Mössbauer neutrinos. Joachim Kopp. Max-Planck-Institut für Kernphysik, Heidelberg. IFIC Valencia, 8 January 2009
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1 Mössbauer neutrinos Joachim Kopp Max-Planck-Institut für Kernphysik, Heidelberg IFIC Valencia, 8 January 2009 in collaboration with E. Kh. Akhmedov and M. Lindner based on JHEP 0805 (2008) 005 (arxiv: ), arxiv: , and work in progress Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 1
2 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 2
3 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 3
4 Classical Mössbauer effect Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. R. L. Mössbauer, Z. Phys. 151 (1958) 124 H. Frauenfelder, The Mössbauer effect, W. A. Benjamin Inc., New York, 1962 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 4
5 Classical Mössbauer effect Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. Extremely narrow emission and absorption lines R. L. Mössbauer, Z. Phys. 151 (1958) 124 H. Frauenfelder, The Mössbauer effect, W. A. Benjamin Inc., New York, 1962 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 4
6 Classical Mössbauer effect Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. Extremely narrow emission and absorption lines Observation of gravitational redshift of photons R. L. Mössbauer, Z. Phys. 151 (1958) 124 H. Frauenfelder, The Mössbauer effect, W. A. Benjamin Inc., New York, 1962 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 4
7 Classical Mössbauer effect Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. R. L. Mössbauer, Z. Phys. 151 (1958) 124 H. Frauenfelder, The Mössbauer effect, W. A. Benjamin Inc., New York, 1962 Extremely narrow emission and absorption lines Observation of gravitational redshift of photons Determination of the chemical environment of the emitting nucleus Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 4
8 Classical Mössbauer effect Classical Mössbauer effect: Recoilfree emission and absorption of γ-rays from nuclei bound in a crystal lattice. R. L. Mössbauer, Z. Phys. 151 (1958) 124 H. Frauenfelder, The Mössbauer effect, W. A. Benjamin Inc., New York, 1962 Extremely narrow emission and absorption lines Observation of gravitational redshift of photons Determination of the chemical environment of the emitting nucleus Fe 3 O 4 Fe 2 O 3 Fe 3+ Fe 3 O 4 Fe 2+ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 4
9 Mössbauer neutrinos A similar effect should exist for neutrino emission/absorption in bound state β decay and induced electron capture processes. W. M. Visscher, Phys. Rev. 116 (1959) 1581; W. P. Kells, J. P. Schiffer, Phys. Rev. C28 (1983) 2162 R. S. Raghavan, hep-ph/ ; R. S. Raghavan, hep-ph/ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 5
10 Mössbauer neutrinos A similar effect should exist for neutrino emission/absorption in bound state β decay and induced electron capture processes. Proposed experiment: W. M. Visscher, Phys. Rev. 116 (1959) 1581; W. P. Kells, J. P. Schiffer, Phys. Rev. C28 (1983) 2162 R. S. Raghavan, hep-ph/ ; R. S. Raghavan, hep-ph/ Production: Detection: 3 H 3 He + + ν e + e (bound) 3 He + + e (bound) + ν e 3 H 3 H and 3 He are embedded in metal crystals (metal hydrides). Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 5
11 Mössbauer neutrinos A similar effect should exist for neutrino emission/absorption in bound state β decay and induced electron capture processes. Proposed experiment: W. M. Visscher, Phys. Rev. 116 (1959) 1581; W. P. Kells, J. P. Schiffer, Phys. Rev. C28 (1983) 2162 R. S. Raghavan, hep-ph/ ; R. S. Raghavan, hep-ph/ Production: Detection: 3 H 3 He + + ν e + e (bound) 3 He + + e (bound) + ν e 3 H 3 H and 3 He are embedded in metal crystals (metal hydrides). Physics goals: Neutrino oscillations on a laboratory scale: E = 18.6 kev, L osc atm 20 m. Gravitational interactions of neutrinos Study of solid state effects with unprecedented precision Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 5
12 Mössbauer neutrinos (2) Mössbauer neutrinos have very special properties: Neutrino receives full decay energy: Q = 18.6 kev Natural line width: γ ev Atucal line width: γ ev Inhomogeneous broadening (Impurities, lattice defects) Homogeneous broadening (Spin interactions) R. S. Raghavan, W. Potzel Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 6
13 Mössbauer neutrinos (2) Mössbauer neutrinos have very special properties: Neutrino receives full decay energy: Q = 18.6 kev Natural line width: γ ev Atucal line width: γ ev Inhomogeneous broadening (Impurities, lattice defects) Homogeneous broadening (Spin interactions) R. S. Raghavan, W. Potzel Experimental challenges: Is the Lamb-Mössbauer factor (fraction of recoil-free emissions/absorptions) large enough? Can a linewidth γ ev be achieved? Can the resonance condition be fulfilled? Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 6
14 Mössbauer neutrinos (3) Recent controversy: Does the small energy uncertainty prohibit oscillations of Mössbauer neutrinos? Do oscillating neutrinos need to have equal energies resp. equal momenta? S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G34 (2007) 987, hep-ph/ Does the time-energy uncertainty relation prevent oscillations? S. M. Bilenky, arxiv: , S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G35 (2008) (arxiv: ), arxiv: E. Kh. Akhmedov, JK, M. Lindner, arxiv: Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 7
15 Mössbauer neutrinos (3) Recent controversy: Does the small energy uncertainty prohibit oscillations of Mössbauer neutrinos? Do oscillating neutrinos need to have equal energies resp. equal momenta? S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G34 (2007) 987, hep-ph/ Does the time-energy uncertainty relation prevent oscillations? S. M. Bilenky, arxiv: , S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G35 (2008) (arxiv: ), arxiv: E. Kh. Akhmedov, JK, M. Lindner, arxiv: Careful treatment with as few assumptions as possible is needed Answer to the above questions will be No. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 7
16 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 8
17 Textbook derivation of the oscillation formula Diagonalization of the mass terms of the charged leptons and neutrinos gives L g 2 (ē αl γ µ U αj ν jl ) W µ + diag. mass terms + h.c. (flavour eigenstates: α = e, µ, τ, mass eigenstates: j = 1, 2, 3) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 9
18 Textbook derivation of the oscillation formula Diagonalization of the mass terms of the charged leptons and neutrinos gives L g 2 (ē αl γ µ U αj ν jl ) W µ + diag. mass terms + h.c. (flavour eigenstates: α = e, µ, τ, mass eigenstates: j = 1, 2, 3) Assume, at time t = 0 and location x = 0, a flavour eigenstate ν(0, 0) = ν α = i U αj ν j is produced. At time t and position x, it has evolved into ν(t, x) = i U αje ie j t+i p j x ν i Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 9
19 Textbook derivation of the oscillation formula Diagonalization of the mass terms of the charged leptons and neutrinos gives L g 2 (ē αl γ µ U αj ν jl ) W µ + diag. mass terms + h.c. (flavour eigenstates: α = e, µ, τ, mass eigenstates: j = 1, 2, 3) Assume, at time t = 0 and location x = 0, a flavour eigenstate ν(0, 0) = ν α = i U αj ν j is produced. At time t and position x, it has evolved into ν(t, x) = i U αje ie j t+i p j x ν i Oscillation probability: P(ν α ν β ) = ν β ν(t, x) 2 = UαjU βj U αk Uβke i(e j E k )t+i( p j p k ) x j,k Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 9
20 Equal energies or equal momenta? Typical assumptions in the textbook derivation of the oscillation formula: Different mass eigenstates have equal energies: E j = E k E ( Evolution only in space, Stationary evolution ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 10
21 Equal energies or equal momenta? Typical assumptions in the textbook derivation of the oscillation formula: Different mass eigenstates have equal energies: E j = E k E ( Evolution only in space, Stationary evolution ) p j = E 2 mj 2 E m2 j 2E P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 10
22 Equal energies or equal momenta? Typical assumptions in the textbook derivation of the oscillation formula: Different mass eigenstates have equal energies: E j = E k E ( Evolution only in space, Stationary evolution ) p j = E 2 mj 2 E m2 j 2E P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E Different mass eigenstates have equal momenta: p j = p k p ( Evolution only in time, Non-stationary evolution ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 10
23 Equal energies or equal momenta? Typical assumptions in the textbook derivation of the oscillation formula: Different mass eigenstates have equal energies: E j = E k E ( Evolution only in space, Stationary evolution ) p j = E 2 mj 2 E m2 j 2E P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E Different mass eigenstates have equal momenta: p j = p k p ( Evolution only in time, Non-stationary evolution ) E j = p 2 + mj 2 p + m2 j 2p P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i T 2p Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 10
24 Equal energies or equal momenta? Typical assumptions in the textbook derivation of the oscillation formula: Different mass eigenstates have equal energies: E j = E k E ( Evolution only in space, Stationary evolution ) p j = E 2 mj 2 E m2 j 2E P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E Different mass eigenstates have equal momenta: p j = p k p ( Evolution only in time, Non-stationary evolution ) E j = p 2 + mj 2 p + m2 j 2p P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i T 2p These are assumptions or approximations, not fundamental principles! Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 10
25 Equal energies or equal momenta? (2) In general, neither the equal energy assumption nor the equal momentum assumption is physically justified because both violate energy-momentum conservation in the production and detection processes. R. G. Winter, Lett. Nuovo Cim. 30 (1981) 101 C. Giunti, W. Kim, Found. Phys. Lett. 14 (2001) 213, hep-ph/ C. Giunti, Mod. Phys. Lett. A16 (2001) 2363, hep-ph/ , C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 11
26 Equal energies or equal momenta? (2) In general, neither the equal energy assumption nor the equal momentum assumption is physically justified because both violate energy-momentum conservation in the production and detection processes. R. G. Winter, Lett. Nuovo Cim. 30 (1981) 101 C. Giunti, W. Kim, Found. Phys. Lett. 14 (2001) 213, hep-ph/ C. Giunti, Mod. Phys. Lett. A16 (2001) 2363, hep-ph/ , C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/ Example: Pion decay at rest: π + µ + + ν µ, π µ + ν µ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 11
27 Equal energies or equal momenta? (2) In general, neither the equal energy assumption nor the equal momentum assumption is physically justified because both violate energy-momentum conservation in the production and detection processes. R. G. Winter, Lett. Nuovo Cim. 30 (1981) 101 C. Giunti, W. Kim, Found. Phys. Lett. 14 (2001) 213, hep-ph/ C. Giunti, Mod. Phys. Lett. A16 (2001) 2363, hep-ph/ , C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/ Example: Pion decay at rest: π + µ + + ν µ, π µ + ν µ Energy-momentum conservation for emission of mass eigenstate ν i : ( ) 2 ( ) Ej 2 = m2 π 1 m2 µ + m2 j 1 m2 µ + m4 j 4 2 p 2 j = m2 π 4 m 2 π ( 1 m2 µ m 2 π ) 2 m2 j 2 m 2 π ( 1 m2 µ m 2 π For massless neutrinos: E j = p j = E mπ 2 To first order in m 2 i : E j E + ξ m2 j 2E, ) 4m 2 π + m4 j 4mπ 2 ( 1 m2 µ m 2 π ) 30 MeV. p j E (1 ξ) m2 j 2E, ξ 1 ( ) 1 m2 µ 2 mπ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 11
28 Oscillation probability for Mössbauer neutrinos Mössbauer neutrinos are the only realistic case, where E j E k holds approximately, due to the tiny energy uncertainty, σ E ev. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 12
29 Oscillation probability for Mössbauer neutrinos Mössbauer neutrinos are the only realistic case, where E j E k holds approximately, due to the tiny energy uncertainty, σ E ev. We expect: P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 12
30 Oscillation probability for Mössbauer neutrinos Mössbauer neutrinos are the only realistic case, where E j E k holds approximately, due to the tiny energy uncertainty, σ E ev. We expect: P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E More realistic treatment desirable: Wave packet model Requires neither equal E nor equal p Takes into account finite resolutions of the source and the detector Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 12
31 Oscillation probability for Mössbauer neutrinos Mössbauer neutrinos are the only realistic case, where E j E k holds approximately, due to the tiny energy uncertainty, σ E ev. We expect: P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E More realistic treatment desirable: Wave packet model Requires neither equal E nor equal p Takes into account finite resolutions of the source and the detector Beuthe, Giunti, Grimus, Kiers, Kim, Lee, Mohanty, Nussinov, Stockinger, Weiss,... Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 12
32 Conditions for oscillations in a wave packet approach Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
33 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
34 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Neutrino oscillations are caused by the superposition of different mass eigenstates. If an experiment can distinguish different mass eigenstates, oscillations will vanish. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
35 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Neutrino oscillations are caused by the superposition of different mass eigenstates. If an experiment can distinguish different mass eigenstates, oscillations will vanish. Requirement for mass resolution σ m : σm 2 = (2Eσ E ) 2 + (2pσ p ) 2 > m 2 B. Kayser, Phys. Rev. D24 (1981) 110 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
36 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Neutrino oscillations are caused by the superposition of different mass eigenstates. If an experiment can distinguish different mass eigenstates, oscillations will vanish. Requirement for mass resolution σ m : σm 2 = (2Eσ E ) 2 + (2pσ p ) 2 > m 2 B. Kayser, Phys. Rev. D24 (1981) 110 This is easily fulfilled for Mössbauer neutrinos, since σ E ev σ p = 1/2σ x 1/interatomic distance 10 kev E = p = 18.6 kev Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
37 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Coherence maintained during propagation Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
38 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Coherence maintained during propagation Decoherence could be caused by wave packet separation Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
39 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Coherence maintained during propagation Decoherence could be caused by wave packet separation It can be shown that, for Mössbauer neutrinos, σ p is small enough, so that L osc L coh. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
40 Conditions for oscillations in a wave packet approach Coherence in production and detection processes Coherence maintained during propagation Decoherence could be caused by wave packet separation It can be shown that, for Mössbauer neutrinos, σ p is small enough, so that L osc L coh. Stanard oscillation formula is approximately recovered: P ee = j,k L osc jk = 4πE m 2 jk U ej 2 U ek 2 exp [ 2πi L L osc jk ] Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 13
41 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 14
42 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 15
43 Quantum field theoretical treatment Aim: Properties of the neutrino should be automatically determined from properties of the source and the detector. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 16
44 Quantum field theoretical treatment Aim: Properties of the neutrino should be automatically determined from properties of the source and the detector. Idea: Treat neutrino as an internal line in a tree level Feynman diagram: Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 16
45 Quantum field theoretical treatment Aim: Properties of the neutrino should be automatically determined from properties of the source and the detector. Idea: Treat neutrino as an internal line in a tree level Feynman diagram: External particles reside in harmonic oscillator potentials. E.g. for 3 H atoms in the source: [ mh ω H,S ψ H,S ( x, t) = π ] 3 4 exp [ 1 2 m Hω H,S x x S 2 ] e ie H,St Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 16
46 Oscillation amplitude Z Z ia = d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ M ν U ej 2 j ū e,s γ µ(1 γ 5 ) «3 4 exp» 12 m Hω H,S x 1 x S 2 e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 e +ie H,Dt 2 d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) i(/p + m j ) p 2 0 p2 m 2 j + iɛ (1 + γ5 )γ νu e,d. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 17
47 Oscillation amplitude Z Z ia = d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ M ν U ej 2 j ū e,s γ µ(1 γ 5 ) «3 4 exp» 12 m Hω H,S x 1 x S 2 e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 e +ie H,Dt 2 d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) i(/p + m j ) p 2 0 p2 m 2 j + iɛ (1 + γ5 )γ νu e,d. Evaluation: dt 1 dt 2 -integrals energy-conserving δ functions p 0 -integral trivial d 3 x 1 d 3 x 2 -integrals are Gaussian d 3 p-integral: Use Grimus-Stockinger theorem (for large L = x D x S ). W. Grimus, P. Stockinger, Phys. Rev. D54 (1996) 3414, hep-ph/ Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 17
48 The Grimus-Stockinger theorem Let ψ( p) be a three times continuously differentiable function on R 3, such that ψ itself and all its first and second derivatives decrease at least like 1/ p 2 for p. Then, for any real number A > 0, d 3 p ψ( p) e i p L L 2π2 A p 2 + iɛ L ψ( A L L )ei AL + O(L 3 2 ). Quantification of requirement of on-shellness for large L = L. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 18
49 From the amplitude to the transition rate Amplitude: ia = i [ 2L N δ(e S E D ) exp E S 2 ] m2 j q 2σp 2 M µ M ν U ej 2 e i E 2 S m2 L j j σ 2 p = (m H ω H,S + m He ω He,S ) 1 + (m H ω H,D + m He ω He,D ) 1 ū e,s γ µ 1 γ 5 2 (/p j + m j ) 1+γ5 2 γ ν u e,d, Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 19
50 From the amplitude to the transition rate Amplitude: ia = i [ 2L N δ(e S E D ) exp E S 2 ] m2 j q 2σp 2 M µ M ν U ej 2 e i E 2 S m2 L j j σ 2 p = (m H ω H,S + m He ω He,S ) 1 + (m H ω H,D + m He ω He,D ) 1 ū e,s γ µ 1 γ 5 2 (/p j + m j ) 1+γ5 2 γ ν u e,d, Transition rate: Integrate A 2 over densities of initial and final states Γ 0 de H,S de He,S de He,D de H,D δ(e S E D )ρ H,S (E H,S ) ρ He,D (E He,D ) ρ He,S (E He,S ) ρ H,D (E H,D ) [ U ej 2 U ek 2 exp 2E S 2 m2 j mk 2 ] (q e i E 2 S m2 E j 2 S m2 k j,k 2σ 2 p }{{} Analogue of Lamb-Mössbauer factor (Recoil-free fraction) ) L } {{ } Oscillation phase Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 19
51 The Lamb-Mössbauer factor The Lamb-Mössbauer factor is the relative probability of recoil-free emission and absorption, compared to the total emission and absorption probability. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 20
52 The Lamb-Mössbauer factor The Lamb-Mössbauer factor is the relative probability of recoil-free emission and absorption, compared to the total emission and absorption probability. Difference to the standard Mössbauer effect: Appearance of neutrino masses Emission and absorption of lighter mass eigenstates is suppressed compared to that of heavy mass eigenstates Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 20
53 The Lamb-Mössbauer factor The Lamb-Mössbauer factor is the relative probability of recoil-free emission and absorption, compared to the total emission and absorption probability. Difference to the standard Mössbauer effect: Appearance of neutrino masses Emission and absorption of lighter mass eigenstates is suppressed compared to that of heavy mass eigenstates Convenient reformulation: [ exp 2E S 2 m2 j mk 2 ] 2σ 2 p where (p min jk )2 = E 2 S max(m2 j, m2 k ). [ = exp (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 20
54 The Lamb-Mössbauer factor The Lamb-Mössbauer factor is the relative probability of recoil-free emission and absorption, compared to the total emission and absorption probability. Difference to the standard Mössbauer effect: Appearance of neutrino masses Emission and absorption of lighter mass eigenstates is suppressed compared to that of heavy mass eigenstates Convenient reformulation: [ exp 2E S 2 m2 j mk 2 ] 2σ 2 p where (p min jk )2 = E 2 S max(m2 j, m2 k ). Localization condition 4πσ x E/σ p L osc jk, [ = exp (pmin jk )2 σ 2 p ] [ exp m2 jk ] 2σp 2 (with σ x = 1/2σ p ) is satisfied if L osc jk 2πσ x, which is easily fulfilled in realistics situations. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 20
55 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 21
56 Inhomogeneous line broadening Energy levels of 3 H and 3 He in the source and detector are smeared e.g. due to crystal impurities, lattice defects, etc. R. S. Raghavan, hep-ph/ W. Potzel, Phys. Scripta T127 (2006) 85 B. Balko, I. W. Kay, J. Nicoll, J. D. Silk, G. Herling, Hyperfine Int. 107 (1997) 283 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 22
57 Inhomogeneous line broadening Energy levels of 3 H and 3 He in the source and detector are smeared e.g. due to crystal impurities, lattice defects, etc. Good approximation: ρ A,B (E A,B ) = γ A,B /2π (E A,B E A,B,0 ) 2 + γ 2 A,B /4 R. S. Raghavan, hep-ph/ W. Potzel, Phys. Scripta T127 (2006) 85 B. Balko, I. W. Kay, J. Nicoll, J. D. Silk, G. Herling, Hyperfine Int. 107 (1997) 283 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 22
58 Inhomogeneous line broadening Energy levels of 3 H and 3 He in the source and detector are smeared e.g. due to crystal impurities, lattice defects, etc. Good approximation: ρ A,B (E A,B ) = γ A,B /2π (E A,B E A,B,0 ) 2 + γ 2 A,B /4 R. S. Raghavan, hep-ph/ W. Potzel, Phys. Scripta T127 (2006) 85 B. Balko, I. W. Kay, J. Nicoll, J. D. Silk, G. Herling, Hyperfine Int. 107 (1997) 283 Result for two neutrino flavours (m 2 > m 1 ): [ Γ exp E S,0 2 ] [ ] m2 2 σp 2 exp m2 (γ S + γ D )/2π 2σp 2 (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 [ {1 2s 2 c ( 2 (e L/Lcoh S + e L/Lcoh D ) cos π L )]} L osc L coh S,D = 4Ē 2 / m 2 γ S,D Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 22
59 Inhomogeneous line broadening Energy levels of 3 H and 3 He in the source and detector are smeared e.g. due to crystal impurities, lattice defects, etc. Good approximation: ρ A,B (E A,B ) = γ A,B /2π (E A,B E A,B,0 ) 2 + γ 2 A,B /4 R. S. Raghavan, hep-ph/ W. Potzel, Phys. Scripta T127 (2006) 85 B. Balko, I. W. Kay, J. Nicoll, J. D. Silk, G. Herling, Hyperfine Int. 107 (1997) 283 Result for two neutrino flavours (m 2 > m 1 ): [ Γ exp E S,0 2 ] [ ] m2 2 σp 2 exp m2 (γ S + γ D )/2π 2σp 2 (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 [ {1 2s 2 c ( 2 (e L/Lcoh S + e L/Lcoh D ) cos π L )]} L osc L coh S,D = 4Ē 2 / m 2 γ S,D In realistic cases: L coh S,D Losc Decoherence is not an issue. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 22
60 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 23
61 Homogeneous line broadening Fluctuating electromagnetic fields in solid state crystal Fluctuating energy levels of 3 H and 3 He. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 24
62 Homogeneous line broadening Fluctuating electromagnetic fields in solid state crystal Fluctuating energy levels of 3 H and 3 He. Classical Mössbauer effect: Homogeneous and inhomogeneous broadening both lead to Lorentzian line shapes Experimentally indistinguishable We expect a result similar to that for the case of inhomogeneous broadening Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 24
63 Homogeneous line broadening Fluctuating electromagnetic fields in solid state crystal Fluctuating energy levels of 3 H and 3 He. Classical Mössbauer effect: Homogeneous and inhomogeneous broadening both lead to Lorentzian line shapes Experimentally indistinguishable We expect a result similar to that for the case of inhomogeneous broadening Ansatz: Introduce modulation factors of the form [ t f A,B (t) = exp i dt [ ] E A,B (t ] ) E A,B,0 t 0 in the 3 H and 3 He wave functions (A = H, He, B = S, D). J. Odeurs, Phys. Rev. B52 (1995) 6166 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 24
64 Transition amplitude for homogeneous line broadening Z ia = Z «3 d 3 x 1 dt 1 d 3 mh ω 4 H,S x 2 dt 2 exp» 12 π m Hω H,S x 1 x S 2 «3 mhe ω 4 He,S exp» 12 π m Heω He,S x 1 x S 2 e +ie He,S t 1 «3 mhe ω 4 He,D exp» 12 π m Heω He,D x 2 x D 2 e ie He,Dt 2 «3 mh ω 4 H,D exp» 12 π m Hω H,D x 2 x D 2 e +ie H,Dt 2 X Z M µ S Mν D U ej 2 j d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d e ie H,S t 1 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 25
65 Transition amplitude for homogeneous line broadening Z ia = Z d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ S Mν D U ej 2 j «3 4 exp» 12 m Hω H,S x 1 x S 2 f H,S (t 1 ) e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 f He,S(t 1 ) e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 f He,D (t 2 ) e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 f H,D(t 2 ) e +ie H,Dt 2 d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 25
66 Transition amplitude for homogeneous line broadening Z ia = Z d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt 2 π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ S Mν D U ej 2 j «3 4 exp» 12 m Hω H,S x 1 x S 2 f H,S (t 1 ) e ie H,S t 1 «3 4 exp» 12 m Heω He,S x 1 x S 2 f He,S(t 1 ) e +ie He,S t 1 «3 4 exp» 12 m Heω He,D x 2 x D 2 f He,D (t 2 ) e ie He,Dt 2 «3 4 exp» 12 m Hω H,D x 2 x D 2 f H,D(t 2 ) e +ie H,Dt 2 d 4 p (2π) exp ˆ ip 4 0 (t 2 t 1 ) + i p( x 2 x 1 ) ū e,s γ µ(1 γ 5 i(/p + m j ) ) p0 2 p2 mj 2 + iɛ (1 + γ5 )γ νu e,d Evaluation: d 3 x 1 d 3 x 2 -integrals are Gaussian d 3 p-integral: Use Grimus-Stockinger theorem. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 25
67 Transition rate for homogeneous line broadening Transition rate Γ AA (statistical average of AA over all possible 3 H and 3 He states). Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 26
68 Transition rate for homogeneous line broadening Transition rate Γ AA (statistical average of AA over all possible 3 H and 3 He states). We encounter the quantity B S (t 1, t 1 ) f H,S (t 1 ) fhe,s (t 1) fh,s ( t 1 ) f He,S ( t 1 ) [ t1 ] = exp i dt E S (t ), t 1 where E S (t ) [E H,S (t ) E He,S (t )] [E H,S,0 (t ) E He,S,0 (t )]. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 26
69 Transition rate for homogeneous line broadening Transition rate Γ AA (statistical average of AA over all possible 3 H and 3 He states). We encounter the quantity B S (t 1, t 1 ) f H,S (t 1 ) fhe,s (t 1) fh,s ( t 1 ) f He,S ( t 1 ) [ t1 ] = exp i dt E S (t ), t 1 where E S (t ) [E H,S (t ) E He,S (t )] [E H,S,0 (t ) E He,S,0 (t )]. By definition, E S (t ) = 0 Markov approximation: E S (t ) E S (t ) = γ S δ(t t ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 26
70 Transition rate for homogeneous line broadening Transition rate Γ AA (statistical average of AA over all possible 3 H and 3 He states). We encounter the quantity B S (t 1, t 1 ) f H,S (t 1 ) fhe,s (t 1) fh,s ( t 1 ) f He,S ( t 1 ) [ t1 ] = exp i dt E S (t ), t 1 where E S (t ) [E H,S (t ) E He,S (t )] [E H,S,0 (t ) E He,S,0 (t )]. By definition, E S (t ) = 0 Markov approximation: E S (t ) E S (t ) = γ S δ(t t ) [ ] B S (t 1, t 1 ) = exp 1 2 γ S t 1 t 1. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 26
71 Transition rate for homogeneous line broadening (2) Result: [ Γ exp E S,0 2 ] [ ] m2 2 σp 2 exp m2 (γ S + γ D )/2π 2σp 2 (E S,0 E D,0 ) 2 + (γ S+γ D ) 2 4 [ {1 2s 2 c ( 2 (e L/Lcoh S + e L/Lcoh D ) cos π L )]} L osc... identical to the result for inhomogeneous line broadening. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 27
72 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 28
73 Amplitude for broadening by natural line width Take into account the instability of 3 H in the source and the detector. Z ia = Z T Z Z T d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ M ν U ej 2 j ū e,s γ µ 1 γ 5 2 «3 4exp h 1 2 m Hω H,S x 1 x S 2i e ie H,S t 1 «3 4 exp h 1 2 m Heω He,S x 1 x S 2i e +ie He,S t 1 «3 4 exp h 1 2 m Heω He,D x 2 x D 2i e ie He,Dt 2 «3 4 exp h 1 2 m Hω H,D x 2 x D 2i e +ie H,Dt 2 i(/p + m j ) p0 2 p2 mj 2 + iɛ d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) 1+γ 5 2 γ νu e,d (correctness of this formula can be verified in the Wigner-Weisskopf approach) E. Akhmedov, J. Kopp, M. Lindner, JHEP 0805 (2008) 005 (arxiv: ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 29
74 Amplitude for broadening by natural line width Take into account the instability of 3 H in the source and the detector. Z ia = Z T Z Z T d 3 x 1 dt 1 d 3 mh ω H,S x 2 dt π mhe ω He,S π mhe ω He,D π mh ω H,D π X Z M µ M ν U ej 2 j ū e,s γ µ 1 γ 5 2 «3 4exp h 1 2 m Hω H,S x 1 x S 2i e ie H,S t γt 1 «3 4 exp h 1 2 m Heω He,S x 1 x S 2i e +ie He,S t 1 «3 4 exp h 1 2 m Heω He,D x 2 x D 2i e ie He,Dt 2 «3 4 exp h 1 2 m Hω H,D x 2 x D 2i e +ie H,Dt γ(t t 2) i(/p + m j ) p0 2 p2 mj 2 + iɛ d 4 p (2π) 4 e ip 0(t 2 t 1 )+i p( x 2 x 1 ) 1+γ 5 2 γ νu e,d (correctness of this formula can be verified in the Wigner-Weisskopf approach) E. Akhmedov, J. Kopp, M. Lindner, JHEP 0805 (2008) 005 (arxiv: ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 29
75 Probability for broadening by natural line width P j,k θ(t jk ) U ej 2 U ek 2 exp [ (pmin jk )2 σ 2 p e γt jk e L/Lcoh jk ] [ exp m2 jk ] (qē ) 2σp 2 e i 2 mj 2 Ē 2 mk 2 L sin [ 1 2 (E S E D )(T L v j ) ] sin [ 1 2 (E S E D )(T L v k ) ] (E S E D ) 2 «where T jk = min T Lvj, T Lvk and L coh jk = 4Ē 2 γ m 2 jk Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 30
76 Probability for broadening by natural line width P j,k θ(t jk ) U ej 2 U ek 2 exp [ (pmin jk )2 σ 2 p e γt jk e L/Lcoh jk ] [ exp m2 jk ] (qē ) 2σp 2 e i 2 mj 2 Ē 2 mk 2 L sin [ 1 2 (E S E D )(T L v j ) ] sin [ 1 2 (E S E D )(T L v k ) ] (E S E D ) 2 «where T jk = min T Lvj, T Lvk and L coh jk = 4Ē 2 γ m 2 jk ) L (qē Oscillation term: e i 2 mj 2 Ē 2 mk 2 Lamb-Mössbauer factor: exp [ (pjk min Localization term: exp [ ] mjk 2 /2σ2 p Coherence term: e L/Lcoh jk )2 /σ 2 p Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 30 ]
77 Probability for broadening by natural line width (2) Resonance term sin [ 1 2 (E S E D )(T L v j ) ] sin [ 1 2 (E S E D )(T L v k ) ] (E S E D ) 2 does not depend on γ, but only on the total measurement time T (Heisenberg principle). Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 31
78 Probability for broadening by natural line width (2) Resonance term sin [ 1 2 (E S E D )(T L v j ) ] sin [ 1 2 (E S E D )(T L v k ) ] (E S E D ) 2 does not depend on γ, but only on the total measurement time T (Heisenberg principle). Analogy: subnatural spectroscopy in quantum optics Atom is excited instantaneously to state b. Continuous irradition with frequency ν. Probability for exiciting state a is proportional to [(ν ν res) 2 + (γ a γ b ) 2 /4] 1, not [(ν ν res) 2 + (γ a + γ b ) 2 /4] 1. P. Meystre, O. Scully, H. Walther, Optics Communications 33 (1980) 153 Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 31
79 Probability for broadening by natural line width (2) Resonance term sin [ 1 2 (E S E D )(T L v j ) ] sin [ 1 2 (E S E D )(T L v k ) ] (E S E D ) 2 does not depend on γ, but only on the total measurement time T (Heisenberg principle). Analogy: subnatural spectroscopy in quantum optics Atom is excited instantaneously to state b. Continuous irradition with frequency ν. Probability for exiciting state a is proportional to [(ν ν res) 2 + (γ a γ b ) 2 /4] 1, not [(ν ν res) 2 + (γ a + γ b ) 2 /4] 1. P. Meystre, O. Scully, H. Walther, Optics Communications 33 (1980) 153 Here: b 3 H atom in the source, 3 He atom in the detector a 3 He atom in the source, 3 H atom in the detector Excitation of b Production of source Transition b a neutrino production, propagation and absorption Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 31
80 Probability for broadening by natural line width (3) Time dependence for E S = E D : T 2 e γt /4. Classical argument for this behaviour: Numbers of 3 H nuclei in the detector, N D, and in the source, N S, obey Ṅ D = ṄSN 0 P ee σ(t ) 4πL 2 γn D, where N 0 is the number of 3 He atoms in the detector. Absorption cross section σ(t ) s0 T, because Heisenberg principle requires resonance condition to be fulfilled to an accuracy T 1. For Lorentzian emission an absorption lines, the overlap integral is then proportional to T. Using NS = N S,0 exp( γt ), the solution to the above equation is N D = N S,0N 0 γp ees 0 8πL 2 T 2 e γt. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 32
81 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 33
82 The time-energy uncertainty relation Mandelstam-Tamm relation: E O 1 d 2 dt O(t). Here, O(t) = ψ(t) O ψ(t), for any operator O and QFT Fock state ψ(t). Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 34
83 The time-energy uncertainty relation Mandelstam-Tamm relation: E O 1 d 2 dt O(t). Here, O(t) = ψ(t) O ψ(t), for any operator O and QFT Fock state ψ(t). Choose O ν α ν α (projection in 3d flavour space) and ψ(t) a neutrino state E 1 d dt P(t) 2 P(t) P2 (t), with P(t) = ν α ψ(t) 2. S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G35 (2008) (arxiv: ) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 34
84 The time-energy uncertainty relation Mandelstam-Tamm relation: E O 1 d 2 dt O(t). Here, O(t) = ψ(t) O ψ(t), for any operator O and QFT Fock state ψ(t). Choose O ν α ν α (projection in 3d flavour space) and ψ(t) a neutrino state E 1 d dt P(t) 2 P(t) P2 (t), with P(t) = ν α ψ(t) 2. S. M. Bilenky, F. v. Feilitzsch, W. Potzel, J. Phys. G35 (2008) (arxiv: ) Problem: Interpretation of P(t). Would be the oscillation probability for a completely delocalized detector. Imagine wave packet which is large compared to L osc : Delocalized detector averages out oscillations. More realistic: Projection in flavour space and in coordinate space: O x ν α x x ν α Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 34
85 The time-energy uncertainty relation (2) Mandelstam-Tamm relation now reads: E 1 d dt P( x, t) 2 P( x, t) P 2 ( x, t), with P( x, t) = x ν β Ψ(t) 2 (now indeed an oscillation probability). Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 35
86 The time-energy uncertainty relation (2) Mandelstam-Tamm relation now reads: E 1 d dt P( x, t) 2 P( x, t) P 2 ( x, t), with P( x, t) = x ν β Ψ(t) 2 (now indeed an oscillation probability). Two-flavour approximation: P( x, t) = 1 sin 2 2θ sin 2 φ( x, t) sin 2θ cos φ( x, t) E E 1 E 2. 1 sin 2 2θ sin 2 φ( x, t) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 35
87 The time-energy uncertainty relation (2) Mandelstam-Tamm relation now reads: E 1 d dt P( x, t) 2 P( x, t) P 2 ( x, t), with P( x, t) = x ν β Ψ(t) 2 (now indeed an oscillation probability). Two-flavour approximation: P( x, t) = 1 sin 2 2θ sin 2 φ( x, t) sin 2θ cos φ( x, t) E E 1 E 2. 1 sin 2 2θ sin 2 φ( x, t) Certainly fulfilled, if fulfilled for sin 2 2θ = 1. E E 1 E 2. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 35
88 The time-energy uncertainty relation (2) Mandelstam-Tamm relation now reads: E 1 d dt P( x, t) 2 P( x, t) P 2 ( x, t), with P( x, t) = x ν β Ψ(t) 2 (now indeed an oscillation probability). Two-flavour approximation: P( x, t) = 1 sin 2 2θ sin 2 φ( x, t) sin 2θ cos φ( x, t) E E 1 E 2. 1 sin 2 2θ sin 2 φ( x, t) Certainly fulfilled, if fulfilled for sin 2 2θ = 1. E E 1 E 2. Energy difference of mass eigenstates must be smaller than energy uncertainty. Easily fulfilled for Mössbauer neutrinos due to large momentum uncertainty. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 35
89 Outline 1 The Mössbauer neutrino experiment 2 Oscillations of Mössbauer neutrinos: Qualitative arguments 3 Mössbauer neutrinos in QFT The formalism Inhomogeneous line broadening Homogeneous line broadening Natural line broadening 4 The time-energy uncertainty relation 5 Conclusions Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 36
90 Conclusions Mössbauer neutrinos do oscillate. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
91 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
92 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
93 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
94 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
95 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
96 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
97 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Generalized Lamb-Mössbauer factor leads to localization condition. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
98 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Generalized Lamb-Mössbauer factor leads to localization condition. Nonzero line width (due to homogeneous and inhomogeneous line broadening) leads to coherence condition. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
99 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Generalized Lamb-Mössbauer factor leads to localization condition. Nonzero line width (due to homogeneous and inhomogeneous line broadening) leads to coherence condition. Both conditions are easily fulfilled in realistic experiments. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
100 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Generalized Lamb-Mössbauer factor leads to localization condition. Nonzero line width (due to homogeneous and inhomogeneous line broadening) leads to coherence condition. Both conditions are easily fulfilled in realistic experiments. Natural line width dominance unrealistic, but interesting analogy to subnatural spectroscopy Resolution not limited by line width! Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
101 Conclusions Mössbauer neutrinos do oscillate. Plane wave treatment: Mössbauer neutrinos are the only case where the equal energy assumption is justified. Wave packet treatment: Coherence and localization conditions are irrelevant for realistic experiments. Properties of the neutrino wave packets have to be put in by hand. QFT treatment: Only properties of the source and the detector are put in by hand. Generalized Lamb-Mössbauer factor leads to localization condition. Nonzero line width (due to homogeneous and inhomogeneous line broadening) leads to coherence condition. Both conditions are easily fulfilled in realistic experiments. Natural line width dominance unrealistic, but interesting analogy to subnatural spectroscopy Resolution not limited by line width! The time energy uncertainty relation does not inhibit oscillations of Mössbauer neutrinos. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 37
102 Thank you!
103 Oscillation formula for neutrino wave packets Assume Gaussian wave packets: 1 ν α (x, t) = (2πσpS 2 )1/4 j U αj q dp e (p p js) 2 /4σ 2 i p ps e 2 +m 2 t+ipx j νj 2π Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 39
104 Oscillation formula for neutrino wave packets Assume Gaussian wave packets: 1 ν α (x, t) = (2πσpS 2 )1/4 j U αj Oscillation formula: P ee = [ U ej 2 U ek 2 exp 2πi j,k with L coh q dp e (p p js) 2 /4σ 2 i p ps e 2 +m 2 t+ipx j νj 2π L L osc jk ( ) 2 ( ) 2 ] L 1 L coh 2π 2 ξ 2 jk 2σ p L osc jk C. Giunti, C. W. Kim, U. W. Lee, Phys. Rev. D44 (1991) 3635; C. Giunti, C. W. Kim, Phys. Lett. B274 (1992) 87 K. Kiers, S. Nussinov, N. Weiss, Phys. Rev. D53 (1996) 537, hep-ph/ C. Giunti, C. W. Kim, Phys. Rev. D58 (1998) , hep-ph/ , C. Giunti, Found. Phys. Lett. 17 (2004) 103, hep-ph/ L osc jk = 4πE/ m 2 jk Oscillation length jk = 2 2E 2 /σ p mjk 2 Coherence length E Energy that a massless neutrino would have ξ quantifies the deviation from E (tiny for Mössbauer neutrinos) σ p Effective wave packet width (tiny for Mössbauer neutrinos) Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 39
105 Results from the wave packet treatment Decoherence term [ exp L ] L coh jk cannot inhibit oscillations because L coh jk /Losc jk E/σ p Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 40
106 Results from the wave packet treatment Decoherence term [ exp L ] L coh jk cannot inhibit oscillations because L coh jk /Losc jk E/σ p Localization term [ ( ) 2 ] 1 exp 2π 2 ξ 2 2σ p L osc jk cannot inhibit oscillations, because ξ/σ p L osc jk 1. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 40
107 Results from the wave packet treatment Decoherence term [ exp L ] L coh jk cannot inhibit oscillations because L coh jk /Losc jk E/σ p Localization term [ ( ) 2 ] 1 exp 2π 2 ξ 2 2σ p L osc jk cannot inhibit oscillations, because ξ/σ p L osc jk 1. Our expectation is confirmed: P(ν α ν β ) = j,k U αju βj U αk U βke m 2 jk i L 2E. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 40
108 Problems of the wave packet treatment Assumptions had to be made on σ p Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 41
109 Problems of the wave packet treatment Assumptions had to be made on σ p Assumptions had to be made on ξ p Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 41
110 Problems of the wave packet treatment Assumptions had to be made on σ p Assumptions had to be made on ξ p We are looking for a formalism, in which these quantities are automatically determined from the properties of the source and the detector. Joachim Kopp (MPI Heidelberg) Mössbauer neutrinos 41
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