Majorana Spinors Research Articles

Interacting non-Abelian vector spinor and nilpotent spinor charges

We formulate an interacting theory of a vector-spinor field associated with anti-commuting Majorana spinor charges in arbitrary spacetime dimensions. The field content of the system is , where is a vector spinor in the adjoint representation of an arbitrary gauge group, and is its gauge field, while χαIJ is an extra spinor with antisymmetric adjoint indices IJ. Contrary to common wisdom, the consistency of the vector-spinor field equation is maintained, despite its non-trivial interactions and nilpotency of the spinor charges. After establishing the classical system, we perform its quantization with gauge-fixing and Faddeev–Popov ghost terms, confirming the BRST invariance of the total action.

M-theory on eight-manifolds revisited: Script N = 1 supersymmetry and generalized Spin(7) structures

The requirement of ${\cal N}=1$ supersymmetry for M-theory backgrounds of the form of a warped product ${\cal M}\times_{w}X$, where $X$ is an eight-manifold and ${\cal M}$ is three-dimensional Minkowski or AdS space, implies the existence of a nowhere-vanishing Majorana spinor $\xi$ on $X$. $\xi$ lifts to a nowhere-vanishing spinor on the auxiliary nine-manifold $Y:=X\times S^1$, where $S^1$ is a circle of constant radius, implying the reduction of the structure group of $Y$ to $Spin(7)$. In general, however, there is no reduction of the structure group of $X$ itself. This situation can be described in the language of generalized $Spin(7)$ structures, defined in terms of certain spinors of $Spin(TY\oplus T^*Y)$. We express the condition for ${\cal N}=1$ supersymmetry in terms of differential equations for these spinors. In an equivalent formulation, working locally in the vicinity of any point in $X$ in terms of a `preferred' $Spin(7)$ structure, we show that the requirement of ${\cal N}=1$ supersymmetry amounts to solving for the intrinsic torsion and all irreducible flux components, except for the one lying in the $\bf{27}$ of $Spin(7)$, in terms of the warp factor and a one-form $L$ on $X$ (not necessarily nowhere-vanishing) constructed as a $\xi$ bilinear; in addition, $L$ is constrained to satisfy a pair of differential equations. The formalism based on the group $Spin(7)$ is the most suitable language in which to describe supersymmetric compactifications on eight-manifolds of $Spin(7)$ structure, and/or small-flux perturbations around supersymmetric compactifications on manifolds of $Spin(7)$ holonomy.

Graded Majorana spinors

In many mathematical and physical contexts, spinors are treated as Grassmann odd valued fields. We show that it is possible to extend the classification of reality conditions on such spinors by a new type of Majorana condition. In order to define this graded Majorana condition we make use of pseudo-conjugation, a rather unfamiliar extension of complex conjugation to supernumbers. Like the symplectic Majorana condition, the graded Majorana condition may be imposed, for example, in spacetimes in which the standard Majorana condition is inconsistent. However, in contrast to the symplectic condition, which requires duplicating the number of spinor fields, the graded condition can be imposed on a single Dirac spinor. We illustrate how graded Majorana spinors can be applied to supersymmetry by constructing a globally supersymmetric field theory in three-dimensional Euclidean space, an example of a spacetime where standard Majorana spinors do not exist.

WHERE ARE ELKO SPINOR FIELDS IN LOUNESTO SPINOR FIELD CLASSIFICATION?

This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. We show moreover that algebraic constraints imply that any class 5 spinor field is such that the 2-component spinor fields entering its structure have opposite helicities. The proof of our statement is based on Lounesto general classification of all spinor fields, according to the relations and values taken by their associated bilinear covariants, and can eventually shed some new light on the algebraic investigations concerning dark matter.

VAN DER WAERDEN CALCULUS WITH COMMUTING SPINOR VARIABLES AND APPLICATIONS TO SUPERSYMMETRY

Working with anticommuting Weyl (or Majorana) spinors in the frame of the van der Waerden calculus is standard in supersymmetry. The natural framework of rigorous supersymmetric quantum field theory makes use of operator-valued superdistributions defined on supersymmetric test functions. This makes necessary a van der Waerden calculus in which the Grassmann variables anticommute but the fermionic components commute. We work out such a calculus in view of applications to the rigorous conceptual problems of supersymmetric quantum field theory.

Continuous spin representations from group contraction

We consider how the continuous spin representation (CSR) of the Poincaré group in four dimensions can be generated by dimensional reduction. The analysis uses the front-form little group in five dimensions, which must yield the Euclidean group E(2), the little group of the CSR. We consider two cases, one is the single spin massless representation of the Poincaré group in five dimensions, the other is the infinite component Majorana equation, which describes an infinite tower of massive states in five dimensions. In the first case, the double singular limit j, R→∞, with j∕R fixed, where R is the Kaluza–Klein radius of the fifth dimension, and j is the spin of the particle in five dimensions, yields the CSR in four dimensions. It amounts to the Inönü–Wigner contraction, with the inverse Kaluza–Klein radius as contraction parameter. In the second case, the CSR appears only by taking a triple singular limit, where an internal coordinate of the Majorana theory goes to infinity, while leaving its ratio to the Kaluza–Klein radius fixed.

The construction of spinors in geometric algebra

The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap among algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U (1), SU (2), and SL ( 2 , C ) spinors. The physical observables in Schrödinger–Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches.

Beta decays with momentum space Majorana spinors

We construct and apply to beta decays a truly neutral local quantum field that is entirely based upon momentum space Majorana spinors. We make the observation that theory with momentum space Majorana spinors of real C parities is equivalent to Dirac's theory. For imaginary C parities, the neutrino mass can drop from the single beta decay trace and reappear in 0\nu \beta \beta, a curious and in principle experimentally testable signature for a non-trivial impact of Majorana framework in experiments with polarized sources.

LITTLE GROUPS OF PREON BRANES

Little groups for preon branes (i.e. configurations of branes with maximal (n-1)/n fraction of survived supersymmetry) for dimensions d=2,3,…,11 are calculated for all massless, and partially for massive orbits. For massless orbits little groups are semidirect product of d-2 translational group Td-2 on a subgroup of ( SO (d-2) × R-invariance) group. E.g. at d=9 the subgroup is exceptional G2 group. It is also argued, that 11D Majorana spinor invariants, which distinguish orbits, are actually invariant under d=2+10 Lorentz group. Possible applications of these results include construction of field theories in generalized spacetimes with brane charges coordinates, different problems of group's representations decompositions, spin-statistics issues.

Superspace models for S3

The simplest supersymmetric extension of the group SO(4) is discussed. The superalgebra is realized in a superspace whose Bosonic subspace is the surface of a sphere S3 embedded in four-dimensional Euclidean space. By using Fermionic coordinates in this superspace, which are chiral symplectic Majorana spinors, it proves possible to devise superfield models involving a complex scalar, a pair of chiral symplectic Majorana spinors, and a complex auxiliary scalar. Kinetic terms involve operators that are isometry generators on S3.PACS No.: 11.30.Pb

Bulk quantum effects for de Sitter branes inAdS5

We investigate some issues regarding quantum corrections for de Sitter branes in a bulk AdS(5) spacetime. The one-loop effective action for a Majorana spinor field is evaluated and compared with the scalar field result. We also evaluate the cocycle function for various boundary conditions, finding that the quantum corrections naturally induce higher order curvature terms in the original action and, in general, it is not possible to eliminate the cocycle function by renormalisation. In the one brane limit care must be taken on how one extracts physical results. The effective potential is found to be zero on the conformally related cylinder. However, using the actual metric, the contribution from the cocycle function is non-zero and must be included. Subtleties with any zero modes are also discussed.

SUPERSYMMETRY ON THE SURFACE S2

By using symplectic Majorana spinors as Grassmann coordinates in a superspace associated with the supersymmetric extension of the isometry group on the spherical surface S2, it proves possible to formulate supersymmetric models on S2 using superspace techniques.

SUPERSYMMETRY IN 2+2 DIMENSIONS

The analysis of spinors in 2+2 dimensions is reviewed. The two most basic supersymmetry algebras in this space are considered. In the simpler case, the supersymmetry generators are Majorana spinors. In the other algebra we consider, the supersymmetry generators are Dirac spinors. Consistency with having a positive definite Hilbert space is shown to be impossible in both of these particular cases.

Induced Chern–Simons and WZW action in noncommutative spacetime

We consider noncommutative gauge theory with Dirac or Majorana fermions in odd dimensional spacetime and compute the induced noncommutative Chern-Simons action generated at 1-loop. We observe that the Chern-Simons term induced by a Dirac fermion has a smooth limit when $\th ---> 0$, but there is a finite jump for the Chern-Simons term induced by a Majorana spinor. The induced Chern-Simons action from a Majorana spinor is nonvanishing even in the $\th ---> 0$ limit, a discontinuity that shares a similar characteristic as the UV/IR singularity discovered originally by Minwalla, Raamsdonk and Seiberg. Properties of the noncommutative WZW action are also discussed.

Majorana spinors on unoriented surfaces

In this paper we find the conditions for the existence of Majorana spinors on unoriented surfaces. We prove that a Majorana spinor exists on these surfaces if and only if its genus is an odd integer. We enumerate the different real spin structures and we comment on their parity classification. Potentially interesting applications are pointed out.

Leptogenesis in theories with large extra dimensions

We study the scenario of baryogenesis through leptogenesis in higher-dimensional theories, in which the scale of quantum gravity is many orders of magnitude smaller than the usual Planck mass. The minimal realization of these theories includes an isosinglet neutrino which feels the presence of large compact dimensions, whereas all the SM particles are localized on a $(1+3)$-dimensional subspace. In the formulation of minimal leptogenesis models, we pay particular attention to the existence of Majorana spinors in higher dimensions. After compactification of the extra dimensions, we obtain a tower of Majorana Kaluza-Klein excitations which act as an infinite series of CP-violating resonators and derive the necessary conditions for their constructive interference. Based on this CP-violating mechanism, we find that the decays of the heavy Majorana excitations can produce a leptonic asymmetry which is reprocessed into the observed baryonic asymmetry of the universe by means of out-of-equilibrium sphaleron interactions, provided the reheat temperature is above 5 GeV.

Yukawa coupling in three dimensions. II

The effective potential for the simplest Yukawa model in $2+1$ dimensions discussed in a previous paper is computed to two-loop order. The radiative corrections serve to destabilize the vacuum, both when the fermion is a Dirac spinor and when it is a Majorana spinor but with the instability occurring at values of the external field where perturbation theory is unreliable. If the Yukawa coupling and the self-coupling of the scalar are related by supersymmetry, stability is present. Renormalization of the supersymmetric version of this model presents special difficulties when the auxiliary field is not eliminated using the field equations. In an appendix, the renormalization group functions for a three dimensional Euclidean space model with two Yukawa couplings is analyzed, showing that one of the two couplings is asymptotically free.

Anomaly-induced effective actions in even dimensions and reliability of s-wave approximation

The reduction of 4d matter-gravity theory to S 2, H 2 or R 2 leads to effective 2d dilatonic gravity with dilaton coupled matter. Spinors give the exceptional example of the theory which is conformally invariant in 4d as well as in 2d, after reduction. We find 4d and 2d conformal anomaly induced effective action (EA) for Majorana spinor. It is expected for some time that s-wave EA (i.e. the one for dilaton coupled 2d matter) is some ( s-wave) approximation to 4d EA. We compare such 2d and 4d spinor EAs on the same gravitational background and argue that s-wave EA indeed qualitatively corresponds to no higher derivatives approximation for 4d EA.

Killing spinors and supersymmetry on AdS

In this paper we construct several supersymmetric theories on AdS 5 background. We discuss the proper definition of the Killing equation for the symplectic Majorana spinors required in AdS 5 supersymmetric theories. We find that the symplectic Killing spinor equation involves a matrix M in the USp(2 N) indices whose role was not recognized previously. Using the correct Killing spinors we explicitly confirm that the particle masses in the constructed theories agree with the predictions of the AdS/CFT correspondence. Finally, we establish correct O( d − 1,2) isometry transformations required to keep the Lagrangian invariant on AdS d .

CLASSICAL DESCRIPTION OF SPINNING DEGREES OF FREEDOM OF RELATIVISTIC PARTICLES BY MEANS OF COMMUTING SPINORS

We consider a possibility of describing spin one-half and higher spins of massive relativistic particles by means of commuting spinors. We present two classical gauge models with the variables xμ, ξα, χα, where ξ, χ are commuting Majorana spinors. In the course of quantization both models reproduce Dirac equation. We analyze the possibility of introducing an interaction with an external electromagnetic background into the models and generalizing them to higher spin description. The first model admits a minimal interaction with the external electromagnetic field, but leads to reducible representations of the Poincaré group being generalized for higher spins. The second model turns out to be appropriate for description of the massive higher spins. However, it seems to be difficult to introduce a minimal interaction with an external electromagnetic field into this model. We compare our approach with one, which uses Grassmann variables, and establish a relation between them.