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Universal features of high-energy scattering of Laguerre-Gaussian states

Yaoqi Yang, Igor P. Ivanov

Abstract

Vortex states of photons, electrons, and other particles are wave packets that carry intrinsic orbital angular momentum (OAM) and exhibit other features unavailable for plane waves. Collisions of high-energy vortex states can become a promising tool for nuclear and particle physics, once experimental challenges are overcome. An extensive literature exists on scattering processes involving vortex states; however, most works rely on assumptions that will be challenging to achieve in experiment. In this work, we initiate a systematic re-analysis of vortex-state scattering processes using paraxial Laguerre-Gaussian (LG) wave packets colliding at a non-zero impact parameter $b$. Since the total final transverse momentum $P_\perp$ is no longer fixed, we focus on how the differential cross section depends on $P_\perp$. We emphasize that non-trivial $P_\perp$-dependent features can originate either from the shape of the LG wave packets or from the dynamics of the scattering process under interest. Here, we focus on the former source and explore in detail these universal kinematic features, while the study of process-specific modifications, along with the novel insights they may bring, is delegated to a future work. Interestingly, the non-zero impact parameter $b$ plays a key role in many $P_\perp$-dependent effects, making it a useful probe of vortex states, not a nuisance factor as often assumed.

Universal features of high-energy scattering of Laguerre-Gaussian states

Abstract

Vortex states of photons, electrons, and other particles are wave packets that carry intrinsic orbital angular momentum (OAM) and exhibit other features unavailable for plane waves. Collisions of high-energy vortex states can become a promising tool for nuclear and particle physics, once experimental challenges are overcome. An extensive literature exists on scattering processes involving vortex states; however, most works rely on assumptions that will be challenging to achieve in experiment. In this work, we initiate a systematic re-analysis of vortex-state scattering processes using paraxial Laguerre-Gaussian (LG) wave packets colliding at a non-zero impact parameter . Since the total final transverse momentum is no longer fixed, we focus on how the differential cross section depends on . We emphasize that non-trivial -dependent features can originate either from the shape of the LG wave packets or from the dynamics of the scattering process under interest. Here, we focus on the former source and explore in detail these universal kinematic features, while the study of process-specific modifications, along with the novel insights they may bring, is delegated to a future work. Interestingly, the non-zero impact parameter plays a key role in many -dependent effects, making it a useful probe of vortex states, not a nuisance factor as often assumed.

Paper Structure

This paper contains 25 sections, 78 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Vortex state scattering as a novel tool in subatomic physics
  3. A brief summary of vortex-state calculations
  4. Goals of the present work
  5. Assumptions adopted
  6. Scattering of two vortex states: an overview of the formalism
  7. General expressions
  8. Increased dimensionality of the final state phase space
  9. Transverse momentum differential cross section
  10. Universal kinematic properties of LG-LG scattering
  11. General properties of the transverse integral
  12. Transverse momentum distribution: analytical results for $b = 0$
  13. Transverse momentum distribution: analytical results for $b \neq 0$
  14. Transverse momentum distribution: numerical examples
  15. Consequences of a non-zero impact parameter
  16. ...and 10 more sections

Figures (8)

  • Figure 1: Schematic layout of average center of mass scattering of two initial LG states into two final plane waves with momenta ${\bf k}_1'$ and ${\bf k}_2'$. The transverse part of the total final state momentum ${\bf k}_{1\perp}' + {\bf k}_{2\perp}' = {\bf P}_\perp$ is non-zero and lies in a region of the order of $1/\sigma_{i\perp}$.
  • Figure 2: The auxiliary vectors ${\bf A}_{1\perp}$ and ${\bf A}_{2\perp}$ defined in Eq. \ref{['vectors-A']} that define the positions of the phase vortices in the ${\bf P}_\perp$-plane for the same-sign case. For a generic ${\bf P}_\perp$, the vectors ${\bf P}_\perp - {\bf A}_{1\perp}$ and ${\bf P}_\perp - {\bf A}_{2\perp}$ are also shown.
  • Figure 3: The function $W_0({\bf P}_\perp)$ shown by the shade intensity for $\Sigma_\perp = 1~\mathrm{keV}^{-1}$ at zero impact parameter, $b=0$. Top row: $\ell_1 = 0$ and $\ell_2 = 0, 1, 5$. Bottom row: $\ell_1 = 5$, and $\ell_2 = -5, -1, 5$.
  • Figure 4: The function $W_0({\bf P}_\perp)$ for non-zero $b=1\,\mathrm{keV}^{-1}$ (top row) and $3\,\mathrm{keV}^{-1}$ (bottom row) and for $\ell_1 = 5$ and $\ell_1 = -5$ (left), 0 (middle), and 5 (right column). In all cases, $\sigma_{1\perp} = \sigma_{2\perp} = 1.41\,\mathrm{keV}^{-1}$.
  • Figure 5: Schematic coordinate-space overlap of two LG wave packets offset by impact parameter $b$. The overlap region leads to most intense scattering and leads to a non-zero total final state momentum $P_y$.
  • ...and 3 more figures