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A compact analytical approximation for a light sterile neutrino oscillation
in matter
To cite this article before publication: Baobiao Yue et al 2020 Chinese Phys. C in press https://doi.org/10.1088/16741137/ababfa
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For Review Only
A compact analytical approximation for a light sterile
neutrino oscillation in matter
Journal:
Chinese Physics C
Manuscript ID
CPC20200160.R2
Manuscript Type:
Paper
Date Submitted by the
Author:
23Jun2020
Complete List of Authors:
Yue, Baobiao; Sun YatSen University
Li, Wei; Jinan University; National Taiwan University
Ling, Jiajie; Sun YatSen University
Xu, Fanrong; Jinan University, Department of Physics
Keywords:
Neutrino oscillation, Sterile neutrinos, Matter effect
Subject:
Neutrino/dark matter < Experiments:, Lepton /neutrino physics <
Particle theory:
http://cpc.ihep.ac.cn
Chinese Physics C
Accepted Manuscript
For Review Only
A compact analytical approximation for a light
sterile neutrino oscillation in matter
BaoBiao Yuea, Wei Lib,c, JiaJie Ling ∗a, and FanRong Xu∗b
aSchool of Physics, Sun Yatsen University, No. 135, Xingang Xi Road, Guangzhou, 510275, P. R. China
bDepartment of physics, Jinan University, No. 601, Huang Pu Road, Guangzhou, 510632, P. R. China
cDepartment of Physics and Astronomy, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, P. R. China
June 28, 2020
Abstract
The existence of light sterile neutrinos is a long standing question for
particle physics. Several experimental “anomalies” could be explained by
introducing eV mass scaled light sterile neutrinos. Many experiments are
actively hunting for such light sterile neutrinos through neutrino oscillation.
For long baseline experiments, matter eﬀect needs to be treated carefully
for precise neutrino oscillation probability calculation. However, it is usually
timeconsuming or analytical complexity. In this manuscript we adopt the
Jacobilike method to diagonalize the Hermitian Hamiltonian matrix and de
rive analytically simpliﬁed neutrino oscillation probabilities for 3 (active) + 1
(sterile)neutrino mixing for a constant matter density. These approximations
can reach quite high numerical accuracy while keeping its analytical simplicity
and fast computing speed. It would be useful for the current and future long
baseline neutrino oscillation experiments.
Keywords: Neutrino oscillation, Sterile neutrinos, MSW
1 Introduction
Neutrino oscillation has been indisputably established by atmospheric, solar, reactor
and accelerator experimental results [1]. After the recent reactor experiments [2, 3, 4]
discovered the last unknown mixing angle θ13 in the 3neutrino mixing framework,
neutrino oscillation measurement enters a precision era. Nonzero neutrino mass pro
vides convincing evidence of new physics beyond the standard model. Introducing
righthanded neutrinos is a natural way to introduce neutrino mass. In the stan
dard electroweak VA theory, righthanded neutrinos cannot couple with W±and
Z0bosons. Electron collider experimental data [5] constrain the number of active
∗Corresponding authors: lingjj5@mail.sysu.edu.cn, fanrongxu@jnu.edu.cn
1
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light neutrino ﬂavors to three, other new types of light neutrinos must be sterile.
Currently there is no theoretical constraint on sterile neutrino mass. They could be
very massive (1015 GeV) as suggested by the seesaw mechanism; They also could
be dark matter in the keV mass range; And they also might be as light as subeV
which would explain the CMB measurement.
If light sterile neutrinos mix with the active neutrinos, their signature could be
observed by neutrino oscillation experiments. LSND observed 87.9±22.4±6.0νe
signal events from νµsource from µ+decay at rest, which suggests a sterile neutrino
with mass greater than 0.4 eV [6, 7]. Recently MiniBooNE experiments reported a
4.7σexcess of electronlike events when combining both the νµand νµbeam con
ﬁgurations. The signiﬁcance of the combined LSND and MiniBooNE excesses can
even reach 6σ[8], although the source of the low energy excess from MiniBooNE is
still unclear. Experimental hints of the existence of eV mass scaled sterile neutrinos
also come from short baseline reactor neutrino experiments [9, 10, 11]. However, the
uncertainties with theoretical reactor antineutrino ﬂux calculation might be under
estimated, giving an observed excess of antineutrino events at 46 MeV relative to
predictions [12, 13, 14, 15, 16]. Therefore, whether the reactor anomaly is completely
cased by the theoretical modeling or sterile neutrinos is still up in the air.
It is worth mentioning, although eVscale sterile neutrinos could help to ex
plain several experimental anomalies, they are not quite theoretical motivated. And
they are also in tension with the muon neutrino disappearance results, especially
for recent results from IceCube [17] and MINOS/MINOS+ [18]. The most recent
combined analysis with MINOS+, Bugey and Daya Bay experiments set a very
strong limit on sterile neutrino mixing [19], which can almost completely exclude
LSND and MiniBooNE sterile neutrino hypothesis at eVscale region. However, a
more convincing and direct testing would come from muon decay at rest experi
ments, such as the proposal of JSNS2 [20]. Certainly, the existence of eV mass
scale sterile neutrinos therefore needs further evidence. Many reactor and accelera
tor neutrino experiments are actively searching for sterile neutrinos at various mass
scales [21, 22, 23, 24, 25].
For long baseline accelerator neutrino experiments [26, 27], neutrino matter ef
fect plays an important role in neutrino mass hierarchy [28, 29] and CP violation [30]
measurements. As ﬁrst pointed out by Wolfenstein, neutrinos propagating in mat
ter will oscillate diﬀerently from those in a vacuum [31]. The presence of electrons
in matter changes the energy levels of propagation eigenstates of neutrinos due to
charged current coherent forward scattering of the electron neutrinos. Later on,
Mikheyev and Smirnov [32] further noticed the matter eﬀect can produce resonant
maximal ﬂavor transition when neutrinos propagate through matter at certain elec
tron densities. SuperKamiokande observes an indication of diﬀerent solar neutrino
ﬂux during the night and day for solar neutrinos passing through additional terres
trial matter in the earth at diﬀerent periods [33]. For sterile neutrino and other new
physics searches, matter eﬀect has to be calculated carefully and precisely, especially
for long baseline neutrino oscillation experiments.
Neutrino oscillation in matter can be solved accurately using numerical or ana
lytical calculation [34] with a complex matrix diagonalization algorithm. In practice,
analytic approximations are more commonly used in neutrino experiments and use
2
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ful to understand the oscillation features. High precision analytical expressions for
3neutrino oscillation in matter has been thoroughly studied [35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48]. Some of them utilize perturbation theory and rely on
expansions in parameter θ13. Given the large θ13 observed, higher order corrections
associated with θ13 are needed to achieve numerical accuracy. Thus the oscilla
tion expressions usually become quite complicated. Ref. [49] introduces the Jacobi
method to diagonalize the real Hermitian matrix. It maintains the same analytic
expressions for neutrinos propagating in matter as they have in vacuum in terms of
the eﬀective neutrino mixing angles and masssquared diﬀerences in matter.
For sterile neutrinos, the oscillation expressions will be very complicated if addi
tional light sterile neutrinos exist [50]. Compared with standard 3neutrino mixing,
the simplest 3 (active) + 1 (sterile)neutrino mixing has 3 additional mixing angles
(i.e. θ14,θ24 and θ34) and 2 additional CP phases (i.e. δ24 and δ34). Furthermore,
since sterile neutrinos do not interact with matter, the neutral current potential for
active neutrinos also needs to be taken into account. N. Klop et. al. [51] provides
a method to covert 3+1neutrino mixing with matter eﬀects into a NonStandard
Interaction (NSI) problem in the 3neutrino mixing case. Here we follow the ro
tation strategy introduced in Ref. [49] and adopt the Jacobilike method [52, 53],
which is able to diagonalize the Hermitian complex matrix, to derive analytical ap
proximations for the 3+1neutrino oscillation in matter. While keeping the formula
simplicity, the expressions can also achieve very good numerical accuracy and fast
calculation speed. This could be very useful for the current and near future neutrino
oscillation experiments.
This paper starts with the section 2 and introduces the fundamental theory of
neutrino mixing and oscillation, including sterile neutrinos and matter eﬀect. The
basic idea of the Jacobilike method and the derivation of analytical approximations
for sterile neutrino oscillation probabilities are presented in section 3. In the end,
the accuracy of this work is shown in section 4 with two long baseline accelerator
neutrino experiments as demonstrations. More details about the Jacobilike method
and formula derivation are listed in the appendix.
2 Theoretical framework
2.1 Neutrino oscillation
In the standard neutrino mixing paradigm, three neutrino ﬂavor eigenstates (νe,νµ,
ντ) are superpositions of three neutrino mass eigenstates (ν1,ν2,ν3).
νe
νµ
ντ
=U
ν1
ν2
ν3
.(1)
3
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Here Uis the socalled PMNS (PontecorvoMakiNakawagaSakata) mixing matrix
[54, 55, 56], which can be parametrized as
U=R23(θ23,0)R13(θ13 , δ13)R12 (θ12,0)
=
1 0 0
0c23 s23
0−s23 c23
c13 0s13e−iδ13
0 1 0
−s13eiδ13 0c13
c12 s12 0
−s12 c12 0
0 0 1
,(2)
where Rij (θij , δij ) denotes a counterclockwise rotation in the complex ijplane through
a mixing angle θij and a CP phase δij with cij = cos θij and sij = sin θij . This work
adopts the conventions 0 ≤θij ≤π/2 and 0 ≤δij ≤2π.
Under the plane wave assumption, the general oscillation probability from α
ﬂavor type neutrinos to βﬂavor type neutrinos can be expressed as
Pνα→νβ=δαβ −4X
i>j <UβiU∗
αiU∗
βj Uαj sin2∆ij
±2X
i>j =UβiU∗
αiU∗
βj Uαj sin 2∆ij
,(i, j = 1,2,3) (3)
where the upper and lower sign is for the neutrino and antineutrino cases respec
tively. ∆ij stands for
∆ij ≡∆m2
ij L
4E= 1.267 ∆m2
ij
eV2GeV
EL
km,(4)
where ∆m2
ij =m2
i−m2
jis the masssquared diﬀerence between neutrino mass eigen
states νiand νj.
According to eq. (2) and eq. (3), 3ﬂavor neutrino oscillation is described with six
parameters, including two independent neutrino mass squared diﬀerences (∆m2
21 and
∆m2
32), three mixing angles (θ12 ,θ13 and θ23) and one leptonic CP phase (δ13 ). Fol
lowing the same convention, the 4ﬂavor neutrino mixing matrix can be parametrized
as
U=R34(θ34, δ34)R24 (θ24, δ24 )R14(θ14 ,0)R23(θ23,0)R13(θ13, δ13)R12(θ12 ,0) ,(5)
with six additional neutrino oscillation parameters: θ14 ,θ24,θ34,δ24,δ34 and ∆m2
41
∗. The exact parameterization expression for each mixing element is listed in the
appendix A. The general expression for the neutrino oscillation probabilities still
follow eq. (3) by simply increasing the total number of neutrino ﬂavors and mass
eigenstates to 4.
In practice, when sterile neutrinos are much heavier than active neutrinos (∆m2
41
∆m2
31), due to ﬁnite detector space and energy resolution, the rapid oscillation fre
quency associated with large masssquared diﬀerences between the 4th and the other
mass eigenstates ∆m2
4k(k= 1,2,3) will be averaged out, leading to hsin2∆4ki ≈ 1
2.
∗This is equivalent to use δ14 and δ24 , or δ14 and δ34 for the additional CP phases.
4
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The neutrino oscillation equation can then be simpliﬁed to
Pνα→νβ=δαβ −4X
i>j <UβiU∗
αiU∗
βj Uαj sin2∆ij
±2X
i>j =UβiU∗
αiU∗
βj Uαj sin 2∆ij
−1
2sin22θαβ
(i, j = 1,2,3) (6)
with sin22θαβ = 4Uα42(δαβ − Uβ42). In this paper we prefer to use the full oscil
lation formula to preserve the rapid oscillation induced by sterile neutrinos.
2.2 Matter eﬀect
When active neutrinos propagate through matter, the evolution equation is modiﬁed
by coherent interaction potentials, which are generated through coherent forward
elastic weak chargedcurrent (CC) and the neutralcurrent (NC) scattering in a
medium. All active neutrinos can interact with electrons, neutrons and protons in
matter through the exchange of a Zboson in the NC process. However, only electron
neutrinos participate in the CC process with electrons through the exchange of W±.
For electron neutrinos, CC potential is proportional to electron number density.
VCC =√2GFNe, where GFis the Fermi coupling constant, Neis the electron number
density. The NC potentials caused by electrons and protons will cancel each other
because they have opposite signs and the number densities of electrons and protons
are basically the same in the earth. The net NC potential, VNC =−√2
2GFNn, is
only sensitive to the neutron number density, Nn. Both VCC and VNC need to swap
signs for antineutrinos.
For 3ﬂavor neutrino oscillation, only CC potential needs to be considered for
the electron neutrino eigenstate, while the NC potential is a common term for all
neutrino ﬂavors and has no net eﬀect on neutrino oscillation. However, the NC
potential cannot be neglected in 3+1ﬂavor neutrino case, since sterile neutrinos
do not interact with matter. The eﬀective Hamiltonian in the ﬂavor eigenstate
representation for 3+1ﬂavor neutrino mixing is
H=Hv+V=1
2E
U
0 0 0 0
0 ∆m2
21 0 0
0 0 ∆m2
31 0
0 0 0 ∆m2
41
U†+
ACC 0 0 0
0 0 0 0
0 0 0 0
0 0 0 ANC
,
(7)
where Hvis the neutrino Hamiltonian in vacuum and Vis the matter eﬀect potential.
ACC and ANC for neutrinos are given by
ACC = 2EVCC = 7.63 ×10−5(eV2)( ρ
g/cm3)( E
GeV),(8a)
ANC =−2EVNC = 3.815 ×10−5(eV2)( ρ
g/cm3)( E
GeV),(8b)
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respectively, where ρis the mass density. Similarly to VCC and VNC , both ACC and
ANC have to swap signs for antineutrinos. In this work, we assume a constant ρ. If
there is no special declaration, ρwill be set to 2.6 g/cm3as default.
The evolution of neutrino ﬂavor state Ψαcan be calculated using Schr¨odinger
equation id
dt Ψα=HΨα. After diagonalizing the eﬀective Hamiltonian matrix H, we
can calculate neutrino oscillation probability in matter through the equation
Pνα→νβ=δαβ −4X
i>j <e
Uβi e
U∗
αi e
U∗
βj e
Uαj sin2e
∆ij
±2X
i>j =e
Uβi e
U∗
αi e
U∗
βj e
Uαj sin 2 e
∆ij
,(i, j = 1,2,3,4) (9)
with the eﬀective mixing matrix e
Uand eﬀective masssquared diﬀerences ∆em2
ij (i, j =
1,2,3,4). In the following approximations, we will rotate the Hamiltonian from mass
eigenstate. For simplicity, write the eﬀective Hamiltonian in mass eigenstate as
H=U†HU=1
2E
H11 H12 H13 H14
H21 H22 H23 H24
H31 H32 H33 H34
H41 H42 H43 H44
,(10)
where the Hermitian matrix element Hij yields
Hij =(ACCU∗
eiUej +ANCU∗
siUsj (i6=j)
∆m2
i1+ACCUei 2+ANCUsi 2(i=j).(11)
In this case, the eﬀective mixing e
Uyields e
U=UR, in which Ris the diagonalization
matrix on H.
3 The analytical approximation
As shown in ref. [34], the exact solution for the eﬀective mixing matrix e
Uand ef
fective masssquared diﬀerences ∆em2
ij (i, j = 1,2,3,4) can be obtained analytically.
However, to obtain higher precision analytical approximations for neutrino oscilla
tion in matter would be more convenient and timesaving. Here we would like to
introduce a Jacobilike method, which is a unitary transformation operation method
to diagonalize the complex Hermitian matrix. Then we present the eﬀective mix
ing matrix and eﬀective masssquared diﬀerences of the 3+1ﬂavor neutrino mixing
framework for both neutrinos and antineutrinos. As a result, high accuracy can be
obtained for the calculation of neutrino oscillation probabilities in matter.
3.1 Jacobilike method: Diagonalization of a 2 ×2 Hermi
tian matrix
The Jacobilike method, which originates from the Jacobi eigenvalue algorithm, is a
eﬀective matrix rotation approach to a diagonalize complex Hermitian matrix. Here
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we start with an example of solving a 2 ×2 Hermitian matrix. A Hermitian matrix
M=α β
β∗γ(α , γ ∈R, β ∈C) (12)
can be diagonalized as
M0=R†(ω, φ)M R(ω, φ) = λ−0
0λ+(13)
with a rotation matrix
R(ω, φ) = cos ωsin ωe−iφ
−sin ωeiφ cos ω,(ω, φ ∈R) (14)
where φ= Arg(sign(A)β∗), A=±βand tan ω=2A
γ−α±√(γ−α)2+4A2. The choice
of ±sign for Ais optional. For simplicity, we choose it to be the same sign as
ACC and ANC in eq. (8) for matter eﬀect in the 3+1 framework. The ±sign in
the denominator of tan ωis correlated with the exchange of the values of λ−and
λ+in eq. (13). In this work we adopt +(−) for the ij submatrix diagonalization if
∆m2
ij >0 (∆m2
ij <0). After rotation, the eigenvalues of Mcan be obtained as
λ−=α+γtan2ω−2Atan ω
1 + tan2ω, λ+=αtan2ω+γ+ 2Atan ω
1 + tan2ω.(15)
In a summary, this method is easily used to diagonalize a complex Hermitian matrix
through rotation, in which the complex factor φis used to deal with the complex
diagonalization.
3.2 The application of Jacobilike method on 3+1ﬂavor
neutrino mixing
To accurately diagonalize the 4 ×4 neutrino Hamiltonian Hermitian matrix using
the Jacobilike method, in principle, we need to perform inﬁnite iterations of 2 ×2
submatrix rotation. However, in practice, with only two continuous rotations on
the eﬀective Hamiltonian, we already can get analytical approximations for neutrino
oscillation in matter with very high accuracy. The diagonalized Hamiltonian yields
ˆ
H=R†HR ≈R2,†R1,†HR1R2=e
U†(UHvU†+V)e
U , (16)
where e
U=UR1R2, and R1and R2are the rotation matrices. After some mathe
matical simpliﬁcations, e
Ucan be expressed as R34R24R14R23R13R12, which has the
same form as standard neutrino mixing U. For simplicity, we just show the major
results of e
Uand ∆ em2
ij (i, j = 1,2,3,4) in this section. The complete derivations are
shown in appendix B.1 and B.2.
With two continuous rotations on the eﬀective Hamiltonian H, we can obtain
the eﬀective neutrino mixing matrix e
U
e
U≈R34(θ34, δ34)R24 (θ24, δ24 )R14(θ14 ,0)R23(θ23,0)R13(e
θ13,e
δ13)R12(e
θ12,e
δ12).(17)
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It is very similar to the one in vacuum (i.e eq. (5)), except that there is one additional
eﬀective phase e
δ12 in the submatrix R12.e
θ12,e
θ13,e
δ13 and e
δ12 are the eﬀective angles
and phases as functions of Ein R13 and R12. And R34,R24,R14 and R23 are the
same as in vacuum. In the diagonalization process, it is always better to ﬁrst apply
a rotation to the submatrix which has the largest absolute ratio of the oﬀdiagonal
element to the diﬀerence of diagonal ones. Since ∆m2
21 is the smallest masssquared
diﬀerence compared with others, we can start with R12 submatrix rotation ﬁrst.
After the ﬁrst rotation with R1=R12(ω1, φ1) submatrix, we can obtain the
eﬀective angle e
θ12 and eﬀective phase e
δ12 represented as functions of ω1and φ1
through the combination of R12(θ12,0)R12 (ω1, φ1):
sin e
θ12 ≈c12 tan ω1eiφ1+s12
p1 + tan2ω1
,cos e
θ12 ≈c12 −s12 tan ω1eiφ1
p1 + tan2ω1
,(18a)
ei
e
δ12 ≈(c12 tan ω1eiφ1+s12)(c12 −s12 tan ω1e−iφ1)
cos e
θ12 sin e
θ12(1 + tan2ω1),(18b)
in which tan ω1=2Aω1
(H22−H11 )+√(H22−H11 )2+4A2
ω1
,Aω1=±H12and φ1= Arg(sign(Aω1)H∗
12).
The + and −signs in Aω1are for the neutrino and antineutrino cases respectively.
After the ﬁrst rotation ((30) and (64)), we can obtain the eigenvalues of the eﬀective
Hamiltonian submatrix
λ−=H11 +H22 tan2ω1−2Aω1tan ω1
1 + tan2ω1
, λ+=H11 tan2ω1+H22 + 2Aω1tan ω1
1 + tan2ω1
.
(19)
After partial diagonalization on the 12 submatrix, the oﬀdiagonal elements of 1
3 and 23 submatrices become the relatively largest of the rest of the submatrices for
both neutrinos and antineutrinos cases due to the smallness of the sterile neutrinos
mixing angles (i.e. θ14,θ24,θ34). In the second rotation, we adopt R2=R23(ω2, φ2)
(R2=R13(ω2, φ2)) †rotation matrix for the neutrino (antineutrino) case. After the
second rotation, we obtain the e
θ13 and e
δ13 as the functions of ω2and φ2:
sin e
θ13 ≈c13 tan ω2eiφ2+s13eiδ13 
p1 + tan2ω2
,cos e
θ13 ≈c13 −s13 tan ω2ei(δ13−φ2)
p1 + tan2ω2
,(20a)
ei
e
δ13 ≈(c13 tan ω2eiφ2+s13eiδ13 )(c13 −s13 tan ω2ei(δ13 −φ2))
cos e
θ13 sin e
θ13(1 + tan2ω2),(20b)
in which tan ω2=2Aω2
(H33−λ±)±√(H33 −λ±)2+4A2
ω2
. In the equation for tan ω2, the upper
(lower) sign in front of p(H33 −λ±)2+ 4A2
ω2is for NH (IH) (i.e. normal hierarchy
(inverted hierarchy)) case, and λ+(λ−) is for neutrino (antineutrino) case. In the
above equations, Aω2and eiφ2have diﬀerent expressions for neutrinos and antineu
trinos. For the neutrino case,
Aω2=H0
23, φ2= Arg(sign(Aω2)H0∗
23), H0
23 =H13 tan ω1eiφ1+H23
p1 + tan2ω1
.(21)
†Rotation is chosen by considering convenience of calculations shown in B.1.2 and B.2.2.
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While for the antineutrino case,
Aω2=−H0
13, φ2= Arg(sign(Aω2)H0∗
13), H0
13 =H13 −H23 tan ω1e−iφ1
p1 + tan2ω1
.(22)
In this rotation, we can diagonalize 23 (13) submatrix for neutrinos (antineutrinos)
in eq. (38) (eq. (72)), resulting in two eigenvalues λ0
±. The formula for λ0
±are
λ0
−=λ++H33 tan2ω2−2Aω2tan ω2
1 + tan2ω2
, λ0
+=λ+tan2ω2+H33 + 2Aω2tan ω2
1 + tan2ω2
.
(23)
for the neutrino case, and
λ0
−=λ−+H33 tan2ω2−2Aω2tan ω2
1 + tan2ω2
, λ0
+=λ−tan2ω2+H33 + 2Aω2tan ω2
1 + tan2ω2
.
(24)
for the antineutrino case.
When the mixing between sterile neutrinos and active neutrinos is relatively
small and neutrino beam energy is E < 100 GeV, the oﬀdiagonal elements in
the eﬀective Hamiltonian will be very small compared with the diagonal ones after
two of the above rotations are performed. Namely the eﬀective Hamiltonian is
approximately diagonalized. So far, all of the eﬀective parameters (i.e. e
θ12,e
δ12,e
θ13
and e
δ13) in e
Uare presented. The diagonal terms in the eﬀective Hamiltonian in
the new representation can be treated as em2
i(i= 1,2,3,4). After subtracting the
smallest neutrino (antineutrino) mass λ−(λ0
−), we can get the eﬀective neutrino
(antineutrino) masssquared diﬀerence ∆em2
ij as
∆em2
21 ≈λ0
−−λ−,∆em2
31 ≈λ0
+−λ−,∆em2
41 ≈H44 −λ−.(25)
for the neutrino case, and
∆em2
21 ≈λ+−λ0
−,∆em2
31 ≈λ0
+−λ0
−,∆em2
41 ≈H44 −λ0
−.(26)
for the antineutrino case.
Up to now all the eﬀective parameters in 4ﬂavor neutrino oscillation have been
provided, and hence the neutrino oscillation probabilities can be easily calculated
using eq. 9. Since both CC and NC potentials in matter are proportional to neutrino
energy, the values of those eﬀective parameters in e
Uand em2
i(i= 1,2,3,4) are also
energy dependent, as shown in ﬁgure 1, 2.
3.3 Discussion
The eﬀective matrix e
Uin matter has introduced two eﬀective mixing angles e
θ12 and
e
θ13, two eﬀective CP phases e
δ12 and e
δ13, and eﬀective masssquared diﬀerences ∆ em2
ij ,
in which e
δ12 is an additional parameter introduced from the Jacobilike method.
These eﬀective parameters are clearly energy dependent, as shown in ﬁgure 1, 2.
In ﬁgure 1, when E < 100 MeV, e
θ12 and e
θ13 are very close to θ12 and θ13 values in
vacuum. The value of e
θ12 increases (decreases) rapidly up to the maximum π
2( the
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100101102103104105
Neutrino Energy (MeV)
0
2
3
2
2
Radian
12
13 (NH)
13 (IH)
12
13 (NH)
13 (IH)
100101102103104105
Antineutrino Energy (MeV)
0
2
3
2
2
Radian
['Counts/0.1MeV']
12
13 (NH)
13 (IH)
12
13 (NH)
13 (IH)
Figure 1: The values of e
θ12,e
θ13,e
δ12 and e
δ13 with respect to neutrino energy. In
this ﬁgure, we assume ∆m2
41 = 0.1 eV2, sin2θ14 = 0.019, sin2θ24 = 0.015, sin2θ34 =
0 [57], δ13 = 218◦[58] and δ24 =δ34 = 0◦. The solid and dashed lines are the eﬀective
angles and phases respectively. The shift of e
δ12 with a factor of πis caused by the
transmission of the sign “” from sin e
θ12 and cos e
θ12, in which we set e
θ12 within [0,π
2],
in eq.(18). As the energy rising in the case of the right plot, e
θ12 tends to be negative.
At that time, we shift the negative sign of it to the phase e
δ12 to make sure that e
θ12
is in [0,π
2], resulting in a πshift.
100101102103104105
Neutrino Energy (MeV)
10 5
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10 3
10 2
10 1
100
eV
2
m
2
21
m
2
21
m
2
31

m
2
31
m
2
41
m
2
41
100101102103104105
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100
eV
2
m
2
21
m
2
21
m
2
31

m
2
31
m
2
41
m
2
41
Figure 2: The values of ∆ em2
i1(i= 2,3,4) with respect to neutrino energy, assuming
∆m2
41 = 0.1 eV2, sin2θ14 = 0.019, sin2θ24 = 0.015, sin2θ34 = 0 [57], δ13 = 218◦[58]
and δ24 =δ34 = 0◦. The solid and dashed lines are for NH and IH.
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minimum 0) in the neutrino (antineutrino) energy range from 100 MeV to 10 GeV,
leading to sin e
θ12 →1 (sin e
θ12 →0). While e
θ13 starts to change after E > 1GeV. It
can go up to π
2assuming NH for neutrinos and IH for antineutrinos when E > 100
GeV; While it will go down to 0 for the other two combinations. When E < 1 GeV,
both eﬀective CP phases are close to their corresponding vacuum oscillation values
(e
δ12 →0 and e
δ13 →δ13). When energy increases above 1 GeV, the inﬂuence of
matter eﬀect on e
δ12 and e
δ13 is not negligible.
In ﬁgure 2, the eﬀect of matter also changes the values of eﬀective neutrino mass
squared diﬀerences ∆ em2
ij . When E < 100 MeV, ∆em2
21, ∆ em2
31 and ∆ em2
41 are close to
their vacuum values. ∆ em2
21 begins to vary when E > 100 MeV, while ∆ em2
31 starts
to change values after E > 1 GeV. In the case of ∆m2
41 = 0.1 eV2with current
sterile neutrino limits, ∆ em2
41 is insensitive to matter eﬀect when E < 100 GeV.
As neutrino energy increases, matter eﬀect shifts the values of eﬀective ∆ em2
21 more
than ∆ em2
31 and ∆ em2
41 when E < 100 GeV. It should be noticed that ∆ ˜m2
31has a
dip structure around 10 GeV for the antineutrino IH case. This feature also shows
up in the 3ﬂavor neutrino case.
In general, as shown in ﬁgure 1 and 2, matter eﬀect is negligible on both ∆ em2
21
and 12 neutrino mixing when ACC (ANC)∆m2
21 ∆m2
31 (or equivalently
E100 MeV). When energy increases, it is clear that the 12 neutrino mixing
submatrix is aﬀected from matter much more than other submatrices, as well as
∆em2
21. However, ∆ em2
31 and 13 mixing neutrino mixing still hold stable when ACC
(ANC)∆m2
31 (or equivalently E1 GeV). Furthermore, mixing between active
and sterile neutrinos has little impact. From a mathematical point of view, in the
function of rotation angles yielding tan θ=2A
γ−α±√(γ−α)2+4A2,Ais proportional to
the values of θ14,θ24 and θ34, and γ−αis inversely proportional to ∆m2
41. Hence, the
smallness of those mixing angles and large ∆m2
41(>0.1eV 2) can eﬀectively suppress
the values of the corresponding rotation angles to a negligible level in the submatri
ces. Therefore, after the rotations on 12 and 23 (13) submatrices of the neutrino
Hamiltonian, the eﬀective Hamiltonian matrix is approximately diagonal.
What we have discussed is for the general feature of our derived oscillation for
mula. In some particular cases, the oscillation formula can be simpliﬁed:
•No CP violations (δ13 =δ24 =δ34 = 0/π)
In such case, the neutrino mixing matrix is real and not necessary to introduce
an extra phase e
δ12 in eq. (17) for the matrix diagonalization. Thus, it falls back
to the original Jacobi method. The neutrino oscillation forms are identical to
the ones in vacuum with e
θ12 =θ12 +ω1,e
θ13 =θ13 +ω2and e
δ13 =δ13.
•No activesterile neutrino mixing (θ14 =θ24 =θ34 =δ24 =δ34 = 0)
The analytical approximations will reduce to 3ﬂavor neutrino oscillations.
4 The accuracy of the approximations
All the neutrino oscillation probabilities can be expressed with eq. (9) based on
eﬀective e
Uand ∆ em2
ij calculated in section 3.2. In this section, we ﬁrst check the
accuracy of these approximations. Then, we will highlight the accuracy of this work
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for two speciﬁc longbaseline accelerator neutrino experiments, T2HK and DUNE
respectively.
4.1 The general accuracy
The accuracy of our approximations can be quantiﬁed with ∆Pνα→νβ, which is de
ﬁned as the numerical diﬀerence between the approximations and exact solutions
for neutrino with αﬂavor type converting to βtype.
∆Pνα→νβ=PExact
να→νβ−PApproximate
να→νβ.(27)
For checking the validity of our approximations, Figure 3 presents the results, as
a function of neutrino energy and travel baseline, on four major neutrino oscillation
channels, including νe→νe,νµ→νµdisappearance, and νµ→νeand νe→νµ
appearance. The input values for the oscillation parameters are listed in table 1.
Here we conservatively assume the unknown sterile neutrino associated mixing angles
to be as large as 20◦and unknown phases to be maximal, 90◦.
active sterile
NH IH
sin2θ12 0.307 θ14 20◦
sin2θ13 0.0212 θ24 20◦
sin2θ23 0.417 0.421 θ34 20◦
∆m2
21 7.53 ×10−5eV2∆m2
41 0.1 eV2
∆m2
32 2.51 ×10−3eV2−2.56 ×10−3eV2δ24 π/2
δ13 π/2δ34 π/2
Table 1: Input values for the oscillation parameters. The ones associated with active
neutrino except δ13 are taken from the recent results [58]. The sterile neutrino values
are conservatively using relatively large values.
As shown in Figure 3, when neutrino energy below 20 GeV, the accuracy is
better than 10−3and 10−4for neutrino disappearance and appearance channels
respectively. The accuracy will get one or two orders better when neutrino energy is
smaller. Such numerical accuracy of our approximations is pretty good for most of
the long baseline neutrino oscillation experiments, since the diﬀerence is about one
order smaller than the oscillation probabilities. The inaccuracy of our approach is
mainly caused by the limited rotation iterations we applied. One approach to further
improve the accuracy is to apply some corrections based on perturbative method
after matrix rotation, as shown in Ref [59]. Of course, this will add the complexity of
the analytical expression. Our current approximations give a good balance between
the numerical accuracy and simplicity of the approximation expressions. To be
noted, the exact accuracy of the approximations depends on the exact neutrino
mixing parameter inputs. Once the sterile neutrino mixing angles to be smaller
than 20◦and ∆m2
41 be larger than 0.1 eV2, the accuracy of our expressions will
become even better than the current evaluation.
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100
101
102
103
104
Baseline (Km)
Pe e
100101102103104105
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Baseline (Km)
P
100101102103104105
Neutrino Energy (MeV)
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Baseline (Km)
Pe
100101102103104105
Neutrino Energy (MeV)
100
101
102
103
104
Baseline (Km)
Pe
10 610 510 410 310 210 1100
Figure 3: The accuracies of approximations in diﬀerent oscillation channels with
∆Pνα→νβ.
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0.00
0.02
0.04
0.06
0.08
0.10
Probability
DUNE
2 4 6 8 10
Neutrino Energy (GeV)
10 6
10 5
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0.0
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1.0
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Probability
DUNE
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Exact
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Exact
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0.06
0.08
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Probability
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0.2 0.4 0.6 0.8 1.0
Neutrino Energy (GeV)
10 6
10 5
10 4
P
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Probability
T2HK
0.2 0.4 0.6 0.8 1.0
Neutrino Energy (GeV)
10 5
10 4
10 3
P
e
Approx.
Exact
e
Approx.
Exact
Approx.
Exact
Approx.
Exact
Figure 4: The left plots are appearance channels and the right plots are disap
pearance channels in the case of NH. The appendant plots are accuracies of their
corresponding channels.
4.2 T2HK and DUNE
As shown in eq. (7) and (8), matter eﬀect is proportional to neutrino beam energy
and its propagation distance. Hence matter eﬀect can signiﬁcantly modify neutrino
oscillation features for longbaseline neutrino oscillation experiments, such as T2HK
and DUNE. Those experiments have relative high energy beams at ∼GeV and base
lines of hundreds and thousands of kilometers respectively. Here we would like to use
them as examples to demonstrate 3+1neutrino oscillation and check the accuracy
of our approximations.
Deep Underground Neutrino Experiment (DUNE) is the next generation onaxis
longbaseline accelerator neutrino experiment. It is proposed to use Liquid Argon
(LAr) detectors located deep underground 1300 km away from the beam source.
Its main physics goals are to solve three challenging issues in the neutrino sector,
neutrino mass hierarchy, CP asymmetry and the octant of θ23. It can look for elec
tron and tau neutrino (antineutrino) appearance and muon neutrino (antineutrino)
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disappearance channels from both neutrino and antineutrino beam modes.
0.2 0.4 0.6 0.8 1.0 1.2
Neutrino Energy (GeV)
10 5
10 4
10 3
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10 1
100
P
e
T2HK
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Neutrino Energy (GeV)
10 5
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10 3
10 2
10 1
100
P
e
DUNE
N.Klop et.al. + NCNSI Approximation
C.S.Fong et.al.
This work
Figure 5: The comparison of P¯νµ→¯νefor DUNE and T2HK in the case of NH with
the the values in table 1 except the phases. We adopt δ13 = 90◦and δ14 = 90◦
due to the convention N. Klop et. al. [51] use. Similar to the examples shown in
their paper, we combine it with a Neutral Current NSI solution [60] to produce the
oscillation probability.
TokaitoHyperKamiokande (T2HK) is a proposed longbaseline experiment,
which has a primary objective of measuring CP asymmetry. The far detector is 295
km away and 2.5◦oﬀaxis from the JPARC beam in Japan using water Cherenkov
detector.
Suppose the existence of the sterile neutrinos with relatively large masssquared
diﬀerence ∆m2
41 = 0.1eV2, the high frequency oscillation feature is clearly shown
in the muon neutrino disappearance and electron neutrino appearance modes in
ﬁgure 4. Given matter eﬀect and CPviolation phases, the electron neutrino and
antineutrino appearance probabilities Pνµ→νeand P¯νµ→¯νeare very diﬀerent. Com
pared with numerical calculation, the accuracy of the analytical approximations can
reach 10−5in the case of table 1. for neutrinos and antineutrinos respectively for
appearance mode. For disappearance mode, the accuracies of Pνµ→νµand P¯νµ→¯νµ
can reach 10−4in the case of table 1.
In ﬁgure 5, we compare the accuracy of this work with two previous ones for the
T2HK and DUNE experiments. Our work clearly shows about an order of better
accuracy compared with the other two, especially for the DUNE experiment. For a
higher accuracy of the approximation, we can continue making a perturbation cor
rection afterwards on the eﬀective neutrino mixing and masssquared diﬀerences as
the way [59] adopts. However, given the accuracy of this work is already pretty good
for the current and near future experiments, we don’t think it is really necessary.
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5 Summary
The search for light sterile neutrinos is an area of great interest in the neutrino ﬁeld.
Many long baseline neutrino experiments are very actively searching for light sterile
neutrinos in various mass regions. Both chargedcurrent and neutralcurrent induced
matter eﬀects are quite important for those experiments. Analytical approximations
of neutrino oscillation are more preferred in experimental neutrino research because
of its considerable timesaving and helpful for understanding neutrino oscillation
features.
In this manuscript we introduced a Jacobilike method to derive simpliﬁed ana
lytical expressions with good accuracy for neutrino oscillation in matter. The com
pact expressions of the eﬀective mixing matrix e
Uand the eﬀective masssquared
diﬀerences ∆ em2
ij (i, j = 1,2,3,4) are presented. The accuracy of this work is suﬃ
cient for the majority of long baseline neutrino experiments.
In addition, the Jacobilike method is a general method to diagonalize complex
Hermitian matrices. It also can be extended into other physics topics, such as 3
(active) + N (sterile)neutrino mixing, Neutrino NonStandard Interactions, etc.
6 Acknowledgments
We would like to extend our thanks for valuable discussions with YuFeng Li and
help with editing from Neill Raper. Jiajie Ling acknowledges the support from
National Key R&D program of China under Grant No. 2018YFA0404103 and Na
tional Natural Science Foundation of China under Grant No. 11775315. F. Xu
is supported partially by NSFC under Grant No. 11605076, as well as the FR
FCU (Fundamental Research Funds for the Central Universities in China) under
the Grant No. 21616309.
References
[1] Particle Data Group Collaboration, C. Patrignani et al., “Review of
Particle Physics,” Chin. Phys. C 40 no. 10, (2016) 100001.
[2] Daya Bay Collaboration, F. An et al., “Observation of electronantineutrino
disappearance at Daya Bay,” Phys. Rev. Lett. 108 (2012) 171803,
arXiv:1203.1669 [hepex].
[3] RENO Collaboration, J. Ahn et al., “Observation of Reactor Electron
Antineutrino Disappearance in the RENO Experiment,” Phys. Rev. Lett. 108
(2012) 191802, arXiv:1204.0626 [hepex].
[4] Double Chooz Collaboration, Y. Abe et al., “Indication of Reactor ¯νe
Disappearance in the Double Chooz Experiment,” Phys. Rev. Lett. 108
(2012) 131801, arXiv:1112.6353 [hepex].
[5] ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working
Group, SLD Electroweak Group, SLD Heavy Flavour Group
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A The parameterization of mixing matrix in vac
uum
In the 3 + 1 framework, neutrino mixing can be written as a 4 ×4 matrix (5). 6
rotation angles with 3 Dirac phases‡are found in this matrix. All the elements of
the mixing matrix are listed in table 2. Indeed, if we set the angles and Dirac phases
introduced by sterile neutrinos to 0, the 4×4 matrix will reduce to 3ﬂavor neutrino
mixing.
B Jacobilike method
Matter eﬀect for the 3+1 framework is more diﬃcult to calculate than for the 3
neutrino framework because additional parameters are involved in neutrino Hamil
tonian, which is a 4 ×4 complex Hermitian matrix and diﬃcult to diagonalize. In
this work, we adopt a rotation technique known as the Jacobilike method to solve
the diagonalization for complex Hermitian matrices. In the following, we provide all
technical details for the neutrino and antineutrino cases separately.
B.1 Neutrino case
In this subsection, we show how to diagonalize the eﬀective Hamiltonian with mat
ter eﬀect and simplify the expressions of the eﬀective mixing and masssquared
diﬀerences for the neutrino case.
B.1.1 Diagonalization process
In the case of neutrinos, we ﬁnd the absolute values of elements H12 and H21 in
eq. (10) are the relatively largest oﬀdiagonal ones because of the smallness of ∆m2
21.
Hence, we should diagonalize the 12 submatrix ﬁrst.
‡In this table, we adopt δ24 and δ34 as the sterile phases. It is equivalent to using δ14 and δ24 ,
or δ14 and δ34 for additional CP phases.
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α Uαi 
e
Ue1c12c13c14
Ue2c13c14s12
Ue3c14s13e−iδ13
Ue4s14
µ
Uµ1−s12c23c24 −c12(s13 c24s23 eiδ13 +c13s14 s24e−iδ24 )
Uµ2c12c23c24 −s12(s13 c24s23 eiδ13 +c13s14 s24e−iδ24 )
Uµ3c13c24s23 −s13s14 s24e−iδ13 e−iδ24
Uµ4c14s24e−iδ24
τ
Uτ1
c12[s13(s23s24 s34eiδ24 e−iδ34 −c23 c34)eiδ13 −c13 c24s14s34e−iδ34 ]
+s12(c34s23 +c23s24 s34eiδ24 e−iδ34 )
Uτ2
s12[s13(s23s24 s34eiδ24 e−iδ34 −c23 c34)eiδ13 −c13 c24s14s34e−iδ34 ]
−c12(c34s23 +c23s24 s34eiδ24 e−iδ34 )
Uτ3c13(c23c34 −s23s24 s34eiδ24 e−iδ34 )−s13 c24s14 s34e−iδ13 e−iδ34
Uτ4c14c24s34e−iδ34
s
Us1
c12[s13(c34s23 s24eiδ24 +c23 s34eiδ34 )eiδ13 −c13 c24c34s14]
+s12(c23c34s24 eiδ24 −s23s34 eiδ34 )
Us2
s12[s13(c34s23 s24eiδ24 +c23 s34eiδ34 )eiδ13 −c13 c24c34s14]
+c12(s23s34eiδ34 −c23 c34s24 eiδ24 )
Us3−c13(c34s23s24 eiδ24 +c23s34 eiδ34 )−c24c34 s14e−iδ13
Us4c14c24c34
Table 2: The elements of the 4ﬂavor mixing matrix.
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First rotation: The rotation matrix can be written as
R1=R12(ω1, φ1)≡
cω1sω1e−iφ10 0
−sω1eiφ1cω10 0
0 0 1 0
0 0 0 1
.(cω1= cos ω1, sω1= sin ω1)
(28)
We utilize the Jacobilike method to derive ω1yielding
tan ω1=2Aω1
(H22 −H11) + p(H22 −H11 )2+ 4A2
ω1
,(0 < ω1<π
2−θ12) (29)
with Aω1=H12and φ1= Arg(sign(Aω1)H∗
12). Here Aω1is the amplitude of
H12. After the rotation by R12(ω1, φ1), we rewrite the Hamiltonian in the new
representation as
H0=R†
12(ω1, φ1)HR12 (ω1, φ1) = 1
2E
λ−0H0
13 H0
14
0λ+H0
23 H0
24
H0
31 H0
32 H33 H34
H0
41 H0
42 H43 H44
,(30)
with the eigenvalues of the 12 submatrix
λ−=H11 +H22 tan2ω1−2Aω1tan ω1
1 + tan2ω1
, λ+=H11 tan2ω1+H22 + 2Aω1tan ω1
1 + tan2ω1
.
(31)
The corresponding oﬀdiagonal terms in H0are
H0
13 =H0∗
31 =cω1H13 −sω1H23e−iφ1,(32)
H0
23 =H0∗
32 =cω1H23 +sω1H13eiφ1,(33)
H0
14 =H0∗
41 =cω1H14 −sω1H24e−iφ1,(34)
H0
24 =H0∗
42 =cω1H24 +sω1H14eiφ1.(35)
After ﬁrst rotation, we can diagonalize the 23 submatrix afterward since it has
a relatively big oﬀdiagonal element and is useful to simplify the eﬀective matrix U
through eq. (49).
Second rotation: The second rotation matrix yields
R2=R23(ω2, φ2)≡
1 0 0 0
0cω2sω2e−iφ20
0−sω2eiφ2cω20
0 0 0 1