How Does the Majorana Neutrino Behave in Matter?

[TroyElliott]

For a Majorana neutrino in matter we have the equation $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})\nu_{L} = m\overline{\nu_{L}}.$$ A is to be considered constant.

Squaring, in the ultra-relativistic limit one obtains the dispersion relation

$$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ i.e.

$$p \simeq E -(\frac{mm^{\dagger}}{2E}+A).$$

What I have is $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})(-i(\gamma^{\mu}\partial_{\mu})^{\dagger}-A(\gamma_{0})^{\dagger})$$ and I know $$\gamma_{0}^{\dagger} = \gamma_{0}$$ and $$(\gamma^{\mu})^{\dagger} = \gamma^{0}\gamma^{\mu}\gamma^{0}.$$

But I am not seeing how to get $$(E-A)^{2}-p^{2} \simeq mm^{\dagger}$$ from this. Any hints would be greatly appreciated!

 

[Orodruin]

For a Majorana neutrino in matter we have the equation $$(i\gamma^{\mu}\partial_{\mu}-A\gamma_{0})\nu_{L} = m\overline{\nu_{L}}.$$ A is to be considered constant.

No, that does not make sense. You are essentially saying column vector = row vector. Please give a reference to where you are looking this up.

 

[TroyElliott]

No, that does not make sense. You are essentially saying column vector = row vector. Please give a reference to where you are looking this up.

I uploaded a screen shot. The pdf is called "Neutrino masses and mixings and..." by Alessandro Strumia and Francesco Vissani. The exact page that screen shot comes from is page 31.

 

[Vanadium 50]

m is a matrix. That turns a column vector into a row vector.

 

[Orodruin]

m is a matrix. That turns a column vector into a row vector.

This is not how matrix multiplication works. Square matrix x column vector = new column vector.

 

[Vanadium 50]

You're right. I am not sure what I was thinking.

 

[Orodruin]

I uploaded a screen shot. The pdf is called "Neutrino masses and mixings and..." by Alessandro Strumia and Francesco Vissani. The exact page that screen shot comes from is page 31.

So, the proper way of referring to a paper is giving the link, not providing a screenshot of a single page. This paper is found on the arXiv: https://arxiv.org/pdf/hep-ph/0606054.pdf
Generally, it seems to me that Strumia and Vissani are being rather sloppy (or non-standard) with their notation. They are certainly people I know know better than that.

 

[George Jones]

So, the proper way of referring to a paper is giving the link, not providing a screenshot of a single page. This paper is found on the arXiv: https://arxiv.org/pdf/hep-ph/0606054.pdf

I prefer

https://arxiv.org/abs/hep-ph/0606054

It is easy to click on the pdf, and I (and the Mentors) can see at a a glance if the paper has been submitted/accepted to/by a a particular journal, or if the paper is a report, set of summer school lectures, etc.

 

[George Jones]

So, the proper way of referring to a paper is giving the link, not providing a screenshot of a single page. This paper is found on the arXiv: https://arxiv.org/pdf/hep-ph/0606054.pdf

I prefer

https://arxiv.org/abs/hep-ph/0606054

It is easy to click on the pdf, and I (and the Mentors) can see at a a glance if the paper has been submitted/accepted to/by a a particular journal, or if the paper is a report, set of summer school lectures, etc.

 

[vanhees71]

Well, the equation (why the authors do not use equation numbers is an enigma to me) at the very beginning of 3.4 (a) is correct, if the bar simply means conjugate complex. Maybe one has to read the notation in this way. On the other hand it seems as if the bar is the standard notation for Dirac fermions (i.e., bi-spinors of the $(1/2,0) \oplus (0,1/2)$ representation), i.e., $\bar{\psi}=\psi^{\dagger} \gamma^0$. It's at least bad notation...