Post on 17-Jan-2020
GraduiertentagKepler Center for Astro and Particle PhysicsNeutrino Theory - I: Neutrino Oscillation Phenomenology
Thomas Schwetz-Mangold
Max-Planck-Institut für Kernphysik, Heidelberg
13 May 2011
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 1 / 52
Neutrinos oscillate...
... and have mass ⇒ physics beyond the Standard Model
I Lecture I: Neutrino Oscillation phenomenologyI Lecture II: Neutrinos and physics beyond the Standard Model
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 2 / 52
Neutrinos oscillate...
... and have mass ⇒ physics beyond the Standard Model
I Lecture I: Neutrino Oscillation phenomenologyI Lecture II: Neutrinos and physics beyond the Standard Model
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 2 / 52
Outline
Lepton mixing
Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW
Global data and 3-flavour oscillations
Outlookθ13CPV, mass hierarchy
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 3 / 52
Lepton mixing
Outline
Lepton mixing
Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW
Global data and 3-flavour oscillations
Outlookθ13CPV, mass hierarchy
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 4 / 52
Lepton mixing
The Standard Model
Flavours: 1 2 3
u c tQuarks: d s b
νe νµ ντLeptons: e µ τ
Fermions in the Standard Model come in three generations (“Flavours”)
Neutrinos are the “partners” of the charged leptonsmore precisely: they form a doublet under the SU(2) gauge symmetry
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 5 / 52
Lepton mixing
Flavour neutrinos
A neutrino of flavour α is defined by the charged current interaction withthe corresponding charged lepton:
LCC = − g√2
W ρ∑
α=e,µ,τ
ναLγρ`αL + h.c.
for example
π+ → µ+νµ
the muon neutrino νµ comes together with the charged muon µ+
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 6 / 52
Lepton mixing
Flavour neutrinos
A neutrino of flavour α is defined by the charged current interaction withthe corresponding charged lepton:
LCC = − g√2
W ρ∑
α=e,µ,τ
ναLγρ`αL + h.c.
for example
π+ → µ+νµ
the muon neutrino νµ comes together with the charged muon µ+
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 6 / 52
Lepton mixing
Flavour neutrinos
W
lα
W
lα
νανα
detectorneutrino source
"short" distance
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 7 / 52
Lepton mixing
Let’s give mass to the neutrinos
Majorana mass term:
LM = −12
∑α,β=e,µ,τ
νTαLC−1MαβνβL + h.c.
M: symmetric mass matrix
In the basis where the CC interaction is diagonal the mass matrix is ingeneral not a diagonal matrix
any complex symmetric matrix M can be diagonalised by a unitary matrix
UTν MUν = m , m : diagonal, mi ≥ 0
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 8 / 52
Lepton mixing
Lepton mixing
LCC = − g√2
W ρ∑
α=e,µ,τ
3∑i=1
νiLU∗αiγρ`αL + h.c.
LM = −12
3∑i=1
νTiL C−1νiLmν
i −∑
α=e,µ,τ
¯αR`αLm`
α + h.c.
The unitary lepton mixing matrix:
(Uαi ) ≡ UPMNS = V Dirac DMaj
DMaj = diag(e iαi /2)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 9 / 52
Neutrino oscillations
Outline
Lepton mixing
Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW
Global data and 3-flavour oscillations
Outlookθ13CPV, mass hierarchy
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 10 / 52
Neutrino oscillations
Neutrino oscillations
W
lα
W
lβ
νβνα
detectorneutrino source
"long" distance
neutrino oscillations
|να〉 = U∗αi |νi 〉 e−i(Ei t−pi x) |νβ〉 = U∗
βi |νi 〉
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 11 / 52
Neutrino oscillations Oscillations in vacuum
Neutrino oscillations in vacuum
oscillation amplitude:
Aνα→νβ= 〈νβ| propagation |να〉
=∑i ,j
Uβj〈νj | e−i(Ei t−pi x) |νi 〉U∗αi =
∑i
UβiU∗αie
−i(Ei t−pi x)
oscillation probability:Pνα→νβ
=∣∣Aνα→νβ
∣∣2
To derive the oscillation probability rigorously one needs either awave-packet treatment or field theory Akhmedov, Kopp, JHEP 1004:008 (2010) [1001.4815]
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 12 / 52
Neutrino oscillations Oscillations in vacuum
Neutrino oscillations in vacuum
oscillation amplitude:
Aνα→νβ= 〈νβ| propagation |να〉
=∑i ,j
Uβj〈νj | e−i(Ei t−pi x) |νi 〉U∗αi =
∑i
UβiU∗αie
−i(Ei t−pi x)
oscillation probability:Pνα→νβ
=∣∣Aνα→νβ
∣∣2
To derive the oscillation probability rigorously one needs either awave-packet treatment or field theory Akhmedov, Kopp, JHEP 1004:008 (2010) [1001.4815]
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 12 / 52
Neutrino oscillations Oscillations in vacuum
Neutrino oscillations in vacuum
oscillation amplitude:
Aνα→νβ= 〈νβ| propagation |να〉
=∑i ,j
Uβj〈νj | e−i(Ei t−pi x) |νi 〉U∗αi =
∑i
UβiU∗αie
−i(Ei t−pi x)
oscillation probability:Pνα→νβ
=∣∣Aνα→νβ
∣∣2
To derive the oscillation probability rigorously one needs either awave-packet treatment or field theory Akhmedov, Kopp, JHEP 1004:008 (2010) [1001.4815]
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 12 / 52
Neutrino oscillations Oscillations in vacuum
A hand-waving derivation for two flavoursoscillation phase:
φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2
i + m2i
define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2
1 , E = (E1 + E2)/2
then: ∆E 2 = 2E∆E (similar for p and m)
φ = ∆Et − ∆p2
2px = ∆Et − ∆E 2 −∆m2
2px
= ∆Et − 2E2p
∆Ex +∆m2
2px
use “average velocity” of the neutrino v = p/E and x ≈ vt:
φ ≈ ∆m2
2px ≈ ∆m2
2Ex
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 13 / 52
Neutrino oscillations Oscillations in vacuum
A hand-waving derivation for two flavoursoscillation phase:
φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2
i + m2i
define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2
1 , E = (E1 + E2)/2
then: ∆E 2 = 2E∆E (similar for p and m)
φ = ∆Et − ∆p2
2px = ∆Et − ∆E 2 −∆m2
2px
= ∆Et − 2E2p
∆Ex +∆m2
2px
use “average velocity” of the neutrino v = p/E and x ≈ vt:
φ ≈ ∆m2
2px ≈ ∆m2
2Ex
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 13 / 52
Neutrino oscillations Oscillations in vacuum
A hand-waving derivation for two flavoursoscillation phase:
φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2
i + m2i
define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2
1 , E = (E1 + E2)/2
then: ∆E 2 = 2E∆E (similar for p and m)
φ = ∆Et − ∆p2
2px = ∆Et − ∆E 2 −∆m2
2px
= ∆Et − 2E2p
∆Ex +∆m2
2px
use “average velocity” of the neutrino v = p/E and x ≈ vt:
φ ≈ ∆m2
2px ≈ ∆m2
2Ex
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 13 / 52
Neutrino oscillations Oscillations in vacuum
A hand-waving derivation for two flavoursoscillation phase:
φ = (E2 − E1)t − (p2 − p1)x with E 2i = p2
i + m2i
define: ∆E = E2 − E1, ∆E 2 = E 22 − E 2
1 , E = (E1 + E2)/2
then: ∆E 2 = 2E∆E (similar for p and m)
φ = ∆Et − ∆p2
2px = ∆Et − ∆E 2 −∆m2
2px
= ∆Et − 2E2p
∆Ex +∆m2
2px
use “average velocity” of the neutrino v = p/E and x ≈ vt:
φ ≈ ∆m2
2px ≈ ∆m2
2Ex
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 13 / 52
Neutrino oscillations Oscillations in vacuum
The oscillation probability in vacuum
Aνα→νβ=
∑i
UβiU∗αie
−i(Ei t−pi x)
Pνα→νβ(L) =
∑jk
UαjU∗βjU
∗αkUβk exp
[−i
∆m2kjL
2 Eν
]
∆m2kj ≡ m2
k −m2j : oscillations are sensitive only to mass-squared
differences (not to absolute mass!)
to observe oscillations one needsI non-trivial mixing Uαi
I non-zero mass-squared differences ∆m2kj
I a suitable value for L/Eν
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 14 / 52
Neutrino oscillations Oscillations in vacuum
The oscillation phase
φ =∆m2L4Eν
= 1.27∆m2[eV2] L[km]
Eν [GeV]
I “short” distance: φ � 1: no oscillations can develop and Pνα→νβ= δαβ
because of∑
j UαjU∗βj = δαβ .
I “long” distance: φ & π/2: oscillations are observable
I “very long” distance: φ � 2π: oscillations are averaged out(indep. of L and Eν):
Pνα→νβ=∑
j
|Uαj |2|Uβj |2
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 15 / 52
Neutrino oscillations Oscillations in vacuum
2-neutrino oscillations
Two-flavour limit:
U =
(cos θ sin θ− sin θ cos θ
), P = sin2 2θ sin2 ∆m2L
4Eν
0.1 1 10 100L / Eν (arb. units)
0
0.2
0.4
0.6
0.8
1
Pαβ
4π / ∆m2
sin2 2θ
"short"distance
"long"distance
"very long"distance
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 16 / 52
Neutrino oscillations Oscillations in vacuum
Appearance vs. disappearance
I appearance experiments:
Pνα→νβ, α 6= β
“appearance” of a neutrino of a new flavour β 6= α in a beam of να
I disappearance experiments:
Pνα→να
measurement of the “survival” probability of a neutrino of givenflavour
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 17 / 52
Neutrino oscillations Oscillations in vacuum
General properties of vacuum oscillations
Pνα→νβ=∑jk
UαjU∗βjU
∗αkUβk exp
[−i
∆m2kjL
2 Eν
]
I Unitarity:∑
β Pνα→νβ= 1
I For anti-neutrinos replace Uαi → U∗αi
I Pνα→νβ= Pνβ→να (CPT invariance)
I Phases in U induce CP violation: Pνα→νβ6= Pνα→νβ
I there is no CP violation in disappearance experiments:
Pνα→να = Pνα→να =∑k,j
|Uαk |2|Uαj |2e−i∆m2kj L/2E
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 18 / 52
Neutrino oscillations Oscillations in vacuum
Eff. Schrödinger equation
The evolution of the flavour state can be described by an effectiveSchrödinger equation:
iddt
(aeaµaτ
)= Hvac
(aeaµaτ
)
where
Hνvac = Udiag
(0,
∆m221
2Eν,∆m2
312Eν
)U†
H νvac = U∗diag
(0,
∆m221
2Eν,∆m2
312Eν
)UT
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 19 / 52
Neutrino oscillations Oscillations in matter
The matter effect
When neutrinos pass through matter the interactions with the particles inthe background induce an effective potential for the neutrinos
The coherent forward scattering amplitude leads to an index of refractionfor neutrinos
L. Wolfenstein, Phys. Rev. D 17, 2369 (1978); ibid. D 20, 2634 (1979)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 20 / 52
Neutrino oscillations Oscillations in matter
Effective Hamiltonian in matter
Hνmat = Udiag
(0,
∆m221
2Eν,∆m2
31
2Eν
)U† + diag(
√2GF Ne , 0, 0)
H νmat = U∗diag
(0,
∆m221
2Eν,∆m2
31
2Eν
)UT︸ ︷︷ ︸
Hvac
− diag(√
2GF Ne , 0, 0)︸ ︷︷ ︸Vmat
Ne(x): electron density along the neutrino path
for non-constant matter the Hamiltonian depends on time:
iddt
a = Hmat(t)a
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 21 / 52
Neutrino oscillations Oscillations in matter
Effective matter potential - 1
Effective 4-point interaction Hamiltonian in the SM
Hναint =
GF√2
ναγµ(1− γ5)να
∑f
f γµ(gα,fV − gα,f
A γ5)f︸ ︷︷ ︸Jµ
mat
ordinary matter: e−, p, n
non-relativistic: 〈f γµf 〉 = 12Nf δµ0
unpolarised: 〈f γ5γµf 〉 = 0
neutral: Ne = Np
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 22 / 52
Neutrino oscillations Oscillations in matter
Effective matter potential - 2
Jµmat =
12δµ0
∑f =e,p,n
Nf gα,fV
=12δµ0[Ne(gα,e
V + gα,pV)
+ Nngα,nV]
gV e− p nνe 2 sin2 ΘW + 1
2 −2 sin2 ΘW + 12 −1
2νµ,τ 2 sin2 ΘW − 1
2 −2 sin2 ΘW + 12 −1
2
⇒ Vmat ∝(
Ne −12
Nn,−12
Nn,−12
Nn
)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 23 / 52
Neutrino oscillations Oscillations in matter
Effective matter potential - 3
Vmat =√
2GF diag (Ne − Nn/2,−Nn/2,−Nn/2)
ν l
l ν
W
ν ν
f f
Z0
CC NC
I only νe fell CC (there are no µ, τ in normal matter)
I NC is the same for all flavours ⇒ potential proportional to identiy has noeffect on the evolution
I NC has no effect for 3-flavour active neutrinos, but is important in thepresence of sterile neutrinos
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 24 / 52
Neutrino oscillations Oscillations in matter
Neutrino oscillations in constant matter
diagonalize the Hamiltonian in matter:
Hνmat = Udiag
(0,
∆m221
2Eν,∆m2
31
2Eν
)U† + diag(
√2GF Ne , 0, 0)
= Umdiag (λ1, λ2, λ3) U†m
Same expression for oscillation probability, but replace “vacuum”parameters by “matter” parameters
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 25 / 52
Neutrino oscillations Oscillations in matter
2-neutrino oscillations in constant matter
Two-flavour case:
Pmat = sin2 2θmat sin2 ∆m2matL
4E
with
sin2 2θmat =sin2 2θ
sin2 2θ + (cos 2θ − A)2
∆m2mat = ∆m2
√sin2 2θ + (cos 2θ − A)2
A ≡ 2EV∆m2
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 26 / 52
Neutrino oscillations Oscillations in matter
2-neutrino oscillations in constant matter
sin2 2θmat =sin2 2θ
sin2 2θ + (cos 2θ − A)2A ≡ 2EV
∆m2
resonance for cos 2θ = A: “MSW resonance” Mikheev, Smirnov, Sov. J. Nucl. Phys. 42, 913 (1985)
0.01 0.1 1 10 100A
0
0.2
0.4
0.6
0.8
1
sin2 2θ
mat
small matter effectvacuum osc.
strong matter effectosc. are suppressed
reso
nanc
e
sin22θ
vac = 0.3
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 27 / 52
Neutrino oscillations Varying matter density and MSW
Varying matter density: example solar neutrinos
The electron density in the sun:
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 28 / 52
Neutrino oscillations Varying matter density and MSW
The LMA-MSW mechanism
evolution is adiabatic if(
1θm
dθm
dx
)−1
� Losc
using ∆m2 = 8× 10−5 eV2 the oscillation length is
Losc =4πE∆m2 ' 30 km
(E
MeV
)for large mixing angles (sin2 θ12 ' 0.3):(
1θm
dθm
dx
)−1
∼(
1V
dVdx
)−1
∼ size of sun � 30 km
⇒ adiabatic evolution
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 29 / 52
Neutrino oscillations Varying matter density and MSW
The LMA-MSW mechanism
the electron neutrino is born at the center of the sun as
|νe〉 = cos θm|ν1〉+ sin θm|ν2〉
then |ν1〉 and |ν2〉 evolve adiabatically to the Earth
Pee = Pprode1 Pdet
1e + Pprode2 Pdet
2e
Pprode3 ≈ sin2 θ13 ≈ 0, interference term averages out
Pprode1 = cos2 θm , Pdet
1e = cos2 θ
Pprode2 = sin2 θm , Pdet
2e = sin2 θ
⇒ Pee = cos2 θm cos2 θ + sin2 θm sin2 θ
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 30 / 52
Neutrino oscillations Varying matter density and MSW
The LMA-MSW mechanism
in the center of the sun we have
A ≡ 2EV∆m2 ' 0.2
(Eν
MeV
)(8× 10−5 eV2
∆m2
)
resonance occurs forA = cos 2θ = 0.4
⇒ Eres ' 2 MeV
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 31 / 52
Neutrino oscillations Varying matter density and MSW
The LMA-MSW mechanism
Pee = cos2 θm cos2 θ
+ sin2 θm sin2 θ
0.1 1 10Eν [MeV]
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 - 1/2 sin22θ
sin2θ
Ere
s
strongmatter
vacuum
Pee
sin
2 θ m
Pee =
{c4 + s4 = 1− 1
2 sin2 2θ vacuum (θm = θ)sin2 θ strong matter (θm = π/2)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 32 / 52
Neutrino oscillations Varying matter density and MSW
LMA-MSW and the solar neutrino spectrum
0.2
0.3
0.4
0.5
0.6
Pee
1 - 1/2 sin22θ
sin2θ
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 33 / 52
Neutrino oscillations Varying matter density and MSW
Evidence for LMA-MSW
Measurements of the solar neutrino rate at SNO and Borexino
0.1 1 10Eν [MeV]
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Pee
1 - 1/2 sin22θ
sin2θ
Ere
s
strongmatter
vacuum
SNO CC/NC
Borexino
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 34 / 52
Global data and 3-flavour oscillations
Outline
Lepton mixing
Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW
Global data and 3-flavour oscillations
Outlookθ13CPV, mass hierarchy
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 35 / 52
Global data and 3-flavour oscillations
Evidences for neutrino oscillationsI solar neutrinos Homestake, SAGE+GNO, Super-K, SNO, Borexino
νe → νµ,τ LMA-MSW with ∆m2 ∼ 7× 10−5eV2
I Kamland reactor neutrino experiment (180 km)νe disappearance with Eν/L ∼ 7× 10−5eV2
I atmospheric neutrinos Super-Kamiokandelong-baseline accelerator experiments K2K (250 km), MINOS (735 km)νµ → ντ with Eν/L ∼ 2× 10−3eV2
• fits beautifully in 3-flavour framework with∆m2
21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2
• no oscillations of νe seen so far with Eν/L ∼ 2× 10−3eV2
• controversial hints for oscillations with Eν/L ∼ eV2
(LSND, MiniBooNE, SBL reactor exps) "sterile neutrinos"?
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 36 / 52
Global data and 3-flavour oscillations
Evidences for neutrino oscillationsI solar neutrinos Homestake, SAGE+GNO, Super-K, SNO, Borexino
νe → νµ,τ LMA-MSW with ∆m2 ∼ 7× 10−5eV2
I Kamland reactor neutrino experiment (180 km)νe disappearance with Eν/L ∼ 7× 10−5eV2
I atmospheric neutrinos Super-Kamiokandelong-baseline accelerator experiments K2K (250 km), MINOS (735 km)νµ → ντ with Eν/L ∼ 2× 10−3eV2
• fits beautifully in 3-flavour framework with∆m2
21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2
• no oscillations of νe seen so far with Eν/L ∼ 2× 10−3eV2
• controversial hints for oscillations with Eν/L ∼ eV2
(LSND, MiniBooNE, SBL reactor exps) "sterile neutrinos"?
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 36 / 52
Global data and 3-flavour oscillations
Evidences for neutrino oscillationsI solar neutrinos Homestake, SAGE+GNO, Super-K, SNO, Borexino
νe → νµ,τ LMA-MSW with ∆m2 ∼ 7× 10−5eV2
I Kamland reactor neutrino experiment (180 km)νe disappearance with Eν/L ∼ 7× 10−5eV2
I atmospheric neutrinos Super-Kamiokandelong-baseline accelerator experiments K2K (250 km), MINOS (735 km)νµ → ντ with Eν/L ∼ 2× 10−3eV2
• fits beautifully in 3-flavour framework with∆m2
21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2
• no oscillations of νe seen so far with Eν/L ∼ 2× 10−3eV2
• controversial hints for oscillations with Eν/L ∼ eV2
(LSND, MiniBooNE, SBL reactor exps) "sterile neutrinos"?
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 36 / 52
Global data and 3-flavour oscillations
Evidences for neutrino oscillationsI solar neutrinos Homestake, SAGE+GNO, Super-K, SNO, Borexino
νe → νµ,τ LMA-MSW with ∆m2 ∼ 7× 10−5eV2
I Kamland reactor neutrino experiment (180 km)νe disappearance with Eν/L ∼ 7× 10−5eV2
I atmospheric neutrinos Super-Kamiokandelong-baseline accelerator experiments K2K (250 km), MINOS (735 km)νµ → ντ with Eν/L ∼ 2× 10−3eV2
• fits beautifully in 3-flavour framework with∆m2
21 ≈ 7× 10−5eV2 and ∆m231 ≈ 2× 10−3eV2
• no oscillations of νe seen so far with Eν/L ∼ 2× 10−3eV2
• controversial hints for oscillations with Eν/L ∼ eV2
(LSND, MiniBooNE, SBL reactor exps) "sterile neutrinos"?
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 36 / 52
Global data and 3-flavour oscillations
3-flavour oscillation parameters
νeνµ
ντ
=
Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3
ν1ν2ν3
∆m231 ∆m2
21
U =
1 0 00 c23 s230 −s23 c23
c13 0 e−iδs130 1 0
−e iδs13 0 c13
c12 s12 0−s12 c12 0
0 0 1
atmospheric+LBL Chooz solar+KamLAND
3-flavour effects are suppressed: ∆m221 � ∆m2
31 and θ13 � 1 (Ue3 = s13e−iδ)
⇒ dominant oscillations are well described by effective two-flavour oscillations⇒ CP-violation is suppressed by θ13
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 37 / 52
Global data and 3-flavour oscillations
3-flavour oscillation parameters
νeνµ
ντ
=
Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3Uτ1 Uτ2 Uτ3
ν1ν2ν3
∆m2
31 ∆m221
U =
1 0 00 c23 s230 −s23 c23
c13 0 e−iδs130 1 0
−e iδs13 0 c13
c12 s12 0−s12 c12 0
0 0 1
atmospheric+LBL Chooz solar+KamLAND
3-flavour effects are suppressed: ∆m221 � ∆m2
31 and θ13 � 1 (Ue3 = s13e−iδ)
⇒ dominant oscillations are well described by effective two-flavour oscillations⇒ CP-violation is suppressed by θ13
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 37 / 52
Global data and 3-flavour oscillations
Three flavour oscillation parameters summary
TS, Tortola, Valle, 11
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 38 / 52
Global data and 3-flavour oscillations
Three flavour oscillation parameters summary
★
★
0 0.25 0.5 0.75 1
{sin2θ12, sin
2θ23}
10-4
10-3
{∆m
2 21, ∆
m2 31
} [e
V2 ]
Solar+KamLAND
Atmospheric+LBL
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 39 / 52
Global data and 3-flavour oscillations
Three flavour oscillation parameters summary
two possibilities for the neutrino mass spectrum
INVERTEDNORMAL
[mas
s]2
3ν
ν2
ν1
ν2ν1
ν3
νe
µν
ντ
∆m231 > 0 ∆m2
31 < 0
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 40 / 52
Global data and 3-flavour oscillations
The 1-2 mass ordering
We know that the mass state containing most of νe is the lighter of thetwo “solar mass” states
∆m221 ≡ m2
2 −m21 > 0 and θ12 < 45o
thanks to the observation of the matter effect in the sun:
resonance condition:
∆m221 cos 2θ12 = 2EνV ⇒ ∆m2
21 cos 2θ12 > 0
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 41 / 52
Global data and 3-flavour oscillations
The 1-3 mass ordering
We do not know the sign of ∆m231
(normal or inverted mass ordering)
No matter effect has been observed for oscillations with ∆m231, only
“vacuum” νµ → νµ(ντ ) oscillations:
Pµµ ≈ 1− sin2 2θ23 sin2 ∆m231L
4E
Has to look for matter effect in νe ↔ νµ oscillations due to ∆m231, θ13
⇒ future long-baseline experiments
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 42 / 52
Global data and 3-flavour oscillations
Pνe→νµ
10-2
10-1
100
101
E [GeV]
101
102
103
104
L [
km]
< 0.1%
0.1% − 1%
1% − 5%
5% − 10%
> 10%
Contours of constant Peµ
1st osc m
ax with
∆m2
31
1st osc m
ax with
∆m2
21
LV = π
sin22θ
13 = 0.05 , ∆m
2
31 = 0.002 eV
2 , α = 0.026 , δ = 0
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 43 / 52
Global data and 3-flavour oscillations
Pνe→νµ
10-2 10-1 100 101
E [GeV]
101
102
103
104
L [k
m]
Reactor-θ13
KamLANDT2K
WBB FNL-HS
MINOS
NOνA
CNGS
NuFact
βB100SPL
βB350
< 0.1%0.1% − 1%1% − 5%5% − 10%> 10%
Contours of constant Peµ
1st osc m
ax with
∆m2
31
1st osc m
ax with
∆m2
21
LV = π
sin22θ13 = 0.05 , ∆m231 = 0.002 eV2 , α = 0.026 , δ = 0
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 43 / 52
Outlook
Outline
Lepton mixing
Neutrino oscillationsOscillations in vacuumOscillations in matterVarying matter density and MSW
Global data and 3-flavour oscillations
Outlookθ13CPV, mass hierarchy
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 44 / 52
Outlook θ13
Measuring θ13
two complementary approaches towards θ13:
I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO
“clean” measurement of θ13:
Pee ≈ 1− sin2 2θ13 sin2 ∆m231L
4Eν+O
(∆m2
21∆m2
31
)2
⇒ bring stat. and syst. errors below % level
I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνA
Pµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2
31)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 45 / 52
Outlook θ13
Measuring θ13
two complementary approaches towards θ13:
I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO“clean” measurement of θ13:
Pee ≈ 1− sin2 2θ13 sin2 ∆m231L
4Eν+O
(∆m2
21∆m2
31
)2
⇒ bring stat. and syst. errors below % level
I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνA
Pµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2
31)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 45 / 52
Outlook θ13
Measuring θ13
two complementary approaches towards θ13:
I νe → νe disappearance reactor experiments:Double Chooz, Daya Bay, RENO“clean” measurement of θ13:
Pee ≈ 1− sin2 2θ13 sin2 ∆m231L
4Eν+O
(∆m2
21∆m2
31
)2
⇒ bring stat. and syst. errors below % level
I long-baseline accelerator experiments looking forνµ → νe appearance: T2K, NOνAPµe is a complicated function of all 3-flavour parametersθ13 is correlated with other parameters (CP-phase δ, sign of ∆m2
31)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 45 / 52
Outlook θ13
The LBL appearance oscillation probability
Pµe ' sin2 2θ13 sin2 θ23sin2(1− A)∆
(1− A)2
+ sin 2θ13 α sin 2θ23sin(1− A)∆
1− Asin A∆
Acos(∆ + δCP)
+ α2 cos2 θ23sin2 A∆
A2
with ∆ ≡ ∆m231L
4Eν, α ≡ ∆m2
21
∆m231
sin 2θ12 , A ≡ 2EνV∆m2
31
anti-ν: δCP → −δCP, A → −A, Peµ: δCP → −δCPother hierarchy: ∆ → −∆, A → −A, α → −α
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 46 / 52
Outlook θ13
Reactor vs Beam
assume sin2 2θ13 = 0.1, δ = π/2
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 47 / 52
Outlook θ13
The race for θ13
2010 2012 2014 2016 2018year
10-2
10-1
sin22
θ13
Discovery potential at 3 σ for NH
T2K
NOνADayaBay
DoubleChoozRENO
1
3
4
5
2
1 3 4 52
Mezzetto, TS 10
about 1 order of magnitude improvement within ∼5 yearsT. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 48 / 52
Outlook CPV, mass hierarchy
The ultimate goals
I measure the value of δCP ⇒ establish CP violation
measure Pνα→νβvs Pνα→νβ
cross section and fluxes are different for ν and ν,matter effect is CP violating
I determine the neutrino mass hierarchy, i.e. sgn(∆m231)
observe matter effect
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 49 / 52
Outlook CPV, mass hierarchy
The ultimate goals
I measure the value of δCP ⇒ establish CP violationmeasure Pνα→νβ
vs Pνα→νβ
cross section and fluxes are different for ν and ν,matter effect is CP violating
I determine the neutrino mass hierarchy, i.e. sgn(∆m231)
observe matter effect
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 49 / 52
Outlook CPV, mass hierarchy
The ultimate goals
I measure the value of δCP ⇒ establish CP violationmeasure Pνα→νβ
vs Pνα→νβ
cross section and fluxes are different for ν and ν,matter effect is CP violating
I determine the neutrino mass hierarchy, i.e. sgn(∆m231)
observe matter effect
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 49 / 52
Outlook CPV, mass hierarchy
Determination of the mass hierarchy
the vacuum oscillation probability is invariant under
∆m231 → −∆m2
31 δCP → π − δCP
→ the key to resolve the hierarchy degeneracy is the matter effectresonance condition for νµ → νe oscillations:
± 2EV∆m2
31= cos 2θ13 ≈ 1
can be fulfilled forneutrinos if ∆m2
31 > 0 (normal hierarchy)anti-neutrinos if ∆m2
31 < 0 (inverted hierarchy)
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 50 / 52
Outlook CPV, mass hierarchy
The size of the matter effect
A ≡∣∣∣∣ 2EV∆m2
31
∣∣∣∣ ' 0.09(
EGeV
)(|∆m2
31|2.5× 10−3 eV2
)−1
for experiments at the 1st osc. max, |∆m231|L/2E ' π, and
A ' 0.02(
L100 km
)
need L & 1000 km and Eν & 3 GeV in order to reach the regime of strongmatter effect A & 0.2.
terms linear in A do not break the degeneracy →have to be sensitive to higher order terms in A TS, hep-ph/0703279
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 51 / 52
Outlook CPV, mass hierarchy
Subsequent generation of LBL experiments
I superbeam upgardesI beta beamsI neutrino factory
under intense study e.g.
EUROν http://www.euronu.org
NF-IDS http://www.ids-nf.org
T. Schwetz (MPIK) Neutrino Theory - I 13 May 2011 52 / 52