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Spinor Representations for Fields with any Spin: Lorentz Tensor Basis for Operators and Covariant Multipole Decomposition

Wim Cosyn, Frank Vera

TL;DR

By introducing maximally chiral Lorentz representations $(j,0)\oplus(0,j)$ and rank-$2j$ $t$-tensors, the paper provides a constraint-free, covariant parametrization of operator matrix elements for massive particles of arbitrary spin. It develops a generalized Dirac basis and a covariant multipole (SL(2, C)) decomposition of on-shell bilinears, with explicit formulas valid for any spin and spinor type, and offers efficient algorithms to construct the $t$-tensors. The formalism connects to the massive spinor-helicity framework, enabling seamless translation between chiral spinors and on-shell amplitude techniques while preserving a clear physical multipole interpretation. Practically, this unified approach simplifies the non-perturbative QCD analysis of hadrons and nuclei across spins by identifying covariant multipoles that encode the spin structure of matrix elements in a frame-independent way.

Abstract

This paper discusses a framework to parametrize and decompose operator matrix elements for particles with higher spin $(j > 1/2)$ using chiral representations of the Lorentz group, i.e. the $(j,0)$ and $(0,j)$ representations and their parity-invariant direct sum. Unlike traditional approaches that require imposing constraints to eliminate spurious degrees of freedom, these chiral representations contain exactly the $2j+1$ components needed to describe a spin-$j$ particle. The central objects in the construction are the $t$-tensors, which are generalizations of the Pauli four-vector $σ^μ$ for higher spin. For the generalized spinors of these representations, we demonstrate how the algebra of the $t$-tensors allows to formulate a generalization of the Dirac matrix basis for any spin. For on-shell bilinears, we show that a set consisting exclusively of covariant multipoles of order $0\leq m \leq 2j$ forms a complete basis. We provide explicit expressions for all bilinears of the generalized Dirac matrix basis, which are valid for any spin value. As a byproduct of our derivations we present an efficient algorithm to compute the $t$-tensor matrix elements. The formalism presented here paves the way to use a more unified approach to analyze the non-perturbative QCD structure of hadrons and nuclei across different spin values, with clear physical interpretation of the resulting distributions as covariant multipoles.

Spinor Representations for Fields with any Spin: Lorentz Tensor Basis for Operators and Covariant Multipole Decomposition

TL;DR

By introducing maximally chiral Lorentz representations and rank- -tensors, the paper provides a constraint-free, covariant parametrization of operator matrix elements for massive particles of arbitrary spin. It develops a generalized Dirac basis and a covariant multipole (SL(2, C)) decomposition of on-shell bilinears, with explicit formulas valid for any spin and spinor type, and offers efficient algorithms to construct the -tensors. The formalism connects to the massive spinor-helicity framework, enabling seamless translation between chiral spinors and on-shell amplitude techniques while preserving a clear physical multipole interpretation. Practically, this unified approach simplifies the non-perturbative QCD analysis of hadrons and nuclei across spins by identifying covariant multipoles that encode the spin structure of matrix elements in a frame-independent way.

Abstract

This paper discusses a framework to parametrize and decompose operator matrix elements for particles with higher spin using chiral representations of the Lorentz group, i.e. the and representations and their parity-invariant direct sum. Unlike traditional approaches that require imposing constraints to eliminate spurious degrees of freedom, these chiral representations contain exactly the components needed to describe a spin- particle. The central objects in the construction are the -tensors, which are generalizations of the Pauli four-vector for higher spin. For the generalized spinors of these representations, we demonstrate how the algebra of the -tensors allows to formulate a generalization of the Dirac matrix basis for any spin. For on-shell bilinears, we show that a set consisting exclusively of covariant multipoles of order forms a complete basis. We provide explicit expressions for all bilinears of the generalized Dirac matrix basis, which are valid for any spin value. As a byproduct of our derivations we present an efficient algorithm to compute the -tensor matrix elements. The formalism presented here paves the way to use a more unified approach to analyze the non-perturbative QCD structure of hadrons and nuclei across different spin values, with clear physical interpretation of the resulting distributions as covariant multipoles.

Paper Structure

This paper contains 30 sections, 191 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of the construction for general spinors
  3. Irreducible representations of the Lorentz group
  4. Chiral representations
  5. Representation of boosts and spinors
  6. Bispinors and generalized Dirac gamma matrices
  7. The t-tensors contain the independent generators of u(2j+1)
  8. Correspondence with massive spinor-helicity formalism
  9. Algebra of the t- and gamma-tensors
  10. Cubic products of t-tensors
  11. Quadratic products of t-tensors
  12. The tensor coefficient in the quadratic reduction
  13. Quadratic products are reducible
  14. Covariantly independent terms in the quadratic products
  15. Orthogonalization of the covariantly independent terms
  16. ...and 15 more sections

Figures (3)

  • Figure 1: Flowchart illustrating the structure of the paper and formalism. Equations are meant to be read as schematic here. Equation numbers link to the full expressions. Arrows labeled with $\mathsf{P}$ stand for parity-invariant extensions that combine a left- and right-chiral representation. From the way the oval blob connects to many other parts of the flowchart, one can appreciate the central role the $t$-tensors play in the construction.
  • Figure 2: The figure shows schematically how to obtain the light-front or helicity spinors from canonical ones using rest frame rotations before applying the canonical boost. Here, the Melosh rotations are represented through their Euler angles, where $\phi$ is the azimuthal angle of the momentum, see Eq. (\ref{['eq:spinor_RF_meloshrotated']}).
  • Figure 3: Illustration of how the recursion in the relation between $t$-tensors for different spins works in the $\{+-R\,L\}$-basis. The black grid positions illustrate part of a spin-$j$$t$-tensor element in the $\{+-R\,L\}$-basis which has its single non-zero element on row $c$ and column $\dot d$ (black dot). Using the recursion formula to build a spin-($j+\frac{1}{2}$) $t$-tensor element (part of which is shown in the red grid) then corresponds to adding a Lorentz index: $t^{\mu_1\cdots \mu_{2j}} \to t^{\mu_1\cdots \mu_{2j},\mu_{2j+1}}$. The position of the non-zero element in that rank-($2j+1$) $t$-tensor (nearest neighbor red dot) then depends on the choice $\{+-R\,L\}$ for the extra Lorentz index and is illustrated with the action of the arrows and the corresponding choice of the spin-1/2 matrix which results in the 4 red dots shown. The action of the arrow corresponds to multiplication by the CG-coefficients in Eq. (\ref{['eq:singlet_fullreduction']}) and the non-zero element of the Pauli matrix (which is 2 in the basis we use).