Joint Theoretical Physics Colloquium
Neutrinos
Thomas Gajdosik
Neutrinos and their Oscillations
· History / What we know today · Meaning of Oscillations · "Our" model
Thomas Gajdosik Faculty of Physics Department of Theoretical Physics
Historical benchmarks 1930 Wolfgang Pauli proposed an ”undetectable” particle: the Neutrino • in 1930 the neutrino was written as a two component Weyl spinor – it had to be massless 1957 Bruno Pontecorvo predicted neutrino oscillations • . . . and was ignored for 30+ years . . . by most of the particle physics community 1998 Super Kamiokande (SK) • first experimental evidence for atmospheric neutrino oscillations 2001 Sudbury Neutrino Observatory (SNO) • clear evidence of neutrino flavor change in solar neutrinos ⇒ the three neutrinos νe, νµ, ντ mix Better: there are three massive states of neutrinos ( with a tiny mass ) • that have a mixed coupling to electron, muon and tau • one explanation of the tiny mass: seesaw mechanism Thomas Gajdosik
Neutrinos
JTPC
. . . needs Majorana spinors
2016 / 04 / 25
2
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
How do we distinguish particles? • according to Special Relativity with mass and spin – particles with the same mass and the same spin are the same particles
• by spin: a boson is different from a fermion – 4He behaves differently than 3He
• by mass: a muon has a differnt mass than an electron – proton and neutron have nearly the same mass ∗ π + and π − have exactly the same mass
• by charge: – proton has a positive charge, the neutron is neutral ∗ π + and π − have opposite charge
F for neutrinos this is all
(approximately)
the same !
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Ordering principle: discreet symmetries •
Parity P –
•
− e L u
.
left-handed or right-handed
Charge Conjugation C –
•
− e u
0
possible values:
ν u
.
•
Generation –
first – second – third
.u
-1 − e u
.
− e u
.
.u
− e R u
.u
+ e u
.u
.
.u
2 3 .u
− µ u
.
.u
.
.
particle or antiparticle
Charge Q or Flavour –
.u
.u
u .u
- 31 .u
− τ u
.
d .u
.u
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Weak Interactions: modern explanation • weak interactions couple a pair of fermions with another pair
−
.
µ
• the Fermi coupling constant √ 2 g2 GF = 8 m2W
.
is independent of energy F only if the energy is
smaller than the mass of the W -boson
.
(much)
(80 GeV)
.
..
uL
d¯R
p
.
pW p p
– via vector bosons
.
.
.
.
p
pW p p
νµ . .
ν ¯e . −
.
e
..
µ
+
.
.
.
νµ
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Additional ordering principle: according to the charged weak interactions
• particles of the same spin but of
0
different mass and different charge
u
. .u
are grouped together into doublets F these are the objects that
ν
2 3
∗ make the charged current .u
∗ couple to the W ±-bosons I ”flavours” for neutrinos:
-1
u
.u
νe
u
.u
.u
u
u
. .u
−
e
- 31
.
.u
νµ
d
.u
u
.
. .u
ντ
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Observed fermions of the Standard Model: Fermions left .
particles
. .
.
e−L .
. .
νµ
.
µ−L .
. .
antiparticles
νe
ντ
right
.
τL−
. .
.
.
uL .
cL .
tL
. .
.
.
e+L .
µ+L .
τL+
. .
.
. .
.
.
.
u ¯L .
c¯L .
t¯L
.
.
.
.
dL
.
.
sL
.
.
bL d¯L
. .
ν ¯e
.
¯bL
.
µ−R .
τR−
.
.
.
.
.
µ+R .
. .
ν ¯τ
.
.
e+R
.
ν ¯µ
.
uR .
cR .
tR
.
. .
s¯L
e−R
.
.
τR+
. .
.
. .
.
.
.
u ¯R .
c¯R .
t¯R
.
.
.
dR .
sR .
bR d¯R
.
.
s¯R ¯bR
.
Joint Theoretical Physics Colloquium
Neutrinos
Meaning of Oscillations in the context of quantum mechanics
Thomas Gajdosik
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
From quarks we know • the charged current (i.e. W ±) mixes the generations – parametrized by the quark mixing or CKM matrix
• quarks are not observed as free particles – the mixing is also seen in boundstates: K − → ∗ without mixing this decay would not happen
p
pp
pu ¯p p p ps p pp
p
π− + π0
I
♣
♣♣
♣u ¯♣ ♣ ♣ ♣d ♣ ♣♣
♣
+
♣
♣♣
♣ d¯ ♣ ♣♣ ♣d ♣ ♣♣
♣
• mesons (i.e. boundstates of quarks) exhibit oscillations ¯ 0 instead of B 0 changes with time – the probability to find B s s 0 0 – they form mass-eigenstates BsH and BsL
∗ which have slightly different masses: 0 0 ∆mBs0 = mBsH − mBsL ≈ 0.0734 eV
∗
for comparison mBs0 ≈ 5.367 GeV
¯0 B s
♣
♣♣
♣s ¯♣ ♣ ♣ ♣b ♣ ♣ ♣
Bs0
↔ ♣
J I
♣
♣ ♣¯ b♣ ♣ ♣ ♣
♣s ♣ ♣♣
♣
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
What about neutrinos ? • like for the quarks:
♣
ν1 ♣
♣
ν3 ♣
♣
the charged current can mix the generations – generation means now the mass eigenstate ∗ a mass eigenstate is what we usually just call a particle
• as long as neutrinos have the same mass – ∗ if they are massless (like in the SM) they have the same mass . . . .
– we see no effect ∗ the state produced in the interaction
.
νe
.
I
.
νe
. .
νµ
.
I
.
νµ
.
stays the same .
ντ
.
I
.
ντ
♣ ♣ ♣
ν2 ♣ ♣ ♣ ♣
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
When neutrinos have different masses ? • again like for the quarks: we describe the . Ue1 . νe . ν = Uµ1 . µ . ντ Uτ 1 .
interactions with a mixing matrix ♣ ♣ ♣ν ♣ ♣ν ♣ Ue2 Ue3 ♣♣ ♣♣ ♣ ♣ ♣ ♣ ν Uµ2 Uµ3 · ♣ = UPMNS · ν♣ ♣ ♣ ♣ ♣ ♣ν ♣ ν Uτ 2 Uτ 3 1
1
2
2
♣3
♣3
F propagation of the state produced in the interaction . .
.
.
νe .
νµ
♣ ♣ν ♣ ♣1 ♣
♣ν = U · PMNS ♣
♣
♣ ♣ ν3 ♣ 2
.
ντ
♣
I
♣
♣ν ♣ 1 ♣ ♣
I ♣ ν2 ♣ ♣ ♣ I ♣ ν3 ♣ ♣
· U†
PMNS
. .
νe .
= . νµ
.
ντ
.
n
LyEn , the nm have presumably undergone numero Joint Theoretical Physics Colloquium Thomas Gajdosik Neutrinos cillations and have averaged out to roughly ha initial rate. The asymmetry A of the e-like events in the prese is consistent with expectations without neutrino o tions and two-flavor ne $ nm oscillations are not fa This is in agreement with recent results from the C experiment [22]. The LSND experiment has repor FIG. 2. The 68%, 90%, and 99% confidence intervals are 2 2 appearance of ne in a beam of nm produced by s shown for sin 2u and Dm for nm $ nt two-neutrino oscilpions [23]. The LSND results do not contrad lations based on 33.0 kton yr of Super-Kamiokande data. The • a pion, produced90%by a cosmic ray the experiupperpresent atmosphere, confidence interval obtained by thein Kamiokande results if they are observing small mixing ment is also shown. With the best-fit parameters for nm $ nt oscillatio decays into muon and muon-neutrino expect a total of only 15–20 events from nt ch 23 2 23 2 case overlapped at 1 3 10 , Dm , 4 3 10 eV current interactions in the data sample. Using the 2 sin 2u electron, 1. sample, oscillations between nm and nt are indistin – the muon decaysforinto electron-neutrino, and muon-neutrino As a cross-check of .the above analyses, able from oscillations between nm and a noninte . we have resterile neutrino. constructed the best ν estimate of the ratio νe LyE n for each I we get twice as many than produced µ . . Figure 2 shows the Super-Kamiokande results o event. The neutrino energy is estimated by applying a with the allowed region obtained by the Kamio correction to the final state lepton momentum. Typi-
What we see from the atmosphere
• Super-Kamiokande . ν sees fewer . µ s than expected – depending on the traveldistance
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
What we see from the sun ♣
• the sun produces only low energetic νe♣ ♣ s – highest energy ≤ 18 MeV
♣
I in a detector they can only produce electrons • the Homestake experiment measured only ∼1/3 of the flux predicted by the standard solar model • SNO confirmed the result from the Homestake experiment – but measured also the neutral current (= all ν s) ∗ consistent with the standard solar model I confirmed the explanation: oscillations are responsible
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Oscillations in vacuum • combining mixing UPMNS with propagation ei(Et−px) – we get the approximate amplitude Aν – and the approximate probability Pν
α
α →νβ
→νβ
=
P
k
∗ −i(Ek t−px) Uαk Uβk e
= |Aνα→νβ |2
• taking ultra-relativistic neutrinos with average energy E m – we get t ≡ L and
E` − Ek ≈
m2` − m2k
=
2E
∆m2`k 2E
αβ
∗ U∗ U I with Jk` ≡ Uαk Uβk α` β`
Pνα→νβ = δαβ − 4
X
h
αβ
Re Jk`
i
sin2
k>`
+2
X k>`
Im
h
αβ Jk`
i
sin
∆m2k`L
4E ! 2 ∆mk`L 2E
!
Joint Theoretical Physics Colloquium
Neutrinos
Oscillations in vacuum • are parametrized by – the distance L between production and detection – the mass differences ∆m2k` between the neutrinos αβ
∗ U∗ U – the ”oscillation angles” (θk`) in Jk` ≡ Uαk Uβk α` β`
• trying to fit all data from neutrino measurements
∆m221 ≈ 7.6 · 10−5 eV2
” ”
∆m231 ≈ 2.5 · 10−3 eV2
I this hierarchy allows the separation into ∗ atmospheric paramters: ∆m231 and θ23 ∗ solar paramters: ∆m221 and θ12 ∗ reactor angle: θ13
Thomas Gajdosik
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Oscillations in matter ♣
♣
• only νe♣ ♣ has a charged current interaction with normal matter – this interaction changes the dispersion relation ∗ like when light travels though matter
– this changes the propagation picture:
♣
∗ each mass eigenstate feels a ”drag” proportional to its ♣ νe♣ ♣ component I the mass is changed to an effective mass
∗ one gets different effective mass differences and effective mixing angles
F MSW-effect: Wolfenstein (1978), and Mikheyev and Smirnov (1986) • the MSW-effect is needed to explain the solar neutrino deficit – that only 1/3 of the higher energetic neutrinos from the sun are seen as electron-type neutrinos
– it does not affect the atmospheric oscillations
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos
Consequences for the theoretical description F we have to add additional singlet fermions the the SM:
left .
particles
. .
.
e−L .
. .
νµ
.
µ−L .
. .
antiparticles
νe
ντ
.
τL−
.
.
.
.
.
µ+L .
. .
ν ¯τL
.
.
.
e+L
.
ν ¯µL
.
uL .
cL .
tL
.
.
ν ¯eL
right
.
τL+
. .
.
. .
.
.
.
u ¯L .
c¯L .
t¯L
.
.
.
dL
.
νeR
.
sL
d¯L
.
νµR
.
ντR
.
µ−R .
.
τR−
.
ν ¯e
.
.
.
.
.
µ+R .
. .
ν ¯τ
.
.
.
e+R
.
ν ¯µ
.
uR .
cR .
tR
.
.
.
¯bL
.
. .
s¯L
.
e−R
.
.
bL
.
.
.
τR+
. .
.
. .
.
.
.
u ¯R .
c¯R .
t¯R
.
.
.
dR .
sR .
bR d¯R
.
.
s¯R ¯bR
.
Joint Theoretical Physics Colloquium
Thomas Gajdosik
Neutrinos .
What are these
.
νeR
?
• they do not interact with any other particle of the SM ♣
♣
– except the Higgs and the corresponding νe♣ I they are really invisible !
.
• the effective coupling between Higgs, – why so small, but not zero ?
.
νeR
♣
♣
and ♣ νe♣ ♣ is < 10−11
∗ a possible explanation is the seesaw mechanism • adding a very large Majorana .
mass term M me for
.
– gives the mass mν = ♣
♣
νeR
♣ m2e ♣ νe ♣ for ♣ M
I νe♣ ♣ becomes Majorana, too ! i.e.: its own antiparticle
measured parameters in the neutrino sector masses • differences of squares of the mass parameters are measured: ∆m2 – indirect limits come from ∗ neutrinoless double beta decay ∗ Cosmolgy
∆m221
≈
7.6 × 10−5 eV2
∆m231
≈
2.4 × 10−3 eV2
assuming the ”standard” Big Bang model
couplings to the • W ±-boson and charged leptons (flavour eigenstates) – neutrino mixing matrix UPMNS • Z-boson and (other) neutrinos – fixed at tree-level in the SM
with three mixing angles and the CP phase
sin2 2θ12
=
0.846 ± 0.021
sin2 2θ23
=
0.999+0.001 −0.018
sin2 2θ13
=
0.085 ± 0.005
δCP
∈
[0, 2π)
• Higgs boson(s) and leptons
– in the SM fixed by the Dirac mass term mD ∗ used in the seesaw mechanism Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
19
Joint Theoretical Physics Colloquium
Neutrinos
Thomas Gajdosik
Dmitrij Chomčik, Vytautas Dudenas, T.G., Andrius Juodagalvis, Darius Jurčiukonis, Anton Kunchin, Tomas Sabonis
Our approach
. . . we did not invent it, but we work on it:
the 312-νSM: Standard Model (SM) + 1 fermionic singlet + 2 Higgs doublets • is not a new idea: [G-N]
W. Grimus and H. Neufeld, Nucl. Phys. B 325 (1989) 18. .
• instead of adding
.
νeR
. .
νµR
. .
ντR
– and the corresponding coupling matrix to the Higgs .
• we set
.
νeR
.
=
.
νµR
.
=
.
ντR
=:
♣ ♣ νR ♣ ♣
– so we need only a coupling vector to the Higgs – and only a single Majorana mass term MR ∗ this implements the seesaw mechanism but we have to add a second Higgs doublet – this predicts additional Higgs bosons •
LHC searches for these additional Higgs bosons – we can constrain the Higgs parameter space with our model . . .
The bare Lagrangian of the 312-νSM ¯ iγ µDµψ + LF-H + LM L = V + (Dµφa)†(Dµφa) + LG + ψ
(1)
with the • Higgs sector: two Higgs doublets φa in the Higgs potential †
†
†
V = Ya¯bφ¯aφb + 1 a φb )(φ¯ c φd ) bcd¯(φ¯ 2 Za¯
(2)
∗ with Ya¯b = (Yb¯a)∗ and Za¯bcd¯ = Zcda ¯¯ cb¯ a) b = (Zd¯ [H-ON]
H. E. Haber and D. O’Neil, Phys. Rev. D 83 (2011) 055017 [arXiv:1011.6188 [hep-ph]]
• Gauge sector of the SM: U (1)Y ⊗ SU (2)L ⊗ SU (3)color µν − 1 W j W j µν − 1 Gb Gb µν LG = − 1 B B 4 µν 4 µν 4 µν
(3)
• Gauge-Higgs sector with the gauge covariant derivative 1 φ + igW j 1 σ j φ Dµφa = ∂µφa + ig 0Bµ 2 a a µ2
Thomas Gajdosik
Neutrinos
JTPC
(4)
2016 / 04 / 25
21
• Fermion-Gauge sector with the gauge covariant derivative k ψ + ig Gb 1 λb ψ Dµψ = ∂µψ + ig 0Bµyψ ψ + igWµk 1 σ s µ2 2
(5)
• Fermion-Higgs sector with the Yukawa couplings (ignoring quarks) 0 φ Y¯ a 0 0 0 ˜ Y ea LF−H = −`¯Lj a Ljk eRk − `¯Lj φ ¯ a LjnNn + h.c. ˜¯a defined as with the adjoint Higgs doublet φ ˜¯a = φ∗a = φ
0
1
−1 0
! ·
∗ (φ+ a ) (φ0a )∗
! =:
φ¯0∗ a −φ¯− a
(6)
! (7)
• Majorana sector with N 0 obeying the Majorana condition c0 := η γ 0CN 0∗ N 0 = ηN N N
(8)
b 0 the Lorentz covariant conjugate of N 0. with the Majorana phase ηN , and N ∗N 0 CN 0> + h.c. = − 1 M ∗ N 0N c0N 0 c0 − 1 M N ¯ ¯ ¯ LM = 1 M R 2 R 2 R 2
(9)
where N 0 = PR N 0 is the chiral projection of N 0. Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
22
The 312-νSM has fields additionally to the ”original” SM • the singlet Majorana fermion NR • the second Higgs doublet φ2 – in general, the neutral scalars will mix and violate CP – we assume a CP conserving scalar potential and get additionally ∗ a charged scalar H ± ∗ a neutral scalar H 0, mixing with the SM Higgs boson h0 ∗ a pseudo-scalar A0
Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
23
The 312-νSM has parameters additionally to the ”original” SM • the singlet Majorana mass term MR • parameters due to the second Higgs doublet – neutrino Yukawa coupling of the first Higgs doublet √ (1) (YN )k = v2 (MD )k
. . . the ”Dirac mass” term
– Yukawa couplings of the second Higgs doublet (2)
to lepton doublets and charged lepton singlets `Rj
(2)
to lepton doublets and neutral fermionic singlet NR
(YE )jk (YN )k
– additional parameters in the Higgs sector
see [H-ON]
2 , m2 ∗ m2 , m masses of the additional Higgs bosons H2 H3 H± ∗ θ12, θ13 mixing angles between the neutral Higgs fields
∗ Z2, Z3, Z7 . . . parameters of the Higgs potential, not fixed by tree level mass relations Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
24
312-νSM tree level predictions for the neutral fermions (1)
• the Yukawa coupling (YN )k mixes the neutral leptons νj with NR • the mixing gives a (3 + 1) × (3 + 1) symmetric mass matrix
Mν =
ML
> MD
ML = 03×3
with
MD = (mN e, mN µ, mN τ )
MD MR
– Mν has rank 2
⇒
(10)
only two masses are non-zero
• diagonalizing Mν U (ν) Mν = diag(m1, m2, m3, m4)U (ν)∗
with
m1 = m2 = 0
(11)
by the unitary matrix
U (ν)
U1e U1µ U1τ U2e U2µ U2τ = icU 3e icU3µ icU3τ sU3e sU3µ sU3τ
0
0 −is c
where
4 c2 = m m +m 4 3
s2 =
m3 m4 +m3
(12)
– Uαk is the neutrino mixing matrix (with α, k = 1 . . . 3, α mass, k flavour) Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
25
√ (1) • from the condition eq.(??) and (YN )k = v2 (MD )k we get (1) (1) U1k (YN )k = U2k (YN )k = 0
(13)
• the two tree-level massless ”neutrinos” ζ1,2 are degenerate (2)
• use the second Higgs coupling (YN )k to distinguish them: (2)
U1k (YN )k = 0
(2)
U2k (YN )k =: d 6= 0
and
(14)
⇒ U1k is defined to be orthogonal to both Yukawa couplings (1)
∗ U2k has to be orthogonal to (YN )k and to U1k (1)
⇒ U3k has to be parallel to (YN )k • we can construct the mixing matrix Uαk from the Yukawa couplings or • we can read off the directions of the Yukawa couplings from the mixing matrix Uαk Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
26
Loop corrections in the νSM Only with Renormalization the tree level has a well defined meaning • in QED we see the ”protection” of fermion mass terms: δm ∝ m – we expect the same for the massless neutrinos • in the SM all particle ”start” massless – the vev of the Higgs gives the mass m – loop corrections to the charged fermion masses obey δm ∝ m ∗ the Dirac mass term MD in Mν should do the same • the Majorana mass term is zero at tree level: ML|tree = 0 – its loop correction is not: δML|loop 6= 0 ⇒ radiative mass generation seems possible BUT:
−1 > −1 > δML ∝ Y MR Y ∝ MD MR MD = Mlight (seesaw mass matrix)
– with a single Higgs doublet MD = √v Y 2
⇒ only correction to already non-zero masses Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
27
Loop corrections in the 312-νSM Extension of the SM to include nH = 2 Higgs doublets • Higgs fields with the same quantum numbers mix – orthogonal superpositions are equivalent – we (re)parametrise the Higgs doublets ∗ that only the first doublet has a vev: vk = vδk1
( Higgsbasis )
• the Yukawa matrices are still nL × nR = 3 × 1 – we parametrise them by
(1)
√
2 mD U3k v = d U2k − d0 U3k
(YN )k = (2)
(YN )k
(15) (16)
– Dirac mass term like in the nH = 1 toy model: MD = mD U3k −1 −1 • (δML)jk ∝ f (H1)(MD )j MR (MD )k + f (H2)U2j MR U2k
⇒ δMlight 6 ∝ Mlight ⇒ radiative mass generation becomes possible Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
28
Loop corrections in the 312-νSM • the neutrino mass matrix at one loop can be written Mν[1] =
0 + δML MD + δMD
> MD
+
> δMD
MR + δMR
! ∼
δML
> MD
MD
MR
– with only δML being of relative importance
!
. . . more work needed
[1]
• Mν has to be diagonalised, giving – one heavy state – one seesaw neutrino – one radiatively generated massive neutrino – one massless neutrino – and a 4 × 4 PMNS matrix ∗ which assumes Me = √v2 YE(1) to be diagonal: charged lepton basis • since still one neutrino stays massless – one Majorana phase becomes irrelevant (unphysical)
Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
29
Tasks in the 312-νSM • fully renormalizing the part connected to neutrinos – ∗ Vytautas Dudenas, Tomas Sabonis – with special attention to gauge invariance ∗ Dmitrij Chomčik • restricting parameters due to the measurements – from the neutrino measurements (∆m2 and UPMNS ) ∗ Darius Jurčiukonis – from the Higgs sector ∗ Anton Kunchin • connecting to measurements – scalar resonance around 750 GeV at LHC? ∗ TG, Andrius Juodagalvis
Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
30
Thank you for discussion and comments Acknowledgments The authors thank the Lithuanian Academy of Sciences for the support (the projects DaFi2014 and DaFi2015). References [G-N] W. Grimus and H. Neufeld, Nucl. Phys. B 325 (1989) 18. [H-ON] H. E. Haber and D. O’Neil, Phys. Rev. D 83 (2011) 055017 [arXiv:1011.6188 [hep-ph]]. [G-L] W. Grimus and L. Lavoura, Phys. Lett. B 546 (2002) 86 [arXiv:hep-ph/0207229]. [A.D.] A. Denner, Fortsch. Phys. 41 (1993) 307 [arXiv:0709.1075 [hep-ph]]. [FTV] D. V. Forero, M. Tortola and J. W. F. Valle, Phys. Rev. D 90 (2014) 9, 093006 [arXiv:1405.7540 [hep-ph]]. [we] T. Gajdosik, A. Juodagalvis, D. Jurčiukonis and T. Sabonis, Acta Phys. Polon. B 46 (2015) 11, 2323 doi:10.5506/APhysPolB.46.2323
Thomas Gajdosik
Neutrinos
JTPC
2016 / 04 / 25
31
