# Neutral intermediate boson to neutrinos are flavor preserving

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1. Mar 9, 2015

### physciencer

Hey guys!

How can this sentence be explained and what does it mean?

If the couplings $$Z\nu_{\alpha}\bar{\nu}_{\beta}$$ of the neutral intermediate boson to neutrinos are flavor preserving, they are also diagonal when expressed in terms of mass eigenstates $$Z\nu_{i}\bar{\nu}_{j}.$$

2. Mar 9, 2015

### Simon Bridge

Welcome to PF;
where did you see this sentence?

But it pretty much means what it says. What's the problem.

3. Mar 9, 2015

### physciencer

Last edited by a moderator: May 7, 2017
4. Mar 9, 2015

### Simon Bridge

Note: the context has to be accessible to me. Presumably you don't expect me to buy the book and skim through 300+ pages just to understand your question?
There's a second motivation in getting you to spell out the context - the act of trying to explain a problem to someone else so they understand it often leads to an insight that solves the problem.

Do you not understand some of the words?
What level of education does the explanation need to be pitched at?

5. Mar 9, 2015

### physciencer

I can't find available pdf online. You said that it says what it says, so I assumed that this is clearly written. So if I want to restate my question, why is it that if the couplings of the neutral intermediate boson to neutrinos are flavor preserving, then they are also diagonal when expressed in terms of mass eigenstates?

Edit: Explanation of whatever you feel comfortable at. Graduate level explanation perhaps?

6. Mar 9, 2015

### Simon Bridge

The sentence, by itself, describes an interaction - it does not say that one of the properties leads to the other one, just that the properties are properties of the interaction that will also be described in the resulting lagrangian.

Basically it means that you get the neutrinos out that you put in, and that the interaction provides a basis in which the M (in $\phi^\dagger M \phi$) is diagonal.

If it means anything else, that will be in the context.
You have the text, you can tel me what the chapter is about (i.e. neutrino oscillations) and you can reproduce the paragraph that it appears in just by typing it out ... whatever you think best describes the context that the author has in mind. Otherwise it is impossible to go into detail.

7. Mar 10, 2015

### physciencer

So this was written at the end of the chapter and was part of questions the author thought was intuitive to answer because he didn't refer it to any paragraph or section in the book. In other words, nothing was attached to this question. So the Chapter was Electroweak Interactions of Leptons and I guess the author is thinking that one can answer this according to total understanding of the chapter (perhaps). I will list the subtitles of chapter 6 hoping they would clear what is the chapter handling.
An Effective Lagrangian for the weak interactions
Intermediate vector bosons: A first look
The standard electroweak theory of leptons
neutral-current interactions among leptons
The higgs Boson: A first Look
The Higgs boson, Asymptotic behavior, and the 1 TeV scale
Neutrino Mixing and Neutrino Mass
Renormalizability of the theory

I hope this will help in clearing things up.

8. Mar 10, 2015

### Orodruin

Staff Emeritus
This is a trivial matter of a basis change. If the interaction is diagonal with the same strength for all flavours, then it is proportional to the unit matrix. Changing basis will not affect the unit matrix. If the strengths are different, this is no longer true.

9. Mar 10, 2015

### physciencer

May you please explain this using mathematical terms and how mass eigenstates practically enter? Why is it if they are flavor preserving they are also diagonal when expressed in terms of mass eigenstates? @Orodruin

Edit: Is there a passage in some books that explains what is going on here because as I mentioned the book I am using does not really explain this or where did it pop up from. If you could suggest a read for me to understand this better?

Last edited: Mar 10, 2015
10. Mar 10, 2015

### Orodruin

Staff Emeritus
There really is not much more to understand than the fact that the identity operator is left invariant under a unitary change of basis. This should be in any basic textbook on (complex) linear algebra.

11. Mar 10, 2015

### physciencer

How does this relate to mass eigenstates or mass operator?It is probably this that I can't link? @Orodruin

12. Mar 10, 2015

### ChrisVer

Because you can make a change in your basis and go from flavor eigenstates to mass eigenstates.

13. Mar 10, 2015

### physciencer

Then why is that when flavors are of the same strength then the matrix should be proportional to identity?

14. Mar 10, 2015

### Orodruin

Staff Emeritus
Because the interaction is based on electroweak gauge theory, which is where the coupling strength comes from.

15. Mar 10, 2015

### physciencer

So electroweak theory says if flavors are of the same strength then the representative matrix is identity? If so, why does it say so?

16. Mar 10, 2015

### Orodruin

Staff Emeritus
Basically because all of the neutrinos are in the same type of SU(2) multiplet and the coupling strength comes from the covariant derivative.

17. Mar 10, 2015

### ChrisVer

Also if flavor is conserved, can't you think of the interaction Hamiltonian commutes with the mass matrix? so they both can be simultaneously diagonalized?

18. Mar 10, 2015

### physciencer

Please elaborate, was that a question or an answer? If an answer, why would the interaction Hamiltonian commute with the mass matrix if the flavor is conserved. I can't relate.

19. Mar 11, 2015

### Orodruin

Staff Emeritus
Not necessarily, it would not be the case if flavour was conserved but different flavours had different coupling strengths. If they have the same coupling strength, then yes, everything commutes with the identity operator.

20. Mar 11, 2015

### physciencer

But you were already talking about strength rather than flavors. I am confused now.. What happened? Is it that conserved flavors make the matrix proportional to identity matrix or is it not? :(

21. Mar 11, 2015

### TEFLing

Maybe it would be worthwhile to plod through and define every word...

Neutrinos have mass eigenstates with well defined masses...

But the neutrinos that come out of a Weak Force interaction are NOT in mass eigenstates ... They are in FLAVOR eigenstates ( electron muon tauon ) depending upon the other leptons in said interaction

The flavor eigenstates are superpositions of the mass eigenstates

Note I think the opposite is true as well, if the neutral boson interaction is not diagonal in flavor then it is also not diagonal in mass ?

22. Mar 11, 2015

### Orodruin

Staff Emeritus
No, conserved flavours does not imply proportional to unity, conserved flavour along with equal coupling does. My last post was a response to Chris' question.

23. Mar 11, 2015

### Orodruin

Staff Emeritus
No, this is not true. You could perfectly well have an interaction which is diagonal in one basis but not in another, just not the normal electroweak gauge interaction, which is proportional to the unit matrix in both bases.

24. Mar 11, 2015

### physciencer

But the problem doesn't talk about equal couplings?

25. Mar 11, 2015

### physciencer

The question was: Why is it if the couplings of the neutral intermediate boson to neutrinos are flavor preserving then ....
They did not say any thing about the "same strength" you mentioned in what I quoted. Did you assume that to make the sentence hold?