Neutral Pion and Right-Handed Neutrinos

[JK423]

Hello, i'd really need some help with the following questions.

1) Neutral Pion
It's quark content is written as: $$\pi^0=(u\overline{u}-d\overline{d})/\sqrt{2}$$
But the u-quark have a (quite) different mass than the d-quarks. This means that the neutral pion is a superposition of states of different mass??

2) Right-Handed (RH) Neutrinos
First of all, I am talking about SM neutrinos with zero mass! We know that neutrinos interact only via the weak interactions and only with their Left-Handed (LH) component. So, the theory actually predicts that we cannot observe RH neutrinos. Does this mean that they do not exist? I ask this because my proffessor made that assertion, and also because in Quarks&Leptons of Halzen&Martin they say
<<There is no empirical evidence for the existense of RH neutrinos and it could well be that they do not exist in nature.>>
I have an argument via which i make the assertion that RH neutrinos (and LH antineutrinos) MUST exist, even thought we cannot interact with them and see them. It goes like this:
A neutrino satisfies the dirac equation with zero mass. Solving it, we get four solutions.
2 spinors for the neutrino (RH and LH neutrino)
2 spinors for the antineutrino (RH and LH antineutrino).
If we want to have a complete set of solutions, all 4 spinors must hold (that is, exist). We cannot say that i keep the LH neutrino and throw away the RH one. This argument is the same with the one we make about the negative energy solutions. We cannot just simply say that "E<0 solutions are unacceptable" since the E>0 by themselves don't make a complete set of solutions.
(To be honest i haven't really understood why we MUST have a complete set of solution, but the point is that its true)
So i conclude that RH neutrinos MUST exist.


What do you think?

 

[humanino]

1) Neutral Pion
It's quark content is written as: $$\pi^0=(u\overline{u}-d\overline{d})/\sqrt{2}$$
But the u-quark have a (quite) different mass than the d-quarks. This means that the neutral pion is a superposition of states of different mass??

No : it is not a superposition of quark, it is a superposition of quark-antiquark pairs. Each pair has the same mass, the pion mass.

What do you think?

Have you heard of Weyl neutrinos ?

 

[JK423]

No : it is not a superposition of quark, it is a superposition of quark-antiquark pairs. Each pair has the same mass, the pion mass.

But if u and d have different masses, then shouldn't the u,u-bar, d,d-bar pairs have different masses?

Have you heard of Weyl neutrinos ?

No i havent. Looked for it in wikipedia but there is not.
What are they? And what's the flaw in my argument?

 

[humanino]

But if u and d have different masses, then shouldn't the u,u-bar, d,d-bar pairs have different masses?

Well, no : if you are given a u,u-bar pair, it will not be stable : it has enough uncertainty to fluctuate into a d,d-bar pair, and in fact it is a neutral pion (at least in its ground state). Notice that most of the pion mass is not in the quark masses anyway, but in the glue field binding it.

No i havent. Looked for it in wikipedia but there is not.
What are they? And what's the flaw in my argument?

Dirac, Majorana and Weyl fermions
The flaw is in the premise that neutrinos satisfy the Dirac equation. If they did, you are right that you need both the left and the right. But in fact, the left could be the anti-particle of the right (physical interpretation), which gives you the Majorana-Weyl equation (mathematical premise).

The Dirac equation is a tensorial product of two Weyl fermions. The Dirac neutrino belongs to the (1/2,1/2) representation which is (1/2,0)x(0,1/2).

 

[arivero]

The intuition of JK423 is close to reality: these mixings, when formulated in the framework of SU(3) flavour and introducing the values of the masses, just because of the "tunneling" that humanino is describing. It is more noticeable when you put the strange quark in the game. Isospin, alone, can be masked by other effects.

 

[JK423]

Thanks a lot, you were perfectly clear about the neutrinos, i`ll study the (very good) reference you gave me.

But i think that i lost you on the pion's mass question. I asked whether the u,u-bar, d,d-bar pairs have different masses (since u and d quarks are different) and you said no.
How is that possible? By definition u and d are different! I assume that your 'no' has something to do with the fact that u,u-bar is unstable.
Can you elaborate a little more on this please??

Arivero i think that i lost you too..

 

[arivero]

JK, think two separate wells in 1D QM, similar depth so that if there is an infinite barrier between them, you just have two degenerated states of equal energy, particle in the left or particle in the right well. Now enable some mix across the barrier, and what happens is that now you have two eigenstates of different energy, a symmetric one and a higher, asymmetric (in wave function, but not in wave probability, which is the square). This situation is similar to the pion with equal mass for u and d.

But if you put now only slight different masses in u and d, it is to say slight different wells before the opening of the tunnel, the situation still works in a very similar way; the two states mix because of the tunnel, and the "mostly symmetric" and the "mostly antisymmetric" part get a mass difference change from the tunnel effect. Or the mix, if you wish.

In mesons, this is irrelevant for d and u, but becomes important when you add the s quark into play. So what matters is the mixing between dd, uu and ss.