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我们研究了非标准相互作用(NSI)和真正的CP相δD在低E v≲1 GeV和非常低的E v≲0.1 GeV能量的中微子振荡中的影响。 对于将基线和中微子能量调节至第一个1-3振动最大值的实验装置,我们开发了一种简单的分析形式主义来展示NSI的影响。 模仿真空和违反真空以及使用分离基础在我们的形式主义中起着核心作用。 影响CP相位测量的NSI校正主要来自对真空模拟的违反以及对1-3混合角和质量平方差的校正。 我们发现,在T2(H)K时,对νμ-νe概率Pμe的总NSI校正可以达到20%-30%(1σ)。 相应地,对CP相位的校正可以大到50°,因此大大降低了T2(H)K下的CP灵敏度。 拟议的TNT2K实验,T2(H)K和短基线实验μKam的组合,使用Kamioka的Super-K / Hyper-K检测器来测量静止中的μ子衰减引起的中微子的振荡(μDAR), 可以大大降低NSI和真正的CP相δD之间的简并性,从而提供高CP灵敏度。 对于μDAR中微子,NSI对Pμe的校正仅为2%(1σ)。
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JHEP10(2016)138
Published for SISSA by Springer
Received: August 8, 2016
Revised: September 28, 2016
Accepted: October 6, 2016
Published: October 25, 2016
Non-standard interactions and the CP phase
measurements in neutrino oscillations at low energies
Shao-Feng Ge and Alexei Yu. Smirnov
Max-Planck-Institut f¨ur Kernphysik,
Saupfercheckweg 1, Heidelberg 69117, Germany
E-mail: gesf02@gmail.com, smirnov@mpi-hd.mpg.de
Abstract: We study the effects of non-standard interactions (NSI) and the genuine CP
phase δ
D
in neutrino oscillations at low, E
ν
. 1 GeV, and very low, E
ν
. 0.1 GeV, energies.
For experimental setup with baseline and neutrino energy tuned to the first 1-3 oscillation
maximum, we develop a simple analytic formalism to show the effects of NSI. The vacuum
mimicking and its violation as well as the use of the separation basis play a central role in
our formalism. The NSI corrections that affect the CP phase measurement mainly come
from the violation of vacuum mimicking as well as from the corrections to the 1-3 mixing
angle and mass-squared difference. We find that the total NSI correction to the ν
µ
− ν
e
probability P
µe
can reach 20% −30% (1σ) at T2(H)K. Correspondingly, the correction to
the CP phase can be as large as 50
◦
and hence significantly deteriorates the CP sensitivity
at T2(H)K. The proposed TNT2K experiment, a combination of T2(H)K and the short
baseline experiment µKam that uses the Super-K/Hyper-K detector at Kamioka to measure
the oscillation of the antineutrinos from muon decay at rest (µDAR), can substantially
reduce the degeneracy between NSI and the genuine CP phase δ
D
to provide high CP
sensitivity. The NSI correction to P
µe
is only 2% (1σ) for µDAR neutrinos.
Keywords: CP violation, Neutrino Physics
ArXiv ePrint: 1607.08513
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
doi:10.1007/JHEP10(2016)138
JHEP10(2016)138
Contents
1 Introduction 1
2 Neutrino oscillation in matter at low energies 3
2.1 Generalities and physics setup 3
2.2 Oscillation probability 5
2.3 On vacuum mimicking 7
3 Standard interaction and vacuum mimicking 9
3.1 Oscillation parameters 9
3.2 The oscillation probability 10
4 Effects of non-standard interactions 12
4.1 Oscillation parameters in the presence of NSI 12
4.2 Matter corrections and the violation of vacuum mimicking 15
4.3 Corrections to the total oscillation probability 20
5 CP measurement in the presence of non-standard interactions 22
5.1 The CP sensitivity at T2K in the presence of NSI 24
5.2 Improving the CP sensitivity with neutrinos from µDAR 25
6 Conclusions 28
1 Introduction
The determination of the Dirac CP phase, δ
D
, and searches for new (non-standard) neutrino
interactions [1–3] are among the main objectives of present-day neutrino physics (see [4, 5]
for reviews). There are certain connections between the two unknowns:
- NSI can be additional source of CP violation on top of the PMNS matrix [6, 7].
- The effects of NSI and CP violation are entangled in oscillation experiments, thus
creating the δ
D
-NSI degeneracy problem [7, 8].
Within the standard 3ν paradigm (no NSI, no sterile neutrinos, etc.), the global fit
of oscillation data gives more than 2σ indication of CP violation with the best-fit value
δ
D
≈ 3π/2. This indication follows mainly from the results of T2K [9] and NOνA [10]
in combination with the results of reactor and some other experiments. The confirmation
of this hint and measurement of the CP phase are associated to future experiments with
neutrinos of different origins. The current T2K experiment can be extended to T2K-II [11,
12] by upgrading the flux and eventually to T2HK with a much larger Hyper-K detector [13],
– 1 –
JHEP10(2016)138
or even T2KK/T2KO [14–21] with longer baseline. The DUNE [22–24] experiment should
provide precision measurement of δ
D
. In addition to accelerator experiments, the CP
phase δ
D
can also have large observable effect in the oscillation of atmospheric neutrinos
below 1 GeV. This can be explored using the future possible upgrades of PINGU and
ORCA detectors with 3 ∼ 4 times denser instrumentation [25]. At even lower energy,
E
ν
. 0.1 GeV, one can also employ neutrinos from the muon decay at rest (µDAR) in
proposed experiments like DAEδALUS [26] and CI-ADS [27], or in-flight in experiments
like MOMENT [28, 29] to measure δ
D
. Measuring the antineutrino oscillation ¯ν
µ
→ ¯ν
e
from the µDAR source with the detector of JUNO/RENO-50, which is originally designed
to establish the neutrino mass hierarchy, can also determine the CP phase [30].
Interpretation of the present results on CP violation can be affected by the possible
presence of non-standard interactions [31] or sterile neutrinos [32–34]. The effects of NSI
substantially reduce the sensitivity to δ
D
at the ongoing T2K and NOνA experiments [31,
35, 36], and the future long-baseline experiments, in particular, DUNE [31, 35–43]. So, the
problem is to disentangle the genuine CP violation and the NSI effects in a way that can
guarantee high sensitivity to the CP phase. To achieve this, one can use the fact that the
effects of NSI, in particular the effects on CP violation, are proportional to the neutrino
energy E
ν
. To disentangle the genuine CP phase from NSI, one can use two or more
experiments with different neutrino energies. The effects of NSI can be as large as O(1) at
accelerator neutrino experiments with E
ν
> 500 MeV like DUNE. The neutrino spectrum
from the MOMENT source peaks around (200 ∼ 300) MeV [28], being 10 times smaller
than the peak energy at DUNE. Hence, a combination of the future MOMENT and DUNE
results could reduce the degeneracy between δ
D
and the NSI parameters [40]. Still, the
NSI effects at the MOMENT energy are large. Rescaled from the estimation for T2K, the
relative correction of NSI to the oscillation probability P
µe
is roughly 10% at MOMENT.
For neutrinos from µDAR with energies ∼ (30 −50) MeV the NSI corrections are only 2%.
Various possible experiments with µDAR sources have been explored. In particular,
it was proposed [44] to use the Super-K detector (and later Hyper-K) to detect the an-
tineutrino oscillation ¯ν
µ
→ ¯ν
e
from µDAR source to Kamioka (dubbed by µKam, which
can be either µSK with the Super-K detector or µHK with the Hyper-K detector). The
combination of T2K running solely in the neutrino (νT2K) and µSK in the antineutrino
mode, referred as TNT2K, provides even higher sensitivity to the CP phase δ
D
[44].
In most papers, the NSI effects have been computed numerically although several
analytical studies can be found in [6, 35, 45, 46]. These earlier studies provide analytical
expressions of the oscillation probabilities in the ¯ν
e
→ ¯ν
e
[45, 46], ν
e
→ ν
µ
[6, 46], ν
µ
→
ν
e
[35, 45, 46], and ν
µ
→ ν
µ
[45, 46] channels. In particular, [46] provides the probability
formula for all oscillation channels. In this paper, we present an analytical formalism which
allows us to analyze the effects of NSI in a simple way. The vacuum mimicking
1
in the
1-2 sector and its violation plays a central role in our formalism. We show that NSI can
induce large correction via the violation of vacuum mimicking and apply our results to the
CP measurement at TNT2K.
1
The effect of vacuum mimicking was noticed in [1, 47, 48], explained in [49], further studied in [50–52]
and dubbed as such in [53].
– 2 –
JHEP10(2016)138
The paper is organized as follows. In section 2 we present a general formalism of matter
effects and discuss the role of vacuum mimicking in the sector of 1-2 mass splitting and
mixing. We then apply this formalism to the case of standard interactions in section 3 and
non-standard interactions in section 4. The vacuum mimicking can be significantly violated
by NSI and the Dirac CP phase δ
D
can receive O(1) correction. In section 5 we explore
the sensitivity of T2K and µSK experiments to the genuine CP phase in the presence of
NSI. Especially, the improvement on the sensitivity due to the µSK component of TNT2K
is shown. Our conclusions are given in section 6.
2 Neutrino oscillation in matter at low energies
2.1 Generalities and physics setup
In the flavor basis, the Hamiltonian H that describes the neutrino propagation is a sum of
the vacuum term and the matrix of matter potential V:
H ≡
1
2E
ν
U
PMNS
0
∆m
2
21
∆m
2
31
U
†
PMNS
+ V , (2.1)
where ∆m
2
21
≡ m
2
2
− m
2
1
, ∆m
2
31
≡ m
2
3
− m
2
1
, and E
ν
is the neutrino energy. The matrix
of matter potential V in general has contributions from both standard and non-standard
interactions. For the mixing matrix in vacuum U
PMNS
we use the standard parametrization
U
PMNS
≡ U
23
(θ
23
)Γ
δ
D
U
13
(θ
13
)Γ
†
δ
D
U
12
(θ
12
), Γ
δ
D
= diag(1, 1, e
iδ
D
).
Here θ
ij
are the vacuum mixing angles and δ
D
is the Dirac CP phase. In the case of
standard interaction only, the matrix of matter potential takes the form as
V = diag{V, 0, 0} and V =
√
2G
F
n
e
, (2.2)
where G
F
is the Fermi coupling and n
e
is the electron number density. For the standard
interaction with a constant density 3g/cm
3
of isotopically neutral matter and sin
2
θ
12
≈
0.31, the resonance neutrino energy due to the 1-2 mass splitting equals
E
R
= cos 2θ
12
∆m
2
21
2V
= 122 MeV. (2.3)
At low energy, E
ν
. 1 GeV, the three mass-energy scales (∆m
2
21
, ∆m
2
31
, 2E
ν
V ) in the
Hamiltonian (2.1) satisfy the following conditions:
∆m
2
21
2E
ν
∼ V
∆m
2
31
2E
ν
. (2.4)
This allows us to introduce two small parameters
r
∆
≡
∆m
2
21
∆m
2
31
≈ 0.03, x
31
≡
2EV
∆m
2
31
= 0.01
E
ν
100MeV
(2.5)
– 3 –
JHEP10(2016)138
which will be used for perturbative diagonalization of the Hamiltonian. For typical energies
E
ν
= 50 MeV at µDAR and E
ν
= 600 MeV at T2K, we obtain x
31
= 5·10
−3
and x
31
= 0.06,
respectively.
At low energies the matter corrections to the 1-3 mixing angle and mass splitting are
small. It is more convenient to consider the neutrino oscillation in a different basis ν
0
that
is related to the flavor basis ν
f
as, [54, 55],
ν
f
= Rν
0
, (2.6)
where
R ≡ U
23
(θ
23
)Γ
δ
U
13
(θ
13
) . (2.7)
The matrix R depends only on the vacuum mixing angles (θ
23
, θ
13
) and the phase δ
D
.
Using (2.1) and (2.7), we obtain the Hamiltonian in the ν
0
basis
H
0
≡ R
†
HR =
∆m
2
31
2E
ν
0
0
1
+
∆m
2
21
2E
ν
s
2
12
c
12
s
12
c
12
s
12
c
2
12
0
+ V
sep
, (2.8)
where
V
sep
≡ U
T
13
Γ
†
δ
U
T
23
VU
23
Γ
δ
U
13
, (2.9)
and we use the notation (c
ij
, s
ij
) ≡ (cos θ
ij
, sin θ
ij
). In this basis, the first term with the
largest mass scale ∆m
2
31
is diagonal and separated from the 1-2 sector in vacuum. The
second term in (2.8) depends on the vacuum parameters of the 1-2 sector only while the last
term is the matter term that provides all other (in particular 1-3) mixings. For convenience,
we call this basis the separation basis. It is related to the usual propagation basis [56–58]
by an additional rotation Γ
δ
U
13
(θ
13
) [54, 55].
Let us introduce a unitary mixing matrix U
0
to diagonalize the Hamiltonian H
0
(2.8):
U
0†
H
0
U
0
= diag(H
1
, H
2
, H
3
), (2.10)
where H
i
are the eigenvalues of H
0
. Then the total mixing matrix in matter, U
m
, which
connects the flavor states and the eigenstates of the Hamiltonian H, ν
f
= U
m
ν
m
, is given
by the product
U
m
= RU
0
. (2.11)
In (2.11), the vacuum and matter parts are factorized. Without matter effect, we obtain
U
0
= U
12
(θ
12
).
In matter with constant density, solving the evolution equation is straightforward. For
the eigenstates of the Hamiltonian, ν
m
, we obtain according to (2.10) the evolution matrix
S
d
=
1
e
−i∆H
21
L
e
−i∆H
31
L
, (2.12)
– 4 –
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