The Physics of Neutrino Oscillation Boris Kayser PASI March, 2012 1
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
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
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
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
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
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
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
For Antineutrinos We assume the world is CPT invariant. Our formalism assumes this. 9
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
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
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
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
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 21 2 0?? 14
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
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
P (ν α ν β )= ( ) ( ) e im2 1 L 2E Amp ( ) ( ) (ν α ν β ) 2 =4[ U α3 U β3 2 sin 2 31 + U α2 U β2 2 sin 2 21 +2 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