# 2D reconstruction of magnetotail electron diffusion region measured by MMS

###### Abstract

Models for collisionless magnetic reconnection in near-Earth space are
distinctly characterized as 2D or 3D. In 2D kinetic models, the frozen-in law
for the electron fluid is usually broken by laminar dynamics involving
structures set by the electron orbit size, while in 3D models the width of the
electron diffusion region is broadened by turbulent effects. We present an
analysis of in situ spacecraft observations from the Earth’s magnetotail of a
fortuitous encounter with an active reconnection region, mapping the
observations onto a 2D spatial domain. While the event likely was perturbed by
low-frequency 3D dynamics, the structure of the electron diffusion region
remains consistent with results from a 2D kinetic simulation. As such, the
event represents a unique validation of 2D kinetic, and laminar reconnection
models.

Geophysical Research Letters

Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin
53706, USA Department of Physics, University of Helsinki, Helsinki, Finland
Swedish Institute of Space Physics, Uppsala 75121, Sweden Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA Department of Physics and
Astronomy, Dartmouth College, Hanover, NH, USA Southwest Research Institute,
San Antonio, TX, USA

J. M<EMAIL_ADDRESS>
The fluctuating measurements are consistent with a 2D reconnecting geometry,
permitting a detailed spacecraft trajectory to be determined.

The MMS data is projected onto a 2D spatial domain, revealing the fine-scale
structure of the electron diffusion region (EDR).

The EDR includes profiles with strong gradients in fields and flows,
consistent with those observed in a matched 2D kinetic simulation.

## Plain Language Summary

Magnetic reconnection is a fundamental process that occurs in the near-Earth
space environment with implications for the safety and longevity of space-
borne electronics in which magnetic field lines rearrange and release energy.
To understand whether reconnection is better described as occurring in a
2D-plane without variation in the third direction versus 3D with variation in
all directions, we analyze spacecraft data from the night-side of Earth’s
magnetic field. We conclude for the considered event that the innermost
region, where the field lines reconnect, remains consistent with results from
a 2D simulation.

##  1 Introduction

Magnetic reconnection [Dungey (1953)] is a fundamental physical process in
plasmas in which magnetic field lines rearrange their topology and convert
magnetic energy into particle thermal and kinetic energies [Zweibel Yamada
(2009)]. Despite a large body of research on this topic, the precise physics
of what occurs at the smallest scales in these reconnection events is yet to
be fully understood. One outstanding question is whether structures of the
innermost reconnection region where the electron fluid decouples from the
magnetic field, called the electron diffusion region (EDR), are best described
by laminar, 2D kinetic models [Vasyliunas (1975), Pritchett (2001)] or by 3D
models that predict instabilities that broaden features of the reconnection
site [Papadopoulos (1977), Huba . (1977), Hoshino (1991)]. To address this
issue, NASA recently launched the Magnetospheric Multiscale Mission (MMS),
which orbits about Earth’s magnetosphere and is designed specifically to probe
the small-scale structure of magnetic reconnection in situ [Burch . (2016)]. .
We here revisit one out of several electron diffusion regions observed by MMS
in the Earth’s magnetotail, with the aim to compare the recorded dynamics to
numerical results obtained with a 2D kinetic simulation model.

##  2 Summary of MMS event

On August 10, 2017 MMS observed a reconnection event in Earth’s magnetotail.
During this event the spacecraft flew fortuitously along a topological
boundary between plasma inflow and outflow, also known as a magnetic
separatrix, and straight through the center of a reconnection EDR. The event
has been the subject of previous investigations [Zhou . (2019), Hasegawa .
(2022), Denton . (2020), Cozzani . (2021)]. In cozzani2021, the frequency of
strong fluctuations in fields and flows are found to be consistent with
kinking of the current sheet in the direction normal to the reconnection plane
at the lower-hybrid frequency. In addition, the signals could also be
influenced by the reconnection exhaust separatrices being rippled by small
magnetic islands [Denton . (2020)]. While 3D geometry and time-dependent
dynamics are suggested to describe the observation, whether the nature of the
event can be accounted for by a 2D laminar framework is still not fully
determined. Considering the same event, we here explore in greater detail the
evidence for 2D or 3D reconnection dynamics. We seek to address the extent to
which the observed signals are consistent with a 2D reconnection geometry that
is being “pushed around” as a rigid body by Alfvénic perturbations external to
the EDR. The stronger fluctuations observed in the vicinity of the EDR could
then be interpreted as the result of the spacecraft zigzagging through the
reference frame of a static 2D geometry with strong spatial gradients, rather
than caused by a localized 3D instability.

In Fig. 1(a-e) several key measurements by MMS are shown in the time interval
of the reconnection encounter. The observation of a reversal in the $B_{L}$
magnetic field component (seen near 12:18:33 UTC in panel a) bracketed by
strong electron temperature anisotropy (panel b) indicate an EDR encounter
where the MMS path sampled the EDR as well as the boundaries of the two
opposing inflow regions, since reconnection inflows are typically
characterized by enhanced parallel electron temperature. Additionally, the
normal electric field and electron outflow velocity profiles include
significant fluctuations shortly after the $B_{L}$ reversal. The times within
the grey dashed bars in panels a-e indicate the region of interest and are
chosen to encompass the features described above. We will call this interval
the reconnection region, as MMS samples both the EDR and along the magnetic
separatrix. The reconstructed signals in Fig. 1(f-j) will be discussed in
further detail at the end of the manuscript, but to the extent that they agree
with the corresponding signals in Fig. 1(a-e), they show that a 2D
reconnection geometry is consistent with the observation.

To examine the characteristics of the fluctuations present in the EDR, in Fig.
2(a) the high-pass filtered $B_{L}$ signal is shown for a wider time interval,
where $\Delta t$=0 corresponds to 12:18:30 UTC. The $B_{L}$ fluctuations,
denoted $\delta B_{L}$, exhibit a notable increase in amplitude in the
reconnection region. We first seek to characterize the gradient length scale
typical for various quantities across the reconnection encounter. In turn,
this will allow us to compute the level of displacement $\delta{\bf x}$
required by a hypothetical 2D rigid geometry to account for the observed
fluctuation levels.

In Fig. 2(a) the orange curve represents the magnitude of the spatial gradient
in $B_{L}$ calculated to linear order using low-pass filtered data [Paschmann
Daly (1998)]. Normalizing the fluctuations by this gradient scale yields a
quantity denoted as $\delta{\bf x}=\delta B_{L}/|\nabla B_{L}|$ in Fig. 2(b),
which represents the plasma excursion projected along $\nabla B_{L}$.
Significant to the present analysis, we observe that this displacement
measure, $\delta{\bf x}$, has approximately constant amplitude inside and
outside of the reconnection region. To further address if the displacement
amplitude is uniform or peaked at the reconnection region, we examine the
amplitude Fourier spectrum for the fluctuations in Figs. 2(a,b) at different
time intervals. In panel (c) we show $\langle A_{\delta
B_{L}}\rangle_{\mbox{\scriptsize MMS}}$, the mean amplitude of the $B_{L}$
fluctuations for all four spacecraft, over three 10 second time intervals. The
first of these, beginning at $\Delta t=0$s, covers the crossing of the x-line
and the most significant fluctuation amplitude in $B_{L}$ (between the grey
dashed lines in panels (a,b), approximately matching the grey reconnection
interval in Fig. 1). The two other time intervals are chosen to start at
$\Delta t=25$s and $\Delta t=45$s, which we estimate to be about 5$d_{i}$ and
9$d_{i}$ from the x-line, respectively, based on extrapolating the velocity of
the spacecraft trajectory discussed in the next section ($d_{i}=c/\omega_{pi}$
is the ion inertial length). In panel (d) we compute $\langle A_{\delta{\bf
x}}\rangle_{\mbox{\scriptsize MMS}}$ over the same time intervals. All spectra
exhibit a noisy profile of spectral amplitude peaking around 1-2 Hz. While the
$B_{L}$ fluctuations have considerably stronger power during the central
current sheet crossing, the excursion $\delta{\bf x}$ appears larger outside
the EDR. It should be noted that the second time interval occurs at times with
significant $B_{N}$ variation and the third over a quieter period. Time
intervals over different field structures display different spectral features
from those chosen here, but importantly the $\delta{\bf x}$ fluctuation
amplitude at the current sheet crossing does not dominate over other intervals
in the same way as $\delta B_{L}$. We additionally repeated this analysis with
other components of the magnetic field and found that gradients in those
quantities are much smaller and lead to unrealistically high values of
$\delta{\bf x}$; however, other quantities such as fluctuations in electron
flow speed yield similar results.

The observations based on Fig. 2 provide evidence that the fluctuations
observed within the EDR are influenced by large-scale fluctuations of the
magnetotail current sheet not generated by the reconnection event itself. From
Fig. 2(d), using the peak frequency $f\simeq 1.5$Hz and the mean amplitude
$\langle A_{\delta{\bf x}}\rangle_{\mbox{\scriptsize MMS}}\simeq 2.75$km/Hz,
we estimate a typical perturbation speed as $v_{fluct}=2\pi f\langle
A_{\delta{\bf x}}\rangle_{\mbox{\scriptsize MMS}}\simeq 25$km/s. This speed is
small compared to the upstream Alfvén speed, $V_{A\infty}\simeq 400$km/s,
providing additional evidence that the fluctuations observed within the EDR
are imposed by modest Alfvénic perturbations existing throughout the
reconnecting current sheet. These perturbations likely include kinking of the
current sheet at scales larger than the 2D plane as discussed in cozzani2021
and possibly other types of Alfvénic waves. The present small level of
activity is also in contrast to the event in li_egedal_2021, in which
fluctuations of $\delta{\bf E}$ near the EDR were elevated by orders of
magnitudes above the background level of upper hybrid waves. In addition to
externally imposed fluctuations, below we find that features in the $B_{N}$
measurement indicate that magnetic islands as identified in denton2020 may
also be a source for ripples in the separator.

##  3 Comparison to kinetic simulation

We now explore the extent to which the observed signals are compatible with
results from a 2D kinetic simulation. We apply VPIC particle-in-cell code
[Bowers . (2009)], which is used extensively to study magnetic reconnection.
As in cozzani2021, the numerical run initialized with a normalized upstream
plasma beta $\beta_{e,\infty}=0.09$, temperature ratios
$T_{i,\infty}/T_{e,\infty}=5$, and a guide-field $B_{g}/B_{0}=0.1$. However,
compared to cozzani2021 we use an earlier time slice, which has a 30% lower
reconnection rate and reduced separator angles yielding better agreement with
the observations. The simulation uses the full proton to electron mass ratio,
$m_{i}/m_{e}=1836$, which allows for a direct quantitative mapping between
VPIC and MMS units [Egedal . (2019)]. As discussed in le:2013, the guide
magnetic field $0.05<B_{g}/B_{0}<0.15$ renders this a Regime II event with no
extended electron jets expected to emanate beyond a localized EDR.

In Fig. 3 the inferred spacecraft paths are plotted over VPIC data for the
electron temperature anisotropy, characterized by the ratio of electron
temperature parallel and perpendicular to magnetic field lines. The paths are
found through a least-squares optimization procedure, similar to that laid out
in egedal2019. However, for the present analysis, the MMS1 $B_{L}$ signal
directly determines the simulation $B_{L}$ contour along which the spacecraft
position is optimized. Further details of this procedure are discussed in
Section 1 of the Supporting Information accompanying this text.

Figure 1:  Time series data from the magnetotail reconnection event measured
on August 10, 2017. Shown from top to bottom are (a) temperature anisotropy,
(b/c) magnetic field in the L/N-direction, (d) electric field in the
N-direction, and (e) electron velocity in the L-direction. We use the
previously determined event basis of denton2020, where unit vectors in
geocentric solar ecliptic (GSE) coordinates are [L;M;N] = [0.9872, -0.1305,
-0.0915; 0.1580, 0.8782, 0.4515; 0.0214, -0.4601, 0.8876], aligning the
separatrix in the MMS event with the corresponding separatrix in the VPIC
simulation. (f-j) show the result of interpolation through 2D maps of Fig. 4
along the spacecraft paths shown in Fig. 3. Shaded in pink in is the
reconstruction error, taken as the root-mean-square of the differences between
the measured signal and the reconstructed signal for MMS1-4.  Figure 2:  (a)
Fluctuations in the magnetic field in the L-direction, $\delta B_{L}$, for
each spacecraft are shown. A peak in fluctuations can be seen near $\Delta
t$=5. We compute the high-pass filtered signal by subtracting the Gaussian-
smoothed mean signal from the measurement, using a Gaussian with a width of
1s. (b) The plasma excursion, $\delta{\bf x}=\delta B_{L}/|\nabla B_{L}|$. (c)
Amplitude spectra of fluctuations in $B_{L}$ over 10 second time intervals
averaged across all four spacecraft, $\langle A_{\delta
B_{L}}\rangle_{\mbox{\scriptsize MMS}}$. Shown are three spectra, one starting
at the current sheet crossing $\Delta$t=0s and the other two at later times of
$\Delta$t=25s and $\Delta$t=45s. (d) $\langle A_{\delta{\bf
x}}\rangle_{\mbox{\scriptsize MMS}}$ for the same three time intervals
considered in (c). Figure 3:  Upper panel: The spacecraft paths are plotted
over VPIC simulation data of the ratio of electron temperatures parallel and
perpendicular to the magnetic field. One of the two magnetic separatrices,
plotted as a cyan curve, is used as the basis of a coordinate transform in 2D
MMS plots shown in Fig. 4. Lower panel: Zoomed in view of the domain and
coordinates used for 2D reconstruction.

With the paths of the spacecraft determined, the raw time series data from all
four can be spatially binned and averaged to construct 2D maps of the event in
the LN-plane. Technical details of this procedure are laid out in Section 2 of
the Supporting Information document. The result is shown in the leftmost
column of Fig. 4, where MMS reconstructions for twelve different spacecraft
measurement fields are shown. The corresponding VPIC simulation data are shown
in the middle and right columns. The middle and right columns contain the same
VPIC data values, but the middle column is restricted to the same domain as
the MMS data for better visual comparison. The cyan contour of Fig. 3 becomes
the nearly horizontal black line in the reconstructed maps. The other
simulation separatrix, not shown in Fig. 3, is included to highlight inflow
and outflow regions distinctly. These contours represent the exact simulation
separatrices and are laid over the MMS panels for reference, not to indicate
the physical separatrices for the observed event. Lastly, the spacecraft paths
in the rotated coordinates are superimposed on the first VPIC panel in the
second column.

Figure 4:  The left column shows fields measured with MMS constructed into a
2D map based on the spacecraft trajectory. The map is given in rotated
coordinates ($L^{\prime},N^{\prime})/d_{e}$, where $L^{\prime}$ approximately
represents the distance along and $N^{\prime}$ the distance away from the
separatrix highlighted in cyan in Fig. 3 (shown here as a horizontal black
curve). The middle column shows VPIC data of the same region of the
reconnection geometry for comparison, and the right column shows VPIC data
over the entire domain plotted. The top panel of the VPIC data in the middle
column is overlaid with the spacecraft paths in the rotated coordinates. The
first four fields of VPIC data are multiplied by a factor of 1.3 for better
visual comparison to the MMS data. Figure 5:  MMS maps from Fig. 4 in
L’N’-coordinates. Overlaid are magnetic field lines in the LN-plane calculated
from $A_{M}$ contours, where $A_{M}$ is obtained directly from MMS 2D maps of
$B_{L}$ and $B_{N}$. Contours that fall outside of underlying color plot come
from interpolation of magnetic field maps along the L’-direction.

Visual comparison of the reconstructed maps to the simulation data shows
striking similarity in many key features in each quantity. Sharp gradients
across the separator around $L^{\prime}/d_{e}\sim$ 10-25 are seen in multiple
fields including $E_{N}$ where a variation along N’ of $\sim 25$mV/m over
$\sim 2d_{e}$ matches excellently with the simulation data. As described
above, at each time-point, the optimization for the four spacecraft locations
only includes a single degree of freedom. Therefore, the good match between
multiple MMS maps and VPIC profiles is direct evidence that the EDR has a
rigid 2D structure, without strong 3D effects or fast temporal variations. For
example, had the event included significant 3D dynamics that broaden
separatrix layers, sharp gradients in multiple quantities (including the
elements of ${\bf E}$ and ${\bf u_{e}}$) would be smoothed out by the
overlapping LN-projected spacecraft paths. Good quantitative agreement is also
observed between the MMS and VPIC quantities, but to fit a common color map,
the VPIC profiles of $n_{e}$, $T_{e\perp}$, $P_{eLM}$, and $P_{eMN}$ in Fig. 4
are each multiplied by a factor of 1.3. This suggests that the value of
$\beta_{e\infty}$ applied in the simulation could still be optimized further.
However, this simulation parameter is not expected to influence the accuracy
of the presented 2D MMS data profiles.

To further illustrate how the 2D reconstructed maps well represent the
observation, we construct time series data directly comparable to the MMS
signals by interpolating the 2D maps along each respective spacecraft path.
This is presented in Fig. 1(f-j), where the measured MMS signals of four key
fields are in the left column while the interpolated signals are shown in the
right column. Good agreement is seen for each field, confirming the accuracy
of the reconstructed 2D maps of the event. The pink shaded regions in the
second column are error estimates for the reconstruction, the root-mean-square
of the difference between MMS measurement and the reconstructed signal across
all four spacecraft. The error estimate for $B_{L}$ is particularly small
because that quantity is fixed to the MMS1 measurement in the trajectory
optimization (see Supporting Information, Section 1). In other fields,
however, the error remains bounded to a small value for all times and thus
indicates the merit of the 2D reconstruction method.

Having shown that a 2D geometry well suits the event considered, we now
consider the in-plane magnetic field profile and separatrix locations for the
spacecraft observation. The in-plane magnetic field fulfills ${\bf
B}_{LN}=\nabla\times A_{M}{\bf e}_{M}$, where $A_{M}$ is the $M$-component of
the magnetic vector potential, so it follows that
$A_{M}(L,N)=\int_{L_{0},N_{0}}^{L,N}{\bf B}_{LN}\times\vec{d\ell}_{LN}$
[Kesich . (2008)]. Here the integral can be carried out along an arbitrary
path from an arbitrary starting point $(L_{0},N_{0})$. In Fig. 4 the MMS
profiles of $B_{L}$ and $B_{N}$ are relatively smooth and we can interpolate
where possible in the $L^{\prime}$ direction to obtain an enhanced area of
coverage. $A_{M}(L,N)$ is then obtained by first integrating from
$L^{\prime}/d_{e}=-30$ to $L^{\prime}/d_{e}=30$ along the center of the domain
where ${\bf B}_{LN}$ data is available, characterizing
$A_{M}(L^{\prime},N^{\prime})$ along this center-line. Any point in the
$(L^{\prime},N^{\prime})$-plane is then reached by integrating in
$N^{\prime}$-direction from the center-line.

The result is shown in Fig. 5, where black lines represent contours of
constant $A_{M}$ and the red lines represent the topological separatricies. To
the best of our knowledge, this is the first map of the field lines in a
region of an EDR obtained directly by MMS data without relying on any
extrapolation techniques. The present magnetic map is based on all the data of
the EDR encounter and covers a spatial domain of about 60$d_{e}$ along the
separatrix.

We observe how the separatricies mark the locations of the strongest gradients
in $E_{N}$, $U_{eL}$ and $U_{eM}$, where the gradient length scale for these
quantities is at or below $1d_{e}$. Such fine-scale structures are not
expected in scenarios including strong instabilities and/or 3D dynamics, which
have been shown to broaden reconnection current layers by providing an
additional source of electron diffusion [Graham . (2022)]. Their presence thus
provides additional validation of the 2D structure assumed in the the
analysis.

However, the spatial resolution is limited by the bin-size $(\simeq 2d_{e})$
applied in the $L^{\prime}$-direction, and this resolution does not allow us
to identify magnetic islands at this scale. This can also be seen in Fig.
1(c/h), where the MMS measurements of $B_{N}$ display oscillations including
sign reversals around times 12:18:32-12:18:37 (circled in red). These
oscillations in the N-direction are likely due to island structures previously
reported [Hasegawa . (2022), Denton . (2020)]. Meanwhile, the $B_{N}$
oscillations are not as prominent in the reconstructed signals averaged out by
the finite resolution of our analysis. The oscillations not captured in the
reconstruction are observed to have a magnitude of about $\Delta B_{N}\simeq
0.5$ nT, which should be compared to $B_{L}\simeq 5$nT. In the
$L^{\prime}$-direction the “wave-length” of fluctuations about the separatrix
is about $\lambda\simeq 5d_{e}$, such that the $B_{N}$ fluctuations correspond
to a $(0.5/5)5d_{e}=0.5d_{e}$ ripple in the separatrix, not captured by the
reconstruction. Thus, this effect alone is unlikely to account for the
inferred magnitude of oscillations $(\simeq 4d_{e})$ in the spacecraft
trajectories.

##  4 Conclusions

In summary, we apply particle-in-cell simulation to analyze the Aug. 10, 2017
EDR event. Direct quantitative comparison provides optimized spacecraft
trajectories that give spatial information in the 2D reconnection plane.
Reconstruction of measured quantities in this plane provide a unique and
detailed picture of the electron diffusion region with sub-electron scale
resolution. The 2D reconstructions fit the underlying simulation with good
accuracy suggesting that the present event has an inherently 2D-nature in
agreement with results from 2D kinetic simulations. Our results are also
consistent with results of zhou2019, which conclude the event is mostly
laminar with weak wave activity. The fluctuations in signals observed are well
accounted for by the short gradient length-scale $(<1d_{e})$ of the EDR, which
we characterize as a rigid body being jostled by a bath of large-scale, low-
amplitude fluctuations. The spatial displacements associated with these
fluctuations are consistent with Alfvénic fluctuations observed external to
the EDR, and while magnetic islands likely are present within the EDR, the
data suggest that external fluctuations are responsible for the oscillatory
motion of the spacecraft through the rigid 2D reconnection geometry. The
reconstructed flux-function as well as the agreement of the 2D reconnection
structure with those of the simulation provide a unique confirmation of
laminar 2D kinetic models for reconnection.

###### Acknowledgements.

The work was funded by an H. I. Romnes Faculty Fellowship by the UW-Madison
Office of the Vice Chancellor for Research and Graduate Education.

## Open Research

The Magnetospheric Multiscale Mission (MMS) data used in this paper is
publicly available from the CU-Boulder Laboratory for Atmospheric and Space
Physics MMS Data Center webpage (https://lasp.colorado.edu/mms/sdc/public/).
Using the initial conditions specified in the text, the numerical data can be
reproduced with the open source VPIC code available at
https://github.com/lanl/vpic, as well as at
https://zenodo.org/record/4041845#.X2kA1x17kuY (doi - 10.5281/zenodo.4041845).

## References

  * Bowers . (2009) bowers:2009Bowers, K., Albright, B., Yin, L., Daughton, W., Roytershteyn, V., Bergen, B. Kwan, T.  20092009\.  Advances in petascale kinetic plasma simulation with VPIC and Roadrunner Advances in petascale kinetic plasma simulation with VPIC and Roadrunner [Journal Paper].  Journal of Physics: Conference Series180012055 (10 pp.).
  * Burch . (2016) burch:2016bBurch, JL., Moore, TE., Torbert, RB. Giles, BL.  2016MAR.  Magnetospheric Multiscale Overview and Science Objectives Magnetospheric Multiscale Overview and Science Objectives.  Space Science Reviews1991-45-21.  10.1007/s11214-015-0164-9
  * Cozzani . (2021) cozzani2021Cozzani, G., Khotyaintsev, YV., Graham, DB., Egedal, J., André, M., Vaivads, A.Burch, JL.  2021Nov.  Structure of a Perturbed Magnetic Reconnection Electron Diffusion Region in the Earth’s Magnetotail Structure of a perturbed magnetic reconnection electron diffusion region in the earth’s magnetotail.  Phys. Rev. Lett.127215101.  https://link.aps.org/doi/10.1103/PhysRevLett.127.215101 10.1103/PhysRevLett.127.215101
  * Denton . (2020) denton2020Denton, RE., Torbert, RB., Hasegawa, H., Dors, I., Genestreti, KJ., Argall, MR.Fischer, D.  2020\.  Polynomial Reconstruction of the Reconnection Magnetic Field Observed by Multiple Spacecraft Polynomial reconstruction of the reconnection magnetic field observed by multiple spacecraft.  Journal of Geophysical Research: Space Physics1252e2019JA027481.  https://doi.org/10.1029/2019JA027481
  * Dungey (1953) dungey:1953Dungey, J.  1953\.  Conditions for the occurence of electrical discharges in astrophysical systems Conditions for the occurence of electrical discharges in astrophysical systems.  Philosophical Magazine44725.
  * Egedal . (2019) egedal2019Egedal, J., Ng, J., Le, A., Daughton, W., Wetherton, B., Dorelli, J.Rager, A.  2019Nov.  Pressure Tensor Elements Breaking the Frozen-In Law During Reconnection in Earth’s Magnetotail Pressure tensor elements breaking the frozen-in law during reconnection in earth’s magnetotail.  Phys. Rev. Lett.123225101.  https://link.aps.org/doi/10.1103/PhysRevLett.123.225101 10.1103/PhysRevLett.123.225101
  * Graham . (2022) graham_LHDIGraham, D., Khotyaintsev, Y., André, M., Vaivads, A., Divin, A., Drake, J.Dokgo, K.  202205\.  Direct observations of anomalous resistivity and diffusion in collisionless plasma Direct observations of anomalous resistivity and diffusion in collisionless plasma.  Nature Communications1312954.  https://hal.archives-ouvertes.fr/hal-03697620 10.1038/s41467-022-30561-8
  * Hasegawa . (2022) hasegawa2022Hasegawa, H., Denton, RE., Nakamura, TKM., Genestreti, KJ., Phan, TD., Nakamura, R.Saito, Y.  2022\.  Magnetic field annihilation in a magnetotail electron diffusion region with electron-scale magnetic islands Magnetic field annihilation in a magnetotail electron diffusion region with electron-scale magnetic islands.  Journal of Geophysical Research: Space Physics127.
  * Hoshino (1991) hoshino:1991Hoshino, M.  1991\.  Forced magnetic reconnection in a plasma sheet with localized resistivity profile excited by lower hybrid drift type instability Forced magnetic reconnection in a plasma sheet with localized resistivity profile excited by lower hybrid drift type instability.  Journal of Geophysical Research: Space Physics96A711555-11567.  https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/91JA00984 10.1029/91JA00984
  * Huba . (1977) huba:1977Huba, JD., Gladd, NT. Papadopoulos, K.  1977\.  The lower-hybrid-drift instability as a source of anomalous resistivity for magnetic field line reconnection The lower-hybrid-drift instability as a source of anomalous resistivity for magnetic field line reconnection.  Geophysical Research Letters43125-128.  https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/GL004i003p00125 10.1029/GL004i003p00125
  * Kesich . (2008) kesich:2008Kesich, A., Bonde, J., Egedal, J., Fox, W., Goodwin, R., Katz, N. Le, A.  2008JUN.  Magnetic flux array for spontaneous magnetic reconnection experiments Magnetic flux array for spontaneous magnetic reconnection experiments.  REVIEW OF SCIENTIFIC INSTRUMENTS796.  10.1063/1.2937193
  * Le . (2013) le:2013Le, A., Egedal, J., Ohia, O., Daughton, W., Karimabadi, H. Lukin, VS.  2013MAR 28.  Regimes of the Electron Diffusion Region in Magnetic Reconnection Regimes of the Electron Diffusion Region in Magnetic Reconnection.  11013\.  10.1103/PhysRevLett.110.135004
  * Li . (2021) li_egedal_2021Li, WY., Khotyaintsev, YV., Tang, BB., Graham, DB., Norgren, C., Vaivads, A.Wang, C.  2021\.  Upper-Hybrid Waves Driven by Meandering Electrons Around Magnetic Reconnection X Line Upper-hybrid waves driven by meandering electrons around magnetic reconnection x line.  Geophysical Research Letters4816e2021GL093164.  https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021GL093164 e2021GL093164 2021GL093164  https://doi.org/10.1029/2021GL093164
  * Papadopoulos (1977) papadopoulos:1977Papadopoulos, K.  1977\.  A review of anomalous resistivity for the ionosphere A review of anomalous resistivity for the ionosphere.  Reviews of Geophysics151113-127.  https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/RG015i001p00113 10.1029/RG015i001p00113
  * Paschmann Daly (1998) MutlispacecraftAnalysisPaschmann, G. Daly, PW.  199801\.  Analysis Methods for Multi-Spacecraft Data. ISSI Scientific Reports Series SR-001, ESA/ISSI, Vol. 1. ISBN 1608-280X, 1998 Analysis Methods for Multi-Spacecraft Data. ISSI Scientific Reports Series SR-001, ESA/ISSI, Vol. 1. ISBN 1608-280X, 1998.  ISSI Scientific Reports Series1.
  * Pritchett (2001) pritchett:2001Pritchett, P.  2001MAR 1.  Geospace Environment Modeling magnetic reconnection challenge: Simulations with a full particle electromagnetic code Geospace environment modeling magnetic reconnection challenge: Simulations with a full particle electromagnetic code.  106A33783-3798.
  * Vasyliunas (1975) vasyliunas:1975Vasyliunas, VM.  1975\.  Theoretical models of magnetic-field line merging .1. Theoretical models of magnetic-field line merging .1.  Reviews of Geophysics131303-336.
  * Zhou . (2019) zhou2019Zhou, M., Deng, XH., Zhong, ZH., Pang, Y., Tang, RX., El-Alaoui, M.Lindqvist, PA.  2019jan.  Observations of an Electron Diffusion Region in Symmetric Reconnection with Weak Guide Field Observations of an electron diffusion region in symmetric reconnection with weak guide field.  The Astrophysical Journal870134.  https://doi.org/10.3847/1538-4357/aaf16f 10.3847/1538-4357/aaf16f
  * Zweibel Yamada (2009) zweibel2009Zweibel, EG. Yamada, M.  2009\.  Magnetic Reconnection in Astrophysical and Laboratory Plasmas Magnetic reconnection in astrophysical and laboratory plasmas.  Annual Review of Astronomy and Astrophysics471291-332.  https://doi.org/10.1146/annurev-astro-082708-101726 10.1146/annurev-astro-082708-101726


# Searching for heavy neutrino in terms of tau lepton at future hadron
collider

Chengcheng Han<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen
University, Guangzhou 510275, China  Tong Li<EMAIL_ADDRESS>School of
Physics, Nankai University, Tianjin 300071, China  Chang-Yuan Yao
<EMAIL_ADDRESS>School of Physics, Nankai University, Tianjin 300071,
China

###### Abstract

The tau lepton plays important role in the correlation between the low-energy
neutrino oscillation data and the lepton flavor structure in heavy neutrino
decay. We investigate the lepton flavor signatures with tau lepton at hadron
collider through lepton number violating (LNV) processes. In the Type I Seesaw
with U$(1)_{\rm B-L}$ extension, we study the pair production of heavy
neutrinos via a $Z^{\prime}$ resonance. We present a detailed assessment of
the search sensitivity to the channels with tau lepton in the subsequent decay
of heavy neutrinos. For the benchmark model with $Z^{\prime}$ only coupled to
the third generation fermions, we find that the future circular collider (FCC-
hh) can discover the LNV signal with tau lepton for $M_{Z^{\prime}}$ up to 2.2
(3) TeV with the gauge coupling $g^{\prime}=0.6$ and the integrated luminosity
of 3 (30) ab-1. The test on the flavor combinations of SM charged leptons
would reveal the specific nature of different heavy neutrinos.

##  I Introduction

It is well-known that, in the context of the Standard Model (SM), the non-zero
but small Majorana neutrino masses can be realized at leading order through a
dimension-five operator Weinberg (1979)

$\displaystyle{\kappa\over\Lambda}l_{L}l_{L}HH,$ (1)

where $l_{L}$ and $H$ stand for the SM lepton doublet and the Higgs doublet,
respectively. To receive the above “Weinberg operator”, the minimal extension
of the SM content permits only the Type I, Type II or Type III Seesaw
mechanism at tree level Ma (1998). In particular, in Type I Seesaw model
Minkowski (1977); Yanagida (1979); Gell-Mann  _et al._ (1979); Glashow (1980);
Mohapatra and Senjanovic (1980); Shrock (1981); Schechter and Valle (1980);
Davidson and Wali (1987), the neutrino mass matrix is generated by at least
two generations of fermionic multiplets as SU$(2)_{L}$ singlet. The new
physics scale $\Lambda$ in Eq. (1) is replaced by the mass of the new
multiplets and can be as low as TeV if a small Yukawa coupling $\kappa$ is
allowed.

The new fermionic multiplets in TeV Seesaw mechanisms can be experimentally
accessible at the Large Hadron Collider (LHC) (see Ref. Cai  _et al._ (2018)
and the references therein). Up to now, no significant evidence of such heavy
fermions was observed at the LHC and the lower limits on their masses are
highly dependent on the mixing between the heavy Majorana neutrinos and the SM
leptons. A majority of search results released by ATLAS Aad  _et al._ (2015a,
b) and CMS Khachatryan  _et al._ (2015, 2016); Sirunyan  _et al._ (2018) were
defined by the analysis of electron and/or muon final states from heavy
neutrino decay 111There were searches for heavy neutrinos using events with
two tau leptons and two jets in a left-right symmetric model Sirunyan  _et
al._ (2017, 2019).. However, governed by the constraints from neutrino
oscillation experiments, the channel with tau lepton in heavy neutrino decay
also plays an important role in confirming neutrino mass hierarchies and
determining mixing parameters Fileviez Perez  _et al._ (2008a, b); Li and He
(2009); Cai  _et al._ (2018); Li (2018); Pascoli  _et al._ (2018); Cid Vidal
_et al._ (2019); Pascoli  _et al._ (2019); Cheung  _et al._ (2020);
Ashanujjaman and Ghosh (2021); Das and Mandal (2021). Recently, several
precise reactor measurements lead to more accurate lepton flavor predictions
about the Seesaw models. 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. T2K and NOvA reported on indications
of a non-zero leptonic CP phase Abe  _et al._ (2014); Adamson  _et al._
(2016); Abe  _et al._ (2017). These experiments provide us up-to-date neutrino
oscillation results to investigate the impact on neutrino mass models and
consequently examine the lepton flavor signatures to be searched at colliders.
Thus, from the experimental point of view, a detailed assessment of the search
sensitivity to the channels with tau lepton is in demand at the LHC upgrades.

On the other hand, 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, lepton number is nearly conserved in low-scale seesaw
models with fermionic singlets in light of non-zero neutrino masses. Tiny
neutrino mass is proportional to small lepton number violation (LNV)
parameters. Thus, for low-scale Type I Seesaw, the observable lepton number
violation is decoupled in high energy processes. Any observation of lepton
number violation at colliders implies an extension of the canonical Type I
Seesaw. Below we consider the introduction of a new U$(1)_{\rm B-L}$ gauge
symmetry in which the SM matter fields and heavy neutrinos are charged under
the B-L (baryon minus lepton number) symmetry. However, the minimal B-L model
is highly constrained by the di-lepton search at the LHC. The lower
$Z^{\prime}$ mass is pushed up to 4.2 TeV for the minimal U$(1)_{\rm B-L}$
model Aaboud  _et al._ (2017). To evade the di-lepton constraint, the minimal
B-L model should be extended. In a flavored U$(1)_{\rm B-L}$ model (denoted by
U$(1)_{\rm(B-L)_{3}}$) where only the third generation fermions couple to the
new gauge boson $Z^{\prime}$ Alonso  _et al._ (2017); Cox  _et al._ (2018),
the constraint on the new gauge interaction is much weak and a single heavy
neutrino preferably decays into the tau lepton. In addition, leptogenesis
prefers much heavy right-handed neutrinos and the usual B-L breaking scale
should be high. However, it is sufficient to have two heavy right-handed
neutrinos to explain the baryon asymmetry of the Universe, which leaves a
possibility that the third right-handed neutrino could be light enough to be
probed at future colliders. It is thus plausible to consider this non-
universal U$(1)_{\rm(B-L)_{3}}$ model. We study the pair production of heavy
neutrino via gauge interaction in the U$(1)_{\rm(B-L)_{3}}$ extended Type I
Seesaw Fileviez Perez  _et al._ (2009); Cox  _et al._ (2018); Chiang  _et al._
(2019)

$\displaystyle pp\rightarrow Z^{\prime}\rightarrow NN\;,$ (2)

where $Z^{\prime}$ is the new gauge boson of the U$(1)_{\rm(B-L)_{3}}$
symmetry and $N$ denotes the heavy neutrino followed by the decay into tau
lepton in LNV final states. Suppose a new $Z^{\prime}$ can be discovered
through the di-lepton channel in future, one can look for the LNV signal and
test the specific nature of heavy neutrinos through this channel. We will show
the sensitivity of the search for the above LNV channels in both the high-
luminosity LHC (HL-LHC) and future circular hadron collider (FCC-hh).

This paper is organized as follows. In Sec. II, we first describe the
U$(1)_{\rm B-L}$ extension and discuss the realization of Type I Seesaw in
this model and the constraint from neutrino oscillation experiments on heavy
neutrino decay patterns. In Sec. III, we simulate the pair production of heavy
neutrinos via $Z^{\prime}$ and its LNV signature with tau lepton at the LHC
upgrade. The results of projected sensitivity for heavy neutrinos using tau
lepton are also given. Finally, in Sec. IV we summarize our conclusions.

##  II Type I Seesaw in U$(1)_{\rm B-L}$ extension model

###  II.1 U$(1)_{\rm B-L}$ extension model

To reveal the origins of the B-L accidental symmetry in the SM and Majorana
neutrino masses, one considers the extension of the SM with U$(1)^{\prime}$ as
a local symmetry. In the well-studied U$(1)^{\prime}$ Abelian gauge extension
of the SM Appelquist  _et al._ (2003), U$(1)^{\prime}$ is a linear combination
of U$(1)_{\rm Y}$ and U$(1)_{\rm B-L}$ after the spontaneous breaking of
electroweak symmetry and B-L symmetry. In this model, right-handed neutrinos
(RH) are introduced to cancel gauge anomalies and realize the Type I Seesaw
mechanism. The relevant covariant derivatives can be generally written as

$\displaystyle D_{\mu}\ni
ig_{1}YB_{\mu}+i(g_{1}^{\prime}Y+g^{\prime}Y_{BL})B^{\prime}_{\mu}=ig_{1}YB_{\mu}+ig_{X}(x_{1}^{\prime}Y+x^{\prime}Y_{BL})B^{\prime}_{\mu},$
(3)

where $B_{\mu}(Y)$ and $B^{\prime}_{\mu}(Y_{BL})$ are the gauge fields
(quantum numbers) of U$(1)_{\rm Y}$ and U$(1)_{\rm B-L}$, respectively. Here
we redefine the gauge coupling of $B^{\prime}_{\mu}$ as $g_{X}\equiv
g_{1}^{\prime}/x_{1}^{\prime}=g^{\prime}/x^{\prime}$ with U$(1)^{\prime}$
charge $x_{1}^{\prime}Y+x^{\prime}Y_{BL}$. Besides the gauge field
$B^{\prime}_{\mu}$, a new scalar field $S\sim(1,1,0,2x^{\prime})$ is
introduced to break the local U$(1)_{\rm B-L}$ symmetry and generate the
masses of right-handed neutrinos $N_{R}\sim(1,1,0,-x^{\prime})$. When
$x^{\prime}=1,x_{1}^{\prime}=0$, the mixing between U$(1)_{\rm Y}$ and
U$(1)_{\rm B-L}$ vanishes and the most economical extension emerges Davidson
(1979); Mohapatra and Marshak (1980); Marshak and Mohapatra (1980); Wetterich
(1981); Masiero  _et al._ (1982); Mohapatra and Senjanovic (1983); Buchmuller
_et al._ (1991), i.e. U$(1)^{\prime}=$U$(1)_{\rm B-L}$ and $g_{X}=g^{\prime}$.
In the following studies, we consider this economical U$(1)_{\rm B-L}$
extension. Once $S$ gets the vacuum expectation value (vev) $\langle
S\rangle=v_{S}/\sqrt{2}$, the B-L symmetry is broken and the new gauge boson
$Z^{\prime}=Z_{\rm X}$ can also get mass $M_{Z^{\prime}}=2g^{\prime}v_{S}$
from the kinetic term $(D_{\mu}S)^{\dagger}(D^{\mu}S)$ with
$D_{\mu}S=\partial_{\mu}S+i2g^{\prime}B^{\prime}_{\mu}S$. LEP-II sets the
lower bound on the $Z^{\prime}$ boson mass in U$(1)_{\rm B-L}$ gauge model,
i.e. $M_{Z^{\prime}}/g^{\prime}\gtrsim 6$ TeV Carena  _et al._ (2004). The
most stringent limit comes from the search for dilepton resonance at the LHC
with $\sqrt{s}=13$ TeV and the luminosity of 36 fb-1 Aaboud  _et al._ (2017)
or 139 fb-1 Aad  _et al._ (2019). Whereas, lower mass limit is set up to 4.2
TeV for the minimal U$(1)_{\rm B-L}$ model Aaboud  _et al._ (2017) and the
constraint is also given in the plane of coupling strength $g^{\prime}/g_{Z}$
vs. $M_{Z^{\prime}}$.

On the other hand, in this model, the anomaly-free U$(1)^{\prime}$ extension
allows non-universal U$(1)^{\prime}$ charges for the SM fermions. With the
presence of three right-handed neutrinos, the most general requirement of the
U$(1)^{\prime}$ charges is Kownacki  _et al._ (2017)

$\displaystyle
3(Q^{\prime}_{1}+Q^{\prime}_{2}+Q^{\prime}_{3})+Q^{\prime}_{e}+Q^{\prime}_{\mu}+Q^{\prime}_{\tau}=0\;,$
(4)

where $Q^{\prime}_{1,2,3}$ are the quark charges and $Q^{\prime}_{e,\mu,\tau}$
are the lepton charges. To relax the constraint from dilepton channel, there
proposed a flavored model under the U$(1)_{\rm(B-L)_{3}}$ symmetry where only
the third generation fermions are charged with
$3Q^{\prime}_{3}=-Q^{\prime}_{\tau}=1$ and
$Q^{\prime}_{1}=Q^{\prime}_{2}=Q^{\prime}_{e}=Q^{\prime}_{\mu}=0$ Alonso  _et
al._ (2017); Cox  _et al._ (2018). The $Z^{\prime}$ can only be produced
through $b\bar{b}\rightarrow Z^{\prime}$ and decay into $\tau^{+}\tau^{-}$ and
a pair of one single heavy neutrino. As a result, the above dilepton
constraint can be evaded. The constraint from di-tau search Aaboud  _et al._
(2018a) is expected to be rather weak Han  _et al._ (2019). Note that the
constraint can also be relaxed by maximizing the ratio
$\Gamma(Z^{\prime}_{BL}\rightarrow\ell^{+}\ell^{-})/\Gamma(Z^{\prime}_{X}\rightarrow\ell^{+}\ell^{-})$
Das  _et al._ (2018a, b, 2019) with $Z^{\prime}_{BL}$ and $Z^{\prime}_{X}$
being the $Z^{\prime}$ gauge bosons under the minimal U$(1)_{\rm B-L}$ model
and the general U$(1)^{\prime}$ model, respectively. Below we take the
U$(1)_{\rm(B-L)_{3}}$ model for illustration although our result below is
generic and applicable to any U$(1)^{\prime}$ model with heavy neutrinos.

###  II.2 Type I Seesaw and heavy neutrino decay

The neutrino Yukawa interactions are

$\displaystyle-\mathcal{L}_{Y}^{I}=Y^{D}_{\nu}\bar{l}_{L}\tilde{H}N_{R}+{1\over
2}Y^{M}_{\nu}\overline{(N^{c})_{L}}N_{R}S+h.c.,$ (5)

where $\tilde{H}=i\sigma_{2}H^{\ast}$ with $H$ being the SM Higgs doublet. The
right-handed neutrino mass matrix is spontaneously generated to be
$M_{R}=Y^{M}_{\nu}v_{S}/\sqrt{2}$. The interaction for $Z^{\prime}$ and heavy
neutrinos is obtained from the kinetic term
$i\overline{N_{R}}\gamma^{\mu}D_{\mu}N_{R}$ with
$D_{\mu}N_{R}=\partial_{\mu}N_{R}-ig^{\prime}B^{\prime}_{\mu}N_{R}$. With the
right-handed neutrino mass matrix $M_{R}$ obtained above, the Seesaw formula
and the mixing between the SM charged leptons and heavy neutrinos in this
model follow the same as those in the canonical Type I Seesaw. After $H$
receives the vev $\langle H\rangle=v/\sqrt{2}$ and the Dirac masses
$m_{D}=Y^{D}_{\nu}v/\sqrt{2}$ appear, one has the diagonalization of the
neutrino mass matrix

$\displaystyle\mathbb{N}^{\dagger}\left(\begin{array}[]{cc}0&m_{D}\\\
m_{D}^{T}&M_{R}\\\ \end{array}\right)\mathbb{N}^{\ast}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}m_{\nu}&0\\\ 0&M_{N}\\\
\end{array}\right),$ (10)

and the eigenstate transformation

$\displaystyle\left(\begin{array}[]{c}\nu_{L}\\\ (N^{c})_{L}\\\
\end{array}\right)=\mathbb{N}\left(\begin{array}[]{c}\nu_{L}\\\ (N^{c})_{L}\\\
\end{array}\right)_{mass},\ \ \ \mathbb{N}=\left(\begin{array}[]{cc}U&V\\\
X&Y\\\ \end{array}\right)\;.$ (17)

With a matrix $E$ diagonalizing the charged lepton mass matrix, we can define

$\displaystyle E^{\dagger}U\equiv U_{PMNS}\;,~{}~{}~{}E^{\dagger}V\equiv
V_{\ell N}\;,$ (18)

where $U_{PMNS}$ is the approximate Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
neutrino mass mixing matrix and the matrix $V_{\ell N}$ transits heavy
neutrinos to charged leptons Atre  _et al._ (2009).

One can derive a relation between diagonalized neutrino mass matrices and
mixing matrices

$\displaystyle U^{\ast}_{PMNS}m_{\nu}^{diag}U^{\dagger}_{PMNS}+V^{\ast}_{\ell
N}M_{N}^{diag}V^{\dagger}_{\ell N}=0\ ,$ (19)

with mass eigenvalues $m_{\nu}^{diag}=(m_{1},m_{2},m_{3})$ 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, and the second term contains the masses and mixing of
the new heavy neutrinos. The general solution of the $V_{\ell N}$ in Eq. (19)
can be parameterized in terms of an arbitrary orthogonal complex matrix
$\Omega$, i.e. the Casas-Ibarra parametrization Casas and Ibarra (2001)

$\displaystyle V_{\ell
N}=U_{PMNS}(m_{\nu}^{diag})^{1/2}\Omega(M_{N}^{diag})^{-1/2},$ (20)

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, we can obtain the partial width of their decay into charged
lepton

$\displaystyle\Gamma(N_{i}\rightarrow\ell^{\pm}W^{\mp})={G_{F}\over
8\sqrt{2}\pi}|V_{\ell
N_{i}}|^{2}M_{N_{i}}(M_{N_{i}}^{2}+2M_{W}^{2})\left(1-{M_{W}^{2}\over
M_{N_{i}}^{2}}\right)^{2},$ (21)

and the asymptotic behavior holds when $M_{N_{i}}\gg M_{W},M_{Z},M_{h}$.

$\displaystyle\Gamma(N_{i}\rightarrow\sum_{\ell}\ell^{\pm}W^{\mp})\approx\Gamma(N_{i}\rightarrow\sum_{\nu}\nu
Z+\bar{\nu}Z)\approx\Gamma(N_{i}\rightarrow\sum_{\nu}\nu h+\bar{\nu}h)\;,$
(22)

where $\ell=e,\mu,\tau$. Note that, more generally, the Dirac mass matrix can
be quite arbitrary with three complex angles parameterizing the orthogonal
matrix $\Omega$ Fileviez Perez  _et al._ (2009); Fileviez Pérez and Plascencia
(2020). The above asymptotic behavior of heavy neutrino decay branching ratios
is consistent with the Goldstone equivalence theorem and indicates
BR$(N_{i}\rightarrow\sum_{\ell}\ell^{\pm}W^{\mp})$ is nearly equal to $25\%$
for any heavy neutrino $N_{i}$. The branching fraction of $N_{i}~{}(i=1,2,3)$
decays into each 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 results of $|V_{\ell N}|^{2}$ are proportional to one and only
one light neutrino mass Fileviez Perez  _et al._ (2009). The branching ratio
of $N_{i}\rightarrow\ell^{\pm}W^{\mp}$ for each lepton flavor is thus
independent of neutrino mass Fileviez Perez  _et al._ (2009). The
representative choice of $\Omega=I$ can also lead to both the leading and sub-
leading decay rate cases in each heavy neutrino decay into certain charged
lepton flavor, so one can distinguish different heavy neutrinos Fileviez Pérez
and Plascencia (2020). 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.

In order to understand 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

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

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 $\Phi_{i}$ the Majorana phases. The size of the
mass-squared splitting between three neutrino states is extracted from
neutrino oscillation experiments. The sign of $\Delta
m_{31}^{2}=m_{3}^{2}-m_{1}^{2}$, however, still remains unknown, which can be
either positive, the Normal Hierarchy (NH), or negative, the Inverted
Hierarchy (IH), for the spectrum of the neutrino masses. Taking into account
the data on atmospheric neutrinos provided by the Super-Kamiokande
collaboration, the latest best global fit results of the neutrino masses and
mixing parameters are as below Esteban  _et al._ (2020); nuf (2020)

$\displaystyle\Delta m_{21}^{2}$ $\displaystyle=$ $\displaystyle 7.42\times
10^{-5}\,{\rm eV}^{2}\;,$ $\displaystyle\Delta m_{31}^{2}$ $\displaystyle=$
$\displaystyle 2.517\times 10^{-3}\,{\rm eV}^{2}\ (\Delta
m_{32}^{2}=-2.498\times 10^{-3}\,{\rm eV}^{2})\;,$
$\displaystyle\sin^{2}{\theta_{12}}$ $\displaystyle=$ $\displaystyle
0.304\;,~{}~{}\sin^{2}{\theta_{23}}=0.573\ (0.575)\;,$
$\displaystyle\sin^{2}{\theta_{13}}$ $\displaystyle=$ $\displaystyle 0.02219\
(0.02238)\;,~{}~{}\delta_{\rm CP}=197^{\circ}(282^{\circ})\;,$ (24)

for NH (IH). In addition, the sum of neutrino masses is constrained by
combining the Planck, WMAP, highL and BAO data Ade  _et al._ (2016) at $95\%$
confidence level (CL) as,

$\sum_{i=1}^{3}m_{i}<\ 0.230\ \,{\rm eV}.$ (25)

By inputting the above experimental constraints to the transiting matrix
$V_{\ell N}$ and the above decay width formulas, we can obtain the preferred
values for heavy neutrino decay patterns below. In our numerical calculations
below, we take the benchmark decay branching ratios of heavy neutrinos with
$\Omega=I$ for NH and IH as shown in Table 1.

BR$(N_{i})$ | $e^{\pm}W^{\mp}$ | $\mu^{\pm}W^{\mp}$ | $\tau^{\pm}W^{\mp}$
---|---|---|---
$N_{1}$ | 17% (17%) | 1.85% (3.8%) | 6.15% (4.2%)
$N_{2}$ | 7.43% (7.43%) | 9.15% (7.14%) | 8.42% (10.43%)
$N_{3}$ | 0.55% (0.56%) | 14% (14.04%) | 10.45% (10.4%)
Table 1: Benchmark decay branching ratios of $N_{i}$ in $\Omega=I$ case for NH
(IH). Here we assume the lowest light neutrino mass as $10^{-4}$ eV and
$M_{N_{i}}\gg M_{W}$.

The total decay widths of heavy Majorana neutrinos are proportional to
$|V_{\ell N}|^{2}\sim m_{\nu}/M_{N}$ which is small and may lead to long decay
length. We calculate the decay length of the Majorna neutrinos by the formula
$L=\frac{1}{\Gamma}\frac{E_{N}}{M_{N}}$ where $\frac{E_{N}}{M_{N}}$ is the
boost factor and $E_{N}$ is roughly considered to be $M_{Z^{\prime}}/2$ in the
rest frame of $Z^{\prime}$. Fig. 1 shows the total width (left axis) and decay
length (right axis) versus $M_{N}$ for $N_{i}~{}(i=1,2,3)$ and $\Omega=I$. We
find that in most of cases the decay length is less than $\mathcal{O}(1)$ mm
and the heavy neutrinos can be viewed as decaying promptly. Only for the cases
of $N_{1}$ in NH and $N_{3}$ in IH, the typical decay length is above 2 mm. In
particular, for 250 GeV$<M_{N}<400$ GeV, the decay length could vary from 10
mm to 30 mm. The decay length is sufficiently long to perform the displaced
vertex searches. This is a clear difference between the NH and IH scenarios as
well as different heavy neutrinos. If this is observed in future, it could
serve as an indication to distinguish the neutrino patterns. Despite of this,
such length is still short compared with the size of inner detector of ATLAS
(1.2 meter in radius) and the prompt decay approximation is still valid. We
leave such interesting possibilities for future studies and in the following
analysis we assume the Majorana neutrinos promptly decay.

Figure 1: The total width (left axis) and decay length (right axis) versus
$M_{N}$ for $N_{i}(i=1,2,3)$ and NH (left) and IH (right) scenarios. We assume
$M_{N}=M_{Z^{\prime}}/4$ and $\Omega=I$.

##  III Searching for heavy neutrinos with tau lepton at hadron colliders

The interesting production at hadron colliders for heavy neutrinos in
U$(1)^{\prime}$ extended Type I Seesaw is the pair production of heavy
neutrinos through $Z^{\prime}$ resonance as shown in Fig. 2. In the
U$(1)_{\rm(B-L)_{3}}$ model, the most appealing channel is

$\displaystyle b\bar{b}\rightarrow Z^{\prime}\rightarrow
NN\rightarrow\ell^{\pm}\tau^{\pm}W^{\mp}W^{\mp}\;,$ (26)

where $\ell=e,\mu$ and the heavy neutrino $N$ can be any one of the mass
eigenstates in Table 1 as they can all involve the third generation coupled to
$Z^{\prime}$. The signal channel we consider includes one electron or muon and
one same-sign hadronically decaying tau lepton in the subsequent decay of $N$.
The same-sign $W$ bosons are required to decay hadronically. The
U$(1)_{\rm(B-L)_{3}}$ model file is produced by FeynRules Alloul  _et al._
(2014) and is interfaced with MadGraph5_aMC@NLO Alwall  _et al._ (2014) to
generate signal events. The major irreducible SM background is from
$t\bar{t}W^{\pm}\rightarrow b\bar{b}W^{\pm}W^{\pm}W^{\mp}$ with the two same-
sign $W$ bosons decaying to one charged lepton $\ell=e,\mu$ and one tau
lepton, the opposite-sign $W$ hadronically decays. The SM backgrounds also
include $t\bar{t}Z\rightarrow b\bar{b}W^{+}W^{-}Z$ and $WZ$+jets with the $Z$
boson leptonic decay. Both the signal and background events are then passed to
Pythia 8 Sjöstrand  _et al._ (2015) and Delphes 3 de Favereau  _et al._ (2014)
for parton shower and detector simulation, respectively.

Figure 2: The pair production of heavy neutrino $N$ through $Z^{\prime}$
resonance. The fermion arrows in final states are only for $\mu^{-}\tau^{-}$
and the charge conjugation should be also accounted.

We select the events with exactly one electron or muon and one same-sign tau
lepton satisfying

$\displaystyle|\eta(\ell)|<2.4,~{}p_{T}(\ell)>25~{}{\rm
GeV};~{}~{}~{}|\eta(\tau)|<2.5,~{}p_{T}(\tau)>25~{}{\rm GeV}\;.$ (27)

The jets from $W$ decay are clustered using the anti-$k_{T}$ algorithm in
FastJet Cacciari  _et al._ (2012). If the jets are well isolated, we
reconstruct each $W$ boson from two resolved jets with $|\eta(j)|<2.4$ and
$p_{T}(j)>25$ GeV. As we consider the decay of heavy neutrino, the $W$ boson
can be boosted and the decay products are likely to merge into a single fat
jet $J$ Das  _et al._ (2020). The fat jet is reconstructed via the
anti-$k_{T}$ algorithm with $R=0.8$. The fat jet with $65<M_{J}<95$ GeV,
$p_{T}(J)>250$ GeV and $|\eta(J)|<2.5$ is identified as boosted $W$ candidate.
If there are more than two boosted $W$ candidates, we sort them by
$|M_{J}-M_{W}|$ and select the first two as those from $N$ decay. If there is
only one boosted $W$ boson, the resolved jets are required to satisfy $\Delta
R(W,j)>0.8$ and the other $W$ boson candidate is reconstructed by choosing the
pair of jets which minimizes $|M_{jj}-M_{W}|$. The di-jet invariant mass
should satisfy $50<M_{jj}<110$ GeV in order to be identified as a $W$ boson.
If there is no boosted $W$ boson, there should be at least two pairs of
resolved jets to reconstruct the two $W$ bosons. We sort their combinations by
$|M_{jj}-M_{W}|$ and select the first two to construct $W$ bosons. Given the
same-sign $\ell$ and $\tau$ and the two reconstructed $W$ bosons, we find the
combination that minimizes $\Delta R(\ell,W_{1})+\Delta R(\tau,W_{2})$ and
then construct $N^{\ell}$ and $N^{\tau}$ respectively denoting the heavy
neutrino decay into $\ell$ and $\tau$. In Fig. 3, assuming $M_{Z^{\prime}}=2$
TeV and $M_{N}=M_{Z^{\prime}}/4$, the distributions of $p_{T}$ and invariant
mass are respectively shown for the two reconstructed $N$. We can find that
$M_{N^{\mu}}$ peaks at 500 GeV which is very close to the true value, while
the reconstruction of $N$ from $(\tau,W)$ is not as good as that from
$(\mu,W)$. As a result, in the final selection, we only apply the minimal
$p_{T}$ cut and the mass window for $N^{\mu}$

$\displaystyle p_{T}(N^{\mu})>250~{}{\rm GeV},~{}~{}p_{T}(N^{\tau})>200~{}{\rm
GeV};~{}~{}~{}|M_{N^{\mu}}-M_{N}|<0.1M_{N}\;.$ (28)

The cuts for $M_{N^{\tau}}$ and the reconstructed $Z^{\prime}$ are not
applied.

Figure 3: Normalized distributions of $p_{T}$ (top) and invariant mass
(bottom) for reconstructed $N^{\mu}$ (left) and $N^{\tau}$ (right). We assume
$M_{Z^{\prime}}=2$ TeV and $M_{N}=M_{Z^{\prime}}/4$ for illustration.

Given the characteristic features of the signal and backgrounds discussed
above, we can use the Gaussian method to calculate the significance

$\displaystyle S/\sqrt{B},$ (29)

where $S$ and $B$ are the signal and background event expectations,
respectively. With the proper branching fractions in Table 1, the discovered
signal events for this channel read as

$\displaystyle S=L\times\sigma_{0}\times 2\times{\rm
BR}^{2}(W^{\pm}\rightarrow q\bar{q}^{\prime})\times\epsilon^{\tau}_{NN},$ (30)

where $\sigma_{0}\equiv\sigma(pp\rightarrow Z^{\prime}\rightarrow
NN)\times{\rm BR}(N\rightarrow\tau^{\pm}W^{\mp})\times{\rm
BR}(N\rightarrow\ell^{\pm}W^{\mp})$, $L$ is the integrated luminosity and the
factor of 2 accounts for the charge-conjugation of final states. The
$\epsilon^{\tau}_{NN}$ denotes the selection efficiency for our signal with
one tau lepton. The gray and black curves in Fig. 4 (a) respectively
correspond to the sensitivities of $\sigma(pp\rightarrow Z^{\prime}\rightarrow
NN\rightarrow\mu^{\pm}\tau^{\pm}W^{\mp}W^{\mp})=2\sigma_{0}$ for 5$\sigma$
discovery and 95% C.L. exclusion, assuming an expected luminosity of 3 ab-1 at
14 TeV LHC and $M_{N}=M_{Z^{\prime}}/4$. These sensitivities are applicable to
the search for heavy neutrino pair production in any U$(1)^{\prime}$ extension
model. Given the realistic decay branching fractions of $N$ in Table 1, for
our U$(1)_{\rm(B-L)_{3}}$ model, we find that HL-LHC can produce at most 20
LNV events for $M_{Z^{\prime}}=1$ TeV and cannot reach 5$\sigma$ discovery as
shown in Fig. 4 (a). We then assess the discovery potential of future circular
hadron collider. The recently released Delphes 3.4.2 includes the beta card
for FCC studies. Figs. 4 (b,c) show the projected sensitivities of the
production cross section as a function of $M_{Z^{\prime}}$, with the
integrated luminosity of 3 ab-1 (bottom left) and 30 ab-1 (bottom right) at
$\sqrt{s}=100$ TeV. Compared with the predicted cross sections by our
U$(1)_{\rm(B-L)_{3}}$ model, one can see that the FCC-hh can discover the LNV
signal with tau lepton for $M_{Z^{\prime}}$ up to 2.2 (3) TeV and $N=N_{3}$
with the fixed $g^{\prime}=0.6$ and the integrated luminosity of 3 (30) ab-1.
The projected sensitivities are highly dependent on the benchmark decay
branching fractions for each heavy neutrino but not quite on the neutrino mass
patterns. The test on the flavor combinations of SM charged leptons would
reveal the specific nature of different heavy neutrinos. In Fig. 5 the 95%
C.L. exclusion obtained using our search methodology is depicted on the
parameter space of $g^{\prime}$ versus $M_{Z^{\prime}}$ assuming
$M_{N}=M_{Z^{\prime}}/4$. The 95% C.L. limit from the search
$Z^{\prime}\rightarrow\tau\tau$ Aaboud  _et al._ (2018a) is also shown and one
can see the gauge coupling $g^{\prime}$ greater than 0.35 has been excluded
for $M_{Z^{\prime}}=1$ TeV. The searches for $Z^{\prime}\rightarrow b\bar{b}$
Aaboud  _et al._ (2018b) or $Z^{\prime}\rightarrow t\bar{t}$ Aad  _et al._
(2020) do not place severe constraint due to the suppression by the baryon
number. The FCC-hh with the integrated luminosity of 3 (30) ab-1 can probe the
LNV signature with tau lepton for the gauge coupling $g^{\prime}$ down to 0.12
(0.07) with $M_{Z^{\prime}}=1$ TeV and $N=N_{3}$.

In Fig. 5 we also interpolate the current bounds of
$Z^{\prime}\rightarrow\tau^{+}\tau^{-}$ Aaboud  _et al._ (2018a),
$Z^{\prime}\rightarrow b\bar{b}$ Aaboud  _et al._ (2018b) and
$Z^{\prime}\rightarrow t\bar{t}$ Aad  _et al._ (2020) and estimate their
exclusions at future FCC-hh. One can see that the most optimistic channel to
directly observe $Z^{\prime}$ is from the
$Z^{\prime}\rightarrow\tau^{+}\tau^{-}$ channel. However, in the U(1)B-L
model, the right-handed neutrinos are needed to cancel the gauge anomalies and
the search for LNV signal is important to confirm the existence of the right-
handed neutrinos and explain the neutrino masses. Suppose the $Z^{\prime}$
resonance can be discovered through the di-tau channel in future, the search
for the pair production of heavy neutrinos we explored would provide crucial
information for the LNV nature and mass pattern of the heavy neutrinos.

(a)

(b)

(c)

Figure 4: Projected sensitivity of cross section $\sigma(pp\rightarrow
Z^{\prime}\rightarrow NN\rightarrow\mu^{\pm}\tau^{\pm}W^{\mp}W^{\mp})$ (black
and gray curves) at $\sqrt{s}=14$ TeV (top) and $\sqrt{s}=100$ TeV (bottom)
for the benchmark model with $M_{N}=M_{Z^{\prime}}/4$. The red, green and blue
solid (dashed) curves show the predicted cross sections for
$N=N_{1},N_{2},N_{3}$, respectively, assuming $g^{\prime}=0.6$ and NH (IH)
neutrino mass spectrum.

Figure 5: Projected sensitivity of the gauge coupling $g^{\prime}$ versus
$M_{Z^{\prime}}$ with the integrated luminosity of 3 ab-1 (left) and 30 ab-1
(right) at $\sqrt{s}=100$ TeV for the benchmark model with
$M_{N}=M_{Z^{\prime}}/4$. The red, green and blue solid (dashed) curves
correspond to $N=N_{1},N_{2},N_{3}$, respectively, assuming NH (IH) neutrino
mass spectrum. The gray curve shows the 95% C.L. limit from the search for
$Z^{\prime}\rightarrow\tau\tau$ at LHC Aaboud  _et al._ (2018a). The estimated
exclusions from $Z^{\prime}\rightarrow\tau\tau$ (cyan), $Z^{\prime}\rightarrow
b\bar{b}$ (brown) and $Z^{\prime}\rightarrow t\bar{t}$ (purple) at FCC-hh are
also shown.

##  IV Conclusions

Low energy neutrino oscillation data lead to correlation with the flavor
structure of the heavy neutrino decays and tau lepton plays manifest role in
such correlation. Due to the low tau identification efficiencies, future
colliders with high energy and luminosity enables us to investigate and search
for heavy neutrino decaying to tau lepton. Moreover, the decoupled lepton
number violation in high energy process for low-scale Type I Seesaw encourages
us to consider the extension of the canonical Type I Seesaw. We thus examine
the lepton flavor signatures with tau lepton at FCC-hh, through lepton number
violating processes in the U$(1)_{\rm B-L}$ extended Type I Seesaw. In the
U$(1)_{\rm B-L}$ extended Type I Seesaw, we consider a flavored
U$(1)_{\rm(B-L)_{3}}$ model where only the third generation fermions couple to
the new gauge boson $Z^{\prime}$ and thus the constraint on the new gauge
interaction is much weak. We investigate the impact of up-to-date neutrino
oscillation results on neutrino mass models and consequently examine the
lepton flavor signatures of heavy neutrinos in the Type I Seesaw.

We study the pair production of heavy neutrinos via $Z^{\prime}$ mediation in
the U$(1)_{\rm(B-L)_{3}}$ model, with one tau lepton in the subsequent decays:
$pp\rightarrow Z^{\prime}\rightarrow
NN\rightarrow\ell^{\pm}\tau^{\pm}W^{\mp}W^{\mp}\rightarrow\ell^{\pm}\tau^{\pm}+{\rm
jets}$. The $\tau$ lepton’s hadronic decay is assumed in the above channels.
We take into account the realistic decay branching ratios of heavy neutrinos
determined by neutrino oscillation measurements. The general sensitivities of
LNV production cross sections are shown for 5$\sigma$ discovery and 95% C.L.
exclusion. We find that HL-LHC cannot reach 5$\sigma$ discovery for heavy
neutrinos in the U$(1)_{\rm(B-L)_{3}}$ model. The FCC-hh can discover the LNV
signal with tau lepton for $M_{Z^{\prime}}$ up to 2.2 (3) TeV with
$g^{\prime}=0.6$ and the integrated luminosity of 3 (30) ab-1. The gauge
coupling $g^{\prime}$ down to 0.12 (0.07) can be probed through the LNV
signature with tau lepton by the FCC-hh with the integrated luminosity of 3
(30) ab-1 and $M_{Z^{\prime}}=1$ TeV. The test on the flavor combinations of
SM charged leptons would reveal the specific nature of different heavy
neutrinos and the three heavy neutrinos in the Casas-Ibarra parametrization
with diagonal unity matrix can be distinguished.

###### Acknowledgements.

TL 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). CYY is supported in
part by the Grants No. NSFC-11975130, No. NSFC-12035008, No. NSFC-12047533, by
the National Key Research and Development Program of China under Grant No.
2017YFA0402200 and the China Post-doctoral Science Foundation under Grant No.
2018M641621. CH acknowledges support from the Sun Yat-Sen University Science
Foundation.

## References

  * Weinberg (1979) S. Weinberg, Phys. Rev. Lett. 43, 1566 (1979).
  * Ma (1998) E. Ma, Phys. Rev. Lett. 81, 1171 (1998), arXiv:hep-ph/9805219 .
  * Minkowski (1977) P. Minkowski, Phys. Lett. B 67, 421 (1977).
  * Yanagida (1979) T. Yanagida, Conf. Proc. C 7902131, 95 (1979).
  * Gell-Mann  _et al._ (1979) M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C 790927, 315 (1979), arXiv:1306.4669 [hep-th] .
  * Glashow (1980) S. Glashow, NATO Sci. Ser. B 61, 687 (1980).
  * Mohapatra and Senjanovic (1980) R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
  * Shrock (1981) R. E. Shrock, Phys. Rev. D 24, 1232 (1981).
  * Schechter and Valle (1980) J. Schechter and J. Valle, Phys. Rev. D 22, 2227 (1980).
  * Davidson and Wali (1987) A. Davidson and K. C. Wali, Phys. Rev. Lett. 59, 393 (1987).
  * Cai  _et al._ (2018) Y. Cai, T. Han, T. Li, and R. Ruiz, Front. in Phys. 6, 40 (2018), arXiv:1711.02180 [hep-ph] .
  * Aad  _et al._ (2015a) G. Aad _et al._ (ATLAS), JHEP 07, 162 (2015a), arXiv:1506.06020 [hep-ex] .
  * Aad  _et al._ (2015b) G. Aad _et al._ (ATLAS), Phys. Rev. D 92, 032001 (2015b), arXiv:1506.01839 [hep-ex] .
  * Khachatryan  _et al._ (2015) V. Khachatryan _et al._ (CMS), Phys. Lett. B 748, 144 (2015), arXiv:1501.05566 [hep-ex] .
  * Khachatryan  _et al._ (2016) V. Khachatryan _et al._ (CMS), JHEP 04, 169 (2016), arXiv:1603.02248 [hep-ex] .
  * Sirunyan  _et al._ (2018) A. M. Sirunyan _et al._ (CMS), Phys. Rev. Lett. 120, 221801 (2018), arXiv:1802.02965 [hep-ex] .
  * Sirunyan  _et al._ (2017) A. M. Sirunyan _et al._ (CMS), JHEP 07, 121 (2017), arXiv:1703.03995 [hep-ex] .
  * Sirunyan  _et al._ (2019) A. M. Sirunyan _et al._ (CMS), JHEP 03, 170 (2019), arXiv:1811.00806 [hep-ex] .
  * Fileviez Perez  _et al._ (2008a) P. Fileviez Perez, T. Han, G.-Y. Huang, T. Li, and K. Wang, Phys. Rev. D 78, 071301 (2008a), arXiv:0803.3450 [hep-ph] .
  * Fileviez Perez  _et al._ (2008b) P. Fileviez Perez, T. Han, G.-y. Huang, T. Li, and K. Wang, Phys. Rev. D 78, 015018 (2008b), arXiv:0805.3536 [hep-ph] .
  * Li and He (2009) T. Li and X.-G. He, Phys. Rev. D80, 093003 (2009), arXiv:0907.4193 [hep-ph] .
  * Li (2018) T. Li, JHEP 09, 079 (2018), arXiv:1802.00945 [hep-ph] .
  * Pascoli  _et al._ (2018) S. Pascoli, R. Ruiz, and C. Weiland, Phys. Lett. B 786, 106 (2018), arXiv:1805.09335 [hep-ph] .
  * Cid Vidal  _et al._ (2019) X. Cid Vidal _et al._ , CERN Yellow Rep. Monogr. 7, 585 (2019), arXiv:1812.07831 [hep-ph] .
  * Pascoli  _et al._ (2019) S. Pascoli, R. Ruiz, and C. Weiland, JHEP 06, 049 (2019), arXiv:1812.08750 [hep-ph] .
  * Cheung  _et al._ (2020) K. Cheung, Y.-L. Chung, H. Ishida, and C.-T. Lu, Phys. Rev. D 102, 075038 (2020), arXiv:2004.11537 [hep-ph] .
  * Ashanujjaman and Ghosh (2021) S. Ashanujjaman and K. Ghosh, (2021), arXiv:2102.09536 [hep-ph] .
  * Das and Mandal (2021) A. Das and S. Mandal, Nucl. Phys. B 966, 115374 (2021), arXiv:2006.04123 [hep-ph] .
  * Abe  _et al._ (2012) Y. Abe _et al._ (Double Chooz), Phys. Rev. Lett. 108, 131801 (2012), arXiv:1112.6353 [hep-ex] .
  * Ahn  _et al._ (2012) J. K. Ahn _et al._ (RENO), Phys. Rev. Lett. 108, 191802 (2012), arXiv:1204.0626 [hep-ex] .
  * An  _et al._ (2012) F. P. An _et al._ (Daya Bay), Phys. Rev. Lett. 108, 171803 (2012), arXiv:1203.1669 [hep-ex] .
  * Abe  _et al._ (2014) K. Abe _et al._ (T2K), Phys. Rev. Lett. 112, 061802 (2014), arXiv:1311.4750 [hep-ex] .
  * Adamson  _et al._ (2016) P. Adamson _et al._ (NOvA), Phys. Rev. Lett. 116, 151806 (2016), arXiv:1601.05022 [hep-ex] .
  * Abe  _et al._ (2017) K. Abe _et al._ (T2K), Phys. Rev. Lett. 118, 151801 (2017), arXiv:1701.00432 [hep-ex] .
  * Kersten and Smirnov (2007) J. Kersten and A. Yu. Smirnov, Phys. Rev. D76, 073005 (2007), arXiv:0705.3221 [hep-ph] .
  * Moffat  _et al._ (2017) K. Moffat, S. Pascoli, and C. Weiland, (2017), arXiv:1712.07611 [hep-ph] .
  * Aaboud  _et al._ (2017) M. Aaboud _et al._ (ATLAS), JHEP 10, 182 (2017), arXiv:1707.02424 [hep-ex] .
  * Alonso  _et al._ (2017) R. Alonso, P. Cox, C. Han, and T. T. Yanagida, Phys. Lett. B 774, 643 (2017), arXiv:1705.03858 [hep-ph] .
  * Cox  _et al._ (2018) P. Cox, C. Han, and T. T. Yanagida, JHEP 01, 037 (2018), arXiv:1707.04532 [hep-ph] .
  * Fileviez Perez  _et al._ (2009) P. Fileviez Perez, T. Han, and T. Li, Phys. Rev. D 80, 073015 (2009), arXiv:0907.4186 [hep-ph] .
  * Chiang  _et al._ (2019) C.-W. Chiang, G. Cottin, A. Das, and S. Mandal, JHEP 12, 070 (2019), arXiv:1908.09838 [hep-ph] .
  * Appelquist  _et al._ (2003) T. Appelquist, B. A. Dobrescu, and A. R. Hopper, Phys. Rev. D 68, 035012 (2003), arXiv:hep-ph/0212073 .
  * Davidson (1979) A. Davidson, Phys. Rev. D 20, 776 (1979).
  * Mohapatra and Marshak (1980) R. N. Mohapatra and R. Marshak, Phys. Rev. Lett. 44, 1316 (1980), [Erratum: Phys.Rev.Lett. 44, 1643 (1980)].
  * Marshak and Mohapatra (1980) R. Marshak and R. N. Mohapatra, Phys. Lett. B 91, 222 (1980).
  * Wetterich (1981) C. Wetterich, Nucl. Phys. B 187, 343 (1981).
  * Masiero  _et al._ (1982) A. Masiero, J. Nieves, and T. Yanagida, Phys. Lett. B 116, 11 (1982).
  * Mohapatra and Senjanovic (1983) R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 27, 254 (1983).
  * Buchmuller  _et al._ (1991) W. Buchmuller, C. Greub, and P. Minkowski, Phys. Lett. B 267, 395 (1991).
  * Carena  _et al._ (2004) M. Carena, A. Daleo, B. A. Dobrescu, and T. M. Tait, Phys. Rev. D 70, 093009 (2004), arXiv:hep-ph/0408098 .
  * Aad  _et al._ (2019) G. Aad _et al._ (ATLAS), Phys. Lett. B 796, 68 (2019), arXiv:1903.06248 [hep-ex] .
  * Kownacki  _et al._ (2017) C. Kownacki, E. Ma, N. Pollard, and M. Zakeri, Phys. Lett. B 766, 149 (2017), arXiv:1611.05017 [hep-ph] .
  * Aaboud  _et al._ (2018a) M. Aaboud _et al._ (ATLAS), JHEP 01, 055 (2018a), arXiv:1709.07242 [hep-ex] .
  * Han  _et al._ (2019) T. Han, J. Liao, H. Liu, and D. Marfatia, JHEP 11, 028 (2019), arXiv:1910.03272 [hep-ph] .
  * Das  _et al._ (2018a) A. Das, N. Okada, and D. Raut, Phys. Rev. D 97, 115023 (2018a), arXiv:1710.03377 [hep-ph] .
  * Das  _et al._ (2018b) A. Das, N. Okada, and D. Raut, Eur. Phys. J. C 78, 696 (2018b), arXiv:1711.09896 [hep-ph] .
  * Das  _et al._ (2019) A. Das, P. S. B. Dev, and N. Okada, Phys. Lett. B 799, 135052 (2019), arXiv:1906.04132 [hep-ph] .
  * Atre  _et al._ (2009) A. Atre, T. Han, S. Pascoli, and B. Zhang, JHEP 05, 030 (2009), arXiv:0901.3589 [hep-ph] .
  * Casas and Ibarra (2001) J. Casas and A. Ibarra, Nucl. Phys. B 618, 171 (2001), arXiv:hep-ph/0103065 .
  * Fileviez Pérez and Plascencia (2020) P. Fileviez Pérez and A. D. Plascencia, Phys. Rev. D 102, 015010 (2020), arXiv:2005.04235 [hep-ph] .
  * Esteban  _et al._ (2020) I. Esteban, M. Gonzalez-Garcia, M. Maltoni, T. Schwetz, and A. Zhou, JHEP 09, 178 (2020), arXiv:2007.14792 [hep-ph] .
  * nuf (2020) NuFIT 5.0, www.nu-fit.org (2020).
  * Ade  _et al._ (2016) P. Ade _et al._ (Planck), Astron. Astrophys. 594, A13 (2016), arXiv:1502.01589 [astro-ph.CO] .
  * Alloul  _et al._ (2014) A. Alloul, N. D. Christensen, C. Degrande, C. Duhr, and B. Fuks, Comput. Phys. Commun. 185, 2250 (2014), arXiv:1310.1921 [hep-ph] .
  * Alwall  _et al._ (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), arXiv:1405.0301 [hep-ph] .
  * Sjöstrand  _et al._ (2015) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, and P. Z. Skands, Comput. Phys. Commun. 191, 159 (2015), arXiv:1410.3012 [hep-ph] .
  * de Favereau  _et al._ (2014) J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3), JHEP 02, 057 (2014), arXiv:1307.6346 [hep-ex] .
  * Cacciari  _et al._ (2012) M. Cacciari, G. P. Salam, and G. Soyez, Eur. Phys. J. C 72, 1896 (2012), arXiv:1111.6097 [hep-ph] .
  * Das  _et al._ (2020) A. Das, S. Mandal, and T. Modak, Phys. Rev. D 102, 033001 (2020), arXiv:2005.02267 [hep-ph] .
  * Aaboud  _et al._ (2018b) M. Aaboud _et al._ (ATLAS), Phys. Rev. D 98, 032016 (2018b), arXiv:1805.09299 [hep-ex] .
  * Aad  _et al._ (2020) G. Aad _et al._ (ATLAS), JHEP 10, 061 (2020), arXiv:2005.05138 [hep-ex] .