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Neutrino oscillations, energy-momentum conservation and entanglement Evgeny Akhmedov MPI-K, Heidelberg & Kurchatov Inst., Moscow Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 1 4-mom. conservation and ν production Neutrino oscillations and energy-momentum conservation – an intricate relationship Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 2 4-mom. conservation and ν production Neutrino oscillations and energy-momentum conservation – an intricate relationship Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule): X X 2 d3 pf (2π)4 δ 4 pi M pf − Γ = f i 3 (2π) 2Ef i i f f P P The factor δ 4 ensures energy-momentum conservation. i pi f pf − Y Evgeny Akhmedov 1 (2Ei ) Z Y TAUP-2011 Munich, Sept. 5-9, 2011 – p. 2 4-mom. conservation and ν production Neutrino oscillations and energy-momentum conservation – an intricate relationship Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule): X X 2 d3 pf (2π)4 δ 4 pi M pf − Γ = f i 3 (2π) 2Ef i i f f P P The factor δ 4 ensures energy-momentum conservation. i pi f pf − Y 1 (2Ei ) Z Y Used to calculate neutrino production rates and detection cross sections. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 2 4-mom. conservation and ν production Neutrino oscillations and energy-momentum conservation – an intricate relationship Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule): X X 2 d3 pf (2π)4 δ 4 pi M pf − Γ = f i 3 (2π) 2Ef i i f f P P The factor δ 4 ensures energy-momentum conservation. i pi f pf − Y 1 (2Ei ) Z Y Used to calculate neutrino production rates and detection cross sections. If applied to neutrino production, implies that the neutrino 4-momentum p = (E, p~) can be determined from the 4-momenta of all other particles participating in the production process. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 2 4-mom. conservation and ν production Neutrino oscillations and energy-momentum conservation – an intricate relationship Calculation of rates of processes in quant. theory (gen. Fermi’s Golden rule): X X 2 d3 pf (2π)4 δ 4 pi M pf − Γ = f i 3 (2π) 2Ef i i f f P P The factor δ 4 ensures energy-momentum conservation. i pi f pf − Y 1 (2Ei ) Z Y Used to calculate neutrino production rates and detection cross sections. If applied to neutrino production, implies that the neutrino 4-momentum p = (E, p~) can be determined from the 4-momenta of all other particles participating in the production process. E.g. for π → µ + ν decay: if 4-momenta of π and µ have well-defined values, then the neutrino 4-momentum is fully determined by energy-momentum conservation. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 2 A dichotomy But: Due to the on-shell relation E 2 = p~ 2 + m2 , if the neutrino energy and momentum are exactly known, so is its mass! ⇒ The emitted neutrino is a mass eigenstate rather than a flavor eigenstate ( = coherent superposition of different mass eigenstates). ⇒ Neutrino oscillations cannot occur! (Mass eigenstates do not oscillate in vacuum). A dichotomy: On the one hand, energy-momentum conservation is an exact law of nature. On the other hand, exact energy and momentum conservation at neutrino production or detection would apparently make the oscillations impossible. ⇒ Significant confusion in the literature Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 3 Kinematic entanglement – way out? A suggested way out (an attempt to base the theory of neutrino oscillations on exact energy-momentum conservation): Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 4 Kinematic entanglement – way out? A suggested way out (an attempt to base the theory of neutrino oscillations on exact energy-momentum conservation): Assumption: the neutrino is produced in an entangled state with the accompanying particle(s), so that the 4-momentum of the entangled state obeys exact energy-momentum conservation. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 4 Kinematic entanglement – way out? A suggested way out (an attempt to base the theory of neutrino oscillations on exact energy-momentum conservation): Assumption: the neutrino is produced in an entangled state with the accompanying particle(s), so that the 4-momentum of the entangled state obeys exact energy-momentum conservation. Example: for π → µ + ν decay. Take for the final-state wave function |µ νi = X ∗ Uµj |µ(pµj )i|νj (pνj )i j For every value of neutrino 4-momentum pνj the corresponding value of muon 4-momentum pµj is uniquely defined through pνj + pµj = pπ . Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 4 Kinematic entanglement – way out? A suggested way out (an attempt to base the theory of neutrino oscillations on exact energy-momentum conservation): Assumption: the neutrino is produced in an entangled state with the accompanying particle(s), so that the 4-momentum of the entangled state obeys exact energy-momentum conservation. Example: for π → µ + ν decay. Take for the final-state wave function |µ νi = X ∗ Uµj |µ(pµj )i|νj (pνj )i j For every value of neutrino 4-momentum pνj the corresponding value of muon 4-momentum pµj is uniquely defined through pνj + pµj = pπ . The 4-momenta of mass eigenstates νj are correlated with 4-momenta of the accompanying muon (entanglement). Only pπ is fixed; pµ and pν are allowed to vary. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 4 Entanglement – contd. Energy-mom. conservation is satisfied, yet the produced state contains a superposition of different neutrino mass eigenstates. But: because muon states of different 4-momenta are orthogonal, energy-momentum conservation would still make the interference (oscillatory) terms in Pαβ vanish – disentanglement is necessary. Assumption: the muon is “measured”, either through a direct detection or through interaction with environment, which leads to its localization ⇒ necessary energy and momentum uncertainties are created. – Completely misses the point that in reality the parent pion is always localized and so described by a wave packet rather than by a plane wave. (Btw: without this localization the production coordinate would be completely undefined ⇒ the oscillations would be unobservable). Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 5 Entanglement – contd. The pion momentum has a spread σpπ around the mean momentum pπ ⇒ there is no strict correlation between 4-momenta of ν and µ. For a given value pνj the value of pµ is no longer uniquely determined by pνj + pµ = pπ ; can take any value within a range of width ∼ σpπ . In other words, instead of pνj + pµj = pπ we now have pνj + pµj = pπj where pπj is no longer uniquely fixed. E.g., in the case when the 4-momenta of the components of the muon accompanying different νj are the same, pµ1 = pµ2 = pµ3 ≡ pµ : energy-momentum conservation is satisfied if |pνj − pνk | . σpπ . No entanglement takes place! Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 6 NB: |pνj − pνk | . σpπ is just the condition of coherent neutrino production/detection (necessary for observability of ν oscillations). ⇒ Due to localization of the neutrino production and detection processes, kinematic entanglement is completely irrelevant to neutrino oscillations. Entanglement/disentanglement approach predicts no oscillations for ν’s from π → µ + ν decay if the muons is not detected and do not interact with the environment; wave packet approach predicts that the oscillations should occur. QM energy and momentum uncertainties play a crucial role! Energy and momentum uncertainties are in no contradiction with exact energy-mom. conservation! In describing the oscillations we have to deal with localized states which are not eigenstates of E and p but rather linear superpositions of their eigenstates. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 7 QM energy and mom. uncertainties Neutrino oscillations are a QM interference phenomenon ⇒ owe their very existence to QM uncertainty relations. Coordinate-momentum and time-energy uncertainty relations are implicated in the oscillations phenomenon in a number of ways: It is the E and p uncertainties of the emitted ν state that allow it to be a coherent superposition of the states of well-defined and different mass. Similarly, for neutrino detection to be coherent, the E and p uncertainties inherent in the detection process should be large enough to prevent a determination of the absorbed neutrino’s mass in this process. QM uncertainty relations determine the size of the neutrino wave packets ⇒ are crucial to the issue of the loss of coherence due to the wave packet separation. Are important for understanding how the produced and detected neutrino states are disentangled from the accompanying particles. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 8 When are neutrino oscillations observable? Keyword: Coherence Neutrino flavour eigenstates νe , νµ and ντ are coherent superpositions of ⇒ oscillations are only observable if mass eigenstates ν1 , ν2 and ν3 neutrino production and detection are coherent coherence is not (irreversibly) lost during neutrino propagation. Possible decoherence at production (detection): If by accurate E and p p measurements one can tell (through E = p2 + m2 ) which mass eigenstate is emitted, the coherence is lost and oscillations disappear! Full analogy with electron interference in double slit experiments: if one can establish which slit the detected electron has passed through, the interference fringes are washed out. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 9 When are neutrino oscillations observable? Another source of decoherence: wave packet separation due to the difference of group velocities ∆v of different mass eigenstates. If coherence is lost: Flavour transition can still occur, but in a non-oscillatory way. E.g. for π → µνi decay with a subsequent detection of νi with the emission of e: P ∝ X Pprod (µ νi )Pdet (e νi ) ∝ X |Uµi |2 |Uei |2 i i – the same result as for averaged oscillations. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 10 A consistent approach – WP formalism The evolved produced state: |ναfl (~x, t)i = X ∗ Uαj |νjmass (~x, t)i = X ∗ Uαj ΨSj (~x, t)|νjmass i j j The coordinate-space wave function of the jth mass eigenstate (w. packet): ΨSj (~x, t) = Z d3 p S i~ p~ x−iEj (p)t f (~ p ) e (2π)3 j p): sharp maximum at p~ = P~ (width of the Momentum distribution function fjS (~ ~ peak σpP ≪ P ). Detected state (centered at ~x = L): X fl ∗ |νβ (~x)i = Uβk ΨD x)|νkmass i k (~ k The coordinate-space wave function of the kth mass eigenstate (w. packet): ΨD x) k (~ Evgeny Akhmedov = Z d3 p D ~ i~ p(~ x−L) f (~ p ) e (2π)3 k TAUP-2011 Munich, Sept. 5-9, 2011 – p. 11 Oscillation probability Transition amplitude: ~ = hνβfl |ναfl (T, L)i ~ = Aαβ (T, L) X ∗ ~ Uαj Uβj Aj (T, L) j ~ = Aj (T, L) Z d3 p S ~ D∗ −iEj (p)T +i~ pL f (~ p ) f (~ p ) e j (2π)3 j ~ − ~vj T | . σx . E.g., for Gaussian wave packets: Strongly suppressed unless |L " ~ − ~vj T )2 ( L ~ ∝ exp − Aj (T, L) 4σx2 # , 2 2 σx2 ≡ σxP + σxD Oscillation probability: ♦ 2 X ∗ ∗ ~ = |Aαβ | = ~ ∗k (T, L) ~ P (να → νβ ; T, L) Uαj Uβj Uαk Uβk Aj (T, L)A j,k Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 12 Oscillation probability Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : Z ∆m2 X ∗ ∗ −i 2P̄jk L ˜ Ijk dT P (να → νβ ; T, L) = P (να → νβ ; L) = Uαj Uβj Uαk Uβk e j,k Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 13 Oscillation probability Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : Z ∆m2 X ∗ ∗ −i 2P̄jk L ˜ Ijk dT P (να → νβ ; T, L) = P (να → νβ ; L) = Uαj Uβj Uαk Uβk e j,k I˜jk = N Z dq S fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj ) 2π ×fkS∗ (rj q Here: v≡ vj +vk 2 Evgeny Akhmedov + ∆Ejk /2v + , ∆v ≡ vk − vj , Pk )fkD (rj q rj,k ≡ TAUP-2011 vj,k v , i ∆v v qL + ∆Ejk /2v + Pk ) e N ≡ 1/[2Ei (P )2Ek (P )v] Munich, Sept. 5-9, 2011 – p. 13 Oscillation probability Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : Z ∆m2 X ∗ ∗ −i 2P̄jk L ˜ Ijk dT P (να → νβ ; T, L) = P (να → νβ ; L) = Uαj Uβj Uαk Uβk e j,k I˜jk = N Z dq S fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj ) 2π ×fkS∗ (rj q Here: v≡ vj +vk 2 + ∆Ejk /2v + , ∆v ≡ vk − vj , Pk )fkD (rj q rj,k ≡ vj,k v , i ∆v v qL + ∆Ejk /2v + Pk ) e N ≡ 1/[2Ei (P )2Ek (P )v] For (∆v/v)σp L ≪ 1 (i.e. L ≪ lcoh = (v/∆v)σx ) I˜jk is approximately independent of L; in the opposite case I˜jk is strongly suppressed Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 13 Oscillation probability Neutrino emission and detection times are not measured (or not accurately measured) in most experiments ⇒ integration over T : Z ∆m2 X ∗ ∗ −i 2P̄jk L ˜ Ijk dT P (να → νβ ; T, L) = P (να → νβ ; L) = Uαj Uβj Uαk Uβk e j,k I˜jk = N Z dq S fj (rk q − ∆Ejk /2v + Pj )fjD∗ (rk q − ∆Ejk /2v + Pj ) 2π ×fkS∗ (rj q Here: v≡ vj +vk 2 + ∆Ejk /2v + , ∆v ≡ vk − vj , Pk )fkD (rj q rj,k ≡ vj,k v , i ∆v v qL + ∆Ejk /2v + Pk ) e N ≡ 1/[2Ei (P )2Ek (P )v] For (∆v/v)σp L ≪ 1 (i.e. L ≪ lcoh = (v/∆v)σx ) I˜jk is approximately independent of L; in the opposite case I˜jk is strongly suppressed I˜jk is also strongly suppressed unless ∆Ejk /v ≪ σp , i.e. ∆Ejk ≪ σE – coherent production/detection condition Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 13 Conservation of hEi and h~pi For the initially produced flavour state |να i: Z X |Uαj |2 hEα i ≡ hνα |H|να i = i h~ pα i ≡ hνα |~ p|να i = X |Uαj | 2 j For the evolved state ~ t)i = |ν(L, X ∗ Uαj j Z d3 p S 2 |f (~ p )| Ej (~ p) j (2π)3 d3 p S 2 |f (~ p )| p~ j 3 (2π) d3 p S ~ i~ pL−iE j (p)t f (~ p ) e |νj i j 3 (2π) Z E and p expectation values are hEi ≡ hν|H|νi = hEα i , ⇒ h~ pi ≡ hν|~ p|νi = h~ pα i hEi and h~ pi do not change in the course of the oscillations. In neutrino oscillations energy-momentum conservation manifests itself as conservation of the mean values of neutrino energy and momentum. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 14 Flavor independence of hEi and new physics Ahluwalia & Schritt, arXiv:0911.2965: Assume that mean neutrino energies hEα i measured in ν oscillation experiments are flavor independent: hEe i = hEµ i = hEτ i . Then: either sterile neutrinos must exist, or neutrinos must have some new interactions beyond those of the standard model. (Flavour independence of hEα i is inconsistent with what is known about the usual 3-f neutrino mixing). A 2-f example: hEe i = hE1 i cos2 θ + hE2 i sin2 θ , hEµ i = hE1 i sin2 θ + hE2 i cos2 θ , For m1 6= m2 one has hE1 i = 6 hE2 i ⇒ the condition hEe i = hEµ i would exclude all values of the mixing angle except θ = 45◦ corresponding to the maximal leptonic mixing! Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 15 Flavour independence of hEi – contd. There is no physical reasons to expect flavour independence of hEα i ! ~ can be represented as The oscillated state |ν(t, L)i ~ = Ae (t, L)|ν ~ e i + Aµ (t, L)|ν ~ µ i + Aτ (t, L)|ν ~ τi, |ν(t, L)i ~ the probability amplitudes of finding the flavour states να with Aα (t, L) ~ The expect. value hEi in the state |ν(t, L)i ~ (α = e, µ, τ ) at x = (t, L). remains constant and equal to that in the initially produced flavour state. But: This does not mean that each individual flavour component of the state has the same energy expectation value! The conjecture of Ahluwalia & Schritt actually contradicts the observability conditions for neutrino oscillations. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 16 Flavour independence of hEi – contd. An assumption of Ahluwalia & Schritt: The detection process does not distort the measured neutrino energy, because “by choosing an appropriate kinematical setting, this ‘back reaction’ can be made average to zero”. But: this condition would actually make neutrino oscillations unobservable! In practical terms the condition of “no back reaction” actually means |hEα i − hEβ i| ≫ σE Notation: hEj i ≡ Ēj ; X X 2 2 (|Uαi |2 − |Uβi |2 )(Ēi − Ēk + Ēk ) (|Uαi | − |Uβi | )Ēi = hEα i − hEβ i = i i = X (|Uαi |2 − |Uβi |2 )(Ēi − Ēk ) , i (Ēk is the mean energy of any of the neutrino mass eigenstates; unitarity of U used). Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 17 Flavour independence of hEi – contd. Taking into account |Uai |2 ≤ 1, from |hEa i − hEb i| ≫ σE |Ēi − Ēk | ≫ σE ⇒ (i 6= k) . But: This is just the opposite of the coherent detection condition which is a necessary condition for observability of neutrino oscillations! Coherent det. condition ensures that the detection process cannot discriminate between different neutrino mass eigenstates. If it is violated, the neutrino cannot be detected as a coherent superposition of different mass eigenstates. ⇒ The fact that the oscillations have been observed completely rules out the conjecture of Ahluwalia & Schritt. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 18 Summary Attempts of simple-minded application of energy-momentum conservation to neutrino oscillations are inconsistent and lead to no oscillations at all. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 19 Summary Attempts of simple-minded application of energy-momentum conservation to neutrino oscillations are inconsistent and lead to no oscillations at all. Kinematic entanglement is irrelevant to neutrino oscillations. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 19 Summary Attempts of simple-minded application of energy-momentum conservation to neutrino oscillations are inconsistent and lead to no oscillations at all. Kinematic entanglement is irrelevant to neutrino oscillations. Energy-momentum conservation manifests itself in ν oscillations as conservation of the expectation values of ν energy amd momentum in the course of the oscillations. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 19 Summary Attempts of simple-minded application of energy-momentum conservation to neutrino oscillations are inconsistent and lead to no oscillations at all. Kinematic entanglement is irrelevant to neutrino oscillations. Energy-momentum conservation manifests itself in ν oscillations as conservation of the expectation values of ν energy amd momentum in the course of the oscillations. Neutrino oscillations are a QM interference phenomenon; owe their existence to QM uncertainty relations. The coord.-mom. and time-en. uncertainty relations play a crucial role. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 19 Summary Attempts of simple-minded application of energy-momentum conservation to neutrino oscillations are inconsistent and lead to no oscillations at all. Kinematic entanglement is irrelevant to neutrino oscillations. Energy-momentum conservation manifests itself in ν oscillations as conservation of the expectation values of ν energy amd momentum in the course of the oscillations. Neutrino oscillations are a QM interference phenomenon; owe their existence to QM uncertainty relations. The coord.-mom. and time-en. uncertainty relations play a crucial role. QM uncertainties are in no conflict with energy-momentum conservation! Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 19 Backup slides Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 20 When are neutrino oscillations observable? Another source of decoherence: wave packet separation due to the difference of group velocities ∆v of different mass eigenstates. If coherence is lost: Flavour transition can still occur, but in a non-oscillatory way. E.g. for π → µνi decay with a subsequent detection of νi with the emission of e: P ∝ X Pprod (µ νi )Pdet (e νi ) ∝ X |Uµi |2 |Uei |2 i i – the same result as for averaged oscillations. How are the oscillations destroyed? Suppose by measuring momenta and energies of particles at neutrino production (or detection) we can determine its energy E and momentum p with uncertainties σE and σp . From Ei2 = p2i + m2i : 2 2 1/2 σm2 = (2EσE ) + (2pσp ) Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 21 When are neutrino oscillations observable? If σm2 < ∆m2 = |m2i − m2k | – one can tell which mass eigenstate is emitted. −1 σm2 < ∆m2 implies 2pσp < ∆m2 , or σp < ∆m2 /2p ≃ losc . But: To measure p with the accuracy σp one needs to measure the momenta of particles at production with (at least) the same accuracy ⇒ uncertainty of their coordinates (and the coordinate of ν production point) will be σx, prod & σp−1 > losc ⇒ Oscillations washed out. Similarly for neutrino detection. Natural necessary condition for coherence (observability of oscillations): Lsource ≪ losc , Ldet ≪ losc No averaging of oscillations in the source and detector Satisfied with very large margins in most cases of practical interest Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 22 Wave packet separation Wave packets representing different mass eigenstate components have different group velocities vgi ⇒ after time tcoh (coherence time) they separate ⇒ Neutrinos stop oscillating! (Only averaged effect observable). Coherence time and length: ∆v · tcoh ≃ σx ; lcoh ≃ vtcoh pi pk ∆m2 ∆v = − ≃ Ei Ek 2E 2 lcoh ≃ v σ ∆v x = 2E 2 ∆m2 vσx The standard formula for Posc is obtained when the decoherence effects are negligible. Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 23 A manifestation of neutrino coherence Even non-observation of neutrino oscillations at distances L ≪ losc is a consequence of and an evidence for coherence of neutrino emission and detection! Two-flavour example (e.g. for νe emission and detection): Aprod/det (ν1 ) ∼ cos θ , A(νe → νe ) = X Aprod/det (ν2 ) ∼ sin θ ⇒ Aprod (νi )Adet (νi ) ∼ cos2 θ + e−i∆φ sin2 θ i=1,2 Phase difference ∆φ vanishes at short L ⇒ P (νe → νe ) = (cos2 θ + sin2 θ)2 = 1 If ν1 and ν2 were emitted and absorbed incoherently) to sum probabilities rather than amplitudes: P (νe → νe ) ∼ X ⇒ one would have |Aprod (νi )Adet (νi )|2 ∼ cos4 θ + sin4 θ < 1 i=1,2 Evgeny Akhmedov TAUP-2011 Munich, Sept. 5-9, 2011 – p. 24