# Non-minimal Lorentz invariance violation in light of muon anomalous magnetic
moment and long-baseline neutrino oscillation data

Hai-Xing Lin<EMAIL_ADDRESS>School of Physics, Sun Yat-sen
University, Guangzhou 510275, China  Pedro Pasquini<EMAIL_ADDRESS>Tsung-Dao Lee Institute (TDLI), Shanghai Jiao Tong University (SJTU), Shanghai
200240, China  Jian Tang<EMAIL_ADDRESS>School of Physics, Sun
Yat-sen University, Guangzhou 510275, China  Sampsa Vihonen
<EMAIL_ADDRESS>School of Physics, Sun Yat-sen University, Guangzhou
510275, China

###### Abstract

In light of the increasing hints of new physics at the muon $g-2$ and neutrino
oscillation experiments, we consider the recently observed tension in the
long-baseline neutrino oscillation experiments as a potential indication of
Lorentz invariance violation. For this purpose, the latest data from T2K and
NO$\nu$A is analysed in presence of non-minimal Lorentz invariance violation.
Indeed, we find that isotropic violation in dimensions $D=$ 4, 5 and 6 can
alleviate the tension in neutrino oscillation data by about 0.4–2.4$\sigma$ CL
significance, with the isotropic coefficient $\gamma^{(5)}_{\tau\tau}=$
3.58$\times$10-32GeV-1 yielding the best fit. At the same time, the anomalous
muon $g-2$ result can be reproduced with an additional non-isotropic violation
of $d^{zt}=$ -1.7$\times$10-25. The analysis highlights the possibility of
simultaneous relaxation of experimental tensions with Lorentz invariance
violation of mixed nature.

##  I Introduction

Standard Model (SM) of particle physics is the most successful theory to
describe properties of elementary particles and their interactions. Its
robustness has been tested in numerous experiments, culminating in the
discovery of Higgs boson at the LHC. The conservation of Lorentz invariance
and CPT symmetry are an inseparable part of SM physics, as they ensure that
physics stay the same regardless of the observer. Only recently, the integrity
of SM has started to falter as mounting evidence from electroweak precision
observables, CKM matrix element measurements and neutrino experiments are
showing divergences between experiments and SM predictions. As physicists look
for methods to accommodate SM physics in a more complete theoretical
framework, searching evidence of violation to fundamental symmetries could
provide hints of the underlying theory, such as the formulation of quantum
gravity Kostelecky and Potting (2009); Torri (2020); Antonelli  _et al._
(2020).

CPT is a fundamental symmetry that is conserved in quantum field theories set
in a flat spacetime Schwinger (1951); Luders (1957). It is closely related to
Lorentz invariance Greenberg (2002). Well-known examples of theories that give
rise to Lorentz Invariance Violation (LIV) or CPT violation are found in the
string theory Kostelecky and Samuel (1989); Kostelecky and Potting (1991),
while LIV could also arise in supersymmetry together with CPT violation Berger
and Kostelecky (2002) and even without it Gaete  _et al._ (2021). An example
of a simple non-local field theory of CPT violation can be found in Ref.
Barenboim and Lykken (2003). Generally speaking, theories that uphold any non-
trivial space-time dependence on the vacuum also lead to violation of either
Lorentz invariance, CPT symmetry, or both. Lorentz invariance can furthermore
be violated isotropically or into a specific direction. Effects of LIV and CPT
violation are typically studied in the effective field theory framework, the
most general one being the famous Standard Model Extension Colladay and
Kostelecky (1998) where SM Hamiltonian is expanded to arbitrary dimensions.
Evidence of LIV or CPT-odd operators have been searched in many experiments
from atmospheric neutrino fluxes to gravitational waves Tomar  _et al._
(2015); Stecker (2017); Wei  _et al._ (2019); Rahaman (2021); Wang  _et al._
(2021), which have led to very stringent constraints especially for high-order
operators Coleman and Glashow (1999); Kostelecky and Russell (2008).

In this work, we study the prospects of uncovering LIV physics in neutrino
oscillation experiments. Neutrino oscillations stand out as the first direct
evidence of physics beyond SM, making neutrino experiments an ideal platform
to look for new physics. In recent years, significant advances have been made
in the precision measurements of the standard oscillation parameters:
$\theta_{13}$ has been measured at a few-percent-level at 90% confidence level
(CL) in reactor experiments, $\theta_{23}$ and $|\Delta m_{31}^{2}|$ in
atmospheric and long-baseline neutrino experiments, and $\theta_{12}$ and
$\Delta m_{21}^{2}$ in a combination of solar and reactor neutrino experiments
Esteban  _et al._ (2020). At the same time, observations of neutrino anomalies
and tensions in oscillation data Aguilar-Arevalo  _et al._ (2001); Giunti and
Laveder (2011); Mention  _et al._ (2011); Aguilar-Arevalo  _et al._ (2018)
have seeded an intense discussion over whether new physics is in play Ohlsson
(2013); Denton  _et al._ (2021); Hu  _et al._ (2021); Dasgupta and Kopp
(2021). We investigate the tension in the long-baseline neutrino experiments
T2K and NO$\nu$A Abe  _et al._ (2021a); Acero  _et al._ (2021), where recent
data has shown contradicting results on the parameters $\theta_{23}$ and
$\delta_{CP}$. We study the parameter discrepancies in T2K and NO$\nu$A data
in light of non-minimal LIV, focusing on the lesser constrained dimensions 4,
5 and 6. It is shown in this work that that isotropic LIV could notably
alleviate the tension observed in the three-neutrino mixing. We also show how
the recent muon $g-2$ measurements Abi  _et al._ (2021) could simultaneously
arise from the LIV operators when non-isotropic coefficients are present.

This article is organized as follows: In section II, we provide a brief
overview of the theoretical formalism of LIV effects in neutrino oscillations
and muon $g-2$. In section III, we describe the experimental data and
numerical methods adopted in this work. The results on fitting LIV
coefficients into T2K and NO$\nu$A data are presented in section IV. We leave
the concluding remarks in section V.

##  II Lorentz Invariance Violation in neutrino oscillations

In this section, we review the theoretical framework to study LIV in neutrino
oscillations. We first consider the isotropic LIV effects in section II.1. The
non-isotropic implementation of LIV is then presented in section II.2.
Finally, we discuss the implications to LIV coefficients from recent muon
$g-2$ measurements in section II.3.

###  II.1 Isotropic violation

In the following, we summarize the effective Hamiltonian describing neutrino
oscillations under Lorentz invariance violation. The method described in this
section was originally proposed in Coleman and Glashow (1999) and further
developed in Cohen and Glashow (2006); Kostelecky and Mewes (2012); Antonelli
_et al._ (2018). We focus on LIV terms that appear in the kinetic term. The
generalized kinetic therm of the neutrino Lagrangian can be written as

$\mathcal{L}_{D}=i\nu^{\dagger}_{iL}\bar{\sigma}^{\mu}\partial_{\mu}\left[\delta_{ij}-i^{D-4}\gamma_{ij}^{\mu_{1}...\mu_{D-4}}\partial_{\mu_{1}}...\partial_{\mu_{D-4}}\right]\nu_{jL}$
(1)

where $\gamma_{ij}^{\mu_{1}...\mu_{D-4}}$ are tensors of rank $D-4$ that
parametrizes the size of the LIV and $\sigma^{\mu}$ are the Pauli matrices.

In this work, we mainly focus on the isotropic part of the Lorentz violation.
The isotropic sector is generally confined by less stringent bounds Kostelecky
and Russell (2008), which makes it more interesting to probe in experiments.
Hence, we will set $\mu_{i}=0$ unless stated otherwise. Under the isotropic
hypothesis, there is no violation related to direction of the momentum and the
neutrino propagation Hamiltonian can therefore be written by two terms, that
is,

$\displaystyle H=H_{0}+H_{\rm LIV},$ (2)

where $H_{0}$ stands for the standard Hamiltonian consistent with SM and
$H_{\rm LIV}$ contains the Lorentz-violating terms. In the flavour basis, the
standard Hamiltonian is given by $H_{0}=U^{\dagger}{\rm diag}[0\Delta
m_{21}^{2}/2E,\Delta m_{31}^{2}/2E]U+V(x)$ and LIV part by

$H_{\rm
LIV}=\left(\begin{array}[]{ccc}\gamma^{(D)}_{ee}&\gamma^{(D)}_{e\mu}&\gamma^{(D)}_{e\tau}\\\
\gamma^{(D)*}_{e\mu}&\gamma^{(D)}_{\mu\mu}&\gamma^{(D)}_{\mu\tau}\\\
\gamma^{(D)*}_{e\tau}&\gamma^{(D)*}_{\mu\tau}&\gamma^{(D)}_{\tau\tau}\end{array}\right)E^{D-3}\,,$
(3)

where we simplified our notation to
$\gamma_{\alpha\beta}^{0...0}=\gamma^{(D)}_{\alpha\beta}$. The parameters
$\gamma^{(D)}_{\alpha\beta}$ have mass dimension $4-D$. They are CPT-odd when
$D$ is odd and CPT-even when $D$ is even. The corresponding Hamiltonian for
antineutrinos receives a minus sign when $D$ is an odd number. Parameters
$\gamma_{\ell\ell}^{(D)}$ ($\ell=e$, $\mu$, $\tau$) are real and
$\gamma_{\ell\ell^{\prime}}^{(D)}$ ($\ell\neq\ell^{\prime}$) are complex
parameters with phases $\phi_{\ell\ell^{\prime}}^{(D)}$. In the following, we
mainly study LIV parameters in the flavour basis where
$\gamma^{(D)}_{\ell\ell^{\prime}}=|\gamma^{(D)}_{\ell\ell^{\prime}}|e^{-i\phi_{\ell\ell^{\prime}}^{(D)}}$.

###  II.2 Non-isotropic violation

In case of non-isotropic Lorentz invariance violation, the strength of LIV
effect depends on the direction of the propagating neutrino. In such case, the
equation of state can be decomposed into two parts Kostelecky and Mewes
(2004):

$\displaystyle\left[i\delta_{\alpha\beta}\partial_{t}-H_{0}-\delta
H\right]\nu_{\beta}=0\,,$ (4)

where $H_{0}$ conserves Lorentz invariance and $\delta H$ violates it.
Assuming the neutrino propagates into direction $\bf x$, the conserving part
can be written as

$\displaystyle
H_{0}=-\gamma^{0}\left(i\delta_{\alpha\beta}\boldsymbol{\gamma}\cdot\nabla_{\bf
x}-M_{\alpha\beta}\right),$ (5)

where the neutrino mass matrix is given by $M_{\alpha\beta}$ ($\alpha$,
$\beta=e$, $\mu$, $\tau$). The perturbation introducing Lorentz invariance
violation on the other hand can be presented as

$\delta H=-\frac{1}{2}d_{\alpha\beta}^{\mu
0}\left(\gamma^{0}\gamma_{5}\gamma_{\mu}H_{0}+H_{0}\gamma^{0}\gamma_{5}\gamma_{\mu}\right)\\\
-id_{\alpha\beta}^{\mu j}\gamma^{0}\gamma_{5}\gamma_{\mu}\partial_{j},$ (6)

where $d_{\alpha\beta}^{\mu j}$ is the strength of the Lorentz violation into
the direction $\vec{x}_{j}$ ($j=$ 1, 2, 3). We assume neutrino propagation
align with direction $\vec{x}_{3}$ and keep only the axial-current operator
related to $\gamma_{5}\gamma_{\mu}$. As we shall see in the following section,
parameter $d_{\alpha\beta}^{\mu j}$ could have profound implications in the
calculation of muon anomalous magnetic moment.

In order to derive the evolution equation for neutrinos, the neutrino mass
matrix $M$ must be diagonalized. This can be accomplished within the
relativistic limit $|{\bf p}|\gg m_{i}$, where $p_{\mu}\approx(|{\bf p}|,-{\bf
p})$. In such a case the evolution equation is given by 111The full derivation
of equation (7) can be found in Appendix A of Ref. Kostelecky and Mewes
(2004).

$\displaystyle i\partial_{t}\psi_{i}=\left(|{\bf p}|+\frac{m_{i}^{2}}{2|{\bf
p}|}-\frac{1}{|{\bf p}|}d^{\mu\nu}_{\alpha\beta}p_{\mu}p_{\nu}\right)\psi_{i}$
(7)

where $\psi_{i}$ is the mass eigenstate of the Hamiltonian $H=H_{0}+\delta H$.
It is also convenient to express coordinates in the Sun-centered inertial
frame Kostelecky and Mewes (2004),

$\displaystyle i\partial_{t}\psi_{i}=\left(|{\bf p}|+\frac{m_{i}^{2}}{2|{\bf
p}|}-d^{zt}_{ij}|{\bf p}|\cos\Theta\right)\psi_{i},$ (8)

where $\Theta$ denotes colatitude 222Colatitute is expressed as complement of
latitude, that is, $\Phi$ = $\pi-\Theta$. of the Earth. Neutrino experiments
with large variation in colatitude can therefore provide sensitive probes to
LIV of this kind.

In neutrino oscillations, we introduce non-isotropic LIV to oscillation
probabilities. To do this, we include the non-isotropic coefficient $d^{zt}$
from equation (8) in addition to the isotropic LIV parameters
$\gamma_{\ell\ell^{\prime}}^{(D)}$ in the Hamiltonian (3).

###  II.3 Implications of $(g-2)_{\mu}$ measurement

In this subsection, we calculate the contribution to anomalous muon magnetic
moment from Lorentz invariance violation. It was recently reported by $g-2$
collaboration that the muon magnetic moment is different from the Standard
Model prediction by 4.2$\,\sigma$ CL Abi  _et al._ (2021). We show in the
following that this value can be accommodated with the non-isotropic LIV
coefficient $d_{zt}$ in mass dimension $D=$ 5.

The general Lagrangian responsible for LIV in muonic sector can be written as
Bluhm  _et al._ (2000)

$\mathcal{L}_{LIV}=\overline{\mu}\left[-a_{\mu}\gamma^{\mu}+b_{\mu}\gamma_{5}\gamma^{\mu}+\frac{i}{2}c^{\alpha\beta}\gamma_{\alpha}D_{\beta}+\frac{i}{2}d^{\alpha\beta}\gamma_{5}\gamma_{\alpha}D_{\beta}-\frac{1}{2}H^{\alpha\beta}\sigma_{\alpha\beta}\right]\mu$
(9)

where $a_{\alpha}$, $b_{\alpha}$, $c^{\alpha\beta}$, $d^{\alpha\beta}$ and
$H^{\alpha\beta}$ parameterize LIV for muons and
$D^{\alpha}=\partial^{\mu}+A^{\mu}$ is the covariant derivative. Whereas
$a_{\alpha}$ and $b_{\alpha}$ are CPT-odd, other coefficients are CPT-even.
Most of these parameters are strictly constrained by previous experiments
Kostelecky and Russell (2008).

The general contribution to the muon $g-2$ frequency is given by Gomes  _et
al._ (2014)

$\displaystyle\delta\omega^{\pm}=2\sum_{dnjm}E^{D-3}e^{im\omega_{\oplus}t}G_{jm}\left[\check{H}^{(D)}_{nlm}\pm\check{g}^{(D)}_{nlm}\right],$
(10)

where $E^{D-3}$ denotes neutrino energy in dimension $D=$ 3, 4, 5…,
$\omega_{\oplus}$ is the Earth’s frequency around the Sun and factors
$\check{H}^{D}_{nlm}$ and $\check{g}^{D}_{nlm}$ are combinations of parameters
$b_{\mu}$, $d^{\alpha\beta}$ and $H_{\alpha\beta}$. Function
$G_{jm}=G_{jm}(\Theta)$ on the other hand depends on colatitude $\Theta$ and
vanishes for the isotropic case $j=$ 0\. LIV must therefore be non-isotropic
to explain the anomalous muon $g-2$ value.

In case of non-isotropic LIV, the contribution to muon $g-2$ arises from mass
dimension $D=$ 5\. When $j=$ 1, $m=$ 0, the $G_{jm}$ function yields

$\displaystyle\delta\omega=-2\frac{E^{2}}{m_{\mu}}G_{10}\sqrt{\frac{4\pi}{3}}d^{zt}=-2\frac{E^{2}}{m_{\mu}}d^{zt}\cos\Theta,$
(11)

where $G_{10}(\Theta)=0.5\sqrt{3/\pi}\cos\Theta$. The total correction to SM
value of muon magnetic moment $a_{\mu}$ is therefore

$\displaystyle\Delta a_{\mu}=-\frac{2E^{2}\cos\Theta}{eB}d^{zt},$ (12)

where $B$ is the magnetic field. The $g-2$ collaboration Abi  _et al._ (2021)
reported the value of $\Delta a_{\mu}=251\times 10^{-11}$ in their recent
measurement, which was acquired in magnetic field $B=1.45$ T and colatitude
$\Theta=$ 48.2∘ Keshavarzi (2019). This correction to muonic $g-2$ can be also
obtained with non-isotropic LIV coefficient $d^{zt}=$ -1.7$\times$10-25 from
equation (12). Such a coefficient is well within the present experimental
bounds from the neutrino sector Kostelecky and Russell (2008). For more
details about the $g-2$ measurement, see also Refs. Davier  _et al._ (2011);
Colangelo  _et al._ (2017); Davier  _et al._ (2017, 2020); Aoyama  _et al._
(2020).

##  III Description of the neutrino oscillation data

In the present work, we focus on the analysis of the neutrino oscillation data
from the presently running Tokai-to-Kamioka (T2K) and NuMI Off-axis $\nu_{e}$
Appearance (NO$\nu$A) experiments. T2K and NO$\nu$A are long-baseline
accelerator based experiments where intensive beams of neutrinos and
antineutrinos are created by colliding protons on a fixed target. In this
work, we focus on the recent data releases from the T2K and NO$\nu$A
collaborations reported in Refs. Abe  _et al._ (2021a) and Acero  _et al._
(2021), respectively. A particular area of interest is the observed
discrepancy in the $\theta_{23}$ measurement, which has shown a tension in
neutrino oscillation data between T2K and NO$\nu$A according to an analysis in
the standard neutrino mixing picture. In this section, we briefly review the
experimental data and the analysis methods that are used in our work.

###  III.1 T2K experiment

Tokai-to-Kamioka (T2K) experiment is one of the two presently running long-
baseline neutrino oscillation facilities. T2K uses a proton accelerator of 750
kW average output to generate muon neutrino and antineutrino beams. Based on
the J-PARC campus in Tokai, Japan, the neutrino beam traverses 295 km away to
Kamioka, where it is met by SuperKamiokande neutrino detector. The neutrino
beam is also monitored at the near and intermediate detector facilities ND280
and INGRID, respectively. Super-Kamiokande and ND280 are both located 2.5∘ off
the beam axis, where the observed neutrinos are mainly of about 600 MeV
energy. The beam polarity can be switched between muon neutrino and
antineutrino modes, with the initial strategy of dividing the operational time
into 2 years in neutrino mode and 6 years in antineutrino mode, respectively.
A second run is planned for T2K experiment (T2K-II), with the aim to continue
the successful run of the first stage.

The neutrino beam used in T2K consists predominantly of muon neutrinos
(96%-98%), accompanied by smaller components of beam-related backgrounds. The
corresponding antineutrino beam has similar composition with muon
antineutrinos taking the majority. Most neutrinos and antineutrinos interact
via charged-current quasi-elastic (CCQE) interaction with a small but
observable chance for resonant charged-current pion production (CC1$\pi$). The
neutrino data collected in SuperKamiokande is typically reported in five
different samples: two appearance channels measuring
$\nu_{\mu}\rightarrow\nu_{e}$ and $\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$
oscillations via CCQE interaction and two disappearance channels
$\nu_{\mu}\rightarrow\nu_{\mu}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$. There is also a third appearance
channel dedicated to $\nu_{\mu}\rightarrow\nu_{e}$ oscillations observed via
$\nu_{e}$CC1$\pi^{+}$ interaction. All neutrino events are reconstructed from
the information coming from the Cherenkov light when charged particles
interact with the water content of the neutrino detectors.

In this work we analyse the neutrino data that has been collected in the first
phase of T2K between years 2009 and 2018. In Refs. Abe  _et al._ (2021a, b,
2020), T2K collaboration reported neutrino oscillation data from
3.13$\times$1021 protons-on-target (POT). The collected data contains the
information about the reconstructed neutrino and antineutrino events from the
combined run of 1.49$\times$1021 POT in neutrino beam and 1.64$\times$1021 POT
in antineutrino beam, respectively. The $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ CCQE
samples are divided into 28 and 19 energy bins in the interval [0.2, 3.0] GeV.
Correspondingly, $\nu_{e}$ and $\bar{\nu}_{e}$ CCQE are scattered over 23
same-size energy bins within [0.1, 1.25] GeV interval and
$\nu_{e}$CC1$\pi^{+}$ in 16 bins within [0.45, 1.25] GeV respectively. All
neutrino and antineutrino events were recorded in SuperKamiokande, which is
assumed to have 22.5 kton of fiducial mass.

###  III.2 NO$\nu$A experiment

NuMI Off-axis $\nu_{e}$ Appearance (NO$\nu$A) experiment is the second of the
two long-baseline neutrino oscillation experiment currently collecting data.
Based in the United States, NO$\nu$A generates beams of muon neutrinos and
antineutrinos and sends them to traverse 810 km underground. The neutrino
source is based on the NuMI beamline in Fermilab, Illinois, which produces
neutrinos with an average beam power of 700 kW. The detector facilities in
NOvA are a near detector located 1 km from the beam facility and a far
detector stationed at an underground laboratory in Ash River, Minnesota. Both
detector facilities are placed 0.8∘ off-axis from the source. In contrast to
T2K, neutrinos and antineutrinos produced in NO$\nu$A spread over a wide range
of energies around 2 GeV. Neutrino interactions observed in NO$\nu$A therefore
consist of various types of charged-current (CC) interactions.

The neutrino and antineutrino data analyzed in this work is based on the
events that were collected in NO$\nu$A far detector in 2014-2020 Acero  _et
al._ (2021). The considered data sets were acquired in the far detector, which
is a segmented detector consisting of alternating planes of PVC scintillator.
The neutrino detection in NO$\nu$A is based on scintillation light that is
emitted by the charged particles created in neutrino-nucleus interactions. The
data consist of six different samples which correspond to 12.5$\times$1020 POT
in neutrino mode and 13.6$\times$1020 POT in antineutrino modes. There are
four electron-like characterising $\nu_{\mu}\rightarrow\nu_{e}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ oscillations and two muon-like
samples describing $\nu_{\mu}\rightarrow\nu_{\mu}$ and
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{\mu}$ disappearance. The electron-like
samples are assigned into two categories based on the purity of each event:
low-CNNevt and high-CNNevt. Here CNNevt stands for the convolutional neural
network used in particle identification. The muon-like events on are split
into 19 unequally spaced energy bins in the range [0.75, 4.0] GeV, whereas the
electron-like events are distributed in 6 same-size bins over [1.0, 4.0] GeV
interval. There is also the so-called peripheral sample included in NO$\nu$A,
which is used to increase the number of pure electron-like events. In this
work, we consider the samples for the electron-like, muon-like and peripheral
events while assuming 14 kton fiducial mass in the far detector.

###  III.3 Numerical analysis

The analysis of T2K and NO$\nu$A data is based on the $\chi^{2}$ method. The
numerical analysis conducted in this work is done with General Long Baseline
Experiment Simulator (GLoBES) Huber  _et al._ (2005, 2007), which has been
modified to calculate neutrino evolution with Lorentz invariance violation.
The essential parameters as well as the collected neutrino events are
summarized for the T2K and NO$\nu$A experiments in table 1.

The neutrino data is analysed with the following $\chi^{2}$ function:

$\begin{split}\chi^{2}&=\sum_{i}2\left[T_{i,d}-O_{i,d}\left(1+\log\frac{O_{i,d}}{T_{i,d}}\right)\right]\\\
&+\frac{\zeta_{\text{sg}}^{2}}{\sigma_{\zeta_{\text{sg}}}^{2}}+\frac{\zeta_{\text{bg}}^{2}}{\sigma_{\zeta_{\text{bg}}}^{2}}+{\rm
priors},\end{split}$ (13)

where index $i=$ 1, 2, … runs over the energy bins. Here $O_{i}$ and $T_{i}$
stand for the observed and theoretical (predicted) events in the far detectors
of T2K and NO$\nu$A. Nuisance parameters $\zeta_{\text{sg}}$ and
$\zeta_{\text{bg}}$ are used in the calculation to reflect systematic
uncertainties in the signal and background events, respectively. The
systematic uncertainties are addressed with the so-called pull-method Fogli
_et al._ (2002). The systematic uncertainties considered in this work
influence the predicted events $T_{i}$ in neutrino detectors with a simple
shift:
$T_{i,d}=(1+\zeta_{\text{sg}})N^{\text{sg}}_{i,d}+(1+\zeta_{\text{bg}})N^{\text{bg}}_{i,d}$,
where $N^{\text{sg}}_{i,d}$ and $N^{\text{bg}}_{i,d}$ denote the predicted
signal and background events, respectively. The prior function is defined as
the Gaussian distributions of each of the standard neutrino oscillation
parameters.

Calculation of theoretically predicted events $T_{i}$ is performed entirely by
GLoBES. The number of events can be described with the formula

$T_{i}=N_{\rm nucl}\,T\,\epsilon\int_{E_{\rm min}}^{E_{\rm max}}\int_{E_{\rm
min}^{{}^{\prime}}}^{E_{\rm
max}^{{}^{\prime}}}dE\,dE^{{}^{\prime}}\,\phi(E)\,\sigma(E)\,R(E,E^{{}^{\prime}})\,P_{\nu_{\ell}\rightarrow\nu_{\ell^{\prime}}}(E),$
(14)

where properties of the neutrino experiment are integrated over the true
energy $E$ and reconstructed energy $E^{\prime}$ of the incident energy. The
first part, $N_{\rm nucl}\,T\,\epsilon$, is independent of energy and is
defined by the number of nucleons in the neutrino detector, operational time
of the experiment and detector efficiency, respectively. The integrand on the
other hand is formed by the neutrino flux $\phi(E)$, cross-section
$\sigma(E)$, energy resolution function $R(E,E^{{}^{\prime}})$ and oscillation
probability $P_{\nu_{\ell}\rightarrow\nu_{\ell^{\prime}}}(E)$. In this work,
we adopt the neutrino fluxes and cross-sections for T2K and NO$\nu$A from
Refs. Cao (2018); Abe  _et al._ (2018) and Hewes ; Messier (1999); Paschos and
Yu (2002), respectively.

Table 1:  Summary of the experimental configurations considered in the numerical analysis. Both experiments describe superbeam configurations of either Water Cherenkov (WC) or Totally Active Scintillator Detector (TASD) technology. Experiment | T2K | NO$\nu$A
---|---|---
Source location | Japan | USA
Beam power | 750 kW | 700 kW
Protons-on-target | 3.13$\times$1021 | 12.33$\times$1020
Detector type | WC | TASD
Detector mass | 22.5 kt | 14 kt
Baseline length | 295 km | 810 km
Off-axis angle | 2.5∘ | 0.8∘
Data source | Ref. Abe  _et al._ (2021a) | Ref. Acero  _et al._ (2021)

One of very important elements in the analysis of the neutrino oscillation
data is the detector response associated with T2K and NO$\nu$A experiments. In
the event calculation, the detector response is mainly represented by function
$R(E,E^{\prime})$ which relates the incident and reconstructed neutrino
energies $E$ and $E^{\prime}$ with a Gaussian function of width $\sigma_{\rm
res}(E)$. We analyse the T2K and NO$\nu$A far detector data using a modified
energy resolution function, where the Gaussian width is given by $\sigma_{\rm
res}(E)=\alpha E+\beta\sqrt{E}$. We furthermore introduce an additional phase
shift $\gamma$ in detector response function $R(E,E^{\prime})$. The energy
resolution function is determined for T2K and NO$\nu$A by fitting parameters
$\alpha$, $\beta$ and $\gamma$ recursively for each channel until the correct
spectral shape is achieved. Remaining inconsistencies between the official
data and prediction of GLoBES are mitigated with channel and bin-based
efficiencies.

In order to compute the probabilities
$P_{\nu_{\ell}\rightarrow\nu_{\ell^{\prime}}}$ with general Lorentz invariance
violation and $l$, $l^{\prime}=e$, $\mu$, $\tau$, a custom-made probability
code is adopted to include the calculation of isotropic and non-isotropic LIV
effects in GLoBES. It should be noted that Lorentz invariance violation of
higher dimensions could in principle influence the neutrino fluxes $\phi(E)$.
In this work, however, we assume the effect on fluxes to be small and fall
within the present uncertainties of the neutrino fluxes.

The systematic uncertainties used in the $\chi^{2}$ function (13) are one of
the key characteristics in the analysis of the neutrino oscillation data. In
the analysis of experiment data from T2K experiment, we impose 5% systematic
uncertainty on the signal events undergoing CCQE interaction. The
corresponding background events are addressed with 10% systematic uncertainty.
Identical systematic uncertainties are used for the events associated with
CC1$\pi$. This choice of priors for the nuisance parameters are found to
reproduce the fit results reported by the T2K Collaboration in Ref. Abe  _et
al._ (2021a) with sufficient accuracy. In a similar manner, we implement
systematic uncertainties on the each channel considered in the NO$\nu$A
experiment. In the analysis of NO$\nu$A far detector data, systematic
uncertainties are driven by detector calibration, which amounts to about 5%
systematic uncertainty Acero  _et al._ (2021). We treat electron-like samples
with low and high CNNevt with the same pull parameters. The systematic
uncertainties used here are found to be adequate to reproduce the official
results reported by the NO$\nu$A Collaboration in Ref. Acero  _et al._ (2021).

##  IV Numerical results

We present the results of our numerical analysis in this section. In this
work, we investigated effects of general Lorentz invariance violation (LIV) in
dimensions $D=$ 4, 5 and 6. The effects were studied in the context of the
$\theta_{23}$ discrepancy recently observed in T2K and NO$\nu$A. In the
following, we examine whether LIV could alleviate the observed tension between
the recent data in T2K and NO$\nu$A and improve the fit to the standard
neutrino oscillation parameters. At the same time, we study the effect of LIV
in the anomalous muon magnetic moment. As we shall see in this section,
isotropic LIV with dimension $D=$ 5 provides the best fit result to the T2K
and NO$\nu$A data while its non-isotropic version is simultaneously able to
resolve $(g-2)_{\mu}$.

We first begin investigation with the isotropic Lorentz invariance violation.
The analysis of the T2K and NO$\nu$A far detector data is carried out in
dimensions $D=$ 4, 5 and 6. The first goal of this study is to identify the
LIV parameters that lead to significant improvement on the goodness-of-fit in
the T2K and NO$\nu$A data. The second goal is to determine the resulting
effect on the fit values of $\sin^{2}\theta_{23}$ and $\delta_{CP}$.

Table 2: Fit results from the scan of isotropic Lorentz invariance violation in dimension $D=$ 4, 5 and 6. The results are obtained from a combined fit to T2K and NO$\nu$A far detector data. The consequent improvement to the fit is shown in the $\Delta\chi^{2}$ and significance columns. Results are obtained assuming normal ordering for neutrino masses. Parameter | Fit value ($\sin^{2}\theta_{23}$) | Fit value ($\delta_{CP}/\pi$) | Fit value ($|\gamma_{\ell\ell^{\prime}}|$) | $\Delta\chi^{2}$ | Significance (C.L.)
---|---|---|---|---|---
$\gamma_{ee}^{(4)}$ | 0.562 | 1.33 | -4.62$\times$10-23 | 0.89 | 0.95$\sigma$
$\gamma_{e\mu}^{(4)}$ | 0.562 | 1.48 | 0.88$\times$10-23 | 3.96 | 1.48$\sigma$
$\gamma_{e\tau}^{(4)}$ | 0.563 | 1.49 | 0.95$\times$10-23 | 4.24 | 1.56$\sigma$
$\gamma_{\mu\mu}^{(4)}$ | 0.522 | 1.30 | 5.58$\times$10-23 | 2.47 | 1.57$\sigma$
$\gamma_{\mu\tau}^{(4)}$ | 0.549 | 1.43 | -3.75$\times$10-23 | 4.80 | 1.69$\sigma$
$\gamma_{\tau\tau}^{(4)}$ | 0.543 | 1.22 | -2.65$\times$10-23 | 0.57 | 0.75$\sigma$
$\gamma_{ee}^{(5)}$ | 0.561 | 1.48 | -4.65$\times$10-32 GeV-1 | 4.34 | 2.08$\sigma$
$\gamma_{e\mu}^{(5)}$ | 0.561 | 1.50 | 0.28$\times$10-32 GeV-1 | 3.85 | 1.45$\sigma$
$\gamma_{e\tau}^{(5)}$ | 0.559 | 1.49 | 1.17$\times$10-32 GeV-1 | 4.41 | 1.60$\sigma$
$\gamma_{\mu\mu}^{(5)}$ | 0.524 | 1.31 | -3.32$\times$10-32 GeV-1 | 2.73 | 1.65$\sigma$
$\gamma_{\mu\tau}^{(5)}$ | 0.560 | 1.26 | -0.45$\times$10-32 GeV-1 | 0.65 | 0.36$\sigma$
$\gamma_{\tau\tau}^{(5)}$ | 0.526 | 1.45 | 3.58$\times$10-32 GeV-1 | 5.79 | 2.41$\sigma$
$\gamma_{ee}^{(6)}$ | 0.562 | 1.34 | -1.15$\times$10-41 GeV-2 | 1.00 | 1.00$\sigma$
$\gamma_{e\mu}^{(6)}$ | 0.560 | 1.43 | 0.11$\times$10-41 GeV-2 | 3.05 | 1.23$\sigma$
$\gamma_{e\tau}^{(6)}$ | 0.560 | 1.46 | 0.31$\times$10-41 GeV-2 | 4.90 | 1.72$\sigma$
$\gamma_{\mu\mu}^{(6)}$ | 0.548 | 1.25 | 0.81$\times$10-41 GeV-2 | 1.32 | 1.15$\sigma$
$\gamma_{\mu\tau}^{(6)}$ | 0.554 | 1.27 | -0.17$\times$10-41 GeV-2 | 1.29 | 0.64$\sigma$
$\gamma_{\tau\tau}^{(6)}$ | 0.544 | 1.22 | -0.28$\times$10-41 GeV-2 | 0.15 | 0.39$\sigma$

The analysis of the T2K and NO$\nu$A data is carried out as follows. Using the
methods described in section III, the far detector data in the two experiments
is fitted with a $\chi^{2}$ function. We fix the solar parameters
$\theta_{12}$ and $\Delta m_{21}^{2}$ to their respective best-fit values
33.4∘ and 7.4$\times$10-5eV2 following the most recent global fit Esteban  _et
al._ (2020). We also impose a prior from the reactor experiments:
$\sin^{2}2\theta_{13}=$ 0.0857$\pm$0.0046. The $\chi^{2}$ distribution is
computed for parameters
$\gamma^{(D)}_{\ell\ell^{\prime}}=|\gamma^{(D)}_{\ell\ell^{\prime}}|\exp{(-i\phi^{(D)}_{\ell\ell^{\prime}})}$
keeping one LIV parameter free each time. To see the difference to the
standard fit, we also obtain the $\chi^{2}$ distribution corresponding to the
case where all LIV parameters are kept at zero.

The fit results to $\sin^{2}\theta_{23}$ and $\delta_{CP}$ from the joint
analysis is T2K and NO$\nu$A data are presented in table 2. The LIV parameters
$\gamma_{\ell\ell^{\prime}}^{(D)}$ were analysed one by one in dimensions $D=$
4, 5 and 6. In each row, the denoted parameter was let to run free in order to
obtain the fit result. The resulting improvement to the fit result is listed
in the $\Delta\chi^{2}$ column, where
$\Delta\chi^{2}=\chi^{2}_{SM}-\chi^{2}_{LIV}$ is calculated from $\chi^{2}$
values attributed to the standard and LIV-influenced fits, respectively.
Finally, the significance column indicates the confidence level (C.L.) at
which the fit obtained in presence of the LIV parameter is favoured over the
standard fit result. The best fit result is obtained with parameter
$\gamma^{(5)}_{\tau\tau}$, which can yield an enhancement of about
2.4$\,\sigma$ C.L. in statistical significance. All results are consistent
with the experimental bounds reported in the literature Kostelecky and Russell
(2008), including the stringent bounds from IceCube Aartsen  _et al._ (2018).
The statistical significance is computed using Wilks’ theorem Wilks (1938).
For the caveats concerning the statistical analysis in T2K and NO$\nu$A data,
see Ref. Algeri  _et al._ (2020).

The effects of the Lorentz invariance violation on the $\theta_{23}$ and
$\delta_{CP}$ measurements in T2K and NO$\nu$A experiments are shown in Fig.
1. The standard fits on T2K and NO$\nu$A data are illustrated with solid red
and blue lines, respectively, whereas the fits obtained with Lorentz
invariance violation are indicated with dashed lines. The discrepancy in the
$\theta_{23}$ measurements is evident in the standard fits, which are clearly
separated in T2K and NO$\nu$A. This tension is partially removed with the
introduction of LIV coefficient $\gamma^{(5)}_{\tau\tau}$. Imposing LIV into
the fit leaves the T2K fit mostly unaffected but shifts the NO$\nu$A fit
significantly towards the $\theta_{23}$ value that is preferred by the T2K
data. This results in an alleviation of tension in the T2K and NO$\nu$A
measurements. The tension between the T2K and NO$\nu$A data can also be seen
the fit results on $\delta_{CP}$, which show nearly $\Delta\chi^{2}\simeq$ 5
difference at the value that is currently preferred by T2K data. Marginalizing
over $\gamma^{(D)}_{\tau\tau}$ reduces this difference to
$\Delta\chi^{2}\simeq$ 2\. This amounts to about 1.8$\,\sigma$ C.L.
improvement.  Figure 1: The effect of Lorentz invariance violation on the
determination of $\theta_{23}$ (left panel) and $\delta_{CP}$ (right panel) in
T2K and NO$\nu$A experiments. Fit results are shown for T2K and NO$\nu$A
experiments with red and blue lines, respectively. Solid lines correspond to
the standard fit whereas dashed lines represent fit results where
$\gamma^{(5)}_{\tau\tau}$ is taken as a free parameter.

The impact of each LIV parameter on the T2K and NO$\nu$A analysis is further
elaborated in Fig. 2. Each panel portrays the fit to
$\delta_{CP}-\sin^{2}\theta_{23}$ plane as obtained from the joint analysis of
the two experiments. The red curves represent the edges of the enclosed
parameter space in the standard neutrino oscillations, as allowed by 90% C.L.
in T2K and NO$\nu$A. The best-fit point is marked with the red dot. On the
other hand, the corresponding limits obtained with the freely-varying LIV
parameters are presented with blue colour. The results show how the best-fit
points shift upon the introduction of the single LIV coefficient for $D=$ 5\.
The fits reveal that the best fit values for $\sin^{2}\theta_{23}$ and
$\delta_{CP}$ shift significantly for parameters $\gamma^{(D)}_{\mu\mu}$ and
$\gamma^{(D)}_{\tau\tau}$, whereas the inclusion of $\gamma^{D}_{ee}$,
$\gamma^{D}_{e\mu}$ or $\gamma^{D}_{e\tau}$ only has notable effect on the fit
value of $\delta_{CP}$. The last off-diagonal parameter $\gamma^{D}_{\mu\tau}$
on the other hand leads to no significant changes in the fit values of
$\sin^{2}\theta_{23}$ and $\delta_{CP}$. The fit results obtained for
dimensions $D=$ 4 and 6 are analogous and are not shown here.

We finally make a remark on the effect of non-isotropic Lorentz invariance
violation in neutrino experiments. As we pointed out in section II.2, the
recently measured anomalous muon magnetic moment Abi  _et al._ (2021) could be
explained with non-isotropic LIV coefficient $d^{zt}=-$1.7$\times$10-25. We
investigated the potential effect of this solution in the long-baseline
experiments T2K and NO$\nu$A. Our results show non-isotropic coefficient too
small to induce notable effects in T2K and NO$\nu$A 333The smallness of the
non-isotropic effect is mainly owed to the relatively small difference in
colatitudes between T2K and NO$\nu$A. In T2K, colatitude is given by
$\cos\Theta=$ 0.993 whereas in NO$\nu$A it is $\cos\Theta=$ 1.000. Future
experiments with longer baseline lengths could see more profound effect from
non-isotropic Lorentz invariance violation., where the value of $d^{zt}$
required to satisfy $(g-2)_{\mu}$ leads only to $\chi^{2}_{\rm min}\sim$ 10-3
correction to the fit result. It is therefore possible to recover the measured
$(g-2)_{\mu}$ value with non-isotropic LIV and alleviate the tension on
$\theta_{23}$ in T2K and NO$\nu$A at the same time.

##  V Conclusions

Lorentz invariance violation could have profound effects in the interpretation
of physical observations in laboratories and astrophysical environments. In
the present work, we have investigated non-minimal LIV as a potential solution
to the recently observed tension in the measurement of the atmospheric mixing
angle $\theta_{23}$ and Dirac CP phase $\delta_{CP}$ in the long-baseline
neutrino experiments T2K and NO$\nu$A. To this extent, we interpreted the
recently published experiment data in terms of isotropic and non-isotropic LIV
effects.

In contrast to the previous studies conducted on the topic, we studied the
relatively unexplored Lorentz invariance violation in dimensions $D=$ 4, 5 and
6. Investigating the isotropic effect on the fits to T2K and NO$\nu$A data, we
found that diagonal parameters $\gamma^{(D)}_{\ell\ell^{\prime}}$ ($\ell$,
$\ell^{\prime}=e$, $\mu$ and $\tau$) could resolve the tension by about
0.4–2.4$\,\sigma$ confidence levels when one parameter is considered at a
time. The extracted fit results are consistent the existing bounds on Lorentz
invariance violation in the neutrino sector Kostelecky and Russell (2008). We
found the best fit with $\gamma^{(5)}_{\tau\tau}=$ 3.58$\times$10-32GeV-1,
which leads to an improvement of 2.41$\,\sigma$ C.L. from the standard
scenario where Lorentz invariance remains conserved. In this case, the
favoured values for the atmospheric mixing and Dirac CP phase are
$\sin^{2}\theta_{23}=$ 0.526 and $\delta_{CP}=$ 1.45$\pi$, respectively.

Figure 2: Allowed values on the $\delta_{CP}-\sin^{2}\theta_{23}$ plane
according to the neutrino oscillation data in T2K and NO$\nu$A experiments.
Isotropic Lorentz invariance violation of dimension $D=5$ is included in the
parameter region highlighted with blue colour, whereas standard oscillations
are shown with red colour. The best-fit values identified in the two scenarios
are indicated with blue and red stars, respectively. The results are shown by
90% C.L. while assuming normally ordered neutrino masses.

Lorentz invariance violation is also able to explain the recent results on the
anomalous muon magnetic moment as reported by the Muon $g-2$ Collaboration Abi
_et al._ (2021). The measured value of $(g-2)_{\mu}$ could be generated by
non-isotropic Lorentz invariance violation with directional coefficient
$d^{zt}\simeq-$1.7$\times$10-25. Isotropic Lorentz invariance violation on the
other hand has no effect on the muon magnetic moment. We estimated the impact
of this specific non-isotropic Lorentz invariance violation on the neutrino
oscillations in T2K and NO$\nu$A. We find the effect of the non-isotropic LIV
parameter to be indistinguishable in T2K and NO$\nu$A due to relatively small
change in colatitude in their experimental setups. Though neutrino
oscillations and muon $g-2$ do not favour the same type of LIV, it is
noteworthy that both solutions can exist simultaneously.

In summary, we have shown that non-minimal Lorentz invariance violation can
notably alleviate the $\theta_{23}$ discrepancy in T2K and NO$\nu$A data and
simultaneously give rise to the anomalous muon magnetic moment. Respecting the
present experimental bounds, we note that the significance for isotropic LIV
can be as large as 2.41$\sigma$ C.L. Our results place a mild preference on
dimension-5 LIV. We are looking forward to the future long-baseline neutrino
oscillation data to check whether the observed discrepancy is due to the
violation of Lorentz invariance.

###### Acknowledgements.

This project was supported in part by National Natural Science Foundation of
China under Grant Nos. 12075326, 12090064, 11505301 and 11881240247. PP is
additionally supported by the Shanghai Pujian Program under Grant No.
20PJ1407800 and SV by China Postdoctoral Science Foundation under Grant No.
2020M672930. JT and PP appreciate the support from CAS Center for Excellence
in Particle Physics (CCEPP).

##  Appendix A Implications on mass ordering

In the present work, we have conducted the analysis on T2K and NO$\nu$A data
assuming normal ordering for neutrino masses. For completeness, we now
consider the results in the case of inverted ordering and discuss the
implications on the sensitivity to neutrino mass ordering.

Figure 3 provides the fit results on the $\delta_{CP}-\sin^{2}\theta_{23}$
plane at 90% C.L. in T2K and NO$\nu$A experiments when  Figure 3: Fit results
on the $\delta_{CP}-\sin^{2}\theta_{23}$ plane when inverted ordering is
assumed. Neutrino data from the T2K and NO$\nu$A experiments are analysed in
the standard three-neutrino picture (red colour) and also letting
$\gamma^{(D)}_{\ell\ell^{\prime}}$ run free, whereby $D=$ 5 and $\ell$,
$\ell^{\prime}=e$, $\mu$ and $\tau$. Results are shown at 90% C.L. while the
best-ft values are marked by stars. inverted mass ordering is assumed. The
standard three-neutrino oscillation scenario is taken into account in the red
regions, which correspond to the scenario where no Lorentz invariance
violation takes place. Isotropic LIV is taken into account in the blue
regions, which were obtained by letting $\gamma^{(D)}_{\ell\ell^{\prime}}$ run
free for $D=$ 5 and $\ell$, $\ell^{\prime}=e$, $\mu$ and $\tau$. Each panel
corresponds to the running of one  Figure 4: The effect of Lorentz invariance
violation on the determination of mass hierarchy in T2K and NO$\nu$A. The fits
to the far detector data in T2K (left panel) and NO$\nu$A (right panel) are
presented for normal ordering (NO) and inverted ordering (IO). In the standard
scenario (solid) no violation is taken into account, while
$\gamma_{\tau\tau}^{(5)}$ is marginalized in the LIV scenario (dashed). LIV
parameter, whilst other LIV parameters are fixed at zero. We have similarly
performed the analysis for LIV parameters in dimensions $D=$ 4 and 6, finding
analogous results.

The effects of LIV is studied in the sensitivity to neutrino mass ordering by
comparing the fit results in both neutrino mass orderings. The results are
presented in Figure 4, where the fit results are shown separately for T2K and
NO$\nu$A in left and right panels, respectively. While the fit results
associated with normal ordering (NO) are shown in blue colour, the results
corresponding to inverted ordering (IO) are indicated with red colour. The
sensitivity to the neutrino mass ordering can be seen from the relative
difference of $\Delta\chi^{2}$ between the NO and IO curves. In the standard
oscillation scenario (solid curves), the difference between NO and IO is more
significant in NO$\nu$A than in T2K. The higher sensitivity in NO$\nu$A arises
mainly from the longer baseline length in the experiment. When the LIV effect
is taken into account (dashed curves), the LIV parameter
$\gamma_{\tau\tau}^{(5)}$ is let to vary freely. In such case, the difference
between NO and IO is nearly halved for most values of $\delta_{CP}$,
indicating significant loss in mass hierarchy discrimination.

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  * Note (3) The smallness of the non-isotropic effect is mainly owed to the relatively small difference in colatitudes between T2K and NO$\nu$A. In T2K, colatitude is given by $\mathop{cos}\nolimits\Theta=$ 0.993 whereas in NO$\nu$A it is $\mathop{cos}\nolimits\Theta=$ 1.000. Future experiments with longer baseline lengths could see more profound effect from non-isotropic Lorentz invariance violation.