A Paradox on QFT A Paradox on QFT of Neutrino Mixing of Neutrino Mixing

and Oscillationsand OscillationsYu-Feng Li Qiu-Yu LiuYu-Feng Li Qiu-Yu Liu

( based on Hep-ph/0604069 )( based on Hep-ph/0604069 )

University of Science and University of Science and Technology of ChinaTechnology of China

How to define weak states How to define weak states in QFT ?in QFT ?

Two approaches to define flavor neutrinos:Two approaches to define flavor neutrinos: Defined from quantization of mixing fields: Defined from quantization of mixing fields:

construct the Fock space from creation and construct the Fock space from creation and annihilation operators of flavor neutrinos. But a annihilation operators of flavor neutrinos. But a problem exists for mixing fields.problem exists for mixing fields.

Defined form weak interactions: it is on the basis Defined form weak interactions: it is on the basis

of diagonal form of charge leptons and of diagonal form of charge leptons and corresponding flavor neutrinos.corresponding flavor neutrinos.

Weak states or Weak process states ?

In-equivalent Vacua Model In-equivalent Vacua Model (I)(I)

constructed by M. Blasone & G. Vitielloconstructed by M. Blasone & G. Vitiello The expansions of mixing fields:The expansions of mixing fields:

Bogliubov Transformation between flavor vacuum Bogliubov Transformation between flavor vacuum and mass vacuum:and mass vacuum:

The flavor vacuum and operators are time-The flavor vacuum and operators are time-dependent. So they will violate Lorentz invariance.dependent. So they will violate Lorentz invariance.

In their model, the flavor operators must act on the In their model, the flavor operators must act on the flavor vacuumflavor vacuum

In-equivalent Vacua In-equivalent Vacua Model (II)Model (II)

The relation of flavor operators and those for mass neutrinos:

Oscillation formula for two-generation case

Corrections to Pontecovo’sdescription of neutrino weak

states.

Flavor Changing Effect of Flavor Changing Effect of this modelthis model

We compute We compute anti-commutations of flavor operators for different time:

And so on.And so on. Same flavor operators give right Fermi-Dirac Same flavor operators give right Fermi-Dirac

statistics:statistics:

But for different flavors there are flavor But for different flavors there are flavor changing effects:changing effects:

Calculations of W boson Calculations of W boson decaydecay

Foundations of our calculationsFoundations of our calculations Hamiltonian described by SM:Hamiltonian described by SM:

Neutrino weak states proposed in their Neutrino weak states proposed in their model:model:

Notice that this state is defined at t=0.Notice that this state is defined at t=0.

Negative Energy Neutrinos Negative Energy Neutrinos in the mode of in the mode of

We assume interaction happens at t=0, and give the We assume interaction happens at t=0, and give the final result of this amplitude:final result of this amplitude:

Two terms give special expressions about energy Two terms give special expressions about energy conservation: there will be Negative Energy conservation: there will be Negative Energy Neutrinos!Neutrinos!

Only in the case of entirely degenerate neutrino Only in the case of entirely degenerate neutrino masses, these terms vanish.masses, these terms vanish.

Amplitude of Amplitude of decay modedecay mode

Two reasons to make this mode non-zero:Two reasons to make this mode non-zero: The different poles of delta functions with respect The different poles of delta functions with respect

to the index of i.to the index of i. The different values of The different values of rhorho and and lambdalambda with with

respect to i.respect to i.

Branching Ratio Branching Ratio (I) :(I) :SimplificationSimplification

For real processes, we omit the differences For real processes, we omit the differences of delta functions with respect to i.of delta functions with respect to i.

We expand parameter We expand parameter rhorho and and lambdalambda to to high order: high order:

For on-shell particles, we will omit the For on-shell particles, we will omit the contributions of terms with negative contributions of terms with negative energy neutrinos.energy neutrinos.

Branching Ratio (II) : Branching Ratio (II) : estimated valueestimated value

Then the leading term of the Branching Then the leading term of the Branching Ratio: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 And the order is the same as the

ccorrections of this model to usual Pontecovo’s theory.

Paradox about weak Paradox about weak statesstates

The emergence of the flavor changing current The emergence of the flavor changing current tell us that:tell us that:

It will spoil the tree-level diagonal form of the It will spoil the tree-level diagonal form of the CC interactions.CC interactions.

The identification of neutrino flavor will be The identification of neutrino flavor will be invalid.invalid.

Neutrino weak states defined from mixing field Neutrino weak states defined from mixing field quantization is improper to describe weak quantization is improper to describe weak interactions.interactions.

Discussion (I)Discussion (I)

About this model:About this model: There is another problem about this There is another problem about this

model. (FHY 1999~2001; C. Giunti, 2003)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 It can’t describe weak interaction

properly.properly. We want to get the unified description of We want to get the unified description of

oscillations and interactions in QFT.oscillations and interactions in QFT.

where is it ?where is it ?

Discussion (II)Discussion (II)

To answer:To answer: The first attitude of weak states definition: the The first attitude of weak states definition: the

in-equivalent vacua model has been ruled out.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.

Discussion (III)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.

The endThe end

Thank you !Thank you !