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Majorana and Weyl fermions

Last updated Dec 29, 2024

# Majorana mass terms

We found the Dirac mass term Ldirac mass=m(ψLψR+ψRψL),\mathcal{L}_\text{dirac mass}=m(\psi^\dagger_L\psi_R+\psi^\dagger_R\psi_L), to be Lorentz invariant, while terms like ψRψ\psi^\dagger_R\psi are not.

It turns out there is another Lorentz invariant mass term

Majorana mass term

The Majorana mass term for Spin 12\frac{1}{2} particles is LMaj=mψRTσ2ψR.\mathcal{L}_\text{Maj}=m\psi^T_R\sigma_2\psi_R.

Lorentz invariance of Majorana mass term

The transformation of the Spinors ψR\psi_R and ψR\psi^\dagger_R was given as δψR=12(iΘj+βj)σjψR,δψR=12(iΘj+βj)ψRσj\delta\psi_R = \frac{1}{2}(i\Theta_j+\beta_j)\sigma_j\psi_R,\quad \delta\psi^\dagger_R=\frac{1}{2}(-i\Theta_j+\beta_j)\psi^\dagger_R\sigma_j We can now calculate the transformation property of the majorana mass term δ(ψRTσ2ψR)=δ(ψRTσ2)ψR+ψRTσ2δψR\delta(\psi^T_R\sigma_2\psi_R) = \delta(\psi^T_R\sigma_2)\psi_R + \psi^T_R\sigma_2\delta\psi_R using σjTσ2=σ2σj\sigma^T_j\sigma_2 = -\sigma_2\sigma_j to be δ(ψRTσ2ψR)=12(iΘj+βj)ψRTσ2σjψR+12(iΘj+βj)ψRTσ2σjψR=0\delta(\psi^T_R\sigma_2\psi_R)=-\frac{1}{2}(i\Theta_j+\beta_j)\psi^T_R\sigma_2\sigma_j\psi_R + \frac{1}{2}(i\Theta_j+\beta_j)\psi^T_R\sigma_2\sigma_j\psi_R=0

Explicitly plugging in σ2\sigma_2 one can write ψRTσ2ψR=i(ψ1ψ2ψ2ψ1).\psi^T_R\sigma_2\psi_R=-i(\psi_1\psi_2-\psi_2\psi_1).

However this term is only non-trivial if ψ1ψ2ψ2ψ1,\psi_1\psi_2\neq\psi_2\psi_1, meaning the fermion components must not commute!

One finds that the fermion components must anti-commute, meaning they are Grassmann numbers.

We can also write the mass term in terms of a four component spinor ψ=(ψLiσ2ψL)\psi=\begin{pmatrix}\psi_L\\ i\sigma_2\psi^*_L\end{pmatrix} as m2ψˉψ=im2(ψLσ2ψLψLTσ2ψL)\frac{m}{2}\bar{\psi}\psi=i\frac{m}{2}(\psi^\dagger_L\sigma_2\psi^*_L- \psi^T_L\sigma_2\psi_L)

These Majorana spinors are invariant under Charge conjugation

: iγ2ψ=i(0σ2\-σ20)(ψL\iσ2ψL)=(ψL\iσ2ψL)=ψ-i\gamma_2\psi^*=-i\begin{pmatrix}0&\sigma_2\-\sigma_2&0\end{pmatrix}\begin{pmatrix}\psi_L\i\sigma_2\psi^*_L\end{pmatrix}=\begin{pmatrix}\psi_L\i\sigma_2\psi^*_L\end{pmatrix}=\psi Because of this, Majorana fermions cant be charged under any U(1)U(1) gauge or any global symmetry, because ψeiαψ\psi\to e^{i\alpha}\psi but ψceiαψc\psi_c\to e^{-i\alpha}\psi_c which is a contradiction to ψ=ψc\psi=\psi_c for α0\alpha\neq 0.

# Notation for Weyl spinors

One writes ψ=ψL,ψ~=ψR,\psi=\psi_L, \quad \tilde{\psi}=\psi_R, And the indices are included as ψα\psi^\alpha and ψβ˙\psi_{\dot{\beta}} so that a Dirac Spinor can be written as ψ=(ψαψ~β˙)\psi=\begin{pmatrix}\psi^{\alpha}\\ \tilde{\psi}_{\dot{\beta}}\end{pmatrix} Using this notation the Majorana mass term can be written as Lmaj=ψRTσ2ψR=iψαϵαβψβ,\mathcal{L}_\text{maj}=\psi^T_R\sigma_2\psi_R=i\psi_\alpha\epsilon^{\alpha\beta}\psi_\beta, with the ϵ\epsilon tensor ϵαβ=iσ2αβ\epsilon^{\alpha\beta}=-i\sigma^{\alpha\beta}_2.

The ϵ\epsilon tensor can raise and lower indices for Weyl spinors, like gμνg_{\mu\nu} can do for four vectors. ψχ=ψαχα=ψαϵαβχβ=χβϵαβψα=χβϵβαψα=χψ\psi\chi = \psi_\alpha\chi^\alpha = \psi_\alpha\epsilon^{\alpha\beta}\chi_\beta=-\chi_\beta\epsilon^{\alpha\beta}\psi_\alpha=\chi_\beta\epsilon^{\beta\alpha}\psi_\alpha=\chi_\psi \tilde{\psi}\tilde{\chi}=\tilde{\psi}^{\dot{\alpha}}\tilde{\chi}_\dot{\alpha}=-\tilde{\chi}_\dot{\alpha}\tilde{\psi}^{\dot{\alpha}}=\tilde{\chi}^\dot{\alpha}\tilde{\psi}_\dot{\alpha}=\tilde{\chi}\tilde{\psi}