Majorana neutrino in matter

In summary: 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.
  • #1
TroyElliott
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3
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!
 
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  • #2
TroyElliott said:
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.
 
  • #3
Orodruin said:
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.
 

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  • #4
m is a matrix. That turns a column vector into a row vector.
 
  • #5
Vanadium 50 said:
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.
 
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  • #6
You're right. I am not sure what I was thinking.
 
  • #7
TroyElliott said:
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.
 
  • #8
Orodruin said:
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.
 
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  • #9
Orodruin said:
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.
 
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Likes Vanadium 50
  • #10
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...
 

1. What is a Majorana neutrino?

A Majorana neutrino is a hypothetical type of neutrino that is its own antiparticle. This means that it has the same mass and other properties as its antiparticle, and they can potentially annihilate each other.

2. How is a Majorana neutrino different from a Dirac neutrino?

A Dirac neutrino is a type of neutrino that has a distinct antiparticle, while a Majorana neutrino is its own antiparticle. This means that a Majorana neutrino has only half the number of degrees of freedom as a Dirac neutrino.

3. Can Majorana neutrinos exist in matter?

Yes, Majorana neutrinos can exist in matter. In fact, some theories suggest that the neutrino may be a Majorana particle, and experiments are currently underway to try to detect this phenomenon.

4. What is the significance of Majorana neutrinos in the study of particle physics?

The existence of Majorana neutrinos would have significant implications for our understanding of the fundamental particles and interactions in the universe. It could also help explain the matter-antimatter asymmetry problem and provide insight into the nature of dark matter.

5. How are scientists attempting to detect Majorana neutrinos in matter?

Scientists are using a variety of methods, such as neutrinoless double beta decay experiments and searches for neutrino oscillations, to try to detect Majorana neutrinos in matter. These experiments involve observing rare interactions between neutrinos and other particles, which could provide evidence for the existence of Majorana neutrinos.

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