Abstract
The parity transformation law of the fermion field $\psi(x)$ is usually defined by the "$\gamma^{0}$-parity" $\psi^{p}(t,-\vec{x}) = \gamma^{0}\psi(t,-\vec{x})$ with eigenvalues $\pm 1$, while the "$i\gamma^{0}$-parity" $\psi^{p}(t,-\vec{x})=i\gamma^{0}\psi(t,-\vec{x})$ is required for the Majorana fermion. The compatibility issues of these two parity laws arise in generic fermion number violating theories where a general class of Majorana fermions appear. In the case of Majorana neutrinos constructed from chiral neutrinos in an extension of the Standard Model, the Majorana neutrinos can be characterized by CP symmetry although C and P are separately broken. It is then shown that either choice of the parity operation, $\gamma^{0}$ or $i\gamma^{0}$, in the level of the starting fermions gives rise to the consistent and physically equivalent descriptions of emergent Majorana neutrinos both for Weinberg's model of neutrinos and for a general class of seesaw models. The mechanism of this equivalence is that the Majorana neutrino constructed from a chiral neutrino, which satisfies the classical Majorana condition $\psi(x)=C\overline{\psi(x)}^{T}$, allows the phase freedom $\psi(x)=e^{i\alpha}\nu_{L}(x) + e^{-i\alpha}C\overline{\nu_{L}(x)}^{T}$ with $\alpha=0\ {\rm or}\ \pi/4$ that accounts for the phase coming from the different definitions of parity for $\nu_{L}(x)$ and ensures the consistent definitions of CP symmetry $({\cal CP})\psi(x)({\cal CP})^{\dagger}= \pm i\gamma^{0}\psi(t,-\vec{x})$.
Highlights
The parity symmetry, which is a disconnected component of the Lorentz transformation, is defined for the Dirac fermion with real m ZS 1⁄4 d4xψðxÞ1⁄2iγμ∂μ − mψðxÞ ð1Þ by the substitution rule [1]ψðxÞ → γ0ψðt; −x⃗ Þ 1⁄4 ψpðt; −x⃗ Þ; ψpðt; −x⃗ Þ → ψðxÞ; ð2Þ which is the symmetry of (1)
It should be noted that our use of the CP symmetry for the Majorana fermion, which is constructed from chiral fermions, is very different from the proposal of the use of CP or CPT symmetries in the definition of general Majorana neutrinos by taking into account of the possible symmetry breaking by weak interactions, as was discussed, for example, in [8]
We have demonstrated the physical equivalence of γ0 parity and iγ0 parity in the description of emergent Majorana neutrinos formed from chiral fermions using Weinberg’s model of neutrinos in an extension of the Standard Model
Summary
The parity symmetry, which is a disconnected component of the Lorentz transformation, is defined for the Dirac fermion with real m. In the generic representation of the Dirac matrices [1], the “iγ0 parity” satisfies the condition iγ0ψðt; −x⃗ Þ 1⁄4 Ciγ0ψðt; −x⃗ ÞT ð9Þ for the field which satisfies the classical Majorana condition ψðxÞ 1⁄4 CψðxÞT; ð10Þ and iγ0 parity is a natural choice of the parity for the Majorana fermion in this generic representation . The parity ψpðt; −x⃗ Þ 1⁄4 iγ0ψðt; −x⃗ Þ of a fermion and the parity of an antifermion ψcpðt; −x⃗ Þ 1⁄4 iγ0ψcðt; −x⃗ Þ have symmetric forms, which is natural since we do not distinguish the particle and its antiparticle in the case of the Majorana fermion These two choices of parity, γ0 and iγ0, are equivalent for the Dirac fermion with the fermion number Uð1Þ freedom since iγ0 is regarded as a composition of γ0 and Uð1Þ transformations, but their physical equivalence in generic theories with fermion number nonconservation is not obvious. It should be noted that our use of the CP symmetry for the Majorana fermion, which is constructed from chiral fermions, is very different from the proposal of the use of CP or CPT (which is defined as a product of CP and the time reversal T) symmetries in the definition of general Majorana neutrinos by taking into account of the possible symmetry breaking by weak interactions, as was discussed, for example, in [8]
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