pseudo-C Symmetry Research Articles

Operatorial characterization of Majorana neutrinos

The Majorana neutrino psi _{M}(x) when constructed as a superposition of chiral fermions such as nu _{L} + Coverline{nu _{L}}^{T} is characterized by (mathcal{C}mathcal{P}) psi _{M}(x)(mathcal{C}mathcal{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 mathcal{C} psi _{M}(x) mathcal{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” nu _{L,R}(x)rightarrow Coverline{nu _{L,R}(x)}^{T} (and associated “pseudo P-symmetry”), that has been often used in both the seesaw model and Weinberg’s 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.

Seesaw mechanism and pseudo C-symmetry

It is shown that the specific “charge conjugation” transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely (nu _{R})^{C}=Coverline{nu _{R}}^{T} for a chiral fermion nu _{R} (and similarly for nu _{L}), is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. When carefully examined, it also fails as a classical transformation law in a Lorentz invariant field theory. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. The pseudo C-symmetry is effective to identify Majorana neutrinos analogously to the classical Majorana condition. The analysis of CP breaking in weak interactions is performed using the conventional CP transformation, which is defined independently of the pseudo C-transformation, in the seesaw model after mass diagonalization. A way to ensure an operatorially consistent formulation of C-conjugation is to formulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.

Formation of Mono(dithiolene)−Thiocarboxamido Complexes in Reactions of Thio(dithiocarbamato)−Mo/W Complexes and Dimethyl Acetylenedicarboxylate

Reactions of TpMS(S(2)CNEt(2)) with dimethyl acetylenedicarboxylate in dichloromethane produce olive green/black TpM{S(2)C(2)(CO(2)Me)(2)}(SCNEt(2)-kappa(2)S,C) (M = Mo (1), W (2); Tp = hydrotris(3,5-dimethylpyrazol-1-yl)borate). The seven-coordinate complexes exhibit pseudo-octahedral (1) and distorted pentagonal bipyramidal (2) coordination spheres comprised of tridentate fac-Tp, bidentate dithiolene, and thiocarboxamido-kappa(2)S,C ligands. In the solid state, molecules of 1 exhibit pseudo-C(s)() symmetry, with the thiocarboxamide NEt(2) group in a cleft in the Tp ligand. Molecules of 2 have C(1) symmetry in the solid state; here, the thiocarboxamide unit is orientated along one of the W-S(dithiolene) bonds with its NEt(2) group projecting away from the Tp ligand. Both complexes possess effective C(s)() symmetry in solution. Reaction of TpMoI(CO)(3) with AgS(2)CNEt(2) affords olive green TpMo(S(2)CNEt(2))(CO)(2) (3), which reacts with propylene sulfide in a new synthesis for TpMoS(S(2)CNEt(2)), the starting material for 1. Complex 3 exhibits a distorted pentagonal bipyramidal structure, the axial sites being defined by a Tp nitrogen atom and a carbonyl ligand, the pentagonal plane by the remaining nitrogen and carbonyl donors and the two sulfur atoms of the bidentate dithiocarbamate ligand.

Influence of steric symmetry and electronic dissymmetry on the enantioselectivity in palladium-catalyzed allylic substitutions.

Chiral P,N-ligands with pseudo-C2 and pseudo-Cs symmetry based on chiral pyrrolidine and phospholane rings or on dinaphthatodihydroazepino and dinaphthatodihydrophosphepino moieties were prepared and assessed in the palladium-catalyzed allylic substitutions of allylic acetates. Higher selectivity was achieved with pseudo-C2-symmetric ligands based on the binaphthyl skeleton than with the analogous C2-symmetric P,P- and N,N-analogues. Pseudo-C2-type ligands had properties superior to those of pseudo-meso-type ligands when 1,3-diphenyl-2-propenyl acetate was used as a substrate, whereas the reverse situation was found for 3-cyclohexenyl acetate. Chirally flexible ligands, prepared by substitution of one of the rigid binaphthyl skeletons for a flexible biphenyl system, were found to induce chirality to the same extent as a 1:1 mixture of the rigid ligands.

Ab initio calculation and spectroscopic analysis of the intramolecular vibrational redistribution in 1,1,1,2-tetrafluoroiodoethane CF3CHFI

We report a new mechanism for intramolecular vibrational redistribution (IVR) in CF3CHFI which couples the CH chromophore vibrations through a strong Fermi resonance to the formal CF stretching normal mode (a heavy atom frame mode) involving the trans F-atom across the CC bond. The analysis is made possible by comparing spectroscopic results with extensive ab initio calculations of the vibrational fundamental and overtone spectra in the range extending to 12 000 cm−1. Potential energy and electric dipole moment hypersurfaces are calculated ab initio by second order Møller–Plesset perturbation theory (MP2) on a grid involving the CH stretching, two CH bending modes and one high frequency CF stretching normal mode. The potentials are scaled to obtain agreement between the experimental spectrum and the theoretical spectrum calculated by a discrete variable representation technique on this grid. Both spectra are then analyzed in terms of three-dimensional (3D) and four-dimensional (4D) effective vibrational Hamiltonians including Fermi- and Darling–Dennison-type resonances between the CH stretching mode and the CH bending modes and the CF stretching mode. The interaction between the CH modes and the CF mode is clearly visible in the experimental and calculated (4D) spectra. The effective Fermi resonance coupling constants [ksff′≃(40±10) cm−1 and ksaf′≃(55±10) cm−1] coupling the CH and CF mode subspaces are of about the same magnitude as the intra-CH chromophore Fermi resonances (ksaa′≃56 cm−1 and ksbb′≃42 cm−1, coupling CH stretching mode “s” with the two CH bending modes “a” and “b”). The chiral, pseudo-Cs symmetry breaking coupling (ksab′≃11 cm−1) is complemented by an equally strong coupling through the CF mode (ksfb′≃15 cm−1). It is demonstrated that low order perturbation theoretical analysis using potential constants from a polynomial expansion to represent effective coupling constants gives inadequate results with discrepancies ranging about from factors of 2–5. Time dependent population and wave packet analysis shows essentially complete IVR among the CH chromophore modes within about 100 fs, the 3D and 4D evolutions being similar up to about that time. At longer times of about 250 fs, there is substantial excitation of the CF stretching mode (with initial pure CH stretching excitation). The 4D treatment is then essential for a correct description of the dynamics.