1. The GSI anomaly Alexander Merle Max-Planck-Institute for Nuclear Physics Heidelberg Based on: H. Kienert, J. Kopp, M. Lindner, AM The GSI anomaly 0808.2389 [hep-ph] Neutrino 2008 Conf. Proc. Trento, 18.11.2008

2. Contents: 1.The Observation at GSI 2.The Experiment 3.Problems & Errors 4.Our more formal Treatment 5.One question 6.Conclusions

3. 1. The Observation at GSI: Periodic modula- tion of the expect- ed exponential law in EC-decays of different highly charged ions (Pm-142 & Pr- 140) Litvinov et al: Phys. Lett. B664, 162 (2008)

4. 1. The Observation at GSI: Periodic modula- tion of the expect- ed exponential law in EC-decays of different highly charged ions (Pm-142 & Pr- 140) exponential law Litvinov et al: Phys. Lett. B664, 162 (2008)

5. 1. The Observation at GSI: Periodic modula- tion of the expect- ed exponential law in EC-decays of different highly charged ions (Pm-142 & Pr- 140) exponential law periodic modulation Litvinov et al: Phys. Lett. B664, 162 (2008)

6. 1. The Observation at GSI: Periodic modula- tion of the expect- ed exponential law in EC-decays of different highly charged ions (Pm-142 & Pr- 140) Litvinov et al: Phys. Lett. B664, 162 (2008)

7. 2. The Experiment:

8. See previous talk by Yuri Litvinov!

9. 2. The Experiment: See previous talk by Yuri Litvinov! → I will only give a short summary.

10. 2. The Experiment:

11. Injection of a single type of ions

12. 2. The Experiment: Injection of a single type of ions ⇓ Storage in the Experimental Storage Ring (ESR)

13. 2. The Experiment: Injection of a single type of ions ⇓ Storage in the Experimental Storage Ring (ESR) ⇓ Cooling (stochastic & electron)

14. 2. The Experiment: Injection of a single type of ions ⇓ Storage in the Experimental Storage Ring (ESR) ⇓ Cooling (stochastic & electron) ⇓ Frenquency measurement (by Schottky-Pickups)

15. 2. The Experiment: Injection of a single type of ions ⇓ Storage in the Experimental Storage Ring (ESR) ⇓ Cooling (stochastic & electron) ⇓ Frenquency measurement (by Schottky-Pickups) → due to cooling (Δv/v → 0), the fre- quency only depends on the mass over charge ratio M/Q

16. Lifetime determination:

19. the lifetimes of individual ions are determined

20. Lifetime determination: the lifetimes of individual ions are determined their distribution is plotted

21. Lifetime determination: the lifetimes of individual ions are determined their distribution is plotted the result is NOT only an exponential law…

22. 3. Problems & Errors:

23. Experimental problems & oddities:

24. 3. Problems & Errors: Experimental problems & oddities: low statistics:

25. 3. Problems & Errors: Experimental problems & oddities: low statistics: only 2650 decays of Pr and 2740 of Pm → both fits, with the modified and pure exponential curve, are not so different (e.g. for Pm: χ 2 /D.O.F.=0.91 vs. 1.68)

26. 3. Problems & Errors: Experimental problems & oddities: low statistics: only 2650 decays of Pr and 2740 of Pm → both fits, with the modified and pure exponential curve, are not so different (e.g. for Pm: χ 2 /D.O.F.=0.91 vs. 1.68) unexplained statistical features (pointed out by us):

27. 3. Problems & Errors: Experimental problems & oddities: low statistics: only 2650 decays of Pr and 2740 of Pm → both fits, with the modified and pure exponential curve, are not so different (e.g. for Pm: χ 2 /D.O.F.=0.91 vs. 1.68) unexplained statistical features (pointed out by us): If we take the data and subtract the best-fit function, the res- ulting errors are significantly SMALLER than the statistical error √N for N events.

28. 3. Problems & Errors: Experimental problems & oddities: low statistics: only 2650 decays of Pr and 2740 of Pm → both fits, with the modified and pure exponential curve, are not so different (e.g. for Pm: χ 2 /D.O.F.=0.91 vs. 1.68) unexplained statistical features (pointed out by us): If we take the data and subtract the best-fit function, the res- ulting errors are significantly SMALLER than the statistical error √N for N events. → “Mann-Whitney-Test”: The probability that the remaining fluctuations are random is about 5% (a truly random list would give about 30% or so).

29. 3. Problems & Errors: Experimental problems & oddities: low statistics: only 2650 decays of Pr and 2740 of Pm → both fits, with the modified and pure exponential curve, are not so different (e.g. for Pm: χ 2 /D.O.F.=0.91 vs. 1.68) unexplained statistical features (pointed out by us): If we take the data and subtract the best-fit function, the res- ulting errors are significantly SMALLER than the statistical error √N for N events. → “Mann-Whitney-Test”: The probability that the remaining fluctuations are random is about 5% (a truly random list would give about 30% or so). → the fit function seems to confuse some fluctuations with real data

30. 3. Problems & Errors:

31. Physical errors:

32. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations!

33. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations:

34. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations:

35. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ), then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ), and is then detected as FLAVOUR eigenstate

36. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ), then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ), and is then detected as FLAVOUR eigenstate → more than one way to reach THE SAME final state v e

37. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ), then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ), and is then detected as FLAVOUR eigenstate → more than one way to reach THE SAME final state v e → amplitude is given by a COHERENT SUM:

38. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -neutrino oscillations: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ), then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ), and is then detected as FLAVOUR eigenstate → more than one way to reach THE SAME final state v e → amplitude is given by a COHERENT SUM:

39. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment:

40. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment:

41. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ) and then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei )

42. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ) and then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ) → BUT: there is no second FLAVOUR measurement

43. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ) and then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ) → BUT: there is no second FLAVOUR measurement → amplitude is given by an INCOHERENT SUM:

44. 3. Problems & Errors: Physical errors: The process is NOT analogous to neutrino oscillations! -GSI experiment: the neutrino is produced as FLAVOUR eigenstate (e.g. v e ) and then propagates as superposition of MASS eigenstates (v i with i=1,2,3, and admixtures U ei ) → BUT: there is no second FLAVOUR measurement → amplitude is given by an INCOHERENT SUM:

45. 3. Problems & Errors: Physical errors: This has been done differently in:

46. 3. Problems & Errors: Physical errors: This has been done differently in: - Ivanov, Reda, Kienle: 0801.2121 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: 0804.1311 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: Phys. Rev. Lett. 101, 182501 (2008) - Faber: 0801.3262 [nucl-th] - Lipkin: 0801.1465 [hep-ph] - Lipkin: 0805.0435 [hep-ph] - Walker: Nature 453, 864 (2008)

47. 3. Problems & Errors: Physical errors: This has been done differently in: - Ivanov, Reda, Kienle: 0801.2121 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: 0804.1311 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: Phys. Rev. Lett. 101, 182501 (2008) - Faber: 0801.3262 [nucl-th] - Lipkin: 0801.1465 [hep-ph] - Lipkin: 0805.0435 [hep-ph] - Walker: Nature 453, 864 (2008) Works that agree with us:

48. 3. Problems & Errors: Physical errors: This has been done differently in: - Ivanov, Reda, Kienle: 0801.2121 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: 0804.1311 [nucl-th] - Ivanov, Kryshen, Pitschmann, Kienle: Phys. Rev. Lett. 101, 182501 (2008) - Faber: 0801.3262 [nucl-th] - Lipkin: 0801.1465 [hep-ph] - Lipkin: 0805.0435 [hep-ph] - Walker: Nature 453, 864 (2008) Works that agree with us: - Giunti: 0801.4639 [hep-ph] - Giunti: Phys. Lett. B665, 92 (2008) - Burkhardt et al.: 0804.1099 [hep-ph] - Peshkin: 0804.4891 [hep-ph] - Peshkin: 0811.1765 [hep-ph] - Gal: 0809.1213 [nucl-th] - Cohen, Glashow, Ligeti: 0810.4602 [hep-ph]

49. 3. Problems & Errors: Further points:

50. 3. Problems & Errors: Further points: wrong Δm 2 ~10 -4 eV 2 → neither solar nor atmospheric Δm 2

51. 3. Problems & Errors: Further points: wrong Δm 2 ~10 -4 eV 2 → neither solar nor atmospheric Δm 2 necessary energy splitting ΔE~10 -15 eV → not (yet) explained, coherence over the experiment time doubtful

52. 3. Problems & Errors: Further points: wrong Δm 2 ~10 -4 eV 2 → neither solar nor atmospheric Δm 2 necessary energy splitting ΔE~10 -15 eV → not (yet) explained, coherence over the experiment time doubtful other (but different!) experiments have not found the oscila- tory behavior: Vetter et al.: 0807.0649 [nucl-ex] Faestermann et al.: 0807.3297 [nucl-ex]

53. 3. Problems & Errors: Further points: wrong Δm 2 ~10 -4 eV 2 → neither solar nor atmospheric Δm 2 necessary energy splitting ΔE~10 -15 eV → not (yet) explained, coherence over the experiment time doubtful other (but different!) experiments have not found the oscila- tory behavior: Vetter et al.: 0807.0649 [nucl-ex] Faestermann et al.: 0807.3297 [nucl-ex] wrong statement: v e and v μ are called „mass eigenstates“ by Walker, Nature 453, 864 (2008) → OBVIOUSLY WRONG!!!

54. 4. Our more formal treatment:

55. Several works have tried to relate the GSI-oscillations to neutrino mixing.

56. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate.

57. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate. Our formalism:

58. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate. Our formalism: We describe both, mother (A=M) and daughter (D=M) nuclear state by Gaussian wave packets with central momentum p A0 and spread σ A :

59. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate. Our formalism: We describe both, mother (A=M) and daughter (D=M) nuclear state by Gaussian wave packets with central momentum p A0 and spread σ A :

60. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate. Our formalism: We describe both, mother (A=M) and daughter (D=M) nuclear state by Gaussian wave packets with central momentum p A0 and spread σ A : The neutrino mass eigenstate ν j is described by a plane wave:

61. 4. Our more formal treatment: Several works have tried to relate the GSI-oscillations to neutrino mixing. We have shown, that, even when using wave packets, this is not the case and neutrino mixing is not related to any oscilla- tions in the decay rate. Our formalism: We describe both, mother (A=M) and daughter (D=M) nuclear state by Gaussian wave packets with central momentum p A0 and spread σ A : The neutrino mass eigenstate ν j is described by a plane wave:

62. 4. Our more formal treatment: There is one initial state:

63. 4. Our more formal treatment: There is one initial state:

64. 4. Our more formal treatment: There is one initial state: There are three distinct final states (the different neutrino mass eigenstates v j are orthogonal vectors in Hilbert space) with j=1,2,3:

65. 4. Our more formal treatment: There is one initial state: There are three distinct final states (the different neutrino mass eigenstates v j are orthogonal vectors in Hilbert space) with j=1,2,3:

66. 4. Our more formal treatment: There is one initial state: There are three distinct final states (the different neutrino mass eigenstates v j are orthogonal vectors in Hilbert space) with j=1,2,3: Then, the Feynman rules in coordinate space tell us unambi- guously how to write down the decay amplitude:

67. 4. Our more formal treatment: There is one initial state: There are three distinct final states (the different neutrino mass eigenstates v j are orthogonal vectors in Hilbert space) with j=1,2,3: Then, the Feynman rules in coordinate space tell us unambi- guously how to write down the decay amplitude:

68. 4. Our more formal treatment: We adopt the following approximations:

69. 4. Our more formal treatment: We adopt the following approximations: - we expand E M =(p M 2 +m M 2 ) 1/2 to first order in (p M -p M0 ) → this approximation neglects the wave packet spreading

70. 4. Our more formal treatment: We adopt the following approximations: - we expand E M =(p M 2 +m M 2 ) 1/2 to first order in (p M -p M0 ) → this approximation neglects the wave packet spreading - we neglect the energy dependence of the pre-factors for mother and daughter (1/√E A → 1/√E 0A ) → this is okay, because these factors varies much more slowly than the Gaussian exponentials

71. 4. Our more formal treatment: We adopt the following approximations: - we expand E M =(p M 2 +m M 2 ) 1/2 to first order in (p M -p M0 ) → this approximation neglects the wave packet spreading - we neglect the energy dependence of the pre-factors for mother and daughter (1/√E A → 1/√E 0A ) → this is okay, because these factors varies much more slowly than the Gaussian exponentials - we also neglect the energy dependence of the matrix element (also because of slow variation)

72. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet):

73. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet):

74. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet): the result is:

75. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet): the result is:

76. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet): the result is: the same can be done for the daughter and one finally gets, after solving the time-integrals, too, an easy solution:

77. 4. Our more formal treatment: one then has to evaluate Gaussian integrals like the following (with the group velocity v 0M =p 0M /E 0M of the wave packet): the result is: the same can be done for the daughter and one finally gets, after solving the time-integrals, too, an easy solution:

78. 4. Our more formal treatment: here, we have used some abbreviations:

79. 4. Our more formal treatment: here, we have used some abbreviations:

80. 4. Our more formal treatment: but let‘s go back to the point of the result:

81. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely:

82. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely:

83. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely:

84. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely: dependences on the neutrino mass eigenstates j=1,2,3

85. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely: dependences on the neutrino mass eigenstates j=1,2,3 → will be summed incoherently (because the three mass eigenstates v 1, v 2, and v 3 are distinct!):

86. 4. Our more formal treatment: but let‘s go back to the point of the result: and look more closely: dependences on the neutrino mass eigenstates j=1,2,3 → will be summed incoherently (because the three mass eigenstates v 1, v 2, and v 3 are distinct!):

87. 4. Our more formal treatment: of course, the phases cancel out due to the absolute value:

88. 4. Our more formal treatment: of course, the phases cancel out due to the absolute value:

89. 4. Our more formal treatment: of course, the phases cancel out due to the absolute value:

90. 4. Our more formal treatment: of course, the phases cancel out due to the absolute value: This seems to be easy, but has inspite of that caused a lot of confusion in the community…

91. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies

92. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies

93. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies then, also the phases Φ get a dependence on n:

94. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies then, also the phases Φ get a dependence on n:

95. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies then, also the phases Φ get a dependence on n: then, the absolute squares show indeed oscillatory behavior:

96. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies then, also the phases Φ get a dependence on n: then, the absolute squares show indeed oscillatory behavior:

97. 4. Our more formal treatment: the only possibility for oscillations: if the initial state is a superposition of several states n of different energies then, also the phases Φ get a dependence on n: then, the absolute squares show indeed oscillatory behavior:

98. 4. Our more formal treatment: HOWEVER:

99. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations:

100. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations:

101. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations: this would require an energy splitting of:

102. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations: this would require an energy splitting of:

103. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations: this would require an energy splitting of: ⇓

104. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations: this would require an energy splitting of: ⇓ → no know mechanism that could produce such a tiny splitting

105. 4. Our more formal treatment: HOWEVER: duration of the GSI-oscillations: this would require an energy splitting of: ⇓ → no know mechanism that could produce such a tiny splitting → no reason for production of a superposition of such states

106. 4. Our more formal treatment: FURTHERMORE:

107. 4. Our more formal treatment: FURTHERMORE: it was objected in 0811.0922 [nucl-th] (Faber et al.) and in the talk by Andrei Ivanov at the EXA08-Meeting, Vienna, Sept- ember 2008 that this level splitting would also lead to slow oscillations in β + -decays

108. 4. Our more formal treatment: FURTHERMORE: it was objected in 0811.0922 [nucl-th] (Faber et al.) and in the talk by Andrei Ivanov at the EXA08-Meeting, Vienna, Sept- ember 2008 that this level splitting would also lead to slow oscillations in β + -decays this does not happen in the β + -decays of the same ions as used for the EC-measurements (Faber et al.)

109. 4. Our more formal treatment: FURTHERMORE: it was objected in 0811.0922 [nucl-th] (Faber et al.) and in the talk by Andrei Ivanov at the EXA08-Meeting, Vienna, Sept- ember 2008 that this level splitting would also lead to slow oscillations in β + -decays this does not happen in the β + -decays of the same ions as used for the EC-measurements (Faber et al.) we were not aware of this data when we wrote our paper

110. 4. Our more formal treatment: FURTHERMORE: it was objected in 0811.0922 [nucl-th] (Faber et al.) and in the talk by Andrei Ivanov at the EXA08-Meeting, Vienna, Sept- ember 2008 that this level splitting would also lead to slow oscillations in β + -decays this does not happen in the β + -decays of the same ions as used for the EC-measurements (Faber et al.) we were not aware of this data when we wrote our paper BUT: we also did not claim to be able to explain the GSI- oscillations

111. 4. Our more formal treatment: FURTHERMORE: it was objected in 0811.0922 [nucl-th] (Faber et al.) and in the talk by Andrei Ivanov at the EXA08-Meeting, Vienna, Sept- ember 2008 that this level splitting would also lead to slow oscillations in β + -decays this does not happen in the β + -decays of the same ions as used for the EC-measurements (Faber et al.) we were not aware of this data when we wrote our paper BUT: we also did not claim to be able to explain the GSI- oscillations at the moment, we have no objection against the above argument

112. 5. One question:

113. Let us assume for a moment that the COHERENT summation is correct.

114. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment?

115. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment? tritium beta decay: 3 H → 3 He + e - + v e ˉ

116. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment? tritium beta decay: 3 H → 3 He + e - + v e the energy spectrum of the electron is given by (Farzan & Smirnov, Phys. Lett. B557, 224 (2003)): ˉ

117. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment? tritium beta decay: 3 H → 3 He + e - + v e the energy spectrum of the electron is given by (Farzan & Smirnov, Phys. Lett. B557, 224 (2003)): ˉ

118. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment? tritium beta decay: 3 H → 3 He + e - + v e the energy spectrum of the electron is given by (Farzan & Smirnov, Phys. Lett. B557, 224 (2003)): → this is an INCOHERENT sum over the contributions from the different mass eigenstates (Vissani, Nucl. Phys. Proc. Suppl.100, 273 (2001)): ˉ

119. 5. One question: Let us assume for a moment that the COHERENT summation is correct. → What about the effective mass in the KATRIN-experiment? tritium beta decay: 3 H → 3 He + e - + v e the energy spectrum of the electron is given by (Farzan & Smirnov, Phys. Lett. B557, 224 (2003)): → this is an INCOHERENT sum over the contributions from the different mass eigenstates (Vissani, Nucl. Phys. Proc. Suppl.100, 273 (2001)): ˉ

120. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is:

121. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is: → this is the expression mostly used

122. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is: → this is the expression mostly used my questions:

123. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is: → this is the expression mostly used my questions: Should the definition of the „effective electron neutrino mass“ then be modified???

124. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is: → this is the expression mostly used my questions: Should the definition of the „effective electron neutrino mass“ then be modified??? Would the planned KATRIN-analysis be in- correct???

125. 5. One question: for (E 0 -E)>>m j, this can be parametrized by a single para- meter, the „effective mass“ of the electron-neutrino, which is: → this is the expression mostly used my questions: Should the definition of the „effective electron neutrino mass“ then be modified??? Would the planned KATRIN-analysis be in- correct??? What about MAINZ & TROITSK???

126. 5. One question: I don‘t think so!!!

127. 6. Conclusions:

128. the oscillations at GSI are NOT YET EXPLAINED

129. 6. Conclusions: the oscillations at GSI are NOT YET EXPLAINED they are definitely NOT related to neutrino mixing

130. 6. Conclusions: the oscillations at GSI are NOT YET EXPLAINED they are definitely NOT related to neutrino mixing of course, people (including us) had a careful look at all sorts of systematics

131. 6. Conclusions: the oscillations at GSI are NOT YET EXPLAINED they are definitely NOT related to neutrino mixing of course, people (including us) had a careful look at all sorts of systematics HOWEVER: there are some unexplained strange statistical properties of the data

132. 6. Conclusions: the oscillations at GSI are NOT YET EXPLAINED they are definitely NOT related to neutrino mixing of course, people (including us) had a careful look at all sorts of systematics HOWEVER: there are some unexplained strange statistical properties of the data that all has caused some confusion in the community

133. 6. Conclusions: the oscillations at GSI are NOT YET EXPLAINED they are definitely NOT related to neutrino mixing of course, people (including us) had a careful look at all sorts of systematics HOWEVER: there are some unexplained strange statistical properties of the data that all has caused some confusion in the community the new run using I-122 will hopefully clarify some issues

134. THANKS TO MY COLLABORATORS!!!!

135. THANKS TO MY COLLABORATORS!!!! … AND, OF COURSE, TO YOU ALL FOR YOUR ATTENTION!

136. References: "The GSI-Anomaly": Talk by Manfred Lindner, Neutrino 2008 Conference, Christchurch/New Zealand, 30th May 2008 & Proceedings "Observation of Non-Exponential Orbital Electron Capture Decays of Hydrogen-Like $^{140}$Pr and $^{142}$Pm Ions": Yu.A. Litvinov et al.; Phys.Lett.B664:162-168,2008; e-Print: arXiv:0801.2079 [nucl-ex] "Observation of non-exponential two-body beta decays of highly-charged, stored ions": Talks by Fritz Bosch & Yuri Litvinov, Transregio 27 "Neutrinos and Beyond"-Meeting, Heidelberg, 30th January 2008; Milos, 21st May 2008