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Register a new userGeneralized Pauli-Gursey transformation and Majorana neutrinos
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Kazuo Fujikawa
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We discuss a generalization of the Pauli-Gursey transformation, which is motivated by the Autonne-Takagi factorization, to an arbitrary $n$ number of generations of neutrinos using $U(2n)$ that defines general canonical transformations and diagonalizes symmetric complex Majorana mass matrices in special cases. The Pauli-Gursey transformation mixes particles and antiparticles and thus changes the definition of the vacuum and C. We define C, P and CP symmetries at each Pauli frame specified by a generalized Pauli-Gursey transformation. The Majorana neutrinos in the C and P violating seesaw model are then naturally defined by a suitable choice of the Pauli frame, where only Dirac-type fermions appear with well-defined C, P and CP, and thus the C symmetry for Majorana neutrinos agrees with the C symmetry for Dirac-type fermions. This fully symmetric setting corresponds to the idea of Majorana neutrinos as Bogoliubov quasi-particles. In contrast, the conventional direct construction of Majorana neutrinos in the seesaw model, where CP is well-defined but C and P are violated, encounters the mismatch of C symmetry for Majorana neutrinos and C symmetry for chiral fermions; this mismatch is recognized as the inevitable appearance of the singlet (trivial) representation of C symmetry for chiral fermions.
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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 $igamma^{0}$-parity $psi^{p}(t,-vec{x})=igamma^{0}psi(t,-vec{x})$ 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 $igamma^{0}$, in the level of the starting fermions gives rise to the consistent and physically equivalent descriptions of emergent Majorana neutrinos both for Weinbergs 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 $psi(x)=Coverline{psi(x)}^{T}$, allows the phase freedom $psi(x)=e^{ialpha} u_{L}(x) + e^{-ialpha}Coverline{ u_{L}(x)}^{T}$ with $alpha=0 {rm or} pi/4$ that accounts for the phase coming from the different definitions of parity for $ u_{L}(x)$ and ensures the consistent definitions of CP symmetry $({cal CP})psi(x)({cal CP})^{dagger}= pm igamma^{0}psi(t,-vec{x})$.
The Majorana neutrino $psi_{M}(x)$ when constructed as a superposition of chiral fermions such as $ u_{L} + Coverline{ u_{L}}^{T}$ is characterized by $ ({cal C}{cal P}) psi_{M}(x)({cal C}{cal P})^{dagger} =igamma^{0}psi_{M}(t,-vec{x})$, and the CP symmetry describes the entire physics contents of Majorana neutrinos. Further specifications of C and P separately could lead to difficulties depending on the choice of C and P. The conventional $ {cal C} psi_{M}(x) {cal C}^{dagger} = psi_{M}(x)$ with well-defined P is naturally defined when one constructs the Majorana neutrino from the Dirac-type fermion. In the seesaw model of Type I or Type I+II where the same number of left- and right-handed chiral fermions appear, it is possible to use the generalized Pauli-Gursey transformation to rewrite the seesaw Lagrangian in terms of Dirac-type fermions only; the conventional C symmetry then works to define Majorana neutrinos. In contrast, the pseudo C-symmetry $ u_{L,R}(x)rightarrow Coverline{ u_{L,R}(x)}^{T}$ (and associated pseudo P-symmetry), that has been often used in both the seesaw model and Weinbergs model to describe Majorana neutrinos, attempts to assign a nontrivial charge conjugation transformation rule to each chiral fermion separately. But this common construction is known to be operatorially ill-defined and, for example, the amplitude of the neutrinoless double beta decay vanishes if the vacuum is assumed to be invariant under the pseudo C-symmetry.
The texture zero mass matrices for the leptons and the seesaw mechanism are used to derive relations between the matrix elements of the lepton mixing matrix and the ratios of the neutrino masses.
We propose a new mechanism producing a non-vanishing lepton number asymmetry, based on decays of heavy Majorana neutrinos. If they are produced out of equilibrium, as occurs in preheating scenario, and are superpositions of mass eigenstates rapidly decaying, their decay rates contains interference terms provided the mass differences $Delta m$ are small compared to widths $Gamma$. The resulting lepton asymmetry, which is the analogue of the time-integrated CP asymmetry in $B^0-bar{B}^0$ system, is found to be proportional to $Delta m/Gamma$.
In this paper we reply to the comment presented in [1]. In that work the author raises several points about the geometric phase for neutrinos discussed in [2]. He affirms that the calculation is flawed due to incorrect application of the definition of noncyclic geometric phase and the omission of one term in Wolfenstein effective Hamiltonian. He claims that the results are neither gauge invariant nor lepton field rephasing invariant and presents an alternative calculation, solely in order to demonstrate that the Majorana CP-violating phase enters the geometric phase essentially by lepton field rephasing transformation. Finally he claims that the nontrivial dependence of geometric phase on Majorana CP-violating phase presented in [2] is unphysical and thus unmeasurable. We discuss each of the points raised in [1] and show that they are incorrect. In particular, we introduce geometric invariants which are gauge and reparametrization invariants and show that the omitted term in the Wolfenstein effective Hamiltonian has no effect on them. We prove that the appearance of the Majorana phase cannot be ascribed to a lepton field rephasing transformation and thus the incorrectness of the claim of unphysicality and unmeasurability of the geometric phase. In the end we show that the calculation presented in [1] is inconsistent and based on the erroneous assumption and implementation of the wavefunction collapse. We remark that the geometric invariants defined in the present paper show a difference between Dirac and Majorana neutrinos, since they depend on the CP-violating Majorana phase.
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