PHYSICAL REVIEW D 91, 052003 (2015)

Test of Lorentz invariance with atmospheric neutrinos
K. Abe,1,29 Y. Haga,1 Y. Hayato,1,29 M. Ikeda,1 K. Iyogi,1 J. Kameda,1,29 Y. Kishimoto,1,29 M. Miura,1,29 S. Moriyama,1,29
M. Nakahata,1,29 Y. Nakano,1 S. Nakayama,1,29 H. Sekiya,1,29 M. Shiozawa,1,29 Y. Suzuki,1,29 A. Takeda,1,29 H. Tanaka,1
T. Tomura,1,29 K. Ueno,1 R. A. Wendell,1,29 T. Yokozawa,1 T. Irvine,2 T. Kajita,2,29 I. Kametani,2 K. Kaneyuki,2,29,*
K. P. Lee,2 T. McLachlan,2 Y. Nishimura,2 E. Richard,2 K. Okumura,2,29 L. Labarga,3 P. Fernandez,3 J. Gustafson,4
E. Kearns,4,29 J. L. Raaf,4 J. L. Stone,4,29 L. R. Sulak,4 S. Berkman,5 H. A. Tanaka,5 S. Tobayama,5 M. Goldhaber,6,*
G. Carminati,7 W. R. Kropp,7 S. Mine,7 P. Weatherly,7 A. Renshaw,7 M. B. Smy,7,29 H. W. Sobel,7,29 V. Takhistov,7K. S. Ganezer,8 B. L. Hartfiel,8 J. Hill,8 W. E. Keig,8 N. Hong,9 J. Y. Kim,9 I. T. Lim,9 T. Akiri,10 A. Himmel,10
K. Scholberg,10,29 C. W. Walter,10,29 T. Wongjirad,10 T. Ishizuka,11 S. Tasaka,12 J. S. Jang,13 J. G. Learned,14 S. Matsuno,14
S. N. Smith,14 T. Hasegawa,15 T. Ishida,15 T. Ishii,15 T. Kobayashi,15 T. Nakadaira,15 K. Nakamura,15,29 Y. Oyama,15
K. Sakashita,15 T. Sekiguchi,15 T. Tsukamoto,15 A. T. Suzuki,16 Y. Takeuchi,16 C. Bronner,17 S. Hirota,17 K. Huang,17
K. Ieki,17 T. Kikawa,17 A. Minamino,17 A. Murakami,17 T. Nakaya,17,29 K. Suzuki,17 S. Takahashi,17 K. Tateishi,17
Y. Fukuda,18 K. Choi,19 Y. Itow,19 G. Mitsuka,19 P. Mijakowski,34 J. Hignight,20 J. Imber,20 C. K. Jung,20 C. Yanagisawa,20
H. Ishino,21 A. Kibayashi,21 Y. Koshio,21 T. Mori,21 M. Sakuda,21 R. Yamaguchi,21 T. Yano,21 Y. Kuno,22
R. Tacik,23,31 S. B. Kim,24 H. Okazawa,25 Y. Choi,26 K. Nishijima,27 M. Koshiba,28 Y. Suda,28 Y. Totsuka,28,*
M. Yokoyama,28,29 K. Martens,29 Ll. Marti,29 M. R. Vagins,29,7 J. F. Martin,30 P. de Perio,30 A. Konaka,31
M. J. Wilking,31 S. Chen,32 Y. Zhang,32 K. Connolly,33 and R. J. Wilkes33(Super-Kamiokande Collaboration)
1

Kamioka Observatory, Institute for Cosmic Ray Research, University of Tokyo,
Kamioka, Gifu 506-1205, Japan
2
Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo,
Kashiwa, Chiba 277-8582, Japan3
Department of Theoretical Physics, University Autonoma Madrid, 28049 Madrid, Spain
4
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
5
Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia V6T1Z4, Canada
6
Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
7
Department of Physics and Astronomy, University of California,
Irvine, Irvine, California 92697-4575, USA
8
Department of Physics, California State University, Dominguez Hills, Carson, California 90747, USA
9
Department of Physics, Chonnam National University, Kwangju 500-757, Korea
10
Department of Physics, Duke University, Durham, North Carolina 27708, USA
11
Junior College, Fukuoka Institute of Technology, Fukuoka, Fukuoka 811-0295, Japan
12
Department of Physics, Gifu University, Gifu, Gifu 501-1193, Japan
13
GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea
14
Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822, USA
15High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
16
Department of Physics, Kobe University, Kobe, Hyogo 657-8501, Japan17
Department of Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan18
Department of Physics, Miyagi University of Education, Sendai, Miyagi 980-0845, Japan
19
Solar Terrestrial Environment Laboratory, Nagoya University, Nagoya, Aichi 464-8602, Japan20
Department of Physics and Astronomy, State University of New York at Stony Brook,
New York 11794-3800, USA
21
Department of Physics, Okayama University, Okayama, Okayama 700-8530, Japan
22
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
23
Department of Physics, University of Regina,
3737 Wascana Parkway, Regina, Saskatchewan S4SOA2, Canada
24
Department of Physics, Seoul National University, Seoul 151-742, Korea
25
Department of Informatics in Social Welfare, Shizuoka University of Welfare,
Yaizu, Shizuoka 425-8611, Japan
26
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
27
Department of Physics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
28
The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
29
Kavli Institute for the Physics and Mathematics of the Universe (WPI),
Todai Institutes for Advanced Study, University of Tokyo,
Kashiwa, Chiba 277-8582, Japan

1550-7998=2015=91(5)=052003(16)052003-1© 2015 American Physical Society

PHYSICAL REVIEW D 91, 052003 (2015)

K. ABE et al.
30

Department of Physics, University of Toronto,
60 St. George Street, Toronto, Ontario M5S1A7, Canada
31
TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T2A3, Canada
32
Department of Engineering Physics, Tsinghua University, Beijing 100084, China
33
Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA
34
National Centre For Nuclear Research, 00-681 Warsaw, Poland(Received 16 October 2014; published 3 March 2015)
A search for neutrino oscillations induced by Lorentz violation has been performed using 4,438 livedays of Super-Kamiokande atmospheric neutrino data. The Lorentz violation is included in addition to
standard three-flavor oscillations using the nonperturbative standard model extension (SME), allowing
the use of the full range of neutrino path lengths, ranging from 15 to 12,800 km, and energies ranging
from 100 MeV to more than 100 TeV in the search. No evidence of Lorentz violation was observed, so
limits are set on the renormalizable isotropic SME coefficients in the eμ, μτ, and eτ sectors, improving theexisting limits by up to 7 orders of magnitude and setting limits for the first time in the neutrino μτ sectorof the SME.
DOI: 10.1103/PhysRevD.91.052003

PACS numbers: 11.30.Cp, 14.60.Pq

I. INTRODUCTION
Symmetry under Lorentz transformations is a fundamental feature of both the standard model of particle
physics and the general theory of relativity, but violationsof this symmetry at or below the Planck scale,mP ≈ 1019 GeV, have been predicted in a variety ofmodels, including discrete spacetime structure and spacetime foam interactions [1,2]. The direct observation ofLorentz violation (LV) would provide access to thisPlanck-scale physics [3–6]. The standard model extension[4–8] (SME) is an observer-independent effective field
theory with all the features of the standard model andgeneral relativity plus all possible LV terms. Lorentz
violation can also include violation of charge-paritytime reversal (CPT) symmetry [9,10]. At experimentally
accessible energies, these LV signatures are strongly
suppressed by a factor of the order of mW =mP ≈ 10−17 ,the relative magnitudes of the electroweak and Planckscales [11]. Despite this suppression, numerous experimental techniques have been employed to search for LV
phenomena [12–14].
Neutrino oscillations, as an interferometric effect, are asensitive probe of LV with two possible signatures: sidereal
variations, which would be evidence of a preferred spatial
direction, and spectral anomalies [15–17]. Previous
searches within the SME framework have generally
focused on sidereal variations, though they sometimes
include time-independent components that only modify
the spectrum. They have been performed in short-baselinemuon (anti)neutrino beams [18–21], the long-baseline
NuMI neutrino beam [22,23], the Double-Chooz reactor
experiment [24,25], and in atmospheric neutrinos at Ice
Cube [26]. These experiments have generally used either*

Deceased.

the short-baseline [11] or perturbative [27] approximationsof the SME to set limits on LV parameters and report limits
for each parameter independently.
The Super-Kamiokande experiment (SK) [28] is acylindrical, underground, water-Cherenkov detector, witha fiducial mass of 22.5 kton in the inner detector and an
active veto outer detector for tagging cosmic ray muons
entering the detector. The atmospheric neutrinos are incident from all directions with path lengths and energies
spanning 3 and 6 orders of magnitude, respectively. Thiswide range in L and E makes atmospheric neutrinos asensitive tool to probe the coefficients which produce
spectral anomalies. The SK data cover such a wide rangeof lengths and energies that the perturbative SME can nolonger be used and the exact Hamiltonian must be diagonalized [29]. Since the oscillation of massive neutrinos
has been well established in numerous experiments[30–47], the exact Hamiltonian includes three-flavor oscillations and Mikheyev-Smirnov-Wolfenstein (MSW) matter
effects [48] in addition to LV. We investigate the real andimaginary parts of the lowest-order effective CPT-evenand CPT-odd LV coefficients in the eμ, μτ, and eτ sectors.
We are the first neutrino experiment to study isotropic LV
in the μτ sector of the SME; all previous experiments sensitive to this sector searched only for sidereal
variations.
The neutrino oscillation probability, whether fromstandard oscillations or Lorentz violation, depends on
the initial neutrino flavor, the distance the neutrino travels,L, and the neutrino energy, E. We separate our data into
samples correlated with energy and with enhanced νμ , νeor neutral current (NC) flavor content. The events fully
contained (FC) within the inner detector have the lowest
energies. Events that start in the inner detector but thenexit before depositing all their energy are classified as
partially contained (PC) and have generally have higher

052003-2

TEST OF LORENTZ INVARIANCE WITH ATMOSPHERIC …

energies. The up-going muon events (UP-μ ) that enter thedetector having deposited some of their energy in thesurrounding rock beforehand are the highest energy
sample. This sample only contains up-going events toavoid the overwhelming background of down-going cosmic ray muons. While the energy of the muons cannot be
determined event by event, the highest-energy muons will
shower inside the detector and can be identified using themethod described in [49].
We then further bin the data using observables correlatedwith L and E. Instead of path length, we bin the data inzenith angle, cos θz , defined as the angle between the event
direction and the downward vertical direction. The neutrinos with the shortest path lengths are downward-going(cos θz near 1) and the neutrinos with the longest path
lengths are upward-going (cos θz near −1). The simulation
which predicts the number of neutrino events in each bin
includes a distribution of neutrino production heights basedon a model of the atmosphere described in more detail in[50]. This range of production heights introduces a smearing of the oscillation probability for a given zenith angle fordownward-going and horizontal events but is negligible
for upward-going events which cross most of the Earth. For
events with one visible Cherenkov ring, we bin in momentum, which is reconstructed using the total amount of lightwith a 70° cone, and then refined using templates fromsimulation. For multiring events, partially contained events,and stopping UP-μ events, we bin in visible energy, definedas the energy of an electron that would produce the same
total amount of light seen in the detector. The data are
divided into a total of 480 bins for each run period, which
are then combined across run periods before fitting. The
binning is chosen so that enough events are expected ineach bin for the fit to be stable. The binning scheme islargely the same as that used for the standard three-flavor
oscillation analysis [51], with some upgrades describedin [52].
The various data samples and the SK-I through SK-IV
data used in this analysis are described in detail in [52],the event generator, Monte Carlo simulation (MC), andreconstruction are described in [50], and recent improvements are described in [53].
II. LORENTZ VIOLATION IN NEUTRINO
OSCILLATIONSPHYSICAL REVIEW D 91, 052003 (2015)
TABLE I. Lorentz-violating coefficients and their properties.
The last row includes all possible combinations of X; Y; Z,
and T except TT. d refers to the dimension of the operator.
α and β range over the neutrino flavors, e, μ, and τ. The X, Y,
and Z indicate coefficients which introduce effects in aparticular direction in a Lorentz-violating preferred reference
frame. The T and TT terms are not associated with any
direction and thus introduce isotropic distortions in the oscillation pattern.
Coefficient

Unit

d

CPT

Oscillation effect

aTαβ

GeV

3

Odd

∝L

cTT
αβ

…

4

Even

∝ LE

aXαβ ; aYαβ ; aZαβ

GeV

3

Odd

Sidereal variation

YZcXX
αβ ; cαβ ; …

…

4

Even

Sidereal variation

Isotropic

Directional

0
M¼

H ¼ UMU þ V e þ H LV ;

1 B
@0
2E
0

1

0

0

Δm221

0

0

Δm231

C
A;

ð2:2Þ

and V e is the electron potential which introduces matter
effects [48],
1
0
N 0 0
pﬃﬃﬃ B eC
V e ¼  2GF @ 0 0 0 A;
ð2:3Þ
0

0 0

where GF is the Fermi constant and N e is the average
electron density along the neutrino’s path, calculated using
the four-layer Primary Reference Earth Model (PREM) of
the density profile of the Earth [55].
The Lorentz-violating Hamiltonian, HLV , has many
possible terms with complex coefficients summarizedin Table I, broadly categorized as isotropic or directional.
While in principle atmospheric neutrinos are sensitive tothe sidereal variations induced by the directional terms[26], in this analysis we focus only on the isotropic terms
which introduce spectral variations which oscillate depending on L and LE (as opposed to the L=E dependence ofstandard oscillations [56]). The diagonal elements of HLVhave also been neglected since they cannot be observed inoscillations, giving

In the SME, Lorentz violation is included with neutrino
oscillations by adding an LV term, HLV , to the standard
neutrino Hamiltonian,†00052003-3

aTeμB
HLV ¼ @ ðaTeμ ÞðaTeτ Þ0

ð2:1Þwhere U is the Pontecorvo–Maki–Nakagawa–Sakata mixing (PMNS) matrix [54], M is the neutrino mass matrix,

0

−

aTeτ

C
aTμτ A0ðaTμτ Þ
0

4E B TT @ ðceμ Þ
3

ðcTT
eτ Þ

1

0
cTT
eμ
0


ðcTT
μτ Þ

cTT
eτ

1

C
cTT
μτ A
0

ð2:4Þ

PHYSICAL REVIEW D 91, 052003 (2015)

K. ABE et al.
Path Length (km)

10

0.8
3

10

0.6
0.4

102
0.2
10-1

1

10

102

3

10

0.0

Energy (GeV)

1.0

Path Length (km)

104

0.8
3

10

0.6
0.4

102
0.2
10-1

1

10

102

3

10

0.0

Energy (GeV)

1.0

104

Path Length (km)for neutrinos. For antineutrinos, the a parameters go to−ðaT Þ and the cTT parameters go to ðcTT Þ , which isequivalent to ReðaT Þ→−ReðaT Þ and ImðcTT Þ→−ImðcTT Þ.
A common approach in neutrino oscillation experiments is to treat HLV as a small perturbation δh on thestandard model Hamiltonian [27] and calculate the lowest
order nonzero variations in the oscillation probability(first order when standard oscillations are present, second
order when they are not, such as at short baselines). The
sensitivity of Super-K to this model was evaluated [29].
However, for this approach to be valid the perturbationmust be small, defined as jδhj ≪ 1=L. If we take thecondition as jδhj < 10% × ð1=LÞ, then more than 30% of
the events in SK fail this perturbative condition foraT ¼ 5 × 10−24 GeV, resulting in unphysical oscillation
probabilities greater than one and less than zero. Since
the events failing the condition belong to the samples
most sensitive to Lorentz-violation effects, this model
was deemed inappropriate for the Super-K atmospheric
neutrino analysis. Instead, we use an exact diagonalizationof H which produces bounded oscillation probabilities inSuper-K samples for all values of the Lorentz-violating
coefficients, shown in detail in Appendix A. The accuracy of this calculation was ensured by confirming thatthe oscillation probabilities from the full diagonalization
matched the standard three-flavor oscillation calculation
used in SK (based on [57]) and the perturbative calculations for parameter values that were valid in theperturbative scheme.
Figure 1 shows examples of the νμ survival probability vsenergy and path length for the aT and cTT parameters in theμτ sector. Standard oscillations appear as lines of constant
L=E, which have slope one on these log-log plots. The aTμτ
and aTeμ coefficients create oscillation patterns in νμ disappearance that depend only on length. These oscillations
will appear as horizontal lines, which can be seen in
Fig. 1(b) at high energies where there are no L=E
oscillations. The distance (or equivalently cos θz ) at which
the LV oscillation begin is set by the value of aT . The cTTμτ
and cTTeμ coefficients introduce LE oscillations which willappear as lines with slope minus one, which can be seen athigh energies in Fig. 1(c). The value of cTT controls theenergy the new oscillations begin at.
The samples most sensitive to the high-energy μτsignatures are the UP-μ samples. Figure 2 shows thezenith-angle distributions of the three UP-μ data samples,as ratios relative to standard oscillations, compared with
the MC predictions corresponding aTμτ ¼ 10−22 GeV and−23
cTT(the same as the examples in Fig. 1).
μτ ¼ 7.5 × 10
The length-only oscillations from aTμτ appear as large,
zenith-dependent oscillations in the nonshowering andshowering UP-μ samples since cos θz is monotonically(though not linearly) related to distance. The fast oscillations at high energy from cTTμτ create significant extra νμ

1.0

4

T

0.8
103

0.6
0.4

102
0.2
10-1

1

10

102

3

10

0.0

Energy (GeV)

FIG. 1 (color online). The νμ → νμ oscillation probabilities,
plotted in path length vs neutrino energy. (a) Standard oscillations
appear as lines of constant L=E which have slope 1 on this loglog scale. Standard three-flavor oscillations are concentrated in
the upper-left portion in all three oscillograms, corresponding to
low energy and long distance. (b) The aTμτ coefficient introduces
oscillations proportional to L, which appear as horizontal lines(constant L) at high energies. (c) The cTTμτ coefficient introduces
LE oscillations which appear as lines with slope minus one.
Oscillograms for all sectors, as well as the μ → e probabilities areshown in Appendix B.

disappearance at all cos θz ’s in the same through-goingsamples.
Plots of the νμ survival probabilities and the νμ → νeoscillation probabilities for all the aT and cTT parameters
can be seen in Appendix B and the zenith distributions of
all the samples compared with the data can be seen in

052003-4

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

0.0

2.0

PHYSICAL REVIEW D 91, 052003 (2015)

Up-μ Non-Showering
Ratio to 3ν Oscillations

2.0

Up-μ Stopping
Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

TEST OF LORENTZ INVARIANCE WITH ATMOSPHERIC …

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

cos θz0.0

2.0

Up-μ Showering

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

cos θz0.0

cos θzFIG. 2 (color online). Ratios of the summed SK-I through SK-IV cos θz distributions relative to standard three-flavor oscillations for
the UP-μ subsamples, which are the most sensitive to the effects of LV. The stopping subsample (left) contains neutrinos with energies
peaking around 10 GeV, the nonshowering subsample (center) peaks around 100 GeV, and the showering subsample (right) peaks
around 1 TeV. The black points represent the data with statistical errors. The lines correspond to the MC prediction including Lorentz−23 in dashed red.
violating effects, with aTμτ ¼ 10−22 GeV in solid blue and cTTμτ ¼ 7.5 × 10

Appendix C. Both aTeμ and cTTeμ behave much like their μτ
counterpart in the highest energy samples, but also introduce some smaller but significant changes in the lower
energy e-like and μ-like samples that would allow theeffects of the two sectors to be distinguished from one
another. The aTeτ and cTTeτ terms, on the other hand, behave
quite differently from the other sectors: they reduce or
eliminate L=E oscillations that should otherwise occur
at medium and higher energies. So, instead of extra νμdisappearance there is less. They also enhance the νeappearance signal at lower energies. Oscillograms are only
shown for nonzero real parts of the parameters, but thereal and imaginary parts produce similar oscillation effects
in the high-energy regions where LV-induced oscillations
are dominant. The influence of the imaginary parts isonly in this high-energy region while the real parts also
introduce small modifications in the low-energy oscillation
probability.
III. LORENTZ-VIOLATING OSCILLATIONANALYSIS
The three-flavor plus SME oscillation model describedin Sec. II is fit to the data samples described above using thetechniques from [52]. The fitter minimizes a “pulled” χ 2[58] assuming Poisson statistics between the MC expectation, calculated for a particular value of the complex
coefficient, and the data:
XX
X~ ~ϵÞ −
χ ¼2
ðθ;
OSKn
E~ SKn
i
i
2

i

n

n

P SKn Xn Oi
þ χ 2penalty ð~ϵÞ;
þ
OSKn
ln
P SKn
i
~
~
~
ð
θ;
ϵ
Þ
E
n
i
n

ð3:1Þ

where n indexes the four SK run periods, i indexes theanalysis bins, OSKn
is the number of observed events ini~ ~ϵÞ is the MC expectation
bin i during SKn, and E~ SKn
ðθ;
i
~ and
in bin i in SKn with the coefficients being tested, θ,systematic parameters, ~ϵ. The expectation in each bin is
calculated separately for each run period and then the run
periods are summed for the comparison between data
and MC.
The systematic uncertainties are approximated as linear
effects on the analysis bins,


X
ϵj
SKn
SKn
~
~
~ESKn
;
ðθ; ~ϵÞ ¼ Ei ðθÞ 1 þ
f i;j
i
σj
j

ð3:2Þ

~ is thewhere j indexes the systematic errors, ESKnðθÞi
MC expectation in bin i in SKn without systematic shifts,
and f SKnis the fractional change in bin i in SKn duei;jto σ j , the 1-sigma change in systematic j. The constraints
on these parameters are included as a penalty term in
Eq. (3.1):
χ 2penalty ð~ϵÞ ¼

X ϵj 2
j

σj

:

ð3:3Þ

The analysis includes 155 systematic error parameters. The
uncertainties in the atmospheric neutrino flux, neutrino
interaction cross sections, particle production within nuclei,
and the standard PMNS oscillation parameters are sharedacross all run periods so f SKni;j is the same in SK-I through
SK-IV. The uncertainties related to detector performance—
reconstruction, particle identification, energy scale, andfiducial volume uncertainties—differ between run periods
since they depend on the specific detector geometry and052003-5

PHYSICAL REVIEW D 91, 052003 (2015)

K. ABE et al.
Δχ2

9
6
3

Im(aT) (GeV)

0 -28
-22
10-27 10-26 10-25 10-24 10-23 10
10-22
10-22
10-23

10-23

10-24

10-24

10-25

10-25

10-26
10-27

10-26

eτ
eμ
μτ

10-27

-28
10-28 -28
10-27 10-26 10-25 10-24 10-23 10-220
10Re(aT) (GeV)

3

6

9

Δχ2

Δχ2

9
6
3
0
10-22

Im(cTT)

10-23

eτ
eμ
μτ

10-24
10-25
10-26
10-27
10-28
10-28 10-27 10-26 10-25 10-24 10-23 10-220

Re(cTT)

3

6

9

Δχ2

FIG. 3 (color online). Two-dimensional contours at the 95%
confidence level for the real and imaginary parts of aTeτ , aTeμ , andTTTT
aTμτ in (a) and cTTeτ , ceμ , and cμτ in (b). The hashed areas indicate
the side of the contour that is excluded. The best-fit points fromthe three fits are also shown as markers. The one-dimensionalΔχ 2 curves are shown in the top and right side plots with thealternate variable profiled out.

hardware. For these uncertainties, f SKni;j will be nonzero in
only one run period. A table of all systematic uncertainties
included in the analysis can be found in the appendix
to [52].
In order to focus the analysis on the LV coefficients,
the standard oscillation parameters are constrained toexternal measurements and their uncertainties are taken as

systematic errors. The T2K measurement of νμ disappearance, jΔm232 j¼ð2.510.10Þ×10−3 eV2 and sin2 ðθ23 Þ ¼
0.514  0.055 [59], is used because its narrow-band beamand shorter fixed baseline make it less sensitive to theLorentz-violating spectral distortions considered in thisanalysis. The mixing angle sin2 ð2θ13 Þ ¼ 0.095  0.01 is
taken from the 2013 PDG world average [60], the solar terms
are taken from the global fit performed by the SK solar+KamLAND analysis, Δm221 ¼ ð7.46  0.19Þ × 10−5 eV2 ,sin2 ðθ12 Þ ¼ 0.305  0.021 [32]. The CP-violating phase δand the mass hierarchy (the sign of Δm2 ) are not yet known
and so are allowed to float unconstrained.
Equation (3.1) is minimized with respect to the ~ϵ for eachchoice of θ~ in a fit’s parameter space. A set of linear
equations in ϵj ’s is derived from Eq. (3.1) using the fact thatthe derivative ∂χ 2 =∂ϵj is zero at the minimum [58]. Theseequations can then be solved iteratively to find the minimum profile likelihood for that set of oscillation param~ The best fit point iseters, building up a map of χ 2 vs θ.
defined as the global minimum of this map.
Six fits are performed for the real and imaginary parts ofaT and cTT in the three sectors, eμ, eτ, and μτ. The real andimaginary parts of each coefficient are fit simultaneously,
but otherwise the coefficients are fit independently following the procedure typical for SME analyses [12]. Tests
with fits to high-statistics fake data sets reliably find no
LV when none is present and correctly extract the best fit
point if a fake data set with an LV signal is used. However,there is generally some ambiguity between the real andimaginary parts since they produce similar oscillation
effects at the energies where LV-oscillations dominate.
The low-energy differences allow the correct parameter tobe chosen in fits to simulated data with high statistics, but
small fluctuations can easily move the best fit point to justthe real part, just the imaginary part, or a combination of
the two.
No significant evidence of Lorentz violation is seen in
any of the fits. The most significant exclusion of no LV is
for aTeμ, and it has a Δχ 2 ¼ 1.4, less than 1σ with 2 degrees
of freedom. The absolute χ 2 for the fits ranges from 538.6
to 540.0 with 480 bins (477 degrees of freedom), corresponding to goodness-of-fit p-values around 2.5%. The
best-fit momentum and zenith distributions for the aT andcTT fits are shown, compared with the data, in Appendix C.
A summary of the fit results, including upper limits at the
95% confidence level, best-fit values, and levels of agreement with no Lorentz violation can be seen in Table II. The
two-dimensional contours at the 95% confidence level onaT and cTT are shown in Figs. 3(a) and 3(b) respectively.
The limits on the real and imaginary parts of the parameterare slightly different in the eτ and μτ sectors because thesefits found best fit points with different values for the realand imaginary components.

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TABLE II. Summary of the results of the six fits for Lorentz-violating parameters (the real and imaginary parts of each parameter are
fit simultaneously). The upper limits at the 95% confidence level (C.L.) and best fits are shown, as well as the Δχ 2 between the best fit
and the hypothesis of no Lorentz violation. The most significant exclusion of Lorentz invariance is in the aTeμ fit, which is still consistent
with no LV at the 68% confidence level. Since the parameters are scanned on a logarithmic scale and only positive parameter values areused, 10−28 is the minimum value considered and is equivalent to no LV.
LV parameter

Limit at 95% C.L.

eτμτ

No LV Δχ 2

Previous limit

1.4

4.2 × 10−20 GeV [61]0.0

9.6 × 10−20 [61]0.0

7.8 × 10−20 GeV [62]0.3

1.3 × 10−17 [62]

0.9

…

0.1

…

−23

Reða Þ
ImðaT ÞReðcTT ÞImðcTT Þ

1.8 × 10GeV
1.8 × 10−23 GeV
8.0 × 10−27
8.0 × 10−27

1.0 × 10GeV
4.6 × 10−24 GeV
1.0 × 10−28
1.0 × 10−28ReðaT ÞImðaT ÞReðcTT ÞImðcTT Þ

4.1 × 10−23 GeV
2.8 × 10−23 GeV
9.3 × 10−25
1.0 × 10−24

2.2 × 10−24 GeV
1.0 × 10−28 GeV
1.0 × 10−28
3.5 × 10−25ReðaT ÞImðaT ÞReðcTT ÞImðcTT Þ

6.5 × 10−24 GeV
5.1 × 10−24 GeV
4.4 × 10−27
4.2 × 10−27

3.2 × 10−24 GeV
1.0 × 10−28 GeV
1.0 × 10−28
7.5 × 10−28

T

eμ

−23

Best fit

IV. CONCLUSIONSThe large range of energies and path lengths in theatmospheric neutrino sample make it sensitive to a varietyof spectral distortions introduced by violations of Lorentz
invariance as parametrized by the standard model extension. However, the long distances and high energies makethe perturbative approach used in other experiments inappropriate, so we present the first analysis of Lorentz
violation in neutrino oscillations where the full, nonperturbative Hamiltonian is used, combined with three-flavor
neutrino oscillations. No evidence of LV is seen, so we set
limits on the isotropic parameters aT and cTT in the eμ, μτ,and eτ sectors. These are the first limits on the isotropic
parameters in the μτ sector, and we improve the existing
limits [12] on aT by 3 orders of magnitude and on cTT by 7
orders of magnitude thanks to the wide range of energies
and path lengths of the neutrinos in the atmospheric
neutrino samples. Future studies of SK atmospheric neutrino data could also set limits on the directional parameters
by searching for sidereal variations in the atmospheric
neutrino data.
ACKNOWLEDGMENTS
We would like to thank A. Kostelecky for his advice andsupport and we are grateful to J. S. Diaz for working closelywith us to calculate and implement the Lorentz-violating
oscillation probabilities. The authors gratefully acknowledge the cooperation of the Kamioka Mining and Smelting
Company. Super-K has been built and operated from funds
provided by the Japanese Ministry of Education, Culture,
Sports, Science and Technology, the U.S. Department ofEnergy, and the U.S. National Science Foundation. This
work was partially supported by the Research Foundationof Korea (BK21 and KNRC), the Korean Ministry of

Science and Technology, the National Science Foundationof China, the European Union FP7 (DS laguna-lbno PN284518 and ITN invisibles GA-2011-289442), the National
Science and Engineering Research Council (NSERC) of
Canada, and the Scinet and Westgrid consortia of Compute
Canada.
APPENDIX A: NEUTRINO OSCILLATIONSWITH LORENTZ VIOLATIONThis Appendix shows the full calculation of the neutrino
oscillation probabilities with both three-neutrino mixing
and Lorentz Violation, without assuming that the baseline
is short or that the LV Hamiltonian is small. Neither of theseapproximations is valid for SK because of its wide range ofpath lengths and energies.
The oscillation probabilities for Lorentz violation plus
three-flavor oscillations including matter effects are calculated by diagonalizing the Hamiltonian which includes allthese pieces, following the method from [63]. Combining
the parts described individually in Sec. II,
0
1
1
0
0 0
0
Ne 0 02
p
ﬃﬃ
ﬃ
B Δm21CC
00 CU†  2GF B
H ¼ UB@ 0 0 0A
@ 2E
A
Δm2
0 0 0
0 0 2E31
0
0
1
1
0
cTT
0
aTeμ aTeτ
cTT
eμ
eτ
B T 
C 4E B TT CBC
0 aTμτ C
0
cTT
B
μτ A;
@ ðaeμ Þ
A − 3 @ ðceμ ÞTT ðaTeτ Þ ðaTμτ Þ 0ðcTT0eτ Þ ðcμτ ÞðA1Þwhere U is the PMNS mixing matrix, E is the neutrinoenergy, GF is Fermi’s constant, and N e is the average

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electron density along the neutrino’s path. For antineutrinos,
the complex conjugates of all terms are taken (though inpractice this only affects δcp in U and the a and c parameters)
and the signs of the νe matter effect and a matrices arenegative.
The next step is to diagonalize this 3 × 3 matrix to
calculate the new eigenvalues and mixing matrix. Since the
Hamiltonian is Hermitian the eigenvalues are guaranteed
to be real. They can be calculated below as the roots of a
cubic equation,
 
pﬃﬃﬃﬃ
θ
a
Ei ¼ −2 Q cos i − ;
3
3

ðA2Þ

where
Ai ¼ Hμτ ðHee − Ei Þ − Hμe H eτ ;ðA12ÞBi ¼ Hτe ðHμμ − Ei Þ − Hτμ Hμe ;ðA13Þ

Ci ¼ Hμe ðHττ − Ei Þ − Hμτ Hτe ;ðA14Þ

N 2i ¼ jAi Bi j2 þ jAi Ci j2 þ jBi Ci j2 :ðA15ÞThe oscillation probabilities can then be calculated
fromwith i ¼ 0, 1, 2 and where the components Q and θi ,2

a − 3b;Q¼
93

θ0 ¼ cos−1 ðRQ−2 Þ;Pαβ ¼ jhνβ je−iHL jνα ij2ðA3Þwhich expands toðA4Þ

X
Pαβ ¼ δαβ − 4 ReðU βj Uβi U αj U αi Þsin2 ðLΔEji =2ÞðA5ÞXþ 2 ImðUβj U βi Uαj Uαi Þsin2 ðLΔEji Þ;

ðA16Þ

j>i

θ1 ¼ θ0 þ 2π;
θ2 ¼ θ0 − 2π;ðA6Þ

can be calculated from the trace and determinant of H:a ¼ − TrðHÞ;
b¼

ðA7Þ

TrðHÞ2 − TrðH2 Þ;
2R¼

2a3 − 9ab þ 27c:
54AC
Uμi ¼ i i ;Ni

Pμμ ¼ 1 −

ðA9Þ
ðA10Þ

The diagonalization also produces a mixing matrix U,
B C
Uei ¼ i i ;Niwhere ΔEji ¼ Ej − Ei are the differences between the
eigenvalues.
We can expand further, taking two examples particularly relevant for atmospheric neutrinos. The νμ survival
probability,

ðA8Þ

c ¼ − detðHÞ;AB
Uτi ¼ i i ;
Ni

ðA17Þ

j>i

X jAj Ai Cj Ci j2j>i

N 2j N 2i

sin2 ðLΔEji =2Þ;ðA18Þand the νe appearance probability,
X ReðAj Ai Bj Bi ÞjCj Ci j2
sin2 ðLΔEji =2Þ
Pμe ¼ − 4
2 2
N
N
j i
j>i

ðA11Þ

þ2

X ImðAj Ai Bj Bi ÞjCj Ci j2j>i

N 2j N 2i

sin ðLΔEji Þ:ðA19Þ

APPENDIX B: OSCILLOGRAMSThis Appendix includes plots of the νμ →νμ and νμ → νe oscillation probabilities vs neutrino path length and neutrinoenergy for standard three-flavor oscillations (Fig. 4) and for large (10−22 GeV and 7.5 × 10−23 ) values of the LV parameters(Figs. 5-6) to show the effects of the six different coefficients. Oscillograms for standard three-flavor oscillations areincluded for comparison at the end. For both aT and cTT , the eμ and μτ sectors have similar νμ → νμ probabilities but
different νμ → νe probabilities. For both parameters the eτ sector has the opposite effect of the other sectors: eliminating
standard oscillations instead of introducing nonstandard oscillations.

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TEST OF LORENTZ INVARIANCE WITH ATMOSPHERIC …
0.8
3

10

0.6
0.4

102
0.2
10-1

1

102

10

103

1.0

4

10

Path Length (km)10Path Length (km)PHYSICAL REVIEW D 91, 052003 (2015)

1.0

4

0.8
3

10

0.6
0.4

102

0.0

0.2
10-1

Energy (GeV)

1

0.0

103

102

10

Energy (GeV)

FIG. 4 (color online). For comparison, the νμ → νμ (left) and νμ → νe (right) oscillation probabilities, plotted in path length vs neutrinoenergy for standard three-flavor oscillations.

1.0
0.8

103

0.6
0.4

102
0.2
10-1

1

10

102

3

10

1.0

104

Path Length (km)Path Length (km)

104

0.0

0.8
103

0.6
0.4

102
0.2
10-1

1

Energy (GeV)

1.0

103

0.6
0.4

102
0.2

10

102

103

0.0

0.8
103

0.6
0.4

102
0.2
10-1

1

1.0

10

0.6
0.4

102
0.2
102

102

103

0.0

3

10

1.0

104

Path Length (km)Path Length (km)

0.8
3

10

10

Energy (GeV)

104

1

0.0

1.0

Energy (GeV)

10-1

3

10

104

Path Length (km)Path Length (km)

0.8

1

102

Energy (GeV)

104

10-1

10

0.0

Energy (GeV)

0.8
3

10

0.6
0.4

102
0.2
10-1

1

10

102

3

10

0.0

Energy (GeV)

FIG. 5 (color online). The νμ → νμ (left) and νμ → νe (right) oscillation probabilities, plotted in path length vs neutrino energy for theaT parameter in the (top to bottom) eμ, μτ, and eτ sectors. The aT coefficients scale terms proportional to L, so the distortions getstronger as cos θz approaches −1.

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1.0

Path Length (km)

10

0.8
103

0.6
0.4

102
0.2
10-1

1

10

102

103

1.0

4

10

Path Length (km)

4

0.0

0.8
103

0.6
0.4

102
0.2
10-1

1

Energy (GeV)

1.0

103

0.6
0.4

2

10

0.2

10

102

103

0.0

0.8
103

0.6
0.4

2

10

0.2
10-1

1

1.0

103

0.6
0.4

102
0.2
102

102

103

0.0

103

1.0

104

Path Length (km)Path Length (km)

0.8

10

10

Energy (GeV)

104

1

0.0

1.0

Energy (GeV)

10-1

103

104

Path Length (km)Path Length (km)

0.8

1

102

Energy (GeV)

104

10-1

10

0.0

Energy (GeV)

0.8
103

0.6
0.4

102
0.2
10-1

1

10

102

103

0.0

Energy (GeV)

FIG. 6 (color online). The νμ → νμ (left) and νμ → νe (right) oscillation probabilities, plotted in path length vs neutrino energy for thecTT parameter in the (top to bottom) eμ, μτ, and eτ sectors. The cTT coefficients scale terms proportional to LE, so the distortions getstronger at higher energies.

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PHYSICAL REVIEW D 91, 052003 (2015)

APPENDIX C: ZENITH ANGLE DISTRIBUTIONSThis Appendix includes ratios relative to standard three-flavor oscillations for all of the subsamples included in theanalysis. The ratios are generally plotted vs zenith angle (cos θz ), except for samples which are binned only in energy.
The data is shown as points with statistical error bars. The dashed lines represent the best fits for each of the six fits while thesolid lines represent examples of large LV parameters (10−22 GeV and 7.5 × 10−23 ), the same as the oscillograms in
Appendix B). The plots are divided into e- and NC-like samples and μ-like samples, and the aT and cTT fits are shown
separately (Figs. 7-8 and Figs. 9-10, respectively), and are divided into e-and NC-like samples (Figs. 7 and 9) and π 0 -likesamples (Figs. 8 and 10).

Ratio to 3 ν Oscillations

Ratio to 3 ν Oscillations

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

2.0

Ratio to 3 ν Oscillations

Multi-GeV νe-like

Sub-GeV e-like no decay-e
2.0

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

cos θz

1.0
0.5

1500

2.0

Ratio to 3 ν Oscillations

Ratio to 3 ν Oscillations

Ratio to 3 ν Oscillations

1.5

1000

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

Ratio to 3 ν Oscillations

Ratio to 3 ν Oscillations

1-ring π0

1.5
1.0
0.5
0.0

500

1000

1.0
0.5
0.0
-1.0

-0.5

1500

e momentum (MeV/c)

0.5

1.0

2.0

0.0

0.5

1.0

Multi-Ring νe-like

1.5
1.0
0.5
0.0
-1.0

-0.5

cos θze momentum (MeV/c)
2.0

1.5

cos θzMulti-GeV νe-like

Sub-GeV e-like 1 decay-e

500

1.0

Multi-Ring νe-likecos θz

2.0

0.0

0.5

2.0

0.0

0.5

1.0

cos θz2-ring π0

2.0

SKI-IV Dataaeμ Best Fit

1.5

aeμ 10-22 GeV Example
1.0aμτ Best Fit
aμτ 10-22 GeV Example

0.5
0.0

aeτ Best Fit
aeτ 10-22 GeV Example
200

400

600

800 1000π0 momentum (MeV/c)

FIG. 7 (color online). Ratios of the summed SK-I through SK-IV cos θz or momentum distributions relative to standard three-flavor
oscillations of the e- and NCπ 0 -like FC subsamples. They are projected into cos θz when binned in both momentum and angle. The
black points represent the data with statistical errors. The dashed lines represent the best fits from the three sectors for the aT parameters
and the solid lines represent examples of large Lorentz violation (aT ¼ 10−22 GeV, equivalent to Fig. 5). Significant deviations fromunity would indicate Lorentz violation.

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1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

Ratio to 3ν Oscillations

1.5

2.0

Sub-GeV μ-like 1 decay-e

1.5
1.0
0.5
0.0
-1.0

-0.5

2.0

Multi-GeV μ-like

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

2.0

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

1.0
0.5

0.5

1.0

500

1000

1500

SKI-IV Dataaeμ Best Fitaμτ Best Fit

1.0

aμτ 10-22 GeV Example

0.5
0.0
-1.0

aeτ Best Fit
aeτ 10-22 GeV Example
-0.5

0.0

0.5

1.0

2.0
1.5
1.0
0.5
0.0
-1.0

Up-μ Stopping

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

-0.5

0.0

0.5

1.0

cos θzRatio to 3ν Oscillations

Ratio to 3ν Oscillations

0.0

aeμ 10-22 GeV Examplecos θzcos θz0.5

PC Through-going

1.5

2.0

1.0

μ momentum (MeV/c)

1.5

PC Stopping

0.0

1.5

cos θz

2.0

-0.5

1.0

Sub-GeV μ-like 2 decay-e

Multi-Ring μ-likecos θz

0.0
-1.0

0.5

2.0

cos θzRatio to 3ν Oscillations

Ratio to 3ν Oscillationscos θz

0.0

0.0

2.0

Up-μ Non-Showering
Ratio to 3ν Oscillations

2.0

Sub-GeV μ-like no decay-e
Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

K. ABE et al.

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

cos θz0.0

2.0

Up-μ Showering

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

0.0

cos θzFIG. 8 (color online). Ratios of the summed SK-I through SK-IV cos θz distributions relative to standard three-flavor oscillations forthe μ-like FC, PC, and UP-μ subsamples. They are projected into cos θz when binned in both momentum and angle and the Sub-GeV 2
decay-e sample is binned only in momentum. The black points represent the data with statistical errors. The dashed lines represent thebest fits from the three sectors for the aT parameters and the solid lines represent examples of large Lorentz violation (aT ¼ 10−22 GeV,
equivalent to Fig. 5). Significant deviations from unity would indicate Lorentz violation.

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TEST OF LORENTZ INVARIANCE WITH ATMOSPHERIC …

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

2.0

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

2.0

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

cos θz

1.0
0.5

1500

2.0
1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

1-ring π0

1.5
1.0
0.5
0.0

500

1000

1.5
1.0
0.5
0.0
-1.0

-0.5

1500

e momentum (MeV/c)

0.5

1.0

2.0

0.0

0.5

1.0

Multi-Ring νe-like

1.5
1.0
0.5
0.0
-1.0

-0.5

cos θze momentum (MeV/c)
2.0

Multi-Ring νe-likecos θzRatio to 3ν Oscillations

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

1.5

1000

1.0

Multi-GeV νe-like

Sub-GeV e-like 1 decay-e

500

0.5

2.0

cos θz2.0

0.0

PHYSICAL REVIEW D 91, 052003 (2015)

Multi-GeV νe-like

Sub-GeV e-like no decay-e

0.0

0.5

1.0

cos θz2-ring π0

2.0

SKI-IV Data
ceμ Best Fit

1.5ceμ 10-22 Example
1.0cμτ Best Fit
cμτ 10-22 Example

0.5
0.0

ceτ Best Fit
ceτ 10-22 Example
200

400

600

800 1000π0 momentum (MeV/c)

FIG. 9 (color online). Ratios of the summed SK-I through SK-IV cos θz or momentum distributions relative to standard three-flavor
oscillations of the e- and NCπ 0 -like FC subsamples. They are projected into cos θz when binned in momentum and angle. The black
points represent the data with statistical errors. The dashed lines represent the best fits from the three sectors for the cTT parameters andthe solid lines represent examples of large Lorentz violation (cTT ¼ 7.5 × 10−23 , equivalent to Fig. 6). Significant deviations from unity
would indicate Lorentz violation.

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1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

Ratio to 3ν Oscillations

1.5

2.0

Sub-GeV μ-like 1 decay-e

1.5
1.0
0.5
0.0
-1.0

-0.5

2.0

Multi-GeV μ-like

1.5
1.0
0.5
0.0
-1.0

-0.5

0.0

0.5

1.0

2.0

Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

1.0
0.5

0.5

1.0

500

1000

1500

SKI-IV Data
ceμ Best Fit
cμτ Best Fit

1.0

cμτ 10-22 Example

0.5
0.0
-1.0

ceτ Best Fit
ceτ 10-22 Example
-0.5

0.0

0.5

1.0

2.0
1.5
1.0
0.5
0.0
-1.0

Up-μ Stopping

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

-0.5

0.0

0.5

1.0

cos θzRatio to 3ν Oscillations

Ratio to 3ν Oscillations

0.0

ceμ 10-22 Examplecos θzcos θz0.5

PC Through-going

1.5

2.0

1.0

μ momentum (MeV/c)

1.5

PC Stopping

0.0

1.5

cos θz

2.0

-0.5

1.0

Sub-GeV μ-like 2 decay-e

Multi-Ring μ-likecos θz

0.0
-1.0

0.5

2.0

cos θzRatio to 3ν Oscillations

Ratio to 3ν Oscillationscos θz

0.0

0.0

2.0

Up-μ Non-Showering
Ratio to 3ν Oscillations

2.0

Sub-GeV μ-like no decay-e
Ratio to 3ν Oscillations

Ratio to 3ν Oscillations

K. ABE et al.

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

cos θz0.0

2.0

Up-μ Showering

1.5
1.0
0.5
0.0
-1.0 -0.8 -0.6 -0.4 -0.2

cos θz0.0

FIG. 10 (color online). Ratios of the summed SK-I through SK-IV cos θz distributions relative to standard three-flavor oscillations forthe μ-like FC, PC, and UP-μ subsamples. They are projected into cos θz when binned in both and the Sub-GeV 2 decay-e sample isbinned only in momentum. The black points represent the data with statistical errors. The dashed lines represent the best fits from thethree sectors for the cTT parameters and the solid lines represent examples of large Lorentz violation (cTT ¼ 7.5 × 10−23 , equivalent to
Fig. 6). Significant deviations from unity would indicate Lorentz violation.

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TEST OF LORENTZ INVARIANCE WITH ATMOSPHERIC …
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