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I have a few questions about Majorana fermions.

  1. What is Majorana mass? Does it have a different value compared to the mass in the Dirac equation for an arbitrary fermion? How exactly do they differ?

  2. Can the Majorana equation be rewritten in form two-component spinor equation? Or it is two-component already?

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Please clarify: What do you mean by "different value" for the mass? Do you mean observed masses in nature? Furthermore, what do you mean by "two-component at once"? –  Frederic Brünner Nov 20 '13 at 21:11
    
@FredericBrünner : "Do you mean observed masses in nature?", - yes, I do. "...Furthermore, what do you mean by "two-component at once"? ..", - there are two forms of Majorana equation: for real and complex spinor. The second refer to two-component spinor, the first refers for four-component spinor. But generally Majorana fermion has only two components, doesn't it? –  Andrew McAddams Nov 20 '13 at 21:40
    
You might want to break this up into two questions. –  Dan Nov 21 '13 at 23:04

1 Answer 1

up vote 10 down vote accepted

We know that we can describe a spin $1/2$ massless particle using only a single Weyl field (lets say left-handed $\psi_{L}$). To introduce a mass term we have to use two spinor fields (one left-handed and one right-handed) and this gives the Dirac mass term. The question is now that if we can describe a massive particle with a single Weyl field. Well yes, due to the fact that given a left-handed Weyl spinor, it is possible to construct a right-handed spinor $\psi_{R}=i\sigma^{2}\psi_{L}^{*}$. Thus, we can write the Dirac equation using $i\sigma^{2}\psi_{L}^{*}$

$$\hspace{43mm} \bar{\sigma}^{\mu}i\partial_{\mu}\psi_{L}=im\sigma^{2}\psi_{L}^{*} \hspace{30mm}(1)$$

The known algebraic methods performed for the Dirac equation to prove that it implies a massive Klein-Gordon equation can be performed here without any problems. Thus, the above equation implies $(\Box+m^{2})\psi_{L}=0$. Here we have constructed a mass term using only $\psi_{L}$ and this is known as Majorana mass. The similarity with the Dirac mass can be seen by writting $(1)$ in terms of the four component Majorana spinor $\psi_{m}$ in the chiral representation

$$\psi_{m}=\begin{pmatrix} \psi_{L}\\i\sigma^{2}\psi_{L}^{*} \end{pmatrix}$$

Now, equation $(1)$ becomes

$$(i\gamma_{\mu}\partial^{\mu}-m)\psi_{m}=0$$

The Majorana mass has a very important physical difference when compared to the Dirac mass. We know that the Dirac action with a mass term is invariant under a global $U(1)$ transformations of $\psi_{L}$ and $\psi_{R}$ (i.e. $\psi_{L}\rightarrow e^{i\alpha}\psi_{L},\hspace{2mm}\psi_{R}\rightarrow e^{i\alpha}\psi_{R}$).But for Majorana spinors, $\psi_{L}$ and $\psi_{R}$ are not independent, they are related by complex conjugation. So, if $\psi_{L}$ transforms as $\psi_{L}\rightarrow e^{i\alpha}\psi_{L}$then $\psi_{R}$ transforms like $\psi_{R}\rightarrow e^{-i\alpha}\psi_{R}$. The Majorana equation $(1)$ is not invariant under global $U(1)$ symmetries. This fact implies that a spin $1/2$ particle with a $U(1)$ conserved charge cannot have a Majorana mass. All spin $1/2$ particles with an electric charge cannot have a Majorana mass. Also leptons that have a Majorana mass violate the lepton number (because this is a $U(1)$ symmetry).

One possible particle that could have a Majorana mass is the neutrino. But this is yet to be determined. (I didn't answer your questions point by point but I hope this clarifies some of them).

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Thank you for your answer. It is useful for me. Can you also answer what is relations between expressions for Dirac and Majorana masses (if they may be compared with each other)? By the other words, does exist relation like $$ m_{Dirac} = m_{Majorana} - A, \quad A = const? $$ –  Andrew McAddams Nov 22 '13 at 15:19
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The short answer is no. A relation like that does not exist. But, there are some "connections" for the masses in the See-Saw mechanism. Not knowing much about this mechanism, here are some good (I hope) articles :ias.ac.in/pramana/v72/p217/fulltext.pdf kvi.nl/~loehner/saf_seminar/2010/NeutrinoMassAndNewPhysics.pdf –  nijankowski Nov 22 '13 at 16:07
    
I return to your great answer again and want to ask: how to associate lepton number conservation with the $U(1)$-symmetry? This is not a gauge symmetry, so it must be something global. But how to build leptonic current (or, by the other words, charge conservation law) in this case? –  Andrew McAddams Feb 6 at 19:04

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