Review of Neutrino Oscillations With Sterile and Active Neutrinos

0.19 MB·

by  Leonard S. Kisslinger

Review of Neutrino Oscillations With Sterile and Active
Neutrinos
6 Leonard S. Kisslinger
1
Department of Physics, Carnegie Mellon University, Pittsburgh PA 15213
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g August 12, 2016
u
A

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PACS Indices:11.30.Er,14.60.Lm,13.15.+g
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] Keywords: sterile neutrinos, neutrino oscillations, U-matrix
h
p
Abstract
-
p
e Recently neutrino oscillation experiments have shown that it is very likely that
h
[ there are one or two sterile neutrinos. In this review neutrino oscillations with one,

two, three sterile and three active neutrinos, and parameters that are consistent with
2
v experiments, are reviewed.
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3 1 Introduction
5
0
1. ThisisareviewofthemethodintroducedbySatoandcollaboratorsforthreeactiveneutrinos[1,
0 2] extended to three active neutrinos plus one, two, or three sterile neutrinos. The transition
6
probability for a neutrino of flavor f1 to oscillate to a neutrino of flavor f2, P(ν → ν ), is
1 f1 f2
: derived using S-Matrix theory, which is discussed in the next section with f1,f2 → muon,
v
i electron neutrinos.
X
In the following three sections the derivation of P(ν → ν ) is described for one, two,
r µ e
a three sterile neutrinos, with predictions using parameters of four recent neutrino oscillation
experiments. In all three sections a U-matrix approach is used, introduced with a 3x3 U-
martix[1], and extended to a 4x4 U-martix with three active and one sterile neutrino a 5x5
U-martix with three active and two sterile neutrinos and a 6x6 U-martix with three active
and three sterile neutrinos.
From these sections the dependence of the P(ν → ν ) neutrino oscillation probability
µ e
on the number of sterile neutrinos and oscillation parameters will be shown.
1
2 P(ν → ν ) for Active Neutrinos Derived Using Im-
µ e
proved S-Matrix Theory
Neutrinos are produced as ν , with f =flavor=e,µ,τ. They do not have definite mass,
f
which is the cause of neutrino oscillatios, which we now discuss. Active neutrinos with
flavors ν ,ν ,ν are related to neutrinos with definite mass ν , m=1,2,3 by the 3×3 unitary
e µ τ m
matrix, U,
ν = Uν , (1)
f m
where ν ,ν are 3×1 column vectors and U a 3x3 matrix. Therefore the electron state
f m
produced at time t=0 is
3
|ν > = U |ν > . (2)
e 1i mj
X
i=1
Making use of quantum theory, a state with energy E satisfies
id/dt|E(t) > = E|E(t) > giving
|E(t) > = e−iEt|E(t = 0) > , (3)
or the electron neutrino state of Eq(2) at time t for the neutrino at rest (e=m, with c ≡ 1)
is
3
|ν ,t >= = U e−imit|m ,t = 0 > or
e 1i i
X
i=1
3
|ν ,t > = c |ν > , (4)
e f f
X
i=1
whichshowsthatanelectronneutrinoproducedattimet=0oscillatestoneutrinosofdifferent
flavors at time t. Therefore an electron neutrino produced at t=0 when it travels a distance
L ≃ t (as the velocity of the very low mass neutrinos is almost the speed of light) oscillates
intoe,µ,τ neutrinos. Byplacingdetectors atadistance Lthisoscillationhasbeenmeasured.
The ν neutrino has a similar relationship, and can also oscillate to a sterile neutrino as we
µ
discuss in sections below.
The 3x3 active neutrino U-matrix is (sinθ ≡ s , etc).
ij ij
c c s c s e−iδCP
12 13 12 13 13
U=  −s c −c s s eiδCP c c −s s s eiδCP s c 
12 23 12 23 13 12 23 12 23 13 23 13
s s −c s s eiδCP −c s −s c s eiδCP c c
 12 23 12 23 13 12 23 12 23 13 23 13 
where c = .83, s = .56, s = c = .7071, s = 0.19, and δ =0 are used.
12 12 23 23 13 CP
Given the Hamiltonian, H(t), for neutrinos, the neutrino state at time = t is obtained
from the state at time = t from the S-matrix, S(t,t ), by
0 0
|ν(t) > = S(t,t )|ν(t ) > (5)
0 0
d
i S(t,t ) = H(t)S(t,t ) . (6)
0 0
dt
2
In the vacuum the S-matrix is obtained from
3
S (t,t ) = U expiEj(t−t0)U∗ , (7)
ab 0 aj bj
X
j=1
while for neutrinos travelling through the earth the potential V = 1.13×10−13 ev is included.
The transition probability P(ν → ν ) is obtained from the S-Matrix element S :
µ e 12
P(ν → ν ) = (Re[S ])2 +(Im[S ])2 . (8)
µ e 12 12
From Ref[4]
Re[S12] = s23a[cos(∆¯L)Im[Iα∗]−sin(∆¯L)Re[Iα∗]]
Im[S12] = −c23sin2θsinωL−s23a[cos(∆¯L)Re[Iα∗]
+sin(∆¯L)Im[Iα∗] , (9)
with∆¯ = ∆−(V+δ)/2,∆ = δm2 /(2E),δ = δm2 /(2E),wheretheneutrinomassdifferences
13 12
are δm2 = 7.6x10−5(eV)2, δm2 = 2.4x10−3(eV)2, sin2θ = s c δ, a = s (∆−s2 δ), and
12 13 12 12ω 13 12
E is the neutrino energy. Note that t → L, where L is the baseline, for v ≃ c. The
ν
neutrino-matter potential V = 1.13×10−13 eV.
An important quantity is Iα∗
t
Iα∗ = Z dt′α∗(t′)e−i∆¯t′ , (10)
0
with α(t) = cos(ωt) − icos2θsin(ωt), ω = δ2 +V2 −2δVcos(2θ )/2. In Ref.[4], as in
12
Ref.[5], one used δ,ω ≪ ∆ to obtain p
Re[Iα∗] ≃ sin∆¯L/∆¯
Im[Iα∗] ≃ (1−cos∆¯L)/∆¯ . (11)
In an improved theory[6] it was shown that
Re[Iα∗] = [(ω −∆¯cos2θ)cos∆¯LsinωL
−(∆¯ −ωcos2θ)sin∆¯LcosωL]/(ω2−∆¯2) (12)
¯ ¯ ¯
Im[Iα∗] = [∆+ωcos2θ−(∆+ωcos2θ)cos∆LcosωL
−(ω +∆¯cos2θ)sin∆¯LsinωL]/(ω2 −∆¯2) .
From Eqs(9,12) P(ν → ν ) = (Re[S ])2 + (Im[S ])2 is obtained, giving the results
µ e 12 12
shown in Figure 1.
3
P(νµ → νe) with old and precise Iα∗(E,L).
0.10
0.08
−e) 0.06
µP( 0.04
0.02
1 2 3 4 5 6 7 8E ν (GeV) MINOS
8x10−7
6x10−7
µP( −e) 4x10−7
2x10−7
E ν (GeV) MiniBoone
0.5 0.7 0.9 1.1 1.3 1.5
0.10
0.08
−e)
µP(
0.06
0.040.4 0.5 0.6 0.7 0.8 0.9 1.0E ν (GeV) Kamioka
0.6
0.5
µP( −e) 0.4
0.3
0.2 E ν (MeV) CHOOZ
2.999 3.000 3.001
Figure 1: P(ν → ν ) for MINOS(L=735 km), MiniBooNE(L=500m), JHF-Kamioka(L=295
µ e
km), and CHOOZ(L=1.03 km) using the improved 3×3 mixing matrix. Solid curve for
precise Iα∗(E,L) and dashed curve for approximate Iα∗(E,L), s13=0.19
4
3 P(ν → ν ) With Three Active and One Sterile Neu-
µ e
trino
Active neutrinos have only weak and gravitational interactions, and are therefore difficult
to detect in neutrino oscillation experiments. Sterile neutrinos have no interaction except
gravity and therefore cannot be detected via the apparatus used in neutrino oscillation or
other experiments. For an overview of sterile neutrinos and neutrino oscillations see Ref[8] in
which sterile neutrino states are investigated using neutrino oscillation data. These authors,
J. Koppe et. al., considered one and two sterile neutrinos and discussed both experimental
and theoretical publications.
In the present section we discuss neutrino oscillations with one sterile neutrino, while in
the next section two sterile neutrinos are discussed.
Motivated by an experiment measuring neutrino oscillations[9], which suggested the ex-
istence of at least one sterile neutrino and estimated the mass differences and mixing angles
with active neutrinos, estimates of P(ν → ν ) were made[10]. We now review this article.
µ e
This is an exension of the method introduced by Sato and collaborators for three active
neutrino oscillations[1, 2] to three active neutrinos plus onesterile neutrino. Active neutrinos
with flavors ν ,ν ,ν and a sterile neutrino ν are related to neutrinos with definite mass by
e µ τ s
ν = Uν , (13)
f m
where U is a 4x4 matrix and ν ,ν are 4x1 column vectors.
f m
U = O23φO13O12O14O24O34 with (14)
1 0 0 0 c 0 s 0
13 13
 0 c s 0   0 1 0 0 
O23= 23 23 , O13= ,
0 −s c 0 −s 0 c 0
 23 23   13 13 
 0 0 0 1   0 0 0 1 
   
c s 0 0 c 0 0 s
12 12 α α
 −s c 0 0   0 1 0 0 
O12= 12 12 , O14= ,
0 0 1 0 0 0 1 0
   
 0 0 0 1   −s 0 0 c 
   α α 
1 0 0 0 1 0 0 0
 0 c 0 s   0 1 0 0 
O24= α α , O34=
0 0 1 0 0 0 c s
   α α 
 0 −s 0 c   0 0 −s c 
 α α   α α 
1 0 0 0
 0 1 0 0 
φ=
0 0 eiδCP 0
 
 0 0 0 1 
 
5
with c = .83, s = .56, s = c = .7071. We use s = .15 from the Daya Bay
12 12 23 23 13
Collaboration[7]. In our present work we assume the angles θ ≡ α for all three j, and
j4
s ,c = sinα,cosα. Animportant aspect of thisworkwas to findthedependence ofneutrino
α α
oscillation probabilities on s ,c .
α α
From Eq(14) the 4x4 U matrix is
c c c c (s c −c s2) −c s2(c c +s )+s c c s c (c c +s )+s s
12 13 α 13 12 α 12 α 13 α 12 α 12 13 α 13 α α 12 α 12 13 α
 Ac −As2 +Bc −As2c −Bs2 +c s eiδCPc As c2 +Bs c +c s eiδCPs 
α α α α α α 13 23 α α α α α 13 23 α
Cc −Cs2 +Dc −Cs2c −Ds2 +c c eiδCPc Cs c2 +Ds c +c s eiδCPs
 α α α α α α 13 23 α α α α α 13 23 α 
 −s −s c −s c2 c3 
 α α α α α α 
with
A = −(c s +c s s eiδCP)
23 12 12 13 23
B = (c c −s s s eiδCP) (15)
23 12 12 13 23
C = (s s −c s c eiδCP)
23 12 12 13 23
D = −(s c +s s c eiδCP) .
23 12 12 13 23
Using the formalism of Refs.[1, 2] extended to four neutrinos, the transition probability
P(ν → ν ) is obtained from the 4x4 U matrix and the neutrino mass differences δm2 =
µ e ij
m2 −m2 for a neutrino beam with energy E and baseline L by[1]
i j
4 4
P(νµ → νe) = U1iU1∗jU2∗iU2je−i(δm2ij/E)L , (16)
XX
i=1 j=1
∗
or, since with δ = 0 U = U ,
CP ij ij
P(ν → ν ) = U U [U U +U U e−iδL +U U e−i∆L +U U e−iγL]+
µ e 11 21 11 21 12 22 13 23 14 24
U U [U U e−iδL +U U +U U e−i∆L +U U e−iγL]+
12 22 11 21 12 22 13 23 14 24
U U [U U e−i∆L +U U e−i∆L +U U +U U e−iγL]+
13 23 11 21 12 22 13 23 14 24
U U [(U U +U U +U U )e−iγL +U U ] , (17)
14 24 11 21 12 22 13 23 14 24
with δ = δm2 /2E, ∆ = δm2 /2E, γ = δm2 /2E (j=1,2,3). The neutrino mass differences
12 13 j4
are δm2 = 7.6×10−5(eV)2, δm2 = 2.4×10−3(eV)2; and we use both δm2 = 0.9(eV)2 and
12 13 j4
δm2 = 0.043(eV)2, since δm2 = 0.043(eV)2 was the best fit parameter found via the 2013
j4 j4
MiniBooNE analysis, while δm2 = 0.9(eV)2 is the best fit using the 2013 MiniBooNE data
j4
and previous experimental fits[9].
Note that in Refs[3, 4] P(ν → ν ) = |S |2, with S obtained from the 3x3 U-matrix
µ e 12 12
and the δm parameters. Therefore our formalism, given by Eq(17), is quite different, and
ij
as will be shown for the same L,E the magnitude of P(ν → ν ) is also different. Since
µ e
the S-matrix formalism was not used in Refs.[1, 2] for the 3x3 study, P(ν → ν ) was quite
µ e
different from Refs[3, 4].
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From Eq(17),
P(ν → ν ) = U2 U2 +U2 U2 +U2 U2 +
µ e 11 21 12 22 13 23
U2 U2 +2U U U U cosδL+ (18)
14 24 11 21 12 22
2(U U U U +U U U U )cos∆L+
11 21 13 22 12 22 13 23
2U U (U U +U U +U U )cosγL .
14 24 11 21 12 22 13 23
Using the parameters given above,
U = .822c U = −.554s2 +0.084c
11 α 12 α α
U = −.822s2c −.554s2 +.15c U = .822s c2 +.554s c +.15s
13 α α α α 14 α α α α α
U = −.484c U = .484s2 +.527c (19)
21 α 22 α α
U = .484c −.527s2 +.7c U = −.484s c2 +.527s c +.7s .
23 α α α 24 α α α α α
With the addition of a sterile neutrino, the 4th neutrino, there are three new angles, θ , θ ,
14 24
and θ . The main assumption is that these three angles are the same, θ = α. The angle
34 j4
α is the main parameter that is being studyied.
Two values for the sterile-active mass differences are used. The most widely accepted
value for m2 − m2 is 0.9(eV)2[9], but we also use m2 − m2 = .043(eV)2 from the 2013
4 1 4 1
MiniBoonEresult totestthesensitivity ofP(ν → ν )tothesterileneutrino-activeneutrinos
µ e
mass differences. Since m2 − m2 >> m2 − m2 for (i,j)=1,2,3, we assume that m2 −m2 =
4 1 j i 4 j
m2 −m2.
4 1
Figure 2 shows the results for P(ν → ν ) for the four experiments with m2 − m2 =
µ e 4 1
0.9(eV)2 and α = 45o,60o,30o. As one can see, P(ν → ν ) is very strongly dependent on α.
µ e
Next m2−m2 = 0.043(eV)2 was used, as found in the recent MiniBooNE experiment, to
4 1
study the effects of m2 −m2 on P(ν → ν ), with α = 45o,30o,60o, as shown in Figure 3.
4 1 µ e
Note for α = 0 (no sterile-active mixing) U = 0. Therefore, P(ν → ν ) is a 3x3 theory;
14 µ e
however, we find that P(ν → ν ) is different with the model of Refs.[1, 2].
µ e
An article Neutrino Oscillations With Recently Measured Sterile-Active Neutrino Mixing
Angle sin(α) = 0.16 was recently published[11], with estimates of P(ν → ν ), is shown in
µ e
Figure 4.
7
1.00
(a)
0.75 (b)
e)
− 0.50 (c)
µP(
0.25
0.0 E ν (GeV) MINOS
5.002 5.006 5.010
0.75
(a)
e) 0.50 (b)
−
µP( (c)
0.25
0.0 E ν (GeV) MiniBooNE
0.5 0.7 0.9 1.1 1.3 1.5
(a)
0.50 (b)
(c)
e)
µP( − 0.25
0.0 E ν (GeV) Kamioka
0.7010 0.7015 0.7020
1.00
(a)
e) 0.75 (b)
−
µP(
0.50
(c)
0.25
0.0 E ν (GeV) CHOOZ
0.00300001 0.003000015 0.00300002
Figure 2: The ordinate is P(ν → ν ) for MINOS(L=735 km), MiniBooNE(L=500m),
µ e
JHF-Kamioka(L=295 km), and CHOOZ(L=1.03 km) using the 4x4 U matrix with δm2 =
j4
0.9(eV)2, s = 0.15, and (a),(b),(c) for α = 45o,60o,30o. The dashed curves are for α = 0
13
(3x3).
1.00 (a)
(b)
0.75
(c)
0.50
e)
−
µP( 0.25
0.0 E ν (GeV) MINOS
5.002 5.006 5.010
0.100
0.075
e) (b)
−
µP( 0.050 (a)
0.025 (c)
0.0 Eν (GeV) MiniBooNE
0.5 0.7 0.9 1.1 1.3 1.5
0.20
0.15
0.10
e)
− (c)
µP( 0.05 (b)
(a)
0.0 E ν (GeV) Kamioka
0.7010 0.7015 0.7020
0.20
e) 0.15
−
µP(
0.10
(c) (b)
0.05
(a)
0.0 E ν (MeV) CHOOZ
.000300000 .00300001 .00300002
Figure 3: The ordinate is P(ν → ν ) for MINOS(L=735 km), MiniBooNE(L=500m),
µ e
JHF-Kamioka(L=295 km), and CHOOZ(L=1.03 km) using the 4x4 U matrix with δm2 =
j4
0.043(eV)2, s = 0.15, and (a),(b),(c) for α = 45o,60o,30o. The dashed curves are for α = 0
13
(3x3).
9
0.010
0.009
0.008
e)
µP( − 0.007
0.006 E ν (GeV) MINOS
5.002 5.004 5.006 5.008
0.003
0.002
e)
−
µP( 0.001
1.3x10−6 E ν (GeV) MiniBoone
0.5 0.7 0.9 1.1 1.3 1.5
0.045
e)
−
µP( 0.040
0.035
0.030 E ν (GeV) T2K
0.7012 0.7016 0.7020
0.40
e)
−
µP( 0.35
0.30 E ν (MeV) Double CHOOZ
3.00000 3.00001 3.00002
Figure 4: The ordinate is P(ν → ν ) for MINOS(L=735 km), MiniBooNE(L=500m),
µ e
JHF-Kamioka(L=295 km), and CHOOZ(L=1.03 km) using the 4x4 U matrix with δm2 =
j4
0.9(eV)2 and sin(α) ≃ 0.16. The dashed curves are for α = 0 (3x3).
10