Neutrino Oscillations from Quantum Field Theory

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1 Neutrino Oscillations from Quantum Field Theory Partly based on: EA, J. Kopp & M. Lindner, JHEP 0805:005,2008 [arxiv: ] and EA & J. Kopp, work in progress Evgeny Akhmedov MPI-K, Heidelberg & Kurchatov Inst., Moscow Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 1

2 Why QFT? Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 2

3 Two standard approaches to ν oscillations Evolution of the lavor eigenstate ν fl a = i Uai νi mass i φ i = E i t p i x U ai e iφ i ν mass i, Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 3

4 Two standard approaches to ν oscillations Evolution of the lavor eigenstate ν fl a = i U ai ν mass i i U ai e iφ i ν mass i, Oscillation phase: φ i = E i t p i x φ = φ i φ k = E t p x Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 3

5 Two standard approaches to ν oscillations Evolution of the lavor eigenstate ν fl a = i U ai ν mass i i U ai e iφ i ν mass i, Oscillation phase: φ i = E i t p i x φ = φ i φ k = E t p x I. Same momentum prescription: Assume the emitted neutrino state has a well defined momentum (plane wave) p = 0. For ultra-relativistic neutrinos E i = p 2 + m 2 i p + m2 i 2p Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 3

6 Two standard approaches to ν oscillations Evolution of the lavor eigenstate ν fl a = i U ai ν mass i i U ai e iφ i ν mass i, Oscillation phase: φ i = E i t p i x φ = φ i φ k = E t p x I. Same momentum prescription: Assume the emitted neutrino state has a well defined momentum (plane wave) p = 0. For ultra-relativistic neutrinos E i = p 2 + m 2 i p + m2 i 2p E m2 2 m 2 1 2p m2 2p ; m2 φ = 2p t Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 3

7 Two standard approaches to ν oscillations Evolution of the lavor eigenstate ν fl a = i U ai ν mass i i U ai e iφ i ν mass i, Oscillation phase: φ i = E i t p i x φ = φ i φ k = E t p x I. Same momentum prescription: Assume the emitted neutrino state has a well defined momentum (plane wave) p = 0. For ultra-relativistic neutrinos E i = p 2 + m 2 i p + m2 i 2p E m2 2 m 2 1 2p m2 2p ; m2 φ = 2p t Also, assume L t ( time-to-space conversion ) The standard formula is obtained Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 3

8 II. Same energy prescription: Assume the emitted neutrino state has a well defined energy (stationary state) E = 0. φ = E t p L p L For ultra-relativistic neutrinos p i = E 2 m 2 i E m2 i 2E p p 1 p 2 m2 2E ; The standard formula is obtained Stand. phase (l osc ) = 4πp m m p (MeV) m 2 ev 2 No time-to-space conversion necessary Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 4

9 Problems with plane waves and stat. states I. Plane waves: have the same probability throughout the whole space Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 5

10 Problems with plane waves and stat. states I. Plane waves: have the same probability throughout the whole space L - dependence: only through time-to-space conversion (dubious at least!) valid only for pointlike particles Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 5

11 Problems with plane waves and stat. states I. Plane waves: have the same probability throughout the whole space L - dependence: only through time-to-space conversion (dubious at least!) valid only for pointlike particles For on-shell free particles: Ei 2 = p 2 + m 2 i pσ p = Eσ E fixed momentum (σ p = 0) means σ E = 0 for each ν i no production/deterction coherence Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 5

12 Problems with plane waves and stat. states I. Plane waves: have the same probability throughout the whole space L - dependence: only through time-to-space conversion (dubious at least!) valid only for pointlike particles For on-shell free particles: Ei 2 = p 2 + m 2 i pσ p = Eσ E fixed momentum (σ p = 0) means σ E = 0 for each ν i no production/deterction coherence II. Stationary states: no time evolution Time-to-space conversion not necessary Cannot describe decoherence by wave packet separation For on-shell free particles σ E = 0 means σ p = 0 for each ν i Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 5

13 The wave packet approach In quantum theory free propagating particles are describes by wave packets. The evolved produced state: x ν a (t) = i U ai d 3 p (2π) 3/2 f is( p p i ) e i p x ie(p)t ν i The detected state: x ν b = i U bi d 3 p (2π) 3/2 f id( p p i ) e i p( x L) ν i The oscillation amplitude: A ab (L, T) = i U aiu bi d 3 x ν b x x ν a (T) = i U aiu bi d 3 pf id ( p p i ) f is ( p p i )e ie i( p)t+i p L Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 6

14 The wave packet approach contd. Neutrino production and detection times usually not measured the oscillation probability obtained upon integration over T : P ab (L) = dt A ab (L,T) 2 Must satisfy the unitarity conditions P ab (L) = a b P ab (L) = 1 Not automatically satisfied for standard normalization of the wave packets the proper normalization has to be imposed by hand. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 7

15 The wave packet approach contd. Result for Gaussian wave packets: P(ν a ν b ; L) = i,k UaiU ak U bi Ubk e i m2 ik 2E L [ [ ] 2 [ ] ] 2 L exp 2π 2 ξ 2 σ x (l coh ) ik (l osc ) ik First exponential: loss of coherence due to wave packet separation for L l coh : l coh = 2 2 σ x = 4 2E 2 σ x v g m 2 ik Second exponential: suppression of oscillations due to averaging when ξσ x is large compared to the oscillation length l osc. [ ξ : E ξ(m 2 i /2E) ] Accounts for possible coherence violation at neutrino production and detection. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 8

16 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

17 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Accounts for possible decoherence effects due to the wave packet separation and/or lack of production and detection coherence. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

18 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Accounts for possible decoherence effects due to the wave packet separation and/or lack of production and detection coherence. Problems: Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

19 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Accounts for possible decoherence effects due to the wave packet separation and/or lack of production and detection coherence. Problems: The shape and width of neutrino wave packets have to be postulated; the parameter ξ cannot be calculated Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

20 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Accounts for possible decoherence effects due to the wave packet separation and/or lack of production and detection coherence. Problems: The shape and width of neutrino wave packets have to be postulated; the parameter ξ cannot be calculated Production and detection processes are not properly taken into account; neutrinos are assumed to be always on-shell Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

21 The wave packet approach contd. Avoids problems of plane-wave and stationary state ( same momentum and same energy ) approaches. Accounts for possible decoherence effects due to the wave packet separation and/or lack of production and detection coherence. Problems: The shape and width of neutrino wave packets have to be postulated; the parameter ξ cannot be calculated Production and detection processes are not properly taken into account; neutrinos are assumed to be always on-shell Normalization by hand is invoked Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 9

22 QFT approach Most rigorous and consistent approach to neutrino oscillations Fully takes into account neutrino production and detection processes Treats neutrino production, propagation and detection as a single process Avoids unjustified assumptions about the neutrino wave function ( same energy, same momentum ), shape and width of the wave packets, etc.; automatic normalization Requires knowledge of the wave functions of particles accompanying neutrino production and detection Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 10

23 QFT approach contd. Neutrino production, propagation and detection described by a single Feynman diagram: Coordinate-space Feynman rules have to be used Propagation over macroscopic distances neutrinos are essentially on the mass shell. Described by propagators rather than by wave functions no questions about the properties of neutrino w. functions The rate of the overall prod. - propag. - detect. process calculated. The oscillation probability is obtained by dividing by the rates of the production and detection processes Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 11

24 QFT approach contd. The amplitude of the overall process: ia ab = UaiU bi d 4 x 1 d 4 x 2 M ip (x 1 ) S Fi (x 1 x 2 ) M id (x 2 ) i Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 12

25 QFT approach contd. The amplitude of the overall process: UaiU bi d 4 x 1 d 4 x 2 M ip (x 1 ) S Fi (x 1 x 2 ) M id (x 2 ) = i ia ab = i U aiu bi d 4 x 1 d 4 x 2 MiP (x 1 ) M id (x 2 ) d 4 p (2π) 4 e ip(x2 x1) (p/ + m i ) p 2 m 2 i + iǫ Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 12

26 QFT approach contd. The amplitude of the overall process: ia ab = UaiU bi d 4 x 1 d 4 x 2 M ip (x 1 ) S Fi (x 1 x 2 ) M id (x 2 ) i = i U aiu bi d 4 x 1 d 4 x 2 MiP (x 1 ) M id (x 2 ) d 4 p (2π) 4 e ip(x2 x1) (p/ + m i ) p 2 m 2 i + iǫ Using p/ + m i = s ūi(p, s)u i (p, s), x 1 = x S + x 1, x 2 = x D + x 2 with L = x D x S, T = t D t S, Ψ(p 0, p) = Ψ(p 0, p) S Ψ(p 0, p) D Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 12

27 QFT approach contd. The amplitude of the overall process: UaiU bi d 4 x 1 d 4 x 2 M ip (x 1 ) S Fi (x 1 x 2 ) M id (x 2 ) = i ia ab = i U aiu bi d 4 x 1 d 4 x 2 MiP (x 1 ) M id (x 2 ) d 4 p (2π) 4 e ip(x2 x1) (p/ + m i ) p 2 m 2 i + iǫ Using p/ + m i = s ūi(p, s)u i (p, s), x 1 = x S + x 1, x 2 = x D + x 2 with L = x D x S, T = t D t S, Ψ(p 0, p) = Ψ(p 0, p) S Ψ(p 0, p) D ia ab = i U aiu bi 0 T+i p L d 4 p (2π) 4 Ψ(p0, p) e ip p 2 m 2 i + iǫ Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 12

28 QFT approach contd. The amplitude of the overall process: UaiU bi d 4 x 1 d 4 x 2 M ip (x 1 ) S Fi (x 1 x 2 ) M id (x 2 ) = i ia ab = i U aiu bi d 4 x 1 d 4 x 2 MiP (x 1 ) M id (x 2 ) d 4 p (2π) 4 e ip(x2 x1) (p/ + m i ) p 2 m 2 i + iǫ Using p/ + m i = s ūi(p, s)u i (p, s), x 1 = x S + x 1, x 2 = x D + x 2 with L = x D x S, T = t D t S, Ψ(p 0, p) = Ψ(p 0, p) S Ψ(p 0, p) D ia ab = i U aiu bi 0 T+i p L d 4 p (2π) 4 Ψ(p0, p) e ip p 2 m 2 i + iǫ Ψ(p 0, p) S = d 4 x 1 M P (x 1) e ipx 1, Ψ(p 0, p) D = d 4 x 2 M D (x 2) e ipx 2 Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 12

29 QFT calculation Grimus-Stockinger f-la. Grimus-Stockinger theorem for p - integration of the neutrino propagator (limit L ): d 3 p ψ( p) ei p L A p 2 + iǫ L 2π2 L ψ( L A L )ei AL + O(L 3 2 ) p (p 0, (p 2 0 m 2 i )1/2 L/L). When is it actually valid? Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 13

30 QFT calculation Grimus-Stockinger f-la. Grimus-Stockinger theorem for p - integration of the neutrino propagator (limit L ): d 3 p ψ( p) ei p L A p 2 + iǫ L 2π2 L ψ( L A L )ei AL + O(L 3 2 ) p (p 0, (p 2 0 m 2 i )1/2 L/L). When is it actually valid? L p σ 2 p. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 13

31 QFT calculation Grimus-Stockinger f-la. Grimus-Stockinger theorem for p - integration of the neutrino propagator (limit L ): d 3 p ψ( p) ei p L A p 2 + iǫ L 2π2 L ψ( L A L )ei AL + O(L 3 2 ) p (p 0, (p 2 0 m 2 i )1/2 L/L). When is it actually valid? L p σ 2 p. In the opposite limit no factor 1/L. Reason: no transverse spreading of the wave packets in this regime; perfectly collimated beam. t transv E/σ 2 p, t long. E 3 /σ 2 pm 2. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 13

32 QFT calculation Grimus-Stockinger f-la. Grimus-Stockinger theorem for p - integration of the neutrino propagator (limit L ): d 3 p ψ( p) ei p L A p 2 + iǫ L 2π2 L ψ( L A L )ei AL + O(L 3 2 ) p (p 0, (p 2 0 m 2 i )1/2 L/L). When is it actually valid? L p σ 2 p. In the opposite limit no factor 1/L. Reason: no transverse spreading of the wave packets in this regime; perfectly collimated beam. t transv E/σ 2 p, t long. E 3 /σ 2 pm 2. explains the strange plane wave behaviour found in this limit by Ioannissian & Pilaftsis (1998) not clear if can be realized in any experimental setting. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 13

33 QFT approach contd. First consistent derivation in a (simplified) QFT framework Kobzarev et al., At production: plane-wave charged leptons collide with infinitely heavy nuclei. At detection: neutrinos collide with infinitely heavy nuclei and charged leptons (and other nuclei) are produced. (σ p ) lept = 0, (σ p ) nucl (nuclei are localized) Ψ nucl (p) = const. Ψ(p 0, p) S δ(p 0 E in )δ(p 0 E out ) The amplitude for process with propagation of ν i : A i δ(e in E out ) d 3 e i p L p Ein 2 p2 m 2 i + iǫ 1 L δ(e in E out )e i p il Leads to the standard formula for the oscillation probability of relativistic neutrinos in vacuum: P(ν a ν b ; L) = i 2 Ubi e i m2 i1 2E L Uai Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 14

34 QFT approach Calculations in a realistic model with Gaussian wave packets: Giunti et al. 1993, Dolgov et al. 2004, Beuthe, Calculation with localized stationary external states: Grimus & Stockinger, 1996; Ioannissian &Pilaftsis (1998);... A good review (up to 2003) M. Beuthe, Phys. Rep. 375 (2003) 105. QFT calculation for Mössbauer neutrinos): EA, J. Kopp & M. Lindner, Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 15

35 QFT calculation for Mössbauer neutrinos The amplitude for zero linewidths: ia = d 3 x 1 dt 1 d 3 x 2 dt 2 Ψ He,S( x 1 )e +ie He,S t 1 Ψ H,S ( x 1 )e ie H,S t 1 Here j Ψ H,D( x 2 )e +ie H,D t 2 Ψ He,S ( x 2 )e ie He,D t 2 M µ S Mν D U ej 2 d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) ū e,s γ µ (1 γ 5 ) i(p/ + m j ) p 2 0 p2 m 2 j + iǫ γ ν(1 γ 5 )u e,d M µ S,D = G F cosθ c 2 ψ e (R) ū He (M V δ µ 0 g AM A σ i δ µ i / 3) u H κ 1/2 S,D Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 16

36 QFT calculation contd. For Lorentzian energy distributions of external particles: ρ A,B (E A,B ) = γ A,B /2π (E A,B E A,B,0 ) 2 + γ 2 A,B /4 (A = {H, He}, B = {S, D}, E A,B,0 = m A ω A,B ) Γ Γ 0 B 0 4πL 2 Y SY D j,k U ej 2 U ek 2 exp 1 [ ] ( ) e L/L coh jk,s + e L/L coh jk,d exp i m2 jk 2 2Ē L [ (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 (γ S + γ D )/2π (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 L coh jk,b coherence lengths: L coh jk,b = 4Ē2 γ B m 2 jk = σ x, σ x = 2 (B = S, D) v g γ B Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 17

37 QFT calculation contd. Generalized Lamb Mössbauer (Debye Waller) factor [ exp p2 j + p2 k 2σ 2 p ] = exp [ (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 First factor suppression of emission and absorption, i.e. a generalized Lamb-Mössbauer factor, second factor suppression of oscillations. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 18

38 QFT calculation contd. Generalized Lamb Mössbauer (Debye Waller) factor [ exp p2 j + p2 k 2σ 2 p ] = exp [ (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 First factor suppression of emission and absorption, i.e. a generalized Lamb-Mössbauer factor, second factor suppression of oscillations. m 2 jk 2σ2 p localization condition: Spatial localization σ x 1/σ p. Oscillations would be suppressed only if m 2 jk 2σ2 p. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 18

39 QFT calculation contd. Generalized Lamb Mössbauer (Debye Waller) factor [ exp p2 j + p2 k 2σ 2 p ] = exp [ (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 First factor suppression of emission and absorption, i.e. a generalized Lamb-Mössbauer factor, second factor suppression of oscillations. m 2 jk 2σ2 p localization condition: Spatial localization σ x 1/σ p. Oscillations would be suppressed only if m 2 jk 2σ2 p. In reality: m 2 jk max ev 2 ; σ 2 p (10 kev) 2 oscillations will not be suppressed. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 18

40 Conclusions For a consistent derivation of the probability of neutrino oscillations one needs either wave packet (QM) or QFT approach QFT has an avantage of fully taking into account the neutrino production and detection processes Does not treat neutrinos to be always on the mass shell allows σ P σ E Does not require postulating the shapes and widths of the wave packets Automatically leads to the correct normalization of P ab (L) Indispensible for accurate description of transition from coherence to decoherence regime Clarifies many subtle issues of the theory of neutrino oscillations Once used to derive the osc. probability, can be forgotten in most situations of practical interest Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 19

41 Backup slides Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 20

42 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

43 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! 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 σ E and σ p. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

44 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! 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 σ E and σ p. E i = p 2 i + m2 i σ m 2 = [ (2Eσ E ) 2 + (2pσ p ) 2] 1/2 Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

45 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! 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 σ E and σ p. E i = p 2 i + m2 i σ m 2 = [ (2Eσ E ) 2 + (2pσ p ) 2] 1/2 If σ m 2 < m 2 = m 2 i m2 k one can tell which mass eigenstate is emitted. σ m 2 < m 2 implies 2pσ p < m 2, or σ p < m 2 /2p losc. 1 Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

46 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! 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 σ E and σ p. E i = p 2 i + m2 i σ m 2 = [ (2Eσ E ) 2 + (2pσ p ) 2] 1/2 If σ m 2 < m 2 = m 2 i m2 k one can tell which mass eigenstate is emitted. σ m 2 < m 2 implies 2pσ p < m 2, or σ p < m 2 /2p losc. 1 But: To measure p with the accuracy σ p 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 σ x, prod σ 1 p l osc Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

47 Coherence at neutrino production If by accurate E and p measurements one can tell which mass eigenstate is emitted, the coherence is lost and oscillations disappear! 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 σ E and σ p. E i = p 2 i + m2 i σ m 2 = [ (2Eσ E ) 2 + (2pσ p ) 2] 1/2 If σ m 2 < m 2 = m 2 i m2 k one can tell which mass eigenstate is emitted. σ m 2 < m 2 implies 2pσ p < m 2, or σ p < m 2 /2p losc. 1 But: To measure p with the accuracy σ p 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 σ x, prod σ 1 p l osc Localization condition violated oscillations washed out (Kayser, 1981) Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 21

48 Longitudinal vs. transversal w.p. dispersion Spreading of the wave packets: consequence of the fact that the there is a spread of momenta inside of the wave packets and of the p-dependence of the group velocity. This gives v i spr v i p j σj p = 1 E (δ ij v i v j ) = 1 E [σi p v i ( v σ p )] v spr. = σ p E, v spr. = σ p E (1 v2 ) = σ p E m 2 E 2 t transv E/σ 2 p, t long. E 3 /σ 2 pm 2. Evgeny Akhmedov SFB day, Heidelberg July 9, 2009 p. 22

Mössbauer effect with neutrinos and neutrino oscillations

Mössbauer effect with neutrinos and neutrino oscillations Based on: EA, J. Kopp & M. Lindner, JHEP 0805:005,2008 [arxiv:0802.2513] and arxiv:0803.1424 Evgeny Akhmedov MPI-K, Heidelberg Evgeny Akhmedov