Introduction to Majorana masses

We present a pedogogical review of Majorana masses and Majorana's theory of two-component massive fermions. We discuss the difference between Majorana and Dirac masses and show that Majorana masses are formion-number violating. We discuss the connection between Majorana and, Weyl spinors and show that the massive Majorana and Weyl field theories are equivalent. We study the second quantization of the massive Weyl theory in detail.     

Extended projection method for massive fermions

Tensor reduction is of considerable importance in calculations of multi-loop amplitudes,and the projection method is one of the most popular approaches for tensor reduction.However,the projection method can be problematic when applied to amplitudes with massive fermions,due to the inconsistency between helicity and chirality.We propose an extended projection method for reducing the loop amplitude which contains a fermion chain with two massive spinors.The extension is achieved by decomposing one of the massive spinors into two massless spinors,the"null spinor"and the"reference spinor".The extended projection method can be effectively applied in all processes,including the production of massive fermions.We present the tensor reduction for a virtual Z boson decaying into a top-quark pair as a demonstration of our approach.     

Collective modes of type-ⅡWeyl fermions with repulsive S-wave interaction

Three-dimensional type-ⅡWeyl fermions possess overtilted cone-like low-energy band dispersion.Unlike the closed ellipsoidal Fermi surface for type-ⅠWeyl fermions,the Fermi surface is an open hyperboloid for type-ⅡWeyl fermions.We evaluate the spin and density susceptibility of type-ⅡWeyl fermions with repulsive S-wave interaction by means of Green’s functions.We obtain the particle–hole continuum along the tilted momentum direction and perpendicular to the tilted momentum direction respectively.We find the zero sound mode in some repulsive interaction strengths by numerically solving the pole equations of the susceptibility within the random-phase approximation.     

Influence of <Emphasis Type="Italic">CP</Emphasis> and <Emphasis Type="Italic">CPT</Emphasis> on production and decay of Dirac and Majorana fermions

The consequences of CP and CPT invariance for production and subsequent decay of Dirac and Majorana fermions in polarized fermion-antifermion annihilation are analytically studied. We derive general symmetry relations for the production spin density matrix and for the three-particle decay matrices and obtain constraints for the polarization and the spin-spin correlations of Dirac and Majorana fermions. We prove that only for Majorana fermions the energy and opening angle distribution factorizes exactly into contributions from production and decay if CP is conserved. Received: 6 November 2001 / Revised version: 23 April 2002 / Published online: 12 July 2002     

A spinor equation of the pure electromagnetic field

In the early history of spinors it became evident that a single undotted covariant elementary spinor can represent a plane wave of light. Further study of that relation shows that plane electromagnetic waves satisfy the Weyl equation, in a way that indicates the correct spin angular momentum. On the subatomic scale the Weyl equation discloses more detail than the vector equations. The spinor and vector equations are equivalent when applied to plane waves, and more generally (in the absence of sources) on the large scale when the spinors are incoherent.     

Dimensional reduction of Weyl,Majorana and Majorana-Weyl spinors

We discuss the dimensional reduction for Weyl, Majorana, or Majorana-Weyl spinors coupled to pure d-dimensional (d ? 4) gravity. The only case where a realistic four-dimensional low-energy spectrum for the fermions may be obtained, is for a Majorana-Weyl spinor in d = 2 mod 8 dimensions. Chiral massless fermions are not excluded in this case. The minimal number of dimensions for the construction of a realistic theory out of pure gravity is d = 18.     

Spinor Factorizations for Relativity

Factorization of 2-component spinors in the mathematical language of General Relativity is often the starting point of classifications of them. Particular factorizations for the Ricci and Weyl spinors are well known. The determination of factorizations of (p, q) spinors is equivalent to the determination of double or bipartite partitions. Firstly a technique for enumeration of such partitions is given, with explicit calculations up to (4, 4). Secondly a method is devised to actually obtain the factorizations, display them for the Ricci, Weyl and Lanczos-Zund spinors, and then reference the full tabulation up to (4, 4) spinors—the most useful for Relativity.     

Weyl invariance and the origins of mass

By a uniform and simple Weyl invariant coupling of scale and matter fields, we construct theories that unify massless, massive, and partially massless excitations. Masses are related to tractor Weyl weights, and Breitenlohner–Freedman stability bounds in anti de Sitter amount to reality of these weights. The method relies on tractor calculus – mathematical machinery allowing Weyl invariance to be kept manifest at all stages. The equivalence between tractor and higher spin systems with arbitrary spins and masses is also considered.     

Radiative four-fermion processes at LEP2

The production of four fermions plus a visible photon in electron–positron collisions is analyzed, with particular emphasis on the LEP2 energy range. The study is based on the calculation of exact matrix elements, including the effect of fermion masses. In the light of the present measurements performed at LEP, triple and quartic anomalous gauge couplings are taken into account. Due to the presence of a visible photon in the final state, particular attention is paid to the treatment of higher-order QED corrections. Explicit results for integrated cross sections and differential distributions are shown and commented on. The features of the Monte Carlo program WRAP, used to perform the calculation and available for experimental analysis, are described. Received: 14 March 2001 / Revised version: 26 June 2001 / Published online: 3 August 2001     

Gravitational anomalies and Schwinger terms

A BRST transformation incorporating general coordinate, Lorentz and Weyl transformations is defined. The variation of the effective action for fermions moving in an external gravitational field is calculated using an extended BRST analysis. InD=2 dimensions the Schwinger terms appearing in the equal time commutators of energy momentum tensors are related to the gravitational and Weyl anomalies for a massless spin 0 and a massless spin 1/2 model, respectively.     

Spin-photon Entanglement in Spontaneous Emission

The two-pathway decay, spin-photon entanglement, and spontaneous generated spin coherence in a Lambdatype three-level system can be understood in a unified picture in terms of quantum entanglement of the electron spin, the photon polarization, and the photon waveform. Within such a unified picture, the spontaneous emission can be controlled to present a designated effect by external field, optical excitation, and quantum measurement. Various experiments are proposed to implement such control, with some of them already realized.     

Asymptotically Flat Conformal Structures

In the first part of this paper we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version of the positive mass theorem on conformal spin manifolds.     

On the quantum few-body problem

We propose a new method for computing the energy of the ground state of a few-body quantum system, using a modified Monte Carlo technique. This technique works for both fermionic and bosonic systems. We present and discuss the preliminary numerical results of a simple lattice computer calculation for a system of few fermions (half spin up, half spin down) with an attractive delta interaction.     

Classical probabilities for Majorana and Weyl spinors

We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function q_τ(t) for the Ising states τ. The time dependent probability distribution of a generalized Ising model obtains as . The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure characteristic for quantum mechanics. Probability distributions of the Ising model which correspond to one or many propagating fermions are discussed explicitly. Expectation values of observables can be computed equivalently in the classical statistical Ising model or in the quantum field theory for fermions.     

Anomalous production of chiral charge from black holes

The Reissner Nordström solution to Einstein equations coupled to a Yang-Mills field is unstable against the emission of light-charged fermions. We study the flux chirality in this decay process and illustrate the consistency of the result with the prediction of the axial anomaly.     

Top quark production in e^+ e^- annihilation

We analyze the four-fermion reactions containing a single top quark and three other fermions, a possible decay product of the resonant anti-top quark, in the final state. This allows us to estimate the contribution of the nonresonant Feynman graphs and effects related to the off mass shell production and decay of the top quark. We test the sensitivity of the total cross section at center of mass energies in the threshold region and far above it to the variation of the top quark width. We perform the calculation in an arbitrary linear gauge in the framework of the standard model and discuss the important issue of gauge symmetry violation by a constant top quark width. Received: 21 February 2001 / Revised version: 27 March 2001 / Published online: 18 May 2001     

Exactly solvable two-dimensional quantum spin models

A method is proposed for constructing an exact ground-state wave function of a two-dimensional model with spin 1/2. The basis of the method is to represent the wave function by a product of fourth-rank spinors associated with the nodes of a lattice and the metric spinors corresponding to bonds between nearest neighbor nodes. The function so constructed is an exact wave function of a 14-parameter model. The special case of this model depending on one parameter is analyzed in detail. The ground state is always a nondegenerate singlet, and the spin correlation functions decay exponentially with distance. The method can be generalized for models with spin 1/2 to other types of lattices. Zh. éksp. Teor. Fiz. 115, 249–267 (January 1999)