The Physics of Neutrino Oscillation. Boris Kayser INSS August, 2013
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1 The Physics of Neutrino Oscillation Boris Kayser INSS August,
2 Neutrino Flavor Change (Oscillation) in + l α (e.g. µ) Vacuum ( Approach of ) B.K. & Stodolsky - l β (e.g. τ) Amp ν W (ν α ) (ν β ) W Source Target l α + l β - = Σ Amp i W U αi * ν i Prop(ν i ) U W βi Source Target 2
3 Amp [ν α ν β ] = ΣU αi * Prop(ν i )U βi What is Propagator (ν i ) Prop(ν i )? In the ν i rest frame, where the proper time is τ i, Thus, i τ i ν i (τ i ) >= m i ν i (τ i ) >. ν i (τ i ) >= e im iτ i ν i (0) >. Then, the amplitude for propagation for time τ i is Prop(ν i ) < ν i (0) ν i (τ i ) >= e im iτ i. 3
4 In the laboratory frame Time t ν Distance L The experimenter chooses L and t. They are common to all components of the beam. For each ν i, by Lorentz invariance, m i τ i = E i t - p i L. (E i, p i ) x (t, L) = m i τ i = E i t p i L. 4
5 Neutrino sources are ~ constant in time. Averaged over time, the e -ie 1t e -ie 2t interference is < e -i(e 1-E 2 )t > t = 0 unless E 2 = E 1. Only neutrino mass eigenstates with a common energy E are coherent. (Stodolsky) 5
6 For each mass eigenstate, p i = E 2 m 2 i = E m2 i 2E. Then the phase in the ν i propagator exp[-im i τ i ] is m i τ i = E i t p i L = ~ Et (E m 2 i /2E)L Irrelevant overall phase = E(t L) + m i 2 L/2E. } 6
7 Amp [ν α ν β ] l α + l β - = Σ Amp i W U αi * ν i e im2 i Prop(ν i ) L 2E U W βi Source Target = i U αie im2 i L 2E Uβi 7
8 Probability for Neutrino Oscillation in Vacuum P (ν α ν β )= Amp(ν α ν β ) 2 = = δ αβ 4 i>j R(U αiu βi U αj U βj) sin 2 ( m 2 ij +2 i>j I(U αiu βi U αj U βj) sin( m 2 ij L 2E ) L 4E ) where m 2 ij m 2 i m 2 j 8
9 For Antineutrinos We assume the world is CPT invariant. Our formalism assumes this. C: Particle Antiparticle P: Helicity Reversed helicity ν(right-handed) = CP[ν(left-handed)] 9
10 P (ν α ν β ) CPT = P (ν β ν α )=P (ν α ν β ; U U ) Thus, ( ) ( ) P (ν α ν β )= = δ αβ 4 i>j R(U αiu βi U αj U βj) sin 2 ( m 2 ij +2 i>j I(U αiu βi U αj U βj) sin( m 2 ij ( ) L 2E ) L 4E ) A complex U would lead to the CP violation P (ν α ν β ) P (ν α ν β ). 10
11 Must we assume all mass eigenstates have the same E? No, we can take entanglement into account, and use energy conservation. The oscillation probabilities are still the same. B.K., arxiv:
12 Comments 1. If all m i = 0, so that all Δm ij 2 = 0, 2. If there is no mixing, ( ) ( ) P (ν α ν β ) = δ αβ Flavor change ν Mass W l α but W l β α ν i ν j i always same ν i U α i U β α,i = 0, so that P (ν ( ) α ( ) ν β ) = δ αβ. Flavor change Mixing 12
13 3. One can detect (ν α ν β ) in two ways: See ν β α in a ν α beam (Appearance) See some of known ν α flux disappear (Disappearance) 4. Including ħ and c m 2 L 4E =1.27 m2 (ev 2 ) L(km) E(GeV) sin 2 [1.27 m 2 (ev) 2 L(km) becomes appreciable when E(GeV) ] its argument reaches O(1). An experiment with given L/E is sensitive to m 2 (ev 2 ) > E(GeV). L(km) 13
14 5. Flavor change in vacuum oscillates with L/E. Hence the name neutrino oscillation. {The L/E is from the proper time τ.} ( ) ( ) 6. P (ν α ν β ) depends only on squared-mass splittings. Oscillation experiments cannot tell us (mass) 2 ν 3 } ν 2 2 ν 1 }Δm 21 2 Δm 32?? 0 14
15 7. Neutrino flavor change does not change the total flux in a beam. It just redistributes it among the flavors. All β ( ) ( ) P (ν α ν β )=1 But some of the flavors β α could be sterile. Then some of the active flux disappears: φ νe + φ νµ + φ ντ <φ Original 15
16 Important Special Cases For, Three Flavors e im2 1 L 2E Amp (ν α ν β )= i U αi U βie im2 i L 2E e im2 1 L 2E = U +U α1 β1 α3 Uβ3e 2i 31 + U α2 Uβ2e 2i 21 (U α3 Uβ3 + U α2 Uβ2) }{{} Unitarity =2i[U α3 U β3e i 31 sin 31 + U α2 U β2e i 21 sin 21 ] where ij m 2 ij L 4E (m2 i m 2 j) L 4E. 16
17 P (ν α ν β )= 2E Amp (ν α ν β ) ( ) ( ) e im2 1 L ( ) ( ) 2 =4[ U α3 U β3 2 sin U α2 U β2 2 sin U α3 U β3 U α2 U β2 sin 31 sin 21 cos( 32 + δ 32 )]. ( ) Here δ 32 arg(u α3 U β3u α2u β2 ), a CP violating phase. Two waves of different frequencies, and their CP interference. 17
18 When the Spectrum Is ν 3 ν 2 ν 1 Invisible if Δm 2 or Δm 2 { m 2 = O(1). LE ν 2 ν 1 ν 3 For, ( ) ( ) P (ν α ν β ) = 4 U α3 U β3 2 sin 2 ( m 2 L 4E ). For no flavor change, ( ) ( ) P (ν α ν α ) = 1 4 U α3 2 (1 U α3 2 ) sin 2 ( m 2 L 4E ). Experiments with flavor content of ν 3. m 2 L E = O(1) can determine the 18
19 When There are Only Two Flavors and Two Mass Eigenstates ν 2 U = Uα1 U β1 U α2 U β2 ν 1 Δm 2 cos θ = sin θ sin θ cos θ Majorana CP phase e iξ Mixing angle ( ) ( ) For, P (ν α ν β ) = sin 2 2θ sin 2 ( m 2 L 4E ). ( ) ( ) For no flavor change, P (ν α ν α )=1 sin 2 2θ sin 2 ( m 2 L 4E ). 19
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