Charge conjugation of Majorana pairs

[Andrea M.]

I was reading this article (http://arxiv.org/abs/1301.0021) about WIMP pair annihilation. At page six the author says that under charge conjugation a state of Majorana particles with orbital angular momentum $L$ and spin angular momentum $S$ take a phase $(-1)^{L+S}$. I understand this for system of particle-antiparticle because under C the particle go into the antiparticle and vice versa so the spatial and spin wave functions give respectively a phase $(-1)^L$ and $(-1)^{S+1}$ on top of this the intrinsic parities give another $(-1)$ factor so we get the total phase factor $(-1)^{L+S}$. But for Majorana two particles system under charge conjugation, being no distinction between particle and antiparticle, I expect than nothing happens under charge conjugation so that the system is CP even independently from the values of $L$ and $S$. Where I'm wrong?

 

[AndyW]

Thank you for bringing up this interesting question. The concept of charge conjugation can be a bit confusing, especially when it comes to Majorana particles. Let me try to clarify it for you.

First of all, let's define what charge conjugation is. Charge conjugation (C) is a symmetry operation that swaps particles with their antiparticles. In other words, it transforms a particle into its antiparticle and vice versa. This operation can be represented mathematically as the complex conjugation of the wave function, i.e. taking the complex conjugate of the spatial and spin parts of the wave function.

Now, let's consider a system of two Majorana particles with orbital angular momentum L and spin angular momentum S. Under charge conjugation, the wave function of this system will be transformed into its complex conjugate, as mentioned earlier. This means that the spatial and spin wave functions will acquire a phase factor of (-1)^(L+S). However, this phase factor is not the only thing that changes under charge conjugation.

In addition to the spatial and spin parts of the wave function, the intrinsic parity of the particles also changes under charge conjugation. Intrinsic parity (P) is a quantum number that describes the transformation properties of a particle under space inversion. In other words, it tells us whether a particle is even or odd under spatial inversion.

Now, for a system of two Majorana particles, the intrinsic parity of each particle is (-1)^(L+S). This means that under charge conjugation, the intrinsic parity of the system will change from (-1)^(L+S) to (-1)^{(-1)^(L+S)} = (-1)^(L+S+1). Combining this with the phase factor of (-1)^(L+S) from the spatial and spin parts of the wave function, we get a total phase factor of (-1)^(L+S+1) = (-1)^(L+S).

In conclusion, under charge conjugation, a system of two Majorana particles will acquire a phase factor of (-1)^(L+S), which is consistent with the statement in the article you mentioned. I hope this helps to clarify your understanding of charge conjugation and its effects on Majorana particles.