XXX 2 U. Munari et al.

11institutetext:  INAF Padova Astronomical Observatory, I-36012 Asiago (VI),
Italy 22institutetext: ANS Collaboration, Astronomical Observatory, 36012
Asiago (VI), Italy

# Photometry and spectroscopy of the new symbiotic star 2SXPS J173508.4-292958

U. Munari 11 P. Valisa 22 A. Vagnozzi 22 S. Dallaporta 22 F. J. Hambsch 22 A.
Frigo 22

(2021; November 30, 2020; December 21, 2020; XXX.X.XXXX)

###### Abstract

We present and discuss the results of our photometric and spectroscopic
monitoring of 2SXPS J173508.4-292958 carried out from April to August 2020.
This X-ray source, in the foreground with respect to the Galactic center,
brightened in X-rays during 2020, prompting our follow-up optical
observations. We found the star to contain a K4III giant with a modest but
highly variable H$\alpha$ emission, composed by a $\sim$470 km s-1 wide
component with superimposed a narrow absorption, offset by a positive velocity
with respect to the giant. No orbital motion is detected for the K4III,
showing an heliocentric radial velocity stable at $-$12$\pm$1 km s-1. No
flickering in excess of 0.005 mag in $B$ band was observed at three separate
visits of 2SXPS J173508.4-292958. While photometrically stable in 2016 through
2018, in 2019 the star developed a limited photometric variability, that in
2020 took the form of a sinusoidal modulation with a period of 38 days and an
amplitude of 0.12 mag in $V$ band. We argue this variability cannot be
ascribed to Roche-lobe filling by the K4III star. No correlation is observed
between the photometric variability and the amount of emission in H$\alpha$,
the latter probably originating directly from the accretion disk around the
accreting companion. While no emission from dust is detected at mid-IR
wavelengths, an excess in $U$-band is probably present and caused by direct
emission from the accretion disk. We conclude that 2SXPS J173508.4-292958 is a
new symbiotic star of the accreting-only variety (AO-SySt).

###### keywords:

binaries : symbiotic

††volume: 51

##  1 Introduction

Attention to the X–ray source 2SXPS J173508.4-292958 has been recently driven
by a report from Heinke et al. (2020) about a brightening of the object they
observed in X–rays with the Swift satellite in April 2020. Noting the
positional coincidence with the cool 2MASS star 17350831-2929580 of Ks=7.4
mag, Heinke et al. suggested that 2SXPS J173508.4-292958 could be a previously
unknown symbiotic star. The optical counterpart was quickly subjected to BVRI
photometry and high- and low-resolution spectroscopy by Munari et al. (2020a)
that found 2SXPS J173508.4-292958 to host a K4 III/II giant, with marked near-
UV excess and H$\alpha$ in emission, the latter showing a broad profile with a
central absorption. Such properties confirm a probable symbiotic nature for
2SXPS J173508.4-292958 (for a recent global review of symbiotic stars see
Munari 2019). In this paper we report about the results of our follow-up
photometric and spectroscopic campaign to monitor 2SXPS J173508.4-292958
(2SXPS for short in the rest of this paper) during the period of seasonal
visibility in 2020.

##  2 Observations

Our observations have been carried out primarily from Italy. At such northern
latitudes, the $-$29∘ declination of 2SXPS implies a very short observing
window, $\sim$2 hours around transit at meridian at just 15 to 20∘ above
horizon. Such a short observing interval coupled with the large atmospheric
extinction and seeing degradation led to a lower-than-usual S/N for our
observations, and argued against employing the highest possible resolution in
the spectroscopic observations.

###  2.1 $B$$V$$R$$I$ photometry

We have obtained optical photometry of 2SXPS in the Landolt photometric system
from late April to early August 2020 with two telescopes operated by ANS
Collaboration. ID 0310 is a 0.30m f/8 Richey-Chretien telescope located in
Cembra (TN, Italy) and equipped with an SBIG ST-8 CCD camera, 1530$\times$1020
array, 9 $\mu$m pixels $\equiv$ 0.77′′/pix, with a field of view of
19′$\times$13′. ID 1301 is a 0.50m f/6 Ritchey-Chretien telescope located in
Stroncone (TR, Italy). It feeds light to an SBIG STL1001E CCD Camera
1024$\times$1024 array, 24 $\mu$m pixels $\equiv$ 1.60′′/pix, field of view of
28′$\times$28′. Both telescopes adopt $B$$V$$R_{\rm C}$$I_{\rm C}$ photometric
filters from Astrodon, in the version corrected for red-leak.

Table 1: Our BVRI photometry on the Landolt system of 2SXPS J173508.4-292958.

Data reduction has involved all the usual steps for bias, dark and flat with
calibration images collected during the same observing nights. We adopted
aperture photometry because the sparse field around 2SXPS did not required
PSF-fitting procedures. The transformation from the local to the Landolt
standard system was carried out via color equations calibrated on a
photometric sequence recorded on the same frames as 2SXPS:

$\displaystyle V$ $\displaystyle=$ $\displaystyle
v+\alpha_{v}\times(v-i)+\gamma_{v}$ $\displaystyle B-V$ $\displaystyle=$
$\displaystyle\beta_{bv}\times(b-v)+\delta_{bv}$ $\displaystyle V-R$
$\displaystyle=$ $\displaystyle\beta_{vr}\times(v-r)+\delta_{vr}$
$\displaystyle V-I$ $\displaystyle=$
$\displaystyle\beta_{vi}\times(v-i)+\delta_{vi}$

where lowercase and uppercase letters denote values in the local and and in
the standard system, respectively. The local photometric sequence has been
extracted from APASS DR8 survey (Henden et al. 2016), ported to the Landolt
system via the transformations calibrated by Munari et al. (2014). Our
photometry of 2SXPS is listed in Table 1. The quoted errors are the quadratic
sum of the Poissonian error on the variable and the error in the
transformation to the standard system via the above color equations.

###  2.2 Spectroscopy

Low resolution spectra of 2SXPS have been obtained with a Shelyak LHIRES
spectrograph + 300 ln/mm grating mounted on the 0.50 Ritchey-Chretien
telescope operated for ANS Collaboration in Stroncone (TR, Italy). The CCD
Camera is an ATIK 460EX (2749$\times$2199 pixels, 4.5 $\mu$m in size). A slit
width of 30 $\mu$m (=2.0 arcsec) has been adopted. Table 2 presents a log of
these low-res observations.

Echelle high resolution spectra of 2SXPS have been obtained with the 0.84m
telescope operated by ANS Collaboration in Varese (Italy), and equipped with a
mark.III Multi-Mode Spectrograph from Astrolight Instruments. The detector is
a SBIG ST10XME CCD camera (2192$\times$1472 array, 6.8 $\mu$m pixel,
KAF-3200ME chip with micro-lenses to boost the quantum efficiency). In the
high resolution mode, an R2 Echelle grating of 79 ln/mm is used in conjunction
with an equilateral 60∘ prims made in high dispersion N-SF11 flint glass, for
a final 18,000 resolving power with a 2arcsec slit width. For 2SXPS
observations a binning 2x2 was adopted leading to a resolving power 10,000.

The spectroscopic observations at both telescopes were obtained in long-slit
mode, with the slit always rotated to the parallactic angle and widened to
2arcsec on the sky. All data have been similarly reduced within IRAF,
carefully involving all steps connected with correction for bias, dark and
flat, sky subtraction, wavelength calibration and heliocentric correction. The
low-res spectra have been flux calibrated using similar observations of the
nearby spectrophotometric standard HR 6698 observed immediately before or
after 2SXPS.

Table 2: Log of spectroscopic observations of 2SXPS J173508.4-292958.

##  3 Results

###  3.1 Spectral classification, energy distribution, and radial velocities

Figure 1: The observed energy distribution of 2SXPS J173508.4-292958 from our
BVRIzY observations complemented with JHKs from 2MASS (Cutri et al. 2003),
W1,W2,W3 from AllWISE (Cutri et al. 2014), and U band from Heinke et al.
(2020). The green line is the intrinsic energy distribution of a K4III star
from Koornneef (1983), Cousins (1980) and Fitzgerald (1970) subjected to an
$E_{B-V}$=0.375 following the reddening law of Fitzpatrick (1999).

Even if projected close to the Galactic Center, 2SXPS is in the foreground at
1.50 kpc distance according to Gaia eDR3 parallax 0.6685$\pm$0.0197 mas (Gaia
Collaboration 2020). The distance would be minimally affected, to 1.53 kpc, by
the application of the mean $-$0.017mas offset derived by Lindegren et al.
(2020) to affect globally the eDR3 parallaxes (they reported this bias to
depend in a non-trivial way on - at least - the magnitude, colour, and
ecliptic latitude of the source, with different dependencies that apply to the
five- and six-parameter solutions in eDR3).

From Green et al. (2019) 3D extinction maps, 2SXPS should suffer from a modest
E(B-V)=0.375. We have dereddened accordingly our BVRI photometry combined with
2MASS JHKs data and found an excellent match with the spectral energy
distribution of a K4III giant (Fig. 1). Fitting instead with K3III or K5III
distributions would require to change E(B-V) to 0.49 or 0.28, respectively.
Combining distance, reddening and observed V=11.825, an absolute magnitude
M(V)=$-$0.55 is derived, intermediate between K4III and K4III/II luminosity
classes for which Sowell et al. (2007) list M(V)=$+$0.20 and M(V)=$-$1.00,
respectively.

The AllWISE mid-IR data in Fig. 1 exclude the presence of circumstellar dust
warmer than $\sim$200 K. Any remnant blown-off by the progenitor of the
present-day degenerate companion to the K4III has therefore already dispersed
at a great distance from the central binary.

The SED in Fig. 1 is suggestive of an excess flux at the shortest wavelengths,
amounting to 0.29 mag at U band. A word of caution is however in order
considering that the U-band Swift observation from Heinke et al. (2020) in
Fig. 1 is not contemporaneous with the BVRIzY data, and 2SXPS is moderately
variable (see below). So the apparent excess at the shortest wavelengths of
Fig. 1 may be (in part or entirely) an artifact. Such an ultraviolet excess is
frequently seen in symbiotic binaries of the accretion-only variety, and
probably originates from the accretion disk around the companion to the cool
giant (Mukai et al. 2016, Munari et al. 2020b). A significant contribution to
the UV excess can also be provided by nebular emission from ionized gas
located somewhere else in the binary system (Skopal 2005a,b), ionized by
either the hard radiation emitted from the accretion disk and its central star
or from some other mechanism, including wind collision. For simplicity, in
this paper we attribute the UV excess to the accretion disk, including
cumulatively in it also the contribution by any nebular region distinct from
the disk itself.

Figure 2: The low resolution spectrum of 2SXPS J173508.4-292958 compared with
those of K3III, K4III and K5III spectral standards observed along with it. The
strong telluric O2 band at 6285 Å is marked, and the arrow points to H$\alpha$
in obvious emission.

The K4III classification for 2SXPS agrees with our low resolution spectra.
Their average is plotted in Fig. 2 and compared with spectra of standards for
spectral types K3III, K4III and K5III selected from the compilation by
Yamashita and Nariai (1977) and observed along 2SXPS with the same
spectrograph set-up. These nearby K giants trace the metallicity and chemical
partition of the Solar neighborhod. The presence of H$\alpha$ in emission in
2SXPS is rather obvious. The very large airmass affecting all our observations
of 2SXPS marks the spectrum in Fig. 2 with signatures of telluric absorption
red-ward of the NaI doublet at 5893 Å, the strongest feature being the O2 band
at 6285 Å. The BaII feature at 6496 Å shows the same strength in 2SXPS and the
field K giants of Fig. 2, suggesting a normal abundance (for a discussion of
Barium enhanced stars vs. symbiotic stars see Jorissen 2003).

Table 3: Heliocentric radial velocities of the K-giant in 2SXPS J173508.4-292958 obtained with the Varese 0.84m telescope. date |  RV⊙ | err
---|---|---
|  | (km/s) | (km/s)
2020 | Apr 23/24 |  $-$11 | 1.5
2020 | May 21 |  $-$15 | 3
2020 | Aug 05 |  $-$13 | 1.5
2020 | Aug 09 |  $-$12 | 1.5
2020 | Aug 20 |  $-$12 | 2

Radial velocities have been obtained from the Echelle spectra by cross-
correlation (fxcor task in IRAF) with spectra of the IAU radial velocity
standard HR 6859 obtained immediately before and after those of 2SXPS on each
visit. This standard is rather convenient: it is located a short angular
distance from 2SXPS at RA=18:21:48 DEC=$-$29:49:18 (minimizing differential
spectrograph flexures), it is very bright at V=2.7 mag, and its K2III spectral
type is an excellent proxy for that of 2SXPS. We adopted the $-$20.0$\pm$0.0
km s-1 heliocentric velocity listed in the ”Standard Radial Velocity Stars”
section of the Nautical Almanac. The epoch radial velocities so obtained for
2SXPS are listed in Table 3. They are fully compatible with a constant
heliocentric radial velocity of $-$12$\pm$1 km s-1 for 2SXPS during the four
months span of our observations. This suggests that the orbital period is
either rather long (and as a consequence the amplitude of orbital motion quite
small) and/or the viewing angle to 2SXPS is oriented high above the orbital
plane. The tangential velocity of 2SXPS is similarly low at $-$26 km s-1 (from
Gaia eDR3 parallax and proper motions), for a combined space velocity of 29 km
s-1.

###  3.2 The variable and structured emission in H$\alpha$

Supporting the association of the K4III giant with the X–ray source 2SXPS is
the presence on optical spectra of emission in H$\alpha$, not expected in
normal and single K4III stars belonging to the field.

In the left panel of Fig. 3 we present small portions around H$\alpha$ from a
sample of our low-resolution spectra (those with the highest S/N). The flux
and equivalent width of H$\alpha$ are observed to change by a large amount
over the course of our observations; no correlation is noted (see phases
marked on the spectra) with the low-amplitude (0.12 mag), 38-day sinusoidal
variability exhibited by 2SXPS, and presumably associated with the K4III giant
(see sect. 3.4 an Eq. 1 below). The absence of correlation, and the low
temperature characterizing a K4III giant, suggest the accretion disk around
the companion as the origin of the H$\alpha$ emission.

The H$\alpha$ profile as recorded on our Echelle spectra is presented on the
right panel of Fig. 3. It is composed by a broad emission, about 470 km s-1 of
width at half intensity, on which it is superimposed a narrow absorption
component (only slightly wider than the instrumental PSF), which heliocentric
velocity averaged over all spectra is $-$4 km s-1. While the observed velocity
of the K4III giant is consistent with a constant value, the velocity of the
narrow absorption component could vary over the range $-$10 to 0 km s-1. Our
spectra lack however the S/N required for a firm conclusion about that.

One thing about the narrow absorption seems however well established: its
velocity is positive with respect to the K4III giant, so it seems unlikely it
may form in the gentle wind out-flowing from it and engulfing the whole binary
system. A typical shift $\Delta$vel $\approx$$-$10/$-$20 km s-1 is generally
observed in symbiotic stars for the narrow absorption superimposed to the
H$\alpha$ emission, matching the typical wind velocity of field cool giants
(Munari et al. 2020b, Shagatova et al. 2020). This leaves open the possibility
that the narrow absorption component seen in 2SXPS may form elsewhere in the
binary system (as the accretion disk itself or an atmosphere engulfing it), or
in alternative its appearance is spurious and the H$\alpha$ profile is
actually composed by two separate emission components. The latter could be
expected for an accretion disk seen at high inclination angle (e.g. Horne and
Marsh 1986), but this condition seems to contrast with the limited (or null)
orbital motion exhibited by the K4III giant over the four months of our
observations.

Figure 3: The variable H$\alpha$ emission observed in 2SXPS J173508.4-292958
during 2020. Left panel: a strong variation in intensity in recorded on our
low-res spectra. Right panel: on Echelle spectra the profile turns out to be
characterized by a broad emission component with superimposed a narrow
absorption. For reference, the spectrum at the bottom show the telluric
absorption lines in the region as derived from observations of a telluric
divider observed along the program star.

###  3.3 No flickering observed

We carried out a search for flickering at optical wavelengths as an indication
of on-going accretion. To search for flickering we used a 50cm telescope, with
a 40arcmin corrected field of view and quality photometric filters, operated
robotically for ANS Collaboration in Atacama (San Pedro Martir, Chile). We
observed in $B$-band with 1-min integration time, with interspersed
observations in $V$-band serving to construct the $B$$-$$V$ color base for the
transformation from the instantaneous local photometric system to the
Landolt’s standard one. The same local photometric sequence and data reduction
procedures as used for the long term $B$$V$$R$$I$ monitoring, were adopted.

Figure 4: Results from a 70min run in $B$ band in search for flickering from
2SXPS J173508.4-292958 (red dots). The regular gaps in the data mark epochs
when $V$-band exposures have been obtained to allow the transformation to the
standard system. The corresponding data for four field stars of similar color
and magnitude are presented as crosses. The progression from $\sigma$=0.007
for field star D to $\sigma$=0.010 for 2SXPS is accounted for by the $\Delta
B$$\sim$0.5mag difference, and exclude the presence of any flickering with an
amplitude in excess of 0.005 mag.

About 50 field stars close on the image to 2SXPS, of a similar magnitude and
well isolated from neighboring stars were also measured on all recorded images
in exactly the same way as 2SXPS. The photometry of these 50 field stars was
then inspected looking for those with a $B$$-$$V$ color as close as possible
to 2SXPS. Four such stars were found, which serve as samplers of the
observational noise above which the flickering has to be detected.

A first run, 70min in duration, was carried out from Sept 2.971 to 3.018 UT,
2020\. The results are presented in Fig. 4: the dispersion of $B$-band
measurements for 2SXPS is the same as for the four field stars of the same
color and magnitude, and if any flickering is present, its amplitude does not
exceed 0.005 mag. Other two runs, each one lasting 35min were carried out from
Oct 24.980 to 25.003, and from Oct 26.981 to 27.003 UT, 2020. They confirmed
the findings for the early September run, with no flickering being detected.

We have seen above how large is the variability displayed by H$\alpha$
emission. It probably parallels a similarly large variability of the accretion
rate onto the compact object in 2SXPS. Therefore, the absence of flickering on
early September and late October observations does not necessarily imply it is
always absent. Unfortunately, at the time of flickering observations, the
object was no more observable from Italy, and therefore we have no
contemporaneous spectroscopy to compare the absence of flickering to the
amount of emission in H$\alpha$.

Figure 5: Our April-to-August 2020 light-curve of 2SXPS J173508.4-292958 built
from data in Table 1 (red dots mark the observations obtained with ANS
Collaboration telescope ID 0310, and black dots those with telescope ID 1301).
The sinusoid going through the $V$-band data has minima according to Ephemeris
(1), an amplitude of 0.12 mag and a period of 38 days.

###  3.4 Photometric evolution

Our $B$$V$$R$$I$ photometry from Table 1 is plotted in Fig. 5. The $V$ band
light-curve is characterized by a mean 11.825 mag value, and a sinusoidal
modulation with a period of 38 days and an amplitude of 0.12 mag, with the
following ephemeris providing times of mimina:

${\rm min(V)}=2458994.5(\pm 1)+38(\pm 0.5)\times E$ (1)

The colors do not appear to change along the cycle. As illustrated by the SED
in Fig. 1, the emission at optical wavelengths is dominated by the K4III
giant, so the sinusoidal variability probably originates from it, either in
the form of some pulsation or the rotation of a spotted surface.

The sinusoidal variability could instead point to an ellipsoidal deformation
of the K4III, should the latter fill its Roche lobe, in which case the orbital
period would be twice the photometric one, or 76 days. The typical mass of a
field K4III is listed as 1.2 M⊙ by Drilling and Landolt (2000). Assuming 1.0
M⊙ for the companion, this leads to a semi-major axis for the binary system of
98 R⊙. The relation between mass ratio $q$, orbital separation $a$, and Roche
lobe radius $R_{RL}$ as given by Eggleton (1983) is

$\frac{R_{RL}}{a}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}$ (2)

and returns a radius of 39 R⊙ for the K4III, twice the tabulated values of
$\sim$20 R⊙ (Drilling and Landolt 2000). The discrepancy would persist also
for different assumption on the mass of the companion: 0.5 M⊙ leads to 41 R⊙
and 2.5 M⊙ to 37 R⊙. Therefore, the origin of the observed sinusoidal light-
curve is probably other that Roche-lobe filling.

Figure 6: Long-term light-curve of 2SXPS J173508.4-292958 built from ASAS-SN
sky-patrol data in $V$ and $g$ bands.

Figure 7: Phased light-curves for the modulated part of Figure 6.

In support of this conclusion comes the absence of detectable radial velocity
changes in Table 3. A 76-day period would correspond to an orbital velocity of
the K4III of 30 km s-1 for a 0.5 M⊙ companion and 38 km s-1 for 2.5 M⊙. The
maximum photometric amplitude (for edge-on systems) of the ellipsoidal
modulation is 0.25/0.30 mag. An amplitude of 0.12 mag for 2SXPS requires a
significant orbital inclination, such that the amplitude of radial velocity
variations would be at least 10 km s-1, incompatible with results from Table
3.

The nail in the coffin about ellipsoidal modulation is however the fact that
the sinusoidal variability is not always present but highly variable with
time, as discussed in the following section.

###  3.5 Long-term light-curve

To reconstruct a longer term light-curve of 2SXPS we have downloaded ASAS-SN
all-sky patrol data (Shappee et al. 2014, Kochanek et al. 2017) and plotted
them in Fig. 6. 2SXPS was observed by ASAS-SN with a $V$-filter over 2016-2018
and with a $g$-filter during 2018-2020.

The median value of the 2016-2018 ASAS-SN $V$ band data in Fugure 6 is 0.076
mag brighter than our data in 2020. We do not regard this as significative:
ASAS-SN data are just differential magnitudes with respect to not-transformed
data for field stars, while our photometry is fully transformed to Landolt’s
system of equatorial standards (cf. sect. 2.1). The rather red color of 2SXPS,
much redder than typical field stars, can easily account for the offset
affecting ASAS-SN data.

The ASAS-SN light-curve in Fig. 6 shows a stable 2SXPS during 2016-2018, and
appearance of the sinusoidal modulation only in 2019, which extended to 2020
with a lower degree of coherence. The phased light-curves of 2SXPS from ASAS-
SN data for 2018, 2019 and 2020 is presented in Fig. 7, with the best-fit
ephemeris given at the top of the panels. The differences between them are
marginally within uncertainties.

While certainly arguing against an interpretation in terms of ellipsoidal
distortion of the K4III giant, the light-curves of Fig. 5, 6 and 7 do not
disentangle a specific cause for the variability observed in 2SXPS, although
some kind of pulsation could be a viable explanation.

##  4 Conclusions

Our observations suggest 2SXPS to be a previously unknown symbiotic star, of
the accreting-only variety (AO-SySt). The low space velocity, absence of cool
circumstellar dust, solar-like metallicity and lack of enhancement in Barium
set 2SXPS aside from other symbiotic stars of the yellow class (i.e. those
harbouring a cool giant of the F,G,K spectral types). As typical of AO-SySt
systems, the signature for 2SXPS of on-going accretion onto a degenerate
companion are feeble (weak and structure emission in H$\alpha$, just a hint of
near-UV excess emission) and highly variable in time, so that the AO-SySt
nature may be not recognized at all times from optical observations alone. The
star clearly deserves further observations, especially accurate radial
velocities over a long time interval in order to derive the orbital period,
orbital inclination and an estimate for the mass of the unseen companion.

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# Optimal search reach for heavy neutral leptons at a muon collider

Krzysztof Mękała<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron
DESY,
Notkestr. 85, 22607 Hamburg, Germany  Faculty of Physics, University of
Warsaw,
Pasteura 5, 02-093 Warszawa, Poland  Jürgen Reuter<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Aleksander Filip Żarnecki
<EMAIL_ADDRESS>Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warszawa, Poland

###### Abstract

Neutrinos are the most elusive particles known. Heavier sterile neutrinos
mixing with the standard neutrinos might solve the mystery of the baryon
asymmetry of the universe. In this letter, we show that among all future
energy frontier accelerators, muon colliders will provide the farthest search
reach for such neutrinos for mass ranges above the $Z$ pole into the multi-TeV
regime. We compare the performance of muon with electron colliders of the same
machine energy and briefly discuss the complementarity in flavor space between
the two types of accelerators.

###### pacs:

13.35.Hb, 13.66.Lm. 14.60.Pq, 14.60.St

### Introduction

Massive neutrinos are considered the first established building blocks of
physics beyond the Standard Model (SM) of particle physics. Their tiny masses
are believed to originate from seesaw-like mixing with heavier sterile
neutrinos whose masses could be all the way from the electroweak (EW) to the
unification scale. While long-distance neutrino oscillation experiments like
DUNE or Hyper-Kamiokande will shed more light on the mass hierarchy and the
mixing parameters, heavier neutrinos can be directly searched for at hadron
colliders such as the LHC and future lepton colliders [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17]. For collider searches, three different
regimes can be considered: light neutrinos which are long-lived and result in
displaced vertices or decay outside the detectors, intermediate-mass neutrinos
that decay promptly and are dominantly produced in $Z$ (and $W$ or Higgs)
decays, and heavy neutrinos with masses $M_{\nu}\gtrsim M_{H}$. In this paper,
building upon an analysis framework similar to earlier studies for searches at
linear $e^{+}e^{-}$ machines, we focus on the third case and show that the
most sensitive searches for direct heavy neutrino production are possible at
high-energy muon colliders. Lepton colliders are, in general, sensitive to
much smaller mixing parameters and hence to much higher scales of UV
completions. In this paper, we will consider a muon collider setup with
energies of 3 and 10 TeV, and integrated luminosities of 1 and 10 ab-1,
respectively [18, 19, 20].

### Model setup and simulation framework

In this letter, we consider the Phenomenological Type I Seesaw Mechanism [21,
22], implemented within the HeavyN model with Dirac neutrinos [23, 5], i.e. we
assume it just as a representative model candidate without any prejudice (our
findings are quite generic, though specific model setups like artificial
flavor mixings could of course lead to singular cases; Refs. [24, 25] provide
an example where such heavy neutrinos appear even at a multi-TeV scale UV
completion). This effective extension of the SM introduces three flavors of
right-handed neutrinos (denoted as $N_{k}$) that are singlets under the SM
gauge groups. The Lagrangian of the model reads:

$\mathcal{L}_{[}]=\mathcal{L}_{[}SM]+\mathcal{L}_{[}N]+\mathcal{L}_{[}WN\ell]+\mathcal{L}_{[}ZN\nu]+\mathcal{L}_{[}HN\nu]$
(1)

where $\mathcal{L}_{[}N]$ is a sum of kinetic and mass terms for heavy
neutrinos (in 4-spinor notation, which combines terms with spinors of dotted
and undotted indices):

$\mathcal{L}_{[}N]=\bar{N}_{k}i\not{\partial}N_{k}-m_{N_{k}}\bar{N}_{k}N_{k}\qquad\textnormal{for}\;k=1,\,2,\,3,$
(2)

$\mathcal{L}_{[}WN\ell]$ yields neutrino interactions with the $W$ boson:

$\mathcal{L}_{[}WN\ell]=-\frac{g}{\sqrt{2}}W^{+}_{\mu}\sum^{3}_{k=1}\sum^{\tau}_{l=e}\bar{N}_{k}V^{*}_{lk}\gamma^{\mu}P_{L}\ell^{-}+\textnormal{
h.c.},$ (3)

$\mathcal{L}_{[}ZN\nu]$ interactions with the $Z$ boson:

$\mathcal{L}_{[}ZN\nu]=-\frac{g}{2\cos\theta_{W}}Z_{\mu}\sum^{3}_{k=1}\sum^{\tau}_{l=e}\bar{N}_{k}V^{*}_{lk}\gamma^{\mu}P_{L}\nu_{l}+\textnormal{
h.c.},$ (4)

and $\mathcal{L}_{[}HN\nu]$ interactions with the Higgs boson:

$\mathcal{L}_{[}HN\nu]=-\frac{gm_{N}}{2M_{W}}h\sum^{3}_{k=1}\sum^{\tau}_{l=e}\bar{N}_{k}V^{*}_{lk}P_{L}\nu_{l}+\textnormal{
h.c.}$ (5)

The UFO library of the model contains 12 free parameters in addition to the SM
parameters, which are three masses of the heavy neutrinos: m${}_{N_{k}}$ and
nine real (no $CP$ violation expected) mixing parameters $V_{lk}$, where
$l=e,\mu,\tau$ and $k=N_{1},N_{2},N_{3}$. For the purpose of this analysis, we
considered a scenario with only one heavy Dirac neutrino $N_{1}\equiv N$ with
a mass below $\mathcal{O}(10\,\text{TeV})$ and equal couplings to all SM
leptons ($|V_{eN_{1}}|^{2}=|V_{\mu N_{1}}|^{2}=|V_{\tau N_{1}}|^{2}\equiv
V_{lN}^{2}$). For sample generation, the mixing parameter $V_{lN}^{2}$ has
been set to 0.0003. Other values for the mixing parameters in the analysis
below were accessed via rescaling with the corresponding cross section.
Although there are many different possible signatures of such particles at
future colliders, for center-of-mass energies above the $Z$ pole, the
t-channel $W$ exchange resulting in the production of a light-heavy neutrino
pair ($\mu^{+}\mu^{-}\rightarrow N\,\nu$) is dominant [11] and the production
cross section is of the order of 1-10 fb for masses of the neutrinos up to the
collision energy. For our choice of the parameter space, the heavy neutrino
has a microscopic lifetime ($c\tau\ll 1$ nm) and no displaced vertices are
expected. Among the possible decay channels of such particles, the signature
of two jets and a lepton ($N\rightarrow qq\ell$) is the most promising because
it allows for direct reconstruction of the mass of the heavy state.

Figure 1: $qq\ell$ mass distribution for a reference scenario assuming the
existence of one Dirac neutrino with a mass of 3 TeV, at a 10 TeV muon
collider. The black solid line stand for the $\mu^{+}\mu^{-}$ background and
the thick green one for the signal scenario.

In the first step, we generated event samples with Whizard 3.0.2 [26, 27, 28]
at leading order (LO) in the SM coupling constants (though recently higher-
order corrections have become available in an automated manner [29] and
simulated detector response with Delphes 3.5.0 [30] using built-in Muon
Collider detector cards. At the generator level, a set of cuts was applied to
remove possible singularities. They included 10-GeV cuts on the energy of
produced jets and leptons, the invariant mass of quark and lepton pairs, and
the four-momentum transfer from the incoming muons. Furthermore, it was
required that at least one lepton could be detected in the central detector
(we assumed $5^{\circ}<\theta<175^{\circ}$, where $\theta$ is the lepton polar
angle). For the detector simulation, the VLC clustering algorithm in the
exclusive two-jet mode (R = 1.5, $\beta$ = 1, $\gamma$ = 1 – see [31]) was
applied. Since the considered Delphes model cannot generate fake lepton
tracks, only 4- and 6-fermion background processes with at least one lepton in
the final state ($qq\ell\nu$, $qq\ell\ell$, $\ell\ell\ell\ell$, $qqqq\ell\nu$,
$qqqq\ell\ell$, $qq\ell\nu\ell\nu$, $qq\ell\nu\nu\nu$) were generated. The
most important channels in terms of cross section ($\mathcal{O}(1\textnormal{
ab})$ at both energy stages) were $qq\ell\nu$ and $\ell\ell\ell\ell$; the
latter could be, however, easily reduced by lepton identification. Background
channels induced by photons from collinear initial-state splittings were
neglected, as their impact on the final results was found to be marginal.

### Analysis procedure

In the next step, a set of selection cuts was applied to reject events
incompatible with the expected topology of two jets and one lepton. To exclude
events with significant contributions of forward deposits assigned to the beam
jets, an upper limit of 20 GeV was applied on the transverse momentum of
objects not included in the reconstructed final state. In Figure 1, we show a
distribution of the invariant mass of two jets and a lepton for a reference
scenario (a 3 TeV neutrino at a 10 TeV muon collider). A peak corresponding to
the mass of the heavy neutrino is clearly visible. The left tail is due to
events with leptonic $\tau$ decays, for which the escaping neutrinos reduce
the reconstructed invariant mass. On the right-hand side, the tail is an
effect of finite detector resolution.

Figure 2: Distribution of the BDT response for the reference scenario (Dirac
neutrino, mN = 3 TeV) with electrons in the final state at a 10 TeV muon
collider. The red line denotes the background, and the green line the signal.

Subsequently, we applied the Boosted Decision Tree (BDT) method implemented in
the TMVA package [32] to discriminate between signal and background events. A
set of eight variables describing event kinematics was chosen to optimize the
classification:

  * •

mqqℓ – invariant mass of the dijet-lepton system,

  * •

$\alpha$ – angle between the dijet system and the lepton,

  * •

$\alpha_{qq}$ – angle between the two jets,

  * •

Eℓ – lepton energy,

  * •

Eqqℓ – energy of the dijet-lepton system,

  * •

p${}^{T}_{\ell}$ – lepton transverse momentum,

  * •

p${}^{T}_{qq}$ – dijet transverse momentum,

  * •

p${}^{T}_{qq\ell}$ – transverse momentum of the dijet-lepton system.

Due to the considerable difference between the numbers of expected background
events, the algorithm was implemented separately for events with reconstructed
electrons and muons in the final state. The BDT response for the reference
scenario is shown in Figure 2. The two distributions are partially separated
and thus, they were used to extract expected limits on the coupling parameter
$V^{2}_{lN}$ within the CLs method, implemented in the RooStats package [33].
This allowed for combining the electron and muon channels. The impact of
systematic uncertainties has been neglected at this stage, as they are not
expected to significantly affect the final conclusions.

### Results

In Figure 3, limits on the coupling $V^{2}_{lN}$ for the two Muon Collider
setups are presented and compared with the current limits coming from the CMS
experiment (Majorana neutrinos, Fig. 2 in [1]), as well as with the results
obtained for future hadron colliders (Dirac neutrinos, Fig. 25b in [5]) and
$e^{+}e^{-}$ colliders (Dirac neutrinos, Fig. 12 in [16]). It should be noted
that in the hadron collider analyses, heavy neutrino decays into taus were not
considered, and thus their sensitivity is enhanced relative to the results
presented for the lepton colliders, where the tau-channel decays are included.
As shown in Figure 3, limits expected from the $e^{+}e^{-}$ colliders, ILC
running at 1 TeV and CLIC running at 3 TeV, are more stringent for masses of
the heavy neutrinos up to about 700 GeV. The fact that the results for CLIC
and a Muon Collider operating at the same energy of 3 TeV do not coincide may
be surprising. However, several effects must be taken into account for a
proper comparison: the most important factors are different integrated
luminosities and beam polarizations. In addition, the beam spectra and the
beam-induced background channels cannot be neglected for $e^{+}e^{-}$
colliders, while their impact is significantly reduced for $\mu^{+}\mu^{-}$
machines due to the larger mass of the muon. It was verified that, for the
same generation setup (no beam polarization, no beam spectrum, no beam-induced
background channels, but different initial-state particles and detector
designs), the expected CLIC limits are consistent with the Muon Collider ones,
giving the analysis precision. The discrepancy visible in Figure 3 could then
be explained as follows: at lower neutrino masses, the expected limits from
CLIC are more stringent due to the higher integrated luminosity and electron
beam polarization, and at higher masses, they are worse because of the impact
of the luminosity spectra and beam-induced backgrounds.

Figure 3: Limits on the coupling $V^{2}_{\ell N}$ for different Muon Collider
setups (solid lines: 3 TeV – turquoise, 10 TeV – orange). Dashed lines
indicate limits from current and future hadron [1, 5] machines, dashed-dotted
for $e^{+}e^{-}$ colliders [16]. See text for details.

In the analysis, we assumed that all the mixing parameters $V_{lN}$ have the
same value. It is important to note that this approach is not unique. Using
data from both electron-positron and muon colliders, one could potentially
loosen this assumption and constrain the parameters $V_{eN}$ and $V_{\mu N}$
separately, by either excluding taus from the physical model or implementing a
proper tau tagging procedure. Such a method would give limits not only on the
couplings themselves but also on their product in the framework where
couplings are treated independently, possibly hinting at a flavor-universality
violation. The details are, however, beyond the scope of this letter.

### Conclusions

Extensions of the Standard Model introducing heavy neutrinos offer interesting
solutions to several of its open questions, e.g. the baryon asymmetry of the
universe, dark matter and flavor. If such particles are at mass scales well
above a GeV, they can be efficiently searched for at future lepton colliders.
Due to the highest achievable energies and the clean experimental
environments, muon colliders would provide the furthest discovery reach for
this kind of particles and models, vastly surpassing high-energy hadron
colliders. By employing the synergy of both different types of lepton
machines, electron-positron and muon colliders, different paths in the flavor
parameter space of the models could be pursued.

## Acknowledgments

The work was partially supported by the National Science Centre (Poland) under
the OPUS research project no. 2021/43/B/ST2/01778. KM and JRR acknowledge the
support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Association) under Germany’s Excellence Strategy-EXC 2121 "Quantum
Universe"-3908333. This work has also been funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – 491245950.

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