$ in Eq. (1) by

${\rm Re}\,b_{0}(\rho)=\frac{{\rm Re}\,b_{0}}{1+\frac{3k_{F}}{2\pi}{\rm
Re}\,b_{0}^{\rm
lab}},\,\,\,\,\,\,\,\,\,\,k_{F}=(3{\pi}^{2}\rho/2)^{\frac{1}{3}},$ (2)

where $k_{F}$ is the Fermi momentum corresponding to nuclear density $\rho$
and $b_{0}^{\rm lab}=(1+\frac{m_{\Xi^{-}}}{m_{N}})b_{0}$ is the lab
transformed form of the c.m. scattering amplitude $b_{0}$. Eq. (2) accounts
for Pauli exclusion correlations in $\Xi N$ in-medium multiple scatterings
DHL71 ; WRW97 . Shorter-range correlations, disregarded here, were shown in
Ref. WH08  to contribute less than $\sim$30% of the long-range Pauli
correlation term. Applying Eq. (2) in the present context,
$B_{\Xi^{-}}^{1p}(^{12}$C)=0.82 MeV is refitted by Re$\,b_{0}$=0.527 fm. The
nuclear-matter $\Xi$-nuclear potential depth $V_{\Xi}$ decreases from
24.3$\pm$0.8 to 21.9$\pm$0.7 MeV, a decrease of merely 10%.

##  3 $1p_{\Xi^{-}}$ states in 14N

Applying Eqs. (1,2) to 14N, with Re$\,b_{0}$ fitted to
$B_{\Xi^{-}}^{1p}(^{12}$C)=0.82$\pm$0.15 MeV, results in
$B_{\Xi^{-}}^{1p}(^{14}$N)=1.96$\pm$0.26 MeV, Pauli correlations included.
This binding energy is considerably higher than the value
$B_{\Xi^{-}}$=1.15$\pm$0.20 MeV obtained from the three events assigned in
Table 1 to $\Xi^{-}$ capture in 14N. To resolve this apparent dicrepancy, we
note that the calculated $B_{\Xi^{-}}^{1p}(^{14}$N) corresponds to a
$(2\Lambda+1)$-average of binding energies for a triplet of states
$\Lambda^{\pi}=(0^{-},1^{-},2^{-})$ obtained by coupling a $1p_{\Xi^{-}}$
state to $J^{\pi}(^{14}$N${}_{\rm g.s.})$=1+, as shown in Fig. 1.

Figure 1: Energies (in MeV) of $\Lambda^{\pi}=(0^{-},1^{-},2^{-})$ triplet of
14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ states, split by a $Q_{N}\cdot Q_{\Xi}$
residual interaction (3). The $(2\Lambda+1)$-averaged energy $-1.96$ MeV was
calculated using the same Pauli-corrected optical potential parameter $b_{0}$
that yields a 12C${}_{\rm g.s.}+1p_{\Xi^{-}}$ state at $-$0.82 MeV,
corresponding to the $\Xi^{-}$ capture events in 12C listed in Table 1. Figure
updating Fig. 2 in Ref. FG21 .

The energy splittings marked in Fig. 1 follow from a shell-model quadrupole-
quadrupole spin independent residual interaction ${\cal V}_{\Xi N}$,

${\cal V}_{\Xi N}=F^{(2)}_{\Xi N}Q_{N}\cdot
Q_{\Xi},\,\,\,\,\,\,Q_{B}=\sqrt{\frac{4\pi}{5}}Y_{2}({\hat{r}}_{B}),$ (3)

where $F^{(2)}$ is the corresponding Slater integral. A representative value
of $F^{(2)}_{\Xi N}=-3$ MeV is used here, smaller than the value
$F^{(2)}_{\Lambda N}=-3.7$ MeV established empirically for $p$-shell $\Lambda$
hypernuclei DalGal81 , in accordance with a $\Xi N$ strong interaction
somewhat weaker than the $\Lambda N$ strong interaction. A single
${}^{3}D_{1}$ 14Ng.s. component providing a good approximation to the full
intermediate-coupling g.s. wavefunction Mill07  was assumed in the present
evaluation.

Fig. 1 shows a triplet of 14N${}_{\rm g.s.}+1p_{\Xi^{-}}$ levels, spread over
more than 1 MeV. The least bound of these states, $\Lambda^{\pi}=0^{-}$, is
shifted upward by 0.84 MeV from the $(2\Lambda+1)$ averaged position at
$-1.96\pm 0.26$ MeV to $E(0^{-})=-1.12\pm 0.26$ MeV. This is consistent with
the averaged position $\bar{E}=-1.15\pm 0.20$ MeV of the three
$\Xi^{-}\,{}^{14}$Ng.s. capture events listed in Table 1. The
$\Lambda^{\pi}=0^{-}$ state assumes spin-parity $J^{\pi}=\frac{1}{2}^{-}$ when
Pauli-spin $s_{\Xi^{-}}=\frac{1}{2}$ is introduced, but its position is
unaffected by spin dependent $\Xi N$ residual interactions in leading order.
We are not aware of any good reason why capture has not been seen from the
other two states with $\Lambda^{\pi}=1^{-},2^{-}$. This may change when more
events are collected at the next stage of the ongoing J-PARC E07 emulsion
experiment.

##  4 $1s_{\Xi^{-}}$ states in 14N?

In addition to the $\Xi^{-}_{1p}$–14N capture events listed as KISO and IBUKI
in Table 1, the J-PARC light-nuclei emulsion experiment E07 reported also two
other events KINKA and IRRAWADDY, assigned as $\Xi^{-}_{1s}$–14N states, see
Fig. 2. We note that $2P\to 1S$ radiative decay rates are of order 1% of
$3D\to 2P$ radiative decay rates Zhu91 ; Koike17  suggesting that $\Xi^{-}$
capture from a nuclear $\Xi^{-}_{1s}$–14N state is suppressed to this order
relative to capture from a nuclear $\Xi^{-}_{1p}$–14N state. Assigning a
$\Xi^{-}_{1s}$–14N bound state to IRRAWADDY, and by default also KINKA
which–given its large uncertainty–is not inconsistent with IRRAWADDY, is
therefore questionable.

Figure 2: $\Xi^{-}_{1s}$ and $\Xi^{-}_{1p}$ nuclear states in 14N, assigned in
KEK-E373 and J-PARC E07 emulsion experiments by interpreting $\Xi^{-}$ capture
events that lead to observed twin-$\Lambda$ hypernuclear decays. Figure
provided by Dr. K. Nakazawa, based on recent results from Refs. E373b ; E07a ;
E07b .

It has been conjectured by us recently FG23  that IRRAWADDY is a near-
threshold $\Xi^{0}_{1p}$–14C bound state that has nothing to do with a
$\Xi^{-}_{1s}$–14N bound state suggested by E07. The mixing induced by the
$\Xi N$ strong interaction between a $\Xi^{-}_{1p}$–14N bound state identified
with IBUKI and a $\Xi^{0}_{1p}$–14C bound state lying about 5 MeV below IBUKU,
within the J-PARC E07 experimental uncertainty of IRRAWADDY, is sufficiently
strong to make the $E1$ radiative deexcitation of the $\Xi^{-}_{3D}$–14N
atomic state populate equally well both $\Xi^{-}_{1p}$–14N and
$\Xi^{0}_{1p}$–14C nuclear states.

##  5 Discussion

Table 2: $\Xi^{-}$–12C and $\Xi^{-}$–14N binding energies in $1s$ and $1p$ states, $B_{\Xi^{-}}^{1s}$ and $B_{\Xi^{-}}^{1p}$, plus $\Xi$ nuclear potential depths $V_{\Xi}(\rho_{0})$ at nuclear-matter density $\rho_{0}=0.17\,$fm-3 calculated using a density-dependent optical potential Eqs. (1,2) with Re$\,b_{0}$ fitted to binding energies underlined for each input choice. All entries are in MeV. Input |  $B_{\Xi^{-}}^{1s}(^{12}$C) |  $B_{\Xi^{-}}^{1p}(^{12}$C) |  $B_{\Xi^{-}}^{1s}(^{14}$N) |  $B_{\Xi^{-}}^{1p}(^{14}$N) | $V_{\Xi}(\rho_{0})$
---|---|---|---|---|---
KEK E176 E176  | 9.82 | $\underline{0.82}$ | 11.78 | 1.96 | 21.9
J-PARC E07 E07b  | 4.94 | 0.31 | $\underline{6.27}$ | 0.50 | 13.8

Two $\Xi$-nuclear scenarios are listed in Table 2. In the first one, two
KEK-E176 12C events E176 , with $B_{\Xi^{-}}^{1p}=0.82\pm 0.15$ MeV, serve as
input for setting up the strength of the $\Xi$-nuclear optical potential.
Other $\Xi^{-}$ binding energies are then predicted, as listed in the first
row of Table 2. We note that a value $V_{\Xi}\gtrsim 20$ MeV implies a
substantially stronger in-medium $\Xi N$ attraction than reported by some
recent free-space model evaluations (HAL-QCD HALQCD19 , EFT@NLO HM19 ; K19
and RMF Gaitanos21 ), all of which satisfy $V_{\Xi}\lesssim 10$ MeV. A notable
exception is provided by versions ESC16*(A,B) of the latest Nijmegen extended-
soft-core $\Xi N$ interaction model ESC16 , in which values of $V_{\Xi}$
higher than 20 MeV are derived. However, these large values are reduced
substantially by $\Xi NN$ three-body contributions within the same ESC16*
model.

Choosing instead the J-PARC E07 14N IRRAWADDY event, with
$B_{\Xi^{-}}^{1s}=6.27\pm 0.27$ MeV E07b  as input, gives rise to different
predictions as listed in the second row of the table. The difference between
the two sets of predictions is striking, particularly for the $\Xi^{-}_{1s}$
binding energies. This large difference is reflected also in the $\Xi$-nuclear
potential depths at nuclear matter density, $V_{\Xi}(\rho_{0})$, listed in the
last column of the table. Equally interesting is the difference between the
two sets with regard to $\Xi^{-}_{1p}$ bound states. In particular, the
$\Xi^{-}_{1p}$–12C binding energy constrained by
$B_{\Xi^{-}}^{1s}(^{14}$N)=6.27$\pm$0.27 MeV comes out in the second row
exceedingly small, substantially disagreeing with that determined from the two
KEK E176 capture events E176  underlined in the first row. As for the
calculated $\Xi^{-}_{1p}$–14N binding energies, since $J({{}^{14}{\rm N}})\neq
0$, see Fig. 1, these cannot be compared directly with IBUKI’s binding energy
of 1.27$\pm$0.21 MeV from Fig. 2.

Recent Skyrme-Hartree-Fock (SHF) calculations GZS21  presented global fits to
comprehensive $\Xi$-nuclear data, including $B_{\Xi^{-}}^{1s}$=8.00 MeV for
KINKA and $B_{\Xi^{-}}^{1p}$=1.13 MeV for the mean of KISO and IBUKI. Apart
from small nonlocal potential terms and effective mass corrections, the SHF
$\Xi$-nuclear mean-field potential $V_{\Xi}(\rho_{N})$ consists of two terms:
$V_{\Xi}^{(2)}(\rho_{N})\propto\rho_{N}$ &
$V_{\Xi}^{(3)}(\rho_{N})\propto\rho^{2}_{N}$. Large-scale SHF fits of the
corresponding $\Xi$ potential depths, and their sum, are listed in the first
row of Table 3. Listed in the lower rows are $\Xi$ potential depths obtained
in the optical potential methodology FG22  when fitting just the two $\Xi^{-}$
states in 14N as used in the SHF calculations GZS21 . Good agreement is
observed between the two methods, with $V_{\Xi}(\rho_{0})\approx\,$14 MeV,
similar to the depth value listed in the second row of Table 2 using IRRAWADDY
alone. We note that the $\Xi^{-}_{1p}$ state in 12C (see Table 2) comes out
unbound in the SHF calculations unless 12C is made artificially deformed GZS21
.

Table 3: $\Xi$-nuclear potential depths (in MeV) from a large-scale SHF fit GZS21  and from our $V_{\rm opt}$ two-parameter fits to just $B_{\Xi^{-}}^{1s}$=8.00 MeV (KINKA) and $B_{\Xi^{-}}^{1p}$=1.13 MeV (KISO,IBUKI) in 14N. ‘Pauli’ refers to Eq. (2). Method | Pauli | $V_{\Xi}^{(2)}(\rho_{0})$ | $V_{\Xi}^{(3)}(\rho_{0})$ | $V_{\Xi}(\rho_{0})$
---|---|---|---|---
SHF | No | 34.1 |  $-$20.4 | 13.7
$V_{\rm opt}$ | No | 27.5 |  $-$12.6 | 14.9
$V_{\rm opt}$ | Yes | 24.6 |  $-$11.0 | 13.6

The solution proposed here to the difficulty of interpreting IRRAWADDY as a
$\Xi^{-}_{1s}$ bound state in 14N is by pointing out that it could correspond
to a $\Xi^{0}_{1p}$–14C bound state, something that cannot occur kinematically
in the other light-emulsion nuclei 12C and 16O. Given that in this nuclear
mass range capture rates from $1s_{\Xi^{-}}$ states are estimated to be two
orders of magnitude below capture rates from $1p_{\Xi^{-}}$ states Zhu91 ;
Koike17 , this $\Xi^{0}_{1p}$–14C new assignment addresses satisfactorily the
capture rate hierarchy.

## Acknowledgments

One of us (A.G.) thanks Jiří Mareš and other members of the HYP2022 organizing
team for their generous hospitality during the Conference. This work is part
of a project funded by the EU Horizon 2020 Research & Innovation Programme
under grant agreement 824093.

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