arXiv subjects

Kazuo Fujikawa

Publications and source records attributed to Kazuo Fujikawa.

At least 37 records · Page 2Linked to original sources

arXiv2018

Generalized Pauli-Gursey transformation and Majorana neutrinos

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.

hep-ph↗

arXiv2018

Majorana neutrino and the vacuum of Bogoliubov quasiparticle

The Lagrangian of the seesaw mechanism is C violating but the same Lagrangian when re-written in terms of Majorana neutrinos is manifestly C invariant. To resolve this puzzling feature, a relativistic analogue of Bogoliubov transformation, which preserves CP but explicitly breaks C and P separately, was introduced together with the notions of a Bogoliubov quasiparticle and an analogue of the energy gap in BCS theory. The idea of Majorana neutrino as Bogoliubov quasiparticle was then suggested. In this paper, we study the vacuum structure of the Bogoliubov quasiparticle which becomes heavy by absorbing the C-breaking. By treating an infinitesimally small C violating term as an analogue of the chiral symmetry breaking nucleon mass in the model of Nambu and Jona-Lasinio, we construct an explicit form of the vacuum of the Bogoliubov quasiparticle which defines Majorana neutrinos in seesaw mechanism. The vacuum of the Bogoliubov quasiparticle thus constructed has an analogous condensate structure as the vacuum of the quasiparticle (nucleon) in the Nambu--Jona-Lasinio model.

hep-th↗

arXiv2018

Majorana neutrinos in seesaw mechanism and Bogoliubov quasiparticles

The idea that the Majorana neutrino should be identified as a Bogoliubov quasiparticle is applied to the seesaw mechanism for the three generations of neutrinos in the Standard Model. A relativistic analogue of the Bogoliubov transformation in the present context is a CP-preserving canonical transformation but modifies charge conjugation properties in such a way that the C-noninvariant fermion number violating term (condensate) is converted to a Dirac mass term. Puzzling aspects associated with the charge conjugation of chiral Weyl fermions are clarified. By invoking the Coleman--Weinberg mechanism in the framework of dimensional regularization, it is also noted that seesaw models become unnatural in some parameter regions which induce the hierarchy problems in the bosonic sector.

hep-ph↗

arXiv2018

Characteristics of Chiral Anomaly in View of Various Applications

In view of the recent applications of chiral anomaly to various fields beyond particle physics, we discuss some basic aspects of chiral anomaly which may help deepen our understanding of chiral anomaly in particle physics also. It is first shown that Berry's phase (and its generalization) for the Weyl model $H =v_{F} \vecσ\cdot \vec{p}(t)$ assumes a monopole form at the exact adiabatic limit but deviates from it off the adiabatic limit and vanishes in the high frequency limit of the Fourier transform of $\vec{p}(t)$ for bounded $|\vec{p}(t)|$. An effective action, which is consistent with the non-adiabatic limit of Berry's phase, combined with the Bjorken-Johnson-Low prescription gives normal equal-time space-time commutators and no chiral anomaly. In contrast, an effective action with a monopole at the origin of the momentum space, which describes Berry's phase in the precise adiabatic limit but fails off the adiabatic limit, gives anomalous space-time commutators and a covariant anomaly to the gauge current. We regard this anomaly as an artifact of the postulated monopole and not a consequence of Berry's phase. As for the recent application of the chiral anomaly to the description of effective Weyl fermions in condensed matter and nuclear physics, which is closely related to the formulation of lattice chiral fermions, we point out that the chiral anomaly for each species doubler separately vanishes for a finite lattice spacing, contrary to the common assumption. Instead a general form of pair creation associated with the spectral flow for the Dirac sea with finite depth takes place. This view is supported by the Ginsparg-Wilson fermion, which defines a single Weyl fermion without doublers on the lattice and gives a well-defined index (anomaly) even for a finite lattice spacing.

hep-th↗

arXiv2017

Majorana Neutrino as Bogoliubov Quasiparticle

We suggest that the Majorana neutrino should be regarded as a Bogoliubov quasiparticle that is consistently understood only by use of a relativistic analogue of the Bogoliubov transformation. The unitary charge conjugation condition ${\cal C}ψ{\cal C}^{\dagger}=ψ$ is not maintained in the definition of a quantum Majorana fermion from a Weyl fermion. This is remedied by the Bogoliubov transformation accompanying a redefinition of the charge conjugation properties of vacuum, such that a C-noninvariant fermion number violating term (condensate) is converted to a Dirac mass. We also comment on the chiral symmetry of a Majorana fermion; a massless Majorana fermion is invariant under a global chiral transformation $ψ\rightarrow \exp[iαγ_{5}]ψ$ and different Majorana fermions are distinguished by different chiral $U(1)$ charge assignments. The reversed process, namely, the definition of a Weyl fermion from a well-defined massless Majorana fermion is also briefly discussed.

hep-ph↗

arXiv2017

Baryon number violation and novel canonical anti-commutation relations

The possible neutron-antineutron oscillation is described by an effective quadratic Lagrangian analogous to the BCS theory. It is shown that the conventional equal-time anti-commutation relations of the neutron variable $n(t,\vec{x})$ are modified by the baryon number violating terms. This is established by the Bjorken--Johnson--Low prescription and also by the canonical quantization combined with equations of motion. This novel canonical behavior can give rise to an important physical effect, which is illustrated by analyzing the Lagrangian that violates the baryon number but gives rise to the degenerate effective Majorana fermions and thus no neutron-antineutron oscillation. Technically, this model is neatly treated using a relativistic analogue of the Bogoliubov transformation.

hep-th↗

arXiv2017

Naturalness in see-saw mechanism and Bogoliubov transformation

We present an alternative perspective on the see-saw mechanism for the neutrino mass, according to which the small neutrino mass is given as a difference of two large masses. This view emerges when an analogue of the Bogoliubov transformation is used to describe Majorana neutrinos in the Lagrangian of the see-saw mechanism, which is analogous to the BCS theory. The Bogoliubov transformation clarifies the natural appearance of Majorana fermions when C is strongly violated by the right-handed neutrino mass term with good CP in the single flavor model. Analyzing typical models with $m_{R}$= $10^{4}$ to $10^{15}$ GeV, it is shown that a hitherto unrecognized fine tuning of the order $m_ν/m_{R}=10^{-15}$ to $10^{-26}$ is required to make the commonly perceived see-saw mechanism work in a natural setting, namely, when none of dimensionless coupling constants are very small.

hep-ph↗

arXiv2016

Parity-doublet representation of Majorana fermions and neutron oscillation

We present a parity-doublet theorem for the representation of the intrinsic parity of Majorana fermions, which is expected to be useful also in condensed matter physics, and it is illustrated to provide a criterion of neutron-antineutron oscillation in a BCS-like effective theory with $ΔB=2$ baryon number violating terms. The CP violation in the present effective theory causes no direct CP violating effects in the oscillation itself, which is demonstrated by the exact solution, although it influences the neutron electric dipole moment in the leading order of small $ΔB=2$ parameters. An analogue of Bogoliubov transformation, which preserves P and CP, is crucial in the analysis.

hep-ph↗

arXiv2016

Exact algebraic separability criterion for two-qubit systems

A conceptually simpler proof of the separability criterion for two-qubit systems, which is referred to as "Hefei inequality" in literature, is presented. This inequality gives a necessary and sufficient separability criterion for any mixed two-qubit system unlike the Bell-CHSH inequality that cannot test the mixed-states such as the Werner state when regarded as a separability criterion. The original derivation of this inequality emphasized the uncertainty relation of complementary observables, but we show that the uncertainty relation does not play any role in the actual derivation and the Peres-Hodrodecki condition is solely responsible for the inequality. Our derivation, which contains technically novel aspects such as an analogy to the Dirac equation, sheds light on this inequality and on the fundamental issue to what extent the uncertainty relation can provide a test of entanglement. This separability criterion is illustrated for an exact treatment of the Werner state.

quant-ph↗

arXiv2016

Lorentz invariant CPT breaking in the Dirac equation

If one modifies the Dirac equation in momentum space to $[γ^μp_μ-m-Δm(θ(p_{0})-θ(-p_{0})) θ(p_μ^{2})]ψ(p)=0$, the symmetry of positive and negative energy eigenvalues is lifted by $m\pm Δm$ for a small $Δm$. The mass degeneracy of the particle and antiparticle is thus lifted in a Lorentz invariant manner since the combinations $θ(\pm p_{0})θ(p_μ^{2})$ with step functions are manifestly Lorentz invariant. We explain an explicit construction of this CPT breaking term in coordinate space, which is Lorentz invariant but non-local at a distance scale of the Planck length. The application of this Lorentz invariant CPT breaking mechanism to the possible mass splitting of the neutrino and antineutrino in the Standard Model is briefly discussed.

hep-ph↗

arXiv2016

Dimensional regularization is generic

The absence of the quadratic divergence in the Higgs sector of the Standard Model in the dimensional regularization is usually regarded to be an exceptional property of a specific regularization. To understand what is going on in the dimensional regularization, we illustrate how to reproduce the results of the dimensional regularization for the $λϕ^{4}$ theory in the more conventional regularization such as the higher derivative regularization; the basic postulate involved is that the quadratically divergent induced mass, which is independent of the scale change of the physical mass, is kinematical and unphysical. This is consistent with the derivation of the Callan-Symanzik equation, which is a comparison of two theories with slightly different masses, for the $λϕ^{4}$ theory without encountering the quadratic divergence. We thus suggest that the dimensional regularization is generic in a bottom-up approach starting with a successful low-energy theory. We also define a modified version of the mass independent renormalization for a scalar field which leads to the homogeneous renormalization group equation. Implications of the present analysis on the Standard Model at high energies and the presence or absence of SUSY at LHC energies are briefly discussed.

hep-th↗

arXiv2016

Separability criteria with angular and Hilbert space averages

The practically useful criteria of separable states $ρ=\sum_{k}w_{k}ρ_{k}$ in $d=2\times2$ are discussed. The equality $G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$ provides a necessary and sufficient separability criterion in the case of a separable pure state $ρ=|ψ\rangle\langleψ|$. We propose the separability criteria of mixed states, which are given by ${\rm Tr}ρ\{{\bf a}\cdot {\bf σ}\otimes {\bf b}\cdot {\bf σ}\}=(1/3)C\cosφ$ for two spin $1/2$ systems and $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/2)C\cos2φ$ for two photon systems, respectively, after taking a geometrical angular average of ${\bf a}$ and ${\bf b}$ with fixed $\cosφ={\bf a}\cdot{\bf b}$. Here $-1\leq C\leq 1$, and the difference in the numerical coefficients $1/2$ and $1/3$ arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the $d=2$ Hilbert space, the criterion for two photon systems is replaced by $4{\rm Tr}ρ\{P({\bf a})\otimes P({\bf b})\}=1+(1/3)C\cos2φ$. Those separability criteria are shown to be very efficient using the existing experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the Werner state is applied to two photon systems, it is shown that the Hilbert space average can judge its inseparability but not the geometrical angular average.

quant-ph↗

arXiv2016

BCS, Nambu-Jona-Lasinio, and Han-Nambu -- A sketch of Nambu's works in 1960-1965

The years of 1960-1965 were a remarkable period for Yoichiro Nambu. Starting with a reformulation of BCS theory with emphasis on gauge invariance, he recognized the realization of spontaneous chiral symmetry breaking in particle physics as is evidenced by the Goldberger-Treiman relation. A concrete model of Nambu and Jona-Lasinio illustrated the essence of the Nambu-Goldstone theorem and the idea of soft pions. After the proposal of the quark model by Gell-Mann, he together with Han constructed an alternative model of integrally charged quarks with possible non-Abelian gluons. All those remarkable works were performed during the years 1960-1965. Here I briefly review those works following the original papers of Nambu chronologically, together with a brief introduction to a formulation of Neother's theorem and Ward-Takahashi identities using path integrals. This article is mostly based on a lecture given at the Nambu Memorial Symposium held at Osaka City University in September 2015, where Nambu started his professional career.

physics.hist-ph↗

arXiv2016

The gradient flow in $λϕ^{4}$ theory

A gradient flow equation for $λϕ^{4}$ theory in $D=4$ is formulated. In this scheme the gradient flow equation is written in terms of the renormalized probe variable $Φ(t,x)$ and renormalized parameters $m^{2}$ and $λ$ in a manner analogous to the higher derivative regularization. No extra divergence is induced in the interaction of the probe variable $Φ(t,x)$ and the 4-dimensional dynamical variable $ϕ(x)$ which is defined in renormalized perturbation theory. The finiteness to all orders in perturbation theory is established by power counting argument in the context of $D+1$ dimensional field theory. This illustrates that one can formulate the gradient flow for the simple but important $λϕ^{4}$ theory in addition to the well-known Yang-Mills flow, and it shows the generality of the gradient flow for a wider class of field theory.

hep-lat↗

arXiv2015

Neutron-antineutron oscillation and parity and CP symmetries

It is shown that the neutron-antineutron oscillation per se does not necessarily imply CP violation in an effective local Lorentz invariant description of the neutron which preserves CPT, contrary to a recent analysis in the literature. A CP- and baryon number-violating term can be transformed into a CP conserving one, thus rendering the CP violation spurious and unusable in the analysis of baryogenesis, for example. It is also shown that the neutron-antineutron oscillation in a $ΔB=2$, Lorentz and CPT invariant interaction can occur only when parity is violated, irrespective of the CP properties; when parity is conserved, it is the ordinary quantum transition from neutron to antineutron which takes place. Those statements are proven by explicitly analyzing all the possible combinations with P, C and CP violation or their conservation. Moreover, a suitable combination of P=odd and P=even $ΔB=2$ interactions in the present model is shown to give rise to the CP preserving mass term for the right-handed neutrino if one replaces the neutron by the neutrino, reinforcing the conclusion.

hep-ph↗

arXiv2015

Neutrino-antineutrino Mass Splitting in the Standard Model: Neutrino Oscillation and Baryogenesis

By adding a neutrino mass term to the Standard Model, which is Lorentz and $SU(2)\times U(1)$ invariant but non-local to evade $CPT$ theorem, it is shown that non-locality within a distance scale of the Planck length, that may not be fatal to unitarity in generic effective theory, can generate the neutrino-antineutrino mass splitting of the order of observed neutrino mass differences, which is tested in oscillation experiments, and non-negligible baryon asymmetry depending on the estimate of sphaleron dynamics. The one-loop order induced electron-positron mass splitting in the Standard Model is shown to be finite and estimated at $\sim 10^{-20}$ eV, well below the experimental bound $< 10^{-2}$ eV. The induced $CPT$ violation in the $K$-meson in the Standard Model is expected to be even smaller and well below the experimental bound $|m_{K}-m_{\bar{K}}|<0.44\times 10^{-18}$ GeV.

hep-ph↗

arXiv2015

Uncertainty principle, Shannon-Nyquist sampling and beyond

Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of view and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.

quant-ph↗

arXiv2015

Bosonization in the path integral formulation

We establish the direct $d=2$ on-shell bosonization $ψ_{L}(x_{+})=e^{iξ(x_{+})}$ and $ψ_{R}^{\dagger}(x_{-})=e^{iξ(x_{-})}$ in path integral formulation by deriving the off-shell relations $ψ_{L}(x)ψ_{R}^{\dagger}(x)=\exp[iξ(x)]$ and $ψ_{R}(x)ψ_{L}^{\dagger}(x)=\exp[-iξ(x)]$. Similarly, the on-shell bosonization of the bosonic commuting spinor, $ϕ_{L}(x_{+})=ie^{-iξ(x_{+})}\partial^{+}e^{-iχ(x_{+})}$, $ϕ^{\dagger}_{R}(x_{-})=e^{-iξ(x_{-})-iχ(x_{-})}$ and $ϕ_{R}(x_{-})=ie^{iξ(x_{-})}\partial^{-}e^{+iχ(x_{-})}$, $ϕ^{\dagger}_{L}(x_{+})=e^{iξ(x_{+})+iχ(x_{+})}$, is established in path integral formulation by deriving the off-shell relations $ϕ_{L}(x)ϕ^{\dagger}_{R}(x)=ie^{-iξ(x)}\partial^{+}e^{-iχ(x)}$ and $ϕ_{R}(x)ϕ^{\dagger}_{L}(x)=ie^{iξ(x)}\partial^{-}e^{iχ(x)}$.

hep-th↗