⚛️ phenomenology

Angular momentum dynamics of vortex particles in accelerators

This paper investigates the radiative and non-radiative orbital angular momentum (OAM) dynamics of relativistic vortex particles in accelerators, revealing that while OAM loss via photon emission is negligible, non-radiative precession induces resonances at lower energies than spin, thereby suggesting the use of linacs for acceleration and adapted Siberian snakes for OAM manipulation to enable high-energy collisions with enhanced magnetic moments.

Original authors: D. Karlovets, D. Grosman, I. Pavlov

Published 2026-03-03

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CC BY 4.0

Original authors: D. Karlovets, D. Grosman, I. Pavlov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to spin a top. Usually, when we think of spinning particles in physics, we talk about spin—an intrinsic property like a tiny, built-in gyroscope inside an electron. Scientists have long used beams of these "spinning" particles to probe the secrets of the universe.

But this paper introduces a new, exotic player: the Vortex Particle.

The Twisted Top vs. The Spinning Top

Think of a standard particle (like an electron in a normal beam) as a spinning top moving in a straight line. It has a little internal spin, but its path is smooth.

Now, imagine a vortex particle. This is like a hurricane or a tornado made of matter. Instead of just spinning in place, the entire wave of the particle swirls around its path like water going down a drain. This swirling motion is called Orbital Angular Momentum (OAM).

Here's the magic trick: While a normal electron has a tiny magnetic "stickiness" (magnetic moment) based on its spin, a vortex particle's magnetic stickiness grows with how hard it swirls. If you make the swirl super intense, the particle becomes a magnetic giant, orders of magnitude stronger than a normal electron. This could let us see things in particle collisions that we've never seen before.

The Problem: The Wobbly Ride

The authors ask: Can we speed these swirling particles up to near the speed of light in a particle accelerator without them losing their twist?

There are two ways a particle can lose its "spin" or "twist":

The Radiative Leak (The Leaky Bucket): As the particle accelerates, it might spit out a photon (a particle of light). If that photon is "twisted," the particle loses its own twist.
- The Good News: The authors did the math and found that for these particles, this leak is incredibly slow. It would take a very long time for them to lose their twist this way. In fact, they will reach their top speed long before they leak any significant energy. Verdict: Safe to accelerate.
The Wobbly Ride (The Resonance Trap): This is the tricky part. In a circular accelerator (like a race track), magnets bend the beam. These magnets make the particle's magnetic moment wobble (precess).
- The Old Way (Spin): For normal spinning particles, this wobble is slow. They only hit a "resonance" (a point where the wobble gets out of control and destroys the polarization) at very high energies (around 440 MeV).
- The New Way (Vortex): For vortex particles, the math is different. Because of a relativistic effect called Thomas Precession (think of it as a weird wobble that happens when you turn a corner at relativistic speeds), the vortex particles wobble much faster.
- The Danger Zone: Because they wobble so fast, they hit these "resonance traps" at very low energies—starting as low as 3 MeV. If you try to run a vortex beam in a standard circular ring, it would lose its twist almost immediately, like a top falling over the moment you start pushing it.

The Solution: The Straight Road and the Magic Snake

So, how do we fix this?

For the Race Track (Circular Accelerators): We need a "Siberian Snake." In normal physics, this is a special magnet arrangement that flips the spin to cancel out the wobble. The authors suggest we can adapt these snakes to flip the twist (OAM) instead. If we do this, we can keep the vortex beam stable in a circular ring.
The Better Option (Linear Accelerators): The authors suggest a simpler solution: Don't use a ring at all. Use a Linear Accelerator (Linac).
- Imagine a straight highway instead of a race track. In a straight line, there are no sharp turns to cause the "wobbly" resonance. The particles just zoom straight ahead.
- This means we can build a straight machine, accelerate these twisted particles to high speeds, and they will arrive at the destination perfectly intact, ready for experiments.

Why Should We Care?

If we can master these "twisted" beams, we open a new door in physics:

New Experiments: We can collide these "magnetic giants" to create new types of particles or study forces in ways impossible with normal beams.
Better Tools: We could use them to take incredibly detailed "photos" of atoms (electron microscopy) or study the quantum entanglement of particles in new ways.
Low Energy, High Impact: Because the "resonance traps" happen at such low energies (3 MeV), we don't need massive, billion-dollar machines to test these ideas. Small, compact labs could start doing this research right now.

The Bottom Line

The paper says: "Yes, we can accelerate these swirling, twisted particles!"
While they are very sensitive to the "wobbles" of circular tracks, they are perfectly happy zooming down a straight line. By using linear accelerators and special magnets to manage their twist, we can finally bring these exotic "hurricane" particles out of the lab and into the real world of high-energy physics.

1. Problem Statement

Conventional particle accelerators primarily utilize plane-wave states with definite momenta. However, "vortex particles" (charged or neutral) possess a quantized orbital angular momentum (OAM) projection ( $\ell$ ) along the propagation axis. These states carry a magnetic moment proportional to $|\ell|$ , which can be orders of magnitude larger than the magnetic moment of spin-polarized plane-wave states.

Despite the potential for vortex beams to enable new physics (e.g., generating spin-polarized particles from unpolarized collisions, probing strong interactions, and enhancing quantum interference), no ultrarelativistic vortex particles have been successfully accelerated in a facility. The primary barriers are:

Radiative Loss: Will the emission of twisted photons during acceleration destroy the OAM state?
Non-Radiative Dynamics: How does the OAM precess in electromagnetic fields compared to spin? Are there resonances that cause rapid depolarization (OAM loss) at energies where spin-polarized beams remain stable?

2. Methodology

The authors employ a combination of Quantum Electrodynamics (QED) and relativistic classical mechanics to model the dynamics of vortex particles in accelerator fields.

Operator Formalism: They define gauge-invariant kinetic momentum and OAM operators, distinguishing between extrinsic and intrinsic OAM. They derive the mean magnetic moment operator for a relativistic fermion with a phase vortex, incorporating the anomalous magnetic moment (AMM) and spin-orbit interaction (SOI).
Generalization of the BMT Equation: The authors generalize the Bargmann–Michel–Telegdi (BMT) equation (which describes spin precession) to include intrinsic OAM. They construct covariant 4-vectors for the magnetic moment contributions from both spin and OAM to derive a relativistic precession equation for the OAM vector.
Radiative Lifetime Calculation: Using first-order QED in the Furry picture (exact solutions of the Dirac equation in magnetic fields), they calculate the transition probabilities for excited Landau states to decay by emitting twisted photons. They account for the acceleration effects by averaging emission probabilities over longitudinal momenta.
Resonance Analysis: They analyze the "tune" (precession frequency relative to the orbital frequency) for both spin and OAM in circular accelerators to identify depolarizing resonances.

3. Key Contributions

A. Radiative Stability of OAM

The paper calculates the effective lifetime of vortex states against radiative OAM loss (emission of twisted photons).

Result: The radiative lifetime ( $\tau$ $τ$ ) is found to be many orders of magnitude longer than typical acceleration times in linear accelerators (linacs).
- For electrons: $\tau \sim 10^{-2}$ to $10^3$ seconds (depending on field strength), whereas transit time in a 1 km linac is $\sim 10^{-5}$ s.
- For ions (e.g., $^{12}\text{C}^{6+}$ ): $\tau \gtrsim 10^7$ seconds due to mass scaling ( $\tau \propto m^3$ ).
Conclusion: Radiative depolarization is negligible. Vortex beams can be accelerated to ultrarelativistic energies without losing their OAM structure via photon emission.

B. Non-Radiative Dynamics and Precession

The authors derive a new precession equation for the intrinsic OAM vector ( $\langle \mathbf{L} \rangle$ ).

Distinct Dynamics: Unlike spin, where the precession is dominated by the anomalous magnetic moment (AMM), the OAM precession is dominated by the Thomas precession term coupled with the orbital g-factor ( $g_L = 1$ ).
The "Magic" Energy: There exists a "magic" energy ( $\gamma = 2$ ) where transverse OAM precession stops ( $\Omega_{L,\perp} \to 0$ ). Above this energy, the precession direction flips relative to the spin precession.
Ratio of Frequencies: The ratio of OAM precession frequency to spin precession frequency varies significantly with energy, reaching values up to $\approx 430$ at high energies (GeV range), compared to the spin-only case.

C. OAM Depolarizing Resonances

The study identifies a critical challenge for circular accelerators (synchrotrons/storage rings):

Spin Resonances: For electrons, the first spin depolarizing resonance occurs at $\approx 440$ MeV (due to AMM).
OAM Resonances: For vortex electrons, the first OAM depolarizing resonance occurs at a much lower energy: $\approx 3.066$ MeV.
Mechanism: The step between resonances is $\Delta \epsilon \approx 1.022$ MeV (much smaller than the 440 MeV step for spin). This means that as a vortex beam is accelerated in a ring, it will encounter frequent resonances that rapidly destroy the OAM polarization unless mitigated.

4. Results and Implications

Feasibility of Acceleration:
- Linear Accelerators (Linacs): Ideal for accelerating vortex beams. They avoid the transverse magnetic fields that cause the frequent OAM resonances found in circular machines. Longitudinal electric fields do not affect OAM precession.
- Circular Accelerators: Vortex beams in rings will suffer rapid OAM depolarization starting at just 3 MeV for electrons.
Mitigation Strategies:
- Siberian Snakes: The authors propose adapting Siberian snakes (magnetic devices used to rotate spin to avoid resonances) to rotate the OAM vector instead of spin. This can shift the beam out of resonance conditions or restore the OAM state.
- Spatiotemporal Vortex Beams: By using snakes, one can deliberately create "spatiotemporal vortex beams" where the OAM is not aligned with the momentum, allowing for novel experimental configurations.
- Spin-Polarized Vortex Particles: Using particles that are both spin-polarized and have OAM can help bypass OAM resonances because the spin precession frequency differs, disrupting the resonance condition.
Experimental Probes:
- The predicted OAM resonances at $\sim 3$ MeV can be probed using compact machines like microtrons.
- OAM analysis can be performed using electron diffraction techniques.
- Current projects, such as the 200 MeV vortex electron source at JINR (aiming for 400 MeV), are directly relevant to testing these dynamics.

5. Significance

This work provides the theoretical foundation for the acceleration and manipulation of vortex particle beams in high-energy physics.

Paradigm Shift: It moves vortex particles from a theoretical curiosity to a viable tool for accelerators, suggesting they could complement or even replace spin-polarized beams in certain high-energy collision experiments.
New Observables: By accessing observables related to OAM, physicists can study strong interactions, quantum entanglement, and interference phenomena previously inaccessible with standard beams.
Technical Roadmap: The paper outlines a clear path forward: use linacs for acceleration to ultrarelativistic energies and employ Siberian snakes adapted for OAM manipulation in circular machines to preserve the beam quality.

In summary, the paper demonstrates that while radiative losses are not a barrier, non-radiative resonances are a critical constraint for circular accelerators. However, with the appropriate use of linear acceleration and OAM-specific spin rotators (snakes), vortex beams can be successfully accelerated to high energies, opening a new frontier in particle physics.