Theory and Phenomenology of Massive Neutrinos Part II: Neutrino Oscillations Carlo Giunti INFN, Sezione di Torino and Dipartimento di Fisica Teorica, Universit` a di Torino [email protected]Neutrino Unbound: http://www.nu.to.infn.it KIAS, Seoul, 30 November – 2 December 2015 http://www.nu.to.infn.it/slides/2015/giunti-151201-kias-2.pdf C. Giunti and C.W. Kim Fundamentals of Neutrino Physics and Astrophysics Oxford University Press 15 March 2007 – 728 pages C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 1/73
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Theory and Phenomenology of Massive Neutrinos
Part II: Neutrino Oscillations
Carlo GiuntiINFN, Sezione di Torino
andDipartimento di Fisica Teorica, Universita di Torino
Mistake: Oscillation Phase Larger by a Factor of 2
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 2/73
Neutrino Oscillations◮ 1957: Pontecorvo proposed Neutrino Oscillations in analogy with
K 0 ⇆ K 0 oscillations (Gell-Mann and Pais, 1955) =⇒ ν ⇆ ν◮ In 1957 only one neutrino ν = νe was known!◮ 1958: Goldhaber, Grodzins and Sunyar measure neutrino helicity:ν(h = −1)
◮ Then, in weak interactions ν(h = −1) and ν(h = +1)◮ Helicity conservation =⇒ ν(h = −1) ⇆ ν(h = −1)◮ ν(h = −1) is a sterile neutrino (Pontecorvo, 1967)◮ 1962: Lederman, Schwartz and Steinberger discover νµ◮ 1962: Maki, Nakagava, Sakata proposed a model with neutrino mixing:
νe = cos ϑ ν1 + sinϑ ν2νµ = − sinϑ ν1 + cos ϑ ν2
”weak neutrinos are not stable due to the occurrence of a virtualtransmutation νe ⇆ νµ”
◮ 1967: Pontecorvo: νe ⇆ νµ oscillations and applications (solarneutrinos)
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 3/73
◮ Flavor Neutrinos: νe , νµ, ντ produced in Weak Interactions
◮ Massive Neutrinos: ν1, ν2, ν3 propagate from Source to Detector
◮ A Flavor Neutrino is a superposition of Massive Neutrinos
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 25/73
Anatomy of Exclusion Plotslog∆m
2
sin2 2ϑ ≃ 2Pmax
∆m2 ≃ 4〈LE〉−1
√Pmax
sin2 2ϑ
∆m2 ≃ 2π〈LE〉−1
sin2 2ϑ & Pmax
log sin2 2ϑ
12 sin
2 2ϑ[1−
⟨cos(
∆m2L2E
)⟩]= Pmax
◮ ∆m2 ≫ 〈L/E 〉−1
Pmax ≃1
2sin2 2ϑ⇒ sin2 2ϑ ≃
2Pmax
◮ Min⟨cos(
∆m2L2E
)⟩≥ −1
sin2 2ϑ =2 Pmax
1−Min⟨
cos(
∆m2L2E
)⟩ ≥ Pmax
∆m2 ≃ 2π〈L/E 〉−1
◮ ∆m2 ≪ 2π〈L/E 〉−1
cos
(∆m2L
2E
)≃ 1− 1
2
(∆m2L
2E
)2
∆m2 ≃ 4
⟨L
E
⟩−1√Pmax
sin2 2ϑ
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 26/73
Off-Axis Experiments
high-intensity WB beamdetector shifted by a small angle from axis of beam
almost monochromatic neutrino energy
π+
cm
µ+
νµ
−~pcm
π+
~pπv = pπ/Eπ
~pcm
lab
µ+
~pνµ
θ
Ecm = pcm = mπ2
(1− m2
µ
m2π
)≃ 29.79MeV
γ =(1− v2
)−1/2= Eπ/mπ ≫ 1
{E = γ (Ecm + v pzcm)pz = γ (v Ecm + pzcm)
pz = p cos θ = E cos θ =⇒ E =Ecm
γ (1− v cos θ)
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 27/73
cos θ ≃ 1− θ2/2 and v ≃ 1
E =Ecm
γ (1− v cos θ)≃ γ (1 + v)
1 + γ2θ2v (1 + v) /2Ecm ≃
2γ
1 + γ2θ2Ecm
E ≃(1−
m2µ
m2π
)Eπ
1 + γ2 θ2=
(1−
m2µ
m2π
)Eπm
2π
m2π + E 2
π θ2
◮ θ = 0 =⇒ E ∝ Eπ WB beam
◮ Eπθ ≫ mπ =⇒ E ∝ m2π
Eπ θ2high-energy π+ give low-energy νµ
dE
dEπ≃(1−
m2µ
m2π
)1− γ2 θ2
(1 + γ2 θ2)2
dE
dEπ≃ 0 for θ = γ−1 =
mπ
Eπ=⇒ E ≃
(1−
m2µ
m2π
)mπ
2θ≃ 29.79MeV
θ
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 28/73
off-axis angle θ ≃ mπ/〈Eπ〉 =⇒ E ≃ 29.79MeV
θ
(a)
Eπ [GeV]
E[G
eV]
4035302520151050
5
4
3
2
1
0
θ = 0.0◦, 0.5◦, 1.0◦, 1.5◦, 2.0◦
◮ E can be tuned on oscillation peak Epeak = ∆m2L/2π
◮ small E =⇒ short Losc =4πE
∆m2=⇒ sensitivity to small values of ∆m2
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 29/73
φ(θ)
φ(0)=
1
4
(2
1 + γ2 θ2
)2
(b)
Eπ [GeV]
φ(θ
)/φ(0
)
4035302520151050
100
10−1
10−2
10−3
10−4
θ = 0.0◦, 0.5◦, 1.0◦, 1.5◦, 2.0◦
flux suppression requires superbeam
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 30/73
Neutrino Oscillations in Matter
Neutrino Oscillations
Neutrino Oscillations in Vacuum
Two-Neutrino Oscillations
Neutrino Oscillations in MatterEffective Potentials in MatterEvolution of Neutrino Flavors in MatterTwo-Neutrino MixingConstant Matter DensityMSW Effect (Resonant Transitions in Matter)
Wave-Packet Theory of NuOsc
Common Question: Do Charged Leptons Oscillate?
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 31/73
Effective Potentials in Mattercoherent interactions with medium: forward elastic CC and NC scattering
e
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W
�
e
; �
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; p; n e
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Z
VCC =√2GFNe V
(e−)NC = −V (p)
NC ⇒ VNC = V(n)NC = −
√2
2GFNn
Ve = VCC + VNC Vµ = Vτ = VNC
only VCC = Ve − Vµ = Ve − Vτ is important for flavor transitions
antineutrinos: V CC = −VCC VNC = −VNC
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 32/73
Evolution of Neutrino Flavors in Matter
◮ Flavor neutrino να with momentum p: |να(p)〉 =∑
k
U∗αk |νk(p)〉
◮ Evolution is determined by Hamiltonian
◮ Hamiltonian in vacuum: H = H0
H0 |νk(p)〉 = Ek |νk(p)〉 Ek =√
p2 +m2k
◮ Hamiltonian in matter: H = H0 +HI HI |να(p)〉 = Vα |να(p)〉
◮ Schrodinger evolution equation: id
dt|ν(p, t)〉 = H|ν(p, t)〉
◮ Initial condition: |ν(p, 0)〉 = |να(p)〉
◮ For t > 0 the state |ν(p, t)〉 is a superposition of all flavors:
|ν(p, t)〉 =∑
β
ϕβ(p, t)|νβ(p)〉
◮ Transition probability: Pνα→νβ = |ϕβ|2
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 33/73
Wave Packets have different velocities and separate
different massive neutrinos can interfere if and only if
wave packets arrive with δtkj .√
(δtP)2 + (δtD)2⇒ L . Lcohkj
|δtkj | ≃ |vk − vj |T ≃|∆m2
kj |2E 2
L =⇒ Lcohkj ∼2E 2
|∆m2kj |
√(δtP)2 + (δtD)2
v1 v1
v2 v2
v1
v2
L0
δtP L ∼ Lcoh
21
(δtD ≪ δtP) (δtD ≪ δtP)
L ≫ Lcoh
21
v1
v2
L0
δtP L ∼ Lcoh
21
δtD ≃ δtP
v1v2 v2
v1
L ≫ Lcoh
21
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 56/73
Quantum Mechanical Wave Packet Model
[CG, Kim, Lee, PRD 44 (1991) 3635] [CG, Kim, PRD 58 (1998) 017301]
neglecting mass effects in amplitudes of production and detection processes
|να〉 =∑
k
U∗αk
∫dp ψP
k (p) |νk(p)〉 |νβ〉 =∑
k
U∗βk
∫dp ψD
k (p) |νk(p)〉
Aαβ(x , t) = 〈νβ | e−i E t+i Px |να〉
=∑
k
U∗αk Uβk
∫dp ψP
k (p)ψD∗k (p) e−iEk (p)t+ipx
Gaussian Approximation of Wave Packets
ψPk (p) =
(2πσ2pP
)−1/4exp
[−(p − pk)
2
4σ2pP
]
ψDk (p) =
(2πσ2pD
)−1/4exp
[−(p − pk)
2
4σ2pD
]
the value of pk is determined by the production process (causality)C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 57/73
Aαβ(x , t) ∝∑
k
U∗αk Uβk
∫dp exp
[−iEk(p)t + ipx − (p − pk)
2
4σ2p
]
global energy-momentum uncertainty:1
σ2p=
1
σ2pP+
1
σ2pD
sharply peaked wave packets
σp ≪ E 2k (pk)/mk =⇒ Ek(p) =
√p2 +m2
k ≃ Ek + vk (p − pk)
Ek = Ek(pk) =√
p2k +m2k vk =
∂Ek(p)
∂p
∣∣∣∣p=pk
=pk
Ek
group velocity
Aαβ(x , t) ∝∑
k
U∗αk Uβk exp
[− iEkt + ipkx −
(x − vkt)2
4σ2x︸ ︷︷ ︸suppression factorfor |x − vkt| & σx
]
σx σp =1
2global space-time uncertainty: σ2x = σ2xP + σ2xD
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 58/73
ReA(k)αβ
x − vkt
νk–νj interferene only if A(k)αβ and A(j)
αβ overlap at detection
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 59/73
−Ek t + pkx = − (Ek − pk) x + Ek (x − t) = −E 2k − p2k
Ek + pkx + Ek (x − t)
= − m2k
Ek + pkx + Ek (x − t) ≃ −m2
k
2Ex + Ek (x − t)
Aαβ(x , t) ∝∑
k
U∗αk Uβk exp
[−i m
2k
2Ex
︸ ︷︷ ︸standardphase
for t = x
+ iEk (x − t)︸ ︷︷ ︸
additionalphase
for t 6= x
−(x − vkt)2
4σ2x
]
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 60/73
Space-Time Flavor Transition Probability
Pαβ(x , t) ∝∑
k,j
U∗αk Uβk Uαj U
∗βj exp
[−i
∆m2kjx
2E︸ ︷︷ ︸standardphase
for t = x
+i (Ek − Ej) (x − t)︸ ︷︷ ︸
additionalphase
for t 6= x
]
× exp
[−(x − vkj t)
2
4σ2x︸ ︷︷ ︸suppressionfactor for
|x − v kjt| & σx
−(vk − vj)2t2
8σ2x︸ ︷︷ ︸suppression
factordue to
separation ofwave packets
]
vk =pk
Ek
≃ 1− m2k
2E 2vkj =
vk + vj
2≃ 1−
m2k +m2
j
4E 2
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 61/73
Oscillations in Space: Pαβ(L) ∝∫
dt Pαβ(L, t)
Gaussian integration over dt
Pαβ(L) ∝∑
k,j
U∗αk Uβk Uαj U
∗βj exp
[−i
∆m2kjL
2E
]
×√
2
v2k + v2j︸ ︷︷ ︸≃1
exp
[− (vk − vj)
2
v2k + v2j︸ ︷︷ ︸≃(∆m2
kj)2/8E4
L2
4σ2x− (Ek − Ej )
2
v2k + v2j︸ ︷︷ ︸≃ξ2(∆m2
kj)2/8E2
σ2x
]
× exp
[i (Ek − Ej)
(1−
2v 2kj
v2k + v2j
)
︸ ︷︷ ︸≪∆m2
kj/2E
L
]
Ultrarelativistic Neutrinos: pk ≃ E − (1− ξ) m2k
2EEk ≃ E + ξ
m2k
2E
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 62/73
Pαβ(L) =∑
k,j
U∗αk Uβk Uαj U
∗βj exp
[−i
∆m2kjL
2E
]
× exp
−
(∆m2
kjL
4√2E 2σx
)2
− 2ξ2
(∆m2
kjσx
4E
)2
OscillationLengths
Losckj =4πE
∆m2kj
CoherenceLengths
Lcohkj =4√2E 2
|∆m2kj |
σx
Pαβ(L) =∑
k,j
U∗αk Uβk Uαj U
∗βj exp
[−2πi L
Losckj
]
× exp
−
(L
Lcohkj
)2
− 2π2ξ2
(σxLosckj
)2
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 63/73
new localization term: exp
−2π2ξ2
(σxLosckj
)2
interference is suppressed for σx & Losckj
equivalent to neutrino mass measurement
uncertainty of neutrino mass measurement:
m2k = E 2
k − p2k =⇒ δm2k ≃
√(2Ek δEk )
2 + (2 pk δpk)2 ∼ 4E σp
σp =1
2σxE =
|∆m2kj |Losckj
4π=⇒ δm2
k ∼|∆m2
kj |Losckj
σx
σx & Losckj =⇒ δm2k . |∆m2
kj | =⇒ only one massive neutrino!
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 64/73
Decoherence in Two-Neutrino Mixing
∆m2 = 10−3eV
2 sin2 2ϑ = 1 E = 1GeV σp = 50MeV
Losc =4πE
∆m2= 2480 km Lcoh =
4√2E 2
|∆m2| σx = 11163 km
L [km℄
P
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!
�
�
10
5
10
4
10
3
10
2
1
0.8
0.6
0.4
0.2
0
Decoherence for L & Lcoh ∼ 104 km
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 65/73
Achievements of the QM Wave Packet Model
◮ Confirmed Standard Oscillation Length: Losckj = 4πE/∆m2kj
refuted in [Lowe et al., PLB 384 (1996) 288] [Burkhardt, Lowe, Stephenson, Goldman, PRD 59 (1999) 054018]
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 69/73
Correct definition of Charged Lepton Oscillations[Pakvasa, Nuovo Cim. Lett. 31 (1981) 497]
P Dν1 ν2e, µ, τ
Analogy
◮ Neutrino Oscillations: massive neutrinos propagate unchanged betweenproduction and detection, with a difference of mass (flavor) of thecharged leptons involved in the production and detection processes.
◮ Charged-Lepton Oscillations: massive charged leptons propagateunchanged between production and detection, with a difference of massof the neutrinos involved in the production and detection processes.
NO FLAVOR CONVERSION!
The propagating charged leptons must be ultrarelativistic, in order to beproduced and detected coherently (if τ is not ultrarelativistic, only e and µcontribute to the phase).
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 70/73
Practical Problems
◮ The initial and final neutrinos must be massive neutrinos of known type:precise neutrino mass measurements.
◮ The energy of the propagating charged leptons must be extremely high,in order to have a measurable oscillation length
the group velocity is the velocity of the factorwhich modulates the amplitude of the wave
packet
Re
x
C. Giunti − Theory and Phenomenology of Massive Neutrinos – II − KIAS − 30 Nov – 2 Dec 2015 − 72/73
in the plane wave approximation the interferenceof different massive neutrino contribution must be calculated
at a definite space distance L and after a definite time interval T[Nieto, hep-ph/9509370] [Kayser, Stodolsky, PLB 359 (1995) 343] [Lowe et al., PLB 384 (1996) 288] [Kayser, hep-ph/9702327]
[CG, Kim, FPL 14 (2001) 213] [CG, Physica Scripta 67 (2003) 29] [Burkhardt et al., PLB 566 (2003) 137]
∆Φkj = (pk − pj)L− (Ek − Ej ) tk WRONG!
∆Φkj = (pk − pj)L− (Ek − Ej )T CORRECT!
no factor of 2 ambiguity claimed in[Lipkin, PLB 348 (1995) 604, hep-ph/9901399] [Grossman, Lipkin, PRD 55 (1997) 2760]