On the Theory of Continuous-Spin Particles: Wavefunctions and Soft-Factor Scattering Amplitudes

TL;DR

This work characterizes continuous-spin particles (CSPs) as massless Poincaré representations labeled by a spin scale $\rho$, generalizing helicity to an infinite tower that mixes under boosts. It introduces covariant auxiliary-space wavefunctions and two classes of CSP wave equations (singular and smooth) that connect to Wigner and Fronsdal formalisms in appropriate limits, and constructs Lorentz-invariant soft factors for CSP emission to build candidate S-matrix amplitudes. The results demonstrate consistent CSP–matter couplings and yield finite cross sections at tree level, with a helicity-correspondence behavior at high energy that aligns CSP amplitudes with helicity-0, ±1, and ±2 modes. Together, these advances provide a first on-shell, covariant framework for CSP dynamics, with implications for infrared physics and potential gauge-theory formulations of CSPs. The analysis sets the stage for exploring CSP self-interactions, couplings to fermions, and deeper questions about locality and analyticity in CSP theories.

Abstract

The most general massless particles allowed by Poincare-invariance are "continuous-spin" particles (CSPs) characterized by a scale ρ, which at ρ=0 reduce to familiar helicity particles. Though known long-range forces are adequately modeled using helicity particles, it is not known whether CSPs can also mediate long-range forces or what consequences such forces might have. We present sharp evidence for consistent interactions of CSPs with matter: new CSP equations of motion, wavefunctions, and covariant radiation amplitudes. In a companion paper, we use these results to resolve old puzzles concerning CSP thermodynamics and exhibit a striking correspondence limit where CSP amplitudes approach helicity-0, 1 or 2 amplitudes.

Figures (5)

• Figure 1: The figure summarizes the Little Group (LG) transformation of massless particle states (see §\ref{['sec:CSR_reps']}). Particle types are characterized by a scale $\rho$. Basis states may be labeled by a tower of integer or half-integer spins, or equivalently by angles on a circle. The two bases are related by Fourier transform. The LG has the structure of the isometries of the Euclidean 2-plane, or $ISO(2)$. The spin basis diagonalizes LG rotations, while the angle basis diagonalizes LG translations. Lorentz boosts induce LG translations (and rotations), which mix states in the spin basis. The scale $\rho$ controls the amount of mixing under boosts, much like the combination $m\times S$ for a spin-$S$ massive particle. When $\rho=0$, spin labels become Lorentz-invariant helicities.
• Figure 2: Irreducible representations of the Poincaré group are labelled by the square of the Pauli-Lubanski vector operator $W^2$, and the square of the momentum operator $P^2=m^2$. The plot illustrates the relation between $W^2$ (y-axis) and $P^2$ (x-axis) imposed by the structure of the Little Group. As $P^2\rightarrow 0$, the massive branches approach the continuous-spin branch as $S(S+1)\rightarrow \rho^2/P^2$. Continuous-spin representations decompose into helicity representations for $\rho \rightarrow 0$ along the $m^2=0$ slice. The helicity-$h$ states $(h\geq 1)$ are not continuously related to the massive branches.
• Figure 3: Three Lorentz transformations that leave a null vector $k^\mu$ invariant are illustrated above. These form a basis for the Little Group of $k^\mu$. For the choice $\bar{k}^\mu=(\omega,0,0,\omega)$, these are $J_{12}$, $(J_{32}+J_{02})$, and $(J_{31}+J_{01})$. In addition to the obvious rotation about the 3-momentum axis (top), two combinations of rotations and transverse boosts (bottom) also leave $\bar{k}$ invaraint. These three transformations form the group of isometries of the Euclidean 2-plane.
• Figure 4: A diagrammatic picture for the factorization of single-emission amplitudes in the limit that the leg $\{k,a\}$ becomes soft. In this limit, radiation of $k$ off external legs grows as $|{\bf k}|\rightarrow 0$ and therefore dominates the amplitude. Each single emission term is proportional to a "parent" amplitude times a "soft factor" depending only on $k$, $a$, and a single external momentum $p_i$. The sum of these single-emission terms must be Lorentz-invariant up to terms of order $|{\bf k}|^0$.
• Figure 5: The construction of a candidate on-shell CSP amplitude is illustrated above. CSPs are attached to a parent amplitude $A_4(p_1,p_2; p_3,p_4)$ using the CSP soft factors, with appropriate matter propagators included. This example presumes that only the outgoing matter legs couple to the CSP. The final result is the 5-point amplitude $A(p_1,p_2; p_3,p_4,\{k, \phi \})$ used below as an example to investigate certain aspects of CSP interactions.