The Physics of Neutrino Oscillation

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1 The Physics of Neutrino Oscillation Boris Kayser PASI March,

2 Neutrino Flavor Change (Oscillation) in Vacuum + l α (e.g. µ) ( ) Approach of B.K. & Stodolsky - l β (e.g. τ) Amp W (ν α ) (ν β ) ν 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 = E(t L) + m i 2 L/2E. Irrelevant overall phase } 6

7 Amp [ν α ν β ] l α + l β - = Σ Amp i Source W U αi * ν i e im2 i Prop(ν i ) L 2E U W βi 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. 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. The write-up of this new way of treating neutrino oscillation will be posted for you. 11

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 αi 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 Δm 32 ν 1 }Δm ?? 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 For, An Important Special Case 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 α3 Uβ3e 2i 31 + U α2 Uβ2e 2i 21 +U (U α1 β1 α3 Uβ3 + U α2 Uβ2) }{{} Unitarity =2i[U α3 Uβ3e i 31 sin 31 + U α2 Uβ2e 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 )= ( ) ( ) e im2 1 L 2E Amp ( ) ( ) (ν α ν β ) 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

The Physics of Neutrino Oscillation. Boris Kayser INSS August, 2013

The Physics of Neutrino Oscillation Boris Kayser INSS August, 2013 1 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 )