# Development of Transient $\mu$SR Method for High-Flux Pulsed Muons

Shoichiro Nishimura<EMAIL_ADDRESS>Hirotaka Okabe  Masatoshi
Hiraishi  Masanori Miyazaki  Jumpei G. Nakamura  Akihiro Koda  Ryosuke Kadono
<EMAIL_ADDRESS>Muon Science Laboratory, Institute of Materials
Structure Science, High Energy Accelerator Research Organization (KEK),
Tsukuba, Ibaraki 305-0801, Japan  Materials and Life Science Division, J-PARC
Center, Tokai, Naka, Ibaraki 319-1106, Japan  Institute for Materials
Research, Tohoku University, Sendai 980-8577, Japan  Graduate School of
Science and Engineering, Ibaraki University, Mito, Ibaraki, 310-8512, Japan
Muroran Institute of Technology, Muroran, Hokkaido 050-8585, Japan  Department
of Materials Structure Science, The Graduate University for Advanced Studies
(Sokendai), Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

In order to expand the applicability of muon spin rotation, relaxation, and
resonance ($\mu$SR) experiments with pulsed muons and to make effective use of
the high-flux beam, we have developed a new experimental method “transient
$\mu$SR”. In this method, $\mu$SR data for each muon pulse are tagged with
external parameters such as temperature and magnetic field in real time,
allowing the sample environment to be changed without interrupting data
collection. As a result, continuous $\mu$SR measurements under sample
conditions that vary on a time scale longer than the beam pulse interval (40
ms) are realized, and efficient beam utilization is achieved by eliminating
the lag time associated with moving between discrete measurement points. The
transient $\mu$SR method was applied to the study of the magnetic properties
of cupric oxide (CuO) and the observation of level-crossing resonance
relaxation in copper, and the method was successfully established by
confirming that the results reproduce those in the previous reports.

###### keywords:

, Muon Spin Rotation, Relaxation, and Resonance , Pulsed High-flux Beam ,
Transient Phenomena

††journal: Nuclear Instruments & Methods in Physics Research A

##  1 Introduction

In recent years, the muon beam intensity provided by the Muon Science
Experimental Facility (MUSE) of the Materials and Life Science Experimental
Facility (MLF) at the Japan Proton Accelerator Research Complex (J-PARC) has
become strong enough to be called high flux ($>$105 $\mu^{+}$ s-1cm-2), and
the data collection time for a single time spectrum in $\mu$SR experiments has
become increasingly short. On the other hand, in conventional $\mu$SR
experiments, measurements are performed by setting up discrete measurement
points with different conditions such as temperature and magnetic field, and
the time required to move to each measurement point with different conditions
often greatly exceeds the data collection time, thus slowing down the overall
experiment time. In particular, rapidly changing temperature and controlling
its disturbances are often difficult and limit the efficiency of experiments;
unexpected fluctuation during data collection can interrupt and delay
measurements (or reduce data reliability). We have developed a next-generation
data acquisition and analysis method to fundamentally solve these difficulties
and take advantage of the high-flux beam.

This method integrates $\mu$SR data with information on sample conditions such
as temperature and magnetic field in real time and sorts them into a multi-
dimensional histogram of positron events for each muon pulse. This enables
continuous $\mu$SR measurements under varying sample conditions on a time
scale longer than the beam pulse interval (40 ms) and efficient beam
utilization is achieved by eliminating the lag time associated with moving
between discrete measurement points in conventional $\mu$SR experiments. It
also opens up the possibility of studying transient phenomena that were
inaccessible with conventional $\mu$SR. However, for such a method to be
practical, it is essential that the beam intensity is high enough to ensure
sufficiently good statistics for positron events in the time region of
interest. In this respect, the high-flux pulsed muon beam at the MUSE is most
suitable for transient $\mu$SR.

Historically, numerous examples are known of attempts to utilize the periodic
time structure of an incident beam to efficiently collect data under a time-
varying sample environment while minimizing systematic errors. To our
knowledge, the oldest example in $\mu$SR is the “stroboscopic technique”,
which performs high-precision spin rotation frequency measurements by
synchronizing the data acquisition (DAQ) with the micro-bunch structure of the
muon beam reflecting the radio frequency (RF) for the proton acceleration in
the cyclotron [1, 2]. Subsequently, the method called “red-green mode” has
been developed in synchrotron-based muon facilities, where data are collected
while irradiating RF/flash lamp/laser pulses or applying a pulsed magnetic
field synchronized with the pulsed beam [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
Recently, the red-green mode has been extended in combination with “event
mode” DAQ, where up to 14 bits (=16384) of different conditions can be pre-
programmed for each pulse, enabling the collection and analysis of $\mu$SR and
neutron data [14, 15, 16, 17]. However, the limitation of these measurements
is that they can only be applied when the sample environment can be controlled
synchronously with the DAQ. The newly developed transient $\mu$SR is novel in
that it removes this limitation and can respond to changes in the sample
environment that are asynchronous to muon pulse and/or DAQ.

In this paper, we detail the principle of the transient $\mu$SR method and
report the results of its application to the study of the magnetic properties
of CuO and the observation of muon level-crossing resonance ($\mu$LCR)
relaxation in Cu to evaluate its effectiveness.

Figure 1: Data acquisition (DAQ) and event analysis flow-chart of the
transient $\mu$SR. The PC’s for DAQ and sample condition monitoring (SCM) are
synchronized by an NTP server that synchronizes the internal clocks. The SCM
information is captured in real time by the DAQ PC and recorded/stored with
positron event information for each beam pulse. The event-analysis software is
invoked for the sorting/histogramming of integrated data.

##  2 Transient $\mu$SR Measurements

###  2.1 Data structure

The purpose of the $\mu$SR experiment is to determine the time evolution of
the spin polarization [$G(t)$] exhibited by muons stopped in a sample. This is
achieved by monitoring the time-dependent forward-backward asymmetry [$A(t)$]
of the positrons emitted from muons with detectors (hodoscopes) and extracting
it using the following relation [18, 19],

$A(t)=A_{0}G(t)=\frac{\alpha N_{+}(t)-N_{-}(t)}{\alpha N_{+}(t)+N_{-}(t)}\>,$
(1)

where $A_{0}$ is the average asymmetry,

$N_{\pm}(t)=N_{0\pm}e^{-t/\tau_{\mu}}[1\pm A_{\pm}G(t)]$

is the time-dependent positron events summed up for the forward ($+$) or
backward ($-$) detectors, $\tau_{\mu}$ is the muon decay lifetime
($=2.197\times 10^{-6}$ s), $A_{\pm}$ is the decay positron asymmetry for the
detector in question ($A_{\pm}\simeq A_{0}$), and $\alpha$ is the instrumental
asymmetry [$\alpha=N_{-}(0)/N_{+}(0)$].

In conventional $\mu$SR experiments, measurements are made under constant
sample conditions such as temperature ($T$), magnetic field ($B$), etc., and
information on the local magnetic field in the sample is obtained by analyzing
the time spectrum obtained by the Eq. (1). The new method monitors these
sample conditions in real time and acquires the data as multidimensional
histograms, namely,

$\tilde{A}(t;T,B,...)=\frac{\alpha\tilde{N}_{+}(t)-\tilde{N}_{-}(t)}{\alpha\tilde{N}_{+}(t)+\tilde{N}_{-}(t)}\>,$
(2)

where

$\tilde{N}_{\pm}(t)=N_{0\pm}e^{-t/\tau_{\mu}}[1\pm
A_{\pm}\tilde{G}(t;T,B,...)].$

Since $\tilde{G}(t;T,B,...)$ also contains information related to the physical
properties of the sample, $\tilde{A}(t)$ is necessarily a measurement quantity
with large dimensions.

###  2.2 Data Acquisition and Analysis

As illustrated in Fig. 1, integration of $\mu$SR data and information on
sample conditions ($T$, $B$, etc.) was achieved by synchronizing the internal
clock of the DAQ-PC and another PC for monitoring these conditions using an
NTP (Network Time Protocol) server. The integrated data were recorded pulse by
pulse, and off-line analysis was performed to extract the data in units of
beam pulses by imposing necessary conditions for data selection, and organize
them as a multidimensional histogram $\tilde{A}(t)$ that includes information
on the sample environment via the Trees in the ROOT data structure. In the
following, this first stage of analysis will be referred to as “event
analysis”.

The event-analysis software developed for the transient $\mu$SR is based on
ROOT (version 6.15/01 or later) [20], and requires an operating system with
the ability to run bash shell scripts. The analysis is performed in three
stages, i.e., the data integration, histogram generation, and drawing parts.
For the sake of speed, a simple confirmation-drawing mode is provided before
creating the histogram. To provide an idea for the capability of the event-
analysis software, snapshots of histogram manipulation using the graphic user
interface (GUI) are shown in Fig. 2 for the CuO data. In the drawing part, by
moving the cursor in the GUI, the time-dependent asymmetry can be displayed by
carving the temperature or magnetic field into slices of arbitrary bin size.
In addition, the time-integrated asymmetry can be plotted as a function of the
temperature or magnetic field [see Fig. 3(a)]. These features are useful in
understanding the global behavior of a large number of spectra.

Figure 2: (a) Multi-dimensional histogram [$\tilde{A}(t)$] generated by event-
analysis software, showing the temperature and time on the vertical and
horizontal axes, respectively, and the colors representing the magnitude of
the asymmetry. (b), (c), and (d) are snapshots showing projections of the
black line in (a) on the horizontal axis; these plots can be created by just
moving the cursor.

The ultimate goal of the $\mu$SR experiment is to extract information on
physical properties of target materials (magnetism, superconductivity, etc.)
based on a physical model from $\tilde{A}(t)$ [$\tilde{G}(t;T,B,...)$], and to
distinguish it from the event analysis, such analysis will be called “physical
analysis” in the following. Although it is attractive to perform physical
analysis on $\tilde{A}(t)$ by “manifold (multi-dimensional surface) fitting”
directly on ROOT, it is yet to be implemented. Therefore, we will show two
examples of analyzing $\tilde{A}(t)$ sliced for a sample condition of
interest, using the batch processing function (chain fitting) of existing
curve fitting programs. For the data slicing, the event-analysis software is
equipped with a one-click function to export spectra in the data format
handled by musrfit [21] or WiMDA [22]. Since the data slicing is such that
each sample condition parameter (e.g., temperature, magnetic field, etc.) is
orthogonal to each other, the potential complexity of error propagation is
reduced to that within each histogram via the least-squares fits. The error
propagation associated with direct analysis of multidimensional histograms,
including the case of slicing in which sample condition parameter varies in a
mixed manner, is a future issue to be incorporated with the development of the
physical models with multidimensional parameters.

###  2.3 $\mu$SR Experiment

The general-purpose $\mu$SR spectrometer, ARTEMIS, installed at the S1 area in
MLF, J-PARC was used for the development of transient $\mu$SR technique; it
accepts a surface muon beam with a total flux of $4.6\times 10^{5}$ muon/s
(double pulse mode) at the current proton beam operation of 0.8 MW and a
repetition rate of 25 Hz [23]. ARTEMIS has two sets of positron hodoscopes
consisting of 20 module detectors, KALLIOPE, arranged along the forward and
backward positions relative to the sample; each KALLIOPE module harnesses 32
plastic scintillators ($12\ {\rm mm}\times 10\ {\rm mm}\times 10\ {\rm mm}$)
with silicon photomultipliers, making up 16 “telescopes” (pairs of
scintillators to identify the position trajectory), and signals are processed
via on-chip readout circuits and Ethernet interfaces (SiTCP), through which
data from 1280 scintillators in total are transferred to the data-acquisition
(DAQ) PC [24].

For the test measurements, CuO (99.99% powder, supplied by Furuuchi Chemical
Co.) and high-purity copper (99.9999% plate, $50\times 50\times 1$ mm3,
supplied by Nilaco Co.) were adopted for temperature and field sweep,
respectively. CuO powder was put into a slab with $25\times 25\times\sim$1 mm3
on a silver plate. The samples were loaded to a He gas-flow cryostat furnished
with the “fly-past” vacuum vessel to eliminate backgrounds from muons stopped
outside the sample. ARTEMIS is also equipped with normal-conducting magnets,
which can apply longitudinal and transverse magnetic fields of up to 0.4 T and
12.5 mT, respectively.

##  3 Examples of Transient $\mu$SR

###  3.1 Temperature sweep: CuO

CuO is an antiferromagnet showing two magnetic transitions, a second-order
transition at $T_{N1}\simeq 230$ K and a first-order transition at
$T_{N2}\simeq 213$–215 K. In the intermediate temperature region between
$T_{N1}$–$T_{N2}$, CuO is considered to exhibit an incommensurate magnetic
ordered phase [25, 26, 27]. Therefore, we tested how accurately we could
capture these successive phase transitions with our new method. The
temperature sweep range was 193 K to 250 K, the sweep rate was 0.5 K/min, and
the proton beam power was about 0.51 MW at the time of measurement. A record
of the temperature change during the measurement is shown in Fig. 3(b). The
spontaneous muon precession frequency due to the internal magnetic field
changes significantly on passing through $T_{N2}$ from commensurate to
incommensurate magnetic ordered phases with increasing temperature, where the
actual time-differential $\mu$SR spectrum changes from partial oscillation
($\propto\frac{2}{3}\cos\omega_{\mu}t+\frac{1}{3}$, whose time-integral is
$\frac{1}{3}$) to that approximated by the Gaussian Kubo-Toyabe relaxation
with relatively large fluctuation rate (which leads to a reduction of the 1/3
component, see below). Therefore, we can observe the reduction of integrated-
asymmetry for $T>T_{N2}$ as shown in Fig. 3(a).

Figure 3: (a) Time-integrated asymmetry (red, blue, and black dots correspond
to heating, cooling, and their mean, respectively), (b) real-time sample
temperature monitored during the transient $\mu$SR measurement. Insets: (a)
the enlarged plots from around 227 K to 232 K and from 211 K to 217 K.

In general, if the heat conduction between the sample and the sample holder is
not sufficient, there will be a difference between the temperature indicated
by the thermometer on the holder and the actual sample temperature. In this
measurement, the fact that the sample was a powder may have also made
temperature control by thermal conduction difficult. However, even in such a
case, the temperature difference could be made sufficiently small by slowing
down the temperature change sufficiently. This was confirmed by the fact that
there was no difference in the time spectrum between heating and cooling, as
shown in Fig. 2(b)–(d).

For effective physical analysis, it is useful to know in advance the general
trend of the time spectra from the event-analysis results. In experiments with
pulsed beams, the spin rotation signal is not observed above $f_{\rm N}$,
which is determined by $1/2\delta$ with respect to the pulse width $\delta$.
The presence of such an internal magnetic field appears as a reduction of the
initial asymmetry. (In the case of MUSE, $\delta$ is FWHM $\simeq$80 ns and
$f_{\rm N}\simeq 6.25$ MHz.) Furthermore, when integrated over time, the
contribution of all spin rotation signals becomes zero, which is observed as a
decrease in the integral asymmetry. Figure 3(a) shows the temperature
dependence of the asymmetry with integration performed for each temperature
change in heating and cooling. Although the expected hysteresis gap of $\sim$2
K near $T_{N2}$ is much greater than the temperature step, it was not
observed. As suggested by this result, it should be noted that the integral
method, which cancels out the contribution of the rotational signal, may not
provide sufficient sensitivity to the magnetic hysteresis.

As is clear from the behavior of $\tilde{A}(t)$ in Fig. 2, the time spectra
are roughly described as a slow relaxation in the paramagnetic (PM) phase
above $T_{\rm N1}$, the sum of two different relaxation components in the
incommensurate antiferromagnetic (IAFM) phase of $T_{\rm N1}$–$T_{\rm N2}$,
and the sum of a rotation component and a relaxation component in the
antiferromagnetic (AFM) phase below $T_{\rm N2}$. Therefore, we generated 120
$\mu$SR time spectra [$A(t)$] by slicing $\tilde{A}(t)$ with a temperature bin
of 0.5 K, and analyzed them by musrfit. As shown in Fig. 4(a), the statistics
(sum of those obtained during temperature rise and fall) for each spectrum was
approximately 12-14 M events. Curve fitting was performed assuming the
following equations for the respective temperature regions,

$A(t)=\left\\{\begin{array}[]{ll}A_{\rm L}e^{-\lambda_{\rm L}t},&({\rm PM})\\\
A_{\rm KT}G_{\rm KT}(t;\Delta,\nu)+A_{\rm L}e^{-(\lambda_{\rm
L}t)^{\beta}},&({\rm IAFM})\\\ A_{\rm T}e^{-\lambda_{\rm
T}t}\cos\omega_{\mu}t+A_{\rm L}e^{-\lambda_{\rm L}t},&({\rm
AFM})\end{array}\right.$ (3)

where $\lambda_{\rm L}$ ($\lambda_{\rm T}$) is the longitudinal (transverse)
relaxation rate with $A_{\rm L}$ ($A_{\rm T}$) being the partial asymmetry,
$\beta$ is the power for the stretched exponential damping,
$\omega_{\mu}\equiv 2\pi f=\gamma_{\mu}B_{\rm loc}$ is the muon Larmor
frequency with $\gamma_{\mu}=2\pi\times 135.53$ MHz/T being the muon
gyromagnetic ratio and $B_{\rm loc}$ the internal field at the muon site,
$G_{\rm KT}(t;\Delta,\nu)$ is the Gaussian Kubo-Toyabe function with $A_{\rm
KT}$ the partial asymmetry, which approximates $B_{\rm loc}$ due to
incommensurate magnetic ordering with a random magnetic field by its rms value
$\langle B_{\rm loc}^{2}\rangle^{1/2}=\Delta/\gamma_{\mu}$ with the
fluctuation rate $\nu$ [28]. Prior to this analysis, the instrumental
asymmetry $\alpha$ was determined by a curve fit that takes into account the
fact that $A(t)$ appears to converge to zero at late times in the IAFM phase,
and was fixed to that value for all time spectra in the subsequent analysis.
The analysis of all 120 spectra by sequential fitting took about 10 minutes,
allowing repeated analysis by trial-and-error of the fitting functions. The
temperature dependence of each parameter obtained by curve fitting is shown in
Fig. 4b) and (c) for the 195–245 K range.

Figure 4: (a) Positron event statistics per sliced $\mu$SR spectrum, (b)
partial asymmetry, and (c) frequency/relaxation/fluctuation rates vs.
temperature. Insets: (b) the power of the stretched exponential decay in
Eq.(3).

As is found in Fig. 4(b), the value of $A_{\rm L}$ in the PM phase is about
0.21, which corresponds to the total asymmetry [$A_{0}$ in Eq. (1)], while in
the AFM phase it is 0.07, about 1/3 of that value. This indicates that the
entire sample volume is in an ordered state; one would expect

$G(t)=\frac{1}{3}e^{-\lambda_{\rm
L}t}+\frac{2}{3}\sum_{i}y_{i}\exp(-\lambda_{\rm
T}^{(i)}t)\cos\omega_{\mu}^{(i)}$

as a powder average, where $\omega_{\mu}^{(i)}$ and $\lambda_{\rm T}^{(i)}$
denote the rotation frequency and relaxation rate at the $i$th muon site
having relative yield $y_{i}$ ($\sum_{i}y_{i}=1$) [18, 19]. Meanwhile, the
amplitude $A_{\rm T}$ of the rotational component in the AFM phase is much
smaller than the expected value of $0.21\times 2/3=0.14$ in such a case. This
can be qualitatively explained by the fact that the observed rotation
frequency $f$ [Fig. 4(c)] is higher than the Nyquist frequency. Previous
studies with DC muon beams have reported that muons are distributed over
several magnetically non-equivalent sites that exhibit greater $B_{\rm loc}$
[27], and the disappearance of the corresponding high-frequency rotation
signals also contributes to the decrease in $A_{\rm T}$.

In contrast to the AFM phase, no clear rotational signal is observed in the
IAFM phase, and the time spectra are reproduced by a component showing
Gaussian-type fast relaxation and a longitudinal relaxation component. The
static linewidth $\Delta$ obtained by fitting with the Kubo-Toyabe function is
about 10–12 MHz near $T_{\rm N2}$, and when this Gaussian damping is
interpreted to be the first quarter period of the rotation signal, the
corresponding frequency is $\Delta/2\pi\sim 1.6$–2 MHz, which is smoothly
extrapolated from $f$ just below $T_{\rm N2}$. Therefore, it is suggested that
$\Delta$ represents the static distribution width determined by the
configuration of Cu magnetic moments, and its fluctuation frequency $\nu$
corresponds to that of the Cu moment fluctuation. In other words, we can
conclude that magnetic fluctuations of 4–8 MHz exist in the IAFM phase. This
is also supported by the behavior of the LF spectra taken in parallel with ZF
data (not shown).

###  3.2 Field sweep: Cu

Resonance spectroscopy is a powerful method because it can focus on any
specific part of the interaction in matter. One such technique in $\mu$SR is
well known as $\mu$LCR, similar to the level crossing (or “cross relaxation”)
method commonly used in solid-state NMR. In non-magnetic metals, $\mu$LCR
occurs between the Zeeman levels of muon spin ${\bm{S}}$ under an external
magnetic field ${\bm{B}}$ and the electric quadrupole levels of the adjacent
host nuclei (spin $I\geq 1$). The local symmetry of the surrounding host
lattice is broken by the electron-lattice distortion caused by the presence of
$\mu^{+}$, and electric quadrupole interaction is induced between nuclear
quadrupoles and electric field gradient (EFG), partly removing the degeneracy
of nuclear spin states:

$\omega_{I}(I_{z})=\frac{1}{2}\omega_{\rm
Q}\left[I_{z}^{2}-\frac{1}{3}I(I+1)+u(I_{z},\eta^{2})\right],\\\ $ (4) $\hskip
20.00003pt\omega_{\rm Q}=\frac{3e^{2}qQ}{2\hbar I(2I-1)},$ (5)

where $eq\equiv V_{zz}$ is the EFG at the nuclear sites, $eQ$ is the nuclear
quadrupole moment, and $\eta$ is the asymmetry of EFG,
$\eta=(V_{xx}-V_{yy})/V_{zz}$ (with $|V_{zz}|\geq|V_{xx}|\geq|V_{yy}|$). The
last term $u$ is of important relevance to $\mu$LCR in that the symmetry of
EFG may be affected by the presence of $\mu^{+}$. In copper crystals, for
example, the nuclear spin levels ($I=3/2$) are split with an energy
$\omega_{I}(\pm\frac{3}{2})-\omega_{I}(\pm\frac{1}{2})=\omega_{\rm Q}$.

In the conventional $\mu$LCR experiment, the resonance is detected as an
enhancement of longitudinal spin relaxation, for which muons polarized
parallel to ${\bm{B}}$ are used. The magnitude of ${\bm{B}}$ is varied to
match the energy splitting of the muon’s Zeeman level with that of the
magnetic level of the host nucleus determined primarily by the electric
quadrupole energy. Usually, the magnetic dipole interaction (described by the
Hamiltonian $H_{\rm D}$) between muon and nucleus is much smaller than the
Zeeman interaction of muons. In this case, the spin relaxation can be
approximated by the $n$th-order parallel moment $M_{n}^{\prime}$ corresponding
to the diagonal term of $H_{\rm D}$ as follows [29, 30]:

$\displaystyle G_{z}(t)$ $\displaystyle\simeq$ $\displaystyle
1-\frac{1}{2}M_{2}^{\prime}t^{2}+\frac{1}{4!}M_{4}^{\prime}t^{4}-...,$ (6)
$\displaystyle M_{4}^{\prime}(B)$ $\displaystyle=$ $\displaystyle
M_{2}^{\prime}\omega_{D}^{2}\sum_{i=1}^{N}[(\omega_{\rm Q}-\gamma_{\mu}B)^{2}$
(7) $\displaystyle+\>C(\Theta_{i})(\omega_{\rm Q}-\gamma_{\mu}B)\gamma_{\rm
Cu}B$ $\displaystyle+\>D(\Theta_{i},I_{z})\gamma_{\rm Cu}^{2}B^{2}],$

where $\gamma_{\rm Cu}$ is the gyromagnetic ratio of Cu nucleus ($=2\pi\times
11.285$ and 12.089 MHz/T for 63Cu and 65Cu, respectively),
$\omega_{D}=\gamma_{\mu}\gamma_{\rm Cu}/r^{3}$ ($r$ being the distance between
muon and Cu nucleus), $C$
[$=(1-3\sin^{2}\Theta_{i})/(1+\frac{3}{2}\sin^{2}\Theta_{i})$] and $D$ are
geometrical factors determined by the polar angle $\Theta_{i}$ between the EFG
and the muon-nuclear axis for the $i$th nucleus, and $N$ is the number of the
nearest neighboring Cu nuclei. Since $M^{\prime}_{2}$ is a constant, the
$\mu$LCR condition is then given as a field $B_{\rm res}$ for minimizing
$M_{4}^{\prime}$ (note that the sign of the coefficient for $M_{4}^{\prime}$
is positive), i.e.,

$B_{\rm res}\simeq\frac{\omega_{\rm
Q}}{\gamma_{\mu}}\left[1+\frac{\overline{C}\gamma_{\rm
Cu}}{2\gamma_{\mu}}+O\left(\frac{\gamma_{\rm
Cu}^{2}}{\gamma_{\mu}^{2}}\right)\right],$ (8)

where $\overline{C}=\sum_{i}C(\Theta_{i})/N$. Since $\overline{C}\leq 1$ and
$D\sim 1$, the above equation is approximated to yield $B_{\rm
res}\simeq\omega_{\rm Q}/\gamma_{\mu}$.

The $\mu$LCR measurements on copper were performed over a magnetic field range
covering $B_{\rm res}\simeq 8$ mT at a few different temperatures. An example
of relatively high-statistics $\tilde{A}(t)$ measured at 50 K is shown in Fig.
5(a) and (b), where the field was swept from 0 to 15.0 mT in 0.2 mT steps
($\sim$80 s/step, with the elapsed time of 1.7 hr) under a proton beam power
of $\sim$0.60 MW. The Kubo-Toyabe function near $B=0$ and the suppression of
relaxation by increasing $B$ can be observed in the contour plot. The number
of positron events per spectrum sliced at 0.2 mT (the same as the $B$ step) is
shown in Fig. 5(c). The physical analysis for 76 spectra were performed using
musrfit, where the curve fits were made (with sequential fitting) to the
following form,

$A(t)=A_{0}G_{\rm KT}(t;\Delta_{\rm n},B)e^{-\lambda_{\rm Q}t},$ (9)

and $G_{\rm KT}(t;\Delta_{\rm n},B)$ is the static Kubo-Toyabe function under
a longitudinal field $B$ with $\Delta_{\rm n}$
[$\simeq(M_{2}^{\prime}/2)^{1/2}$] being the linewidth determined by the
random local fields from Cu nuclear magnetic moments. The curve fits of the
spectra for $B\leq 1.0$ mT yielded $\Delta_{\rm n}=0.374(9)$ MHz in excellent
agreement with the earlier results [31, 32]. (The muon hopping frequency,
which determines the fluctuation rate $\nu_{\rm n}$ of $\Delta_{\rm n}$, is
known to be negligibly small at 50 K in copper [31, 32], and assumed to be
zero.) The obtained $\lambda_{\rm Q}$ versus $B$ is shown in Fig. 5(d). The
peak of $\lambda_{\rm Q}$ is observed at 7.78 mT, which is in excellent
agreement with the previous experimental result [29, 30]. Since we used a
standard DC power supply, continuous sweep of the magnetic field was not
possible. However, the magnetic field was changed asynchronously with the DAQ,
which was a good demonstration of continuous and asynchronous magnetic field
sweep like the magnetic field of a superconducting magnet in the future.

Figure 5: Results of the LCR measurements on a copper plate. (a) A 3D plot of
$\tilde{A}(t)$ with the longitudinal magnetic field $B$ for the direction
crossing the screen surface. The solid pink line represents the static Kubo-
Toyabe function observed under zero field ($B=0$). (b) A contour plot of
$\tilde{A}(t)$ at 50 K, in which the vertical and horizontal axes represent
$B$ and $t$, respectively. (c) Positron event statistics per sliced $\mu$SR
spectrum, and (d) relaxation rate vs. temperature.

##  4 Summary and Prospects

In this paper, we establish a transient $\mu$SR method consisting of (i) data
acquisition in pulse-by-pulse mode that integrates information on the sample
environment in real time and (ii) event-analysis software that generates a
multidimensional asymmetry histogram from the obtained data. The transient
$\mu$SR method was applied to antiferromagnetic CuO and metallic Cu samples to
demonstrate its ability to measure transient time spectra for varying
temperature and magnetic field in each case. Transient $\mu$SR, in combination
with a high-flux pulsed muon beam, enables fast scanning of the sample
environment, which is difficult with conventional measurements, and makes it
possible to measure time-dependent phenomena in materials, such as operando
measurements. The method can also be applied to other pulsed muon beam
experiments such as muon X-ray measurements.

The time scale of change in sample conditions over which this technique is
valid is so far limited by the interval between beam pulses (40 ms). This is
partly due to the relatively long time constant of the sample environment
under normal circumstances. However, it is expected that this limitation can
be overcome by combining this technique with a stroboscopic measurement method
that synchronizes the timing of the beam pulses with the fast changing (or
pulsed) conditions [33, 34]. This will be a future challenge in further
developing transient $\mu$SR.

Another major challenge is to perform direct physical analysis instead of
slicing multidimensional histograms and analyzing them with conventional
physical analysis software. In this regard, we are focusing on the “deep
learning”, which has been rapidly developing in recent years. Deep learning is
revolutionary in that it has shown the possibility of automating the act of
constructing physical models; AI-Feynman, for example, has attempted to derive
physical laws with some success [35]. For $\mu$SR, an attempt has been made to
analyze the time spectra using a simple machine learning code called Principal
Component Analysis (PCA) [36]. While this showed the possibility of
parametrizing some features of the data without relying on a specific physical
model, there still seems to be significant difficulties in bridging these
abstract parameters to a physical model comprehensible to humans. The
development of analysis software that learns the direct correspondence between
multidimensional histograms and physical models through deep learning would be
a great step forward in solving this problem.

##  5 Acknowledgements

The authors would like to acknowledge the help and expertise of the MUSE
S-line staff during experiment at MLF, J-PARC. The muon experiment was
conducted under the support of Inter-University-Research Programs by Institute
of Materials Structure Science, KEK (Proposals No. 2019B0411 and 2020MI21).
This work was partially supported by JSPS KAKENHI (Grant No. 20K05312,
19K15033, and 19K20595) and the MEXT Elements Strategy Initiative to Form Core
Research Center for Electron Materials (Grant No. JPMXP0112101001).

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# Gauged $U(1)_{L_{\mu}-L_{\tau}}$ Symmetry and two-zero Textures of Inverse
Neutrino Mass Matrix in light of Muon ($g-2$)

Labh Singh, Monal Kashav and Surender Verma
sainilabh5@gmail.commonalkashav<EMAIL_ADDRESS>Corresponding
Author

(Department of Physics and Astronomical Science
Central University of Himachal Pradesh
Dharamshala, India 176215 )

###### Abstract

In the framework of anomaly free $U(1)_{L_{\mu}-L_{\tau}}$ model, charged
scalar fields give rise to massive gauge boson ($Z_{\mu\tau}$) through
spontaneous symmetry breaking. $Z_{\mu\tau}$ leads to one loop contribution to
the muon anomalous magnetic moment. These scalar fields may, also, appear in
the structure of right-handed neutrino mass matrix, thus, connecting the
possible explanation of muon ($g-2$) and low energy neutrino phenomenology
through $vevs$ associated with the scalar fields. In the present work, we
consider textures of inverse neutrino mass matrix ($M_{\nu}^{-1}$) wherein any
two elements of the mass matrix are zero. In this ansatz, with Dirac neutrino
mass matrix diagonal, the zero(s) of right-handed Majorana neutrino mass
matrix correspond to zero(s) in the low energy effective neutrino mass matrix
(within Type-I seesaw). We have realized two such textures of $M_{\nu}^{-1}$
accommodating the muon ($g-2$) and low energy neutrino phenomenology. The
requirement of successful explanation of muon ($g-2$), further, constrain the
allowed parameter space of the model and results in sharp correlations amongst
neutrino mixing angles, $CP$ invariants and effective Majorana mass
($M_{ee}$). The model explains muon ($g-2$) for $M_{Z_{\mu\tau}}$ in the range
($0.035$ GeV-$0.100$ GeV) and $g_{\mu\tau}\approx\mathcal{O}(10^{-4}$) which
is found to be consistent with constraints coming from the experiments like
CCFR, COHERENT, BABAR, NA62 and NA64.
Keywords: Muon $(g-2)$; Phenomenology; Neutrino mass matrix; Texture zeros.

##  1 Introduction

The standard model (SM) of particle physics has been very successful in
explaining interactions between fundamental particles and, at the same time,
predicted a wide variety of phenomena. Despite immense success of the SM, it
is facing a growing list of “anomalies”- a significant experimental divergence
from theoretical predictions. For example, unsolved problems like origin of
light neutrino masses, matter-antimatter asymmetry, dark matter, muon
anomalous magnetic moment etc. find no explanation within the SM. The
experimental observations of sub-eV scale neutrino masses and large mixing in
the leptonic sector provide cardinal evidences propounding physics beyond the
standard model (BSM)[1]. The neutrino flavor states
($\nu_{\alpha}(\alpha=e,\mu,\tau)$) are incoherent mixture of mass eigenstates
($\nu_{i}(i=1,2,3)$). The magnitude of mixing is parameterized in terms of
three mixing angles ($\theta_{ij}(i,j=1,2,3;i<j)$) and one Dirac $CP$ phase
($\delta$). The Dirac $CP$ phase has not been observed experimentally,
however, the recent measurements hint $\delta\approx-\pi/2$[2]. Additionally,
two more $CP$ phases($\alpha,\beta$) appears for Majorana nature of neutrinos
which have no influence on neutrino oscillations. Furthermore, the riddle of
octant degeneracy ($\theta_{23}$ above or below 45∘) and mass ordering
(normal($m_{1}<m_{2}<m_{3}$) or inverted hierarchy($m_{3}<m_{1}<m_{2}$)) still
remains unresolved. Furthermore, although absolute scale of neutrino mass is
unknown, we have upper bound on sum of neutrino mass, $\sum m_{i}<0.12$ eV
from cosmological data[3, 4]. In view of the above, the neutrino mass matrix,
in general, contains more free parameters than one can measure experimentally
so phenomenological ansatze are important to fully reconstruct the mass matrix
in terms of less number of parameters to understand the underlying dynamics of
neutrino mass generation.

The recent results from E989 experiment(Run I) [5] at FermiLab for the precise
measurement of muon anomalous magnetic moment $a_{\mu}=(g-2)_{\mu}/2$, shows a
discrepancy with the theoretical [6] prediction of the SM

$a_{\mu}^{\text{FNAL}}=116592040(54)\crossproduct 10^{-11},$ (1)
$a^{\text{SM}}_{\mu}=116591810(43)\crossproduct 10^{-11},$ (2)

which when amalgamated with the previous results of Brookhaven National
Laboratory

$a_{\mu}^{\text{BNL}}=116592089(63)\crossproduct 10^{-11},$ (3)

raises the confidence level from 3.7$\sigma$ to 4.2$\sigma$ such that $\Delta
a_{\mu}=a_{\mu}^{\text{EXP}}-a_{\mu}^{\text{SM}}=(251\pm 59)\crossproduct
10^{-11}$, a compelling evidence of new physics.

The possible implication and interpretation of muon($g-2$) anomaly have been
discussed in different frameworks such as 2HDM[7, 8, 9, 10, 11], model with
axion-like particles (ALP)[12], $A_{4}$ modular symmetry[13], vector-like
leptons (VLL) [14, 15], super-symmetric(SUSY) models[16, 17] $etc.$. In
general, $U(1)_{L}$ and $U(1)_{B}$ are accidental symmetries in the SM leading
to lepton and baryon number conservation, respectively, but are anomalous.
However, the symmetries originating from the difference of any two charged
lepton flavors, $i.e.$ $L_{\alpha}-L_{\beta}$, $(\alpha,\beta=e,\mu,\tau)$ are
anomaly free. Among these $U(1)_{L_{\alpha}-L_{\beta}}$ symmetries,
$U(1)_{L_{\mu}-L_{\tau}}$ gauge symmetry has been explored in different
dimensions of neutrino mass model building scenarios[18, 19, 20].
$L_{\mu}-L_{\tau}$ extension of SM in the framework of Type-I seesaw with
one[21] and two complex scalar singlets[22] have been studied to explain the
muon ($g-2$). In general, spontaneous symmetry breaking of
$U(1)_{L_{\mu}-L_{\tau}}$ symmetry manifests the $L_{\mu}-L_{\tau}$ massive
gauge boson ($Z_{\mu\tau}$). In the framework of $U(1)_{L_{\mu}-L_{\tau}}$
symmetry, $Z_{\mu\tau}$ do not interact with electron and quarks, evading
constraints from LEP[23, 24] and LHC[25]. It interacts only with $\mu$ and
$\tau$ flavors which may contribute significantly to muon magnetic moment.
Also, $Z_{\mu\tau}$ gauge boson contributes to the muon neutrino trident (MNT)
process which constrains the mass of new gauge boson $Z_{\mu\tau}\leq 300$ MeV
for the explanation of muon ($g-2$)[26]. The new developments in muon magnetic
moment measurements compels to investigate theoretical models accommodating
explanation of muon ($g-2$).

The phenomenological ansatze like texture zeroes[27, 28, 29, 30, 31, 32, 33],
hybrid textures[34, 35, 36, 37], magic symmetry[38, 39, 40, 41] lead to
interesting predictions and correlation among low-energy observables. Also,
texture zeroes in inverse neutrino mass matrices($M_{\nu}^{-1}$) are
imperative in the sense, in diagonal charged lepton and Dirac mass basis,
zeroes in right-handed neutrino mass matrix corresponds to zeroes in
$M_{\nu}^{-1}$. The phenomenological implications of inverse neutrino mass
textures have been studied in Refs.[42, 43, 44]. Recently, the authors have
investigated all possible two-zero texture inverse neutrino mass matrices
($M_{\nu}^{-1}$) in light of large mixing angle(LMA) and $dark$-large mixing
angle(DLMA) solutions of neutrino mixing paradigm[45]. Out of fifteen possible
two-zero $M_{\nu}^{-1}$ textures only seven are found to be in consonance with
current neutrino oscillation data. In the present work, we have realized two
such textures $D_{1}$ and $E_{1}$( for notation of textures see Ref.[45])
accommodating muon ($g-2$) anomaly and neutrino oscillation data,
simultaneously. We have employed $U(1)_{L_{\mu}-L_{\tau}}$ symmetry extending
the SM with three right-handed neutrinos and three scalar singlet fields. The
gauge boson contributing to the possible explanation of muon anomalous
magnetic moment, further, constrain the allowed parameter space of these
textures.

The rest of the paper is organised as follows. In Section 2, we have discussed
the $U(1)_{L_{\mu}-L_{\tau}}$ model and corresponding charge assignments
resulting in two-zero $M_{\nu}^{-1}$. The details of numerical analysis and
consequent discussion have been elaborated in Section 3. Finally, in Section
4, we summarize our conclusions.

##  2 $U(1)_{L_{\mu}-L_{\tau}}$ Model

We have extended the SM field content with three heavy right-handed neutrinos
($N_{e},N_{\mu},N_{\tau}$) having $L_{\mu}-L_{\tau}$ charges ($0,1,-1$),
respectively, leading to Type-I seesaw origin of light neutrino masses. In the
scalar sector, three singlet scalar fields $\Phi_{i}$ ($\textit{i}=1,2,3$)
with non-zero $L_{\mu}-L_{\tau}$ charges have been employed. It is known that
zeroes in $M_{R}$ are identical to zeroes in $M_{\nu}^{-1}$ if Dirac and
charged lepton mass matrices ($M_{D}$ and $M_{\ell}$) are diagonal. Therefore,
the charge assignments under $U(1)_{L_{\mu}-L_{\tau}}$ are chosen in such a
way that $M_{\ell}$ and $M_{D}$ are diagonal, in the model. $\Phi_{i}$ breaks
the $U(1)_{L_{\mu}-L_{\tau}}$ symmetry by acquiring vacuum expectation values
($vevs$) $v_{i}$ $(i=1,2,3)$ consequently giving mass to the new
$U(1)_{L_{\mu}-L_{\tau}}$ gauge boson $Z_{\mu\tau}$. Also, the $Z_{4}$
symmetry have been used to constrain the structure of the Yukawa Lagrangian.
The particle content with respective gauge charges under
$SU(2)_{L}\crossproduct U(1)_{Y}\crossproduct
U(1)_{L_{\mu}-L_{\tau}}\crossproduct Z_{4}$ symmetry is shown in Table 1. The
Lagrangian for our model is given by

$\displaystyle\mathcal{L}=\mathcal{L}_{SM}+\mathcal{L}_{N}+\mathcal{L}_{gauge}+\mathcal{L}_{scalar},$
(4)

where $\mathcal{L}_{SM}$ is SM Lagrangian, $\mathcal{L}_{N}$ is Lagrangian for
the right-handed neutrinos (RHN) which contains kinetic and mass terms.
$\mathcal{L}_{gauge}$ includes gauge kinetic terms for new fields whereas
$\mathcal{L}_{scalar}$ includes the form of scalar potential. It is to be
noted that $\mathcal{L}_{SM}$ includes the charged lepton mass terms which is
diagonal under the assignments in Table 1.
The new gauge kinetic terms that appears in the Lagrangian are

$\displaystyle\mathcal{L}_{gauge}=\dfrac{1}{4}\left(Z_{\mu\tau}\right)^{\gamma\eta}\left(Z_{\mu\tau}\right)_{\gamma\eta}-\dfrac{\epsilon}{2}\left(Z_{\mu\tau}\right)^{\gamma\eta}B_{\gamma\eta},$
(5)

where
$\left(Z_{\mu\tau}\right)^{\gamma\eta}=\partial^{\gamma}\left(Z_{\mu\tau}\right)^{\eta}-\partial^{\eta}\left(Z_{\mu\tau}\right)^{\gamma}$
is the field strength tensor for new gauge boson, $Z_{\mu\tau}$, while the
second term in the Eqn.(5) denotes the kinetic mixing of $U(1)_{Y}$ and
$U(1)_{L_{\mu}-L_{\tau}}$ gauge sectors, where $\epsilon$ is kinetic mixing
parameter.
The Lagrangian of the scalar sector can be written as

$\displaystyle\mathcal{L}_{scalar}=(D_{\gamma}\Phi_{i})^{\dagger}(D^{\gamma}\Phi_{i})-V(H,\Phi_{i}),\hskip
8.5359pt\textit{i}=1,2,3.$ (6)

The covariant derivative $D_{\gamma}$ is defined as

$\displaystyle
D_{\gamma}=\partial_{\gamma}-ig\dfrac{\tau}{2}.W_{\gamma}-ig^{{}^{\prime}}\dfrac{Y}{2}B_{\gamma},$
(7)

where $g$ and $g^{{}^{\prime}}$ are the coupling constants associated with
$W_{\gamma}$ and $B_{\gamma}$ gauge fields, respectively.

The scalar potential is given by

$\displaystyle\begin{aligned}
V(H,\Phi_{i})=&-\mu_{\Phi_{i}}^{2}(\Phi_{i}^{\dagger}\Phi_{i})^{2}+\lambda_{\phi_{i}}(\Phi_{i}^{\dagger}\Phi_{i})^{2}+\lambda_{H\Phi_{i}}(H^{\dagger}H)(\Phi_{i}^{\dagger}\Phi_{i})+\lambda_{\Phi_{1}\Phi_{2}}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})\\\
&+[\mu_{12}\Phi_{1}^{2}\Phi_{2}^{\dagger}+H.c.]+\lambda_{\Phi_{1}\Phi_{3}}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{\Phi_{2}\Phi_{3}}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3}).\end{aligned}$
(8)

where $i=1,2,3$. The neutral component of SM Higgs($H$) breaks the electroweak
symmetry spontaneously whereas singlets $\Phi_{1,2,3}$ breaks the
$L_{\mu}-L_{\tau}$ gauge symmetry after acquiring the $vevs$ $v_{1,2,3}$. The
Yukawa Lagrangian for charged leptons is given by

$\displaystyle\mathcal{L_{\ell}}=-Y_{\ell_{e}}\bar{L}_{e}He_{R}-Y_{\ell_{\mu}}\bar{L}_{\mu}H\mu_{R}-Y_{\ell_{\tau}}\bar{L}_{\tau}H\tau_{R}+h.c.,$
(9)

which leads to charged lepton mass matrix diagonal,
$M_{\ell}$=$\frac{v}{\sqrt{2}}$diag($Y_{\ell_{e}},Y_{\ell_{\mu}},Y_{\ell_{\tau}}$),
where $Y_{\ell_{i}}$ with $i=e,\mu,\tau$ are the Yukawa couplings. The
Lagrangian relevant for neutrino mass is given by

$\displaystyle\begin{aligned}
\mathcal{L_{N}}=&\overline{N_{\mu}}i\gamma^{\mu}D_{\mu}N_{\mu}+\overline{N_{\tau}}i\gamma^{\tau}D_{\tau}N_{\tau}-\dfrac{1}{2}MN_{e}N_{e}-Y_{e\tau}\Phi_{1}N_{e}N_{\tau}-Y_{e\mu}\Phi_{3}N_{e}N_{\mu}\\\
&-Y_{\tau\tau}\Phi_{2}N_{\tau}N_{\tau}-Y_{D_{e}}\bar{L}_{e}\tilde{H}N_{e}-Y_{D_{\mu}}\bar{L}_{\mu}\tilde{H}N_{\mu}-Y_{D_{\tau}}\bar{L}_{\tau}\tilde{H}N_{\tau}+h.c.,\end{aligned}$
(10)

where $\tilde{H}=i\sigma_{2}H^{*}$ and $M$ is a constant with dimension of
mass.

Symmetry | $L_{e}$ | $L_{\mu}$ | $L_{\tau}$ | $e_{R}$ | $\mu_{R}$ | $\tau_{R}$ | $N_{e}$ | $N_{\mu}$ | $N_{\tau}$ | H | $\Phi_{1}$ | $\Phi_{2}$ | $\Phi_{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1
$U(1)_{Y}$ | -$\frac{1}{2}$ | -$\frac{1}{2}$ | -$\frac{1}{2}$ | -1 | -1 | -1 | 0 | 0 | 0 | $\frac{1}{2}$ | 0 | 0 | 0
$U(1)_{L_{\mu}-L_{\tau}}$ | 0 | 1 | -1 | 0 | 1 | -1 | 0 | 1 | -1 | 0 | 1 | 2 | -1
$Z_{4}$ | -1 | 1 | i | -1 | 1 | i | -1 | 1 | i | 1 | i | -1 | -1
Table 1: The field content of the model with respective charge assignments
under $SU(2)_{L}\crossproduct U(1)_{Y}\crossproduct
U(1)_{L_{\mu}-L_{\tau}}\crossproduct Z_{4}$.

After expanding the kinetic term in Eqn.(6), the mass of new
$U(1)_{L_{\mu}-L_{\tau}}$ gauge boson can found to be
$M_{Z_{\mu\tau}}$=$g_{\mu\tau}\sqrt{v_{1}^{2}+4v_{2}^{2}+v_{3}^{2}}$, where
$g_{\mu\tau}$ is the $L_{\mu}-L_{\tau}$ gauge coupling.

Using Eqn.(10), the Dirac mass matrix is given by

$\displaystyle M_{D}=\begin{pmatrix}d_{e}&0&0\\\ 0&d_{\mu}&0\\\
0&0&d_{\tau}\\\ \end{pmatrix},$ (11)

where $d_{\alpha}=\frac{Y_{D_{\alpha}}v}{\sqrt{2}}$ with $\alpha=e,\mu,\tau$.
$Y_{D_{\alpha}}$ are real Yukawa couplings and $\frac{v}{\sqrt{2}}$ is the
$vev$ of SM Higgs doublet, $H$. Using Eqn.(10), the right-handed Majorana mass
matrix ($M_{R}$) is given by

$\displaystyle M_{R}=\begin{pmatrix}M&Y_{e\mu}v_{3}&Y_{e\tau}v_{1}e^{i\xi}\\\
Y_{e\mu}v_{3}&0&0\\\ Y_{e\tau}v_{1}e^{i\xi}&0&Y_{\tau\tau}v_{2}\\\
\end{pmatrix},$ (12)

where, in general, the elements of $M_{R}$ are complex. By redefinition of the
fields, $\xi$ is the only remaining irremovable phase. Thus, $M_{R}$ depends
on four real parameters $M$, $Y_{e\mu}$, $Y_{e\tau}$ and $Y_{\tau\tau}$ and a
complex phase $\xi$. As a consequence of diagonal $M_{D}$ and $M_{\ell}$ the
non-trivial neutrino mixing will arise from $M_{R}$.
Within the paradigm of Type-I seesaw, the inverse neutrino mass matrix can be
written as

$\displaystyle M_{\nu}^{-1}=-(M_{D}^{T})^{-1}M_{R}M_{D}^{-1}.$ (13)

Using $M_{D}$ and $M_{R}$ given in Eqns.(11) and (12), respectively, alongwith
Eqn. (13), $M_{\nu}^{-1}$ is given by

$\displaystyle
M_{\nu}^{-1}=\begin{pmatrix}-\dfrac{M}{d_{e}^{2}}&\dfrac{Y_{e\mu}v_{3}e^{i\xi}}{d_{e}d_{\mu}}&-\dfrac{Y_{e\tau}v_{1}}{d_{e}d_{\tau}}\\\
\dfrac{Y_{e\mu}v_{3}e^{i\xi}}{d_{e}d_{\mu}}&0&0\\\
-\dfrac{Y_{e\tau}v_{1}}{d_{e}d_{\tau}}&0&\dfrac{Y_{\tau\tau}v_{2}}{d_{\tau}^{2}}\\\
\end{pmatrix},$ (14)

which corresponds to $D_{1}$ texture of $M_{\nu}^{-1}$ studied in Ref. [45].
Also, if the charge assignment of $\Phi_{2}$ under $U(1)_{L_{\mu}-L_{\tau}}$
and $Z_{4}$ are replaced by $-2$ and $1$, respectively, then we obtain,
$M_{R}$, given by

$\displaystyle M_{R}=\begin{pmatrix}M&Y_{e\mu}v_{3}&Y_{e\tau}v_{1}e^{i\xi}\\\
Y_{e\mu}v_{3}&Y_{\mu\mu}v_{2}&0\\\ Y_{e\tau}v_{1}e^{i\xi}&0&0\\\
\end{pmatrix},$ (15)

while the mass matrices $M_{\ell}$ and $M_{D}$ remains diagonal. The two-zero
texture of $M_{\nu}^{-1}$ obtained using Eqn.(15) corresponds to $E_{1}$
texture with zeroes at ($2,3$) and ($3,3$) place [45].

##  3 Numerical Analysis and Discussion

In this section, as a representative case, we perform the numerical analysis
of texture $D_{1}$ obtained in Eqn.(14) in light of muon ($g-2$) and neutrino
oscillation data (Table 2). The $D_{1}$ texture defined in Eqn.(14)
corresponds to two-zero texture $M_{\nu}$ with zeroes at ($1,1$) and ($1,3$).
$M_{\nu}$ is numerically diagonalised by a unitary matrix $U$ such that
$UM_{\nu}U^{T}=\text{diag}(m_{1},m_{2},m_{3})$ and the neutrino mixing angles
can be obtained using

$\displaystyle\sin^{2}{\theta_{13}}=\left|U_{13}\right|^{2},\hskip
14.22636pt\sin^{2}{\theta_{23}}=\dfrac{\left|U_{23}\right|^{2}}{1-\left|U_{13}\right|^{2}},\hskip
14.22636pt\sin^{2}{\theta_{12}}=\dfrac{\left|U_{12}\right|^{2}}{1-\left|U_{13}\right|^{2}}.$
(16)

Also, the amount of $CP$ violation manifested in Jarlskog invariant
($J_{CP}$)[46, 47] is defined as

$\displaystyle
J_{CP}=\text{Im}\left[U_{11}U_{22}U^{*}_{12}U^{*}_{21}\right]=s_{23}c_{23}s_{12}c_{12}s_{13}c^{2}_{13}\sin{\delta},$
(17)

while other two rephasing invariants $I_{1}$ and $I_{2}$ are given by

$\displaystyle
I_{1}=\text{Im}\left[U^{*}_{11}U_{12}\right]=c_{12}s_{12}c^{2}_{13}\sin\left({\dfrac{\alpha_{1}}{2}}\right),\hskip
14.22636ptI_{2}=\text{Im}\left[U^{*}_{11}U_{13}\right]=c_{12}s_{13}c_{13}\sin\left(\dfrac{\alpha_{2}}{2}-\delta\right),$
(18)

$\mu$$\mu$$Z_{\mu\tau}$$\gamma$ Figure 1: One loop Feynman diagram mediated by
extra gauge boson $Z_{\mu\tau}$ contributing to muon $(g-2)$.

where $\alpha_{1},\alpha_{2}$ are Majorana phases. Furthermore, the
gyromagnetic ratio ($g$-factor) of the muon is the quantity which relates its
spin ($\vec{s}$) to its magnetic moment ($\vec{\mu}$) as given by

$\displaystyle\vec{\mu}=g\left(\dfrac{q}{2m_{\mu}}\right)\vec{s},$ (19)

where, $q$ is muon charge and $m_{\mu}$ is muon mass. In Dirac’s theory of
charged spin-half particles, the gyromagnetic ratio is $g=2$. However, the
recent developments at FermiLab hint towards non-trivial interactions of muon
with BSM fields. The higher-order radiative corrections can generate
additional contributions to magnetic moment of muon parameterised as

$\displaystyle g=2_{\text{Dirac}}(1+a_{\mu})\hskip 28.45274pt\text{and}\hskip
28.45274pta_{\mu}=\dfrac{1}{2}\left(g-2\right).$ (20)

The correction $a_{\mu}$ to the Dirac’s predictions is called the anomalous
magnetic moment. With in SM, the contribution to the anomalous magnetic moment
of muon may comes from: (a) quantum electrodynamic $(\textit{QED})$
contributions (b) electroweak $(\textit{EW})$ contributions (c) hadronic
vacuum polarisation contributions (d) hadronic light-by-light scattering
contributions. As explained earlier, the SM is not consistent with the recent
results on muon $(g-2)$ at FermiLab. Therefore, beyond standard model
contribution is required to explain muon anomalous magnetic moment. In this
model, the additional contribution to muon magnetic moment arises at one-loop
(Fig.(1)) mediated by $U(1)_{L_{\mu}-L_{\tau}}$ gauge boson $Z_{\mu\tau}$ and
is given by[48, 49]

$\displaystyle\Delta
a_{\mu}=\dfrac{\alpha^{\prime}}{2\pi}\int_{0}^{1}dx\dfrac{2m^{2}_{\mu}x^{2}(1-x)}{x^{2}m^{2}_{\mu}+(1-x)M_{Z_{\mu\tau}}},$
(21)

(a) 2(a)

(b) 2(b)

Figure 2: The model predictions of the neutrino mixing angles as correlations
plots between sum of neutrino masses ($\sum m_{i}$) and mixing angles. The
horizontal lines are 3$\sigma$ experimental bounds on the respective mixing
angle. The shaded region is excluded by the cosmological bound on sum of
neutrino masses.

where $\alpha^{\prime}=\dfrac{g^{2}_{\mu\tau}}{4\pi}$ is the structure
constant and $m_{\mu}$ is the mass of muon. The new gauge boson,
$Z_{\mu\tau}$, gets mass after scalar singlet fields $(\Phi_{i})$ acquires
$vevs$. Also, non-trivial neutrino mixing matrix is induced by $vevs$ of
scalar singlet fields (through $M_{R}$), thus, connecting explanation of muon
$(g-2)$ to neutrino phenomenology.

Parameter |  $\pm 1\sigma$ range (NH) |  $\pm 1\sigma$ range (IH) |  $3\sigma$ range (NH) |  $3\sigma$ range (IH)
---|---|---|---|---
$\sin^{2}{\theta_{12}}$ | $0.304^{+0.013}_{-0.012}$ | $0.304^{+0.012}_{-0.012}$ | 0.269-0.343 | 0.269-0.343
$\sin^{2}{\theta_{13}^{2}}$ | $0.02220^{+0.00068}_{-0.00062}$ | $0.02238^{+0.00064}_{-0.00062}$ | 0.02060-0.02435 | 0.02053-0.02434
$\sin^{2}{\theta_{23}^{2}}$ | $0.573^{+0.018}_{-0.023}$ | $0.578^{+0.017}_{-0.021}$ | 0.405-0.624 | 0.410-0.623
$\frac{\Delta m_{12}^{2}}{10^{-5}\text{eV}^{2}}$ | $7.42^{+0.21}_{-0.20}$ | $7.42^{+0.21}_{-0.20}$ | 6.82-8.04 | 6.62-8.04
$\frac{\Delta m_{23}^{2}}{10^{-3}\text{eV}^{2}}$ | $2.515^{+0.028}_{-0.028}$ | $2.498^{+0.028}_{-0.029}$ | 2.431-2.598 | -2.584- -2.413
Table 2:  The neutrino oscillation data from global fit used in the numerical
analysis[1].

There are eight free parameters in inverse neutrino mass matrix
($M_{\nu}^{-1}$), given as, $M_{ee}$, $V_{1}$=$Y_{e\mu}v_{1}$,
$V_{2}=Y_{e\tau}v_{2}$, $V_{3}=Y_{\tau\tau}v_{3}$, $d_{e}$, $d_{\mu}$,
$d_{\tau}$ and $\xi$. In order to obtain the predictions on neutrino
oscillation parameters and muon $(g-2)$ anomaly for $D_{1}$ texture, we have
randomly varied all the free parameters with uniform distribution in the
ranges

$\left.\begin{split}d_{e},d_{\mu},d_{\tau}=\left(10^{-5}-10^{-3}\right)\text{GeV},\\\
V_{1},V_{2},V_{3}=\left(1-280\right)\text{GeV},\\\
M=\left(1-10^{4}\right)\text{GeV},\\\ \xi=\left(0-360\right)^{\circ}.\\\
\end{split}\right\\}$ (22)

We have numerically diagonalized $M_{\nu}$ to obtain the neutrino mixing
matrix $U$. The predictions for neutrino mixing angles obtained from Eqn.(16)
are compared with $3\sigma$ ranges given in Table 2 to ascertain the allowed
parameter space of the model.

Figure 3: The correlation plot between $\left(\sum m_{i}-J_{CP}\right)$. The
shaded region is excluded by the cosmological bound on sum of neutrino masses.

(a) 4(a)

(b) 4(b)

Figure 4: The correlation plots between $\sum m_{i}$ and $CP$ invariants
$I_{1}$(4(a)), $I_{2}$ (4(b)). The shaded region is excluded by the
cosmological bound on sum of neutrino masses.

In Fig.2(a), we have depicted the correlation between sum of neutrino masses
$\sum m_{i}$ versus $\sin^{2}{\theta_{12}}$ and $\sin^{2}{\theta_{23}}$ while
Fig.2(b) shows the correlation between $\sum m_{i}$ versus
$\sin^{2}{\theta_{13}}$ at 3$\sigma$. It is evident that the model is
consistent with the neutrino oscillation data on the mixing angles and
predicts sum of neutrino masses $\sum m_{i}$ to be with in the range
$0.065\lesssim\sum m_{i}(\text{eV})\lesssim 0.075$. In Fig.3, we have given
the correlation plot of $\left(\sum m_{i}-J_{CP}\right)$. $J_{CP}$ lies in the
range $-0.03\leq J_{CP}\leq 0.01$. The predictions for other two $CP$
rephasing invariants $I_{1}$ and $I_{2}$ are shown in Fig.4(a) and (b),
respectively. It is evident from Fig.4(a) that $I_{1}=0$ is disallowed
implying $D_{1}$ texture is necessarily $CP$ violating.

Figure 5: The allowed parameter space of the model in
($M_{Z_{\mu\tau}}-g_{\mu\tau}$) plane accommodating muon ($g-2$) and neutrino
oscillation data. The exclusion regions from various experiments are, also,
shown.

Using the Eqn.(21), we calculate the $Z_{\mu\tau}$ contribution to $\Delta
a_{\mu}$ which has been shown in $M_{Z_{\mu\tau}}-g_{\mu\tau}$ plane in Fig.5.
The gauge coupling is randomly varied in the range $10^{-4}-10^{-3}$. It is
evident from Fig.5 that the model accommodates the observed muon ($g-2$) for
$M_{Z_{\mu\tau}}$ in the range ($0.035$ GeV-$0.100$ GeV) and
$g_{\mu\tau}\approx\mathcal{O}(10^{-4}$), which is consistent with constraints
coming from experiments like COHERENT[50, 51], BABAR[52] and CCFR[53]. The
sensitivities of future experiments NA62[54] and NA64[55, 56] are, also, shown
in Fig.5. The upper left triangular region is excluded by the astrophysical
bound from cooling of white dwarf (WD)[57].

Benchmark point: For the input parameters

$\displaystyle\left(d_{e},d_{\mu},d_{\tau}\right)\crossproduct
10^{-5}=(2.38,1.57,2.19)\text{GeV},$
$\displaystyle\left(V_{1},V_{2},V_{3},M\right)=(65.6,15.0,37.1,28.1)\text{GeV},$
$\displaystyle\xi=309.54^{\circ},$ $\displaystyle g_{\mu\tau}=5.2\crossproduct
10^{-4},$

the corresponding values of mass-squared differences, mixing angles,
$M_{Z_{\mu\tau}}$ and $\Delta a_{\mu}$ are

$\displaystyle\Delta m^{2}_{23}=2.45\crossproduct 10^{-3}\text{eV}^{2};\hskip
2.84544pt\Delta m^{2}_{12}=7.53\crossproduct 10^{-5}\text{eV}^{2},$
$\displaystyle\text{sin}^{2}{\theta_{13}}=0.022;\hskip
2.84544pt\sin^{2}{\theta_{12}}=0.32;\hskip
2.84544pt\sin^{2}{\theta_{23}}=0.58,$ $\displaystyle\hskip
2.84544ptM_{Z_{\mu\tau}}=42.38\text{MeV};\Delta a_{\mu}=3.42\crossproduct
10^{-11}.$

##  4 Conclusions

In this work, we have realised two-zero textures of $M_{\nu}^{-1}$ with
anomaly free gauged $U(1)_{L_{\mu}-L_{\tau}}$ extension of SM in light of the
muon ($g-2$) anomaly. We have extended the SM field content by adding three
scalar singlets ($\Phi_{i}$) and three right-handed neutrinos
$(N_{e},N_{\mu},N_{\tau})$. $U(1)_{L_{\mu}-L_{\tau}}$ symmetry is broken as
the new scalar singlets acquire $vevs$, thereby, giving mass to new
$U(1)_{L_{\mu}-L_{\tau}}$ gauge boson $Z_{\mu\tau}$. The two-zeros in $M_{R}$
corresponds to two-zeros in $M_{\nu}^{-1}$ in the diagonal charged lepton and
Dirac mass basis. Also, the non-trivial neutrino mixing depends on the
structure of $M_{R}$. Thus, the right-handed Majorana neutrino mass matrix
connects the low energy neutrino phenomenology with $M_{Z_{\mu\tau}}$
contributing to muon anomalous magnetic moment. We have scanned the model
parameter space and have found the model consistent with the neutrino
oscillation data within $3\sigma$ ranges. The Jarlskog $CP$ rephasing
invariant, $J_{CP}$, lies in the range $-0.03\leq J_{CP}\leq 0.01$. The
texture is found to be necessarily $CP$ violating as $I_{1}=0$ is disallowed.
The model predicts the mass of new gauge boson $(M_{Z_{\mu\tau}})$ in the
range $0.035\text{GeV}\leq M_{Z_{\mu\tau}}\leq 0.100\text{GeV}$ for gauge
coupling ($g_{\mu\tau}$) between $5\times 10^{-4}\leq g_{\mu\tau}\leq 8\times
10^{-4}$ which is consistent with constraints from experiments such as CCFR,
COHERENT, BABAR, NA62 and NA64.

## Acknowledgments

M. K. acknowledges the financial support provided by Department of Science and
Technology(DST), Government of India vide Grant No. DST/INSPIRE
Fellowship/2018/IF180327. The authors, also, acknowledge Department of Physics
and Astronomical Science for providing necessary facility to carry out this
work.

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