1. A Paradox on QFT of Neutrino Mixing and Oscillations Yu-Feng Li Qiu-Yu Liu ( based on Hep-ph/0604069 ) University of Science and Technology of China

2. How to define weak states in QFT ? Two approaches to define flavor neutrinos: Defined from quantization of mixing fields: construct the Fock space from creation and annihilation operators of flavor neutrinos. But a problem exists for mixing fields. Defined from quantization of mixing fields: construct the Fock space from creation and annihilation operators of flavor neutrinos. But a problem exists for mixing fields. Defined form weak interactions: it is on the basis of diagonal form of charge leptons and corresponding flavor neutrinos. Defined form weak interactions: it is on the basis of diagonal form of charge leptons and corresponding flavor neutrinos. Weak states or Weak process states ?

3. In-equivalent Vacua Model (I) constructed by M. Blasone & G. Vitiello constructed by M. Blasone & G. Vitiello The expansions of mixing fields: The expansions of mixing fields: Bogliubov Transformation between flavor vacuum and mass vacuum: Bogliubov Transformation between flavor vacuum and mass vacuum: The flavor vacuum and operators are time-dependent. So they will violate Lorentz invariance. The flavor vacuum and operators are time-dependent. So they will violate Lorentz invariance. In their model, the flavor operators must act on the flavor vacuum In their model, the flavor operators must act on the flavor vacuum

4. In-equivalent Vacua Model (II) The relation of flavor operators and those for mass neutrinos: Oscillation formula for two-generation case Corrections to Pontecovo ’ s description of neutrino weak states.

5. Flavor Changing Effect of this model We compute We compute anti-commutations of flavor operators for different time: And so on. Same flavor operators give right Fermi-Dirac statistics: Same flavor operators give right Fermi-Dirac statistics: But for different flavors there are flavor changing effects: But for different flavors there are flavor changing effects:

6. Calculations of W boson decay Foundations of our calculations Hamiltonian described by SM: Hamiltonian described by SM: Neutrino weak states proposed in their model: Neutrino weak states proposed in their model: Notice that this state is defined at t=0.

7. Negative Energy Neutrinos in the mode of We assume interaction happens at t=0, and give the final result of this amplitude: We assume interaction happens at t=0, and give the final result of this amplitude: Two terms give special expressions about energy conservation: there will be Negative Energy Neutrinos! Two terms give special expressions about energy conservation: there will be Negative Energy Neutrinos! Only in the case of entirely degenerate neutrino masses, these terms vanish. Only in the case of entirely degenerate neutrino masses, these terms vanish.

8. Amplitude of decay mode Two reasons to make this mode non-zero: The different poles of delta functions with respect to the index of i. The different poles of delta functions with respect to the index of i. The different values of rho and lambda with respect to i. The different values of rho and lambda with respect to i.

9. Branching Ratio (I) : Simplification For real processes, we omit the differences of delta functions with respect to i. For real processes, we omit the differences of delta functions with respect to i. We expand parameter rho and lambda to high order: We expand parameter rho and lambda to high order: For on-shell particles, we will omit the contributions of terms with negative energy neutrinos. For on-shell particles, we will omit the contributions of terms with negative energy neutrinos.

10. Branching Ratio (II) : estimated value Then the leading term of the Branching Ratio: Then the leading term of the Branching Ratio: Then the ratio is at the order of Then the ratio is at the order of And the order is the same as the c And the order is the same as the corrections of this model to usual Pontecovo ’ s theory.

11. Paradox about weak states The emergence of the flavor changing current tell us that: It will spoil the tree-level diagonal form of the CC interactions. It will spoil the tree-level diagonal form of the CC interactions. The identification of neutrino flavor will be invalid. The identification of neutrino flavor will be invalid. Neutrino weak states defined from mixing field quantization is improper to describe weak interactions. Neutrino weak states defined from mixing field quantization is improper to describe weak interactions.

12. Discussion (I) About this model: There is another problem about this model. (FHY 1999~2001; C. Giunti, 2003) There is another problem about this model. (FHY 1999~2001; C. Giunti, 2003) It is a description of vacuum oscillation. It is a description of vacuum oscillation. It can ’ t describe weak interaction properly. It can ’ t describe weak interaction properly. We want to get the unified description of oscillations and interactions in QFT. We want to get the unified description of oscillations and interactions in QFT. where is it ?

13. Discussion (II) To answer: The first attitude of weak states definition: the in- equivalent vacua model has been ruled out. The first attitude of weak states definition: the in- equivalent vacua model has been ruled out. The other definition of weak states: The other definition of weak states: weak process states can also describe neutrino oscillations, such as the external wave-packet model (C.G, C.W.K. et al.) and so on. Coherence length, localization terms and so on.

14. Discussion (III) About About weak process state: It is not an universal weak state but a process- dependent state. Only in relativistic limit, we can omit the process contribution, and an universal weak state emerges: the usual Pontecovo ’ s description of neutrino weak state.

15. The end Thank you !